H a n s
Î . Ge r b e r
I N S U R A N C E
N A T II EN A T I CS
T h i r d
Ed i t i o n
ññ~ñéñé ñ ñ~ñ- —
ÈÈÁ È É ÈÞ Û
© Springer
ñ ôú
H an s U . G er b er
Life Insurance
Mathematics
with exercises contributed by Samuel H. Cox
T h i r d E d i t i o n 19 9 7
Sp r i n g er
Sw i ss A sso c i at i o n o f A c t u a r i e s Z u r i ch
å - m a i l : h g e r b e r @ u n i l .c h
P r o f es s o r H an s U . G e r b er
P r o f e s s o r Sa m u e l Í
. C o x , F S .À .
G e o r g i a St a t e U n iv e r si t y
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D e p t . o f R i sk
M an ag em en t
U n i v e r s i t ts d e L a u s a n n e
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I n su r a n ce
A tlant a, G A
3030 3-3083
Sw i t z e r l a n d
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D i e D e u t sc h e B i b l i o t h e k - C I P - E i n h e i t s a u f n a h m e
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Z u r i c h . [ T r a n s l . î à t h e Ãi r s t e d . : W a l t h e r N e u h a u s l . - 3 . e d .  å ãÈ ï
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T o k y o : S p r i n g e r , 19 9 7
D t . A u sg . u . d .'Ò . : L e b e n sv e r s i c h e r u n g sm a t h e m a t i k
I SB N 3- 54 0 - 6 2 2 4 2 - Õ
M a t h e m a t i c s Su b j e c t C l a s s i fi c a t i o n ( 1 9 9 1 ) : 6 2 P 0 5
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S p r i n g e r - V e r l a g B e r l i n H e i d e l b e r g N ew Y o r k
I SB N 3 - 5 4 0 - 5 8 8 5 8 - 2 2 n d ed i t i o n Sp r i n ger - V er l ag B er l i n H ei d elb er g N ew ãî ã1ã
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r eci t at i o n , b r o ad c ast in g , r ep t o d u ct i o n o n m i cm 6 l m o r i n an y o t h er w ay, an d st o rage i n d at a b an k s.
D u p l i cat i o n o f t h i s p u b l i cat i o n o r p ar t s t h er eo f i s p er m i t t ed o n l y u n d er t h e p r o v i si o n s o f t h e G er m an
C o p y r i gh t L aw o f Sep t em b er 9 , 19 6 5 , i n i t s cu r r en t v er si o n , an d p er m issi o n f o r u se m u st al w ay s b e
o b t ain ed 6 o m Sp r i n g er - V er l ag .V i o l al at io n s are ÁàÛ å f o r p ro secu t io n u n d er t h e G er m an C o p yr i gh t L aw .
Sp r i n g er - V er l ag i st à p ar t o f Sp r i n ger Sci en c e + B u si n ess M ed i a
sp r m g emñ nî ãî
h n e.
© Sp r in ger - V erl ag B erl i n H ei d eIb et g 199 0 , 19 9 5 , 19 9 7
Pr i n t ed
i n G er m any
T h e î ãå î Ã gen er al d escr ip d v e n am es, r eg i st er ed n am es, t r ad em ar k s et c . i n t h i s p u b l i cat i o n d o es n o t
i m p l y, ev en i n t h e ab sen c e o f à sp ec i 6 c st at em en t , t h at su ch n am es ar e eae m p t 6 o m t h e r el ev an t
p ro t ect iv e l aw s an d t e g u l at i o n s an d t h er ef o r e 6 e e f o r g en er al u se .
'Fyp eset t i n g : C am er a- t ead y ñî ð ó 6 o m t h e au t h o r s
SP I N : 1 16 0 2 7 4 3
4 1/ 3 1 1 1 — 10 9 8 7
— P r i n t ed o n d a c i d - f r e e p a p er
Òî Ñ åñå N esbi t t
F o r ew o r d
Hsl ley 's Comet has been prominent ly displayed ø ø àëó newspapers during
t he last few mont hs. For t he fi rst t ime ø 76 years it appeared t his wint er ,
clearly visible against t he noct urnal sky. T his is an appropriat e occasion Ñî
point out t he fact t hat Sir Edmund Halley also const ruct ed t he world's fi rst life
t able ø 1693, t hus creat ing t he scient ifi c foundat ion of life insurance. Halley's
life t able and it s successors were viewed as det erminist ic laws, i .e. t he number
of deat hs ø any given group and year was considered t o Úå à well defi ned
number t hat could be calculat ed by means of à life t able. However , ø reality
t his number is random. T hus any mat hemat ical t reat ment of life insurance
will have t o rely more and more on probability t heory.
Âó sponsoring t his monograph t he Swiss Associat ion of Act uaries wishes
t o support the "modern" probabilist ic view of life cont ingencies. We ar e fort unat e t hat Pr ofessor Gerber , an int ernat ionally renowned expert , has assumed
t he t ask of writ ing t he monograph. We t hank t he Springer-Verlag and hope
t hat t his monograph will be t he fi rst in à successful series of act uarial t ext s.
Ziirich, M arch 1986
»' - . » .
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Hans Buhtmann
President
Swiss Association of Actuaries
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Pr e f a c e
Two maj or development s have infl uenced t he environment of act uarial mat hemat ics. One is t he arrival of powerful and affordable comput ers; t he once
import ant problem of numerical calculat ion has become almost t rivial in many
inst ances. T he ot her is t he fact t hat t oday' s generat ion is quit e familiar wit h
probability theory in an int uit ive sense; t he basic concept s of probability t heory are t aught at many high schools. T hese two fact ors should be t aken int o
account ø t he t eaching and learning of act uarial mat hemat ics. À fi rst consequence is, for example, t hat à recursive algorit hm (for à solut ion) is as useful
as a solut ion expressed in t erms of commut at ion funct ions. In many cases t he
calculat ions are easy; t hus t he quest ion "why" a cal culat ion is done is much
more import ant t han the quest ion "how" it is done. The second consequence
is t hat t he somewhat embarrassing det erminist ic model can be abandoned;
nowadays not hing speaks against t he use of t he st ochast ic model, which bet t er refl ect s t he mechanisms of insurance. Thus t he discussion does not have
t o be limit ed Ñî expect ed values; it can be ext ended t o the deviat ions from
t he expect ed values, t hereby quant ifying t he risk in t he proper sense.
T he book has been writ t en ø t his spirit . It is addressed t o t he young
reader (where "young" should be underst ood ø t he sense of operat ional t ime)
who likes applied mathemat ics and is looking for an int roduct ion int o t he
basic concept s of life insurance mat hemat ics.
In t he fi rst chapt er an overview of t he t heory of compound int erest is given.
In Chapt ers 2—6 various forms of insurance and t heir mechanisms are discussed
in t he basic model . Í åãå t he key element is t he fut ure lifet ime of à È å aged õ,
which is denot ed by Ò and which is (of course!) à random variable. In ChapÑåã 7 t he model is ext ended t o mult iple decrement s, where difFerent causes for
depart ure (for example deat h and disability) are int roduced. In Chapt er 8 ø surance policies are considered where t he benefi t s are cont ingent on ò î ãå t han
one life (for example widows' and orphans' pensions). In àll t hese chapt ers t he
discussion focuses on à single policy, which is possible in t he st ochast ic model,
as opposed t o t he det erminist ic model , where each ðî éñó is considered as à
member of à large group of ident ical policies. In Chapt er 9 t he risk arising
from à group of policies (à por tf oli o) is examined. It is shown how t he dist ribut ion of t he aggregat e claims can be calculat ed recursively. Informat ion about
P r eface
t his dist ribut ion is indispensable when reinsurance is purchased. T he t opic of
Chapt er 10 is of great pract ical import ance; for simplicity of present at ion t he
expense loading is considered only in t his chapt er . Chapt er 11 examines some
st at ist ical problems, for inst ance, how t o est imat e t he dist ribut ion of Ò from
observat ions. T he book has been writ t en wit hout much compromise; however ,
t he appendix should Úå à sign of t he conciliatory nat ure of t he aut hor . For
t he very same reason t he basic probability space (é , Ó , P ) shall be ment ioned
at least once: now!
T he publicat ion of t his book was made possible by t he support of t he Fund
for t he Encouragement of Act uar ial Mat hemat ics of t he Swiss Associat ion of
Act uaries; my sincere t hanks go to t he members of it s commit t ee, not ø
t he least for the freedom grant ed Ñî ø å. 1 would like t o thank ø part icular
Professor Biihlmann and Professor 1ååðø for t heir valuable comment s and
suggest ions. Of course 1 am responsible for any remaining fl aws.
For some years now à t eam of aut hors has been working on à compreherasi've te~ , which was commi ssi oned Úó t he Society of Act uaries and will
be published ø 1987 ø it s defi nit ive form. T he cooperat ion wit h t he coaut hors Professors Bowers, Hickman, Jones and Nesbit t has been an enormously
valuable experience for ø å.
Finally 1would like t o t hank ø ó assist ant , Markus Lienhard, for t he careful
perusal of t he galley proofs and Springer-Verlag for t heir excellent cooperat ion.
Lausanne, March 1986
Hans U. ÑåòÜåò
A ck n o w l e d g e m e n t
1 am indebt ed Ñî ø ó colleague, Dr . Walt her Neuhaus (University of Oslo),
who t ranslat ed t he text int o English and carried out t he proj ect ø à very
compet ent and effi cient way. We are also very grat eful Ñî Professor Hendrik
Âî î ø (University of Manit oba) for his expert advice.
Lausanne and Winnipeg, April 1990
Hans U. Ger ber
A c k n o w l e d g em e n t
T he second edit ion cont ains à rich collect ion of exercises, which have been
prepared Úó Professor Samuel Í . Ñî õ of Georgia St at e University of At lant a,
who is an experienced t eacher of t he subj ect . 1 would like Ñî express ø ó
sincere t hanks t o my American colleague: due Ñî his cont ribut ion, t he book
will not only find readers but it will fi nd user s!
Lausanne, August 1995
Hans U. Gerber
A ck n o w l e d g e m e n t
The second edit ion has been sold out rapidly. T his led t o t he present t hird
edit ion, ø which several misprint s have been correct ed. 1 àø t hankful t o Sam
Cox, Cheng Shixue, Wolfgang Quapp, Andre Dubey and Jean Cochet for t heir
valuable advice.
At t his occasion I would like t o t hank Springer and t he Swiss Associat ion of
Act uaries for aut horising t he Chinese, Slovenian and Russian edit ions of Life
Insurance Mat hemat ics. 1 am indebted Ñî Cheng Shixue, Yan Ying, Darko
Medved and Valery Mishkin. From my own experience I know t hat t ranslat ing
à scient ifi c t ext is à challenging t ask.
Lausanne, January 1997
Hans U. Gerber
C on t ent s
T h e M a t h em a t i c s o f C o m p o u n d I n t e r est
1.9
1.1
1.2
1.3
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1.6
P er p et u i t i es
1.7
1.8
A n nu i t i es
R ep ay m ent of à D eb t . . . . . . . . . . . .
13
11
2
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T h e C u r t at e F ut u r e L i fet i m e of (õ )
L i fe T abl es . . . . . . . . . . . . . . . . . . . .
20
2
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17
16
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3 .2 .1 W h ol e L i f e an d T er m I n su r an ce . . .
3 .2.2
Ðè ãå E n d ow m en t s . . . . . . . . . . .
3 .2.3
E n d ow m ent s . . . . . . . . . . . . . .
3 .3
3 .4
I n su r an ces P ay ab le at t h e M om ent o f D eat h
G en er al T y p es of L i fe I n su r an ce . . . . . . .
3 .5
St an d ar d T y p es of V ar i ab l e L i fe I n su r an ce
23
27
26
25
24
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291
L i f e A n n u i t i es
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8 .2
T h e J o i n t - L i f e St a t u s . . . . . .
8 .3
Si m p l i fi c a t i o n s . . . . . . . . . .
8 .4
T h e L a st - S u r v i v o r St a t u s . . . .
8 .5
T h e G e n er a l Sy m m et r i c St a t u s
8 .6
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8 .7
A sy m m et r i c A n n u i t i es
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9 .2
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9 .3
E x ac t C al c u l a t i o n o f t h e T o t a l C l a i m A m o u n t D i st r i b u t i o n
9 .5
R ec u r si v e C a l c u l a t i o n î 1 t h e C o m p o u n d P o i sso n D i st r i b u t i o n .
9 .6
R ei n su r a n c e
9 .7
St o p - L o ss R ei n su r a n c e
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
3
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1 0 E x p e n se L o a d i n g s
10 .3
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1 1 E st i m a t i n g P r o b a b i l i t i e s o f D e a t h
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1 1 .2 T h e C l assi c a l M e t h o d . . . . . . . .
1 1 .3 A l t er n a t i v e So l u t i o n
1 1 .4 T h e M a x i m u m L i k el i h o o d M e t h o d .
1 1 .5 St a t i st i c al I n f e r en c e . . . . . . . . .
1 1 .6 T h e B ay esi a n A p p r o a ch
1 1 .7 M u l t i p l e C a u ses o f D ec r em en t
. . .
A p p en d ix À . C o m m u t at io n F u n ct io n s
À ..51
INnet
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À .Ç
L i f e A n n u i t i es . . . . . . . . . . . . . . . . . . .
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L i f e I n su r a n c e
A p p e n d i x  . S i m p l e I n t e r est
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A p p e n d i x Ñ . E x e r c i se s
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R e f e r e n ce s . .
2 13
I n d ex
2 15
C h ap t er 1. T h e M at h em at i cs of
C o m p o u n d I n t er est
1 . 1 M a t h e m a t i c a l B a se s o f L i f e C o n t i n g e n c i e s
Òî È å insurance mat hemat ics primarily two areas of mat hemat ics are fundament al : t he t heory of compound interest and probability t heory. T his chapt er
gives an int roduct ion t o t he fi rst t opic. T he probabilist ic model will be int roduced in t he next chapter; however , it is assumed t hat t he reader is familiar
wit h t he basic principles of probability t heory.
1 . 2 E f f e c t i v e I n t e r e st R a t e s
An int erest ãàÑå is always st at ed in conj unct ion wit h à basi c time unit ; for
example, one might speak of an annual ãàÑå of 6%. In addit ion, t he ñî èèåòç÷î ï
peri od has Ñî be st at ed; t his is t he time interval at t he end of which interest is
credited or " compounded" . An int erest rate is called effecti ve if t he conversion
period and t he basic t ime unit are ident ical; in t hat ñàâå interest is credit ed
at t he end of t he basic t ime unit .
Let i be an effect ive annual int erest rate; for simplicity we assume t hat i
is t he âàø å for all years. We consider an account (or fund) where t he init ial
capit al Fs is invest ed, and where at t he end of year k an addit ional amount of
ò» is invest ed, for k = 1,
, è. What is the balance at t he end of è year s? Let
F» be t he balance at t he end of year k, including t he payment of r». Int erest
credit ed on t he previous year's balance is ãл q. T hus
F» = F» ~+ i F» ~+ ò», é = 1,
,è .
We may writ e t his recursive formula as
л —(1 + ã)л 1 —ò»,
( 1.2 .1)
(1.2.2)
if we mult iply t his equat ion by (1+ i )" » and sum over à11values of k, al l but
two t erms on t he left hand side vanish, and we obt ain
FÄ = (1 + ã)" Ã~+ ~ (1 + i )"- " ò».
»= l
C hap t er 1. T he M at hem at ics of Com pound I nt erest
T he power s of ( 1 + i ) ar e called accumulati on f actor s. T he accumulat ed value
of an i nit i al capi t al Ñ aft er h years i s ( 1 + ã)~Ñ . Equat ion ( 1.2.3) il lust r at esan obvious result : t he capit al at t he end of t he int er val is t he accumulat ed
val ue of t he init ial capi t al plus t he sum of t he accumul at ed values of t he
int ermedi at e deposit s.
T he di scount f actor is defi ned as
1
1+ i
( 1 .2 .4 )
E q u at i on ( 1.2 .3) can n ow b e w r i t t en as
( 1 . 2 .5 )
Hence t he present val ue of à capit al Ñ , due at t ime h , is è" Ñ .
If we wr it e equat ion ( 1.2.1) as
Å » — F q | = i F q > + r t,
( 1 . 2 .6 )
an d su m ov er é w e ob t ai n
F Ä — F o = ~» , ãл , + g
rq
( 1 . 2 .7 )
T hus t he increment of t he fund is t he sum of t he t ot al i nt erest credit ed and
t he t ot al deposi t s m ade.
1 .3 N o m i n a l I n t e r e st R a t e s
W hen t he conversion period does not coincide w it h t he basi c t i me unit , t he
int erest ãàÑå is called nomi nal. A n annual int erest r at e of 6% wit h à conversion
per iod of 3 mont hs m eans t hat i nt erest of 6%/ 4 = 1.5% is credit ed at t he end of
each quar t er . T hus an init ial capit al of 1 i ncreases t o (1.015)4 = 1.06136 at t he
end of one year . T herefore, an annual nom inal i nt er est ãàÑå of 6%, convert ible
quar t erly, is equivalent Ñî an annual eff ect ive i nt er est rat e of 6.136%.
Now , let i be à given annual effect ive i nt erest rat e. W e defi ne i < i as t he
nom inal int erest r at e, convert ible ò t im es ð åã year , w hi ch is equival ent t o i .
Equali ty of t he accumul at ion fact ors for one year leads t o t he equat ion
(1+
t (~ )
ò )
= 1+ Ñ,
( 1 .3 . 1 )
w h i ch i m p l i es t h at
t <- ~ = ò [( 1 + ã) ' ~
— 1] .
( 1 .3 .2 )
1.4 . Cont i nuous P ay ment s
T h e l i m i t i n g ñàâå ò - + î î cor r esp on d s t o cont i nu ous com p ou n d i n g . L et
( 1 .3 .3 )
t h i s i s c a l l ed t h e f o r ce of i n t er es t eq u i v a l en t t o i . W r i t i n g ( 1 .3 .2 ) a s
( 1 .+
( 1 + ~) 0
~ ) 1/ ï ç
1/ ò
( 1 .3 .4 )
w e see t h a t î i s t h e d e r i v a t i v e o f t h e f u n c t i o n ( 1 + i ) * a t t h e p o i n t õ = Î . T h u s
w e fi n d t h a t
å = ln (1 + i )
or
å
=
W e c a n v er i f y t h i s r esu l t b y l e t t i n g ò
1 +
( 1 .3 .6 )
i
—~ o o ø
( 1 .3 . 1) a n d u si n g t h e á åé ø Ñþ ï
( 1 .3 .3 ) .
T h u s t h e a c c u m u l a t i o n f ac t o r f o r à p er i o d o f h y ea r s i s ( 1 + i ) " = es" ; t h e
d i sc o u n t f a c t o r f o r t h e sa m e p e r i o d o f t i m e i s u " = å ~" . Í å ãå t h e l en g t h o f
t h e p er i o d h m ay b e a n y r ea l n u m b e r .
I n t u i t i v el y i t i s o b v i o u s t h a t i ~ l i s à d e c r e a si n g f u n c t i o n o f ò . W e c a n g i v e
à f o r m a l p r o o f o f t h i s b y i n t er p r et i n g i ~ l a s t h e sl o p e o f à sec a n t , see ( 1 .3 .4 ) ,
a n d u si n g t h e c o n v ex i t y o f t h e f u n c t i o n ( 1 + t ) .
T h e f o l l o w i n g n u m er i c al
i l l u st r a t i o n i s f o r i = 6 % .
1 .4
ù ,
~(~ )
1
0 .0 60 0 0
2
0 .0 59 13
3
0 .0 5 88 4
4
0 .0 5 8 70
6
0 .05 8 55
12
0 .05 84 1
îî
0 .0 58 27
C o n t in u o u s P ay m e n t s
W e c o n si d er à f u n d as i n Sec t i o n 1 .2 , b u t n o w w e a ssu m e t h a t p ay m e n t s
a r e m ad e c o n t i n u o u sl y w i t h a n a n n u a l i n st a n t a n eo u s r a t e o f p ay m en t o f r ( t ) .
T h u s t h e a m o u n t d e p o si t ed Ñî t h e f u n d d u r i n g t h e i n fi n i t esi m a l t i m e i n t er v al
f r o m t Ñî
t + d t i s r ( t ) d t . L e t F ( t ) d en o t e t h e b a l a n c e o f t h e f u n d at t i m e
t . W e a ssu m e t h a t i n t er est i s c r ed i t ed co n t i n u o u sl y , a c c o r d i n g t o à , p o ssi b l y
Ch apt er 1. T he M at hem at ics of Com p ound I nt er est
t ime-dependent , force of i nt erest b(t ) . Int erest credi t ed ø t he infi nit esim al
t im e int erval from t t o t + dt is F (t )b(t ) dt . T he t ot al incr ease in t he capit aldur ing t his int erval is t hus
d F (t ) = F (t )b (t ) d t + r (t ) dt
( 1 .4 . 1 )
Òî sol v e t he cor r esp on d i n g d i ffer ent i al eq u at i on
F (~) = ~ ( Ì (Ñ) + r (~)
( 1 .4 . 2 )
we wr it e
[ — / ñ á(ç ) Èâ ó ( t ) )
—/
Int egr at ion wit h r espect t o t fr om 0 t o h gives
dt
e—f ~ á(â) "âF (h )
F (0)
(ç ) <Ü .( Ñ)
f e f p á(â) Ûâ~,(Ñ) dt
lo
( 1 .4 . 3 )
( 1 .4 .4 )
T hus t he value at t ime Î of à pay ment Ñî Úå m ade at t im e t (i .å. it s pr esent
value) is obt ained by mult i plicat ion wi t h t he fact or
— /
á ( ) (Ü
( 1 .4 . 5 )
Prom (1.4.4) we fur t her obt ain
F(h) = efoë ~~' F(0) + Jîr h f ë ( )"' r(t)ÈÑ.
( 1 .4 .6 )
T hus t he value at t im e h of à pay m ent m ade at t i me t < h (it s accumulat ed
val ue) is obt ained by mul t iplicat ion w it h t he fact or
]
á (ç ) d a
( 1 .4 . 7 )
I n t he ñàÿå of à const ant for ce of i nt erest , üå. b(t ) = á, t he fact ors (1.4.5) and
( 1.4.7) are reduced t o t he discount fact ors and accumulat ion fact ors i nt roduced
ø Sect ion 1.2.
1 .5 I n t e r est i n A d v a n c e
U nt il now it was assumed t hat int erest was Ñî be credit ed at t he end of each
conversion per iod (or i n ar r ear s) . But somet imes i t is useful Ñî assume t hat
int erest is credit ed at t he beginni ng of each conversi on peri od . I nt erest credit ed in t his way is al so referr ed t o as di scount, and t he corr esponding ãàÑå is
called di scount rate or rate of i nterest-i n-advan ce.
Let È be an annual å1ãåñÑ1÷å discount r at e. À person invest ing an amount of
Ñ w il l be cr edit ed int erest equal t o dC i m medi at ely, and t he invest ed capi t al
1.5. I nt er est in A dvance
Ñ w i l l b e r et u r n e d a t t h e en d o f t h e p er i o d . I n v e st i n g t h e i n t e r est d C a t t h e
âà ò å c o n d i t i o n s , t h e i n v e st o r w i l l r ec ei v e a d d i t i o n a l i n t er est o f d ( d ( ) = Ó Ñ ,
a n d t h e a d d i t i o n a l i n v est ed a m o u n t w i l l b e r e t u r n ed a t t h e en d o f t h e y ea r ;
r ei n v est i n g t h e i n t er est y i el d s a d d i t i o n a l i n t er est o f È ( È~Ñ ) = È~Ñ , a n d âî o n .
R ep ea t i n g t h i s p r o c e ss a d i n fi n i t u m , w e fi n d t h a t t h e i n v est o r w i l l r ec ei v e t h e
t o t a l su m o f
Ñ + d C" + d Ñ + d sÑ + "
=
1
1
—
Ñ
È
( 1 .5 . 1)
a t t h e en d o f t h e y ea r i n r et u r n f o r i n v est i n g t h e i n i t i a l c a p i t a l Ñ . T h e eq u i v a l en t eff ec t i v e i n t e r est r a t e i i s g i v en b y t h e eq u a t i o n
1
= 1+ i ,
( 1 .5 .2 )
w h i ch l e a d s t o
È
( 1 .5 .3 )
=
1 +
i
T h i s r e su l t h a s a n o b v i o u s i n t e r p r et a t i o n : i f à c a p i t a l o f 1 u n i t i s i n v est ed , d
( t h e i n t er e st p a y a b l e a t t h e b eg i n n i n g o f t h e y ea r ) i s t h e d i sc o u n t e d v al u e o f
t h e i n t e r e st i t o b e p a i d a t t h e en d o f t h e y e ar . F u r t h e r m o r e , ( 1 .5 .2 ) i m p l i es
t h at
i =
d
1
—
( 1 .5 .4 )
È
T h u s t h e i n t e r est p ay a b l e a t t h e en d o f t h e y e a r i s t h e a c c u m u l a t ed v a l u e o f
t h e i n t er e st p ay a b l e a t t h e b e gi n n i n g o f t h e y ea r .
Ü åÑ d < ) b e t h e e q u i v a l e n t n o m i n a l r a t e o f i n t er est - i n - a d v a n c e c r ed i t ed ò
t i m es p er y e ar . T h e i n v est o r t h u s o b t a i n s i n t e r est o f —
ó( þâ ) Ñ
at t h e b e g i n n i n g
o f à c o n v e r si o n p er i o d , a n d h i s c a p i t a l Ñ i s r et u r n e d a t t h e e n d o f i t . E q u a l i t y
o f t h e a c c u m u l a t i o n f a c t o r s f o r t h i s m t h p a r t o f à y e a r i s ex p r esse d b y
T h i s l ead s t o
1 — d( )/ ò = 1 + —
;ò
(
È<
ò) ~~ð
m ~ oo
)
= ( 1 + t ) ~/
( 1 + ~) - 1/ m ]
( 1 .5 .5 )
( 1 .5 . 6 )
I n a n a l o g y w i t h ( 1 .5 .3 ) o n e o b t a i n s
<òë )
~<òà )
1 + t <m) /r e
'
( 1 .5 .7 )
r e su l t i n g i n à v er y si m p l e r el a t i o n b et w een i <~ ) a n d d < ) :
1
1
1
È< ,) — ò + ~<„ )
( 1 .5 . 8 )
I t fo llow s t h at
lim
d( ) =
lim
ä
) = á ,
( 1 .5 .9 )
Chapter 1. The Mathematics of Compound Interest
which was to be expect ed: when int erest is compounded cont inuously, t he
difFerence between int erest in advance and int erest in arrears vanishes.
The following numerical illust rat ion is for i = 6%.
ò
d(
1
2
3
4
6
12
oo
0.05660
0.05743
0.05771
0.05785
0.05799
0.05813
0.05827
1 .6 P e r p et u i t i e s
In t his sect ion we int roduce cert ain types of perpet ual payment st reams (ðåãpetuities) and calculat e t heir present values. T he result ing formulae are very
simple and will later be useful for calculat ing t he present value of annuit ies
wit h à fi nit e t erm.
First we consider perpet uit ies consist ing of annual payment s of 1 unit . If
t he fi rst payment occurs at t ime Î , t he perpet uity is called à Iierpetui ty-due,
and it s present value is denot ed by à--~. T hus
à
1 + „ + „ ã+
ã
1 —î
ä
(
1 . 6
. 1 )
If t he fi rst payment is made at t he end of year 1, we call t he perpet uity an
immediate perpetui ty. It s present value is denoted by à—~, and is given by
à- -~— v + vã + î ç +
1 —î
i
(
1
. 6
. 2
)
Let us now consider perpet uit ies where payment s of 1/ ò are made ò t imes
each year . If t he payment s are made in advance (fi rst payment of 1/ ò at t ime
0), t he present value is denot ed by à~ and is
.. (~
—)
= —
1
—ò
—
1
ò
1 rn
—
1
ò
—
m 1 —î ~/, — d(~)
ã ò
(1.6 3)
cf. (1.5.6). If t he payment s are made in arrears (fi rst payment of 1/ ò at t ime
1/ m), t he present value is denot ed by à~ and given by
(~ë)
1
m
l / rn +
1
ò
ã/ ò +
1
ò
3/ òë +
1.6. Per pet ui t ies
ò
1
—
V | ~
m
[( 1
+
à) ~~
—
1 ]
1
~(ò ) '
c f .
( 1 .3 .2 ) .
T
h e
r e s u l t s
( 1 . 5 .8 ) :
p a y m
=
s i n c e
e n t
L e t
r
( 1 .6 .4 )
1
o f
u s
( 1 .6 .3 )
a n d
( 1 .6 .4 )
p e r p e t u i t y - d u e
1 / ò
n o w
a n d
i n
à
a t
t i m
c o n s i d e r
s t a r t i n g
a t
,
a n d
e
Î
à
c o n t i n u o u s
t i m
e
t h e i r
Î
.
l e a d
a n
t o
i m
m
p r e s e n t
I t s
a n
i n t e r p r e t a t i o n
e d i a t e
v a l u e s
d i f
p e r p e t u i t y
p r e s e n t
w
v a l u e
o f
p e r p e t u i t y
e r
it h
i s
b y
1 / ò
e r
id e n t it y
o n l y
b y
à
.
c o n s t a n t
d e n o t e d
t h e
d i f
r a t e
b y
é -
o f
~
p a y m
a n d
e n t
g i v e n
b y
r ~
1
( 1 .6 .5 )
T
h e
s a m
T
h e
À
e
r e s u l t
sy s t e m
c e r t a i n
p a r a m
e t e r s ,
ò
p e r
a n d
4 ,
q
=
p a y m
a t i c
t y p e
i n c r e a s e s
t h e
c a n
b e
p a t t e r n
o f
( t h e
p a y m
o f
n u m
e
e n t s
s u c h
a
d
T ni m
eâ î
ø
b y
f o r m
p e r p e t u i t i e s
y e a r ) ; w
e n t s
o b t a i n e d
b e r
o f
a s s u m
a r e
a n
m
e
u l a e
w
t h a t
a d e
m
21 / q
3
m +
ò
— +
( 1 .6 . 1 ) -
i t h
î î
q
e n t s
i s
à
p e r
a n d
o f
ò
3
4
21 / q
i s
a n d
.
I f
—
( 1 .6 . 4 ) .
is
q
f o r
d e fi
n e d
( t h e
d e fi
i n s t a n c e ,
n e d
b y
n u m
q u a r t e r l y .
a r e
1 / m
ò
o r
e v i d e n t .
e n t s
i n c r e a s e
p e r p e t u i t y - d u e
1 / m
( 1 .6 .3 )
p a y m
y e a r )
f a c t o r
o n t h l y
i n
( 1 .6 . 5 )
i n c r e a s i n g
p a y m
i n c r e a s i n g
o n
l e t t i n g
I n
a s
t w o
b e r
ò
=
o f
1 2
g e n e r a l ,
f o l l o w
s :
P a43
2
1 y/ m
( ø
me n
ä )t
q
0
1 / q
2 / q
3 / q
I n
p a r t i c u l a r ,
t h e
p r e s e n t
r e p r e s e n t i n g
w
a t
it h
t i m
c o n s t a n t
e s
Î
t h e
v a l u e
t h e
l a s t
o f
m
/ q
s u c h
à
s e q u e n c e
p a y m
, 1 / q , 2 / q ,
e n t s
.
p a y m
o f
b y
o f
o f
T
e n t s
p e r p e t u i t y
1 /
h u s
i n c r e a s i n g
( m
w
q )
e
y e a r
p a y m
p a y a b l e
o b t a i n
É
a r e
( 1 ( à ) à ) ~( m
t h e
ò
)
e n t s
t i m
e s
É / ò
.
W
a s
à
p e r
s u r p r i s i n g l y
e a c h .
e
c a n
s u m
o f
y e a r ,
s i m
W
e
d e n o t e
c a l c u l a t e
b y
p e r p e t u i t i e s
a n d
p l e
i t
b e g i n n i n g
f o r m
u l a
( 1 .6 .6 )
Chap t er 1. T he M at hem at ics of Com p ound I nt erest
T h e cor r esp o nd i n g i m m ed i at e an nu i t y d i f er s on l y i n t h at each p ay m ent i s
m ad e one m t h y ear l at er , t h u s gi v i ng
À su p er scr i p t of 1 i s al w ay s om i t t ed . For i n st an ce, t h e p r esen t v al ue of à
p er p et u i t y - d u e w i t h àø ø à1 p ay m ent s of 1, 2,
is
( I ii )
l — (I ~ ~à )- -1 — — .
( 1 .6 .8 )
Å ñ~è àÔþ ï í ( 1.6 .6 ) an d ( 1.6 .7) m ay al so b e used w i t h ò
—~ oo t o cal cu l at e
p r esent val u es of cont i n u ou s p ay m ent st r eam s. O n e ob t ai n s for i n st an ce
—
r oo
t e ~ñ~t =
( 1 .6 .9 )
an d
—
~~ + 1] e áñ~t =
1
w i t h ou t act u al l y cal cu l at i n g t h e i nt egr al s.
W e con cl u d e t h i s sect i on by con si d er i n g à p er p et u i t y w i t h ar b i t r ar y an n u al p ay m ent s of r p, ò1, ò2,
(at t i m es 0 , 1, 2,
) . I t s p r esen t val u e, d en ot ed
si m p l y by à , i s
a = r p + ur q + v r q + .
( 1.6 .11)
Su ch à var i ab l e p er p et u i t y m ay b e r ep r esent ed as à su m of con st ant p er p et u i t i es i n t h e fol l ow i n g w ay :
A n nual p ay m ent
St ar t s at t i m e
r i —r p
ò2 — ò1
13
ò2
and so on
T h e p r esent val u e of t h is p er p et u i t y m ay t h er efor e b e ex p r essed as
à = —
1 { r o ~- þ (ã~ — ãä) + þ 2 (ò~ — ò~) +
) ,
( 1 .6 .12 )
d
w h i ch i s u sefu l i f t h e d i f er en ces of r q ar e si m p l er t h an t h e r q t hem sel v es. I f ,
i n p ar t i cu l ar , r q i s à p o l y n om i al i n É, t h e pr esen t val u e à m ay b e cal cu l at ed by
r ep eat ed d i ff er en ci n g . For i n st an ce, u si n g òö — k + 1 on e m ay v er i fy ( 1.6 .8) .
W e can u se ( 1.6 .11) t o cal cu l at e t he p r esent v alu e of ex p o n en t i al ly gr ow i n g
p ay m ent s. L et t i n g
r q — å " for & = 0 , 1, 2 ,
( 1.6 .13)
1.7. A n nuit ies
on e ob t ai n s
p r ov i d ed t h at ò ( î .
à = 1 . å-1 (á—ò)
( 1.6 .14 )
'
1 .7 A n n u i t i e s
I n pract ice, annui t ies are mor e frequent ly encount ered t han perpet ui t ies.
( 1.7.8
A n)
annuit y is defi ned as à sequence of paym ent s of à li mit ed dur at ion , which we
denot e by è . I n what follow s we consider some st andar d t ypes of annuit ies,
or annuit ies-cert ain as t hey somet imes are ñàÍ åé .
T he pr esent val ue of an annuit y-due wi t h è annual payment s of 1 st art ing
at t i me Î , is denot ed by à-„ -]. It is given by
à
=
1 +
g +
u~ +
.
. +
óâ
Represent i ng t he annui ty as t he éÌ åãåï ñå of two per pet uit ies (one st art ing at
t i me Î , t he ot her at t im e n ) , we fi nd t hat
à~ — à~ — î " à- -~— — — î " — =
â
( 1.7.2)
T his resul t can be verifi ed by direct ly evaluat ing t he geomet ric sum (1.7.1) .
I n à sim il ar way one obt ai ns from ( 1.6.2) ,( 1.6.3) and (1.6.4) t he for mul as
«
( 1.7 .3)
à~
-. ( èç)
% ]
(f (rn )
a„—]
( 1.7.5)
N ot e t hat only t he denom inat or varies, depending on t he pay ment mode (imm edi at e/ due) and frequency. N ot e t hat è must be an int eger i n ( 1.7.2) and
(1.7.3) , and à mult iple of 1/ ò in (1.7.4) and (1.7.5) .
T he fi nal or accumulat ed value of annui t i es is also of int er est . T his is
defi ned as t he accumul at ed val ue of t he pay ment st ream at t im e n , and t he
usual sy mbol used is s. T he fi nal value is obt ained by mult ipl ying t he i nit ial
val ue wit h t he accumulat ion fact or (1 + i') :
(1 + i )" — 1
( 1.7.6)
à ]
(1 + i )" — 1
~ ò~]
)
..( )
(1+ i')" —1
( )
(1+ i )" —1
Ä (èú)
;( )
Ch apt er 1. T he M at hem at ics of Com pound I nt er est
10
A n o t h er
c o n st a n t
si m p le
an n u it y
r el a t io n
m ay
b et w een
e a si ly
b e ÷åï
t h e
in it ia l
v a lu e
a n d
t h e
fi n a l
v a lu e o f
à
ï åá :
1
1
'÷
'÷
+
i
.
( 1 .7 .1 0 )
L et u s n ow con sid er an i n cr easi n g an n u i t y - d u e w i t h p ar am et er s q an d ò :
3 / q — 1/ ò
T im e
2
0 / q
21 // m
q +
1/ ò
1/ q
1/ q +
1/ ò
~ P ay m en t
21 / q — 1 / m
ò
1/ ( m q )
2 / (m q )
3 / (m q )
n, — 1/ q
Su ch
an
st a r t in g
a t
i n c r e a si n g
t im e
è , m in u s à
n — 1/ q + 1/ m
a n n u it y
Î , m in u s
c o n st a n t
a n
a n n u it y
( I (~)a )~
ca n
è — 1/ òï
b e r e p r ese n t e d
id e n t ic a l
st a r t i n g
in cr easin g
a t
t im e è .
n / m
a s a n
in c r ea sin g
p er p et u it y
T h u s w e
p er p et u it y
st a r t i n g
m ay
a t
t im e
w r it e
— ( 1 ® à )- -1 — v " (1 ® à )-, ,- ~ — è" è ~ ~
( 1 .7 . 1 1 )
Su b st i t u t i ng ( 1.6.6) an d ( 1.6.3) an d u si n g ( 1.7 .4 ) , w e ob t ai n t h e eq u at i on
(~) — è è "
S im ila r ly
t h e
p r e se n t
(1 ®
à )
o f
t h e
v a lu e
( ( òí )
( 1 .7 .1 2 )
c o r r e sp o n d i n g
im m ed ia t e
a n n u it y
is
ca lc u -
la t ed :
- ( ß) .
â
( 1 (~) à )
; (N o t e
t h a t
in
t h ese
I m p o r t an t
an d
q
=
fi n a l
T h e
(I ) .
m ad e
=
fo r
a n d
q
t h e
=
a n d
( 1 .7 . 1 3 )
12 , ò
à
=
oo
m u lt i p le o f
a n d
fa c ilit a t e
c o n si d e r e d
d e c r e a si n g
r e v e r se d
it s c o r r e sp o n d i n g
T h is r ela t io n
b e
q
=
t h e
1/ q .
=
1 , a n d
1 a n d
ò
ev a lu a t io n
=
o f
q =
o o
t h e
1 , ò
a n d
q
=
=
p r e se n t
12
oo .
a n d
t h ese c o m b in a t io n s.
a n n u it ies j u st
S ta n d a rd
in
12
( 1 .7 . 1 2 )
v alu es
m u st
sp ec ia l c a ses a r e t h e c o m b in a t io n s o f ò
1 , m
E q u a t io n s
eq u at io n s n
)
o r d er .
st a n d a r d
c a r r ies ov er
a r e
a n n u i t i es
T h e
su m
d ec rea sin g
t o
t h e
k n ow n
(D
)
as
ar e
o f
à
st a n d a r d
an n u ity
p r ese n t
s ta n d a r d
sim ila r ,
i n c r ea s i n g
b u t
t h e
in c r easin g
a n n u ity
is o f c o u r se à c o n st a n t
v a lu es , an d
w e o b t a in
a n n u i ti es
p ay m en t s
a re
a n d
a n n u it y .
1.8. Repayment of à Debt
Usi ng (1.7.12) and ( 1.7.14) and t he ident it y
- (×)
â1
(×) +
â
we obt ain
(1
â)
1
( 1 .7 . 1 5 )
è— ()
(1.7.16)
T he di rect derivat ion of t his ident it y is also inst ruct ive: t he st andard decr easing annui ty-due m ay be int erpret ed as à const ant perpet uity wit h m t hly payment s of è / ò , m inus à ser ies of defer red perpet uit i es-due, each wit h const ant
m t hly payment s of 1/ (m q), and st ar t i ng at t im es 1/ q, 2/ q,
,è.
1 .8 R e p a y m e n t o f à D e b t
Let S be t he value at t i m e 0 of à debt t hat i s t o be repaid by pay ment s
ò~,
, r Ä, mad e at t he end of year s é = 1, 2,
, ï . T hen S must be t he
pr esent value of t he payment s:
S = v r > + v ò~ + ' ' ' + è ò
( 1 .8 . 1 )
L et Sq be t he pr incipal out st anding, i .e. t he r em ai ning debt im m edi at ely aft er
r ), has been pai d . I t consist s of t he prev ious year ' s debt , accumul at ed for one
year , m inus ò~..
Sa = (1 + i )S~ ~ — r „
é = 1,
,è .
(1.8.2)
T his equat ion m ay be writ t en as
òö —i '
r + ( Sg g — Sg) .
( 1 . 8 .3 )
From (1.8.3) it is evident t hat each pay ment consi st s of two com ponent s,
i nter est on t he running debt and r educti on 0f pri nci paL
Subst it ut i ng —Sq for Fq, one sees t hat (1.8.2) is equivalent t o (1.2.2) . T hus
à11 result s of Sect ion 1.2 car ry over wit h t he appropri at e subst it ut ion . From
(1.2.3) one obt ains
k
Sq = (1 + ç)~ß — ~) ( 1 + ã')~ " r p, ,
( 1 .8 .4 )
and one may ver ify t hat SÄ = Î , usi ng ( 1.8.1) . Si m il ar ly, ( 1.2.5) may be used
t o show
ß~ = î ò~+| + ×,èò~+~ + .
+ è" ~ò„ .
(1.8.5)
Formul a ( 1.8.4) is t he r etrospecti ve f or mula, and ( 1.8.5) is t he prospecti ve
f or mula for t he out st andi ng pr incipal .
Chapter 1. The Mathemat ics of Compound Interest
12
T he payment s r q, . •, r » may be chosen arbit rarily, subj ect to t he const raint (1.8.1). Some of t he formulae in Sect ion 1.7 may be derived by proper
choice of t he payment st ream.
For inst ance, a debt of S = 1 can be repaid by t he payment s
r i — ò2 — . = r Ä i = i , ò„ = 1 + i .
(1.8.6)
In t his ñàÿå only int erest is paid for fi rst è — 1 years, and t he ent ire debt ,
t oget her wit h t he last year' s int erest , is repaid at the end of t he nt h year .
From (1.8.1) one fi nds
1 = i a~ + v" ,
(1.8.7)
which is anot her form of (1.7.3).
T he debt of S = 1 may also be repaid by const ant payment s of
ò1 — rq =
—ò„ = 1
% ]
As an alt ernat ive t o repaying t he credit or at t imes 1,
, è —1, one could pay
only t he int erest on S as ø (1.8.6). 1ï order t o cover t he fi nal repayment one
could make equal deposit s t o à fund t hat is Ñî accumulat e Ñî 1 at t he end of
è years; from t his it obvious t hat t he annual deposit must be 1/ â~ . Since
t he tot al annual outgo must be t he same in bot h cases, we arrive once again
at equat ion (1.7.10).
Suppose now t hat we repay à debt of S = è so that t he principal outst anding decreases linearly t o Î , 9» = è —é for k = Î , - , è. From (1.8.3) it
is evident t hat ò» —i (n —/ñ+ 1) + 1. Using (1.8.1) one obt ains t he ident ity
è = i (Ð à)-„-~+ à-„-1,
(1.8.9)
giving
(Ðà)-„-1—
T his result is à special ñàÿå (ò = q = 1) of (1.7.16).
T he loan it self may consist of à series of payment s. Assume t hat equal
payment s of 1 are received by t he debt or at t imes Î , 1,
, è —1. At t he end
of each year int erest on t he received amount s is paid, and, in addit ion, t he
t ot al amount received is repaid at t ime n:
ò» = i k for k = 1, , è —1 , ò„ = i n + è .
( 1 .8 . 1 1 )
From t he equality of t he present values one obt ains
à„-~ —i (I a)~ + nv" .
Equat ion (1.7.13) is obt ained for t he special ñàçå of q = ò = 1.
Many ot her ways of repayment may be t hought up. Present values of
annuit ies-due can be derived if one assumes t hat int erest is paid in advance.
Anot her variant is t he assumpt ion t hat interest is debit ed ò t imes à year , and
t hat t he debt is repaid q t ime à year in equal inst alment s (q à fact or of ò ).
1.9. Internal Kate of Ret urn
1.9 I nt er n al R at e of R et u r n
A n invest or ðàóâ à price ð , which ent it les him Ñî è fut ure payment s. T he
payment s are denot ed by ò~,
, r Ä and paym ent ò~ is due at t ime r q, for
É = 1,
, è . W hat is t he r esult ing r at e of ret urn?
T he present val ue of t he pay ment st ream t o be received by t he i nvest or is
à funct i on of t he force of int er est á. Defi ne
à
= ion
g åõð ( - áò~)ò~ .
L et t be t he sol ut i on of t heà(á)
equat
( 1.9 .1)
k= l
a (t ) = ð .
( 1.9 .2)
T he i nt er nal r ate of return or i nvestm ent yi el d is defi ned as i = å' — 1.
Equat ion (1.9.2) m ay be solved by st andard num er ical met hods, âèñÜ as
int erval bisect ion or t he Newt on-Raphson met hod . W e shal l present à met hod
which is more eff i cient t han t he former and sim pler t han t he lat t er of t hose
m et hods.
Consi der t he funct ion
f (6) = l n ( à (á) / ò ) ,
( 1.9 .3)
(here ò = òä +
+ r Ä denot es t he undiscount ed sum of t he payment s) . It is
not difi cult t o ver ify t hat
f (o) = Î ,
f ' (6) = à' (á)/ à(á) < Î ,
f " (á) = à" (b)/ à(á) — (à' (á)/ à(á) ) ) Î .
( 1.9 .4 )
(T he last i nequal ity m ay Úå ÷åï éåé by i nt er pret i ng f " (6) as à vari ance) . I nt erpr et ing f (b) / 6 as t he slope of à secant and not i ng t hat f is à convex funct ion by ( 1.9.4) , we see t hat f (b)/ 6 is an increasi ng funct ion of á. Hence, for
Î < s < t < è one has t he i nequalit y
~ (â) / â < / (1) / 8 < / (è ) / è ,
gi v i n g
T hu s w e h av e p r oved t h at
f(t),
f (â)
l ï ( à(â)/ ò)
f()„
f (è )
( 1.9 .5)
( 1.9 .6)
1ï (à(è) / ò)
If one has à lower bound s and an upper bound u for t he sol ut ion t of (1.9.2),
t hese bounds m ay immedi at ely be im proved by ( 1.9.7) .
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C h a p t e r 2 . T h e F u t u r e L i f et i m e
o f à L i f e A g ed x
2 .1 T h e M o d el
Let us consider à person aged õ years, al so cal led à lif e aged x and denot ed by
(õ ). W e denot e his or her fut ure li fet ime by Ò or , mor e explicit ly, by Ò(õ) .
T hus x + Ò will be t he age at deat h of t he person .
T he fut ure l ifet i me Ò is à r andom var iable wit h à probabil ity dist ribut ion
funct ion
G (t ) = Ðã( Ò < t ) , t > Î .
(2.1.1)
T he funct i on G (t ) r epr esent s t he probabili ty t hat t he per son wil l die wit hin t
years, for any fi xed Ñ. We assume t hat G , t he probability dist r i but ion of Ò,
is known . W e al so assume t hat G is cont i nuous and has à probabili ty density
g(t ) = G' (t ) . T hus one may wr it e
g (h)ch = Ðã (t < Ò < 1 + ch) ,
(2.1.2)
t his being t he probabi lity t hat deat h will occur in t he infi nit esi m al t ime int erval fr om t t o t + dt (or t hat (x ) '
x + t an d
õ + 1+ É ) .
Probabili t ies and expect ed values of int erest m ay be ex pr essed in t er m s of
t he funct ions ä and G . Nevert heless, t he int ernat ional act uari al com munit y
uses à t im e-honoured not at ion , t o which we shal l adher e. For ex am ple, t he
probabilit y t hat à l ife aged x will die wi t hin t year s, is denot ed by t he symbol
,q . W e have t hus t he relat i on
ü = ~ (~) .
(2 .1.3)
,ð. = 1 —G(t)
(2.1.4)
Si m i l ar l y ,
denot es t he probabi li ty t hat à life aged x will sur vive at least t year s. A not her
com m only used symbol is
â~ñ×õ
=
P r (8 < Ò < s + h)
G (s + t ) — G (s)
ç+ ÔÜ
âààï 1
( 2.1.5 )
Ch ap t er 2. T he F ut u re L ifet i m e of à L i fe A ged r
16
d en ot i n g t h e pr ob ab i l i t y t h at t he l i fe aged x w i l l sur v i v e s y ear s an d su bsequ en t l y d i e w i t h i n t y ear s .
W e d en ot e b y ,ð , + , t h e con d i t i on al p r ob ab i l i t y t h at t h e p er son w i l l su r v i v e
an ot h er t y ear s, aft er h av i n g at t ai n ed t h e age õ + s . T hu s
,ð , = P r (Ò > s + t iT > s) = 1 — G (s + t )
Si m i l ar l y , w e d efi n e
1 — G (s )
ß õò å = P r ( Ò < s + t [T > s ) = G ( s + t ) — G ( s) ,
( 2 . 1 .6 )
(2 .1.7)
t h e con d i t i on al p r ob ab i l i t y of d y i n g w i t hi n t y ear s, gi v en t h at t h e age of õ + â
h as b een at t ai n ed .
I d ent i t i es i n f r eq u ent u se ar e
, +,ð , = 1 — Ñ ( â + 1) = [1 — Ñ ( â)] 1 — G(s + t ) = , ð , ,ð , + , ,
( 2 .1.8)
, ~,ö, = G (s + t ) — G ( s) = [1 — G (s) 1G (s + t ) — G ( s) = , ð , ä + , .
(2 .1.9)
T h ese i d en t i t i es h av e àë obv i ou s i n t er p r et at ion .
T h e ex p ect ed r em ai n i n g l i f et i m e of à l i f e aged x i s Å (Ò ) , an d den ot ed by
î
å, . I t s d efi n i t io n i s
å. = f å,(>=—/G(g))dg
= f,
tg(t)dt
ð,dg.
((22 ..1
1 .10
.1 1))
or , i n t er m s of t he d i st r i b u t i on f u n ct i on ,
I f t = 1, t h e i n d ex t i s u su al l y om i t t ed i n t h e sy m b ol s , ä„ ,ð „ , ~,ä, . T h u s
q, i s t h e p r ob ab i l i t y of d y i n g w it h i n 1 y ear , an d , ~q i s t h e p r ob ab i l i t y of
su r v iv i n g s year s an d su b seq u ent l y d y i n g w i t h i n 1 y ear .
2.2 T he For ce of M or t alit y
T h e f or ce of m or t al i t y of (õ ) at t h e age z + 4 i s d efi ned b y
ð , ~, —
g (t )
— ——
d 1ï [1 — Ñ (~)] .
( 2 .2 . 1 )
2.3. A naly t ical D i st r ibut ions of Ò
17
From (2.1.2) and (2.1.4) one m ay der ive an alt ernat ive expr ession for t he
probabi lit y of dyi ng in t he int erval bet ween t and t + dt :
P r (t < Ò < t + dt ) = ,ð , ð , +~d t .
( 2 .2 .2 )
T h e ex p ect ed f u t u r e l i f et i m e of (õ ) can n ow b e w r i t t en as
( 2 .2 .3 )
T he appr ox im at ion
ç×õ+, — 1~õ+~~
(2.2.4)
is val id for sm al l values of s, as one m ay verify by ex changi ng t he r oles of s
and t in (2.1.9) and com paring t he resul t wi t h (2.2.2) .
T he force of mor t al ity m ay also be defi ned by
( 2 .2 .5 )
I nt egr at i on of (2 .2 .5 ) y i el d s
ðÔ
,ð = å ~î p g q gd 8
( 2 .2 .6 )
2 . 3 A n a l y t i c a l D i st r i b u t i o n s o f Ò
We cal l t he funct ion G an anal yt ical or "m at hem at i cal " pr obabil it y dist ri but ion if i t m ay Úå expressed by à sim ple formula. T here are different r easons
for post ul at i ng an analy t ical dist r ibut ion for Ò.
I n t he past eff ort s have Úååï m ade t o derive univer sally val id anal yt ic
expressions for G (t ) fr om cert ai n basic post ul at es, in analogy w it h t he laws
of physics. T hese eff ort s, seen fr om à 20t h cent ury point of view , now seem
r at her naive and surr ounded wit h à cert ai n myst i que.
A n analy t ical formula has t he advant age t hat G (t ) can readily be calcul at ed from à sm al l number of numer ic par am et ers. St at ist ical i nference in
par t i cular is facilit at ed when only à few par am et er s need Ñî be est im at ed .
T his m ay be an im por t ant considerat ion when t he avail able dat a ar e âñàãñå.
A nalyt ical for mul ae also have some àÑÑãàñÑ|÷å t heoret ical proper t ies. T heir
popul ar ity is akin Ñî t he popul ar ity of t he nor m al dist ri but ion in st at ist ics: À
nor m al model is oft en used , part ly m ot ivat ed by t he Cent r al L i m it T heor em ,
but m ai nly for it s m at hem at ical t r act abi lity.
Áî ò å exam ples of analyt ical dist ribut ions follow , each bearing t he name
of i t s "i nvent or " .
D e Ì î þ ò (1724) post ul at ed t he exist ence of à m ax i mum age ur for hum an
bei ngs and assumed t hat Ò was uni form ly dist r ibut ed bet ween t he ages of 0
Ch apt er 2. T he F ut ur e L ifet i m e of à L ife A ged a:
18
and ø — õ , leading t o g(t ) =
t hen becomes
Èõ+î =
1 for 0 < t < û — õ . T he for ce of mort alit y
1
0 < t < e —õ ,
û — õ —t '
which is an incr easing funct ion of t .
Gomper tz (1824) post ul at ed t hat t he for ce of m ort alit y would grow exponent ially ,
p py g — Â ñ
) t > 0
(2.3.2)
which refl ect s t he aging pr ocess bet t er t han De M oivre's l aw and i n addit ion
removes t he assumpt ion of à m ax i mum age ø .
T he l aw (2.3.2) was gener al i zed by M akeham ( 1860) , who post ulat ed t he
law
ð , ~, — A + Â ñ*+' , t > Î .
(2.3.3)
M akeham 's m ort alit y l aw adds à const ant , age i ndependent com ponent À > 0
t o t he exponent i ally growi ng force of mort al ity of (2.3.2) .
À special ñàÿå of t he mort alit y l aws of Gompert z (by put t ing ñ = 1) and
M akeham (by m aking  = 0) is t hat of à const ant for ce of mor t alit y. T he
probability dist r ibut ion of T t hen becom es t he exponent ial dist r ibut ion . W hile
m at hem at ically very si mple, t his di st r ibut ion does not refl ect hum an m or t al it y
in à r ealist ic way.
From (2.3.3) and (2.2.6) , and put t i ng m = Â / ln ñ, t he sur vival probabil it y
ø M akeham 's m odel m ay be derived :
,ð , = åõð ( —A t — m c* (c' — 1) ) .
( 2 .3 .4 )
Wei bul l (1939) suggest ed t hat t he force of mort al it y grows as à power of
t , inst ead of ex ponent ially :
ð , + ô — é (õ + t ) " ,
( 2 .3 . 5 )
wit h t he fi xed par amet ers k > 0 and è > Î . T he survival pr obabi lity t hen
becom es
,ð = åõð
2 .4
T h e
C u r t at e
— k
F u t u r e
(õ + 1)" +' — õ" +'
L ifet im e o f
( 2 .3 .6 )
(x )
We now ret urn t o t he gener al model int roduced in Sect ions 2.1 and 2.2 and
defi ne t he r andom vari ables Ê = Ê (õ) , S = S (x ) , Si l = ß ~ >(õ) , al l closely
relat ed t o t he original random vari abl e T .
We ï åï ï å Ê = [T ], t he number of com plet ed fut ure years l ived by (s ) , or
t he cur tat e f utur e lif eti m e of (õ) . T he probability dist ribut ion of t he i nt egerval ued random vari abl e Ê is given by
p r (K = k ) = P r ( k < Ò < k + 1) = „ ð î , + „
( 2 .4 . 1 )
2.4. T he Cur t at e F ut ur e L ifet i m e of (õ )
19
for k = Î , 1,
. T he expect ed value of Ê is called t he expect ed curt at e fut ur e
lifet i me of (õ ) and is denot ed by e . T hus
e, = ) , É Ð ã ( Ê = k ) = ~)
o r
k= 1
k Äp , q + ~
( 2 .4 .2 )
k= 1
å, = ~ , Pr (Ê > k ) = ~
k= 1
„ð, .
(2.4.3)
k= 1
Use of t he expect ed curt at e lifet ime has t he advant age t hat (2.4.1) and (2.4.2)
are åàâ1åã Ñî eval uat e t han (2.1.11) and (2.2.3) . A not her advant age is t hat
one only needs t he dist ribut ion of Ê in order t o fi nd e .
L et S be t he fr act ion of à year dur ing whi ch (õ ) is al ive ø t he year of
deat h , |.å.
Ò= K + S.
(2.4.4)
T he random var i able S has à cont inuous di st ri but ion between 0 and 1. A pproxi m at ing i t s ex pect ed value by -' we fi nd , from (2.4.4) , t he approxi mat ion
1
åî = e + —
2'
( 2 .4 .5 )
which m ay be used in pract ice for t he expect ed fut ur e l ifet ime of (õ) .
L et us assume t hat Ê and S are independent r andom vari ables, âî t hat
t he condit ional di st ribut ion of S, given Ê , is independent of Ê ; t hus
PI (S < QIK = É) = " *+k
„ î , +~ — Í (è) î +~
×æ+ü
( 2 .4 .6 )
will not depend on t he argument k , âî t hat one can writ e
( 2 .4 . 7 )
for k = Î , 1, . and 0 < è < 1, and âî ò å funct ion Í (è) .
I f we assum e t hat Í (è ) = è (unifor m dist r ibut ion bet ween 0 and 1) , t hen
t he approx im at ion (2.4.5) is ex act . M oreover , using (2.4.4) and t he assumed
independence, t he vari ance of Ò becomes
Var (T ) = Var (K ) + —
1 .
ò
12
( 2 .4 . 8 )
For posit ive int egers ò we defi ne t he r andom vari able
S<- > = —
1 [m S + 1] .
( 2 .4 . 9 )
T hus Ó l is derived from S by rounding t o t he next higher mult i ple of 1/ ò .
T he dist r i but ion of S< l has it s m ass in t he point s —
' , ~,
, 1. Not e t hat
independence between Ê and S impl ies independence bet ween Ê and S~ l .
Fur t herm ore, i f S has à uni for m dist r ibut ion bet ween 0 and 1, t hen S~ l has
à discret e uniform dist ri but ion .
Chapter 2. The Future Lifet ime of à Life Aged x
20
2 .5 L i f e T a b l e s
In t he previous sect ions of t his chapt er we considered à person of age x. The
probability dist ribut ion of his fut ure lifet ime ñàë be const ructed by adopt ing
à suit able lif e table.
À life t able is essent ially à t able of one-year death probabilit ies qÄ which
complet ely defi nes t he dist ribut ion of Ê . In t he next sect ion we will show how
t o approximate t he dist ribut ion of Ò by int erpolat ion in t he life t able.
Life t ables are const ruct ed from st at ist ical dat a (see Chapter 11). T he
const ruct ion of à life t able involves est imat ion, graduat ion and ext rapolat ion
t echniques (t he lat ter are used Ñî account for changing mort ality pat terns over
time).
Life t ables are const ruct ed for cert ain populat ion groups, diff erent iated by
fact ors such as sex, ãàñå, generation and insurance type. The init ial age x
can have à signifi cant in8uence in âèñÜ t ables. For inst ance, let x denot e the
age when t he person bought life insurance. Since insurance is only off ered t o
individuals of good healt h (somet imes only after à medical test ), it is reasonable to expect t hat à person who has j ust bought insurance, will be of bet ter
healt h t han à person who bought insurance several years ago, ot her factors
(part icularly age) being equal . T his phenomenon is t aken into account by
select li f e tables. In à select life table, t he probabilit ies of death are graded
according t o t he age at ent ry. T hus q[ ]+, is t he one-year probability of deat h
for (õ + t ) wit h x as ent ry age. Select ion leads t o t he inequal it ies
× [õ ] ~
q [s — 1] + 1 ~
(2 .5 .1)
q [s — 2 ] + 2 ~
T he select ion eff ect has usually worn î é aft er some years, say r years aft er
ent ry. We assume t hat
q [õ —ò]+ ò
q [õ - ò —1[ +ò+ 1
q [æ—, —2]~-ã~-2
qõ '
(2 5 2)
T he period ò is called the select peri od, and t he t able used aft er t he select
period has expired, is called an ulti mate lif e table.
Consider à person who buys à life insurance policy at age x . Wit h à select
period of 3 years, t he following probabilit ies are needed in order to det ermine
t he dist ribut ion of Ê :
q [* ] ~ ~ [õ ] + 1 ~ q [s ) + 2 ~ % + 3 ~ q s + 4 ~ q * + 5 ~
(2 .5 .3)
If à life t able varies only wit h t he at t ained age õ, it is called an aggregate
lif e table. It has t he advant age of being single-ent ry, while à select life t able is
double-ent ry. T he one-year probability of deat h at à given att ained age in an
aggregat e life t able will typically be à weight ed average of t he corresponding
probabilit ies in t he select È å t able and in t he ult imat e life t able.
T hough it is easy Ñî use à select life t able, cf. (2.5.3), we shal l, for simplicity, use t he not at ion of t he aggregat e life t able in t he sequel .
2.6.
iesi of
for D
Fract
of à Year
2.6 Probabilit
P r ob ab
l i tDeat
i es hof
eations
h for
Fr act i on s
of à Óåàã
21
T he di st ri but ion of Ê and it s rel at ed quant i t ies may be calculat ed from à life
t able. For example,
„ ð . = ð . ð . + , ð . +,
ð . +„ »
k = 1, 2 , ç , "
,
( 2 .6 . 1 )
cf. (2.1.8) . Òî obt ai n t he di st r ibut ion of Ò by i nt erpol at ion, assumpt ions ar e
m ade regar di ng t he pat t er n of t he probabilit i es of deat h , „ ä„ or t he force of
mort alit y, p +„ , at int er mediat e ages õ + è (õ an i nt eger and 0 < è < 1).
W e shall discuss t hree such assumpt ions.
A ssu m p t i on à : L i n ear i t y of Äq
I f one assumes t hat Äq is à linear funct ion of è , int erpolat ion between è = 0
and è = 1 yields
Äq = u q, .
(2.6.2)
W e have seen in Sect ion 2.4 t hat t his is t he ñàçå where Ê and S are independent , and 8 is uni form ly dist r ibut ed between 0 and 1. T hen
„ ð, = 1 — u q
( 2 .6 .3 )
an d (2 .2 .5 ) gi v es
qõ
1 — u q,
È õ + è
( 2 .6 .4 )
A ssu m p t i o n b : è , + „ ñî ï âÔàï Ñ
IÀt popul
al so fol
arl ow
assum
s t h at
pt ion is t hat t he for ce of mor t ali ty is const ant over each unit
int er val . L et us denot e t he const ant val ue of è, +„ , (Î < è < 1) Úó è, + | .
2
Usi ng (2.2.5) one fi nds
p, + r = —ln p,
(2.6.5)
= å
( 2 .6 .6 )
* +
Fr om (2 .4 .6 ) on e d er i v es
ð
ñ
+ ÿ = ( ð )"
( 2 .6 . 7 )
T he condi t ional dist ri but ion of S , given Ê = k , is t hus à t runcat ed exponent i al
dist r ibut ion, and it depends on k . T he random var iables S and Ê are not
independent i n t his case.
22
Ch ap t er 2. T he F ut ure L ifet i m e of à L ife A ged x
A ssu m p t i o n ñ: L i n ear i t y of , Äq +„
T his hypot hesis, well-known in Nor t h A merica as t he B alducci assumpti on,
st at es
, «q, + = ( 1 — è) ä, .
T his leads t o
From t his and (2.2.5) we obt ain
1 „ð +„
and fi n al l y
1—ü
1 — ( 1 —è ) î
1 — (1qs
— u) q
Pr (S < u ~K = k) =
(2.6.8)
( 2 .6 .9 )
( 2 .6 . 1 0 )
1 — (1 — è ) q +>
T hi s shows t hat t he random vari ables S and Ê are not independent under
t he Balducci hypot hesis,
Under each of t he t hr ee assumpt ions t he for ce of mor t alit y is discont i nuous
at i nt eger val ues. M ore embarr assi ng is t he fact t hat under t he Balducci
assum pt ion t he for ce of mort al ity decreases between consecut ive int eger s, cf .
(2.6.10) .
For î , +~ —~ 0 bot h (2.6.7) and (2.6.11) converge t o è . T hus, if t he pr obabilit ies of deat h are sm all , S is "approxi m at ely" uniform ly dist r ibut ed and
i ndependent of Ê (even under assumpt ions b or ñ) .
C h a p t er
3 . L i f e I n su r a n c e
3 .1 I n t r o d u c t i o n
Under à l i fe i nsur ance cont ract t he benefi t insured consist s of à si ngl e paym ent ,
t he âèò i nsur ed. T he t ime and amount of t his payment m ay be funct ions of
t he random vari able Ò t hat has been int roduced in Chapt er 2. T hus t he t ime
and amount of t he pay m ent m ay be r andom variables t hem sel ves.
T he pr esent value of t he payment is denot ed by Z ; it is cal cul at ed on t he
basis of à fixed r at e of int erest ã' (t he techni cal r at e of int erest ). T he expect ed
present val ue of t he pay ment , E (Z ) , is t he n et si ngle pr emi um of t he cont r act .
T his pr em ium , however , does not in any way refl ect t he r isk t o be carr ied by
t he insur er . I n or der Ñî assess t his one requir es furt her char act eri st ics of t he
di st ribut i on of t he random var i able Z , for ex am ple it s variance.
3 .2 E l em ent ar y I n su r an ce T y p es
3 .2 .1 W h ol e L i fe an d T er m I n su r an ce
Let us consi der à Û î 1å lif e i nsurance; t his provides for paym ent of 1 uni t at
t he end of t he year of deat h . In t his ñàçå t he amount of t he paym ent is fi xed ,
whi le t he t im e of paym ent (Ê + 1) is random . It s present value is
Z = Ð +' .
( 3 .2 . 1 )
T he random var i able Z r anges over t he values v , þ2, è~, . . ., and t he dist ribut ion
of Z is det er m ined by (3.2.1) and t he dist ri but ion of Ê :
P r ( Z = å " + ' ) = P r ( Ê = é ) = ap , q + „
for É = Î , 1, 2 ,
( 3 .2 .2 )
. T h e n et si n gl e p r em i u m is d en ot ed by À , an d gi v en b y
À = Å [å~ +' ) = ,' ) ,' è"+' „ð, î +„ .
Var (Z ) = E (Z ~) — À 2 .
a=o
( 3 .2 .3 )
T he vari ance of Z m ay be calculat ed by t he ident it y
( 3 .2 .4 )
Ch apt er 3. L i fe I nsur ance
24
R ep l aci n g î by e
w e see t h at
E ( g 2)
ö
—2á(Ê + 1) ]
( 3 .2 .5 )
which is t he net single prem i um calcul at ed at t wice t he or iginal for ce of int er est . T hus calculat ing t he var iance i s ï î mor e diffi cult t han calculat ing t he
net si ngle prem i um .
A n insur ance which provides for payment only if deat h occurs wit hin è
years is know n as à ter m i nsur ance of dur at ion è . For ex am ple 1 unit is payable
only if deat h occurs dur i ng t he fi rst è years, t he act ual t im e of pay ment st ill
being t he end of t he year of deat h . One has
î Ê + 1 for Ê = 0>1,
, è —1 ,
for Ê = ï , è + 1, è + 2,
0
( 3 .2 .6 )
T he net si n gl e p r em i u m i s d enot ed by À ' .-„-1. I t is
' ë1
æ:~
q — a=o
.% î k+ 1 up* Ü +à .
( 3 .2 .7 )
A gain t he second moment E (Z 2) equals t he net single prem ium at t wice t he
or igi nal force of int er est , as is seen fr om
z'2=
0
2á'" +» ~
Ê = 0,1" è- 1
for Ê = ï , ï + 1, ï + 2,
)
'I
)
1
( 3 .2 .8 )
3 .2 .2 Ð è ãå E n d ow m en t s
À ðèòå endowm ent of durat ion è prov ides for pay ment of t he sum insur ed only
if t he i nsur ed is alive at t he end of è year s:
0
î"
for Ê = 0, 1,
,n —1 ,
for Ê = è , è + 1, n + 2,
( 3 .2 .9 )
T h e net si n gl e pr em i u m i s d enot ed by À , .—
,';1 an d i s gi v en b y
À .„-'~ = î " „ ð , .
T h e for m ul a for t h e ÷àã1àï ñå of à B er n ou l l i r an do m var i ab l e gi v es
( 3 .2 .10 )
3 .2 . E l em en t ar y I n su r a n ce T y p es
25
3 .2 .3 E n d o w m e n t s
A ssum e t hat t he sum i nsured is payable at t he end of t he year of deat h , if t his
occur s wi t hi n t he fi rst è year s, ot herwise at t he end of t he n t h year :
Ð
Z
=
î"
+'
fo r Ê
=
0 , 1,
,ï
— 1
,
for Ê = ï ,, ï + 1, ï + 2,
(3 .2 .12)
T he net si ngle prem ium is denot ed by À , ,-„ ~. Denot i ng t he pr esent val ue of
(3.2.6) by Z >, and t hat of (3.2.9) by Z z, one may obviously wr it e
Z = Z, + Z, .
(3 .2 .13)
A s à con seq uence,
T h e pr o d u ct ß | 2 ð i s al w ay s zer o , h en ce
(3 .2 .15)
.14)
an d
V ar ( Z ) = × àã( ß ~) + 2 Ñ î ÷ ( Å | , Z z) + V ar ( Z q) .
C ov ( Z ) , Z g) = E ( Z ) Z g) — Å (ß | ) Å (ß ð) = — À ,' .-„-~ À , .~~ .
(3 .2 .16 )
T h e var i an ce of Z i s t hu s gi v en b y
V ar ( Z ) = × û ( Õ, ) + V ar ( Z q) — 2 À ~.-„~ À , .~-~ .
(3 .2 .17)
A s à consequence of t he last ident i ty, t he risk i n selli ng an endowment policy,
measured by t he variance, is less t han t hat in selli ng à t er m insurance t o one
person and à pure endowment t o anot her .
So far , for sim pl icity, we have assumed à sum insured of 1. If t he sum
insur ed is Ñ , t hen t he net single premium is obt ai ned by mult iplying wit h Ñ ,
and t he vari ance by mult iply ing wit h C ~.
L et us fi nall y consi der an ò year def er red whoLe Lif e i nsurance. It s present
val ue is
0
for Ê — 0, 1,
,ò —1 ,
v~ +~ for Ê = ò , ò + 1, ò + 2,
T he net single pr emi um is denot ed by
si ngl e premi um ar e
~À , =
~À , . A lt ernat ive for mulae for it s net
ó,î
À,+
~À , = À — À .~ .
(3.2 .19)
(3.2.20)
T he second moment E (Z ~) agai n equal s t he net single pr em ium at tw ice t he
origi nal force of i nt erest .
Chapt er 3. L ife I nsur ance
26
3 .3 I n su r a n c es P ay a b l e a t t h e M o m e n t o f D e a t h
I n t he previous sect ion it was assumed t hat t he sum insured was payable at t he
end of t he year of deat h. T his assumpt ion does not refl ect insurance pr act ice
in à r eal ist ic way, but has t he advant age t hat t he for mul ae m ay be evaluated
di rect ly from à È å t able.
L et us now assume t hat t he sum insured becomes payable at t he i nst ant of
deat h, |.å. at t ime Ò. T he pr esent value of à paym ent of 1 payable immedi at ely
on deat h is
z
u>
(3.3.1)
T he net si ngle prem ium is denot ed by À . Using (2.2.2) we fi nd t hat
À
=
~î/
6 ÄÐ p ~ gd t .
( 3 .3.2)
À pract ical approx im at ion may be derived under A ssump ti on à of Sect ion 2.6.
Wr it i ng
T = Ê + S = (Ê + 1) — ( 1 — S) ,
(3.3.3)
.Ð
and m aki ng use of t he assumed independence of Ê and S , as well as t he
uni form dist r ibut i on of S, so t hat
w e fi n d
E [( 1 + i )
~] = / r ~( 1 + ã)"„ Ûè = çö = —,
i
(3 .3 .4)
À = Å [è~ +~] Å [(1 + t )1 ~] = —À , .
T hus t he calcul at ion of À, is à simple ext ension of t hat of À , .
À sim ilar formula m ay be derived for t erm insur ances. For endowm ent s
t he fact or i / á is only used ø t he t erm insurance part :
A .~
—
À ,' .-„-1+ À , .~~
i
æâ ] +
À , .~ +
õ:g
e
—— 1
À .-„-1.
(3 .3 .6)
Let us fi nally assume t hat t he sum i nsured is payable at t he end of t he m t h
part of t he year in which deat h occurs, i .e. t i me Ê + ß < > in t he not at ion of
Sect ion 2.4. T he present val ue of à whole li fe insur ance of 1 unit t hen becomes
z =
+" " '
(3 .3 .7)
For calculat ion of t he net single prem ium we again use t he A ssumpti on à of
Sect ion 2.6. We wr it e
K + S - i = (K + 1)
(1- S - >)
(3 .3 .8)
3.4. Gener al T y p es of L i fe I nsur ance
27
i n (3.3.7) and use t he assumed i ndependence of Ê and Ó '" ~, as wel l as t he
equat ion
Å [(1 ~- ã)| ~~ ~] = ~~ ~~ = .
T hen we obt ai n
À ~ ~ = Å [î ~ + ~]E [( 1 + i ,)
~
] =
.< > À , .
( 3 .3 . 10 )
E q u at i on ( 3.3 .5 ) m ay b e v er i fi ed by l et t i n g m - + oo ø (3 .3 .10) .
3 .4
G en er a l T y p es o f L if e I n su r a n ce
We com m ence by consi dering à li fe insur ance wit h benefi t s var ying from year
Ñî year , and we assume t hat t he sum insur ed is payable at t he end of t he year
of deat h . I f c denot es t he sum i nsured dur i ng t he ó'ÑÜ year aft er policy issue,
we have
Z = ñ „ Ð +'
( 3 .4 . 1 )
T he di st ri but ion of Z and , in part icul ar , t he net si ngle prem i um and higher
moment s ar e easy t o calcul at e:
E [Z ë]
E
ë
~% + Î
ä
( 3 .4 .2 )
T he i nsur ance descr i bed m ay be represent ed as à combi nat ion of deferr ed
life insur ances, each of which has à const ant sum insur ed . T hus t he net si ngl e
pr em i um m ay be calculat ed in t he follow ing way :
E (Z ) = ñ~ À + (c~ — ñ1) öÀ + (ñâ — ñ~) ~~À +
(3.4.3)
I n t he ñàâå t hat t he insurance covers only à t erm of è year s, i .e. when ñ„ +1 —
ñ„ +~ —
— Î , t he insur ance m ay al so be represent ed as à combi nat ion of
t erm i nsurances st art ing im mediat ely :
E (Z ) = ñ„ À ' .~ + (ñ„ ~ — ñ„ ) À
~~+ (ñ„ ~ —ñ„ i ) À ' ~ ~-] +
. (3.4.4)
T he alt ernat ive repr esent at ions (3.4.3) and (3.4.4) ar e useful i n calcul at ing
t he net si ngle premi um , but not t he higher order moment s of Z .
If an insurance is payable im m ediat ely on deat h , t he sum insured m ay in
general be à funct ion c(t ) , t > Î , and we have
Z = c (T )v
( 3 .4 .5 )
T he net single premi um i s
E(Z ) = ~î/
c(t )v'.,ð ð , „ é .
( 3 .4 .6 )
C h a p t er 3 . L i f e I n su r a n c e
28
T he act ual cal culat ion of t he net single premi um m ay be reduced t o à calcul at ion in t he discret e model , see (3.4.2) wit h h = 1. From
E(Z )
=
Q E [Z )K = k] Pr (K = /ñ)
«=î
~», Å [ñ(é + ß ) ñè~~)Ê = k] Pr (Ê = k )
ê=î
) , , E [c(k + S) ( 1 + i ) ~ ~)Ê = 1ñ)è~+' Pr ( Ê = é) ,
(3.4.7)
a=o
w e ob t ai n
E (Z ) = ~ ñ~+| è" +' „ð ä, +„ ,
by d efi n i n g
ñ„ + | — E [c ( k + S ) ( 1 + i ) ' ~ )Ê = /ñ] .
( 3 .4 .8 )
«=î
( 3 .4 .9 )
T he condit ional dist r ibut ion of S, given Ê = k , is needed in order t o eval uat e
t he ex pression (3.4.9). T wo assum pt ions about mort al ity at fract ional ages
ar e appropriat e for m aking t his eval uat ion .
A ssumpti on à of Sect ion 2.6 gives
r>
ñá+| — /î c(k + u ) ( 1 + i )
" du ,
( 3 .4 . 1 0 )
w h er eas A ssu m p ti on Ü of t h e sam e sect i on r esu l t s i n
ñ»,+| — ó c(k + è ) ( 1 + i .) , Ä
" Vz+a+) p +a du .
Ð* + ê
( 3 .4 . 1 1 )
A s an ill ust rat ion , consider t he case of an exponent ially increasing sum
i nsur ed , c(t ) = å™. T his reduces formul a (3.4.10) t o
, > åá — åò
ñ~+| — å' á — r
( 3 .4 . 1 2 )
Not e t hat ò = 0 gives us (3.3.5) back . T he al t ernat ive formula (3.4.11) result s
in
ñ»,+| — å'
V +@
+-,'
å — ð +„ å
1 — ð +~ á + ~è, +~+ » — r
(If t he denominat or in (3.4.12) or (3.4.13) should vanish , t he quot ient s become
åá. T his wi ll happen if t he int egrand in (3.4.10) or (3.4.11), respect ively, is
independent of è ).
3.5.
3 .5 StSt
andan
ar dd Tar
y pes
d T
of yVar
p es
iab leofL i fe
V Iar
nsur
i ab
ance
le
L i f e I n su r an ce
29
W e begin by consider ing st andard types where t he sum insured is payable at
t he end of t he year of deat h . T he net si ngle prem ium m ay be readi ly calcul at ed
and is useful al so w hen t he sum i nsured is payable im m edi at ely on deat h .
Let us consider à standard i ncr easi ng whole life insur ance, wit h c
T he pr esent value of t he insur ance is
z = (ê + 1)Ð +' .
( 3 .5 . 1 )
T h e n et si ngl e p r em i u m i s d en ot ed by ( I A ) , an d i s gi ven by
(I A ) = a=o
~ , (é + 1)î ~+' ~ð, î +~ .
( 3 .5 . 2 )
For t he cor r esp ond i n g n -y ear t er m i n su r an ce w e h av e
(Ê + 1) u~ +'
0
for Ê = Î , 1,
,è —1
for Ê = ï , è + 1, è + 2,
( 3 .5 .3 )
It s net single premium is denot ed by (I A ),' ,+ and m ay b e obt ained by li m it i ng
t he sum m at ion in (3.5.2) t o t he fi rst n t erms. I nspired Úó (3.4.3) and (3.4.4)
we m ay writ e
(I A ) , .g — À , + , ~À , + .
+ „ , ~À , — ï „ ~À
( 3 .5 .4 )
an d
(û ).' .„ =
À.' .~ - À'
, — À'
, —" — À' , .
(3.5.5)
Not e t he diff er ence between t he sy mbols (I A ) ' . 1 and (I A ) .-„-1 - t he l at t er
bei ng equal Ñî t he sum of t he former and t he net single pr emi um for à ðø å
endow ment of è .
T he benefi t s of à standard decr easi ng t er m insurance decr ease linearly from
è to Î , hence
(è — Ê ) þ~ +'
Î
for Ê = Î , 1,
,è —1
for Ê = è , ï + 1, ï + 2,
( 3 .5 .6 )
St andar d decreasing insur ance is com monly used t o guarant ee repayment of
à loan , provided t hat t he debt out st anding also decr eases linear ly under t he
amor t isat ion plan of t he loan . T he ident it ies
ï —1
(Ð À ),' .g — ~, (è — é)þ" +' „ð, q +~
a=o
30
C h ap t er 3 . L i f e I n su r an ce
and
(È ),' .+ — À,' .~ + À' ~
+ À'
~+
+ À —
,]
are obvious.
L et us now assum e t hat t he sum insured is payable i mm ediat ely on deat h,
i .e. Z is of t he form (3.4.5) , wit h some funct ion c(t ) . For t hese i nsurances we
shall use Assumpt i on à of Sect ion 2.6 t hroughout t his sect ion .
I f t he sum i nsured is i ncr ement ed annually, we have c(t ) = [t + 1] , hence
Z = ( K + 1) v + .
( 3 .5 .9 )
T h e n et si n gl e p r em i u m i s d en ot ed by (I A ) , . C al cul at i n g t he ex p ect at i on of
Z = (Ê + 1)Ð +1(1 + i ) '~ ç
( 3 .5 . 10 )
and usi ng t he assumed independence of Ê and S as well as (3.3.4) , we obt ai n
t he pr act i cal formula
( 1À)
= —(1A) , .
(ÝÜ .ÚÚ)
Let us now consider t he sit uat ion where t he sum payabl e is increment ed q
t i mes à year , by 1/ q each t ime:
Z = ( Ê + s « ))~'ò .
( 3 .5 .12 )
T he cor responding net single pr em ium is denot ed by (1« )À ) . Not e t hat
(3.5.12) m ay be r ewrit t en as
Z = (Ê + ð ,ò
1~ò + 5'(~) ( 1 + t ) 1-ç à ~ + 1
( 3 .5 .13 )
I n com p u t i n g t h e n et si n gl e p r em i u m w e use i n d ep en d en ce an d t h e r el at i on
-(e)
[S«ain
) ( I + t ) ~ ~] = (I « ) ç)- ~=
Hence weEobt
i — È« )
á
(1« )À ) = (I A ), — À , +
È® á
ã — È« )
À, .
( 3 .5 . 14 )
( 3 .5 .1 5 )
Su b st i t ut i ng fr om (3.3 .5) an d (3.5 .11) , w e fi n d
( 3 .5 .1 6 )
T his l ast expression m ay be evaluat ed direct l y.
I n t he ñàçå of à cont inuously i ncreasing surh insured , c(t ) = t , ( he present
value is
Z = r v' ,
(3.5.17)
3 .6 .
R e c u r s iv e
a n d
t h e
n e t
F o r m
s i n g le
u l a e
p r e m
( ~
3 1
i u m
4 ) ,
=
—
( ~
4 ) , . —
—
A
,
+
ã
( 3 .5 . 1 8 )
is ob t ai n ed by t ak i n g t h e l i m i t q —~ oo ø (3.5 .16 ) .
T he for m u l ae (3 .5.11) , (3 .5 .16 ) an d (3 .5 .18) m ay al so b e ob t ai n ed by su b st i t u t i n g t h e ap p r op r i at e fu n ct i on c (t ) ø
(3 .4.10 ) .
A s an ex am p l e, t ak i n g
c(t ) = t l ead s t o
ñ~+ | — /
rr
—
i
i —á
(Â + è ) ( 1 + ã) ~ " du = k sq + (Is )q = É — + —
,
(3 .5 .19 )
w h i ch gi ves u s (3 .5 .18 ) .
Si m il ar eq u at i on s hol d for t h e cor r esp o n d i n g t er m i n su r an ces, for ex am p l e
( 3 .5 .2 0 )
O b t ai n i n g an el egant d er i v at i on of (3 .5.20) f r om (3.5 .16) i s l eft t o t h e r ead er .
F i n al l y w e con si der an n - y ear cont i n u ou s t er m i n su r an ce w i t h an i n i t i al
su m in su r ed of è , w hi ch i s r ed u ced q t i m es à y ear , Úó 1/ q each t i m e:
z=
(è + 1/ q — Ê — S ® )v T
for Ò ( è
0
fo r
Ò
>
è
( 3 .5 . 2 1 )
T h i s i n su r an ce m ay ob v i ou sl y b e r ep r esen t ed as t h e d i ffer en ce b et w een à t er m
i n su r ance w i t h con st ant su m of è + 1/ q i n su r ed , and à t er m i n su r an ce w i t h
i n cr easi ng su m i n su r ed . T h e n et si n gl e p r em i u m i s gi v en by
(Â ® À ) .-„-~ —
è + —
q
À , .-„-1 — (1® À ) , .~ .
( 3 .5 .2 2 )
3 .6 R e c u r si v e F o r m u l a e
R ecu r sion for m u l ae m ay b e used t o w r i t e al gor i t h m s, b u t t hey al so h ave i nt er est i n g t h eor et i cal i m p l i cat i on s.
W e st ar t by con si d er i n g à w h ol e l i fe i nsu r an ce of 1 p ay ab l e at t h e en d of
t he y ear of d eat h . O n e obv i ou sl y has t h e eq u at io n
À
,
=
v
q
+
î
À
, + ,
ð
.
( 3 .6 . 1 )
T hu s t h e val u es of À , can Úå f ou n d r ecu r si v ely , st ar t i n g w i t h t h e hi gh est p ossi b l e age. T h e r ecu r si ve eq u at ion m ay b e p r ov ed alg eb r ai cal l y by su b st i t u t ion
of
)ñÐ õ =
P z k - 1Ð õ + 1
( 3 .6 .2 )
32
Chapt er 3. L ife I nsur ance
in àll but t he fi r st t erm of t he sum m at ion (3.2.3) . À probabilist ic proof m ay
be bui lt on t he rel at ion
Å [ö~ +~] = ý Ðã (Ê = Î ) + vE [v~ [Ê ) 1] Ðã (Ê > 1) .
(3.6.3)
T he int erpret at i on of (3.6.1) is inst r uct ive. T he net' single premi um at age x
is t he ex pect ed value of à random vari able defi ned as discount ed sum insured
in ñàÿå of deat h , and discount ed net single prem i um at age õ + 1 in ñàÿå of
survival .
A not her inter pret at ion is evident if we wr it e (3.6.1) as
À
= è À , , + v (l — À , , ) ä .
( 3 .6 .4 )
F ir st t he am ount of À , +~ is reserved in any ñàçå (deat h or survival ) . I n ñàÿå
of deat h an addi t i onal 1 — À, +, is needed Ñî cover t he pay ment . T he net
si ngle premium of à one-year t er m i nsur ance of t hi s amount is v (1 — À + ) q .
A pplying (3.6.4) at age õ + k we obt ai n
À + „ — v À + ë+ ~ — þ ( 1 — À , + „ + , ) ä, +ë ,
k = 0 , 1, 2
( 3 .6 .5 )
M ult i ply ing t he above equat ion by v" and sum m i ng over al l values of k we
obt ai n
À , = ~) å~å(1 — À +ë+ ) q +ë ,
(3.6.6)
ë=î
so t hat t he net single prem i um at age x is evident ly t he sum of t he net single
prem iums of à series of one-year t er m insur ances.
Equat ion (3.6.4) m ay also be rewrit t en as
ä À , +, — ( À , +, — À , ) + v ( 1 — À +, ) q, .
(3.6.7)
T hus t he i nt erest ear ned has à dual effect : On t he one hand i t incr eases t he
net si ngle prem ium (from age x t o age õ + 1) , and on t he ot her it fi nances a
fi ct it ious one-year t er m i nsurance.
T he cont inuous count er par t t o à r ecursion formula is à different ial equat ion . Consider t he funct ion À „ t he expect ed value of ox . For h ) 0 we
have
Í å ï ñå
À
=
ö ö~ ]T < h] Ð~(T < h) + ö ~~ ]T > h] Ð~ (T > h )
Å[
IT < h] aq. + v À +„ „ð .
À +ë — .4. = (1
v" ëÐ*
) ,4.
+ë)—
——AÄ
= (b
+ p
ÀE
, [v
— p[" , < h] ëÄ* .
( 3 .6 .8 )
( 3 .6 .9 )
Di vision by h and let t ing h - + 0 y ields
d
( 3 .6 .10 )
3.6. Recu rsive For m ul ae
33
T h i s eq u at ion can b e r ecast i n à for m si m i l ar Ñî (3.6 .7) :
6 A, = —
d À, + p (1 — À, ) .
dz
( 3 .6 . 1 1 )
T he difFerent ial equat ion has à sim il ar int erpret at ion as (3.6.7) for an infi nit esim al ti me i nter val , whi ch is seen by mul ti plyi ng (3.6.11) by dt .
Only t he two sim plest types of insur ance have been form al ly discussed in
t his sect ion . T he int erpret at ions we have given for t he recursion for mul ae
r esp . difFerent ial equat ions above ar e, of cour se, al so valid for t he gener al ñàçå
and m ay t herefore be used t o der ive t he correspondi ng recursion formulae and
di f erenti al equati ons.
C h a p t er
4 . L ife A n n u it ies
4 .1 I n t r o d u ct io n
À li fe annuity consist s of à series of pay ment s w hich ar e m ade w hile t he
benefi ciary (of init i al age õ) lives. T hus à life annui ty m ay be represent ed as
an annuity-cert ai n wit h à t erm dependent on t he remai ni ng lifet ime Ò. I t s
present val ue t hus becomes à r andom vari able, which we shal l denot e by Y .
T he net si ngle prem ium of à l ife annuit y is i t s ex pect ed present value,
E (Y ) . M ore general ly, t he dist r ibut ion of Ó m ay also be of int er est , as wel l
as it s mom ent s.
À li fe annuit y m ay, on t he one hand , be t he benefi t of an insur an ce poli cy
as à combinat ion of pure endowm ent s; on t he ot her hand , periodic pay ment of
prem i um s can al so be considered as à life annuity, of cour se wit h t he algebraic
sign reversed .
4 .2 E l e m en t a r y L i f e A n n u i t i e s
We consider à èÜî 1å li f e annui ty- due whi ch provides for annual pay ment s of
1 unit as long as t he benefi ciary lives. Pay ment s are m ad e at t he t im e point s
Î , 1,
, Ê . T he present val ue of t his pay ment st ream is
Ó = ion
1 + of
î +t his
å~ random
+
+ è~
t he probabil it y di st ribut
var =
iablàe is given by
Pr (Y = à„
, ) = Ðã(Ê = k ) = ~ð, ä, +„ , é = 0, 1, 2>
( 4 .2 .1 )
.
(4.2.2)
T he net single prem ium , denot ed by à„ is t he ex pect ed val ue of (4.2.1) :
É, =also
', ) , be
à„ expressed
, ~ð ä, +„as.
T he present val ue (4.2.1) may
a=o
) ' ~ V I (K ) kj )
( 4 .2 .3 )
( 4 .2 .4 )
Chapt er 4. L ife A nnu it ies
36
w h er e I ~ is t he i nd icat or f u n ct i on of an ev ent À . T he ex p ect at ion of (4 .2 .4) is
à. = a=o
Å Ñ~" up* .
( 4 .2 .5 )
T hus we have found two expressions for t he net single premi um of à whole È å
annuity-due. In ex pression (4.2.3) we consider t he whole annuity as à unit ,
whi le i n (4.2.5) we t hink of t he annuity as a ser ies of pur e endowment s.
T he net single prem ium may also be ex pressed in t erms of t he net single
prem ium for à whole È å insur ance, t he l at ter being given by (3.2.1) and
(3.2.3) . Âó v irt ue of ( 1.7.2) , t he net single pr emi um (4.2.1) equals
Y =
1 —e~ +'
È
1 —Z
d
( 4 .2 .6 )
(T his formul a may also be obt ained by viewing t he li fe annuit y as t he di fference
of two per pet uit ies-due, one st art i ng at t ime Î , t he ot her at t ime Ê + 1.)
T aking ex pect at ions yields
1 — À,
A ft er t r ansfor m ing t his ident i ty t o
1 = d a, + À , ,
( 4 .2 .8 )
we may int erpret it in t erms of à debt of 1 uni t wit h annual int erest in advance,
and à fi nal pay ment of 1 unit at t he end of t he year of deat h. Of course t he
higher order moment s of Ó may also be derived fr om (4.2.6) , âî t hat , for
inst an ce,
àã(Ót)empor
=
T he present val ue of an n-×year
ary li fe annuity-due is
à
,
à-„-1
for Ê = 0, 1,
, ï —1 ,
for Ê = è , è + 1, è + 2,
( 4 .2 .9 )
( 4 .2 . 1 0 )
Sim ilar ly Ñî (4.2.3) and (4.2.5) t he net si ngle prem ium can be ex pressed by
ei t her
â —1
à, .~ — g à„ , ~öð, ä, +~ + à-„-1„ ð
a=o
or
â- 1
(4.2.12)
N ow w e h ave
( 4 .2 . 1 3 )
but here
Z iss def
i ned
by (3.2.12)
à consequence,
4.3.
Payment
made
more
Frequent.lyAtshan
Once à Year
37
1 — À . .~
õ
or
à
]
1
>
1 = d à .~ + À .-„-1.
( 4 .2 .1 4 )
(4.2.15)
T he correspondi ng im medi at e È å annuit ies provide for pay ment s at t imes
1, 2,
,Ê :
Y = v + v~ + .
+ v~ = à—1.
(4.2.16)
T he random var iables (4.2.1) and (4.2.16) differ only by t he const ant t erm 1.
T hus t he net si ngle pr emi um à, is given by
à ,
=
à
—
1
( 4 .2 .1 7 )
From equat ion (1.8.7) , wit h è = Ê + 1, we obt ai n
1 = i à + (1 + ã')è~ +' .
1 = i a, + ( 1 + i ) À ,
Þ
(4 .2
.2.19
.18))
Taking ex pect at ions yields
in analogy t o (4.2.8) .
T he pr esent value of an ò year defer red l ife annuity-due wit h annual payment s of 1 unit is
0
è
+ v~ +1 +
. + v~
for K = 0, 1,
,ò —1 ,
for Ê = ò , ò + 1,
(4 .2 .20 )
T he net single prem i um m ay be obt ai ned from eit her one of t he obvious rel at ions:
à,
ò ~à~
õ
»> -
= òò>à,
;~».
Ð õ è— à
à õ.„--)-òò
(4.2.22)
(4.2.21)
4 .3 P a y m e n t s m a d e m o r e F r e q u e n t l y t h a n O n c e à Y e a r
Consider t he ñàâå where payment s of 1/ ò ar e m ade m t imes à year , |.å. at
t imes Î , 1/ ò , 2/ ò ,
, as long as t he benefi ci ar y, init ial ly aged x , is alive. T he
net single pr em ium of such an annuit y is denot ed by à( ). I n analogy wit h
(4.2.8) we have
~(ò») - (» >) +
õ
À (ò» )
õ
( 4 .3 . 1 )
Ch ap t er 4. L ife A nnuit ies
38
Hence we obt ai n
à~ ~ = —1
— —1 À ~ ~.
d (m )
d (m )
(4.3;2)
T he equat ion m ay be int erpret ed in t he follow ing way : T he li fe annuity
payable ò t imes à year can be viewed as. t he di fference of two per pet uit ies,
one st ar t ing at t im e Î , t he ot her at t ime Ê + S<~ i . T ak ing expect at ions t hen
yields (4.3.2) .
Òî obt ai n ex pressions for a~ i in t erms of à, we use agai n A ssumpti on à
of Sect ion 2.6, so t hat (3.3.10) al lows us t o express A ~
i of (4.3.2) in t erm s
of À , ; if we t hen repl ace À in t ur n by 1 — d à , (4.3.2) becomes
..< >
di
~ ~- 1~~,. 1
..
i —i < >
* — ~ ~,. ~; ~~ )
( 4 .3 .3 )
I n t ro d u cin g
à (ò ) =
di
< ) .<
àï ä
ð (ò ) =
i —i < >
( 4 .3 .4 )
w e can t h en w r i t e (4 .3.2) m or e econ om i cal ly as
à ~ ) = à (ò ) à, — p (m ) .
( 4 .3 .5 )
For i = 5% t he coeffi cient s o (m ) and 9(ò ) are t abul at ed below , wit h ò = 12
(mont hly pay ment s) and w it h ò = î î (cont inuous payment s) .
m
o.(rn)
ð'(ò )
12
oo
1.000197
1.000198
0.46651
0.50823
P r act i cal ap p r ox i m at i on s i n fr eq u en t u se ar e
à (ò ) =
~ 1 , p (m )
2ò
( 4 ,3 .6 )
T hese approx im at ions are obt ai ned from t he T ay lor expansion of t he ñî åé cient s around á = Î , viz.
à (ò ) = 1 +
12ò ~ á +
ò
( 4 .3 .7 )
—1
(4.3.8)
A pparent ly t hese approx im at ions are useful only w hen t he force of int erest is
suffi ci ent ly small .
4 .4 . V a r i a b l e L i f e A n n u i t i e s
39
T he net single prem ium of à t em porar y l ife annuity-due wit h m t hly payment s can now al so be calculat ed wi t h t he help of à (ò ) and 9 (ò ) :
a (m ) à, — ))(m ) — „ ð, à" ( a (m ) à, „ — ))(m ) )
= ~(m) ©.:q —Ð( ) (>—~ ., ") .
(4.3.9)
T he net si ngle prem ium of an i m mediat e l ife annuit y (payment s ø ar rear s)
ò àó be cal culat ed i n t erms of t he cor responding È å annuity-due:
( 4 .3 . 1 0 )
Let us now ret ur n t o t he calcul at ion of à( ). Equat ions (4.2.8) and (4.3.1)
give t he ex act ex pression
à( ) =
((f
à,
.. — —1 ~ À ( ) — À , ) ,
( 4 .3 . 1 1 )
which m ay be int erpret ed in t he followi ng way : T he li fe annuity on t he left
hand side prov ides payment s of 1/ ò at t i mes 0, 1/ ò ,
, Ê + Ô ) — 1/ ò ;
i t ò àó be represent ed as t he di f erence of t wo t em por ary annuit ies, t he fi r st
pr ovi ding payment s at t i mes Î , 1/ ò ,
, Ê + 1 — 1/ ò , t he second providing
ðàóò åï Ì at t im es Ê + Ô ), Ê + S( ) + 1/ ò , . . . , Ê + 1 — 1/ ò . T his second
t em porar y annui ty m ay i n t urn be viewed as t he diff erence of t wo per pet uit ies
(one st ar t i ng at Ê + Ô ), t he ot her at Ê + 1) . T he fi rst t em por ary annuity
has t he âàò å present val ue as an annui ty-due which pr ov ides Ê + 1 annual
paym ent s of d/ d( ). Tak ing ex pect at ions of t he present val ues t hen y ields
(4.3.11) .
U nder A ssumpti on à, we m ay use equat ion (3.3.10) , gi vi ng
( 4 .3 . 1 2 )
t his for mula has an obvious int erpret at ion , whi ch is not t he ñàçå wit h t he
m at hem at ically equivalent formula (4.3.5) .
4 .4 V a r i a b l e L i f e A n n u i t i e s
We st ar t by consideri ng à life annuit y which pr ov ides pay ment s of r s, òä, ò2,
at t he t im e poi nt s Î , 1,
, Ê . T he present val ue is
Y = a=o
g è" ò„ 1(~<>„ ) ,
( 4 .4 . 1 )
Chap t er 4. L ife A nnui t ies
40
an d t h e n et si n gle p r em i u m
Å (Ó ) = ê=î
ð , è" ò„ „ð,
( 4 .4 .2 )
m ay be readily calcul at ed .
Take now à gener al li fe annuit y wit h pay ment s of Zp, þ| ~ , ë~~ ,
àÔt 1me
point s Î , 1/ m , 2/ ò ,
, Ê + Ó ~—1/ m . W e st art by replacing t he ò payment s
of each year by one advance payment w it h t he same present val ue:
r q — ', ~,' Ô
~=î
zq+,.~„ , , é = 0, 1, 2,
( 4 .4 .3 )
T he corr ect ion t er m in t he year of deat h am ount s t o à negat ive life insurance,
t he sum insured at t ime É + è , Î < è < 1 being t he present value of t he
om it t ed payment s:
ñ(Â+ è) = ~,; Ô "þ~+ ~ ,
(4.4.4)
pCJ
here J = J (u ) is t he set of t hose ó' ñ ~1, 2,
, ò — Ö for which j / òï > è . In
or der t o calculat e t he net single prem ium we use A ssumpti on à of Sect ion 2.6
and proceed al ong t he l ines of Sect ion 3.4. Subst it ut ing (4.4.4) in equat ion
(3.4.10) we obt ai n
( 4 .4 .5 )
T he net single prem i um for à gener al li fe annuity wit h payment s ò t imes à
year is t hus
Åv
k
~ .~
r a ~Ð
a=o
3+ 1
ñå+~è
àÐ 9*+a
a=o
wit h t he ñî å1éñ|åï Ôâ defi ned ø (4.4.3) and (4.4.5) .
T he ñàâå of à cont inuously payable annuity is obt ai ned by let t ing ò ~ oo .
L et t he payment rat e at t i me t be r (t ) . T he pr esent val ue is
Y = /î è'r(t)é .
( 4 .4 .7 )
T h e n et si n gl e p r em i u m
E (Y ) =
f
v ' ã (1) ,ð , d t
( 4 .4 .8 )
may St
beandard
eval uat
byof (4.4.6)
wit thy coeffi cient s
4.5.
T yed
p es
L i fe A ,nnui
41
pl
r » — / î v" r (k + è ) du ,
î
rl
ñ~+~ — / è ( 1 + ~) " r (k + è ) du .
( 4 .4 .9 )
( 4 .4 . 1 0 )
W e i llust rat e t he poi nt by à cont inuous life annuit y wi t h ex ponent i al
growt h ,
r (t ) = å" .
(4.4.11)
From (4.4.9) and (4.4.10) we obt ain
ò- á
r» =
a n d
î
á- ò
— ò
1
(î
å
(y
( 4 .4 . 1 2 )
.)
— ò )~
( 4 .4 . 1 3 )
for ò ô á, and
ò» — åb» , c»+i — -1 åá(»+ 1)
(4.4.14)
for ò = b. In t he ñàçå of à const ant pay m ent r at e (ò = Î ) , (4.4.12) and (4.4.13)
become sim ply
d
— —,
which is in accor dance wit hr »(4.3.12)
.ñ»+1 — p (oo) ,
4 .5
St a n d a r d
T y p es o f L ife
( 4 .4 . 1 5 )
A n n u it y
Consider à life annuity of t he for m (4.4.1) wit h r » — É + 1. I t s net single
prem i um , which we denot e by (I a) Ä m ay be readily cal cul at ed by m eans of
(4.4.2) .
À si m ple ident ity connect s (I n), and (I A ) . Replaci ng n by Ê + 1 in t he
ident it y
aq — d (~à )~ + nv
see ( 1.8.12) , and t aking expect at ions we obt ain
à
= d ( I ii ) , + ( I A ) , ,
( 4 .5 . 2 )
which r em inds us of (4.2.8) .
We consider t he ñàçå of ò paym ent s à year wit h annual increment s:
z»+~.g = 1 + 1 , j = 0, 1,
,ò —1.
( 4 .5 .3 )
Ch apt er 4. L i fe A nnu it ies
42
T he net single premi um of t his l ife annuit y is denot ed by (I ii ),i i . Repr esenting t his annuit y as à sum of deferred annuit ies, we obt ai n , wit h (4.3.5)
~=î
ä , ~ð, è ( à (ò ) à +~ —ð'(ò ) )
a=o
a (m ) ~) , „ð, è à, +„ — p (m ) ) , „ð, è
a=o
ê=î
à (ò ) (I ii ), — p (m ) à, .
( 4 .5 .4 )
T his expression m ay Úå eval uat ed di rect ly.
Let t i ng ò —+ oo we obt ai n t he corr esponding cont i nuous annuity wit h
paym ent ãàÑå r (t ) = ft + 1]. I t s net single premium is given by
(l a),
=
f
[I. ~- 1]þ' ,ð, é
à (î î ) (I ii ) — p (oo) à .
( 4 .5 .5 )
T h e p r esent v al u e of à con t i nu o u s l i fe an n u i t y w it h p ay m ent ãàÑå r (t ) = t
is
ò
à
~= / .ú = (è) = ~
T aki ng ex pect at ions y ields t he for mul a
à, — (I A ),
á
ó èò
( 4 .5 .6 )
( 4 .5 .7 )
T his ex pression m ay be eval uat ed using (3.5.18) and (4.3.5) wit h ò = oo.
T he der ivat ion of t he corresponding for mul ae for st andard decreasing l ife
and t em por ary annuit ies is l eft t o t he read er .
4 .6 R e c u r s i o n F o r m u l a e
W e shall rest rict our discussion t o recursion for mul ae for t he funct ion à, .
Repl acing „ð by ð, „ ,ð, +, in al l except t he fi rst t er m in (4.2.5) we éï ä
à, = 1 + è à, + , ð .
( 4 .6 .1 )
T he val ues of à, m ay Úå calcul at ed successively, st art i ng wit h t he highest
possible age.
A n equivalent expr ession is
à, = 1 + è à , + , — è à + , q
( 4 .6 .2 )
4.7. I nequali t ies
43
T he net si ngle prem ium is seen t o cover t he pay ment due at age x and t he
present value of t he net si ngle prem ium at age x + 1, less t he ex pect ed m or t ality
gai n .
A pplicat ion of (4.6.2) at age õ + é y ields
à +„
î àv"+~+|
1 —over
î à, +~+,
+~ain
.
We mult iply t his equat
ion—by
and —
sum
é t o äobt
( 4 .6 .3 )
à, = à-; 1 — a=o
~~ v /ñ+ 1 à- +„ +, q, „ .
( 4 .6 .4 )
T he net single prem i um may t hus be vi ewed as t he present val ue of à perpet uit y, reduced each year by t he expect ed m ort alit y gain .
Fi nal ly we can writ e (4.6.2) as
d a + ~ — 1 + ( à + | — à ) — v a~+ r q ,
( 4 .6 . 5 )
fr om which t he role of t he earned int erest becom es ev ident .
I n analogy wit h (4.6.5) one may der ive t he diff erent ial equat ion
by âè ÜâÈ Ñè é ï ~
d
á à, = 1 + —
dx à, — p, à
( 4 .6 . 6 )
d —
d
À = 1 — á à , — À = —á — à
( 4 .6 . 7 )
i n ( 3.6 .11) .
4 .7 I n eq u a l i t i es
T he net single prem ium à, is occasionally confused wi t h t he present val ue
à . T he values are diff er ent ; in fact one has t he inequalit y
à, (
I n
v iew
o f
( 4 .6 .7 )
a n d
t h e
id en t it y
v '
à,
=
1 — á à , , w it h
ñ )'
( 4 .7 .1 )
t
=
å
, a n
eq u iv a len t
inequal ity m ay be found :
A
> þ'*
( 4 .7 .2 )
Each of t hese inequalit ies is à dir ect consequence of Jensen 's inequal i ty ;
for inst ance t he second inequalit y means
Ch spt er 4. L ife A nnuit ies
44
E (vÒ) >
„ Å (Ò )
( 4 .7 .3 )
which is obvious since v' is à convex funct ion of t .
I n what fol lows we shal l gener ali se t hese inequal it i es. Consider t he net
singl e prem ium À , as à funct ion of t he force of i nt erest á:
A (á)
ð [å- áò] .
( 4 .7 .4 )
t his is t he L apl ace t r ansfor m of t he dist r ibut ion of Ò. We also defi ne t he
funct ion
/ (á) = ( E [e áÒ]} 1/ á
á> 0
(4.7.5)
For small values of á one may approximat e (4.7.4) by 1—á å, . T hus lims o f (á)
exist s, and has t he value
f (0 ) = åõ ð ( — å, ) .
( 4 .7 .6 )
L em m a : The f uncti on f (b) i s m onotone i ncr easi ng.
Òî prove t he lem m a we t ake two posi t ive number s è < to, and dem onst rat e
t hat
J en sen 's i n eq u al i t y i m p l i es
Ê [å- ø Ò ] =
f (v ~) > f (u ) .
E [( e
" Ò } ~è/ è ] >
( 4 .7 .7 )
( E [e
" Ò] p
/è
Í åï ñå
from which (4.7.7) follows. T his
pr oves
f (v/)
> ft he
(u)lem m a.
( 4 .7 .8 )
( 4 .7 .9 )
T he lem m a i mpl ies t hat / (á) > f (0) , hence
f (~) > f (o)' .
( 4 .7 . 10 )
From (4.7.6) one m ay der ive t he i nequal ity (4.7.2) once more.
A n i nt erest ing applicat ion uses t hr ee diff erent forces of int erest , bi < b <
áã. T he lem m a im pl ies t hat
f (bi )á < f (b)'s < ó (áã)
and t hus
( ß (á )}
/ á1 <
A , (b) < ( ß
( 4 .7 . 1 1 )
(á
) } á/ áã
(4.7.12)
which allows us t o est im at e À , (á) if t he values of À (b, ) and À (b2) are
known .
4 .7 . I n eq u a l i t i es
45
For i nst ance, let
À
À
= 0.41272 for i = 4% ,
= '0.34119 for i = 5% .
Bounds for t he net si ngle pr emi um s A ss and à~~ for i = 4-' % m ay now be
found . From (4.7.12) wit h
6r — l n 1.04 , á = ln 1.045 , áã — ln 1.05
we fi nd i m m ediat ely
0.37039 < A s< < 0.37904 .
T he ident it y ass — (1 — A ~ )/ 6 t hen gi ves
14.304 > à~~ > 14.107 .
Repl acing Ò by Ê + 1 and à~ Úó
à~ —
we obt ai n t he i nequal it ies
1 —î '
, t > 0,
(4 .7 .13)
( A. (6,)~'" 'à,
ass
<õ<>> à14
î(6)
' ..157
+<
| (
. A. (6.)~" "
(4 .7 .14)
by sim ilar ar gument s.
T he fi rst two der ivat ives of t he funct ion À , (á) are
À , (á) = —Å [Òî ~] = — (I A ), (6) ,
ä " (6)
Å ð ã„ ò~ > 0
(4.7.17)
T hus À , (6) is à monot onically decreasing, convex funct ion of á. Hence any
curve segm ent lies below t he secant ,
À (á) < áã — 6~ À (61) + áã — 6r À (áã) ,
(4 .7 .18 )
but above t he t angent s
À , (6)
>
À (61) — (6 — 61) (ÕÀ ), (6| ) ,
À , (6)
>
À , (áã) + (áã — 6) (I A ), (áã) .
(4.7.19)
Somet i mes one obt ains nar rower bounds fr om (4.7.18) and (4.7.19) t han from
(4.7.12) . I n t he exam ple above an im proved upper bound is obt ai ned from
(4.7.18) :
À ) < 0.37687 ;
T he lower bound for à „ is also improved :
Ch apt er 4. L ife A nnu i t ies
46
4 .8
P ay m en t s St ar t in g at
N o n -in t egr al A ges
T he i ni t ial age x will i n gener al not be int eger-val ued , unless it is rounded .
We shal l consider cal culat ion of à +„ for int egers x and 0 < è ( 1.
St ar t ing wi t h t he ident i ty
1«Ðç
—kp
u q,
z+«
— u+q,
kp 1
z «Pz
k
( 4 .8 .5
. 21 )
.3
.4
we use A ssumpti on à of Sect ion 2.6 t o fi nd
( 1 — è ä ) „ð +„ — ~ð ( 1 — è ä, +~) .
M ult i ply ing by v" and summ i ng over all /ñ we obt ai n
( 1 — è q, ) à, +„ — à — è ( 1 + i ) À .
Now we replace À
Úó 1 — d à t o obt ain t he desir ed for mul a:
(1 + ut ) à, — è ( 1 + i )
àõ+ «
1 — u q,
Âó means of (4.6.1) we can rew r it e t he above result as
à, +„ = ,
1
è
..
è(1
à, + .
ä ) ..
à +, ,
âî t hat à +„ is à wei ght ed mean of à, and à, +, .
I n pr act ical appl icat ions à, +„ is oft en approxi m at ed by linear int er polat ion , |.å.
à +„
(1 — è ) à + è à +~.
(4.8.6)
T he approxi m at ion is par t icular ly good for sm al l val ues of q , which is i mmedi at ely evident from (4.8.5) .
A s an ill ust rat ion we t ake à~â = 8.0960, G7~ — 7.7364, q~e = 0.05526. T he
result s ar e t abul at ed bel ow .
à 70+ from
(4.8.4),(4.8.5)
1/ 12
2/ 12
3/ 12
4/ 12
5/ 12
6/ 12
7/ 12
8/ 12
9/ 12
10/ 12
11/ 12
8.0676
8.0389
8.0099
7.9806
7.9511
7.9213
7.8912
7.8609
7.8302
7.7992
7.7680
from
(4.8.6)
à 70+
8.0660
8.0361
8.0061
7.9761
7.9462
7.9162
7.8862
7.8563
7.8263
7.7963
7.7664
4.8. P ay m ent s St ar t ing at N on-int egr al A ges
47
I f linear int er polat ion is al so perm it t ed for annuit ies wit h mor e frequent
pay m ent s,
à +„
(1 — u ) à~ ~+ u é +, ,
(4.8.7)
we obt ain from (4.3.5) t he pract ical appr oxi m at ion
à +~
„
à (ò ) ( 1 — è ) à,. + a (ò ) è à + > — ,Â(ò ) .
(4 .8 .8 )
Si m ilar rel at ions m ay be der ived for t he net single prem ium of w hole l ife
insurances st art ing at à fr act ional age. For inst ance, t he follow i ng is an immediat e consequence of (4.8.5) :
( 4 .8 .9 )
C h a p t er
5 . N et P r e m i u m s
5 .1 I n t r o d u ct io n
An insurance policy specifi es on t he one hand t he benefi t s payable by t he
insurer (benefi ts may consist of one payment or à series of payment s, see
Chapters 3 and 4), and on t he ot her hand the premium(s) payable by t he
insured. T hree forms of premium payment can be dist inguished:
1. One single premium,
2. Periodic premiums of à const ant amount (level premiums),
3. Periodic premiums of varying amount s.
For periodic premiums t he durat ion and frequency of premium payments
must be specifi ed in addit ion t o t he premium amount (s). In principle, premiums are paid in advance.
Wit h respect to an insurance policy, we defi ne t he total loss L t o t he insurer
to be t he difference between t he present value of t he benefi t s and t he present
value of t he premium payments. T his loss must be considered in t he algebraic
sense: an accept able choice of t he premiums must result in à range of t he
random variable L t hat includes negat ive as well as posit ive values.
À premium is called à net premi um if it sat isfi es t he åäèò à1åï ñå pri nci ple
E[L] = Î ,
(5 .1.1)
i .e. if t he expect ed value of the loss is zero. If t he insurance policy is fi nanced
Úó à single premium, t he net single premium as defi ned in Chapters 3 and
4 sat isfi es condit ion (5.1.1). If t he premium is to be paid periodically wit h
const ant amount s, equat ion (5.1.1) determines t he net premium uniquely. Of
course, in payment mode 3 (variable premiums), equat ion (5.1.1) is not suff icient for t he determinat ion of t he net premiums.
5 .2 A n E x a m p l e
Let us consider à t erm insurance for à life of age 40 (durat ion: 10 years;
sum insured: Ñ , payable at t he end of t he year of deat h; premium Ï payable
Chapt er 5. N et P rem i um s
50
annually in ad vance w hile t he insured is alive, but not longer t han 10 years) .
T he loss L of t he insurer is given by
C v+ +' — Ï à
—Ï à;—~
% + 1[
for Ê = 0, 1,
for Ê > 10 ;
,9 ,
( 5 .2 . 1 )
here Ê denot es t he curt at e-fut ure-l ifet im e of (40) . T he random var iable L
has à discret e dist r ibut ion concent rat ed in 11 point s:
Pr (L = C v +'~ - Ï à „
, ~) =
Pr (L = —Ï à; 01)
>p 40 q 40+ Ä ,
—
k =
0, 1
9
10Ð 40 .
(5.2.2)
W e shall det erm i ne t he net annual pr em ium . Fr om (5.1.1) one obt ai ns t he
condit ion
Ñ À 40 ~0~ — Ï à40 ~
10 — Î ,
resul t ing in
Ï
Ñ
À'
40 ~o
40:10] )
A s an illust rat ion , we t ake i = 4% and assum e t hat t he mort alit y of (40)
follows D e M oiv re's law wit h t erm inal age û = 100. T his som ewhat unreal ist ic assum pt ion allows t he reader t o check our calcul at ions wit h à pocket
calcul at or . W e have
âî t h at
À 40'1
' 10
— v + — v2 +
+ — v 10 = — à = 0.1352 ,
60
60
60
60 ~î !
5v1
10
î = 0.5630 ,
—
à 4~ î ~ —
(1 — À 40 01)))" = 7.8476 .
6
À 40~10
(5.2.î )
(5.2.6)
— 0.6982 ,
(5.2.4) t hen gives us t he net àø ø à1 prem i um :
Ï = 0.0172Ñ .
( 5 .2 .7 )
T he i nsurer cannot be ex pect ed t o pay benefi t s in ret ur n for net prem iums:
t here should be à safet y loading which refl ect s t he assumed risk . In what
follows à met hod for det ermining pr em ium s will b e demonst r at ed , which t akes
account of t he incur red risk .
Òî t his end prem ium s ar e det er mi ned by à uti hty f uncti on è ( ) ; t his is à
funct ion sat isfyi ng è' (õ ) ) 0 and è" (õ ) ( Î , and m easur ing t he ut i li ty t hat
]
5 .2 . A n E x a m p le
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51
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Chapt er 5. N et Prem ium s
52
overcharged , w hich com pensat es for t he rel at ively high fi xed cost s of t hese
policies.
Net premi um s ar e nevert hel ess of ut m ost im port ance in insurance pr act i ce.
M oreover , t hey are usually calculat ed on conservat ive assum pt ions about fut ur e int er est and mort al ity, t hus cr eat ing an i m pli cit safety load ing.
5.3 E l em ent ar y For m s of I n su r an ce
5 .3 .1 W h o le L i fe an d T er m I n su r an ce
We consider à whole l ife insur ance of 1 unit , payable at t he end of t he year of
deat h , whi ch is t o be fi nanced by net annual pr em i ums, which we denot e by
P, . T he loss of t he insurer is
( 5 .3 . 1 )
Fr o m (5.1.1) i t fol l ow s i m m ed i at el y t h at
Àõ
à,
ð
( 5 .3 . 2 )
Repr esent i ng t he prem ium paym ent s as t he diff er ence of t wo per pet uit ies (one
st ar t ing at t ime Î , t he ot her at t i me Ê + 1) , we obt ain
z
Ê + 1
æ
( 5 .3 .3 )
T h u s
( 5 .3 .4 )
T his equat ion shows t hat t he insur er r uns à gr eat er risk (at least expressed by
t he vari ance of L ) if t he insur ance is fi nanced by net annual pr emi um s rat her
t han by à net si ngle premi um .
Equat ion (5.3.2) can be used t o der ive two form ul ae for P, which can be
gi ven i nst r uct ive int erpret at ions. D iv iding equat ion (4.2.8) by à, we obt ai n
t he ident ity
1
—
à = d + P, .
( 5 .3 .5 )
T his ident ity has t he following i nt erpret at ion : À debt of 1 can be amort ised
by annual advance paym ent s of 1/ à, . A lt ernat ively one can pay advance
int erest (d) on t he debt each year . and t he am ount of 1 at t i me Ê + 1: t he
net annual pr em i um for t he cor responding life i nsur ance is P . T he i dent it y
(5.3.5) m eans t hat t he t he t ot al annual pay ment s are t he sam e i n ei t her way.
5.3. E lem ent ar y For m s of I nsur ance
53
T he ident ity (5.3.5) r emi nds us of anot her ident ity fr om t he t heory of
i nt erest ,
1 — d + —1
T he equivalent ident it y
( 5 . 3 . 67 )
a u]
which also has à si mil ar int er pret at ion (see Sect ion 1.8) .
Repl acing à, by (1 — À )/ d in (5.3.2), we fi nd
dA
1—À
Ð = d A , + Ð, À
(5.3.8)
may be i nt erpret ed as follows: À coverage of 1 unit ñàë be fi nanced by annual
pay ment s of Ð, ; on t he ot her hand , one can im agine t hat an am ount of À is
borr owed Ñî ðàó t he net single prem ium . I nt erest on t he debt of À , is pai d
annually in advance, and t he debt is repaid at t he end of t he year of deat h; t he
annual pr emi um for t he cor responding life insur ance is Ð, À , . T he ident ity
(5.3.8) shows t hat t he t ot al annual payment s are t he sam e ei t her way.
We shall consider à t erm insurance of dur at ion è (sum insured 1 unit ,
payable at t he end of t he year of deat h) . T he net annual prem ium is denot ed
by Ð ~.-„ ~. T he i nsurer 's loss is
v~ +' — Ð ', à
õß
ó ~ | ~
— Ð '.-„ 1à-„-1
for Ê = 0, 1,
for àÊ > è ,
, ï —1 ,
( 5 .3 .9 )
or , as in (5.3.3) ,
Ü = - Ð ~-; 1¸ =-1 + (1 + Ð ~~ ¸
.
, ~)å~ ~ ~1(ê ( - ,
( 5 .3 . 1 0 )
T he net annual premi um is, of course,
Ð,
À' .
*×
( 5 .3 . 1 1 )
5 .3 .2 Ð è ãå E n d o w m e n t s
Let t he sum i nsur ed be 1 unit and t he dur at ion è . T he net annual premium
( 5 .3 .1 3 )
is denot ed by Ð, .+ . T he loss of t he i nsurer is
—Ð , a
é :ra ~
for Ê = 0, 1,
è" — P . ~ à-„-1 for Ê > è .
T he net annual premi um i s obviously
À .'
àæ~
, n —1
Ch apt er 5. N et P rem ium s
54
5 .3 .3 E n d ow m en t s
T he net annual pr emi um is denot ed by Ð; „,~,. T he equat ions
À æ:«
,g
à .~
( 5 .3 . 1 4 )
àþ :« ]
and
~ õ :« ]
~ à :« ] +
p * :« ]
( 5 .3 . 1 5 )
are obvious. T he insurer 's loss is t he sum of (5.3.9) and (5.3.12).
I n analogy wit h (5.3.5) and (5.3.8) we have
1
= 4+ Ð g ,
( 5 .3 . 1 6 )
~. .-„ ~ = < À .~ + >. ,. ~À, .~ ,
(5.3.17)
w it h t he cor respondi ng i nt erpret at ions. Equat ion (5.3.17) can also be obt ai ned
by ad ding t he relat ions
P '.~
— datAion
. .~ sim
+ P.
each of t hese hav i ng an int
erpret
ilarö t o4 t hat of (5.3.8) .
( 5 .3 . 1 8
9 )
Ð,ô = d A , .~ + P, .—
„ ~À , .-,';1,
5 .3 .4 D ef er r e d L i f e A n n u i t i es
T he net annual prem ium payable duri ng t he defer m ent period for à È å annuitydue of 1 ð .à. st ar t i ng at t i me è , is P > à +„ .
5 .4
P r em i u m s P a id ò
T i m es à Y ea r
I f t he net annual prem i um is pai d by ò inst al l ment s of equal size, t he super scri pt " (m )" is is at t ached Ñî t he appr opri at e pr em ium sy mbol . T he net
annual prem iums
ð (ò )
õ ~ "@
p (tn)
p l (m )
õ :« ] '
õ :« ~
~~
ð
t (ò )
:+, y , ,
ã
inat or s of (5.3.2) , (5.3.11) , (5.3.13) , (5.3.14) . T he net an nual premi um of an
endowment payi ng 1 unit is for inst ance
p ( )
À
y -( )
( 5 .4 . 1 )
T h e ex p r essi o n m ay b e r ead i ly ev al u at ed by m ean s of for m ul a (4 .3 .9) .
5.5. À Gener al T yp e of L ife I nsur an ce
55
I n or d er t o co m p ar e Ð ; - w i t h P —
,„ ~
„ w e su b st i t u t e i n ( 5.4 .1)
an d o b t ai n
æ:g
n
õ :à ]
à- (rn
.~)
—
( 5 .4 .3
2 )
õ :g
a
à, ;„-1 — ,(3(ò ) À .-„~
I f w e n ow w r i t e t h e l ast r esu
*(~
:1 )l t i nd t/ hàe( for
)—
Ð
mõ:â
,9. (ò] )
Ð
-( ) Ð( )
t w o r easo n s for t h e r el at i on Ð -„- 1 (
p(
p
l
( 5 .4 .5 )
) Ð (~ ) Ð 1
Ð (ò
-- ) b ecom e ap p ar en t .
A n alo gou s r el at i on s h ol d for o t h er i n su r an ces, å.g .
Ð
à (ò~ ) Ð (ò@)
p ( ~n ) Ð (~ ) Ð
(5 .4.7)
Ð ô ~ —
~
Ð ô ~
— )9(ò ) Ð —
, 11
Ð ~ö .
(5 .4 .8 )
E q u at i on (5 .4 .6) is t h e l i m i t of (5 .4 .5 ) as è —~ î î . E q u at i on ( 5 .4 .5) i s t h e su m
of eq u at i on s (5 .4 .7) an d (5 .4.8) .
5 .5 À G e n er a l T y p e o f L i f e I n su r a n ce
W e r et ur n t o t h e gener al t y p e of l i fe i n sur an ce i n t r od u ced i n Sect i on 3 .4
( 5 .. 5L. 2et)
ñ, b e t h e su m i n su r ed i n t h e j t h y ear aft er p ol i cy i ssu e. W e assu m e t h at t h e
i n su r an ce i s t o b e fi n anced by an n u al p r em i u m s I I o, Ï 1, Ï ð,
p r em i u m d u e at t i m e k . T h e i n su r er 's loss i s
L = ca.+ r v Ê
Ê
+ 1
—%
~
, Ï ~ b ei n g t h e
I I qv é .
a=o
T h e p r em i u m s ar e n et p r em i u m s i f t h ey sat i sfy t h e eq u at io n
Å cq+ r v
a=o
&+ 1
Äp q + ~ — ~
Ï ),è k „ ð .
a=o
T h e m o d el i s m or e gen er al t ha n i t m ay ap p ear at fi r st gl an ce. I f n egat iv e
v al u es ar e p er m i t t ed for t h e Ï ~, i t i n cl u d es ð è ãå en d ow m ent s an d l i fe an n u i t i es.
For i n st an ce , t h e en d ow m ent of Sect i on 5 .3 .3 i s ob t ai ned by set t i n g
ñ1 — ñ2 —
= ñ„ = 1 ,
ñ„ + ~ = ñ„ + ð =
Ï „ = - 1 ,)
. = Î ,
Ï „ + , — Ï „ +, — " • — Î .
(5.5 .3 )
56
C h ap t er
5 .6
P o licies w it h
P r em iu m
5 . N et P r em i u m s
R efu n d
À large variety of insurance forms and payment plans occur in pract ical insurance. T his makes it impract ical to derive t he net single premium explicit ly for
every possible combinat ion. The fundament al rule t o be followed in à given
sit uat ion is t o specify t he insurer 's loss L , and t hen Ñî apply t he condit ion
(5.1.1). T his procedure will be illust rated wit h an example.
À ðèãå endowment wit h 1 unit payable after è, years is issued wit h t he
provision t hat , ø ñàçå of deat h before è, t he premiums paid will be refunded
wit hout interest . What should t he net annual premium be if t he premium
charged is t o exceed t he net annual premium by 40%? (T he 40% loading is
used t o cover expenses).
We let Ð denot e t he net annual premium. T he insurer's loss is obviously
(Ê + 1)(1.4Ð)è~+' —Ð à
Ü=
„
for Ê = 0, 1, . . . , è —1 ,
+ ~
(5.6.1)
for Ê > è .
T he expected loss is
1.4 Ð (I A),~.~ + À .„-' ~ —Ð à, .-„-1,
(5 .6 .2)
and applicat ion of (5.1.1) leads to t he premium
à~,~ —1.4 (1À), .q
5 .7
(5 .6 .3)
S t o c h a st i c I n t e r e st
The int erest rate t hat will apply in fut ure years is of course not known. Thus
it seems reasonable t o ask why fut ure interest rates have not been modelled
as à st ochast ic process. Two reasons have led us t o refrain from such a model:
1) Life insurance is part icularly concerned wit h t he long term development of
int erest rat es and ï î commonly accept ed st ochast ic model exist s for making
long t erm predict ions. 2) À reasonable assumpt ion is t hat t he remaining lifet imes of t he insured lives are, essent ially, independent random variables. With
à fi xed interest assumpt ion, t he insurer 's losses from diff erent policies become
independent random variables. T he probability dist ribut ion of t he aggregat e
loss can t hen simply be obt ained by convolut ion. In part icular, t he variance
of t he aggregat e loss is the sum of t he individual variances, which facilit ates
t he use of t he normal approximat ion. St ochast ic independence between policies would be lost wit h t he int roduct ion of à st ochast ic int erest ãàÑå, since all
policies are aff ect ed by t he same int erest development .
5 .7 .
S t o ch a st ic I n t er est
57
T h u s w e s h a l l c o n t i n u e u s i n g t h e a s s u m p t i o n o f à fi x e d i n t e r e s t r a t e . T h e
p r act ic al
sc en a r i o s .
say
ev a l u a t io n
It
u sin g i ,
i s al so
of
àë
i n su r a n ce
ð î ââ| Û å Ñî
a s t h e i n t er est
l et
co v er
sh o u ld
t h e i n t e r e st
assu m p t io n
fo r
a n a l y se d i f fe r e n t
a ss u m p t i o n v a r y
y ea r j .
T h is w o u ld
i n t er est
ov er
n ot
t im e ,
lead
to
m a t h e m a t i c a l c o m p l i c a t i o n s , b u t w o u l d m a k e t h e n o t a t i o n ò î ãå l a b o r i o u s , âî
t h a t w e sh a l l n o t f o l l ow
in t h is d i r ect io n .
C h a p t er
6 . N e t P r e m i u m R e se r v e s
6 .1 I n t r o d u ct io n
Consider an insurance policy which is fi nanced by net premiums. At t he
t ime of ðî éñó issue, t he expect ed present value of fut ure premiums equals the
expect ed present value of fut ure benefi t payment s, making t he expect ed loss
L of t he insurer zero.
T his equivalence between fut ure payment s and fut ure benefi t s does not ,
in general , exist at à lat er t ime. T hus we defi ne à random variable ,L as t he
diff erence at t ime t between t he present value of fut ure benefi t payments and
t he present value of fut ure premium payment s; we assume t hat ,L is not ident ically equal to í åãî , and we also assume t hat Ò > t . T he net premium reser ve
at time t is denot ed by ,V , and it is defi ned as t he condit ional expect at ion of
,L , given t hat Ò > t .
Life insurance policies are usually designed in such à way t hat t he net
premium reserve is posit ive, or at least non-negat ive, for t he insured should at
all times have an int erest in cont inuing t he insurance. T hus t he expected value
of fut ure benefi t s will always exceed t he expect ed val ue of fut ure premium
payment s. To compensat e for t his liability t he insurer should always reserve
suffi cient funds t o cover t he ñÈ Ååãåàñå of t hese expect ed values, |.å. t he net
premium reserve , V .
6 .2 T w o E x a m p l es
T he net premium reserve at t he end of t he kt h policy year for an endowment
insurance (durat ion: è, sum insured: 1 payable aft er n years or at t he end of
t he year of deat h, annual premiums) is denot ed by „ ~ ,- ~ and given by t he
expression
>V, .„ ~= À
„ ——„-1— Ð .-„-1É
„ , é = Î ,,l ,
,n —1.
Obviously sV , ) —0 because of t he defi nit ion of net premiums.
(6.2.1)
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60
C h ap t er 6 . N et P r em i u m R eser v es
( 6 .2 . 2 )
40:10 )
D ev el o p m en t o f n et p r em i u m r eser v e fo r an en d ow m en t an d a t er m i n su r an ce
40+ k:10 —k I
6.3. R ecur sive Consid er at ions
61
6 .3 R e c u r si v e C o n si d er a t i o n s
W e r et ur n Ñî t he gener al life i nsurance i nt r oduced i n Sect ion 5.5. T he net
prem ium r eserve at t he end of year k is, accor di ng t o t he defi nit ion ,
„ '; = ~~= î c»i ~~r „ „ó+ '1 , ð, +„ ä, +„ +, — ~ß= î, Ï »+, Ñ~ ,ð, +»
( 6 .3 . 1 )
I n or d er t o der i v e à r el at i on b et w een »V and »+ „ ~ , w e su b st i t u t e
« ð
õ ~ - »
=
ë ð
õ ~ - »
ð -
ë ð
õ +
» +
( 6 .3 .2 )
ë
i n ail except t he áãâÑ h t erms of (6.3.1) , and use 1' = ó' — Ü asÄsum m at ion
i ndex . T he result ing rel at ion bet ween »V and ~+ÄV is
»- l
ë- ~
»Ó + ~ Ï ».~ð ' .ð, » — ) ó
~=î
~=î
,~.ô +~ ð, » ä, . »
+ ëð , ð ~ »~,ëÓ . (6.3.3)
I t is not surpr ising t hat t hi s r elat ion has t he followi ng int erpret at ion : If t he
insured is al ive at t he end of year k , t hen t he net premi um reser ve, t oget her
wi t h t he expect ed pr esent val ue of t he pr em iums t o be paid duri ng t he nex t
h years is j ust suffi cient t o pay for t he life insur ance dur ing t hose years, pl us
à ðèãå endowment of ~+ÄV at t he end of year k + h .
À recursive equat ion for t he net premi um reserve is obt ai ned by let t ing
h = 1:
ÄV + Ï » — v [c»+q î , +„ + „+, V ð +Ä] .
(6.3.4)
T hus t he net prem ium reserve m ay be calculat ed r ecursively in t wo direct ions:
1) One m ay calculat e V , gV ,
successively, st art ing wi t h t he init ial val ue
pV = Î . 2) I f t he insurance is of fi nit e dur at ion è , t hen one m ay cal culat e
Ó , „ ~Ê ,
i n t his order , st ar t ing wit h t he known value of ÄV . For exam ple, |ï t he numer ical ex am ple of Sect ion 6.2 we have |î Ó = 1000 for t he
endowm ent , and |î ~ = 0 for t he t erm insur ance.
Equat ion (6.3.4) shows t hat t he sum of t he net prem i um reserve at t i me
k and t he prem ium equals t he expect ed pr esent val ue of t he funds needed at
t he end of t he year (t hese being ñ»+1 i n ñàçå of deat h , else »+| ~ ) . A not her
int erpr et at i on becomes ev ident w hen one wr it es
ÄV + I I » = þ ( »+ ã~ + (ñ»+ ã — ~+ , V ) ä, + „ ] .
(6 .3 .5)
T he amount of „ +, ~ is needed in any ñàâå. T he addit & nal amount needed if
t he i nsur ed dies , ñ»+1 — „+, V , is t he net am ount at r i sk.
Equat ion (6.3.5) shows t hat t he prem i um can be decom posed i nt o t wo
com ponent s, Ï » — Ï » + Ï », where
Ï » — »+ , V v — »V
( 6 .3 .6 )
C h a p t er
62
6 . N e t P r e m i u m R es e r v e s
i s t h e s a vi n g s p r e m i u m u s e d t o i n c r e a s e t h e n e t p r e m i u m
Ï "„ =
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6.4 T T
e viSur
v ival
6.4.
hehSur
val Ri
sk
R isk
63
T he der ivat ions of t he prev ious sect ion ar e vali d also i f ñ»+1 ( »+ V , i .e. if
t he net am ount at risk is negat ive. But i n t his ñàçå t he analysis m ay also be
modifi ed . W e st art by ex pressi ng (6.3.4) ss
»V + Ï „ = c»+ r v + ( „ + , V — ñ»+ | ) è ð , + » .
(6 .4 .1)
T he am ount of ñ»+1 is needed in any ñàçå; i n ñàçå of surv ival , an addit ional
am ount of »+~V —ñ»+1 falls due. T he fi nanci al t r ansact ions during year k + 1
m ay t hus be al locat ed part ly t o ðèãå sav ings, and part ly t o à ðèãå endow ment
wi t h à face am ount of „ +| Ó — ñ»+1. T he prem ium Ï „ m ay be viewed as t he
sum of à modifi ed savi ngs prem ium ,
Ï '„ = c»+ r v — ÄV ,
(6 .4 .2)
an d t h e sur vi val r i sk p r em i u m
Ï » — ( ~+ ~ — ñ»+ | ) è ð + ~ .
(6 .4 .3)
We not e t hat t he sav ings com ponent wil l oft en be negat ive, t oo. Equat ion
(6.4.1) m ay also be ex pressed as
Ï „ + dc»+ > — (ñ~+ ~ — ÄV ) + Ï "„ ,
(6.4 .4 )
à for mula w hi ch rem inds us of (6.3.9) .
T he decom posi t ion of prem ium i nt o (6.4.2) and (6.4.3) is not very comm on ,
and in w hat follows we shal l not use it .
6.5 T he N et P r em ium R eser ve of à
W hole L ife I nsur ance
Consider t he whole li fe i nsurance int roduced in Sect ion 5.3.1. I t s net prem i um
reser ve at t he end of year k is denot ed by ÄV and is by defi nit ion
ÄV, = À , +„ — P à, +„ .
(6 .5 .1)
W e shall derive som e equival ent formulae.
Replaci ng À +Ä by 1 — d à, +», we fi nd
»V = 1 — ( Ð, + d ) à + » .
(6 .5.2)
N ow , r ep l aci n g P + d by 1/ aÄ w e ob t ai n
» õ
à, +»
(6 .5 .3)
64
Chapt er 6. Net P rem i um Reser ves
T h e for m u l a
T h e id en t i t y Ð + „ à, + ~ — À , + ~ (À1 ,—
+~À—
, )À
/ d, an d à + > by ( 1 — À , + ~)
(6 /.5d.4)
.
m ay b e v er i fi ed i f w e r ep l ace à
by
w i t h (6 .5 .1) gi v es
õ
(6 .5 .5 )
an d
)n
à,d+ ~ .
F i n al l y w e r ep l ace à, + ~ ÜózV1/ (= p (, +ÐÄ++ö d—
) tÐ,
o fi
( 6 . 5 .6 )
Ð. +~ — Ð.
/ñ s
Ð
~ ä
T h e m u l t i t u d e of d i f er ent for m u l ae m ay âååò con f u si n g . A p ar t fr om
(6 .5 .1) t h e for m u l ae (6 .5 .2) , (6 .5 .5) sn d (6 .5 .6 ) ar e i m p or t an t b ecau se t h ey
ar e easi l y i nt er p r et ed an d b ecause t h ey m ay b e gener aIised t o ot h er t y p es of
i n su r an ce.
For m u l a (6 .5.2) ex p r esses t h e f act t h at t he net p r em i um r eser v e eq ual s t h e
su m i n su r ed , l ess t h e ex p ect ed p r esen t val u e of f u t u r e p r em i u m s and unu sed
i n t er est . T h i s r em i n d s us of t h e id ent i t y À , = 1 — dpi Ä w h i ch h as à si m i l ar
i n t er pr et at i on .
E qu at i on (6 .5.5) m ay b e i nt er p r et ed by r ecogn isi ng t hat t h e f u t u r e p r em i u m s of Ð m ay ser v e t o fi n an ce à w hol e È å i n su r an ce w i t h face am ou nt
p / p
h
,/
+~,• t h
e net pr em i u m r eserv e is t h en used t o fi n an ce t h e r em ai n in g f ace
am ou n t of 1 — Ð, / Ð, + „ .
I f t he w h ole l i fe i n su r ance wer e Ñî b e b ough t at age õ + /ñ t h e n et an nual
p r em i u m w ou ld b e Ð + ~. T h e ð ãåò ñèò di ff er en ce f or m ul a (6 .5.6 ) show s t h at
t he n et p r em iu m r eser ve is t h e ex p ect ed p r esent val u e of t h e sh or t fal l of t h e
p r em i u m s.
E q u at i on s (6 .5 .3) , (6.5.4 ) and (6 .5.7) ar e of l esser i m p or t an ce an d hav e
ï î obv i ou s i n t er p r et at i on . H ow ev er , t hey al l ow gen er al i sat ion Ñî en d ow m ent
i n su r an ce.
6 .6
N et
P r em iu m
R e se r v e s a t F r a c t i o n a l D u r a t i o n s
W e r et u r n t o t h e gen er al i n su r an ce d iscu ssed i n Sect i on 6.3 . L et u s assum e
t h at t h e i nsu r ed i s al iv e at t i m e /ñ+ è (É an i nt eger , 0 < u < 1) , and d enot e t he
net p r em i u m r eserv e by „ +„ ~ . Si m i l ar l y t o (6 .3 .5) , t he net p r em i u m r eser ve
ñàï b e ex p r essed by
„ +„ ~ = ~+ , ~ è ' " + (c<+ i — ~+ , ~ )è ' " , „ ä, + ~+„ .
( 6 .6 . 1 )
6.7. A ll ocat ion of t he Over al l L oss t o Ðî éñó Y ear s
65
A ssump ti on à of Sect ion 2.6 im plies
( 1 — è) ä, +~
1- è ×õ+ »+ è
— è ä, +»
w hich perm it s dir ect eval uat ion of Ä+ÄV .
We can al so ex press Ä+ÄV in t erms of »V . I n order Ñî do âî we subst i t ut e
(6.6.2) i n (6.6.1) and use (6.3.7) and (6.3.6) . W e obt ai n
»+ „ ~á = ( »V +
Ï ~) ( 1 + i ) " +
Ï ~ ( 1 + ~) " .
( 6 .6 .3 )
1 — u q +»
I n Sect ion 6.3 we saw t hat t he oper at ion in year k + 1 coul d be decom posed ;
equat ion (6.6.3) gives t he cor responding decom posi t ion at à fr act ional durat ion: T he fi r st t er m is t he balance of à fi ct i t ious savi ngs account at t ime k + è ,
and t he second t erm is t he part of t he risk prem i um which is st ill "unear ned"
at t ime k + u .
À t hird possible for mula is
Ä ÄV
" +"
=
( ÄV +
I I ») ( 1 + i )" +
1 —
1 — è ä, + „
Ä+ , V v
" .
( 6 .6 .4 )
1 — è ä, + „
T his shows t hat »+ÄV is à weight ed aver age of t he accumulat ed val ue of
( ÄV + Ï ~) and t he discount ed val ue of Ä+, V ; t he weight s are ident ical t o t he
weight s i n (4.8.5), for k = Î . Òî prove (6.6.4) , we replace Ï » by II » + I I »,
defi nit ion (6.3.6) t hen shows t hat (6.6.4) is equivalent t o (6.6.3) .
I n pract ical applicat ions an approx im at ion based on l inear i nt erpol at ion is
oft en used :
„ +„ Ó æ (1 — è ) ( »Ó + Ï „ ) + è »+, V .
(6.6.5)
Òî see how good t his appr oxi m at ion is, we replace Ï „ Üó II » + Ï '„ àï ñ1 „+, V
by ( „ ~ + Ï '„ ) ( 1 + i ). T he approxim at ion is t hen
»+ÄV = ( »V + Ï '„ ) (1 + ui ) + ( 1 — u ) Ï "„ ,
w hich per m it s di rect com par ison wit h (6.6.3)
(6 .6 .6 )
6 .7 A l l o c a t i o n o f t h e O v er a l l L o ss t o P o l i c y Y ea r s
W e cont inue t he discussion of t he gener al È å i nsurance. For k = Î , 1,
, we
defi ne Ë » Ñî be t he loss i ncurred by t he i nsurer duri ng t he year k + 1; t hus t he
beginni ng of t he year is used as reference poi nt on t he t im e scale. T hree cases
can be di st i nguished : 1) T he insured has died before t ime k , 2) t he insured
dies dur ing year é + 1, 3) t he i nsured survives t o k + 1. T he random vari able
Ë» is t hus defi ned Úó
Ë» —
0
c»+rv
( »V
»+rV v——
( »V+ + Ï Ï~)„ )
if K ( k — 1 ,
iiff Ê
Ê =
> /ñ
k +, 1 .
(6 .7 .1)
Chapt er 6. N et P rem ium Reser ves
66
R ep l aci ng Ï „ Üó Ï ~ + Ï "„ àï ñ1 u si n g (6.3 .6 ) , w e fi n d
Ë~ —
0
if K < k —1 ,
—
II
a
+
(ñ~+~
—
„+,
V
)v
ii ff Ê
—Ï ~
Ê =
> /ñ
1 +, 1 .
( 6 .7 .2 )
T hus, i f t he insured is alive at t ime /ñ, Ë~ is t he loss produced by t he one-year
t erm insurance cover ing t he net amount at r isk .
T he over al l loss of t he insurer is given by equat ion (5.5.1) . T he obvious
result
L = k~
Q Aqv"
( 6 .7 .3 )
may be ver ifi ed dir ect ly t hrough (6.7.1). Of course t he sum is fi nit e, r unning
from 0 t o Ê .
Usi ng (6.7.2) and (6.3.7) we fi nd
Å [Ë ~ )Ê > /ñ] = Î ,
( 6 .7 .4 )
E [A q] = Å [Ë ~[Ê > /ñ] P r ( K > /ñ) = Î .
( 6 .7 .5 )
w h i ch agai n im pl i es
W hile (6.7.3) is general ly valid , t he validity of (6.7.5) r equires t hat each year ' s
payment s ar e off set agai nst t he net premium reserve of t hat year .
T he classical H attendorff â Theorem st at es t hat
Ñ î ÷ (Ë /„ Ë / ) = 0 for /ñ g j ,
( 6 .7 .6 )
Var (L ) = a=o
~ , v~~× àã(A~) .
( 6 .7 .7 )
T he second for mula st at es t hat t he var iance of t he insur er 's over all loss
can be allocat ed t o individual policy years, and it is à di rect consequence of
t he fi rst formul a and (6.7.3) . T he first formula is not directly evident since
t he r andom variables Ao, Ë 1,
ar e not independent .
I n à proof of (6.7.6) we may assume /ñ ( 'j wit hout loss of generalit y. T hen
one has
Ñî ÷(Ë~, Ë )
=
Å [Ë~ Ë ]
Å [Ë~ Ë IK > j ] Ðã(Ê > j )
—Ï ~Å/Ë )Õ >j ~) Pr (K > j )
01
h er e (6.7 .4 ) hss b een used i n t h e l ast st ep .
(6 .7 .8 )
1
5
1
0
4
2
4
8
5
3
3
2
9
2
5
2
3
9
3
7
1
8
1
9
9
0
5
0
9
2
1
9
4
0
4
6
0
0
9
2
3
7
5
1
4
6
7
7
4
7
8
4
5
6
1
9
8
8
9
1
3
6
0
4
4
5
3
2
1
0
0
9
8
8
7
8
1
1
1
1
1
1
0
e
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n
a
r
u
s
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m
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w
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7
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for é = 1, 2 ,
6
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6 .7 . A l lo c a t i o n o f t h e O v er a l l L o ss t o Ðî 1|ñó Y ear s
6 7
T h e v a r i a n c e o f Ë ë m ay b e c a l c u l a t e d a s f o l l o w s :
(~~+~ — ÄÄ V )' ~ ð. +ë ~. +ë Ð~(Ê > a)
ã ã
Su b st i t u t i n g t h i s i n t o ( 6 .7 .7 ) w e fi n al l y fi n d
( 6 .7 . 1 0 )
( 6 .7 . 1 1 )
( 6 .7 . 12 )
68
C h ap t er 6 . N et P r em i u m R eser v es
W e see t hat t he vari ance of L is much sm aller for t he endowm ent (43 229)
t han for t he t erm insurance (108 465) .
Equat ion (6.7.10) is useful in evaluat i ng t he infl uence of t he fi nanci ng
met hod on t he var iance of L , when t he benefi t plan is fi xed . Consi der for inst ance à ðèãå endowment , wit h ñ1 — ñ2 = . . — Î . T he var iance of L increases
wit h t he net pr em i um reserve. T hus fi nancing by à net single prem i um leads
t o à great er var iance t han fi nanci ng by net annual pr em i ums.
6 .8
C o n v e r si o n o f a n I n su r a n c e
I n à t echnical sense t he net premi um reserve "belongs" t o t he i nsur ed and
m ay ø pri nciple be used Ñî help fi nance à modifi cat ion of t he insur ance poli cy
at any t ime.
À classical ex ample is t he conversion of an i nsur ance pol icy i nt o à paidup i nsur ance, |.å. one for which no furt her pr em i um payment s are required .
Consider à whole life i nsurance issued at age x wi t h à sum insured of 1 uni t ,
and fi nanced by annual prem iums of Ð, . A ssum e t hat t he i nsured is al ive at
t im e é, but , for what ever r easons, unable t o pay furt her prem i um s. In such à
sit uat i on t he net prem ium reser ve of ~V, coul d be considered as t he net si ngle
pr em i um for à w hole l ife insurance wi t h à sum insur ed of
„ ~ / À +~ — 1 — Ð, ! Ð +~ ,
( 6 .8 . 1)
see (6.5.5). Such conversions i nt o paid-up insur ance wit h r educed benefi t s are
very com mon for endow ment s (for which t he net premi um reser ve è subst ant iaI) .
À type of i nsurance known as "uni uer sal lif e" or ' f l exi bl e lif e", m ade possi ble by modern dat a processing, off ers t he insur ed à m ax i mum degr ee of
lf exi bi li ty. Í åãå t he insured m ay adj ust t he par am et ers of t he i nsur ance periodically (å.g. annual ly ) . T he insured who "ow ns" t he prem ium reserve of „ ~
at t ime É, m ay change any two of t he following par am et ers:
• Ï », t he next prem i um t o be paid ,
• ñ~+| , t he sum insur ed i n ñàçå of deat h during t he nex t year ,
• „ +, V , t he t ar get val ue of his "savi ngs" one year ahead .
T he t hir d par amet er is t hen det er m ined by t he recursive formula (6.3.4) . In
ot her wor ds, t he i nsured effect ively decides next year 's prem ium , as well as it s
decom posi t ion int o savings pr em ium and r isk prem i um . Cer t ai n r est r ict ions
ar e usually im posed Ñî reduce t he risk of ant iselect ion ; for i nst ance, t he new
sum i nsured (ñ~+| ) should not exceed t he form er sum i nsured (ct ) by mor e t han
à predet er m i ned percent age, which could , possi bly, depend on t he infl at ion
rat e.
6.9 Technical
T ech n Gain
i cal
6.9.
G ai n
69
Consider t he general life insurance of Sect ion 6.3, and let us assume t hat t he
insured is alive at t ime /ñ. We assume furt her t hat t he act ually earned int erest
rat e during year /ñ + 1 is i ' . T he techni cal gai n at t he end of t he year is t hen
"+'
J ( ÄV + Ï ~) (1 + t' ) — ñ~+|
( ( ~Ó + II <) (1 + i ') — k+1V
if Ê = /ñ ,
if Ê > /ñ+ 1
( 6 .9 . 1 )
Essent i al ly t here are two ways in w hich t his t echnical gain can Úå decomposed :
M et h od 1
Replacing 1 + Ã by (i ' —i ) + ( 1 + i ) ø (6.9.1) , one obt ai ns
Ga+> — ( kV + I I >) (i ' —i ) — Ë~(1 + i ) .
(6.9.2)
T he t echni cal gain t hus consist s of an i nvestment gai n and à mor tali ty äàò .
M et h o d 2
Since t he operat ion dur ing year é + 1 may be consider ed as part savings
and part insur ance, à reasonable approach is t o allocat e t he t echnical gain
accordingly :
G k+> Gk+1 + Gk+> .
(6.9.3)
Í åãå
"+'
j
) > /ñ+ 1
Ï "„ ( 1 +Ñi„+,
' ) — ( ÄV + Ï ~) (ã' —
if i Ê
( 6 .9 .4
.5 )
is t he gain fr om savings, and
j
Ï „ ( 1 + ã' ) — (ñ~+ — ÄÄ V ) È Ê = /ñ ,
i s t h e gai n fr om t h e i n su r an ce. T h e l at t er m ay agai n b e d ecom p osed in t o
Ñ ~+ , — Ï "„ (ã' — i ) — Ë ~( 1 + i ) ,
( 6 .9 .6 )
see (6.7.2) . T he l ast equat ion shows t he connect ion t o M ethod 1.
W hen t he t echni cal int erest rat e i is chosen conservat ively, t he t echnical
gain , respect ively Gz+Ä will usually be posit ive. I f t his gain is t o Úå passed on
t o t he insured t hrough increased benefi t s, t hen M ethod ß is preferabl e, since
t he gai n from savings may be wri t t en as
Ñ ~+,
„+, V v (iuniform
' —i ) . ly by
T he fut ure benefi t s may t hen
be —
i ncreased
v (i ' — i ) 100% ,
( 6 .9 .7 )
( 6 .9 .8 )
òî
Chapt er 6. Net Prem i um R eser ves
prov ided t hat t he insured agrees t o fut ure prem ium s being increased by t he
ÿàï äå fact or . A s à result of t hi s profi t shar ing, t he i nsured wi ll obt ai n . à
modifi ed insur ance policy for which
Ck + ä+ ÿ =
5 (1 +
'4 ) ñ ay ä.ä.h
Ï ä,
ä, =
5(1 +
ã~) Ï ~, ä,
( 6 .9 .9 )
for Ü = 0, 1,
. T his will be t he ñàçå if t he i nsured is al ive at t he end of t he
year . I n ñàâå of deat h (Õ = k ) , t he gai n fr om savings Ñ ~+ä m ay be paid in
addi t ion t o t he sum i nsured of ñ~+ä.
6 . 1 0 P r o c e d u r e f o r Ð è ãå E n d o w m e n t s
Consider à pur e endow ment (ñä — ñ~ — '
— Î ) . T he t echni cal gain at t he
end of year É + 1 is
( ÄV + Ï ä,) (1 + þ' )
"+'
( ÄV + Ï „ ) (1 + ã' ) — Ä V
if Ê = k ,
if Ê > 1 + 1 .
(6.10 .1)
Si nce it is desirable t o have an invest ment gai n only in t he ñàÿå of surv ival
(Ê > k + 1) , we decompose t he t echnical gain in à slight ly differ ent way :
~ú +ä = ~ú +ä + ~ à+
w it h
an d
"+'
/
f 0
l
„ +, V u(i ' — i )
ð +ä, „ +, Ó è( 1 + i ' )
(6 .10 .2)
if K = k ,
if Ê > k + 1 ,
(6 .10 .3)
if Ê = É ,
(6 .10 .4)
1 —ü. „„„ ~. (1+ ' ) if ~ ~+ ~.
T he proof of t hi s decom posi t ion follows from (6.10.1) and t he fact t hat
( ä, ~ + Ï „ ) = ð + „ „ + , Ê è ,
( 6 .10 .5)
see (6.3.4) . Not e t hat t he expect at ion of Ñ ~+ä is zero, which i s not t he ñàÿå
wit h t he expect at ion of G>+, .
If t he i nsur ed sur vives, t he gain given by (6.10.3) m ay be used t o i ncrease
t he benefi t s, prov ided fut ure pr emi ums are i ncreased accordingly, by à fact or
det er m ined by (6.9.8).
Si m il ar der i vat ions Ñî t he above m ay be mad e for 1â(å annui ti es, si mply
by equat ing r q, t he cont r act ual ly agreed payment at t im e k , t o —Ï ä,. For
inst ance, if à pension fund has an invest m ent y ield of i ' duri ng à year , t he
int erest gained from t he annuit ies m ay be used t o incr ease al l annuit ies by t he
ÅàñÑî ä ð å ï äï (6.9.8) .
6 . 1 1T he
T hCont
e C
o n t iModel
n u o u s M o d el
6.11.
inuous
71
Let us fi nal ly consider t he cont i nuous counter par t t o t he gener al li fe insur ance
of Sect ion 6.3.
T he insurance is now det erm i ned by t wo funct ions, t he am ount i nsur ed c(t )
and t he prem ium r at e II (t ) , bot h at t he mom ent t , t > Î . T he net prem ium
reserve at t i me t is
V (t ) =
I
c ( t + Ü) î " ëð + ~ ð * + ~+ ëÌ
~î
— (
I I ( t + h ) v " ëð * + ñd h .
(6 .11.1)
~î
T he prem ium r at e can be decom posed i nt o à savi ngs com ponent ,
an d à r i sk co m p onen t ,
Ï ' (t ) = V ' (t ) — áÓ (1) ,
Ï " ( t ) = ( ñ(1) — V (t ) ) ð , +~ .
( 6 . 1 1 ..3
2)
T hat II (t ) is t he sum of t hose two com ponent s est ablishes Thi el e s Di f f erent i al
E quati on:
I I (t ) + áÓ (1) = V ' (t ) + Ï " (t )
(6.11.4)
i t is t he cont i nuous version of (6.3.9) and (6.3.10) and has à si m il ar i nt erpret at ion .
In t he speci al ñàçå t hat
c(t)) == 01, , I II
c(t
I (t(t)) == -Î 1, , VV(t(t) )== Àà +,
+,, ,
( 6 . 1 1 .6
.5 )
equat ion (6.11.4) leads t o (3.6.11) . If
equat ion (6.11.4) confi rms (4.6.6) .
W orking wit hi n t he cont inuous model sim pl ifi es m at t ers. T here is for
inst ance only one m et hod for analysi ng t he t echnical gain , inst ead of t wo, as
i n t he discr et e model of Sect ions 6.9 and 6.10.
We assum e t hat t he i nsured is alive at ti me t , and t hat t he act ual force of
i nt erest at t ime t is á(1) . T he t echnical gain i n t he infi nit esim al t i me int erval
from t t o t + dt , which we denot e be G (t , t + dt ) , can be decom posed int o
G (t , 1 + dt ) = Ñ ' (t , t + dt ) + G (t , t + dt ) ;
here
G' Ï(t",(tt )dt
+ dt ) = (b(t if
)—
T á)>Vt (t+) dt
dt ,
is t he i nvest ment gain , and
f - (c(t ) — V (t ))
if t ( Ò ( t + dt ,
(6.11.7)
( 6 . 1 1 .9
.8 )
72
C hapt er 6. Net P rem i um Reser ves
is t he mort al ity gain . Not e t hat t he probabili ty of deat h is )(á ». dt , and t he
probabi li ty of survival is 1 —)(á, +áé , so t hat t he ex pect ed val ue of G" (t , t + dt )
is zer o. Not e also t hat
(6.11.10)
(6.11.11)
( 6 . 1 1 . 12 )
ø analogy wit h (6.7.7) and (6.7.10) .
Using à li fe annuity as an ex am ple, we shal l demonst r at e how t he invest m ent gain m ay Úå used t o increase t he benefi t s cont inuously. A ssume t hat a
cont inuous life annuity wit h const ant paym ent ãàÑå r (t ) is guar ant eed at t i me
t . T he net premi um r eserve at t ime t is t hus
( 6 . 1 1 . 13 )
At t i me t + dt t he paym ent rat e is t o Úå i ncreased t o r (t + dt ) = r (t ) + r ' (t )dt ,
t he cost of which must be covered by t he i nvest ment gai n . T hi s leads t o t he
condit ion
G' (t , t + dt ) = r ' (t ) dt à +, .
(6.11.14)
Usi ng (6.11.8) and (6.11.13) we obt ain à diff erent i al equat ion for r (t ) , viz.
w i t h so l u t i on
(á(Ñ) = á)ò(Ñ) = ò' ß ,
r (t ) = r (0) exp ( f (á(â) — á)áâ) ,
( 6 . 1 1 .1 5 )
( 6 . 1 1 . 16 )
which is i n accor dance wit h t he result der ived at t he end of Sect ion 6.10.
We have seen in t his and t he last two sect ions how t he invest ment gai n
can be used t o increase t he benefi t s on an i ndiv idually equit able basis. On
t he ot her hand , it is impossi ble t o pass on t he mort ality gain t o t he insured
on an i ndi vidual basis: D eat h of t he i nsured causes à mort ality loss (in ñàÿå
of life insurance) or à mor t ality gain (i n ñàçå of an annui ty ) , which nat urally
cannot be passed on t o t he i nsured .
It is, however , possi ble t o pass on mort ali ty gain (or loss) t o à gr oup of
insureds. T his wi ll be demonst rat ed by an ex am ple which is rem iniscent of
t he hist orical Tont i nes.
Consider à group consist i ng init ially of è persons; all have t he same init ial
age x and are i nit ially guarant eed à È å annui ty of const ant rat e 1. It has
been agr eed Ñî ðàçÿ on any m ort al it y gai n (or loss) Ñî t he annuit ant s i n t he
6.11. T he Cont i nuous M odel
73
f o r m o f i n c r e ased ( o r d e c r e a sed ) f u t u r e p ay m e n t s . W h a t w i l l b e t h e v a l u e o f
r q ( t ) , t h e a n n u i t y r a t e a t t i m e t , i f t h en o n l y É o f t h e i n i t i al l y n p er so n s a r e
st i l l a l i v e ?
A ssu m i n g t h a t lñ p er so n s a r e a l i v e a t t i m e t a n d t h a t al l su r v i v e t o t i m e
t + d t , t h e m o r t a l i t y g a i n w i l l b e n eg a t i v e ; p e r su r v i v o r i t a m o u n t s Ñî
6 (t , t + dt ) = I I ' (t ) dt = —r a(t ) à +,, ð , + , é ,
( 6 . 1 1 .1 7 )
see (6 .11.9) . T h e r ed uct i on i n t h e an n u i t y r at e t h en fol l ow s f r om t h e cond i t i on
( 6 .1 1 .18 )
w h i c h , i n t u r n , i m p l i es t h e d i f er e n t i a l eq u a t i o n
ò „' ( Ñ) = —r a ( t ) ð , „ .
( 6 .1 1 .1 9 )
I f o n e o f t h e é p er so n s d i es at t i m e t , a n i m m e d i a t e m o r t al i t y g a i n o f r q ( t ) à + ,
r esu l t s ; t h i s i s d i st r i b u t ed a m o n g t h e É — 1 su r v i v o r s Ñî i n c r e a se t h e a n n u i t y
r a t e . T h e n ew a n n u i t y r a t e s f o l l o w f r o m t h e c o n d i t i o n t h a t t h e n et p r e m i u m
r eser v e sh o u l d b e u n ch a n g e d :
k r >(t ) à , +, — (É — 1)ò~ ~(Ñ) à +, .
( 6 . 1 1 .2 0 )
T hu s on e m ay w r i t e
r g ~( Ñ) =
é
é — 1r g( t ) ,
É = 2,3,
,è .
( 6 . 1 1 .2 1 )
T h e ex p l i c i t so l u t i o n i s f o u n d u si n g ( 6 .1 1 .19 ) , ( 6 . 1 1 .2 1 ) a n d t h e i n i t i a l c o n d i t i o n r Ä( 0 ) = 1 Ñî b e
( 6 . 1 1 .2 2 )
I s i s e a sy t o ch eck a n d n o t a t a l l su r p r i si n g t h a t t h e o r g a n i ser o f su c h a n
a r r a n g e m e n t m ay i n f a c t b e c o n si d er ed t o b e f u n c t i o n i n g p u r el y a s à b a n k e r
a s l o n g a s at l ea st o n e p e r so n l i v es , a n d fi n a l l y t o b e m a k i n g à p r o fi t o f
ò~(~) à + , — è , ð , à + , ,
i f z den ot es t he t i m e of t h e l ast p er so n ' s d eat h .
( 6 . 1 1 .2 3 )
C
h
a p t e r
7 .
M
u l t i p
l e
D
e c r e m
e n t s
7 . 1 T h e M o d el
I n t h i s c h a p t er w e ex t en d t h e m o d el i n t r o d u c ed i n C h a p t er 2 a n d r ei n t er p r et
t h e r e m a i n i n g l i f et i m e r a n d o m
v ar ia b le Ò .
A s s u m e t h a t t h e p e r s o n u n d e r c o n s i d e r a t i o n i s i n à s p e c i fi c s t a t u s a t a g e
õ . T h e p e r so n l e a v e s t h a t s t a t u s a t t i m e Ò d u e t o o n e o f ò
c a u se s o f d e c r e m e n t
p a ir o f r a n d o m
(n u m b er ed c o n v en i en t l y f r o m
v a r i a b l e s , t h e r e m a i n i n g l i f e t i m e i n t h e s p e c i fi e d s t a t u s Ò a n d
t h e c a u s e of d e c r e m e n t J .
I n à c l a s si c a l ex a m p l e , d i sa b i l i t y
i n s u r a n c e , t h e i n i t i a l st a t u s i s
a n d p o ssi b l e c a u se s o f d e c r em e n t a r e
In
m u t u a lly ex cl u siv e
1 t o ò ) . W e sh a l l st u d y à
an o t h e r set t i n g Ò
"D i s a b l em en t " a n d
is t h e r em ain in g li fet im e of
t w ee n t w o c a u ses o f d e c r e m e n t , d e a t h b y
"A c t i v e " ,
" D ea t h " .
( s ) , d i st i n g u i sh i n g b e-
" A c ci d e n t " a n d b y
" O t h er
ca u s es " .
T h i s m o d el is a p p r o p r i a t e i n c o n n ec t i o n w i t h i n su r a n c es w h i ch p r o v id e d o u b l e
i n d em n it y o n ac ci d en t al d eat h .
T he j oint
p r o b ab ilit y
d i st r i b u t i o n o f Ò
t h e d e n s i t y f u n c t i o n s ä ä ( é) ,
,g
( t ) , âî
and
J
can
b e w r it t en in t er m s o f
t h at
gz(t ) dt = P r (t < Ò < t + dt , J = j )
is t h e p ro b a b ilit y
(t , t +
o f d ecrem en t
by
c a u se j
ø
(7 .1.1)
t h e i n fi n i t e s i m a l t i m e i n t e r v a l
d t ) . O b v i o u sl y
ä ( ~) =
I f t h e d ecr em en t o cc u r s at
ä ä ( ~) +
' ' ' + ~
( ~) .
t im e t , t h e co n d it ion al p ro b ab ility
( 7 . 1 .2 )
of j
b ein g t h e
c a u se o f d e c r e m e n t i s
P r(J =
ß Ò =
t) =
=
ä ( ~)
u (t )
( 7 .1 .3 )
W e i n t r o d u c e t h e sy m b o l s
or , m or e gen er al l y , , î , , +, —
,q, P r =(T P<r (T
) > s) .
s +< t t, , ÓJ== ßj Ò
( 7.1.5)
7 .1.4)
Chapt er 7. M ul t ip le D ecr em ent s
76
T h e l at t er p r ob ab i l i t y i s cal c u l at ed as fol l ow s:
( 7 . 1 .6 )
7 .2 F o r ces o f D ecr em en t
For à life (s ) t he for ce of decrement at age õ + t ø respect of t he cause ó' is
defi ned Úó
) = à,'ð.
(~)
*" = 1-à (~
G,(t)
T he aggr egat e force of decrement is
Ó õ+ 1
Ð' 1,s + t +
'
' +
p m ,z + t >
( 7 .2 .2 )
see (7.1.2) and defi nit ion (2.2.1).
Equat ion (7.1.1) can be ex pr essed as
P r (t < Ò < t + dt , J = ó) = ,ð ð , ~ddt .
( 7 .2 .3 )
Fur t hermor e,
Pr (J = ß Ò = t ) =
/4 +~
I f al l forces of decr em ent ar e known , t he j oint dist ribut ion of Ò and J may
be det erm ined by fi r st using (7.2.2) and (2.2.6) t o det er mi ne ,ð, and t hen
det erm i ning ä .(t ) fr om (7.2.1) .
7.3 T he Cur t at e L ifet im e of (õ)
If t he one-year probabi lit ies of decrement ,
ä * +k — P r (T < é + 1, J = ß Ò ) é )
( 7.3 .1)
ar e known for é = Î , 1,
and ~ = 1,
, ò , t he j oint probability dist ribut ion
of t he cur t at e t ime Ê = [T ] and t he cause of decr ement J m ay be eval uat ed .
St art by observi ng t hat
4 +k =
9 1,õ + ~ñ +
' ' ' +
× ï ,* + k ~
( 7 .3 .2 )
fr om w h i ch ~ð , ñàë b e cal cu l at ed ; t h en
P r (K = é , J = j ') = kp , q,' + „
for k = 0 , 1,
. an d ~ = 1,
,ò .
( 7 .3 .3 )
7.4. À Gener al T y pe of I nsur ance
77
T he j oi nt dist r i but ion of Ò and J can be comput ed under suit able assumpt ions concerning pr obabili t ies of decrement at fr act ional ages. À popular assum pt ion is t hat „ ä +„ is à linear funct ion of è for 0 < è < 1, /ñ an int eger ,
i.e.
„ ä , + „ — è î , +~ .
( 7 .3 .4 )
T his assum pt ion implies A ssumptii on à of Sect ion 2.6, which may Úå verifi ed
by summ at ion over all / . From (7.3.4) follows
ä ( É + " ) = ~ð , à , + ~ .,
( 7 .3 .5 )
t oget h er w it h t h e id ent i t y „ +„ ð, = ~ð , ( 1 — è q + „ ) t h i s y i eld s
qg,*+~
Ð~ä+é+è = 1 —
è ä, +~
( 7 .3 .6 )
A ssum pt i on (7.3.4) has t he obvious advant age known fr om Chapt er 2, t hat Ê
and S become i ndependent random var iables, and t hat S will have à unifor m
dist ri but ion between 0 and 1. In addit ion one has
Ðã( .Ó = ß Ê = /ñ, s = è) =
×õ+ è
( 7 .3 .7 )
à consequence of (7.2.4) and (7.3.6) . T he last r elat ion st at es t hat t he condit ional probabi lity of decrement by cause ó' is const ant duri ng t he year . I n
closing we summarise t hat S has à uniform dist r ibut ion bet ween 0 and 1, independent ly of t he pair (Ê , ,Ó), and t hat t he dist ribut ion of (Ê , .Ó) is given by
(7.3.3) .
7 .4 À G en er a l T y p e o f I n su r a n ce
Consider an insurance which prov ides for payment of t he amount ñ, ~+| at t he
end of year /ñ+ 1, if decrement by cause j occurs during t hat year . T he present
value of t he insured benefi t is t hus
( 7 .4 . 1 )
an d t h e n et si ngl e p r em i u m is
E (Z ) = ~ ,') ñ „ +| è"+' „ð, q,. +„ .
j = s a= o
( 7 .4 .2 )
I f t he i nsurance provides for payment immedi at ely on deat h, t he present( 7val
.4 .3
ue)
of t he insur ed benefi t is
ã = c, (ò)Ð ,
Chapt er 7. M ul t ip le D ecrem ent s
78
an d t h e n et si n gl e p r em i u m i s
E(g) = 1 f
ñÙ
è ä ä { Ñ) é Ñ .
( 7 . 4 .4 )
T his expression may be eval uat ed numerically by split t ing each of t he ò ø t egr al s, viz.
E(g ) = 1
1
f
c (/ñ -~- è )þ~+" ä, ( lñ.~. è )éè .
Use of assum pt ion (7.3.4) al lows us Ñî subst it ut e (7.3.5) in t he ex pr ession
above. T hus (7.4.5) assumes t he for m (7.4.2) if we w r it e
ñ,,ë+| — /rî r ñ (k + u) (1 + i ) ' " du .
( 7 .4 .6 )
I n à p r act i cal cal cu l at io n t h e ap p r ox i m at i on
ñ ë+| =
- ñ (É + -1 ) (1 + i ) i
( 7 .4 .7 )
wil l oft en be sufBcient ly accurat e. T he above der ivat ions show t hat t he eval uat ion of t he net si ngle pr emi um i n t he cont i nuous model (7.4.3) can be reduced
Ñî à calculat ion wit hi n t he discret e model (7.4.1) .
T he insured 's ex it from t he ini t ial st at us will not always r esult i n à single
payment ; anot her possi bilit y is t he init i at ion of à li fe annuity. If , for inst an ce,
t he cause j = 1 denot es disablement , t hen cr (t ) could be t he net 'single premi um of à t em porary l ife annuit y st art ing at age õ + t . T hus in t he gener al
model t he "pay ment s" ñ ë+~ (respect ively ñ, (t ) ) m ay t hemselves be expect ed
val ues; however , t he formulae (7.4.2) and (7.4.4) r em ain val id .
7 .5 T h e N et P r em i u m R eser v e
L et us assume t hat t he gener al insurance benefi t s of Sect ion 7.4 are support ed
by annual premiums of II p, Ï 1, Ï 2,
. T he net prem ium reserve at t he end
of year É is t hen
aV = ~
,) , ñç,g+h+ 1V ~ ëð õ+ê '5 ,* + ê+ ë — ,)
~= | ë=î
I i k+h~ë „ ð + ë .
(7 .5.1)
ë= î
T h e r ecu r si ve eq u at ion
( 7 .5 .2 )
7.5. T he N et Pr em i um Reser ve
79
i s à gen er al i sat io n of (6 .3 .4 ) . I t m ay b e ex p r essed as
Äv + Ï » — „ +, Óå + j', )= ',l (ñ,,»+~ — „ +, v )þ ä, +„ .
( 7 .5 .3 )
T hus t he prem ium m ay again be decomposed int o two com ponent s, t he çàèò äç
premi um
Ï '„ = „ +, V v — ÄV
(7.5.4)
t o incr em ent t he net premi um reserve, and t he ri sk premi um
Ï »= ~
(ñ .,»+ | — »+ >V ) Ä <
+»
( 7 .5 .5 )
t o i nsure t he net amount at risk for one year .
T he i nsur er 's î ÷åãàÈ 1î ÿí
= oc,
may agai n be decomposedL int
Ê
v + — ~ Ï »è
»=o
( 7 .5 .6 )
L = »=î
',) ' Ë »î " ,
w h er e
0
4
ë , =
( 7 .5 .7 )
if K < k — 1 ,
— Ï „ +
( , ,„
, — „, ð
)
if ê
=
k
,
—Ï »
if K > /ñ+ 1 ,
is t he insurer 's loss in year k + 1, eval uat ed at t ime k . H at t endor f 's T heor em (Equat ions (6.7.4)—(6.7.7)) rem ai ns valid . T he var iance of L is most
conveni ent ly eval uat ed by t he formula
n ow w i t h
Var (L ) = »=o
ð Var (A»~K > é)þ~" „ð, ,
( 7 .5 .9 )
m
Var (A»)K > /ñ) = j~= l (ñ »+| — „~, 1~)~î ~ q ., „ — ( Ï "„ )~ .
( 7 .5 . 10 )
T he veri fi cat ion of t he last for mul a is left t o t he r eader .
T he act iv it ies in year k + 1 t hus m ay be regarded as à combinat ion of ðèãå
savi ngs on t he one hand , and à one-year insur ance t ransact ion on t he ot her
hand . T he lat t er can be decomposed int o ò element ar y cover ages, one for
each cause of decr ement . W e may int er pret t he premi um com ponent
Ï ' „ = (ñ; »+ i — „ + , ~ ) î ä, + ~
( 7 .5 . 1 1 )
Chap t er 7. M ul t ip le D ecrem ent s
80
as paying for à one-year insur ance of t he am ount (ñ . ä,+ä — ~+äÓ ) , w hich cover s
t he r isk from decrement cause j . T he insurer 's loss dur ing year k + 1 may be
decom posed accordingly :
A g — Ë ä,ä, + Ë ~ ä, +
+ Ë
ä, ,
( 7 .5 . 1 2 )
i f w e ï åï ï å
0— Ï ," ä, + (ñ,,ä,.+ä — a+r V )v
if Ê
K =
< é
k —
and
1, .7 = ä ,
if Ê = é and J ô ,ä, or Ê > k + 1 .
(7.5.13)
T he t echnical gain at t he end of t he year ,
f ( ÄV + I I q) (1 + i ' ) — ñ~,~+ä if Ê = É ,
( ( ÄV + II ~) (1 + i ' ) — „ +, Ó if K > 1 + 1 ,
m ay si milar ly Úå decom posed i nt o m + 1 component s. For inst ance, t he decom posit ion met hod 1 (Sect ion 6.9) leads t o
Ñ ä,+ä — ( ÄV + I I ~) (i ' — i ) — j~= l Ë, Ä( 1 + i ) .
( 7 .5 . 1 5 )
7 .6 T h e C o n t i n u o u s M o d el
T he model of Sect ion 6.11 can be gener al ised t o t he mul t iple decr ement model
of t his chapt er . A ssume t hat t he i nsured benefi t is defi ned by (7.4.3) and t hat
prem ium i s paid cont inuously, wit h l I (t ) denot ing t he prem i um ãàÑå at t ime
t . T he over all loss of t he insur er is t hus
,ò
= tcg(T
/ IIby(t )v' dt .
T he net pr emi um reserveL at
ime )vò
t is —
given
î
v (t ) =
m
~ä
-
(
ñ (t + h )v ~ ä
+ , ö ,, + ,~ ö d h — /
~î
I I ( t + h ) v ~ ä,ð + , É
( 7 .6 . 1 )
. ( 7 .6 .2 )
.î
T he prem ium rat e II (t ) can be decom posed i nt o à sav ings com ponent Ï ' (t ) ,
see (6.11.2) , and à r isk com ponent
m
Ï ' (Ñ) = j~,
= ,l (ñ (t ) — Ó (ã))ð, , +, ,
T h iel e's d i ffer en t i al equ at i o n (6 .11.4 ) r em ai n s v al i d .
( 7 .6 .3 )
7.6. T he Cont inuous M odel
81
T he t echnical gai n derived from t he i nsurance com ponent i n t he infi nit esim al i nt er val from t t o t + dt is denot ed by G" (t , t + dt ) . I t is obvious t hat
0
if Ò < t ,
G
(t
,
t
+
dt
)
=
—
(cq(t
)
—
V
(t
))
i
f t < Ò < t + dt ,
A s à consequence we have
Ï ' (t )dt
i f T > t + dt .
Var [G' ( Ñ + é ) [Ò > t ] =
( 7 .6 .4 )
E [( G' (t , t + é )j ã[T >
~ (ñ (t ) — Ì ß )~ð), ,dt
(7.6 5)
and
÷àõ[ñ (t, Ñ+ dt)] = î~~ (ñ
(ñ (t(t)) —
—~V(t))'
ð ð,, ).,é .
(Ñ)) ,ð ð, „ .,é .
( 7 .6 .6 )
Finally one obt ai ns
Var (I )
=
j
' Vsr [G' (t , t + dt ))
î
ø
>
( 7 .6 .7 )
Ì î Ñå t hat t his result is sim pler t han i t s discret e count erpar t , see (7.5.9) and
(7.5.10) ; t his is not surpr ising i n view of (7.5.10) : t he risk pr emi um for t he
infi nit esi m al i nt erval is Ï " (t ) dt , so it s square vanishes in t he li mit . From
(7.6.7) it i s also ev ident t hat t he variance of L may be decomposed by causes
of decrem ent .
C h a p t er
8 . M u l t i p l e L i f e I n su r a n c e
8 .1 I n t r o d u ct io n
Consider ò lives wi t h i nit i al ages õ 1, õð,
, õ . For si mplicity we denot e
t he fut ure lifet im e of t he kt h l ife, Ò (õö) i n t he not at ion of Chapt er 2, by
Ò~ (É = 1,
, ò ) . On t he basis of t hese ò element s we shall defi ne à st at us è
wit h à fut ure lifet i me Ò(è ) . W e shall accordi ngly denot e by ,ð„ t he condit ional
probabil ity t hat t he st at us è is st i ll int act at t im e t , given t hat t he st at us
ex ist ed at t ime 0; t he symbols qÄ, pÄ+< et c., ar e defi ned i n à sim il ar way. We
shall also consi der annui t ies which are defi ned in t erms of è . T he symbol à„ ,
for inst ance, denot es t he net si ngle premi um of an annuity-due wit h 1 unit
payable annually, as long as è rem ai ns int act . We shal l also analyse insur ances
wit h à benefi t payable at t he failure of t he st at us è . T he symbol A Ä would
for inst ance denot e t he net single prem i um of an insured benefi t of 1 unit ,
payable im medi at ely upon t he fail ur e of è .
8 .2 T h e J o i n t - L i f e St a t u s
T h e st at u s
(8 .2 .1)
• õ ,„
is defi ned t o ex ist as long as all ò par t ici pat i ng lives sur vive. T he fai lure t im e
of t his j oi nt-lif e status is
Ò (è ) = M i n i m u m (T q, Ò2,
,Ò ) .
(8 .2 .2)
W e shal l assume in what foll ows t hat t he r andom var iabl es Ò1, Ò2,
,Ò
ar e independent . T he probabili ty di st r ibut ion of t he fai lure t i me of st at us
(8.2.1) is t hen given by
k= 1
(8 .2 .3)
C hapt er 8. M u lt i ple L i fe I nsur ance
84
T he i n st ant aneou s f ail u r e r at e of t h e j oi n t - l i fe st at u s is, accor d i n g Ñî (2 .2 .5) :
d
Ñ
™
ïç
p „ +, = —— lï ,ð„ = —— )
ln ð , = ~» ð „ ~~.
k= 1
(8.2.4)
k= 1
T his ident i ty is rem iniscent of (7.2.2) . Not e, however , t hat unl ike t he ident i ty
i n Chapt er 7, t he ident i ty (8.2.4) presupposes t hat T>,
, Ò are independent .
T he pri nciples of Chapt ers 3 and 4 m ay now be applied t o calcul at e, for
ex am pl e, t he net single prem ium for an i nsurance payable on t he fi r st deat h ,
a=o
,~ ~
s y :æð:" ".s ~
Éð õ | :õ ð:" .:s „ „ l z q+ k :õ ð+ É:" ".õ
+É '
( 8 . 2 .5 )
T h e n et si n gle p r em i u m for à j oi nt - l i fe an nu i t y - d u e i s
( 8 . 2 .á )
I d ent i t i es si m i l ar t o t h ose d er i v ed ø C h ap t er 4 w i ll b e val i d , for ex am pl e
1 = d aÄ .Ä .Ä..
+ À „ . , . ..
( 8 .2 .7 )
T he defi ni t ions and der ivat ions of Chapt ers 5 and á can be generalised by
replaci ng (õ ) by (è ) .
I f we denot e by ~ t he st at us which fai ls at t i me è , i .e.
T (ee ) = Ï ,
( 8 .2 .8 )
t hen Ò(õ : é ]) = M i nimum (T (s ) , n ) ; it is t hen evident t hat t he net si ngle
prem ium symbol s À , .—
„ 1 (endowm ent ) and à :„- ~ (t em porary annuit y ) are in
accordance wi t h t he j oi nt -life not at ion .
8 .3
S i m p l i fi c a t i o n s
À signifi cant si m plifi cat ion result s if al l lives are subj ect t o t he same Gom pert z
mort alit y law , i .e.
ð, , + ~ — B c* ' +' ,
t > Î ,
k = 1,
•,ò .
( 8 .3 . 1 )
A ft er sol v i n g t he eq u at i on
ñ" ' + c* ' +
+ ñ' " = ñ"
( 8 .3 .2 )
for è~, t h e i n st ant a neou s j oi n t - l i f e f ai l u r e r at e m ay b e ex p r essed by
I~u + t
p w+ t ,
Ô )
0 •
( 8 .3 .3 )
8 .4 . T h e L a s t - S u r v i v o r S t a t u s
85
T h i s i m p l i es t h at t h e f ai l u r e r at e of t h e j oi nt - l i fe st at u s fol l ow s t h e sam e G om p er t z m or t al i t y l aw as an i n d i v i d u al È å w i t h " i n i t i al age" w . A l l cal cu l at i on s
i n r esp ect of t h e j oi n t -l i fe st at u s m ay t h en b e p er for m ed i n t er m s of t he si n gl e
È å (è ) . A s an ex am pl e w e h av e
À„
, ,...
=
À„ ,
( 8 .3 .4 )
an d
(8 .3 .5)
So m e si m p l i fi cat i on al so r esu l t s i f al l l i ves fol l ow t h e sam e M ak eh am m or t al i t y l aw ,
ð*,+ñ= À + Âñ" +' .
ÜåÑ w b e t h e sol u t i on of t h e eq u at i on
c* ~ + c* ~ +
+ c*
( 8 .3 .6 )
= m î~
( 8 .3 .7 )
t h en ( 8 .2 .4 ) i m p l i es t h a t
~ è+ ~ = my ~ + < = p ~ i c: + ñ:" ". + ñ
T h i s m ean s t h at t h e m l i ves aged õ 1, z ,
,õ
of t h e âàò å "i n i t i al age" ø . A s an ex am p le,
t > Î .
( 8 .3 .8 )
m ay Úå r ep l aced b y ò l i ves
( 8 .3 .9 )
N ot e t h at t h e age w d efi n ed by (8 .3 .7) i s à sor t of m ean of t h e co m p o nen t
ages õ 1, õ 2,
, õ , w h i l e t h e age w d efi n ed Úó (8 .3.2) ex ceed s al l com p onent
ages õ ~, s z,
,õ
T h e si m p l i fi cat i on s p r esent ed i n t h i s sect i on , al b ei t v er y el egan t , hav e l ost
m u ch of t h ei r p r act i cal v al u e. N ow ad ay s for m u l ae l i ke (8 .2 .3 ) , (8 .2 .5) or (8 .2 .6)
m ay b e ev al u at ed d i r ect l y .
8 .4
T h e L a st - S u r v i v o r S t a t u s
T h e 1à ç Ü ç è òè ò î ò s t a t u s
( 8 .4 . 1)
i s d efi ned t o be i nt act w h i l e at l east on e of t h e ò l i ves su r v i ves, so t h at i t f ai l s
w it h t h e l ast d eat h :
T ( u ) = M ax i m u m ( T r , Ò2,
,Ò
) .
( 8 .4 .2 )
T h e j oi n t - l i fe st at u s an d t h e l ast - sur v i vor st at u s m ay b e v i su al i zed by
elect r ic ci r cu i t s: T h e st at us (8 .2 .1) cor r esp on d s t o con n ect i on i n ser ies of t h e
ò com p on ent s, w h i l e t h e st at us (8 .4 .1) cor r esp on d s t o à p ar al lel con n ect i on .
C hapt er 8. M ult ip le L i fe I nsur ance
86
Pr obabilit ies and net single pr emi ums in respect of à l ast -surv ivor st at us
m ay be cal culat ed using cert ai n j oint -li fe st at uses. To see t his, t he reader
should recall t he i nclusion-exclusion for m ula in probability t heory. Let t ing
 1,  ð,
, Â denot e event s, t he pr obabi lity of t heir union is
Pr (B r U B ~ U
U Â „ ,) = Sr — S~ + Ss —
+ ( —1)
'S ;
(8.4.3)
here Sq denot es t he sym met r ic sum
Sq — g Pr (B;, A Â ., 0
Ï Â „) ,
( 8 .4 .4 )
w here t he sum mat ion ranges over al l ™ subset s of É event s.
Denot ing by  » t he event t hat t he kt h l ife st ill l ives at t im e t , we obt ain
from (8.4.3)
Ð
õ 1 .. õð ..
. .. õ„ ,
wi t h t he not at ion
~1
~2 +
~3
+ (
1)
~~â >
SÄ
' ,'> ,ð. ,
( 8 .4 . 5 )
(8.4.6)
M ul t iplyi ng equat ion (8.4.5) by v' and sum m ing over t , we obt ai n an analogous
for mul a for t he net si ngle prem i um of à l ast -sur vi vor annui ty :
S; + Ss
here we have defi ned
S>
+ ( 1)™—1~
( 8 .4 . 7 )
~) , à,
(8.4.8)
Consider now an insured benefi t of 1, payable upon t he l ast deat h . I t s net
single prem ium m ay be cal cul at ed ss follows:
õ 1 . õð .
.õ
1 —È à
1 — È(ß~ — S2 + Sa — . ) .
(8.4.9)
L et us defi ne t he sym met ric sums
S" = Ã À
( 8 .4 . 1 0 )
Subst i t ut ing
i n (8.4.9), we obt ain t he for mula
( )
ó
( 8 .4 . 1 1 )
È
À
= S > — S2 + Sz"ß —
õ 1 . õð .
. õ„,
+ ( —1) " ' ' S
.
(8 .4 .12)
8 .5 . T h e G e n e r a l S y m m e t r i c S t a t u s
87
Not e t he sim ilar ity of equat ions (8.4.5) , (8.4.7) and (8.4.12) . Si mi lar formu1àå m ay be der ived for t he net si ngle prem ium of fr act ional or cont inuous
annuit ies, or i nsur ances payable i mm ediat ely on t he last deat h .
A s an il lust rat ion , consider t he ñàÿå of 3 lives wit h i ni t i al ages õ , ó and z.
In t his case we have, for inst ance,
=
õ :y :z
~ 1 — ~2 +
~3 ,
( 8 .4 . 1 3 )
w it h
S; = à + àä + à, ,
S2 =
— à .,ö + a>: + ov.. .
~ s
'-, ÿ
( 8 .4 . 14 )
à õ :ó :z .
T he net single premiums à .„ , à, ,„ à„ ,Ä as well as à, ,„ ,, m ay be cal culat ed
using equat ions (8.2.3) and (8.2.6) .
8 .5 T h e G en er al Sy m m et r i c St at u s
We defi ne t he st at us
> 1 :. Õ 2
( 8 .5 . 1 )
.' Õ , „
t o l ast as long as at least k of t he init ial ò lives sur vive, i .e. it fails upon t he
(ò —É+ 1)t h deat h . T he j oi nt-li fe st at us (k = ò ) and t he last -survivor st at us
(k = 1) ar e obv iously speci al cases of t his st at us.
T he st at us
è =
[k J
Õ 1 . Õ 2 '. ' ' ' .' Õ ï ç
( 8 .5 .2 )
is defi ned t o be i nt act w hen exact ly k of t he ò l ives sur vive. T he st at us st art s
t o exist at t he (m —k ) t h deat h and fails at t he (ò —1 + 1)ÔÜ deat h . T he st at us
(8.5.2) m ay be of int erest i n t he cont ext of annuit ies, but not for insurances.
À gener al solut ion follows from t he ScAuette-# âÛ 1 f or mul a, w hich is t he
t opic of t he nex t sect ion . For arbit r ari ly chosen coefBcient s ñÎ , ñ1,
has
an d , si m i l ar l y ,
,Ä ñ~,ðÕ 1 • Õ 2 • ' ' ' .' Õ ~ä = ó'=,)0, Ü ~ñÎ ~,'
Â= Î
é= Î
Õ 1 .' Õ 2 .'
ñ~ à
'
.' Õ ~â
,ñ
1'= 0
=
~)
one
( 8 .5 .4
.3 )
Ü 1ñ Î ß ' .
Í åãå t he val ues S' and S~ are ñ1åï ï åñ1by (8.4.6) and (8.4.8) , for 1 = 1, 2,
we also defi ne S0 = 1 and S0 — à—~.
,m ;
Chapt er 8. M ul t i ple L ife I nsur ance
88
For ar b i t r ar i l y ch osen co effi cient s d r , dq,
~m þ„ ð
/ 1
,d
on e al so h as
" = j',m>,
' ë -'d,s,'
=
' õ 1 . õ2 . . • . . õò
( 8 . 5 .5 )
. 1
an d , si m i l ar l y ,
m
òë
) , , ä~ à
õ ~ . õ~ .
= jg= A j
.z
dhS~' .
( 8 .5 .6 )
T h e l ast t w o for m u l ae ar e à co n seq u en ce of t h e f or m er t w o : w i t h
ñî = Î ,
ñ~ = ä ~ +
+ ä~ ,
( 8 .5 .7 )
t he left hand sides of (8.5.5) and (8.5.6) assum e t he for m of (8.5.3) and (8.5.4) .
T he expressions (8.5.5) and (8.5.6) have t he advant age t hat t hey can be
gener al ized t o È å insur ances:
rn
g
dq A
Tl l
= jg= h j ' d>S~" .
õ~
õ ~ '. õã .' '
( 8 .5 .8 )
T hi s equat ion is obt ai ned from (8.5.6) in t he same way as (8.4.12) was obt ai ned
fr om (8.4.7) .
A s an il lust r at i on we consider à cont i nuous annuit y payable Ñî 4 l ives of
init ial ages ø , õ , ó, z. T he pay ment rat e st art s at 8 and is reduced by 50%
for each deat h . T he net single premi um of t hi s annuity is obviously
8 à ur : õ : ó : z
+ 4 à ø
: s : ó : z
+ 2
ø
: õ : ó : z
+
Ñà : õ : ó : z
'
( 8 . 5 .9 )
t hus we have t he coeffi cient s ñî — Î , ñ~ — 1, ñ~ — 2, ñç — 4, ñ~ — 8. T he
diff erence t able is as fol lows:
/~ ñ
0
1
2
3
4
0
1
2
4
8
~ ñ,
~ ~ñ,
Ä çñ,
~ ~ñ,
1
1
2
4
0
1
2
1
1
0
T h e n et si n gl e p r em i um of t h e an nu i t y i s t hus S~ + Ss , w i t h
ß, = à,„ + à, + àä + à, ,
CIO
Ss
— à , ,ä + à .. ., + à ,Ä,, + à ,Ä,,
( 8 .5 . 1 0 )
8 .6 . T h e Sch u et t e- N esb i t t F o r m u l a
89
A s à second i llust rat ion we consider à life i nsur ance for 3 li ves (init ial ages
s , ó, z) , for which t he sum insured is 2 on t he fi rst deat h , 5 on t he second
deat h , and 10 on t he t hi rd deat h , each payable at t he end of t he year . T he
net single premium of t his insurance is
2 À
õ : y : z
+ 5À
õ : ó : z
+ 10 À
.
( 8 .5 .1 1 )
õ : ó : Z
St art i ng wi t h dq — 10, Í ã — 5, ds — 2 we may com plet e t he di f erence t able:
1
2
10
5
3
2
-5
-3
2
T he net single prem ium of t he insur ance is t hus 10 S~~ — 5 Sz + 2 S~ ,( wit
8 .5 .h1 2 )
S~
=
À + À„ + À„
S," = À ,„ + À. .. + À ,.„
~ ãÀ
=
À õ :y ë •
8 .6 T h e Sch u et t e- N esb i t t F o r m u l a
L et  1,  ã,
, Â denot e ar bit rary event s. Let N denot e t he number of
event s t hat occur ; N is à r andom var iable ranging over ô0, 1,
, ò ) . For
ar bit raril y chosen coeffi cient s ñä, ñ1,
, ñ , t he for mul a
m
m
ñ» Ðã(Æ = è ) = ) Ë éñäÁ
»= 0
t =o
holds, wit h ß» defi ned as in (8.4.4) , and ßà — 1.
Òî prove (8.6.1) we use t he shif t operator Å defi ned by
Å ñö — ñ ~+ 1
( 8 .6 . 2 )
T he shift operat or and t he dif f er ence operator are connect ed t hrough t he relat ion Å = 1 + b ,. Since 1 — 1â,. is t he indicat or funct ion of t he complement
of  , it is easy Ñî see t hat
rn
flL
ß, , ~{ê = ô
»= 0
=
Ä ( 1 — ~â,. + ~â ~ )
j= 1
m
= I I (1+ ~a, > )
j =l
rn
1 ( ~ ~â,ð â,, ë- ï â,„ ) ~
t =o
C hapt er 8. M ul t ip le L i fe I nsur ance
9 0
Taking expect at ions we obt ai n t he operat or ident it y
fA
òâ
Pr (N = ï ) Å " = ð
»= O
n = O
ß»Ë » .
( 8 .6 .4 )
A pply ing t hi s oper at or Ñî t he sequence of c» at k = Î , we obt ai n (8.6.1)
T he Schuet t e-Nesbit t formula (8.6.1) is an elegant and useful gener alisat ion
of t he much older Åî ãï ø 1àå of W ar ing, w hich ex press Pr (N = n ) and Pr (N >
n ) in t erm s of Sq, Sz, ' ' ' , S
Equat ion (8.5.3) follows from (8.6.1) when Â, is t aken t o be t he event
Ò > Ô.
Fi nal ly we shall present an applicat ion which li es out side t he fi eld of act uarial m at hem at ics. Let t ing ñ„ = z" in (8.6.1) , we obt ai n an expr ession for
t he gener at ing funct ion of N ,
E [z~ ] = g (z — 1)~ß» .
»= O
( 8 .6 .5 )
Consider as an ill ust rat ion t he following mat ching pr oblem . A ssum e t hat ò
di f erent let t ers are inser t ed int o ò addressed envelopes at random . Let  ,.
be t he event t hat let t er ~ is i nser t ed int o t he corr ect envelope, and let N be
t he number of l et t ers wit h cor rect ad dress. From
Pr (B ),,
n B,, n .
1
Ï Â )„ ) =
( 8 .6 .6 )
i t fol l ow s t h at S» = 1/ É!. T h e gen er at i n g f u n ct i on of N i s t h u s
Å [~~ ] = ',) ,
»= O
( 8 .6 .7 )
For ò —+ î î t his funct ion converges t o å' ' , whi ch is t he generat i ng funct ion
of t he P oi sson di str i buti on wit h param et er 1. For large values of ò , t he
dist r ibut ion of N may t hus be appr oxi m at ed by t he Poisson dist r ibut ion wit h
paramet er 1.
8 .7 A sy m m et r i c A n n u i t i es
I n general à compound st at us is less sym m et r ic. For ex am ple, t he st at us
u r
: õ
: ó
:
z
( 8 .7 . 1 )
is int act , if at least one of (è ) and (õ ) and at least one of (ó) and (z) surv ives.
T he fai lure t i me of t he st at us is
Ò = Ì | ï (Ì àõ (Ò (û ) , Ò ( õ ) ) , Ì àõ (Ò (ó ) , T ( z ) ) ) .
( 8 .7 .2 )
8 .8 . A sy m m et r i c I n su r a n c es
91
For t h i s st at u s t h e n et si ngl e p r em i u m of an an nu i t y can b e cal cu l at ed i n
t er m s of t he net si ngl e p r em i u m s of j oi nt - l i fe st at u ses. T h i s fol l ow s fr om t h e
r el at i on s
CP u:v =
àð è +
tp »
tP uþ ~
(8.7 .3)
r esp ect iv el y
à—
„ .„ = à „ + à„ — à„ .„ ,
( 8 .7 .4 )
w h i ch ar e v al i d for ar b i t r ar y st at u ses è an d v . C on si d er f or ex am p le an an nu i t y
of 1 u n i t w h i l e t h e st at us (8 .7 .1) l ast s. B y r ep eat ed app l i cat i on of (8 .7 .4 ) we
ob t ai n an ex p r essi on for t h e n et si n gl e p r em i u m ,
þ :õ :g :ë
à—
, , + à—
, — à—
. . .
õè:õ:ó
þ :õ:ë
þ :õ:y:z
à ,ä + à ,„ — à , ,„
+ à ,, + à. .. — à .. .
—à,„ ,ä., — à, ,ä:, + à
( 8 .7 .5 )
R ever si on ar y an n u i t i ea ar e r el ev ant w h en st u d y i n g w i dow s' and or p h an s'
i n su r an ce . T h e sy m b ol à, ~ä Éåï î Ôåí t h e n et si n gl e p r em i u m of à cont i nu ou s
p ay m ent st r eam of r at e 1, w hi ch st ar t s at t h e d eat h of (õ ) an d t er m i n at es at
t h e d eat h of (y ) . T h i s net si n gl e p r em i u m can b e cal cu l at ed w i t h t h e ai d of
t h e r el at i on
à,.~„ = à„ — à, ,„
8 .8
( 8 .7 .6 )
A sy m m e t r i c I n su r a n c e s
C on si d er t he ò l i v es of Sect io n 8 .2 an d assu m e i n d ep en d en ce of t hei r fu t u r e
l i fet i m es. À gen er al i n su r an ce on t h e fi r st deat h p r ov i d es à b en efi t of ci (t ) i f
l i fe j d ies fi r st at t i m e t (i .å. t h e j oi n t - l i f e st at us f ai l s d u e t o cau se ó') . Su ch an
i n su r an ce i s m at hem at i cal l y equ iv al ent t o t h e i n su r an ce d i scu ssed i n Sect i on
7 .4 . I n an al ogy t o for m u l a (7 .4 .4) , t he n et si n gle p r em i u m o f t h is fi r st -d eat h
i n su r an ce i s
(8 .8 .1)
T h e r ever si on ar y an nu i t y con si d er ed i n t h e p r ev i ou s sect i on i s of t h i s t y p e.
D efi n i n g
ñ| (~) = àä+ñ,
ñã(1) = Î ,
(8 .8 .2 )
w e obt ai n
à, ~ö — (
î
àä+ ~î ,ð , .„ ð,, +, É .
T h i s ex p r essi on pr esu p p oses i n d ep en d en ce b et w een T (s ) an d T (y ) , i n con t r ast
t o (8 .7 .6) .
92
C h ap t er
8 . M u l t ip le L i fe I n su r a n ce
I n t h e sp eci al case w i t h cq(t ) = 1 an d ñ (t ) = 0 for j ô é , t h e net si ngl e
p r em i u m i s d en ot ed by
an d giv en by t h e ex p r essi on À
À
ä
( 8 .8 .4 )
õ | :" ' õ é q.þ é:æé~. | ." " .õ ,~
,
=
j
õ | :" ".õ é q.õ ~æ ~~. ~." ".é.„ ,
è ' ,ð , .„ .„ ,
ää, „ + éé .
(8 .8 .5)
î
N ot e t h at t h e sy m b ol s i nt r od uced i n C h ap t er 3 t o d en ot e t he net si n gl e p r eò äääò of à p u r e en d ow m ent , and t h at of à t er m i n su r an ce, ar e sp eci al cases of
(8 .8 .4 ) ; t h ese ar e ob t ai ned by i n t er p r et i n g ~ as à st at u s w h i ch fai l s at t i m e è .
T h e n et si n gl e p r em i u m (8 .8 .5) i s v er y easy t o cal cu l at e i f al l l i v es obser v e
t he âàò å G om p er t z m or t al i t y l aw , see for m u l a (8 .3 .1) . I n t h at ñàâå,
~ é
w i t h äè d efi n ed by (8 .3.2)Ð ;Õé+
it Üfol l ow~s Èt Õ1+
hatÈÕ2+ Ü: " :Õò
ñ
À
,
+ä)
= —
~ é À „ .„ ..„ .
» :" .:õ é z:õ é:õ é+ ü:" ".õ „
Ñ
( 8 .8 .6 )
= —
~ é À
.
(8 .8 .7)
Ñ
W e sh al l now consi d er an i n su r an ce w h i ch p ay s à b en efi t of 1 u n i t at t h e
t i m e of d eat h of (õ é) , p r ov i d ed t h at t h i s i s t h e r t h d eat h .
p r em i u m i s d enot ed by
À õ | :-
I t s n et si n gle
:Xa z' æé:é é+ | ." ' >m
( 8 .8 .8 )
I n or d er t h at à p ay m ent b e m ad e at t h e d eat h of (õ é) , ex act l y m — ò of t he
ot h er ò — 1 m u st su r v i ve (õ é) . H en ce we h ave
À
,
=
ë é:- ".õ é q.õ ~.õ ~~.~." ".õ„ ,
/
è' ,ð
óð
,ð , p Ä + ~d t .
(8 .8 .9)
ç ä'õð" " æé q.'ç é». ~.""" ë
Su b st i t u t i n g as ø eq u at i on (8 .5.3) , w e ob t ai n à l i n ear com b i n at i on of n et
si n gl e p r em i u m s of t h e f or m (8 .8 .4 ) is, w hi ch m ak es t h e cal cu l at i on easi er .
C on si d er for i n st an ce
[2]
= ñä
J —Vñâñð— Î ., ñð
. — ß
Ð ã+
W e n ow u se (8 .5 .3 )Àwâi :é.:y
t h :»
ñî —
1 an
d fiÀ n~d• t h at
~î
,Ð .
.
~ã Ç,~â
äè : z : Ó
,ð :* + äð : + Ð. :ä, Ç,Ð :. :
( 8 .8 . 1 0 )
(8'8'11)
S u b st i t u t i n g t h e l a st ex p r essi o n i n ( 8 .8 .10 ) y i e l d s
À
ë õ :ó Ç
=
À þ :õ
+
À
:y .
+
À æ: .
— Ç À „
, „ .ä
( 8 .8 . 1 2 )
C h ap t er 9 . T h e T ot al C l ai m A m o u nt i n à
P or t fol io
9 .1 I n t r o d u ct i o n
W e consider à cert ai n port folio of i nsurance pol icies and t he t ot al amount of
cl ai ms ar ising from it duri ng à given per iod (å.g. à year ) . W e ar e par t icular ly
int erest ed in t he pr obabilit y dist ri but ion of t he t ot al claim am ount , whi ch
will allow us t o est im at e t he risk and show whet her or not t her e is à need for
reinsur ance.
We assume t hat t he port foli o consist s of è i nsurance pol icies. T he clai m
m ad e i n respect of policy h is denot ed by ßë. Let us denot e t he possible-val ues
of t he r andom vari able ßë by Î , â| ë, sqq,
P r (S@= Î ) = Ðë
, s q, and defi ne
P r (Sp, = à~ë) = ßðë
( 9 .1 . 1 )
for j = 1,
, ò and h = 1,
, ï . W it h respect t o t he gener al insur ance type
of Chapt er 7, ù , m ay be t aken t o be t he probabili ty of à decrement due t o
ñàðàå ~, and â, ë m ay be t aken t o be t he correspondi ng amount at risk (i .e. t he
diff erence between t he paym ent t o be m ade and t he avail able net pr emi um
reserve) .
T he t ot al , or aggregat e, amount of cl aim s is
S = S, + S>+
+ SÄ.
( 9 . 1 .2 )
Òî enable us t o calcul at e t he dist ri but ion of S we shall assume t hat t he random
var i ables Sq, Sq,
, SÄ are independent .
9 .2 T h e N o r m a l A p p r o x i m at i o n
T he fi rst and second or der m oment s of S m ay be readily cal cul at ed . One has
Ï
è
E [S] = ë=
~ ;1 Å [5ë] , Var [S] = ~) , ×àõàë ] ,
( 9 .2 .1 )
Chapter 9. The Total Claim Amount ø à Portfolio
94
wit h
Å[5»] = ,' ) Ö»äð» , Var [S»] = ,~ @ ß~ë —Å[~»] .
For à large port folio (large è) it seems reasonable t o approximat e t he
probability dist ribut ion of S by à normal dist ribut ion wit h parameters ð =
E[S] and o~ = ×àã[ß]. However, t he quality of t his approximat ion depends
not only on t he size of t he port folio, but also on it s homogeneity. Moreover,
t his approximat ion is not uniformly good: in general t he results are good
around t he mean E[S] and less sat isfactory in t he "t ails" of t he dist ribut ion.
T hese weaknesses of t he approximat ion by t he normal dist ribut ion may be
part ially relieved Úó sophist icat ed procedures, such as t he Esscher Method or
t he Normal Power Approsi mati'on. However, t hese met hods have lost some of
t heir int erest : if à high-powered comput er is available, t he dist ribut ion of S
can be calculat ed more or less "exact ly" .
9 .3 E x act C al cu l at i on of t h e T ot al C l ai m A m ou nt
D i st r i b u t i on
T he probability dist ribut ion of S is obt ained by t he convolut ion of t he ðäî Üàbility dist ribut ions of ßä, . , SÄ. T he dist ribut ions of Sq+ Sq, ßä+ ß~+ ßç ßä+
Sg+ Ss + S4, , are found successively. If t he dist ribut ion of ßä+
+ S», ä
is known, t he dist ribut ion of ßä+
+ S», may be calculat ed by t he formula
Pr(S~+
+ ÿ„ = õ) =
~ (ÿ +
+ S» i
x —s, )q,»
j=l
+ Pr(S>+
+ ~»—, = ~)л
(9.3.1)
Wit h t his procedure it is desirable t hat t he ç,» are mult iples of some basic
monet ary unit . Of course, in general t his will not be t he ñàçå unless t he basic
monet ary unit is chosen very small. T he original dist ribut ion of ß» is t hen
appropriat ely modifi ed. Two met hods are popular in t his respect .
M et hod 1 (Rounding)
T he met hod st art s by replacing ç, » by à rounded value ç' „ , which is à multiple
of t he chosen monet ary unit . In order to keep t he expect ed t ot al claim amount
t he same t he probabilit ies are adj ust ed accordingly by t he subst it ut ions:
s,.» —' ~ ë Úë
q~» = %»çðël ç;» Ðü
Ðü = 1 —(q~ü + ' ' ' + q~Ä) . (9.3.2)
M et hod 2 (D isp er sion )
L e t
s
Ä
ðë
d e n o t e
t h e
l a r g e s t
m
u l t i p le
( o f
t h e
d e s i r e d
m
o n e t a r y
u n i t )
n o t
e x c e e d -
ing ç », àï é let s Ä denot e t he least mult iple which exceeds ç ». T he original
9.3. Ex act Cal cul at ion of t he Tot al Cl ai m A m ount D ist r i but ion
95
dist ribut ion of S» has à point m ass of q~» at â ». T he disper sion met hod ñî ï sist s of re-allocat ing t his poi nt m ass t o s » and s+» ø such à way t hat t he
ex pect at ion is unchanged . T he new poi nt m asses q,.» and q+Ä must t herefore
sat isfy t he equat ions
×, ë + ×,+ë = q>»
+ + = âç»%» ~
8>»q>» + Ö»Ö»
( 9 .3 .3 )
t h at i s
+
sq»
si »
+
si » sj '»
ë = r at+ion à port
% » fol
> io
×~ëof= t hr+ee policies
— % » wit h , for example:
Consi der ss an Öillust
( 9 .3 .4 )
sq»
â ,.ë
â ë
ÿ ë
Pr (Si — 0) = 0.8 , Ðã(ß| — 0.5) = 0.1 , Pr (Si — 2.5) = 0.1 ,
Pr (S@— 0) = 0.7 , Pr (S@— 1.25) = 0.2 , Ðã(ßç — 2.5) = 0.1 ,
Ðã(ßç = Î ) = 0 6 , Pr (Ss = 1 5) = 0.2 ,
(9.3.5)
Ðã(ßç = 2 75) = 0 2
T he convolut ion of t he t hree dist r ibut ions r anges over t he val ues Î , 0.5,
1.25, 1.5, 1.75, 2, 2.5, 2.75,
, 6.5, 7.75, and it m ay in pr inci ple be calcul at ed . Calculat ing t he convol ut ion of t he m odified dist ribut ions is much
easier , however . W e shall use M et hod 2 t o approx im at e t he dist ribut ion of S»
by à dist ri but ion on t he int egers. T he m odificat ions prescr ibed by M et hod 2
are set out ø t he t able below :
0 .05
0 .05
0 .5
0 .05
1
2
0 .1
1
ôî .~
0 .05
1 25
1
0 .1
1.5
2 5
0 .05
2
0 .1
0 .05
3
0 .05
25
3
0 .05
2
0 .15
2 .75
3
9 .4
Ð ã( 5» — 0 ) = ð ë ,
P r ( S » = j ) = qj »
( 7' = 1 , 2 ,
.
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96
C h ap t er 9 . T h e T o t al C l a i m A m o u n t i n à P o r t fo l i o
H en c e , t h e m o d i fi ed d i st r i b u t i o n s a r e a s f o l l o w s :
õ = 3
s)
T h e C o m p o u n d P o i ss o n A p p r o x i m a t i o n
A ssu m e t h a t t h e d i st r i b u t i o n o f S » i s g i v en b y
( 9 .4 . 1)
T h e g en er a t i n g f u n c t i o n o f t h i s d i st r i b u t i o n i s
( 9 .4 .2 )
9 .4 .
T h e C om p ou n d
P o isso n
A p p r ox im a t io n
9 7
T he dist ribut ion of ßë m ay now be approxi m at ed by t he correspondi ng compound Poisson dist ribut ion whose gener at ing funct ion is
ga(z) = exp
jg= l
q~q(z' — 1)
( 9 .4 .3 )
Âó com par ing (9.4.2) and (9.4.3) one will see t hat t he approxim at ion is best
for sm all values of t he ù ë.
I f we now use t he com pound Poi sson approxi m at ion for all t er ms in (9.1.2),
t he result ing approxim at ion of S will have as gener at ing funct ion
â
g(z) = ë=
Ä | ga(z) = åõð
m
~», ä, (z~ — 1)
j =l
( 9 .4 .4 )
w it h t he not at ion
~ = g qa.
h= l
But t his means t hat t he dist ribut ion of S can also be approx im at ed by à
com pound Poisson dist r ibut ion . I n t he correspondi ng m odel t he t ot al cl aim
amount is
S= x , + x, + ..+ x~;
(9.4.6)
here N denot es t he r andom number of cl ai ms, and Õ ; denot es t he amount of
t he i t h claim . Furt herm or e, t he r andom vari ables N , Õ 1, Õ ã, . ar e independent , N has à Poisson dist ribut ion wit h par am et er
q = R + qz + ' ' ' + q
~
( 9 .4 . 7 )
an d t h e p r ob ab i l i t y t h at t h e am ou nt of an i n d i v i d u al cl ai m i s j i s
ð ( 7') = ä, / q (j = 1, 2,
,ò ) .
( 9 .4 . 8 )
T h e p r ob ab i l i t y d ist r i b u t i on of S i s t h en gi v en by t h e for m u l a
P r ( S = ó ) = ) , " ð ' " ( ó ) å ~ä ë / Ö .
a=o
( 9 .4 .9 )
In t he numerical ex ample in t he previous sect ion we had qq — 0.3, qq — 0.3,
qs — 0.25. T hus q = 0.85, and each of t he r andom vari ables Õ , m ay t ake t he
val ues 1,2 or 3, wit h probabi lit ies ð ( 1) = 30/ 85, ð (2) = 30/ 85, ð (3) = 25/ 85.
T he model (9.4.6) , called t he ñî éåñé î å ri sk òï î éå1, is par t i cularly appr opr iat e if t he port fol io is subj ect Ñî changes during t he year . Even i n such
à dy nam ic port folio i t wil l be possi ble t o est im at e t he expect ed number of
clai m s (q) and t he indiv idual clai m am ount dist ribut ion .
Ì î Ñå t hat (9.4.6) can be wri t t en as
S
=
~
1 +
2 ~ 2 +
3 ~ 3 +
• • •
+ m N ~ ,>
( 9 .4 . 1 0 )
Ch ap t er 9. T he Tot al C laim A m ount ø à Por t fol io
98
if we let È
denot e t he number of clai ms for amount ó'. It can be proved
t hat t he r andom variables N q, N q,
, N are independent , and t hat È has
à Poisson dist ri but ion wit h paramet er q~ (so t hat ä is t he frequency of clai ms
for amount ó') .
T he dist ribut ion of S can in pri nciple be calcul at ed by eit her (9.4.9) or
(9.4.10) . À t hird met hod , t he r ecur si ve method, wi ll be present ed in t he nex t
sect ion .
9 .5 R ecu r si v e C al cu l at io n of t h e C om p ou n d P oi sson
D i st r i b u t i on
L et us denot e t he probabi lit ies Pr (S = õ ) by f (z ) and t he cumul at ive dist ribut ion funct ion by F (x ) = Pr (S < s ) . T hus, for ex am ple,
f (0) = P r (S = 0 ) = P r (N = 0 ) = å ~ .
( 9 .5 . 1 )
P anj er d i r ect ed t h e at t ent i on of act u ar i es t o t he usef u l r ecu r si v e fo r m u l a
( 9 .5 . 2 )
which enables us t o calculat e t he val ues f ( 1) , f (2), f (3) , . successivel y.
I n t he num eri cal exam ple considered above t he calculat i ons ar e ss follows:
f (4)
f (0)
- î .ss
å
f (1)
0.3 f (0) ,
f (2)
- (0.3 f ( 1) + 0.6 f (o))
f (3)
—(0.3 f (2) + 0.6 f (1) + 0.75 f (Î ) ) ,
—
1 (0.3 / (3) + 0.6 / (2) + 0.75 f (1)) ,
( 9 .5 .3 )
T he numerical resul t s have been com pi led i n t he following t able; t he part ial
sums Ð (õ ) could , of course, also have been calcul at ed recursively.
9.5. R ecur sive Calcu lat ion of t he Com p ound Poisson D ist r ib ut ion
T h e
õ
f (æ)
Ð (õ )
õ
f (õ )
Ð (õ )
09
07
3 41 31 56
0 . 40 2
99
0 .4
2 7 45 81 4
5
10
0 .0 0 1 3 0 2
0 .9 9 8 8 8 6
1
0 .1 2 8 2 2 4
0 .5 5 5 6 3 9
11
0 .0 0 0 6 4 5
0 .9 9 9 5 3 1
2
0 .14 7 4 5 8
0 .7 0 3 0 9 8
12
0 .0 0 0 2 7 7
0 .9 9 9 8 0 8
3
0 .14 7 24 4
0 .8 5 0 3 4 2
13
0 .0 0 0 1 1 1
0 .9 9 9 9 2 0
4
0 .0 5 7 2 0 4
0 .9 0 7 5 4 6
14
0 .0 0 0 0 4 9
0 .9 9 9 9 6 9
5
0 .0 4 3 2 2 0
0 .9 5 0 7 6 6
15
0 .0 0 0 0 1 9
0 .9 9 9 9 8 8
6
0 .0 2 6 2 8 7
0 .9 7 7 0 5 3
16
0 .0 0 0 0 0 7
0 .9 9 9 9 9 5
7
0 .0 1 0 9 6 0
0 .9 8 8 0 1 4
17
0 .0 0 0 0 0 3
0 .9 9 9 9 9 8
8
0 .0 0 6 4 3 4
0 .9 9 4 4 4 8
18
0 .0 0 0 0 0 1
0 .9 9 9 9 9 9
g en er a t in g
( 9 .5 .2 ) .
O n
w h ile , o n
fu n c t io n
t h e o n e
t h e o t h er
o f
h a n d , it
S
c a n
b e
u sed
t o
p r o v e
t h e
99
r ec u r siv e
fo r m u la
i s ñl å é ï å ñl b y
g (~ ) =
~)
f (~ ) » *
h a n d , ( 9 .4 .4 ) i m p l i e s t h a t
( 9
. 5
.4
)
õ= Î
( 9 . 5 .5 )
Fr om t h e i d en t i t y
w e o b t a in
—Û g ( z ) =
g ( z ) —d
dz
Ä, , õ / (õ )~
—
=
,)
/
ó'= 1
t h e
co eK
ln g (z )
( 9 .5 .6 )
(ó ) » ~
( 9 .5 .7 )
v=o
õ= 1
C o m p a r in g
dz
cien t s o f
»
' , w e fi n d
õ / (õ )
òï
~
=
/ (õ
—
7' ) ù
( 9 .5 .8 )
w h ich
est a b lish es
U n t il
t h a t
t h e
is
n ow
t h a t
t o t a l
w e
all
t er m s
a m o u n t
c la im s , a n d
S
( 9 .5 .2 )
h a v e t a c it ly
o f
in
cla im s
, t h e su m
a ssu m ed
( 9 .4 . 6 )
o f
c a n
t h e
a re
b e
t h a t
o n ly
p o sit iv e .
p o sit iv e
If
d ec o m p o se d
a b so l u t e
clai m s co u ld
n eg a t iv e
in t o
c la im s
S + , t h e
su m
v al u e s o f t h e n e g a t iv e
o f
c a n
a n d
b e
a r e
sh ow n
t h a t
in d ep en d en t .
s e p a r a t e l y , å .g .
v o lu t io n .
fro m
b o t h
W e
S +
a n d
S
c a n
n o w
co m p u t e
( 9 .5 .2 ) , a n d
fi n a l l y
h av e
o c cu r ,
p o sit iv e
cla im s:
S = S+ —SIt
o c cu r ,
ca n
( 9 . 5 .9 )
co m p o u n d
t h e
o b t a in
P o isso n
d i st r i b u t io n s
t h e
d i st r i b u t i o n s
o f
d ist r ib u t io n
S +
o f
S
an d
b y
S
co n -
10 0
Chapter 9. The Total Claim Amount in à Portfolio
9 .6 R ei n su r a n ce
If inspect ion of t he dist ribut ion of S shows t hat t he risk is too high t he acquisition of proper reinsurance is indicated. Diff erent forms of reinsurance are
available, two of which will be discussed in t his and t he next sect ion.
Quite generally à reinsurance cont ract guarant ees t he insurer t he reimbursement of an amount R (à funct ion of t he individual claims and t hus à
random variable) in ret urn for à reinsurance premium Ï . T he insurer's retenti on is
S = S+ 11- Â .
(9.6.1)
Wit h proper reinsurance t he dist ribut ion of S will be more favourable t han
t he dist ribut ion of S. Let us defi ne f (s) = Pr(S = õ) and F (s) = Pr(S ( õ).
An Åõñåçç of Loss reinsurance wit h priority à reimburses t he excess Õ; —à
for all individual claims which exceed à .
Let us assume in our numerical example t hat excess of loss reinsurance
wit h à = 1 can be purchased for à premium of Ï = 1.2. The original claims
which can assume t he values 1,2,3, are à11 reduced Ñî 1 by t he reinsurance
arrangement . T hus t he insurer's retent ion is
S = 1.2 + #
( 9 .6 .2 )
here N denotes t he number of claims and has à Poisson dist ribut ion wit h
paramet er 0.85. T he dist ribut ion of S is t abulated below:
õ
1.2
2.2
3.2
4.2
5.2
6.2
7.2
8.2
9.2
/ (õ)
0.427415
0.363303
0.154404
0.043748
0.009296
0.001580
0.000224
0.000027
0.000003
Ð(õ)
0.427415
0.790718
0.945121
0.988869
0.998165
0.999746
0.999970
0.999997
1.000000
Since t he reinsurance premium cont ains à loading, Ï > Å[Â], it is clear
from (9.6.1) t hat E[S] > E[S]; in our example we have E[S] = 2.05, while
E[S] = 1.65. T he purpose of reinsurance is to reduce t he probabilit ies of large
t ot al claims; indeed in our example we have F (6.2) = 0.999746, which exceeds
t he corresponding probability wit hout reinsurance by far (F (6) = 0.977053).
In the next sect ion we shall present à reinsurance form which is ext remely
eff ect ive in t his respect .
9.7.
op-L
ossLReinsur
9 .7 StSt
oposs Rance
ei n sur an ce
10 1
Under a st op-loss reinsur ance cont r act wit h deduct ible Â, t he excess  =
(S — p )+ of t he t ot al clai ms over t he specifi ed deduct i ble is rei mbursed . In
t his ñàçå
Ï
—
( 8
—
ð )
[
J
ß
+
Ï
][ ,Î + ï
i f
S
(
p
if s > p .
(9 .7 .1)
L et us now assume t hat à st op-loss cover for t he deduct ible p = 3 has
been bought at à premi um of Ï = 1.1. T he insurer 's por t ion of t he t ot al
cl ai m amount wi ll be limit ed t o 3. T he dist r ibut ion of S can be derived from
t he dist r ibut ion of S:
õ
f (õ)
F (z )
1.1
2.1
3.1
4.1
0.427415
0.128224
0.147458
0.296903
0.427415
0.555639
0.703098
1.000000
T he ex pect ed val ue of S is quit e large, E [S] = 2.41, but t he "risk" has been
reduced t o à minimum .
W e shal l fi nally consider cal cul at ion of t he net st op-loss pr emium , which
we denot e by p(p ) :
ð[ð [ = Åð[ð
[[ ß) —
=—
f F [*
[õ)l—
= ôf )+[ [1
~*ð.[ÂÐ [õ[ .
(9 .7 .2)
.3)
Âó part ial int egr at ion we obt ai n
H ence, for i n t eger val u es of Î , w e m ay w r i t e
or , w r i t t en r ecur si vely ,
~
—[1
Ð (õ
, )]
à(Î ð(~
+ 3)
~) == õ=
~(,,çÎ [1
)—
—)]~"(Î
(9
(9 .7 .4)
.5)
T hus t he values ð(1) , ð(2) , ð(3) ,
can be com put ed successively, st art i ng
wi t h ð(0) = E [S]. Of course, t hese comput at ions can be combined wit h t he
r ecursive cal culat ion of F (x ) (see Sect ion 9.5) .
I n our ex ampl e t he st op-loss premiums assume t he fol lowing values.
10 2
Chapter 9. The Total Claim Amount ø à Portfolio
ð(ß
0
1
2
3
4
5
6
7
8
9
10
1.650000
1.077415
0.633054
0.336152
0.186494
0.094040
0.044807
0.021860
0.009874
0.004322
0.001906
Of course, t he act ual st op-loss premium Ï will exceed t he net premium
p(p ) signifi cant ly. Our example, wit h Ï = 1.1 and ð(3) = 0.336152 corresponds t o à 227% loading. Loadings of t his order of magnit ude are not
uncommon.
The net premium is st ill of int erest , since it allows one Ñî calculate t he
expect ed value of t he retent ion, which is
E[s ] E[S]
= Å[ß]
+ Ï +—1.1
ð(/3)
In our example we have again
= 1.65
—.0.34 = 2.41.
( 9 .7 .6 )
C h ap t er
1 0 . E x p e n se L o a d i n g s
10 .1 I n t r o d u ct io n
T he oper at i ons of an insur ance cont r act wi ll i nvolve cert ain expenses, w het her
under t aken by pension funds or by insurance com panies. I n t he ñàçå of à
pension fund t hese ex penses are most oft en lum ped t oget her and consi dered
separat ely from t he st ri ct ly t echnical insur ance analysis. I n t he ñàÿå of insurance com panies, on t he ot her hand , t he cost element is bui lt int o t he model ,
as explici t ly and equit ably as possi ble. A s we shal l see, however , t he r esul ting prem ium s and reserves are very closely relat ed t o t he net pr emi ums and
reserves we have been discussing so far , and which will t herefore cont inue Ñî
hold our pr im ar y int erest .
Ex penses can be classifi ed int o t hree m ai n groups:
à . A cq u i si t i o n E x p en ses
T hese com prise àll expenses connect ed wi t h à new pol icy issue: agent s' comm ission and t ravel expenses, medical ex am inat ion , poli cy w ri t i ng, advert ising.
T hese ex penses are char ged agai nst t he policy as à single am ount , which is
pr oport ional t o t he sum insur ed . T he cor responding r at e w ill be denot ed by
b . C ol l ect i on E x p en ses
T hese ex penses are charged at t he begi nni ng of every year in which à prem ium
is t o be collect ed . W e assume t hat t hese ex penses are proport ional t o t he
expense-l oaded prem ium (see 10.2) , at à rat e which we wil l denot e by p .
ñ. A d m i n i st r at ion E x p en ses
A ll ot her expenses are included i n t his it em , such as wages, r ent s, dat a processing cost s, i nvest ment cost s, t axes, li cense fees et c. T hese cost s are charged
agai nst t he ðî éñó dur ing it s ent ire cont r act per iod , at t he begi nning of å÷ery pol icy year , usual ly as à propor t ion of t he sum insured , respect ively t he
annuit y level , and t he cor respondi ng rat e is denot ed by y.
104
C h ap t er 10 . E x p en se ? î àñ1| ï ó
T his t r adit ional al locat ion of ex penses is somewhat arbit r ary. Áî ò å expense
it ems will obv iously be fi xed cost s, independent of t he sum insured . Nevert he.less, t he assum pt ion of propor t ionalit y is ret ai ned for t he sake of si m plicity.
T he fact or s à , p and à wi ll , however , depend on t he t ype of insur ance involved . Expenses in respect of an indiv idual insurance ar e relat ively higher
t han expenses in respect of à group insurance; for t he l at t er t he acquisit ion
expense is oft en even wai ved ent ir ely (i .å. à = 0) .
1 0 . 2 T h e E x p e n se - L o a d e d P r e m i u m
T he åõðåèçå-loaded premi um (or adequate pr emi um) , which we will denot e by
Ð ' , is t he am ount of annual prem i um of w hich t he expect ed present val ue is
j ust suffi cient t o fi nance t he insur ed benefi t s, àÿ well as t he incurred cost s in
respect of t he insurance policy. Hence we m ay w ri t e
= Ð + ~ + P~ + P' ,
( 10 .2 .1)
her e Ð denot es t he net annual pr em i um , whi le P , P ~ and P ~ denot e t he
t hree component s of t he expense loadi ng.
W e consider as à fi r st ex am ple an endowment (sum insured : 1, dur at ion :
è , age at issue: õ ) . T he exp ense-loaded annual prem i um must sat isfy t he
condit i on
Ð; .~ à. .~ — À , :q + î + 0 Ð: ,q à. ö + ~ à,.ö ,
so t hat
( 10 .2 .2)
À .~ + a + óà ,-„-1
( 1 — p )a, .~
T he expense-load ed annual prem i um wi ll be ex pressed in t er ms of t he net
àø ø à1 prem ium if we r epl ace à by à (À .~ + da, .q ) i n t he above formul a:
s :é ]
1+ à
1
p
ì :â ] +
cd + ó
1
ð
( 10 .2.4 )
I f w e now d i v i d e ( 10 .2 .2 ) b y à, -„-~, w e ob t ai n ( 10 .2 .1) i n t h e sp eci fi c for m :
Ð: þ — Ð, -„ , + à.. ,q + pP: —
„ 1+ ~ .
( 10 .2 .5)
A s à second ex am ple we consider t he âàò å endowm ent , but wit h à short er
premium payi ng per iod ò ( è . T he expense-loaded annual premi um is obt ained from t he condit ion
Ð à .- .- ~— A .„ ~ + à + )3 Ð ' à, , 1+ àà ,-„ ~ .
( 10 .2.6 )
10 .3 . E x p en se- L oad ed P r em i u m R eser v es
10 5
It s component s are
Ð' = Ð + —
.. Î
~4 :~
~
+ ,9Ð + ó..à, .~
<4 :~
Q
( 1 0 .2 .7 )
wit h, of course, Ð = À, .„-~/ à, .~ .
For deferred annuit ies fi nanced by annual premiums it is cust omary to
charge acquisit ion expenses as à fract ion of t he expense-loaded annual premium, in t he âàò å way as collect ion expenses. Í åãå it is also possible to use
two administ rat ion expense rates, à rat e y~ for t he premium paying period,
and anot her rat e y~.for t he annuity's durat ion.
For simplicity, t he reader may ident ify t he expense-loaded premium wit h
t he gross premium; t he necessary safety loading is t hen t aken t o be implicit
in t he "net " premium, t hrough conservat ive assumpt ions about interest and
mort ality rat es. In pract ice, t he gross premium may also diff er from t he
expense-loaded premium in t hat eit her surcharges for small policies or discount s for large policies are used.
In some count ries t he premium quot ed by t he insurance company consist s
of t he net premium and administ rat ion expenses, but not acquisit ion and
collect ion expenses. T his premium (German: Invent arpramie),
Ð'"" = Ð + Ð ,
(10.2.8)
covers t he act ual cost s of insured benefi t s and internal administ rat ion expenses.
1 0 .3 E x p e n s e - L o a d e d P r e m i u m R e se r v e s
T he expense-loaded premium reserve (or adequat e reserve) at t he end of year
é is denoted Úó «V ' . It is defi ned as t he difference between t he expected
present value of fut ure benefi t s plus expenses, and t hat of fut ure expenseloaded premiums. T he expense-loaded premium reserve can be separat ed int o
components similar to t hose of t he expense-loaded premium:
~/ ~~ —
~/ ' .+
~/ '~ +
V 7
(10.3.1)
Í åãå ÄV denot es t he net premium reserve, «V is t he negat ive of t he expected
present value of fut ure P , and t he òåçåòèå f or administration åõðåï çåç is the
diff erence in expect ed present value between fut ure administ rat ion expenses
and fut ure P~.
For an endowment we have
«
ó à
õ :g
n
ð à õ + « :ë — « )
É
æ+ « à — k ~
—A
àõ :â
. ]
- à (1 — «V, ,q )
( 1 0 .3 .2 )
Chapt er 10. Ex pense L oad i ngs
106
and éà " = 0 for É = 1, 2 ,
, è . T h us
( 1 0 .3 . 3 )
I f t h e p r em i u m p ay i n g p er i o d i s r ed uced t o ò year s, t h en
kV
for é = 1, 2,
= —Ð às +
, ò —1, and „ è'
k w
— é ~
= —à (1 — „ è' , )
( 1 0 .3 .4 )
= 0 for é > ò . ÒÜå ãåçåã÷å Åî ã adm inist r at ion
ex penses is t hen
7 àõ + é :»
- é]
ó à ;„-~
— Ð ~ àõ + é :ò
à* + é :»
—k I
às + k w
—é !
æ~
»
for k = 1, 2,
—é ~
s :g
m
, ò — 1, andó à ;„-~( „ ~, .~ — „ Ó, .~ )
àõ + é :»
( 10 .3 .5 )
( 10 .3 .6 )
—k ~
for é > ò .
T he idea t o include t he negat ive acquisit ion cost reser ve „ ~ in t he preï è èò r eserve is due t o Zi l lm er . I n t he fi rst few years, t he expense-loaded
prem ium r eserve may be negat ive if à is large. Hence t he need for upper
bounds on à ar ose. One suggest ion was t o choose t he value of à at most
equal t o t he one for w hich t he expense-loaded prem ium reserve is í åãî at t he
end of t he fi rst year . Consider an endow ment as an illust r at ion . T he condit ion
V; .—
„ ~ > 0 t oget her wit h (10.3.3) im plies t hat t he acquisit ion expense
ãäå ñàï ï î ðî åõñååñ1
W
i t hupper
t he su
b st i t ubecom
t i on s es à = 7 , .g / ( 1 — , V .~ ) .
t he
bound
and
1
þ :g
»
(
~ ~. ã .»
y !
1 — , Ó .q — à
ç :g
n
,
)
~ + ] .»
((10.3.8)
10.3.7)
10 .3 .9 )
] ~~
~/ à .~ ,
( 1 0 .3 . 1 0 )
T h u s i t is ev i dent t h at
Ð+ Ð
= Ð -„ ~ + É..
—)
= Ð
+
( 1 0 .3 . 1 1 )
10.3. Ex pense-L oad ed P rem iu m R eser ves
10 7
T hi s r esult should not com e as à surpr ise: Since ~~ + , Ó = Î , t he prem i um s
of Ð + Ð paid from age õ + 1 and onward must be suffi cient t o cover t he
fut ur e benefi t s. I t is also clear t hat t hen
( 10 .3 .1 2 )
holds for é = 2, 3,
,è.
I n pr act ical insur ance, t he m aximum val ue of à is usual ly given as à fi xed
percent age (say à = 3-' %) .
I n som e count ries t he ex pense-load ed prem i um reserve does not incl ude
an acquisit ion cost r eserve. T he modifi ed expense-load ed reser ve (Ger man :
I nvent ar deckungskapit al ) t hen becom es
( 10 .3 . 13 )
C h ap t er 11. E st i m at i n g P r ob ab i l i t i es of
D eat h
1 1 .1 P r o b l e m D e sc r i p t i o n
T h e on e- y ear p r ob ab i l i t y of d eat h q, h as t o b e est i m at ed fr om st at ist i cal d at a ;
t h ese d at a w i ll b e gen er at ed by à cer t ai n gr ou p of l i ves (å.g . p ol i cy hol d er s) ,
w h i ch h as b een u n d er ob ser v at i on for à cer t ai n p er io d (on e or m or e cal en d ar
y ear s) , t h e obser vati on p er i od. T he est i m at ed val u e of q
w i l l b e den ot ed by
Ú.
I f al l ob ser v at ion s ar e com p l et e, m ean i n g t h at each l i fe h as b een ob ser v ed
f r om age x u n t il age õ + 1 or p r i or d eat h , t h e st at i st i cal an al y si s i s q ui t e
si m p le . U n for t u n at ely , t h i s i s i n p r act i ce not t h e ñàâå, as w i l l b e i l l ust r at ed
by t h e so-cal l ed L exi s di agr am :
Òè ï å
À
Ê
å
õ
Obser vat ion p er iod
1
1
C h a p t er 1 1. E st i m a t i n g P r ob a b i l i t i es o f D ea t h
I n t his diagram each life under observat ion cor responds t o à di agonal line
segment showing t he t i me i nt erval dur ing which t he È å has been observed .
T he horizont al borders of t he rect angle are m ade up by t he age gr oup under
considerat ion , and t he vert ical borders represent begi nning and end of t he
observat ion per iod . L ives aged s before t he observat ion period begins ar e
incomplet ely observed (some m ay have died wit hout t his being obser ved );
si m ilarly, lives aged x + 1 aft er t he observat ion p eriod ends will be incom pl et ely
observed . A not her sour ce of i ncompl et e observat ions is lives which ent er t he
gr oup between t he ages of x and õ + 1, when t hey buy an insur ance policy ;
as well as l ives leav ing t he group between t he ages of x and õ + 1 for reasons
ot her t han deat h, such as pol icy t ermi nat ion .
L et è lives cont ri but e t o t he observat ions i n t he rect angle. A ssume t hat
life ï î . i is observed between t he ages of õ + t; and õ + s; (Î < t, < s; < 1).
T he sum
Å = (sl — t l ) + (âã — ãã) +
+ (~ — t Ä)
(11.1.1)
is called t he t he exposur e. T he t ot al lengt h of al l li ne segm ent s i n t he Lexis
diagr am is ~/ 2 Å .
L et D denot e t he number of deat hs observed i n t he rect angle (unlike Å„
D is of cour se an int eger ) . Denot e by I t he set of observat ions i which were
t erm i nat ed by deat h , and defi ne, for i Å I ,
8; =s; [ò
+ next
1]/ ò m
, t h ðàãÑ of t he year . ( 11.1.2)
i .å. s~ ~ |à obt ai ned by rounding
t o â;t he
1 1 .2 T h e C l a ssi ca l M et h o d
T he idea behi nd t he classical m et hod. is t o equat e t he ex pect ed number of
deat hs Ñî t he observed number of deat hs in order t o der ive an est im at or q .
T he expect ed number of deat hs is i n çî ãï å sense
i= l
E
1- Üqz + t;
i/i
5
1—a; Ü + ç;
( 1 1 .2 . 1)
T his ex pression is sim plifi ed by À ââèò ðé î è c of Sect ion 2.6, which st at es t hat
, Äq +„ — ( 1 — u )q, . T he expect ed number of deat hs t hen becomes
è
i= l
ð ,.(1 — t ;)q, — ~, , ( 1 — s,)q, = Å , ä + ~ ( 1 — s;)q, .
i iI I
i GI
( 11.2 .2)
Equat i ng t hi s expression t o t he observed number of deat hs, we obt ain t he
classical est im at or
D
E* + ~.-,„ (~ — ,)
11.3 . A l t er n a t i v e So l u t i o n
T his est im at or works well i f t he vol um e of dat a is lar ge. T he denomi nat or
is som et i m es approx im at ed . For inst ance, under t he assum pt ion t hat deat hs,
î ï ÑÜå à÷åãà~å, î ññè àÑ à~å õ + ~, t he est im at or i s si m ply
D
~ . + ~
( 1 1 .2 .4 )
T he est im at or (11.2.3) does not work sat isfact orily w it h spar se dat a. One
problem is t hat t he num er at or m ay exceed t he denom inat or , giving an obviously useless est im at e of ä, ; àï î ÑÜåã is t hat t he est im at or is not am enable Ñî
confi dence est i m at ion or hypot hesis t est ing, since i t s st at ist ical propert ies are
hard t o eval uat e. A lt ernat ive suggest ions will be present ed below .
1 1 .3 A l t er n a t i v e S o l u t i o n
L et ò be à posit ive int eger , and defi ne h = 1/ ò . We shal l est im at e ëä using
t he met hod of t he prev ious sect ion . To t his end we assume t hat ë „ î +„ is à
linear funct ion of è , i .e.
ë „ ä, + „ — ( 1 — m u ) ë ä, . f o r 0 < è < h
( 1 1 .3 .1 )
I n order Ñî m ake use of al l dat a we al so assum e t hat t he force of mor t alit y
between t he ages of a: and õ + 1 is à periodic funct ion wit h period h . T his
assum pt ion im pl ies, for ó' = 1, 2,
, ò — 1, t hat
z —è × õ ~- ~ ë + è
( 11.3 .2)
ë - è × õ ~- è
M aki ng use of t he two assum pt ions, one m ay now ar gue t hat t he ex pect ed
ï ø ï Üåã of deat hs is
m E , I,q
+ ò )
( ç ~ ) — s , ) ,ä , .
( 1 1 .3 .3 )
E q u a t i n g t h i s t o t h e o b ser v ed n u m b e r o f d e a t h s , o n e o b t a i n s t h e est i m a t o r
hD ,
aq* = Å ,. + ~: ;åã(ç;( m ) — ç;)
A ssum pt ion ( 11.3.2) im pl ies t hat ð
( 11.3 .4 )
= ( ëð )™. A n est i m at or of q, is t hus
obt ai ned from (11.3.4) by
(1
aq )
( 1 1 .3 .5 )
T his alt ernat ive procedure does not becom e i nt erest ing unt il we let m —+
oo. I n t he l i mit ing ñàçå t he assumpt ions ( 11.3.1) and (11.3.2) coi ncide wit h
A ssumpti on b in Sect ion 2.6, st at ing p +Ä — ð , + | for 0 < è < 1, and t he
*+ ú
C hapt er 11. Est im at ing P r obabi l i t ies of D eat h
112
ex pect ed number of deat hs (11.3.3) becomes Å ð õ+
, + -~.' T his leads us t o est im at e
t he const ant value of t he force of mort ality by t he r at io
D,
*+ ÿ
Å
( 1 1 ,3 .6 )
T h e p r ob ab i l i t y q i s t h en est i m at ed by
q, = 1 — åõð ( - ð ,~++ rð ) = 1 — ex p ( D ) Å , ) .
( 1 1 .3 .7 )
1 1 .4 T h e M a x i m u m L i k e l i h o o d M e t h o d
T he moment met hod of t he previous sect ions m ay be cri t ici sed on t he gr ounds
t hat equat ing expect ed'" number of deat hs in t he expressions (11.2.1) , ( 11.2.2)
and (11.3.3) Ñî t he observed number of deat hs, is à heurist ic approach . However , t he est im at or s (11.3.6) and ( 11.3.7) can also be der ived by à different
met hod .
W e assum e t hat t he è lives ar e i ndependent . T he l ikelihood funct i on of
t he observat ions is t hen
( 11 À .1)
T h e assu m p t i o n of à p i ecew i se co nst ant for ce of m or t al i t y si m p l i fi es t h i s t o
(ð , + ~) ~ * åõ ð ( - ð + ~Å ) .
T h i s ex p r e s si o n
i s m ax i m i sed b y
ð , + ~ — D , ( Å
+ .2
, so t h a t
( 1 1 .4 .2 )
( 1 1 . 3 .6 ) i s a l s o t h e
m ax imum likel ihood est im at or . T he i nvariance pri nciple t hen implies t hat q,
defi ned by (11.3.7) will also be t he m axi mum likeli hood est im at or of q, .
1 1 .5 S t a t i st i c a l I n f e r e n c e
A ct ual ly, bot h D and Å , are r andom variables. However , i t is convenient t o
t reat Å , as à non-random quant i ty. L et us t her efor e assum e t hat t he r andom
variable D has à Poisson dist ri but ion wi t h m ean
( 1 1 .5 . 1 )
wit h unknown par amet er ð ,~++ ~~.' T he probabilit y of D , deat hs, apart from
à fact or which is independent of ð , + ~, t hen is ident ical wi t h t he l ikelihood
(11.4.2) . T he poi nt est im at ors ( 11.3.6) and (11.3.7) t herefore r et ai n t heir
vali dity.
11.5. St at i st i cal I nfer ence
1 13
I t is also possible t o t reat D as à non-random quant ity, assumi ng t hat Å,
follows à óàò ò à distr i buti on wit h par amet ers o. = D and p = ,è, + ~. T his
approach is also compat ible wit h t he likel i hood ( 11.4.2) ; we shall not pur sue
t his her e.
T he followi ng t able display s confi dence li m it s for t he par am et er of à Poisson dist ri but ion , for an observed value of è . T he lower lim it Ë' is defi ned in
âèñÜ à way t hat t he probabi lit y of an observat ion of è or great er , calcul at ed
for t he val ue Ë' , is equal Ñî è ; si m ilarly, t he probability of observi ng è or less
for Ë" is equal t o ø .
T he confi dence int er val for Ë m ay be read off di rect ly ø t he t able from
t he number of observed deat hs D . D i viding t he confi dence limi t s by E , t he
confi dence i nt erval for y, +i is obt ai ned . F inally t he li mi t s m ay Úå t ransformed
2
t o give à ñî ï éñ1åï ñå i nt erval for q, . A s an ill ust r at ion , assume t hat D = 19
and E = 2000. T he 90% confi dence int er vals ar e t hen 12.44 < Ë < 27.88,
0.00622 < è + ' < 0.01394, 0.00620 < q* < 0.01384.
T he est im at ed pr obabili t ies q, (called "crude" r at es in pract ice) m ay fl uct uat e wi ldly from one age i nt erval Ñî t he next . I n such à sit uat ion one m ay use
one of t he more or less sophist icat ed m et hods of gr ad uat ion t heory i n order Ñî
obt ai n à smoot h funct ion . We shall not discuss t hese met hods in t his book .
I t is also possible t o use an exi st i ng l i fe t able as à st andard and t o post ulat e t hat t he forces of m ort ality i n t he observed group ar e à ñî ï í Ñàï Ñ (àäå
independent ) mult iple of t he for ces of mort ali ty i n t he st andar d È å t able. Denot ing t he forces of mort ality ø t he st andard t able by p '
, we t hus assume
*+ ~'
t hat
(11.5.2)
t he obj ect ive now being Ñî est im at e t he fact or f . Under t he assum pt ion t hat
t he number of deat hs occur ing i n diff erent age groups are i ndependent random
var i ables, we see t hat t he t ot al number of deat hs,
( 1 1 .5 .3 )
fol l ow s à Poi sson d i st r i b u t i on w i t h m ean
( 1 1 .5 .4 )
T h e est i m at or for Ë i s t hen Ë = D , an d w e fi n d
D
ñ
Å
( 1 1 .5 .5 )
t his ex pression is refer red t o as t he mor tali ty ra ti o. À confi dence int erval for
Ë m ay easily be t r ansformed int o à confi dence i nt erval for f .
Ch apt er 11. E st im at i ng Pr obabi li t ies of D eat h
1 14
For i n st an ce, assu m e t h at à t ot al of
D = D 40 + Ð 4| +
+ D 49 — 93 2
( 1 1 .5 .6 )
deat hs have been observed in t he age group between 40 and 50, whi le t he
ex pect ed number of deat hs accordi ng t o à st andard t able is
49
~ , p,' + | Å, = 1145.7 .
z =
4 0
( 1
1
. 5
. 7
)
T hen one obt ai ns f = 932/ 1145.7 = 0.813 = 81.3%. I n order t o const r uct
à confi dence i nt erval for f , we fi nd approx im at e 90% confi dence li mi t s for Ë'
and Ë" by solvi ng
932 — Ë' = 1.645 ,
áë
'
932 — Ë" = - 1.645 ,
á ë=
( 1
1
. 5
. 8
)
(not e t hat we have m ad e use of t he norm al approx i m at ion t o t he Poisson
dist ri but ion ) . One obt ains Ë' = 883.1 and Ë" = 983.6, and aft er di vision by
(11.5.7) t he confi dence int erval t ur ns out t o be 0.771 < / < 0.856.
3
0
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3
3
3
3
3
3
4
4
4
4
4
5
5
6
6
Ë"
7
0 .0 5 )
8
0
4
0
5
5
1
4
5
3
1
6
1
4
7
9
0
0
0
9
8
6
4
2
9
5
2
8
3
9
4
9
0
7
9
9
5
9
0
7
3
7
1
5
8
1
4
7
9
2
4
6
8
1
3
5
6
8
0
2
4
5
7
9
0
2
3
5
6
4
0
6
2
8
3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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.
.
.
.
.
.
.
.
.
.
.
.
3
4
6
7
9
0
1
3
4
5
6
8
9
0
1
3
4
5
6
7
9
0
1
2
3
4
6
7
8
9
0
6
2
7
3
8
4
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
4
4
5
5
=
6
Ë" (þ
6
ï
7
9
8
7
6
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
5
0
5
0
5
= 0 .0 5 )
0
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
4
4
5
5
6
0
5
6
2
7
7
1
9
8
0
3
7
2
9
6
5
4
3
3
4
5
7
9
2
5
8
2
6
0
5
9
7
0
6
6
0
5
0
0
3
8
3
9
6
2
9
7
4
1
9
6
4
2
0
8
6
4
2
0
8
7
5
3
2
0
9
7
5
8
2
5
9
4
8
.
.
.
.
.
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.
.
.
.
.
.
0
0
0
0
1
1
2
3
3
4
5
6
6
7
8
9
0
0
1
2
3
4
4
5
6
7
8
9
9
0
1
5
0
4
8
3
7
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
3
3
3
Ë~ (è
4
Ë ' ( w = 0 .0 1 )
4
6
4
.
3
4
1 1 .5 . S t a t i st i c a l I n fer en ce
1 15
C o n f i d e n c e l i m i t s f o r t h e p a r am e t e r o f à P o i s so n d i s t r i b u t i o n
( w = 0 .0 1 )
Chapt er 11. Est i m at ing Pr obabi l i t ies of D eat h
1 16
1 1 .6 T h e B a y e si a n A p p r o a ch
T he idea behind t he Bayesi an met hod is t o view ð,, + ~ as t he value assumed by
à random variable 9 wi t h prior probabilit y densi ty è (ä ) . Because of ( 11.4.2)
t he post erior densit y t hen is
ä~* åõð ( —ä Å , ) è (ä )
Ä~ Ð * åõð ( —t Å , )u (t ) dt '
( 1 1 .6 . 1)
T he par amet er ð,* +
~ m ay t hen be est im at ed by t he post er ior mean of 9 . T he
+ÿ
uncert aint y at t ached Ñî t he est i m at e m ay be quant ifi ed by t he percent iles of
t he post er ior dist ri but ion of 9 .
À com mon assum pt ion is t hat t he pr ior dist ribut ion of 9 is à gam m a
dist r i but ion wi t h paramet er s à and p . From ( 11.6.1) it is easy t o see t hat
t he post er ior dist ribut ion will again be à gam m a dist r ibut ion , now wit h t he
param et er s
H en ce w e ob t ai n
à = à + D, ,
à
,Î
rr + D ,
,Î + Å
p = p + Å, .
' p
à
Î + Å ,Î
Å,
D,
,Î + Å Å , '
( 1 1 .6 .2 )
( 1 1 .6 .3 )
à resul t t hat rem inds us of credibility t heory . A n est im at or of q, is obt ai ned
by t aki ng t he post er ior expect at ion of
q
=
l
— å
~
,
( 1 1 .6 .4 )
nam ely
'. = - (.-' )
T he percent iles of t he post er ior gam m a dist r ibut ion can be found using t he
t able of confi dence lim it s of t he Poisson par am et er , since it can be shown t hat
À' is t he ts-percent i le of à gamm a dist r ibut ion wit h par amet er s è and 1, and
t hat Ë" is t he (1 — û )-per cent ile of à gam m a dist r i but ion w it h par amet ers
è + 1 and 1. T hus t he post erior probabil ity t hat t he t r ue value of 9 lies
bet ween Ë' / p and Ë" / p , is 1 — 2û . Òî fi nd Ë' we put è = à , and for Ë" we
ðèÑ è = à — 1.
1 1 . 7 M u l t i p l e C a u se s o f D e c r e m e n t
W e ret ur n t o t he model int roduced i n Chapt er 7, where à decrement could be
t he result of any of ò causes. A s before we observe t he exposure Å and t he
number of decrement s D (for si m plicity we shal l refer Ñî t hese as number of
11.7. M ul t ip le Causes of D ecrement
117
deat hs). I n addi t ion we ar e infor med of t he number of deat hs by cause ó, for
7' = 1, 2,
, ò , denot ed by D , , Obv iously
~
1 ,õ +
~ 2 ,õ +
+
Ð
è ,õ
( 1 1 .7 . 1 )
D
T he probabilit y q, can be est imat ed by t he m et hods discussed befor e. We
shall now discuss est im at ion of t he probabilit ies ù , .
Let us assume piecewise const ant forces of decrement , i .e.
è ;, +„ — p , + ~ Ãî ã 0 ( è ( 1
( 1 1 .7 .2 )
Equat ion (7.2.2) shows t hen t hat t he aggregat e for ce of decrement also will
be piecewise const ant . A ssumi ng agai n t hat t he è l ives under observat ion are
i ndependent , we see t hat t he l ikelihood funct ion is gi ven by
m
Ï
j= l
Ì
,*+ , ) ' * ex p( ~. +,, Å . ) .
( 1 1 .7 .3 )
M ax i m u m l i kel i h o od est i m at or s ar e t hu s
~),*+ ~
DÅ,
,
7 = 1, 2, . . . , ò .
( 1 1 .7 .4 )
T h e cor r esp ond i n g est i m at or for
/1
.< 1
ç
/ Õ ~. |
( 1 1 .7 .5 )
õ
( 1 1 .7 .6 )
i s t h en
wi t h q, defi ned by ( 11.3.7) .
I n t he Bayesian set t ing t he ò for ces of decr em ent ar e considered as realisat ions of t he r andom variables B q, Î ð,
, Ý , w hich have à prior pr obability
densit y è (ä 1, äð,
, ä ) . T he post erior probabi lit y density is t hen proport ional Ñî
Ï
ó'= 1
(ä, )~' * åõð (- ä Å )è (ä ~, ä~,
, ä,„ ) ,
( 1 1 .7 .7 )
w it h t he defi nit ion ä = ä 1 + äð +
+ ä . Now ð, , + is t he post er ior m ean
of 9 ., and ù is t he post erior mean of
( 1 1 .7 .8 )
if we w r it e 6 = 6 1 + 6 2 +
+ 6
.
Chapt er 11. Est i m at i ng P r ob ab i li t ies of D eat h
118
T he analysis is part icular ly simple under t he assumpt ion t hat t he r andom
var i ables 9 are independent , 9 , having à gam m a dist ribut ion w it h paramet ers à , and p . I n t hat case t he 9 . ar e also independent à post er ior i , and 9 ,
has à gamm a dist ribut ion wit h paramet er s
à , = à , + Ð ), ,
,9 = p + Å , ,
( 1 1 .7 .9 )
which resul t s i n t he est i m at e
à;
à + Â
Vg.,ä ~ ,' = ó =
( 1 1 .7 . 10 )
Since t he rat io R , / 9 is independent of 9 and has à bet a dist ribut ion , we can
calcul at e t he m ean of (11.7.8) , obt aining
( 11.7 . 11)
h er e à = à ~ + à ð +
+ à ,„ an d q i s d efi n ed by ( 11.6 .5 ) .
1 1 .8 I n t e r p r e t a t i o n o f R e s u l t s
T he probabi lit y of deat h at à given age will oft en be non-st at ionary in t he
sense t hat t he gener al mor t al ity decl ines as t i me proceeds. Let us denot e t he
one-year probabi li ty of deat h of à person aged x at cal endar t i me t by q,' .
On t he basis of st at ist ical dat a from à cert ain observat ion per iod , t he val ues
q,' , q' +Ä q,' + , . ar e est im at ed ; her e t is t aken t o be t he m iddle of t he obser vat ion per iod . À l ife t able const ruct ed in t his way is cal led à cur r ent, or
cr oss-secti onal li fe t able. Such à li fe t able is, of course, an ar t ifi cial const r uct ion .
T he probabi lit ies of ñ1åàÔÜ'àï ñ1 ex pect ed val ues i nt r oduced i n t he preceding
chapt ers àll r efer t o one specifi c life. A ssum i ng t hat t he init i al age of t he
insured is x at t im e t , t he proper pr obabilit ies t o use ar e q' , q,' ++~„ q,' ++~~,
T he correspondi ng life t able is called à l ongi tudi nal or generati on li fe t able,
si nce i t relat es t o t he gener at ion of per sons born at t ime t — õ . T his life
t able defi nes t he probabilit y dist r i but ion of Ê = Ê (õ ) . T he probabil it ies 'of
deat h in à generat ion È å t able must be est im at ed by à sui t able met hod of
ext r apol at ion .
A p p en d ix
À . C om m u t at io n F u n ct ion s
À .1 I n t r o d u ct io n
I n t his appendix we give an int roduct ion t o t he use of com mut at ion funct ions.
T hese funct ions were invent ed in t he 18t h cent ur y and achieved great popul ar it y, whi ch can be ascribed t o two r easons:
R ea so n 1
Tables of com mut at ion funct ions sim pl ify t he cal culat ion of num erical values
for m any act uar ial funct i ons.
R ea s o n 2
Ex pect ed val ues such as net single prem i ums m ay be derived wit hi n a det erm inist ic model closely r el at ed t o com mut at ion funct ions.
Bot h r easons have lost t heir signifi cance, t he fi rst wit h t he advent of powerful
com put er s, t he second wi t h t he growing accept ance of model s based on pr obabili ty t heory, whi ch allows à m ore com plet e under st andi ng of t he essent ials
of insur ance. I t m ay t herefore be t aken for grant ed t hat t he days of glory for
t he com mut at ion funct ions now b elong Ñî t he past .
À . 2 T h e D e t e r m i n i st i c M o d e l
I m agine à cohort of l ives, al l of t he same age, obser ved over t ime, and denot e
by 1 t he number st i ll l iving at age x . T hus d, = 1, — 1, +| is t he number of
deat hs between t he ages of x and õ + 1.
Pr obabilit ies and expect ed values m ay now be derived from sim ple proport ions and aver ages. So is, for inst ance,
(À .2.1)
A ppendix À . Com mut at ion F unct ions
12 0
Ñ
å
p
r
o
p
o
r
t
i
o
n
o
f
p
e
r
s
o
n
s
a
l
i
v
e
a
t
a
g
e
r
+
t
,
r
e
l
a
t
i
v
e
t
o
t
h
e
n
u
m
b
e
r
o
f
p
e
r
s
o
n
s
h
alive at age õ , and t he probabilit y t hat à life aged x w ill die wit hin à year 1s
(À .2.2)
q, = d, / 1.
I n Chapt er 2 we int roduced t he expect ed curt at e fut ure lifet ime of à l ife
aged x . Replacing ~ð, by 1, +~/ 1, i n (2.4.3), we obt ain
1* + 1 + 1õ+ 2 +
å
1,
T he numerat or in t his expression is t he t ot al ï ø ï Üåã of complet e fut ure years
t o be "lived"» by t he 1, lives (õ ) , so t hat å, is t he aver age number of com plet ed
years left .
À
.3
L i f e
A
n n u it ie s
We fi rst consider à È å annui ty-due wit h annual payment s of 1 unit , as int roduced in Sect ion 4.2, t he net single prem ium of which annuity was denot ed
Úó à . Replacing „ð ø (4.2.5) by 1 +1,/ l Ä we obt ain
or
à
1õ +
î 1õ+1 + î
l
~ +
1õ+ 2 +
( À .3 .1 )
1. +1 + î 21. +, +
(À .3.2)
T his result is oft en referred Ñî àâ t he equi valence pr incipl e, and it s i nt erpr et at ion wit hin t he det erm i nist ic model is ev ident : if each of t he 1, persons
livi ng at age x were t o buy an annuit y of t he given type, t he sum of net si ngle
pr em i ums (t he left hand side of (À .3.2) ) would equal t he present val ue of t he
benefi t s (t he right hand side of (À .3.2) ).
M ult i plyi ng bot h num erat or and denom i nat or in (À .3.1) by v* , we fi nd
õ1 ~ , õ+11
+
õ+21
v* l ,
W it h t he abbreviat ions
1õ for
, m
N u
,æl —
w e t h en obt ai n t hDe,õ si m vp l e
a
Ê
+
~ õ+ 1 +
1~õ + 2 +
' '
( À .3 .4 )
N,
õ
D
T hus t he m anual cal cul at ion of à, is ext r emely easy if t ables of t he commut at ion funct ions D , and N are available. T he funct ion D , is called t he
"di scounted num ber of sur vi vor s".
À .4 . L ife I nsur ance
12 1
ãà ãóS il m
i f ei l aarnl ny uoi tnye, m a y o b t a i n f o r m u l a s f o
o rr t h e
e n
n et
et si
si n
n g l e p r em i u m o f à t em p oN , — 1V
É
õ â 1
ð
õ
i m m ed i a t e l i fe a n n u i t i es ,
Ô ,~|
Î õ
( À .3 .7 )
a ne dt rgaennslera taed
b
l a nÑîn u i t i es w i t h a n n u a l p ay m en t s : f o r m u l1a ( 4 .4 .2 ) m ay n a t u r al l y
@(ó )
ÒÎ
õ + Ò1 Ð õ ~- 1 + Ò2 Ð õ ( -2 +
õ + ~ õ+ 1 + h z + 2 +
' ' '
( À( À.3.3
. 10
.8 )
.9
D ,
F o r t h e sp ec i a l ñàçå Ò~ — k + 1 w e o b t ai n t h e f o r m u l a
(I a )
h åãå
er e t h e c o m m u t a t i o n f u n c t i o n S , i s d efi n ed b y
S
À .4
L ife
=
D , + 2 Ð , ~ | + ÇÐ , ä + .
I n su r a n ce
I n a d d i t i o n t o ( À ..3 ..4 ) a n d ( À .3
. . 10 ) w e n ow d efi n e t h e c o m m u t a t i o n f u n c t i o n s
Ñ,
Ì
=
v*+' d
z ~
. = ñ. +Ñ.„ +Ñ.„ +
 ,
=
Ñ , + 2 Ñ , .~ | + ÇÑ , ~. 2 +
Ì
õ + Ì
~| + Ì
õ+ 2 +
R e p l a c i n g „ ð , ä, + ~ i n eq u a t i o n ( 3 .2 .3 ) b y d + q / 1„
æå o b t a i n
l
Ñ. + Ñ.+, + Ñ.+, + " .
D ,
Si m i l ar l y on e ob t ai n s
(1 À ) ,
ÌD
( À .4 .2 )
è é + 2 v 0 + 1 + Çè É
+
I,
Ñ , + 2 Ñ , + 1 + ÇÑ , ~ 2 +
D
( À .4 .3 )
12 2
A p p en d i x À . C o m m u t at io n F u n ct i o n s
Obviously t hese formul ae m ay be der ived wi t hin t he det erm inist i c model by
means of t he equival ence princi ple. I n order t o det er mi ne À , one would st art
wi t h
l ~A ,õ — ' é 1õ + v Í õ+ È + v Éõ+ ~ + • • •
( À .4 .4 )
by im agi ni ng t hat 1, per sons buy à whole l ife i nsurance of 1 unit each , payable
at t he end of t he year of deat h , in ret urn for à net single prem ium .
Corresponding for mul ae for t er m and endowm ent insurances ar e
( I A ) ' .+
Ì
õ ™
D
*+
+ Â õ+ è
D
Ñ + 2Ñ +1 + ÇÑ, +ð + .
Ì
+ ï Ñ, +„ 1'
Â,
+ Ì õ+| + Ì õ+~ +
+ Ì , +„ 1 — ï Ì
+„
D,
~ õ — ~4 + è — ï Ì
 õ
õ+ è
( À .4 .5 )
which speak for t hem selves.
T he com mut at ion funct ions defi ned i n (À .4.1) can be expr essed in t er ms
of t he com mut at ion funct ions defi ned in Sect ion 3. From d = I, —1, +| follows
Ñ, = v D , — D , +~
( À .4 .6 )
Su m m at i o n y iel d s t h e i d ent i t i es
M , = v Æò — ( N õ, — Â õ ) =
D õ
, — d N ,õ
( À .4 . 7 )
and
= ret
N ,rieve
— dst,he
. ideri t it ies
D ivi ding bot h equat ions by D „R,wå
( À .4 .8 )
À
(I .5
A.2)
), . =
see equ at io n s (4 .2 .8 ) an d (4
( À .4 .9 )
1 dà ,
à, — d(I ii ), ,
À . 5 N e t A n n u a l P r e m i u m s a n d P r e m i u m R e se r v e s
Consider à whole li fe insurance wi t h 1 unit payable at t he end of t he year of
deat h , and payable by net annual prem iums. Using (À .3.5) and (À .4.2) we
fi nd
À.. *, = —
Ì *, .
P = —
(À .5.1)
À .5.
. . Net Annual Premiums and Premium Reserves~
( À .5123
.3
.2 )
õ+ É
Of ccourse, t he det er m inist ic approach , |.å.
' . . t h å con ' it ion
P l
,ð ~
~
2
ð õ ~õ + ~
• • • —
äõ + > 2 ~õ+ 1 +
~ Çç
à õ~-2 + • • •
lead s t o t he âàò å result .
T he net prem i um reserve at t he end of year /ñ t hen becom es
Ä
= À +~ — Ð, à, +~ = Ì
T his r esul t m ay also be obt ained
å bó t hå d et erm inist i c condit i on
V l õ» ~ + Ðõ1, +~ + VP~l y+Q+r + V2Ð l
õ+ é +
=
Vd
V ~ õ+ é+ 1 +
V ~ õ+ é+ 2 +
+
' ' '
(À .5.4)
Í åãå one im agines t hat each person alive at t ime É is al lot t ed t he am ount
t he condit ion (À .5.4) st at es t hat t he sum of t he net r em
T he i nt erest ed reader should be
åà
able
å t îo àðð
à 1ó t his t echnique t o ot her , m ore
A p p en d ix
 . S i m p l e I n t e r e st
I n pract ice, t he accumulat ion fact or for à t i me int erval of lengt h h is occasionally appr oxi m at ed by
( 1 + i )"
1 + hi .
( Â .1)
T his appr ox im at i on is obt ained by negl ect i ng àll but t he linear t erm s in t he
T aylor ex pansion of t he left hand side above; alt er nat ively t he right hand side
m ay be obt ai ned by l inear int erpolat ion bet ween h = 0 and h = 1. Sim il arly
an approxi m at ion for t he discount fact or for an int er val of lengt h h is
ê
(1
~) ü
(Â .2 )
T he approxi m at ions (Â .1) and (Â .2) have lit t le pr act ical im port ance since t he
advent of pocket calculat ors.
Int erest on t ransact i ons wit h à savi ngs account is som et im es calculat ed
accor ding Ñî t he followi ng rule: If an amount of ò is deposit ed (dr aw n) at
t i me è (Î < è < 1) , i t is val ued at t ime 0 as
r v" ~ ò( 1 — u d) .
( Â .3)
A t t h e en d of t h e y ear (t i m e 1) t he am ou n t i s v al u ed as
r (1 + i )
"
=
r ( 1 + i )v"
ò(1 + i ) ( 1 — ud)
r j l + ( 1 — u)i ) .
( Â .4 )
T his t echnique am ount s t o accumul at ion from t ime è t o t i me 1 according t o
(Â .1) or discount i ng from u t o 0 accordi ng t o (Â .2) . W it h à sui t ably chosen
vari able force of int erest t he r ule is ex act ; t his var i able force of int erest is
det er mi ned Úó equat ing t he accumul at ion fact ors:
1 + hms
( 1 —giuves
)i =t he
åõðex( pression
f 6(t )dt)
D iffer ent iat ing t he logarit
á(è ) = 1 + (1 — u )i
1 — ud
(Â .5)
(Â .6 )
12 6
A ppend ix  . Si m pl e I nt er est
for 0 ( è ( 1. T he force of int erest t hus increases from á(0) = d t o á( 1) = i
dur ing t he year .
T he t echnique sket ched above is based on t he assumpt ion t hat t he accumul at ion fact or for t he t im e int erval from è t o 1 is à li near funct ion of è ; t his
assum pt ion is analogous t o A ssump ti on c of Sect ion 2.6, concer ning mor t alit y
for fract ional dur at ions. T he sim i lar it y bet ween (Â .6) and (2.6.10) is evident .
A p p en d ix Ñ
E x er ci ses
128
Ñ .Î
A P P E N D I X Ñ . E X E R CI SE S
I n t r o d u ct i on
T hese exer cises provide t wo t ypes of pr act ice. T he fi rst type consists of t heor et i cal ex ercises, some demonst r at ions, and manipulat ion of symbols. Áî ò å of
t hese problems of t he fi rst ki nd are based on Society of Act uar ies quest ions from
ex am i nat ions pr ior t o Ì àó 1990. T he second type of pract ice involves usi ng à
spreadsheet program . M any exercises are sol ved in Appendix D . For t he spreadsheet exercises, we gi ve à gui de t o follow in wr it ing your own program . For t he
t heoret ical exercises, we usual ly gi ve à complet e descr ipt ion . We pr ovi de guides
for solvi ng t he spreadsheet problems, rat her t han comput er codes. T he st udent
should wr it e à program and use t he guide Ñî veri fy it . We use t he t ermi nology
of Excel in t he gui des. T he t erminology of ot her programs is analogous.
1 woul d l ike Ñî t hank Í àë ÿ Gerber for allowing me t o cont r ibut e t hese exercises t o his t ext book . It is à pleasure Ñî acknowledge t he assist ance of Georgi a
St at e Uni versity gr aduat e st udent s, Ì àçà Ozeki and Javier Suar ez who helped
by checking solut ions and proofreading t he exercises.
1 hope t hat st udent s will fi nd t hese exercises challenging and enlight ening.
A t l ant a, Ju ne 1995
Sam u e1 Í . Ñî õ
Ñ.1. MATHEMATI CS OF COMPOUND INTEREST:EXERCISES
Ñ .1
129
M at hem at ics of Com p ou n d I nt er est :
Ex er cises
À 6ond is à cont ract obligat ing one party, t he borrower or bond issuer, Ñî ðàó
to t he ot her party, the lender or bondholder, à series of future payments defi ned
by t he face value, F , and t he coupon rate, ñ. At the end of each future period
t he borrower pays cF to t he lender . The bond mat ures after N periods wit h à
fi nal coupon payment and à simult aneous payment of the redempt ion value Ñ.
Usually Ñ is equal to F . Investors (lenders) require à yield to mat urity of i > 0
effect ive per period. The price, P, is t he present value of future cash fl ows paid
t o t he bondholder . The fi ve values are relat ed by t he following equat ion.
Ð = cF
1 —î N
.
+ CaN
where è = 1/ (1+ i ).
Ñ .1. 1
T h eor y E x er cises
1. Show t hat
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3. À company must retire à bond issue with fi ve annual payments of 15,000.
The fi rst payment is due on December 31, 1999. In order t o accumulate t he
funds, t he company begins making annual payments of Õ on January 1, 1990
into an account paying effect ive annual interest of 6%. The last payment is to
be made on January 1, 1999. Calculat e Õ.
4. At à nominal annual rat e of interest j , convert ible semiannually, t he present
value of à series of payment s of 1 at t he end of every 2 years, which continue
forever, is 5.89. Calculat e j .
5. À perpet uity consist s of yearly increasing payments of (1 + é), (1 + /ñ)ã,
(1 + É)ç, et c., commencing at t he end of t he fi rst year. At an annual åï åñé÷å
interest rat e of 4%, t he present value one year before t he fi rst payment is 51.
Det ermine k.
6. Six months before the fi rst coupon is due à t en-year çåï è-annual coupon
bond sells for 94 per 100 of face value. The rat e of payment of coupons is
10% per year. The yield to maturity for à zero-coupon ten-year bond is 12%.
Calculate t he yield to mat urity of the coupon payments.
APPENDIX Ñ. EXERCISES
130
7. À loan of 1000 at à nominal ãàÑå of 12% convertible mont hly is t o be repaid
by six mont hly payments with the fi rst payment due at t he end of one mont h.
The fi rst t hree payments are z each, and t he fi nal three payments are Çõ each.
Calculate z.
8. À loan of 4000 is being repaid by à 30-year increasing annuity immediat e.
The initial payment Û ñ, each subsequent payment is é larger t han t he preceding
payment . The annual eff ect ive interest rate is 4%. Calculat e t he principal
outst anding immediately aft er t he nint h payment .
9. John pays 98.51 for à bond t hat is due to mat ure for 100 in one year. It
has coupons at 4% convert ible semiannually. Calculat e t he annual yield rate
convertible semiannually.
10. The death benefi t on à life insurance ðî éñó can be paid in four ways. All
have t he âàò å present value:
(i) À perpet uity of 120 at t he end of each mont h, fi rst payment one mont h after
t he moment of deat h;
(È) Payment s of 365.47 at t he end of each mont h for ri years, fi rst payment one
month after t he moment of deat h;
(ø ) À payment of 17,866.32 at the end of n, years after t he moment of death;
and
(iv) À payment of Õ at t he moment of deat h.
Calculat e Õ .
Ñ . 1 .2
Sp r ea d sh e et E x er c i se s
1. À serial bond wit h à face amount of 1000 is priced at 1145. The owner
of the bond receives annual coupons of 12% of t he outst anding principal . The
principal is repaid by t he following schedule:
(i) 100 at t he end of each years 10 t hrough 14, and
(È) 500 at t he end of year 15.
(à) Calculat e t he investment yield using the Üø 1Ñ-|ï Goal Seek procedure.
(Ü) Use t he graphic capability of t he spreadsheet t o illust rat e t he invest ment
yield graphically. To do t his, construct à Dat a Table showing various invest ment
yield values and the corresponding bond prices. B om t he graph, determine
which yield corresponds Ñî à price of 1,145.
2. À deposit of 100,000 is made int o à newly est ablished fund. The fund ðàóâ
nominal interest of 12% convert ible quarterly. At t he end of each six mont hs
à wit hdrawal is made from t he fund. The fi rst withdrawal is Õ, t he second is
2Õ, t he t hird is ÇÕ, and so on. The last is t he sixt h withdrawal which exactly
exhausts t he fund. Calculate Õ.
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A P P E N D I X Ñ . E X E R CI SE S
1 32
Det ermine how many bonds of each mat ur ity t he insurer should buy on July
1, 1994 âî t hat t he aggregate cash fl ow rom t he bonds will exact ly mat ch
t he i nsurer 's obligat ion under t he t erms of t he claim set t lement . Assume t hat
fract ions of bonds may be pur chased.
8 . À loan of 100,000 is repayable over 20 years by semi annual payments of
2500, plus 5% interest (per year convert i ble t wice ðåã year ) on t he out st anding
balance. Immediat ely aft er t he t ent h payment t he lender sel ls t he loan for
65,000. Cal cul at e t he correspondi ng market yiel d t o mat ur ity of t he loan (per
year convert ible t wice per year ).
9. À bond wi th Ãàñå val ue 1000 has 9% àï ï èà! coupons. T he borrower may
call t he bond at t he end of years 10 though 15 by payi ng t he face amount plus
à call premium , accor ding t o t he schedule:
~ Óåàã
! 10 ~ 11 ! 12 ! 13 ~ 14 ~ 15 !
] Premi um [ 100 [ 80 ! 60 ( 40 j 20 ! Î !
For example, i f t he borrower elect s t o repay t he debt at t he end of year 11 (11
year s from now), à payment of 1000 + 80 = 1080 pl us t he coupon t hen due of
90 would be paid t o t he lender . T he debt is pai d; ï î furt her payment s would
be made. Calculat e t he pri ce now, one year befor e t he next coupon payment ,
Ñî be cert ain of à yield of at least 8% Ñî t he call dat e.
10 . Equal deposit s of 200 are made Ñî à bank account at t he beginning of each
quart er of à year for fi ve years. T he bank pays int erest from t he dat e of deposit
at an annual eff ect i ve ãàÑå of i . One quart er year aft er t he l ast deposit t he
account balance is 5000. Cal cul at e i .
Ñ.2. ÒÍ Å F UT UR E L I F E T I M E OF À L I F E À Ñ Å Þ Õ : Å ÕÅ ß Ñ1ÁÅ Á
Ñ .2
133
T h e Fut u r e L ifet im e of à L i fe A ged x :
E x er cises
T hese exer cises somet imes use t he commut at ion funct ion not at ion int roduced
in Appendix À and t he following not at ion wit h regar d t o mort al ity t ables. T he
Il lust rat i ve Li fe Table is gi ven in Appendix Å . It is required for some exer cises.
À mort al ity t able coveri ng t he range of ages õ (Î < õ < ø ) is denot ed by
l , which represent s t he number 1î of t he new-born li ves who sur vive t o age x .
T he probability of surviving t o age x is â(õ ) = l / le. T he r ule for calculat ing
condit ional probabil it i es est ablishes t his rel at ionshi p t o ñð :
,p = Pr (T (0) > õ + t ]T (0) > õ) = â(õ + t )
â(õ )
l +q
l
In t he ñàçå t hat t he condit ioning invol ves more informat ion t han mere survi val ,
t he not at ion ñp~~ is used. T hus if à person age x applies for i nsurance and is
found t o be in good healt h, t he mort ality funct ion is denot ed <p i i rat her t han
ñð . T he not at ion [õ] t ells us t hat some infor mat ion in addi t ion t o Ò(0) > õ
was used in preparing t he sur vi val dist ri but ion . T his gives r ise t o t he select and
ult imat e mort ality t able discussed in t he t ext .
Í åãå are some addit ional mort ality funct ions:
m
L
c e n t r a l d e a t h ãà Ñå =
=
aver age number of sur vi vor s t o (õ, õ + 1)
ê+ 1
d
d
=
=
—
L
lvd ñ. =
1* +ññññ
î
ï ø ï Üåã of deat hs in (õ , õ + 1) = l —1 ~ 1.
Since ~ð ð~+, — —ñ,
d ,ð , t hen in t erms of l we have lz+cpz+c = —óñ4 +ñ or
let t ing y = õ + t, èå have 1„ ññä —
~ 1„cc for all y . T he following àãå useful for
— —~
dy
cal culat ing Var (T ) and Var (K ) :
E [T ]
=
tñ
, p~p~+,dt
î
2t rp~dt
î
E ]I t 2]
~), ~ ~ 2
@
=i
~) (2/ñ+ 1)ñ,+ 1ð .
1 34
A P P E N D I X Ñ . E X E R C I SE S
Ñ .2 . 1
T h e o r y E x er c i se s
1 . G i ven :
100 — õ — t
100 —
for 0 < õ < 100 an d 0 < t < 100 — õ . C al cul at e p 4s .
2 . G iven :
100
for õ = 60 an d 0 < t < 100. C al cul at e Å [Ò (õ )] .
3
3 . G i ven : ð , .+ , — — 1
+
for 0 < t < 85. C al cu lat e sep .
85 — 8
105 — t
á ~+ ~ 3'
4 . G i ven : <p ~ = (
)
~1+ + t)
for t > Î . C al cu l at e t he com plet e È å ex p ect an cy
of à p erson age õ = 4 1.
5.
G i ven : q
= 0 .200 .
C al cu lat e m
=
qz
usi ng assump ti on ñ, t he
3î ' ~ ' ~
B ald u cci assu m pt ion
6 . G i ven :
( i ) p ~ q is const ant for 0 < t < 1 an d
(é ) ~Ü = 0 .16.
C al cu l at e t h e val ue of t for w h i ch ~ð
= 0.95.
7 . G i ven :
(i ) T h e cu r v e of deat h l y
is const ant for 0 < õ < û .
(é ) þ = 100 .
C al cu l at e t h e var i an ce of t h e r em ai ni n g l i fet im e r andom var i ab le Ò (õ ) at õ = 88 .
8 . G i ven :
(i ) W hen t h e for ce of m or t al i t y is p + Ä 0 < t < 1, t hen q = 0.05.
(é ) W hen t h e for ce of m ort al it y i s ð + ~ — ñ, 0 < t < 1, t hen q = 0.07 .
C al cu l at e c.
9 . Pr ove:
(i 3 ,ð
= ex p ( — ) *
( i i ) ùâ ñÐ * =
( ru
ð , d s) an d
Ð * + ñ) ñÐ õ .
10 . You ar e gi ven t he fol lowi ng ex cer pt from à select an d u lt i m at e m or t ali t y
t ab l e w i t h à t wo-year select p er i od .
Ñ .2 .
Ò Í Å F U T U R E L I F E T I M E O F À L I F E A G E D X : E X E R C I SE S
0 .737
1 00 0. 5q 5i . 11 ~ 1 0 0 q ~. 1+ ,
33
~
13 5
100q i s !
30
0 .4 3 8
0 .5 7 4
0 .6 9 9
3 1
0 .4 5 3
0 .5 9 9
0 .7 3 4
32
0 .4 7 2
0 .6 3 4
0 .7 9 0
0 .5 1 0
0 .6 8 0
0 .8 5 6
0 .9 3 7
C alcu lat e
1 1 .
1 0 0 ( ö q is p i + q ) .
G iv en :
f
fo r
0
<
õ
a t t a in in g
1 2 .
<
12 1 .
C al c u l a t e
a ge 4 0 , b u t
G iv en
= ( 12 1 — õ ) ~
t h e
p ro b ab ility
b e fo r e a t t a i n i n g
age
t h e fo ll o w i n g t a b l e o f v a l u e s o f å
A g7 e7 e
!
t h at
à
li fe
ag e
2 1
w ill
d ie
a ft er
57 .
:
1 å0 . 5
7 5
7 6
1 0 .0
9 .5
C a l cu l a t e t h e p r o b a b il it y
t h at
à l i fe a g e 7 5 w i ll su r v i v e t o a g e 7 7 . H i n t :
r ecu r sio n
(1 +
å
r el a t io n
1 3 .
M o r t alit y
1 4 .
G iv en :
e
=
ð
fo l lo w s d e M o i v r e ' s l a w
fo r
0
1 5 .
<
t
<
a n d E [T ( 1 6 ) ] =
7800 70t
ñ ò î
6 0 .
=
C al cu l a t e t h e ex a ct
tã
v a l u e o f qs p — e sp .
G iv en :
0
1 6 .
<
t
<
G iv en :
10 0 — õ .
q
C alcu lat e m
1 7 .
(i ) ð
an d
=
0 .4 2 0
t w o
an d
a s s u m p ti o n
i n d ep e n d e n t
t h e ot h er
1.
— t
6 a p p l i e s Ñî
rat e ex a ct ly .
li v es,
w h ich
t h e y ea r
o f age õ
Ñî
õ
+
1 .
(S ee ex e r c ise 5 .)
are
id en t ic a l
ex cep t
t h at
o n e
is
à
is à n o n - sm o k er . G iv e n :
is t h e fo r ce o f m o r t a l it y
( | | ) cy
õ
C alcu lat e × àã(Ò (õ ) ) .
, t h e ce n t r al d e a t h
C o n si d e r
sm o k er
ñ )
3 6 . C alcu lat e × àã(Ò (16 )).
7
10 0 -
fo r
U se t h e
ù 1) .
i s t h e fo r c e o f m o r t a li t y
fo r
n on -sm o k ers
fo r s m o k e r s fo r 0
fo r
0
<
õ
<
cu .
<
õ
<
û , w h e r e c is à c o n st a n t ,
A P P E N D I X Ñ . E X E R CI SE S
Calcul at e t he probability t hat t he remaini ng l ifet ime of t he smoker exceeds t hat
of t he non-smoker .
18 . Deri ve an expression for t he der ivat ive of q wit h respect t o x in t er ms of
t he force of mort al ity.
19 . Gi ven: p = kx for all õ > 0 where é is à posit ive const ant and qepss = 0.81.
Cal culat e zepqe .
20 . Gi ven:
(i ) l = 1000(ñ ~ —õç) for 0 < õ < ø and
(é) Å [Ò(0)] = Çø/ 4.
Cal cul at e × àã(Ò(0)).
Ñ .2.2
Spr eadsheet Ex er cises
1. Put t he Illust r at i ve Li fe Table I values int o à spreadsheet . Calcul at e d and
1000q for õ = Î , 1, . . . , 99.
2 . Cal cul at e å , õ = 0, 1, 2, . . . , 99 for t he Illust r at ive L i fe Table. Hint : Use
formula (2.4.3) t o get ess — pss = 0 for t his t able. T he ãåñöãÿ ÷å formula
å = ð (1 + å + ~) foll ows from (2.4.3) . Use it Ñî calculat e from t he higher age
t o t he lower .
3 . À sub-st andard mort ality t able is obt ained fr om à st andar d t able by addi ng
à const ant c t o t he force of mort ality. T his results in sub-st andard mor t ality
r at es q' whi ch are r el at ed Ñî t he st andar d r at es q by q' = 1 —å ' ( 1 —q ). Use
t he Illust r at i ve L ife Table for t he st andar d mort alit y. À physi cian ex amines à
li fe age õ = 40 and det ermines t hat t he expect at ion of remai ni ng li fet ime is 10
years. Det ermine t he const ant ñ, and t he result ing subst andard t able. Prepare
à t able and graph of t he mort al ity rat io (sub-st andard q' t o st andar d q ) by
year of age, beginning at age 40.
4 . Draw t he graph of ð
=  ñ' , õ = 0, 1, 2, . . . , 110 for  = 0.0001 and each
value of ñ = 1.01, 1.05, 1.10, 1.20. Calculat e t he corresponding values of l
draw t he graphs. Use ls = 100, 000 and round t o an int eger .
and
5. Let q = 0.10. Draw t he gr aphs of ð +„ for è r unning from Î t o 1 i ncr ements
of 0.05 for each of t he int er polat ion formulas given by assumpti ons à, 6, and c.
6. Subst it ut e Äq for p +Ä in Exercise 5 and rework .
7. Use t he met hod of least squares (and t he spreadsheet Sol ver feat ure) t o fi t à
Gompert z dist ribut ion t o t he Ill ust r at ive Li fe Table values of ~p for õ = 50 and
t = 1, 2, . . . , 50. Dr aw t he gr aph of t he t able values and t he Gompert z values
on t he same àõåÿ.
8. À sub-st andard mort al ity t able is obt ained from à st andard t able by mult iplying t he st andar d q by à const ant é > 1, subj ect to an upper bound of 1.
Ñ .2 .
ÒÍ Å
F U T U R E L IF E T IM E
OF À
L I F E A G E D Õ : Å Õ Å ß Ñ 1Á Å Á
T h u s t h e su b st a n d a r d q' m o r t al i t y r a t es a r e r el a t e d t o t h e st a n d ar d r at es q
13 7
by
q' = m i n ( k q , 1 ) .
( à ) F o r v a l u es o f É r a n g i n g f r o m 1 t o 10 i n i n cr em en t s o f 0 .5 , ca l c u l a t e p o i n t s
o n t h e g r a p h o f ~ð ' fo r ag e õ = 4 5 a n d t r u n n i n g fr o m 0 t o t h e en d o f t h e t ab l e
i n i n c r em e n t s o f o n e y ea r . D r a w t h e g r a p h s i n à si n g l e ch a r t .
( Ü) C al cu l a t e t h e su b - st an d ar d l i fe ex p ect a n c y a t a ge õ = 4 5 fo r e ach v a l u e o f
/ñ i n ( à ) .
A P P E N D I X Ñ . E X E R C I SE S
13 8
Ñ .3
L i f e I n su r a n c e
Ñ .3.1
T heor y Ex er cises
1 . G i ven :
(i ) T h e su r v i val fu n ct i on is ã(õ ) = 1 — õ / 100 for 0 < z < 100.
(i i ) T h e for ce o f int er est is 6 = 0 .10 .
C al cul at e 50,000À Çî .
2 . ßé î ÷ t h at
(I A )
—
À ñ
ñ1
(1À ) + , + À + ~
si m pl i fi es t o ep .
3 . Z q is t he p r esent val ue r an dom var i able for an n -y ear cont i nuous en dowm ent
i n su r an ce o f 1 i ssued t o (z ) . Z q is t he pr esent val ue r an dom v ar i ab l e for an
è -y ear cont i nuous t er m i nsu r an ce o f 1 i ssued Ñî õ . G i ven :
(i ) V ar ( Z @) = 0 .01
(é ) e
(ø ) „ ð
= 0 .30
= 0 .8
(i v ) E [Z q] = 0.04.
C al cu l at e × àã(ß ~) .
4 . U se t he I l lu st r at i ve L i fe T ab l e an d i = 5% t o cal cu l at e À
45 : 201
5 . G i ven :
(i ) À .-„ ~ — è
(é ) À ~.-„ ~ — ó
( i i i ) À ~ + ò, — z .
D et er m i n e t h e val ue of À
in t er m s of è, ó, an d z.
6 . À cont i nuous wh ole l i fe i nsu r an ce i s i ssued t o (50 ) . G i ven :
( i ) M ort al it y fol lows de M oi v r e's l aw w i t h ñî = 100 .
(é ) Sim p le int er est w i t h i = 0 .01.
(i i i ) bq = 1000
0.1Ð .
Ñ .Ç.
I I F E I N S UR A N CE
13 9
C al cu l at e t he ex p ect ed valu e of t he p r esent val u e r an dom var i ab le for t h i s i nsur an ce .
7 . A ssu m e t h at t h e for ces o f m ort al it y and i nt er est ar e each const ant and
den ot ed by p an d 6, resp ect i vely . D et er m i n e × àã(è~ ) in t er m s o f p an d á .
8 . For a sel ect an d u l t im at e m or t al it y t ab l e w i t h à one-ó÷àò select p er i od ,
ù ~ = 0 .5q for al l õ > Î . Show t h at A — À ~ ~ = 0.5eq ( 1 — À » 1) .
9 . À sin gl e p rem i um w hol e l i fe i nsu r an ce i ssued t o (õ) p r ov i des 10 ,000 of i nsu r an ce du r i ng t h e fi r st 20 years an d 20,000 of i n su r an ce t h er eaft er , p l us à ret u r n
w i t h out i nt er est o f t h e n et sin gl e prem i u m i f t he i nsur ed di es d ur i ng t h e fi r st 20
year s. T h e n et si n gle p r em iu m is p aid at t h e b egin n i n g o f t he fi r st y ear . T h e
deat h b enefi t is p ai d at t h e en d of t h e y ear of deat h . Ex pr ess t he net si n gl e
p r em i u m u si ng com m ut at ion fu n ct i ons.
10 . À t en-year t er m i nsur an ce p ol i cy i ssu ed t o (õ) p r ov i des t he fol low i ng deat h
b en efi t s p ayab l e at t h e en d of t h e y ear o f deat h .
Y ear o10
f D eat h ! D eat h B en efi t !
10
1
10
2
3
9
4
9
5
9
8
6
7
8
8
8
9
8
7
E x p r ess t h e net si n gle pr em i u m for t h i s p ol i cy usin g com m u t at i on fun ct ion s.
1 1 . G i ven :
(i ) T h e sur vi v al fu n ct ion is ç(õ ) = 1 — z / 100 for 0 < õ < 100.
(| | ) T h e for ce of i nt er est is á = 0.10 .
(|11) T h e deat h b en efi t is pai d at t h e m om ent of deat h .
C al cul at e t h e net si n gl e pr em iu m for à 10-year en dow m ent i nsu r an ce of 50,000
for à p er son age õ = 50 .
12 . G i ven :
(i ) s (z ) = å e.e~ for õ > 0
(é ) á = 0 .04 .
C al cu l at e t he m edi an o f t he pr esent val ue r an dom var i abl e Z = è~ for à w hol e
l i fe p ol i cy i ssued t o (ó ) .
13 . À 2- year t erm i nsu r an ce p oli cy issued Ñî (õ ) p ay s à deat h b en efi t o f 1 at
t h e en d of t he year o f deat h . G i v en :
14 0
A P P E N D I X Ñ . E X E R C I SE S
(i ) q~ = 0 .50
(é ) â = 0
(ø ) V ar ( Z ) = 0 .1771
w here Z is t he pr esent val ue of fu t ur e b enefi t s. C alcu l at e ä » ä.
14 . À 3-year t er m li fe i nsur an ce Ñî (õ ) i s defi ned by t he foll ow i n g t ab l e:
Y e a r t ] D e at h B en efi t [
0.20
0 .25
0.50
G i ven : v = 0.9, t he deat h b enefi t s ar e p ay ab l e at t h e end of t h e year o f deat h an d
t he ex p ect ed p resent val ue of t he d eat h b enefi t is Ï . C al cu l at e t he pr ob abi li t y
t h at t he p r esent val ue o f t h e b enefi t p ay m ent t h at i s act u all y m ade w il l ex ceed
Ï .
15 . G i ven :
(i ) A r s — 0 .800
(i i ) D ms = 400
(äää) D n = 360
(ää~)] â = 0 .03 .
C al cu l at e A 77 by use o f t he recu r si on for m u l a (3 .6 .1) .
16 . À w hole li fe i nsu r an ce o f 50 i s issued Ñî (õ ) . T he b enefi t is p ay ab l e at t he
m om en t of deat h . T he pr ob ab il it y densit y fu nct ion o f t he fu t u re l i fet i m e, Ò, is
u(t) = ((
(
t / 5000
0
for 0 < t < 100
elsew her e.
T he for ce of i nt er est is const ant : á = 0 .10 . C al cu lat e t h e net si ngle prem i u m .
1 7 . For à cont i nuous w hole l i fe in su r an ce, Å [âä~7 ] = 0 .25 . A ssu m e t he for ces o f
m or t al i t y an d i nt er est ar e each con st ant . C al cul at e E [i i+ ] .
18 . T h er e ar e 100 cl ub m emb ers age z è Üî each cont r i b ut e an am ou nt ø t o
à fun d . T h e fu n d ear n s i nt er est at i = 10% p er y ear . T h e fu nd is ob li gat ed t o
ð àó 1000 at t he m om ent o f deat h of each m em b er . T he p r ob ab i l i t y is 0.95 t h at
t h e fu n d w i l l m eet i t s benefi t ob l ig at ions. G iv en t he fol l ow i ng val ues cal cu l at ed
at â = 10% : A = 0 .06 an d A = 0.01. C al cu l at e ø . A ssu m e t h at t h e fu t ur e
l i fet im es ar e ind ep en dent an d t h at à nor m al di st r i b u t i on m ay b e u sed .
19 . A n i n su r an ce is issued t o (õ ) t h at
(i ) p ay s 10,000 at t he en d of 20 y ear s i f x is al i ve and
(é ) r et u r ns t he net si n gle p rem i u m Ï at t he end o f t h e y ear of deat h i f (õ ) d ies
d ur i ng t he fi rst 20 year s.
Ñ .3 .
E
x
p
2
0
.
b
e n
L I F E I N S UR A N CE
r e s s
Ï
À
e fi
u
w
h
t s
p
s i n
o
g
c o
l e
l i f e
a b
l e
a y
m
m
i n
a t
u
s u
t h
14 1
t a
t i o
r a n
e
e n
n
c e
d
f u
p
î
o
n
c t i o
l i c y
à t h
e
e Ya ce ha r o ot hf
y
n
s
i s s u
e a r
.
e d
o
f
Ñ î
d
( õ
e a t h
eD r e ya et ah r
D
)
p
r o
v
i d
e s
t h
e
f o
l l o
w
i n
g
d
e a t h
.
e a
t h
18
79B
0
e n
e fi
t
)
1
2
3
4
5
6
7
8
9
1 0
C
a
l c u
l a t e
t h
Ñ .3.2
1
.
C
a l c u
l a t
r s i v e
m
v
e s
A
2
.
e
q
o
f
3
T
h
.
À
C
r
a t
i
a g
e s
4
.
a
n
p
a i d
õ
=
C
a t
0
.
o
v
a
5
F
r e m
t h
b
e
,
( 1
i u
t
e n
e
m
.
.
h
e
n
t
o
o
m
m
t h
e
r e m
à
l i f e
y
f o
r
i n
p
h
à
c r e m
.
a g
w
e
h
e n
o
9
f o
r
t h
i s
p
o
l i c y
.
.
o
=
È
o
f
I l l u
g
y
,
i n
1 0
s u
T
a b
l e
i u
o
( 3
e r
i
d
e n
â î
o
a l i z e
a b
l e
a t
s t
r u
c t
à
n
=
a
. 1
i n
b
T
o
õ
=
a n
i f e
C
i n
.6
d
e
L
d
c e s ,
f
a n
g e n
a n
d
t h
e
. 1 ) .
%
a n
m
a t
. 6
r a n
( a n
h
a t i v
( 3
i s
e a r
Ñ î
s t r
l a
h
t
d
u
. 5 %
r e m
i n
I l 'y
—
d
t
h
m
l e
t
i f e
i r
g l e
f
s o
õ
7
l i f e
p
t h
,
s e
L
e
Î
n
a
) )
t o
5 %
.
c r e a s
e fi
n
t
Ñ î
U
à
t
1
l o
g
c a
i n
i s
.
,
,
y
t o
l c u
g
1
s e
a b
, 2
t
l e
.
i
g
.
=
r a
. ,
5
p
h
9 9
.
. 6
. 1 ) ,
e
à
t a
2
0
y
e a r
t
å
ï
ã â é
ó
e
o
f
I l l u
r e m
o
l e
i u
o
U
w
i s
b
s e
i n
( I
l e
o
g
A
f
i n
s u
r
1
+
ä
m
s
L
i f e
f o r
e
t h
e
)
e r m
e
t h
v a l u
å à ã ,
s t r a t i v
p
.
s h
( 3
l a t
Ô Ü
h
%
a n
c e
t h
T
a l l
e s
a b
e
l e
i s s u
e
.
s i n
i u
t s
g
m
t h
f o r m
5 %
l e
e
1 0 0
t
g
f
y
U
a s s u
t h
9
f
t e s
p
s i n
,
e t
l i z e
,
1 ) .
6 %
e n
o
b
. 5 %
»
) z
. ,
e fi
n
s t r a t i v e
=
,
m
c r e a s i n
À
e t
g
ä
2
, 2
2 5
+
d
1 ,
n
=
l u
e s t e d
i n
1
I l l u
z
g
Î
r
e r a
r
r a
e
f o
e n
%
b
t h
l a
g
=
G
l e
g
,
t h
r i a b
e
+
l a t e
.
5
p
c o
s u
~
a n
0
i t i a l
5
g l e
À
d
~ +
e
r
%
a l c u
i n
s i n
e
o
i
r
e a
=
h
r
a g e
5
õ
t
u
l a t
e
y
l a
h
( A
f o
d
c a l c u
2
r m
a l c u
=
e t
f o
~
)
n
e
e t h
f o
p
i s s u
s e c o
t
f
e
v
( 1
f o
o
+
n
Spr eadsheet Ex er cises
r e c u
a l u
e
3
a t
0
5
,
. 5 %
h
x
r
i n
r e m
a t
e a t
f o
å
p
z
i u
U
s e
a n
d
ã
c a l c u
.
n
f o
õ
.
r e a s o
s u
m
a g e
t
a
a b
h
e
f
M
r t a l i t y
d
I l l u
i n
i n
t h
c e
à
e c r e a s i n
e c r e a s i n
r e
r a n
o
r
d
l e
l a t e
o
,
e
1 0 0
p
s t r a t i v
e
L
v a
l u
t
c e l l
v a r i a
.
f o l l o w
w
y
t e r e s t
0
g
b
u
g
T
n
h
s
r
1
a t e s
c e
e
t h
o
t
å ã
i f e
h
d
e
i n
t e r e s t
e
I l l u
y
T
e s ,
a n
f
h
ð
l i f e
e
a b
a n
a r
i n
.
l e
d
y
s u
T
h
a t
o
r
i
u
r
a n
e
b
c e
e
=
s p
n
w
i t h
e fi
5
%
r e a
d
t
i s
a n
s h
d
e e t
a g e s .
p
r e s e n
r a t e
s t r a t
i v
ç
e
L
t
v
v
a l u
a
r i e s
i f e
T
a
e
r a n
f r o
b
l e
m
.
d
o
0
D
m
t o
r a w
A P P E N D I X Ñ. E X E R CI SE S
14 2
Ñ .4
L ife
Ñ .4 .1
A n n u it ie s
T heor y Ex er cises
1. Using assumpti on à and t he Illust rat ive L i fe T able wit h int erest at t he
ef f ect i ve annual rat e of 5%, cal cul at e à.. ( 2)
40 : 30 ]
2 . Ðåï þ ï âÑãàÑå ÑÏ àÑ
*: !
(1à) + + à +1
simpl ifi es Ñî à . ~l .
3. (1-,ö à)
is equal to E[Y ] where
( ~» ) » ~ + » ( „ ~» ò - „ [)
if Ò
Û
0 <> Ò
è < è and
T he force of mort al it y is const ant , p = 0.04 for all õ, and t he for ce of int erest
is const ant , î = 0.06. Cal cul at e ~ — (1—
„ ~à) .
4 . Gi ven t he followi ng informat ion for à 3-year t empor ary li fe annuit y due,
cont ingent on t he life of (õ ) :
Pay m ent (
p a+ t
)
0.80
0.75
0.50
and è = 0.9. Cal culat e t he vari ance of t he present value of t he indicat ed payment s.
5 . G i ven :
(i) t = 100, 000( 100 —z ), 0 < z < 100 and
(é) i = 0.
Cal cul at e
( 1 à ) 95
exact ly.
6 . Cal culat e ö ~à
.. (12)
using t he Illust rat i ve L ife Table, assump ti on à and
| 0 ~ 2 : 0~
i = 5%. (T he sy mbol denot es an annuit y issued on à li fe age 25, t he fi rst
payment deferred 10 years, paid in level mont hly payment s at à ãàÑå of 1 per
year during t he lifet ime of t he annuit ant but not more t han 10 years.)
7. Given:
Ñ .4 .
õ
I
S
( é )
I F E À Õ Õ Ø Ò 1Å Á
[
6 9
[
7 0
[
7 1
[
7 2
[
. . .
[
7 9
[
8 0
[
8 1
[
8 2
]
I
7 7 , 9 3 8
I
6 7 , 1 1 7
I
5 7 , 5 2 0
I
4 9 , 0 4 3
I
. . .
I
1 3 , 4 8 3
I
1 0 , 8 7 5
I
8 , 6 9 1
I
6 , 8 7 5
I
9 ( 1 2 )
=
u
( i i i )
14 3
( 1 2 )
=
1 .0 0 0 2 8
a s s u m p ti o n
à
a n d
a p p l i e s :
0 .4 6 8 1 2
d e a t h s
a r e
d i st r i b u t e d
u n i fo r m
ly
o v e r
e a c h
y e a r
o f
a g e .
C
a l c u l a t e
8 .
S h o w
( Ã à )
Ö
.
:
ð ,1à ù + )
( 1 — e k+ 1) ÿ ð õ î +~ = 1 — A ~,,-ùÄ
k = o
9 .
Y
is
y e a r
à
=
1 0 .
t h e
is s u e d
6 ,
à
p r e s e n t
t o
e v a l u a t e d
.—
,ö
is
v a l u e
( õ ) .
G
w i t h
e q u a l
Ñî
Ì
v a r i a b l e
1 1 .
G
i v e n
=
i n ( K
i
=
Å
( å "
+
1 2 .
G
i v e n
=
[Y
å ~~
]
t h e
t h e
—
1 0 ,
1 .
C
o f
à
w h o l e
È
å
w i t h
i
=
e v a l u a t e d
a l c u la t e
t h e
v a r i a n c e
a n n u it y
1 / 2 4
o f
Ó
=
d u e
e ~
o f
—
1
1 ,
p e r
a n d
.
w h e r e
a -„~ i + ~]
' " <+
0 .0 3
c o m
v a r i a b le
=
×
à ã [Ó
]
+
'
is
" 1)
=
if Ê
0 <> Êï .( è an d
Ì
(
t h e
2 á )
m
o m
—
e n t
Ì
( — á ) ÿ
g e n e r a t i n g
fu n c t io n
o f t h e
r a n d o m
1 , è ) .
I
C a l c u l a t e
à
Y
( è )
m
i
E
Sh o w t h at
w h e r e
r a n d o m
i v e n :
a n d
c o m
m
u t a t i o n
fu n c t io n
v a l u e s :
õ
[
2 7
[
2 8
[
2 9
[
3 0
[
3 1
[
~ *
I
1 , 8 6 8
I
1 , 7 6 7
I
1 , 6 7 0
I
1 , 5 7 7
I
1 , 4 8 8
I
m
u t a t io n
fo ll o w i n g
M
qs .
fu n c t i o n s
v a l u e d
[
7õ 5
[
a t
i
=
0 .0 3 :
8 à. 0 6
7 2
7 3
7 .7 3
7 4
7 .4 3
7 .1 5
C
a l c u l a t e
1 3 .
t h e
G
i v e n
l i fe
o f
p qs .
t h e
( õ ) :
fo l l o w
in g
i n fo r m
a t i o n
fo r
à
3 - y e a r
l i fe
a n n u it y
d u e ,
c o n t i n g e n t
o n
A P P E N D I X Ñ. E X E R CI SE S
14 4
Pay m ent I
0.80
0.75
0.50
Assume t hat i = 0.10. Calcul at e t he probabil ity t hat t he present value of t he
i ndi cat ed payment s exceeds 4.
14 . Given l = 100, 000(100 —õ) , 0 < õ < 100 and i = Î . Calculat e t he present
value of à whole l ife annuity issued t o (80). T he annuity is paid cont i nuously at
an annual r at e of 1 per year t he fi rst year and 2 per year t hereaft er .
15 . As i n exer cise 14, 1 = 100, 000(100 —õ ), 0 < õ < 100 and i = Î . Calculat e
t he present value of à t empor ary 5-year È å annuity issued t o (80). T he annuit y
is pai d cont inuously at an annual r at e of 1 per year t he fi rst year and 2 per year
for four years t hereaft er .
18 . Gi ven î = Î , /
t hp~dt = ä, and × àò(à ð-~) = h , where Ò is t he fut ure
Jo
li fet ime r andom vari able for (õ ). Express E [T ] in t erms of ä and h.
17. Gi ven:
õ ~
69 !
70 !
71 !
72 ! . . . !
79 !
80 !
81 ~
82 ~
I S I 77, 938 I 67, 117 I 57, 520 I 49, 043 I . I 13, 483 I 10, 875 I 8, 691 I 6, 875 I
Calcul at e (Ð à) .~áä whi ch denot es t he pr esent val ue of à decreasing annuity.
T he fi rst payment of 10 is at age 70, t he second of 9 is scheduled for age 71, and
so on. T he last payment of 1 is scheduled for age 79.
18. Show t hat
à ö ßë
à ~ð ß ê + 1 + à ~| ß ì + 2
simpl ifi es t o A .
19. For s for ce î ( i nterest of á > Î , t he ÷à1í å î ( Å ( àó~ ) is equal t o 10. W it h
t he äàò å mort ali ty, Üèñ à for ce of i nt erest of 2á, t he val ue of Å ( à~ ~) is 7.316.
A lso × àã(à ~~) = 50. Cal cul at e A .
20. Calculat e a +Ä using t he Il lust r at i ve L i fe Table at 5% for age x + u = 35.75.
Ass9hmpti oyh à applies.
Ñ .4 .2
S p r ea d sh e et E x er c i se s
1. Cal culate à based on t he Illust rat i ve L i fe Table at i = 5% . Use t he
recursion formul a (4.6.1) . Const ruct à graph showing t he values of à for i =
Î , 2.5%, 5%, 7.5%, 10% and õ = Î , 1, 2, . . . , 99.
Ñ .4 .
L I F E A N N UI T I E S
14 5
2 . C on si d er ag ai n t he st r u ct u r ed set t l em ent annu i t y m ent i on ed i n ex er cise 7
o f Sect i on Ñ .1. I n ad d i t ion t o t he fi n an ci al dat a an d t h e sch edul ed p ay m ent s,
i n cl u de n ow t he i n for m at ion t h at t he p ay m ent s ar e cont ingent u p on t h e su r vi val
o f à l i fe sub j ect t o t h e m ort al it y descr i b ed i n ex er cise 3 o f Sect ion Ñ .2 . C al cu l at e
t h e su m o f m ar ket val u es o f b on ds r equ ir ed Ñî h edge t h e ex p ect ed valu e o f t he
an nu i t y p ay m ent s .
3 . À l i fe age x = 50 i s su b j ect Ñî à for ce of m ort al i t y vse+ < obt ai ned fr om t he
for ce of m or t ali t y st an dar d as fol low s:
~'âî + ÿ =
p so~ ~+ ñ
,èóî ~.~
for 0 < t < 15
ot her w ise
w her e i so+ ~ denot es t h e for ce o f m ort al it y un der l y in g t h e I l lust r at i ve L i fe T ab le.
T he for ce o f i nt er est is const ant á = 4% . C al cul at e t he var i an ce of t he pr esent
val ue o f an an nui t y i m m ed i at e of on e p er annu m issued t o (50 ) for val u es of
ñ = —0 .01, —0 .005, 0 , 0.005, an d 0.01. D r aw t he gr ap h .
4 . C r eat e à spr eadsh eet w h i ch cal cu l at es à.. ( òâ
+ )Ä an d À » „ for à gi ven age, õ + è ,
w i t h x an i nt eger an d 0 < u < 1, an d à gi ven int erest r at e i . A ssu m e t h at
m or t al i t y fol lows t he I l l ust r at i ve L i fe T ab l e. U se for m ul as (4 .8 .5) an d (4 .3.5)
(or (4 .8 .6) an d (4 .3.5) i f you li ke.) for t he annu i t y an d an al ogous on es for t h e
l i fe i nsu r an ce.
5 . U se yo u r sp r eadsheet 's b u i l t - in r an dom nu m b er feat u r e t o si mu l at e 200 val ues
of Ó = 1 + v +
+ v ~ = É + , ~ w h er e Ê = Ê (40) . U se i = 5% an d àâÿø ï å
m or t al i t y fol lows t h e I ll ust r at i ve L i fe T abl e. C om p ar e t he sam p l e m ean an d
var i ance Ñî t h e v al ues g iven by for m u l as (4 .2 .7) an d (4 .2 .9) .
APPENDIX Ñ. EXERCISES
14 6
Ñ .5
N et P r em iu m s
Ñ .5.1
N ot es
The exercises sometimes use the not at ion based on t he syst em of Internat ional
Act uarial Not at ion. Appendix 4 of Actuari al Mathemati cs by Bowers et al.
describes t he syst em. Í åãå are the premium symbols and defi nit ions used in
t hese exercises.
P (A ) denotes t he annual rat e of payment of net premium, paid continuously, for à whole life insurance of 1 issued on the life of (õ), benefi t paid at t he
moment of deat h.
Ð (À .ù ) denot es the annual ãàÑå of payment of net premium for an endowment insurance of 1 issued on t he life of (z). The deat h benefi t is paid at t he
moment of death.
À life insurance ðî éñó is fully continuous if t he deat h benefi t is paid at the
moment of deat h, and the premiums are paid cont inuously over the premium
payment period.
Policies wit h limit ed premium payment periods can be described symbolically wit h à pre-subscript . For example, „ Ð (A ) denot es t he annual rate of
payment of premium, paid once per year, for à whole life insurance of 1 issued
on t he life of (õ), benefi t paid at t he moment of deat h. For à policy wit h t he
deat h benefi t paid at the end of the year of deat h the symbol is simplifi ed to
„ Ð~.
Ñ .5.2
T heor y Ex er cises
1. Given: ~åÐ~~ = 0.046, Ð,~,—
, ] = 0.064, and À45 = 0.640. Calculat e P i
2. À level premium whole È å insurance of 1, payable at t he end of the year of
deat h, is issued to (õ). À premium of G is due at t he beginning of each year,
provided (õ) survives. Given:
(i) L = the insurer's loss when G = P
(é) L' = t he insurer s loss when G is chosen such t hat E[L' ] = —0.20
(|è) Var [L] = 0.30
Calculat e Var [L' ].
3. Use t he Illust rat ive Life Table and i = 5% to calculate t he level net annual
premium payable for ten years for à whole life insurance issued to à person
age 25. The deat h benefi t is 50,000 init ially, and increases by 5,000 at ages
30, 35,40,45 and 50 to an ultimate value of 75,000. Premiums are paid at t he
beginning of the year and t he death benefi ts are paid at t he end of the year .
4. Given t he following values calculated at d = 0.08 for two whole life policies
issued Ñî (z):
Ñ.5. N E T P R EM I UM S
I
Pol icy À
14 7
( Deat h Benefi t ! Premi um ! Variance of Loss (
4
0.18
3.25
Premiums are paid at t he beginning of t he year and t he deat h benefi t øàãå paid
at t he end of t he year . Calculat e t he variance of t he loss for policy  .
5 . À whole È å insur ance issued t o (õ ) provides 10,000 of insurance. Annual
premiums ar e pai d at t he beginning of t he year for 20 years. Deat h clai ms are
pai d at t he end of t he year of deat h . À premium refund feat ure is in effect
during t he premi um payment per iod whi ch provides t hat one hal f of t he last
premium pai d Ñî t he company is refunded as an addi t ional deat h benefi t . Show
t hat t he net annual premium is equal t o
(1 + ,1~2) -
10, Î Î Î À
— ( 1 — ãp
6 . Obt ai n an expression for t he annual premium „ Ð in t erms of net si ngle
pr emiums and t he r at e of discount d. („ Ð denot es t he net annual premium
payable for n years for à whole l ife i nsurance issued t o õ.)
7 . À whole È å insur ance issued t o (õ) provides à deat h benefi t in year j of
Ü; = 1, 000(1.06)~ payable at t he end of t he year . Level annual premiums are
payable for l i fe. Gi ven: 1, 000Ð = 10 and i = 0.06 per year . Calculat e t he net
annual pr emium .
8 . Gi ven:
(i ) À = 0.25
(||) À +ãî = 0 40
(111) À .ãî ~ = 0.55
(i v) i = 0.03
(÷) àçâèòï ðÜî ï à applies
C al cu l at e ! Î Î Î Ð ( À
ù
) .
9. À fully cont inuous whole li fe insurance of 1 is issued t o (õ ). Gi ven:
(i ) T he i nsurer 's loss random variable is L = î ~ —Ð (A ) à ~ .
(é) T he for ce of int erest î is const ant .
(iii ) T he force of mor t ality is const ant : @ +i — à , t > Î .
Show t hat Var (L ) = p / (2á + ,ö).
10 . À ful ly-cont inuous level premi um 10-year t erm insur ance issued t o (õ) ðàóâ
à benefi t at deat h of 1 plus t he ret urn of all premiums paid accumulat ed wit h
i nt erest . T he int erest r at e used i n cal cul at ing t he deat h benefi t is t he same as
14 8
A P PE N D IX
Ñ.
E X E R C I SE S
t h a t u sed t o d et er m i n e t h e p r ese n t v a l u e o f t h e i n su r er ' s l o ss . L et G d e n o t e t h e
ãàÑå o f a n n u a l p r e m i u m p ai d co n t i n u o u sl y .
( à ) W r i t e a n ex p r essi o n fo r t h e i n su r er ' s l o ss r a n d o m v a r i a b l e L .
( Ü ) D e r i v e an ex p r essi on fo r V ar [L ] .
( ñ ) Sh o w t h at , i f G i s d et er m i n e d b y t h e eq u i v a l en ce p r i n ci p l e , t h en
T h e ð ãå- su p e r sc r i p t i n d i c a t es t h a t t h e sy m b o l i s e v a l u at ed at à fo r ce o f i n t er est
o f 2á , w h er e á i s t h e àï î ãåå o f i n t er est u n d er l y i n g t h e u su a l sy m b o l s .
1 1 . G i v en :
( i ) â = 0 .10
(ii ) à
.q
s ~ = 5 .6
( i i i ) e ~e þ ð çî = 0 3 5
C a l cu l at e 10 0 0 Ð~~
çî : i o ]
1 2 . G i v en :
( â) â' = 0 .0 5
( é ) 10 , 00 0 À
= 2 , 000 .
A p p l y à ççè òï ð é î ï
à an d c al cu l at e 10 , 0 0 0 Ð ( A ) — 10 , 0 0 0 P ( À
).
1 3 . Sh o w t h a t
~ 30 . 1â ]
1 —
ld
30 ) ~ 30 , 15]
15 15ð çî
s i m p l i f i es t o A ~s .
1 4 . G i v en :
(i) A
= 0 .3
( é ) á = 0 .0 7 .
À w h o l e l i f e p o l i cy issu e d t o ( õ ) h a s à d ea t h b en e fi t o f 1,0 0 0 p ai d a t t h e m o m en t
o f d e at h . P r e m i u m s a r e p ai d t w i ce p er y ea r . C a l c u l a t e t h e çåï è - an n u a l n et
p r em i u m u si n g à ççè ò ð é î ï à .
1 5 . G i v en t h e fo l l o w i n g i n fo r m a t io n a b o u t à fu l l y co n t i n u o u s w h o l e Í å i n su r a n ce p o l i cy w i t h d ea t h b e n efi t 1 issu e d Ñî ( õ ) :
( i ) T h e n e t si n g le p r em i u m i s A
= 0 .4 .
( é ) á = 0 .0 6
( ø ) V a r [L ] = 0 .25 w h er e L d en o t es t h e i n su r er 's l o ss a sso ci at ed w i t h t h e n et
a n n u a l p r em i u m P ( À
).
Ñ .5 .
N E T Ðß ÅÌ Ø Ì Á
14 9
U nd er t he sam e con di t ions, ex cept t hat t h e i nsu rer r equi res à p r em i um ãàÑå of
G = 0.05 ð åã y ear pai d cont i nuousl y, t he i nsu rer 's loss r an dom vari abl e i s L ' .
C al cu l at e × àã[L ' ] .
16 . À ful l y d iscr et e 20-y ear en dow m ent insur ance o f 1 is i ssued t o (40) . T he
i nsu r an ce also pr ovides for t he refu n d o f all net p r em i u m s p ai d accu mu l at ed at
t he i nt erest ãàÑå i i f d eat h occurs w i t hi n 10 y ears o f issu e. P r esent values ar e
cal cu l at ed at t h e sam e i nt er est ãàÑå ò. U si ng t he equ i valen ce p r i n ci p le, t h e net
ann u al p r em i um p ay ab le for 20 year s for t h is ð î é ñó can b e wr it t en in t h e form :
44î . Ù
D et er m i n e /ñ.
1 7 . L is t he loss r andom var i ab le for à ful ly d iscr et e, 2-y ear t er m i nsur an ce of 1
i ssu ed t o (õ ) . T he n et lev el an nu al pr em iu m is cal cul at ed usin g t h e equ i val en ce
pr i n ci pl e. G i ven :
(1) ~* = 0.1,
(é ) ×~~-~ = 0 .2 an d
(i ii ) v = 0 .9
C al cul at e V ar ( L ) .
18 . G i ven :
(i ) À ~~. 1 — 0 .4275
(é ) á = 0 .055, and
(ø ) y * + t — 0 .045 , t > 0
C al cu l at e 1, Î Î Î Ð ( A ,—
„ 1) .
1 9 . À 4-year au t om o bi le l oan i ssued t o (25) i s t o be rep aid wi t h eq u al an nual
p ay m ent s at t he en d o f each year . À four -year t er m i nsu r an ce h as à deat h
b enefi t w h i ch w i l l ðàó î é t he l oan at t he en d o f t he year of deat h , in cl u di ng t he
p ay m ent t h en d ue. G i ven :
(i ) i = 0.06 for b ot h t he act u ar i al cal cu l at ions an d t h e loan ,
( é ) à ~~ . 4 ~ =
3.667, an d
(ø ) 4~ s = 0.005 .
(à) E x p ress t he i nsu r er 's loss r an dom var i ab l e in t er m s o f Ê , t h e cu rt at e fu t ure
l i fet i m e o f (25) , for à lo an o f 1,000 assu m i ng t h at t he insu r an ce is p ur ch ased
w i t h à si ngle prem i u m o f Ñ .
(Ü) C al cul at e G, t he n et si ngl e pr em iu m ãàÑå ð åã 1,000 o f l oan val ue for t h is
i nsu r an ce.
(ñ) T he aut om obi l e l oan i s 10,000. T he b uyer b orr ow s an ad dit ional am ou nt Ñî
ð àó for t he t er m i n su r an ce. C alcu l at e t he t ot al an nual p ay m ent for t he loan .
A P P E N D I X Ñ . E X E R CI SE S
150
20 . À level premium whole life insur ance issued t o (õ ) pays à benefi t of 1 at
t he end of t he year of deat h. Given :
(i ) À = 0.19
(é) sA = 0.064, àï é
(Ø ) d = 0.057
Let Ñ denot e t he r at e of annual pr emium t o be paid at t he beginning of each
year while (õ ) is alive.
(à) Writ e an expression for t he insurer 's loss r andom vari able À.
(Ü) Cal culat e E [L ] and Var [L ], assuming G = 0.019.
(ñ) Assume t hat t he insurer issues n independent policies, each having G =
0.019. Determi ne t he minimum val ue of n, for which t he probabilit y of à loss
on t he ent ire port folio of policies is less t han or equal Ñî 5%. Use t he nor mal
approximat ion.
Ñ .5 .3
Sp r ea d sh e et E x er c i ses
1. Reproduce t he Illust rat i ve L i fe Table values of Ñ , Calcul at e Ì
recursi vely,
from t he end of t he t able wher e M ss = Css, using t he rel at ion Ì
= Ñ +Ì
i.
Cal cul at e t he values of S~ = Ì + Ì , + 1+ . . . using t he same t echni que.
2 . Use t he Ill ust rat i ve Li fe Table t o cal cul at e t he i nit ial net annual premium
for à whole li fe insurance policy issued at age õ = 30. T he benefi t is infl at ion
prot ect ed: each year t he deat h benefi t and t he annual premi um increase by à
fact or of 1 + ó', where ó' = 0.06. Cal culat e t he init i al premium for int erest rat es
of i = 0.05, 0.06, 0.07 and 0.08. Dr aw t he graph of t he init i al premium as à
funct ion of i .
3 . Use i = 4%, t he Ill ust rat ive L i fe Table, and t he ut ility funct i on è(õ) =
(1 —å
)/ à, à = 10 ~, to calcul ate annual premi ums for 10-year t er m i nsurance,
issue age 40, using formul a (5.2.9) : Å [è( - L )] = è (0). Display your result s in
à t able wit h t he sum insur ed Ñ, t he cal cul at ed premi um , and t he rat io t o t he
net premium (loading). Dr aw t he gr aph of t he loading as à funct ion of t he sum
insured. Ï î t he same for premiums based on à = 10 4, 10 5, 10 7 and 10 8
also. Show al l t he gr aphs on t he same chart .
4 . À whole li fe poli cy is issued at age 10 wit h premiums payable for È å. I f
death occurs b efore age 15, t he deat h benefi t is t he ret ur n of net premiums paid
wit h int erest t o t he end of t he year of deat h . I f deat h occur s aft er age 15, t he
deat h benefi t is 1000. Calculat e t he net annual premi um . Use t he Il lust r at ive
L i fe Table and i = 5%. Convince yoursel f t hat t he net premi um is independent
of q for õ ( 15. (T his problem is based on problem 21 at t he end of Chapt er
4 of f if e Conti ngenci es by Ñ. W . Jor dan .)
5 . À 20-year t erm insur ance is issued at age 45 wit h à face amount of 100,000.
T he net premium is det ermined using i = 5% and t he Ill ust rat ive Li fe Table.
T he benefi t is paid at t he end of t he year . Net, pr emiums are i nvested in à fund
Ñ .5 .
N E T P R E M I UM S
15 1
ea r n i n g j ð åã an n u m a n d r et u r n ed at ag e 6 5 i f t h e i n su r ed su r v i v es . C al cu l at e
t h e n et p r em i u m fo r v a l u e s o f j r u n n i n g fr o m 5 % Ñî 9 % i n i n cr em en t s o f 1% .
6 . D et er m i n e t h e p e r cen t a ge z o f a n n u al sa l ar y à p er so n m u st sav e ea ch y ea r
i n o r d er Ñî
p r o v i d e à r et i r em en t i n co m e w h i ch r e p l a ces 5 0 % o f fi n al sa l a r y .
A ssu m e t h at t h e p er so n i s ag e 30 , t h at sav i n g s ear n 5 % p er a n n u m , t h at sa l a r y
i n cr eases at à r at e î Ö = 6% ð åã y ea r , a n d t h at m o r t al i t y fo l l o w s t h e I l l u st r at i v e
L i fe T a b l e . D r aw t h e g r a p h o f z as à fu n ct i o n o f 1' r u n n i n g fr o m 3 % t o 7 % i n
i n cr em en t s o f 0 .5 %
7 . M o r t a l i t y h i st o r i c a l l y h a s i m p r o v e d w i t h t i m e . L et q
d e n o t e t h e m o r t al i t y
t a b l e w h en à p o l i c y i s i ssu ed . Su p p o se t h at t h e i m p r o v em en t ( d e cr easi n g q )
i s d esc r i b ed b y é ' q w h e r e t i s t h e n u m b er o f y ear s si n ce t h e p o l i c y w as issu ed
an d k is à co n st a n t , 0 ( lñ ( 1 . C al cu l at e t h e r a t i o o f n et p r em i u m s on t h e
i n i t i a l m o r t a l i t y b as is t o n et p r e m i u m s a dj u st e d fo r t = 10 y e ar s o f m or t al i t y
i m p r ov em e nt . U se z =
m o r t a l i t y an d i = 5 % .
3 0 , É = 0 .9 9 , t h e I l l u st r a t i v e L i f e T a b l e fo r t h e i n i t i a l
A P P E N D I X Ñ . E X E R CI SE S
15 2
Ñ .6
N et P r em i u m R eser v es
Í åãå are t he addit ional symbols and defi nit ions for reserves used in t hese exercises. Policies wit h premi ums paid at t he begi nning of t he year and deat h
benefi ts pai d at t he end of t he year of deat h are called fully discret e pol icies.
Policies wit h premiums paid cont inuously and deat h beneFit s paid at t he moment of deat h are called fully cont inuous.
qV ( À ) denot es t he net premium reser ve at t he end of year k for à fully
cont inuous whole li fe poli cy issued t o (õ) .
pV ( À ,ù ) denot es t he net premium reser ve at t he end of year /ñ for à fully
cont inuous è-year endowment insurance ðî éñó issued Ñî (õ) .
Polici es wit h limit ed premium payment periods can be described symbolical ly wit h à ðãå-superscr ipt . For example, "„ ê. (À ) denot es t he net premium
reser ve at t he end of year lñfor an n-payment whole li fe poli cy issued t o (õ ) wit h
t he benefi t of 1 pai d at t he moment of deat h . Not e t hat t he corr esponding net
premium is denot ed ÄP ( A ).
Ñ .6.1
T heor y Ex er cises
1. À 20-year fully discret e endowment pol icy of 1000 is issued at age 35 on t he
basis of t he Illust rat ive L i fe Tabl e and â = 5%. Cal culat e t he amount of reduced
pai d-up i nsurance avai lable at t he end of year 5, j ust before t he sixt h premi um
is due. Assume t hat t he ent i re reser ve i s available t o fund t he paid-up poli cy as
descri bed i n sect ion 6.8.
2 . Given : |î Óì = 0.1 and ipVss = 0.2. Calculat e qpV~ .
3 . Gi ven: ~ V (A yp) = 0.3847, à4î = 20.00, and app = 12.25. Calculat e
òî ~ (Acp) —2pV (À î ) .
4 . Given t he fol lowing infor mat ion for à fully discret e 3-year speci al endowment
insur ance issued t o (õ ) :
ñ~~ 1 ~ ü
+~ 1
0.20
0.25
0.50
Level annual net premiums of 1 are pai d at t he beginning of each year while (õ)
is al ive. T he special endowment amount is equal t o t he net pr emium reserve for
year 3. T he effect ive annual int erest r at e is i = 1/ 9 . Calcul at e t he end of poli cy
year reserves recursively using formula (6.3.4) from year one wit h pV = Î .
5 . Given: i = 0.06, q = 0.65, ä » 1 = 0.85, and q ~q = 1.00. Calculat e i V .
(Hint due Ñî George Car r 1989: Cal cul at e t he annuity values recursi vely from
à +ð back t o à . Use (6.5.3).)
6 . À whole li fe poli cy for 1000 is issued on Ì àó 1, 1978 Ñî (60). Given :
(i ) i = 6%
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154
A P P E N D I X Ñ . EX ERCI SES
Cal cul at e ã× :4~
14 . À fully discret e whole l i fe policy wi t h à deat h benefi t of 1000 is issued t o
(40). Use t he Illust rat ive Li fe Table and i = 0.05 t o cal cul at e t he variance of
t he loss allocat ed t o poli cy year 10.
15 . At an i nt erest rat e of i = 4%, ~ V~s = 0.585 and @Ó|~ — 0.600. Calculate
p38.
16 . À fully discret e whole li fe i nsur ance is issued t o (õ ) . Given: P = 4/ 11,
qV~ = 0.5, and à » ~ — 1.1. Cal culat e i .
17 . For à speci al ful ly discret e whole È å i nsur ance of 1000 issued on t he life
of (75), increasing net premiums, Ï ~, are payable at t i me k, for /Ñ= Î , 1, 2, . . ..
Gi ven :
(i ) Ï ~ = Ï î (1.05)" for k = Î , 1, 2, . . .
(é) Mort ality follows de Moivr e's law wit h ø = 105.
(È ) i = 5%
Cal culat e t he net premi um reser ve at t he end of pol icy year fi ve
18 . Given for à ful ly discret e whole li fe insur ance for 1500 wit h level annual
pr emiums on t he l ife of (a ) :
(i(÷)
) i 1000qqV~
= 0.05 = 340.86
É :À Ì
Å
~= :¸ .
(é) T he reserve at t he end of policy year à is 205.
(ø ) T he reserve at t he end of policy year à —1 is 179.
(iv ) à = 16.2
Cal cul at e 1000ä +à 1
19. Given:
(i ) 1 + i = (1.03)~
(" ) q + i o = 0.08
(~~' ) 1 00i oV~ = 311.00
(iv ) 1000P = 60.00
(à) Approximat e 1000qo sV by use of t he t radit ional r ule: int erpol at e bet ween
reserves at i nt egral durat ions and add t he unearned premium .
(Ü) Assumpti on à applies. Cal cul at e t he ex act val ue of 1000|~ sV .
20 . Gi ven : qsi = 0.002, àä ,~s = 9, and i = 0.05. Calculat e | Ó3 Ù .
Ñ
.6.2N ETSpr
eadsheet
Ex er cises
Ñ.6.
PREMI
UM RESERVES
15 5
1. Calculat e à table of values of ,Vsp for Ñ= 0, 1, 2, . . . , 69, usi ng t he Ill ust rat ive
Li fe Table and i = 4%. Recalculate for i = 6% and 8%. Dr aw t he t hree gr aphs
of i Vse as à funct ion of t, corresponding to i = 4%, 6%, and 8%. Put t he graphs
on à single chart .
2 . À 10-year endowment insurance wit h à Ãàñå amount of 1000 is issued t o (50).
Cal culat e t he savings Ï ~ and risk Ï ;, component s of t he net annual premi um
1000Ð .ù (formul as 6.3.6 and 6.3.7) over t he li fe of the policy. Use t he Illust r at i ve L i fe Table and i = 4%. Dr aw t he graph of II i, as à funct ion of t he ðî éñó
year k. Invest igat e it s sensit i vity t o changes in i by cal cul at ing t he gr aphs for
i = 1% and 7% and showing all t hree on à single chart .
3 . À 10,000 whole 1|Ãå poli cy is issued t o (30) on t he basis of t he Ill ust rat i ve
L i fe Table at 5%. T he act ual int erest earned in policy years 1 - 5 is ã' = 6%.
Assume the poli cyholder is ali ve at age 35 and t he ðî éñó is in for ce.
(à) Calculat e t he t echnical gai n realized in each year usi ng met hod 2 (page 69).
(b) Cal cul at e t he accumulat ed value of t he gains (usi ng a' = 6%) at age 35.
(ñ) Det ermi ne t he value of i ' (level over fi ve years) for which t he accumulat ed
gains are equal t o 400.
4 . T his exer cise concer ns à fl exible l ife pol icy as described in sect ion 6.8. T he
pol icyholder chooses t he benefit level ñ~+ 1 and t he annual premium Ï ~ at t he
beginni ng of each policy year 1 + 1. T he choices are subj ect t o t hese const r aint s:
(i ) Ï å = 100, Î Î Î Ð , t he net level annual premium for whole li fe in t he amount
of 100,000.
(è) 0 < Ï „ + , < 1.2Ï „ for k = 0, 1, . . .
(ø ) ñ1 < ñ~,+1 < 1.2ñ~ for k = 1, 2, . . .
(i v) q+ i V > 0 for k = Î , 1, 2, . . .
T he init i al policyhol der 's account value is eV = Î . T hereaft er t he policyholder 's
account val ues accumul at e according Ñî t he recursion relat ion (6.3.4) wit h t he
int er est r at e speci fi ed in t he policy as i = 5% and mort ali ty following t he Il lust r at i ve Li fe T able wit h õ = 40. Invest igat e t he insurer Ú cumul at i ve gain on t he
pol icy under t wo scenarios:
(Si ) T he policyhol der at t empts t o maxi mize insur ance coverage at mi nimal
cost s over t he fi r st fi ve poli cy years. T he st rat egy is implement ed by choosing
ñ~~. | = 1.2ñ~ for /ñ = 1, 2, . . . and choosing t he level premi um rate whi ch meet s
t he const raint s but has sV = Î . Cal culate t he insurer s annual gai ns assumi ng
a' = 5.5% and t he policyholder dies during year 5.
(Sz) T he policyholder elects Ñî maximize savings by choosing minimum coverage
and maximum premi ums . Calcul at e t he pr esent val ue of t he insurer 's annual
gains assuming i ' = 5.5% and t he poli cyholder sur vives t o t he end of year 5.
A P P E N D I X Ñ . E X E R CI SE S
15 6
Ñ .7
M u l t i p l e D ec r em en t s : E x er c i ses
Ñ .7 . 1
T h e o r y E x e r c i se s
1. In à double decrement t able ~è| +~ — 0.01 for all t > 0 and ðã +i — 0.02 for
all t > Î . Calculate qi , .
2 . Given ð +< — ó/ 150 for ó' = 1, 2, 3 and t > Î . Cal cul at e E [T ) .Ó= 3).
3 . À whole l ife insurance ðî éñó provides t hat upon accident al deat h as à passenger on public t ransport at ion à benefi t of 3000 wi ll be pai d. I f deat h occurs
from ot her accident al causes, à deat h benefi t of 2000 wi ll be pai d. Deat h from
causes ot her t han accidents carries à benefi t of 1000. Given, for all t > 0:
(i ) ð
» ~ — 0.01 where j = 1 indicat es accident al deat h as à passenger on public
t ransport at ion.
(é) ð . +, — 0.03 where 1 = 2 indicates accident al deat h ot her t han as à passenger on publ i c t ransport at ion .
(ø ) p~ +i — 0.03 where 1 = 3 indicat es non-acci dent al deat h.
(i v) á = 0.03.
Calculat e t he net annual premium for issue age x assuming cont inuous premi ums and i mmedi at e payment of claims.
4 . In à double decrement t able, l i +~ — 1 and ð ã +~ — —
,'
Calculat e
for à11 t > Î .
×õ
m~ =
À
5 . À t wo year t er m ðî éñó on (õ) provides à benefi t of 2 i f deat h occurs by
acci dent al means and 1 i f deat h occurs by ot her means. Benefi ts are pai d at t he
moment of deat h. Given for all t > 0:
(i) p i , ~~ — t / 20 where 1 indi cat es accident al deat h.
(é) ð ã,* +ñ = t/ 10 where 2 indicat es ot her t han acci dent al deat h.
(é ) á = Î
Calculate t he net single premium .
6. À mult iple decrement model has 3 causes of decr ement . Each of t he decrement s has à uni for m dist r ibut ion over .each year of age so t hat t he equat ion
(7.3.4) holds for at all ages and durat ions. Given:
(i) p i ,çî +î .ã = 0 20
(ii ) ~èã,çî +î .à = 0 10
(ø ) ðç,çî ~.î .ç = 0.15
Ñ . 7. Ì Ø Ò?ÐÜÅ D E CR E M E N T S: E X E R CI SE S
15 7
Calculate qso.
7. Gi ven for à double decrement t able:
1 I â, 1 e,* 1 ~* I
26
0.02
0.10
0.88
(à) For à group of 10,000 l ives aged x = 25, calculat e t he expect ed number of
lives who sur vive one year and fail due Ñî decrement ~ = 1 i n t he fol lowing year .
(Ü) Calculat e t he effect on t he answer for (à) i f qs ~s changes from 0.15 Ñî 0.25.
8. Gi ven t he fol lowing dat a from à double decrement t able:
0 ) e ,sç = 0.050
(ï ) ×ã,âç = 0.500
(' " ) Ö×âç = 0.070
(' ~) ã~â ,âç = 0.042
(~) çÐâç = Î .
For à group of 500 l ives aged õ = 63, cal cul ate t he expect ed ï ø ï Üåã of li ves
who wi ll fail due t o decrement j = 2 bet ween ages 65 and 66.
9 . Gi ven t he following for à double decrement t able:
0 ) È1,. +î ,s = 0.02
(é) Äã,~ = 0.01
(ø ) Each decrement is uni for mly dist ributed over each year of age, t hus (7.3.4)
Üî 1<Ü Ãî ã åàñî ô ñ1åñãåãï åï Ñ.
Cal cul at e 1000â , .
10 . À mult iple decrement t able has t wo causes of decrement : ( 1) accident and
(2) ot her t han acci dent . À fully cont i nuous whole li fe insurance issued t o (õ)
pays c~ if deat h r esults by accident and c2 i f deat h result s ot her t han by accident .
T he for ce of decrement 1 is à posit ive const ant p i . Show t hat t he net annual
premium for t his insurance is ñãÐ + (ñ~ —ñã)ð ~.
158
A P P E N D I X Ñ . E X E R C I SE S
Ñ .8
M u lt ip le L ife I nsur ance: Ex er cises
Ñ .8 . 1
T h e o r y E x er c i ses
1 . T h e f o l l o w i n g ex cer p t fr o m à m o r t a l i t y t a b l e ap p l i es Ñî ea ch o f t w o i n d ep en d en t l i v es ( 8 0 ) a n d ( 8 1) :
õ
82
0
0 .50
81
0 .75
1 .00
A ssu m p ti on à ap p l i es . C a l cu l at e i so:s
1 1
i so:s2 1, gso:s 1 a n d qeo
~.~1.
2 . G i v en :
( i ) î = 0 .0 55
( é ) ð » , = 0 .0 4 5 , Ô > 0
( 111) p Ä+ , — 0 .0 3 5 , t
0
C a l cu l a t e A 2Ä as d efi n ed Úó f o r m u l a ( 8 .8 .8 ) .
3 . I n à cer t a i n p o p u l a t i o n , sm o k er s h av e à fo r ce o f m o rt al it y t w i ce t h a t o f n o n î
õ < 7 5 . C a l c u l a t e el .ss fo r à
sm o k er s . F o r n o n - sm ok er s , ç ( õ ) = 1 — õ / 75 , 0 (
sm o k er ( 5 5 ) a n d à ï î ï - sm o k er ( 65 ) .
4 . À fu l l y - con t i n u o u s i n su r a n ce p o l i cy i s i ssu ed t o ( õ ) a n d ( y ) . À d ea t h b en efi t
o f 10 ,0 0 0 i s p ay a b l e u p on t h e seco n d d e at h . T h e p r em i u m i s p ay a b l e co n t i n u o u sl y u n t i l t h e l ast d eat h . T h e an n u al r a t e o f p ay m en t o f p r e m i u m is c w h i l e
( õ ) i s a l i v e a n d r ed u ces t o 0 .5 ñ u p o n t h e d e at h o f ( õ ) i f ( õ ) d i es b e fo r e ( y ) . T h e
e q u i v a l e n ce p r i n c i p l e i s u se d Ñî d et e r m i n e ñ . G i v en :
( i ) î = 0 .0 5
(é ) à
(È ) à
= 12
= 15
( i v ) à „ = 10
C a l cu l a t e c .
5 . À fu l l y d i scr et e l ast - su r v i v o r i n su r an ce o f 1 is issu ed o n t w o i n d ep en d en t l i v es
ea ch a g e x . L e v el n et a n n u al p r em i u m s a r e p a i d u n t i l t h e fi r st d e at h . G i v en :
(i) À
(é ) À
= 0 .4
= 0 .5 5
( é 1) à ~ = 9 .0
Ñ.Â. M ULTIPLÅ LIFE I NSURANCE: EXERCISES
15 9
Calculat e the net annual premium.
6. À whole life insurance ðàóâ à death benefi t of 1 upon t he second death of
(õ) and (y). In addit ion, if (õ) dies before (y), à payment of 0.5 is payable at
t he t ime of death. Mort ality for each li fe follows t he Gompertz law with à force
of mort ality given by p, = Â ñ' , z > Î . Show t hat t he net single premium for
t his insurance is equal Ñî
A + Àð —A (1 —0.5ñ™ )
where ñ = ñ" + ñ~.
7. Given:
(i) Male mort ality has à constant force of mort ality y, = 0.04.
(é) Female mort ality follows de Moivre's law wit h ø = 100.
Calculat e t he probability t hat à male age 50 dies after à female age 50.
8. Given:
(i ) Z is the present -value random variable for an insurance on the independent
lives of (õ) and (y) where
î ~<"1
0
if T (y) > Ò(õ)
otherwise
(é) (õ) is subj ect Ñî à const ant force of mort ality of 0.07.
(È ) (y) is subj ect Ñî à const ant force of mort ality of 0.09.
(iv) The force of interest is à const ant á = 0.06.
Calculat e Var[Z].
9. À fully discrete last-survivor insurance of 1000 is issued on two independent
lives each age 25. Net annual premiums are reduced by 40% after the fi rst death.
Use t he Illust rat ive Life Table and i = 0.05 to calculate t he init ial net annual
premium.
10. À life insurance on John and Paul pays deat h benefi ts at the end of the
year of deat h as follows:
(i) 1 at t he deat h of John if Paul is alive,
(é) 2 at t he death of Paul if John is alive,
(ø ) 3 at the death of John if Paul is dead and
(iv) 4 at t he death of Paul if John is dead.
The j oint dist ribution of t he lifet imes of John and Paul is equivalent to the j oint
dist ribution of two independent lifet imes each age x . Show that t he net single
premium of this life insurance is equal t o 7À —2A .
A P P E N D I X Ñ . E X E R CI SE S
160
Ñ .8 .2
S ð ã å à ñÜ Ü å å é E x e r c i s e s
1 . U se t h e I l l u st r at i v e L i fe T a b le a n d i = 5 % t o c al c u l a t e t h e j o i n t l i fe a n n u i t y ,
a .„ , t h e j o i n t - a n d - su r v i v o r a n n u i t y , à ,„ , an d t h e r ev er sio n a r y a n n u i t y , c ~> ,
fo r i n d ep e n d en t l i v es l i v es a g e õ = 6 5 a n d y = 6 0 .
2 . ( 8 .4 .8 ) À j o i n t - an d - su r v i v o r an n u i t y i s p ay ab l e at t h e r at e o f 10 ð åò y e ar at
t h e en d o f ea ch y e a r w h i l e ei t h er o f t w o i n d ep en d en t l i v es ( 6 0 ) a n d ( 5 0 ) i s a l i v e .
G i ven :
( i ) T h e I l l u st r at i v e L i fe T ab l e a p p l i es t o e a ch l i fe .
( é ) i = 0 .0 5
C a l cu l a t e à t a b l e o f su r v i v al p r ob ab i l i t i es fo r t h e j o i n t - an d - su r v i v o r st a t u s . U se
i t Ñî c a l c u l a t e t h e v ar i a n c e o f t h e an n u i t y 's p r esen t v al u e r an d o m v a r i a b l e .
3 . U se t h e I l l u st r a t i v e L i fe T ab l e a n d i = 5 % t o ca l c u l at e t h e n et l ev el an n u a l
p r em i u m fo r à sec o n d - Ñî - d i e l i fe i n su r a n ce o n t w o i n d e p en d en t l i v es a g e ( 3 5 )
a n d ( 4 0 ) . A ssu m e t h at p r em i u m s a r e p ai d a t t h e b eg i n n i n g o f t h e y ea r a s l o n g
as b o t h i n s u r ed l i v es su r v i v e . T h e d e at h b en e fi t i s p a i d a t t h e en d o f t h e y e ar
o f t h e seco n d d e a t h .
4 . C a l c u l at e t h e n et p r e m i u m r eser v e at t h e en d o f y ear s 1 t h r o u g h 10 fo r t h e
ð î é ñó i n e x er c i se 3 . A ssu m e t h at t h e y o u n g er È å su r v i v es 10 y ea r s a n d t h at
t h e o l d e r l i fe d i es i n t h e si x t h p o l i cy y ear .
5 . G i v en :
A p p r o x i m at e àëëî 4e a n d A sp.Ù .
À = 0 .00 4 , Â = 0 .0 0 0 1, ñ = 1 . 15 , a n d
( i ) y = À + Â ñ" fo r æ > 0 w h er e
Ù
Â
( é ) 6 = 5% .
%
.Î å
1Ô ß
;ø
À ß
s n oa e as %If
é
æÌ ýàá
. é 20 ÜÈÝÙ
m s u e n I sN A .Î Õ
Ä ÛÌ
àà ~é ì üÜ 'I o úàå ~
4 1î 4 .Ñàý Ü å 0 Ì
f Ù
à 1î Ñ:Ú Ü à × 1î Í éà åÜ ì È Ýà Ü ( v f )
rri 4i t i f ì é 1î m >fr u d h i aib Ôà Û ( å ï Ò
" ':i f ç6 à î ë ýñô í è' î ì 4 1î å é è Ô ÷~Ì Ü
ã à à å.í ì ì ÿ û ÿ à4ÈÀ Ì
l o è è ï ãï ò ù
,.ú
Ñ
.9.1ÒÍ ÅTTheor
ExAer
Ñ.9.
OTA y
L CL
I Mcises
A M OUNT IN À P ORT FOLI O
Ñ .9
16 1
T h e T ot al C laim A m ount in à P or t fol io
1. T he claim made i n respect of ðî éñó é is denot ed Sq. T he t hree possible
values of Sq are as foll ows:
Sa =
0
i f t he i nsur ed 1Èå (õ ) sur vives,
100
if t he i nsur ed surrenders t he pol icy, and
1000 i f t he i nsur ed É|åç.
T he probabil ity of deat h is qq = 0.001, t he probability of surrender is ù
0.15, and t he probability of sur vival is ð = 1 — qq — qq . Use t he normal
approximat ion t o calculat e t he probabi lity t hat t he aggregat e clai ms of fi ve
i dent i cally dist ribut ed policies S = Sq +
+ Ss exceeds 200.
2. T he aggregat e claims S are approximat ely nor mally dist ri but ed wit h mean
y, and variance o ~. Show t hat t he st op-loss rei nsurance net pr emium p(p ) =
F [(X —p )+] is given by
p (p ) = (p, —,9 )Ô
—
+óô
w here Ô an d ô ar e t he st an d ar d nor m al d ist r ib u t i on an d densi t y fu n ct i ons.
3 . Consider t he compound model described by formula (9.4.6): S = X + .. + Õ ó
where È , Õ; are independent , and Õ; ar e i dent i cal ly dist ribut ed. ß ÂÈ t hat t he
moment gener at i ng funct ion of S is Ì ~(~) = M ~ (l og(M x (t ))) where M ~ (t )
and M x (t ) are t he moment generat ing funct ions of N and Õ . T his provi des à
means of est imat ing moment s of S fr om est imat es of moment s of Õ and # For
exam ple, E [S] = E[N ]E[X ] and
E[S ] = E[N ]ê[õ ] + ê[È] (ê[õ ] - E[X] ) .
4 . À reinsurance cont ract provides à payment of
~ 9 —,9 È,9 < ß < ó
~ .~ —p
if 8 ) ó
E x p r ess E [R ] i n t er ms o f t he cum m u l at i ve d ist r i b u t i on fu n ct ion o f S .
5. (à)
(Ü)
Express F (z ) in terms of t he funct ion p(p ) .
Gi ven t hat p(p ) = (2 + p + ù,9~) „ 9 > Î , fi nd F (x ) and f (x ) .
A P P E N D IX
16 2
6.
Ñ.
E X E R C VS E S
Su p p o se t h at f ( 0 ) , f ( 1 ) , f ( 2 ) , . . . a r e p r o b a b i l i t i es fo r w h i ch t h e fo l l o w i n g
h ol d s:
/ ( 1)
=
3 / ( Î ) , j ( 2 ) = 2 / ( Î ) + 1 .5 / ( 1 ) ,
f (õ )
=
1
— ( 3 / (õ — 3 ) + 4 / ( õ — 2 ) + 3 / ( õ — 1) ) fo r õ = 3 , 4 ,
W h at is t h e v alu e o f f ( 0 ) ?
7 . Su p p o se t h a t l o g S i s n o r m a l l y d i st r i b u t ed w i t h p ar a m et er s, p a n d î . C a l c u l a t e t h e n et st o p - l o ss p r em i u m ð ( ,9 ) = E [ ( S — p ) + ] f o r à d ed u ct i b l e 9 .
8.
(à)
F o r t h e p o r t f o l i o d e fi n ed b y ( 9 .3 .5 ) , ca l cu l at e t h e d i st r i b u t i o n o f
a gg r eg a t e c l a i m s b y a p p l y i n g t h e m et h o d o f d i sp er si o n w i t h à sp an o f 0 .5 .
( Ü)
z at io n .
A p p l y t h e co m p o u n d P o i sso n ap p r o x i m a t i o n w i t h t h e sa m e d i scet i -
Ñ .10
Ex p ense L oad in gs
Ñ . 10 . 1
T h eo r y E x er c i se s
16 3
Ñ . 10. E X P E N SE L OA D I N GS
1 . C onsi der t he en d ow m ent p oli cy o f sect ion 6.2, r est at ed her e for con venien ce:
su m i nsu r ed = 1000, d ur at i on n = 10, i n it i al age õ = 40, D e M oi v re m or t al i t y
w i t h ur = 100 , an d i = 4% .
(i ) T he acquisit i on ex p en se i s 50 . N o ot her ex p enses ar e recogn i zed (ð = ó = 0 ) .
C al cu l at e t he ex p en se-loaded an nu al pr em i u m an d t he ex p ense-lo aded pr em i u m
r eser ves for each ð î é ñó year .
(é ) D et er m i ne t he m ax i m um valu e of acqu i si t ion ex p ense i f negat i ve ex p ensel o aded r eser ves ar e Ñî b e avoi ded .
2 . G i v e à v er b a l i n t er p r et a t i o n o f —q V
.
3 . C on si der t h e t er m i nsu r an ce p ol i cy o f sect i on 6.2, r est at ed her e for conven i en ce: su m insu red = 1000 , du r at i on r», = 10, i nit i al age z = 40 , D e M oi v r e
m or t ali t y w it h þ = 100 , an d i = 4% .
(i ) T h e acqu isit i on ex p en se i s 40. No ot h er ex p enses ar e r ecogni zed (p = ó = 0) .
C al cu l at e t he ex pense-lo aded ann u al p r em ium an d t he ex pense-loaded p r em i u m
r eser ves for each p ol i cy year .
(é ) I f t h e ex p ense-lo aded p rem i um r eser ves are n ot all owed Ñî b e negat i ve, w h at
i s t he i nsu rer 's i nit i al i nvest m ent for sel l i ng su ch à p ol i cy ?
4 . C al cu l at e t he com p onent s 1000Ð , 1000Ð , 1000P > an d 1000Ð'» o f t h e ex p en se-loaded pr em i u m 1000Ð for à w h ol e l i fe i nsu r an ce o f 1000 issued Ñî à l i fe
age 35 . T he p ol i cy h as l evel an nu al pr em iu m s for 30 y ears, b ecom i ng p ai d-u p
at age 65. T h e com p any h as ex p enses as fol l ow s:
acqui si t i on ex p ense
col l ect ion ex p en ses
12 at t he t i m e o f i ssue,
15% o f each ex p ense-loaded pr em i u m , an d
ad m in i st r at i on ex p en s
1 at t he b egi n n in g o f each p ol i cy y ear .
U se t h e I l l u st r a t i v e L i f e T ab le an d i = 5 % .
5 . For t h e ð î é ñó descr i b ed i n ex er ci se 4, cal cu l at e com p onent s 1000@V , 1000»,Ó ,
an d 1000~V » o f t h e ex p ense-loaded p rem i u m r eser ve 1000», Ó for year é = 10.
16 4
Ñ . 1 0 .2
A P P E N D IX
Ñ.
E X E R CI SE S
Sp r ea d sh e e t E x er c i ses
1 . D e v el o p à sp r ea d sh eet t o ca l c u l a t e t h e ex p en se - l o a d ed p r e m i u m co m p o n en t s
an d t h e ex p en se -l o a d ed p r em i u m r ese r v e co m p o n e n t s fo r ea ch ð î é ñó y ea r o f à
2 0 - y ear en d ow m en t i n su r an c e i ssu ed Ñî à l i fe ag e 4 0 . U se t h e I l l u st r at i v e L i f e
T a b l e an d i = 6 % . A ssu m e t h at a c q u i si t i o n ex p e n se i s 2 0 p e r 10 0 0 o f i n su r a n c e,
co l l ec t io n ex p e n se i s 5 % o f t h e ex p e n se- l o a d ed p r em i u m , an d a d m i n i st r at i o n
ex p en se i s 3 a t t h e b eg i n n i n g o f ea ch p o l i cy y ea r .
Ñ
.11ESTE
atPROBAB
in g P rILIob
ab iOF
li t DEAT
ies of
Ñ.11.
I Mst
ATi m
ING
TIES
H D eat h
Ñ .11 .1
16 5
T h eo r y E x e r c i ses
1. Consi der t he fol lowing t wo sets of dat a:
(à)
(Ü)
D = 36
17 = 360
Å = 4820
Å = 48200
For each set , cal culat e à 90% confidence int erval for q .
2 . We model t he uncert ainty about ä (t he unknown value of p + i ~~) by â gam ma
dist ribut ion such t hat Å [ä] = 0.007 and Óàã(ä) = 0.000007. An addit ional 36
deat hs are observed for an addit ional exposure of 4820. Calculat e our post erior
expect at ion and st andard devi at ion of ä, and our est imat e for q .
3 . Wri t e down t he equat ions from whi ch
(i ) Ë' and
(È) Ë" are obt ai ned.
(ø ) Rewrit e t hese equat ions i n t erms of int egrals over f (x ; è ), t he probabil it y
densi ty funct ion of t he gamma dist ribut ion wit h shape par amet er è and scal e
par amet er 1.
4 . In à cl ini cal experi ment , à group of 50 r at s is under observat ion until t he 20t h
r at dies. At t hat t ime t he group has li ved à t ot al of 27.3 r at years. Est imat e
t he for ce of mor t alit y (assumed t o be const ant ) of t his group of rats. W hat is
t heir li fe ex pect ancy?
5 . À cert ain group of li ves hss à tot al exposure of 9758.4 years bet ween ages x
and õ + 1. T here were 357 deat hs by cause one, 218 deat hs by cause t wo, and
528 deat hs by all ot her causes combined. Est imat e t he pr obability t hat à li fe
age x will die by cause one wi t hin à year .
6. There ar e 100 life insur ance pol ici es in force, insur ing li ves age x . An addi tional
60 polices are i ssued at age õ + ~~. Ðî ø deaths ar e observed between age x and
x + 1; we assume that these deaths occur at age õ + 0.5. Cal culate the classical
estimat or given by for mul a (11.2.3), and the maximum likelihood esti mator based
on the àçâèò ðéî ï b, à constant force of mort ali ty (11.4.2).
7. T he force of mort al it y is const ant over t he year age (x , x + 1]. Ten l ives ent er
observat ion at age x . T wo li ves ent er observat ion at age õ + 0.4. T wo lives leave
observat ion at age õ + 0.8, one leaves at age x + 0.2 and one leaves at age õ + 0.5.
T here is one deat h at age õ + 0.6. Cal cul at e t he maximum likelihood est imat e
of t he for ce of mort al ity.
16 6
A P P E N D IX
Ñ .
E X E R C I SE S
8 . À d o u b l e d e cr em en t m o d el i s u sed t o st u d y t w o c au ses o f d ea t h i n t h e i n t er v al
o f ag e ( õ , õ + 1] .
T h e fo r ces o f e a ch cau se ar e co n st an t .
1,0 0 0 l i v es en t e r t h e st u d y at a g e x .
4 0 d ea t h s o c cu r d u e t o ca u se 1 i n ( õ , õ + 1] .
5 0 d ea t h s o cc u r d u e t o ca u se 2 i n (õ , õ + 1] .
C a l cu l at e t h e m a x i m u m l i k el i h o o d est i m a t o r s o f t h e fo r ces o f d e cr em en t .
9 . T h e I l l u st r a t i v e L i fe T a b l e is u sed fo r à st a n d a r d t ab l e i n à m o r t al i t y st u d y .
T h e st u d y r esu l t s i n t h e fo l l o w i n g v a l u es o f ex p o su r es E
an d d e at h s D
o v er
[4 0 , 4 5 )
õ
E
D
40
1 15 0
6
41
900
5
42
12 0 0
12
43
14 00
9
44
130 0
13
C a l cu l at e t h e m o r t al i t y r at i o f a n d t h e 90 % co n fi d en ce i n t er v a l fo r f . C a l c u l at e
t h e est i m at es o f ä~î , qq~, . . . ques co r r esp o n d i n g t o f .
A pp end ix D
Sol u t 1~~~
16 8
D .Î
A P P E N D I X D . SOI UÒI 0 N Á
I n t r o d u ct i o n
We offer solut ions t o most t heory exer cises which we hope st udent s will find
useful . W hen t he sol ut ion is st raight forwar d we simply give t he answer . For
t he spreadsheet exer cises we descr ibe t he solut ion and gi ve some val ues t o use
t o ver i fy your work . We l eave t he j oy of writ ing t he program t o t he st udent .
We have t r ied har d t o avoi d errors. We hope t hat st udent s and ot her users
who discover errors wil l inform us prompt ly. We are also i nt erest ed in seeing
elegant or insight ful solut ions and new problems.
T he solut ions occasionally refer t o t he Ill ust r at ed L i fe Table and it s funct ions. T hey are in Appendix Å .
 .1.
H E M Ai ons
T I CSt OF
OM P y
O UN
T E R E ST
D
.1.1M A TSolut
o TCheor
ExDerI Ncises
Î .1
16 9
M at h em at ics of Com p oun d I nt er est
1 . T h is fol low s easil y from equ at ion ( 1.5.8) .
2 . F ix i ) 0 an d consi der t h e fun ct ion f (x ) = [( 1 + ~)
— 1]x
1 = (åá — 1)/ õ .
Fr om t he p ower ser i es ex p an sion
f (z ) = 6 + —62 õ + —
1 á3 õ 2 + . . . ,
2
3!
i t i s easy t o see t h at f ' (õ ) ) 0 for al l õ ) Î . I t fol low s t h at f (z ) i ncr eases fr om
f (0+ ) = á Ñî f ( 1) = i . T her efor e ä (õ ) = f (x 1) decr eases on [1, î î ) fr om i t o
6. T hus, i < > = g (m ) decr eases t o 6 as m in cr eases. Si m i lar l y, d < > i n cr eases t o
6 as m i ncr eases.
3 . T h e accum u l at ed val ue o f t h e dep osit s as o f Janu ar y 1, 1999 is Õ ÿ —
, 10
T he pr esen t val ue o f t he b on d p ay m ent s as of J anu ar y 1,1999 i s 15, 000à ~
E qu at e t he t wo val u es an d sol ve for Õ = 4794 .
4 . L et i b e t h e effect i ve an nu al int er est ãàÑå. T hen 1 + ç = ( 1 + ó/ 2) 2. T he
equ at ion o f val u e i s
5 .89
=
è ~ + è~ + . .
2
1
è2 '
Í åï ñå ( 1 + ó'/ 2)4 = 1 + 1/ 5 .89 an d so j = 8% .
5 . L et è = -'-ß
an d w r i t e t h e equ at i on of val u e as fol lows:
è + è + ..
è
1 —è
1+ k
0 .04 — /ñ
Sol ve for /ñ = 2% .
6 . U se equ at i on ( 1.9 .8) w it h à st art i ng val ue o f 6 = 12% . T h e p r i ce o f t he
cou p on p ay m ent s is ð = 94 — 100 ( 1.12) ' 0 = 6 1.80 . T he su m of t h e p ay m ent s
i s ò = 100 an d
—10 á
à (á ) =
5
á/ 2
T he solu t i on i s á = 9 .94% . T his is equ i valent t o 10.19% per y ear conver t ib l e
sem i an nu al ly .
17 0
A P P E N D I X D . SOL U T I ON S
7. T he equat ion of value is
1000
=
õ (å + ýÿ + èç) + 3z (v4 + vs + vâ)
õ (Çà~~ —2à ç~)
õ (11.504459)
where t he symbols correspond t o à values of i = 1%. So õ = 86.92.
8 . At t he t ime of t he loan,
4000
=
kv + 2kv2 + ÇÜ ~ + . . . + ÇÎ ÄÅÐ
É (1à) ù
à — —30v~
çî ]
0.04
so é = 18.32. Õî Ñå t hat t he init ial payment is less t han t he int er est (160)
r equired on t he loan so t he loan bal ance increases. Immediat ely aft er t he nint h
payment t he outst anding payments, valued at t he original loan int erest ãàÑå, is
found as follows:
10kv + l l kv2 +
+ ÇÎ Ü ~~ =
9éà ~| , + é (1à) -.
(18.32) (9(14.02916) + 134.37051)
4774.80.
9. Let ó' = i ® / 2 and solve 98.51 = 2(1 + ó') 1 + 102(1 + ó') z for (1 + ó')
0.9729882. T his corresponds t o ç<~> = 5.55%.
10 . From (i) and (é) we get 12( 120) à —1 — 12(365.47) à ã
î " = 0.6716557. Now use (ø ) and (i v) t o sol ve for Õ = 12000.
D .1.2
and solve for
Sol ut i ons t o Sp r eadsheet Ex er cises
1. (à) T he invest ment yield is 9.986%.
2. Gui de: Set up à spreadsheet wit h à t ri al val ue of Õ. Since à t ot al of Õ +- 2Õ +
ÇÕ + . + 6Õ = 21Õ is wit hdrawn, à good t ri al val ue is about 100, 000/ 20 =
5, 000. Use t he fundament al formul a (1.2.1) t o calculat e t he fund balance at t he
end of each hal f-year . T hen experiment wi t h Õ unt il an end-of- period six bal ance of zero is found. Õ = 6, 128.(Alternat i vely, in t he last st age, use Goal
Seek t o det er mine t he value of X whi ch makes t he t arget balance zero.) Not e
t hat t he end-of-period six balance is t he fund balance beginning t he sevent h
hal f- year . Adapt i ng not at ion of t he t ext t o hal f-years we have Fe = 100, 000,
F i = Fc ( 1.03)ã Õ , Fz = F i ( 1 03)ãF ~ —2Õ , and so on.
3 . Gui de: Set up an amort izat ion t able using à spreadsheet and à t rial value
of i = 0.03. In à cel l apart from t he t able, calcul at e t he t arget P — 6I for t he
Î . 1. M A T H E M A T I CS OF C OM P O IJN D 1Õ ÒÅ ß Å ÁÒ
17 1
if ft h year . T hen use Goal Seek t o det ermine i so t hat t he t arget cell is í åãî .
i = 10.93%.
4 . Gui de: T he fund deposit Õ sat isfi es Õ þ-ù .„ î â = 10, 000. In effect , t he
company accept s 10,000 now in exchange for 10 semi annual payments of 300+ Õ .
Calculat e Õ using t he spreadsheet financi al funct ions. T he int ernal ãàÑå of
ret urn j equates t he fut ure cash fl ows 300+ Õ Ñî 10,000. Set up your spreadsheet
wit h à t r ial value of ó'. Use t he Goal Seek feat ure t o det emine t he value of ~.
ó' = 7.80%
5. Gui de: Put i = 10% and à t r ial value of Õ int o cells. Cal culat e t he net
pr esent val ue of t he payment s of 100 minus t he payment s of Õ in anot her cell
as follows:
100 —vi p100 —@~âÕ
100à-,~~ —Õ þ ¸
Use t he Goal Seek feat ure t o det er mine t he valaue of Õ for which t he resent
value is zero. Õ = 375.80
6. Gui de: Set up a spr eadsheet Ñî amort ize t he loan using à t ri al val ue of
Õ = 30, 000. T he interest cr edit ed in year /ñis
0 .08 m i n ( 100000 , Â ) + 0.09 m ax (0, Â — 100000 )
where Bi is t he beginning year balance. Be = 300, 000, Â ~ = 300, 000 + 8, 000 +
18, 000 — Õ , and âî on recursi vely. Use t he Goal Seek feat ur e t o det ermine Õ
so t hat B i i = 0 (beginning year 11 = end of year 10). Õ = 45, 797,09
7. Gui de: Work from t he 1àçÑ year back t o t he present . T he requir ed cash
lf ow for t he last year is known and so is t he coupon, so you can cal culat e t he
number of longest mat urity bonds t o buy. T hen work on t he next t o the last
year , knowing t he requi red cash flow and t he ï èãï Üåã of bonds paying à ñî èðî ï
(but mat uri ng in t he following year ). And so on.
T he t ot al market value is 450,179. You need 1.87 bonds mat uring in 1995,
åÑñ.
8. Gui de: Use t he Goal Seek feat ure. T he mar ket yield is 7.46
9. Gui de: Use t he Goak Seek feat ur e to fi nd t he pri ce for each cal l dat e to yield
8%. T he pr ice is t he minimum of t hese: 1,085.59.
10. Guide: W it h à t ri al val ue for t he interest ãàÑå, use t he fut ure val ue funct ion
(F V ) t o fi nd t he balance aft er 20 quart ers. Use Goal Seek to set t he fut ure val ue
t o 5000. T he solut ion is i = 8.58%
1 72
A P P E N D Ix
D .
S OL U T I ON S
Î .2
T h e Fut ur e Li fet i m e of à L i fe A ged x
D .2 . 1
S o l u t i o n s t o T h e o r y E x er c i se
1 . U se eq u a t i o n ( 2 .2 .5 ) . p qs = — ~, 1ï ( ~ð ) ev a l u a t ed a t t = 4 5 — õ . T h u s
1
4—ù~- ~
2 . U se eq u a t i o n ( 2 . 1 . 1 1) .
5 5
cpzdt
Å [Ò ( õ ) )
1 —
—
dt
60 .
3 . U se ( 2 .2 .6 ) . F i r st : f p
— ' ~ ( 44 ( Òåå ) ) .
ð + ññé =
Ò Ëåë þ ð
— l n ( 8 5 — t ) — 3 1n ( 10 5 — é) 1î
= ss ( å„, )
= 0.400 7
4 . U se ( 2 .1 . 1 1 ) .
Î
å<,
=
ð ~| à
= ~ (.: : ,)'
(4 2 ) ç (4 2 + t ) - '
—2
î
2 1.
5 . T h e sy m b o l m
d e n o t es t h e cen t r a l d e at h r a t e : D eat h s d = l — l + q an d
à+ 1
1 47
a v er ag e p o p u l at i o n =
l Ädy = l
* ~ ' d t . D i v i d e ea ch o f d e a t h s an d
a v er a g e p o p u l a t i o n b y l
î
t o ob t ain m
cp s d t
.ð.î
=
f ' ð* d t .
U se ( 2 .6 .9 ) .
1
ä
1 — ( 1 — ~) ~*
D .2. ÒÍ Å F U T UR E L I F E T I M E O F À L I F E A GE D Õ
17 3
T he formula for ò is t he reciprocal of t his quant ity mult i plied by q . To wor k
t he exer cise subst it ut e q = 0.2 and ð = 0.8. T he answer is rn = 0.224.
6. Use equat ion (2.2.6) . å " = Ð = 1 —0.16 = 0.84 so <p = å ' " = (0.84)' =
0.95 and, t = 1ï (0.95)/ ln(0.84) = 0.294.
7. Since l p is const ant , l is linear . T hus T (88) is uni for m on (0,12) . T herefore
Var (T ) = ( 12)ã/ 12 = 12
8. Before: 0.08 = åõð ( — / p, +, é ) = ð, . A lt er: 0.03 = åõð ( - / o (p, +, —efr8)
= ð å~ = 0.95å' . T herefore å' = 93/ 95, ñ = log(93/ 95) = —0.0213.
9. M ake t he change of vari ables õ + s = y in equat i on (2.2.6) t o prove (i ). Use
t he rul es for di f f er ent i at i ng int egr als Ñî prove (é).
10 .
100 ö ×[çî )+ s =
100 (p[soi+ z) (ß[çî )+ã)
( 1 —q<~ >+ , ) ( 100ä„ )
( 1 —0.00574) (0.699)
0.695
4 î —4üò
121
(81) ' / — (64) ' /
(100) ' ~ã
12. Use t his r el at ion :
å = Å [Ê (õ )] = Ð Å [Ê (õ ) ~Ò(õ) > 1] + q Å [Ê (õ) ~T (õ) ( 1] = ð (1 + å + 1).
T hus ð = å / (1 + e~+ s) ãð òü = Ðòüðòü = ä+î
10.ü ,+10j oüî = 0.909.
13 . Ò(16) is uni form on (Î , ì — 16) since we have à de Moivre mort ality t able.
Í åï ñå Å [Ò( 16)] = (û - 16)/ 2 and × àã[Ò(16)] = (û - 16)ã/ 12. T herefore û —16 =
2(36) = 72 and Var [T ] = (72)ã/ 12 = (72) (6) = 432.
14 . óþ = 1 —srpso/ þ ðþ = 111/ 8000. Àï 4 l rso+r = — ~'„ e rpso = ~
r soo ß ..r .
T herefore, qn —àäüî = 1/ 6000.
15. E [T ] = /î
4ð* à = [î ( ' ' ) fJt = ç where à = 100 — õ . E [T ã] =
Ä - * 284ð <é = 2 / Ü,ð ñé . Use i nt egrat ion by par t s Ñî î ÜÑà ï Å [Ò~] = ~ .
Hence Var (T ) = Å (Ò ) —Å(Ò)ã = àã(1/ 6 — 1/ 9) = (100 —z )~/ 18.
A P P E N D I X Î . SOL UT I ONS
174
16. m
=
,~
and, because of t he const ant for ce of mort ality, gp = å " ~
î «Ðà ! àà
where è = —1ï (ð ). Í åï ñå, /~ ,ð ñé = q / )è and ò = )è = 0.545.
/
17. Let Ò = Ò(õ ) be t he li fet i me of t he non-smoker and Ò' = Ò' (õ ) be t he li fet ime of t he smoker . Use formul a (2.2.6) : Ðã(Ò' > ! ) = exp ( — ( ct« .«„ Í è) =
(,ð ) where ð = Pr (T > t ). Hence Pr (T ' > Ò) = /; Pr (T ' > t )g(t )dt =
,/~ (ñð~) 9(é)Í Å = —f () ø (é)~ø (t )dt where u)(t ) = ó õ. Í åï ñå, Ðã(7 > Ò) =
[ ( )]" ' ~"
! 8 . See exercise 9. q = ! —ó, = ! —åõð ( —f
ó Í ó) which we gct hy à
change of var iable of int egrat ion in formula (2.2.6). Now apply t he r ules for
diff erent iat ion of i ntegrals:
dq
— = —åõð
dz
19 . /
Ã+
—/
ð,„ ô
(- è~+| + )è ) = ð (ð +| —,è ) .
/g ch = 400k and so 0.81 = yppss = exp( — /
Si m i l ar l y
=
(
f ~p
~ ) =
fg dz ) = åõð ( —400k).
—10001 = ( ( Î 8 1 ) 1/ 400 ) 1000 = ( 0 .8 1) 5/ 2
(0.9)5 = 0.59
20. E [X 2] = 2 /
9/ 16) = 3 z/ 80.
D .2.2
õ ð()Í õ = 3(.)~/ 5. Var [X ] = (ÇàÐ / 5) — (Çì / 4)~ = (.Ð (3/ 5 —
Solut i ons t o Sp r eadsheet Ex er cises
1. See appendix E.
2 . Check value: e() = 71.29,
3 . ñ = 0.09226. Assume t hat "expect at ion of remaining l i fe" refers t o complet e
0
l i fe expect at ion and t hat assumpti on à appl ies, so t hat å = e + 0.5.
4 . Use formula (2.3.4) wit h À = Î . Check val ues: 84o = 99, 510 when ñ = 1.01
and l s() = 680 when ñ = 1.20.
5 . Under assuyygp ti org à, y +o s = 0.10638 for example.
Â. Under assump ti org b, () 4q = 0.04127 for example.
7. Use t rial val ues such as  = 0.0001 and ñ = 1 Ñî calcul at e Gompert z
values, and t he sum of t heir squared differences from t he t able values. Use t he
opt imizat ion feat ure t o det ermi ne val ues of  and c which minimize t he sum .
Sol ut ion: Â = 2.69210 ~ and ñ = 1.105261.
8 . For /ñ = 7.5, å45 = 12.924. For é = 1, å45 = 30.890.
D.3.
.3 .1
o l SURAN
u t i o n sCE
t o T h e o r y E x e r c i ses
D
LI FESIN
D .3
175
L ife I n sur ance
1. T he issue age is z = 30. From (i), Ò = Ò(30) is uni formly dist ri but ed on
(0, 70). T he present value random variable is Z = 50, 000vr . Hence, As~ =
E [Z ] = 50 000 Õî èä odt = (50 000/ 70)(1 —å 7)/ (0.10) = 7, 136.
2 . Use t he recur sion relat ion :
(I A) = A .q + èð. (À +ä+ (~ 4)*+,)
An alternat i ve solut ion in t erms of commut at ion funct ions goes like t his: T he
numer at or can be writ t en as follows:
ë. —ñ.
ì . + ë.+, —ñ.
D
D
ì .+, +ë.+,
D
T he denominat or is ~ +'ä™ +' . Hence t he r at io is Ð~+ , / Ð = èð .
0
+ä
3. Let Zs be t he present val ue random vari able for t he ðèãå endowment , so
Z = ã , + ã, . It follows t hat × àã(ã ä) = Var (Z@
) + 2Cov (Zz, ã ü) î × àã(ã ~). Now
use t he fact t hat ã àãà — 0 t o obt ain Ñî ÷(ã ~, Zs) = —E [Zq]E [Zs]. Zs is è" t imes
t he Bernoull i random variable which is 1 wit h probabil it y „ ðõ, zero ot her wise.
Í åï ñå × àã(ã ä) = 0.01 + 2( —E [Zz]E[Zs]) + Var (Zs) = 0.01 — 2(0.04) (0.24) +
(0.30)2(0.8)(1 —0.8) = 0.0052.
4 . À 4ü 2Ù = (Ì ~ü — Ì üü + Ð âü) / Ð àü = 0.40822.
5. Use t he recursion relat ion À = À ' .—
„ 1+ è" „ ð À » „ àï 4 t he relat ion À ,—
À ä. „ ä + è" „ ð . Iï t erms of t he gi ven rel at ions t hese are À = y + è" „ ð ë and
è = ó + è" „ ð . Í åï ñå À = y + (è —y)z = (1 —z)y + uz.
B. From (ää), t he discount funct ion is èä — 1/ (1 + 0.01t ) = 100/ (100 + t ) .
T he benefi t funct ion is: Üä — (10, 000 —Ð )/ 10 = (100 + t )(100 —t )/ 10. Í åï ñå
Z = èò Üò = 10(100 — Ò) and so E[Z ] = 10(100 — E [T ]) . Now use it em (i ):
Ò = Ò(50) is uni form on (Î , 50] so E [T ] = 25 and E [Z ] = 750.
7
E [vò] =
( e û å—,à ð ô = A
=
T herefore, Var [v~] = À —À 2 = . . . =
/
and Å[(èò )~] = çÀ
=
È
~2
(È + 2é) (È + b)z
8. Consider t he recursion rel at ion À = vq (1 — À + ä) + èÀ +ä. T he anal og
for select mort al ity wi t h à one year select period goes l ike t his: Since t he select
period is one year , Ê ( [õ] + 1) and Ê (õ + 1) are ident ically dist ribut ed. Hence,
using t he t heorem on condi t ional expect at ions, we have A l~i = Å [èãã~~+ä] =
è×[õ] + E [v
](1 — ×[õ]) = è×~õ) + E [v
]v( 1 — ×(õ)) = è×[õ~ +
À ».äè(1 — ql l) . Í åï ñå, À ~~ = è×~~(1 — À ».ä) + èÀ » ä. Âó combining t he
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Î .3.
177
f0(~)
.3 1 ~ f (~) 1
A P P E N D I X Â . SO L U T I ON S
178
T hen t he n et si n gle pr em i u m i s gi ven as foll ows:
10M
D .3 .2
— Ì . + 2 — ì . +5 — ì . +ä
S o l u t i o n t o S p r ea d sh e et E x er c i se s
1 . For i = 2.5% , 5.0% , an d 7 .5% , t he singl e p r em iu m l i fe i nsu r ance at age zer o
i s À î = 0 19629, 0 .06463 an d 0.03717.
2 . A t i = 5%, (I A )p = 2.18345.
3 . G u i de: Set u p à t abl e w i t h benefi t s an d p r ob abi l i t i es o f sur v i val t o get t hem .
T h e net si ngl e p rem i um i s 0.0445 .
4 . G u i de: U se t he V L O O K U P () fun ct ion Ñî const r u ct t he ar r ay o f sur v i al
p r ob ab i li t es for à gi ven issue age. C heck values: (Â À ) ,- ~~ = 9 .23 at i = 5% .
( D ® 25 ~~~ = 3 .50 at i = 6% .
5 . G ui de: Set up à sp r eadsheet t o cal cul at e t he val ues of À
an d 2A . A ccor d i ng
t o for m u la (3 .2 .4) , t he secon d mom ent can b e cal cul at ed by ch angi ng t h e for ce
of i nt er est fr om b t o 2á. P ut á i n à cel l and let it dr i ve t he int erest cal cu l at i ons.
U se t he D at a T ab le (or W h at i f ?) feat u re Ñî fi n d t h e t wo valu es o f Å [è~ + | ],
cor resp on d i n g t o á an d 2á. Ch eck values: V ar (v + + ~) = 20, 190 w hen i = 5% ,
and V ar (v + + ) = 17, 175 w hen ç = 2% .
D
Ð . .4.1
4 . L I F ESolut
À È È iØons
Ò 1Å tÁo T heor y Ex er cises
D .4
1.
179
L ife A n nu it ies
U se fo r m u l a (4 .3 .9 ) w i t h r n =
N r p) =
15 . 14 0 4 , D r p/ D gp =
2,õ =
è çî ð 4î =
p ( 2 ) = 0 .2 5 6 17 . T h e an sw er i s à
4 0, n =
(I a)
q u en t l y Ñî v p
3 . ( I >~à )
=
à ,q
+ vp
à 4î .óö
=
11~ô ~ —
0 .1 6 44 > o ( 2 ) =
D 4p ( N 4p 1.0 0 0 15 , an d
= 14 .9 2 8 6 .
2 . U se t h e r ec u r si o n fo r m u l a (4 .6 .2 ) fo r à .q
fo r m u l a :
30 .
t o d ev el o p t h e fo l l o w i n g r ecu r si o n
( à .~ | + ( I i i ) + | ) .
T h e r at i o si m p l i fi es co n se-
= à . 11 .
= f p t v ' i p d t + f Ä n v ' <d d t . D i f e r e n t i a t e b efo r e m ak i n g t h e su b -
st i t u t i o n s v i = e - î .î å' an d
ð
= å —p.~ ' . U se L i eb n it z 's r u le f o r d i f er en t i at i n g
i n t eg r a l s :
—
a~
—
' (I
(. ~ a,
-)'l
= ï è „ ð + ( v ~ð ~Ñ—nv ~p~
ó ~ ð
Çï
J»
4 . A r r an ge t h e cal cul at ions i n à t ab l e:
7 .9
× 4) ) ( P V ) P r [E 4v .7
en64
t ] ] ( P V ) ~P r [E v en t ) )
KE v)en2t [ P r [E v e0n .6
t ] ] P r esen t V al u e ( Ð
Ê = Î
0 .8 0 0
2 .0 0
0 .4 0 0
0 .2
Ê = 1
0 .2
4 .7 0
0 .9 4 0
4 .4 18
3 7 .8 2 6
E [P V ) = 6 .104 an d E [P V ~] = 4 3 .0 4 4 . H e n ce , t h e v a r i a n ce i s 4 3 .04 4 — ( 6 .104 )
5 .7 8 5 .
( I < ) 95
àî ü + 1/àî ü + ã/o ps + 3/à î ü + . . . . Si n ce û = 10 0 , i = 0 an d Ò ( 9 5 ) is
u n i fo r m o n ( Î , 5 ) , t h en t h e fi v e n o n - z er o t er m s a r e àäü = Å [Ò ( 95 ) ] = 2 .5 , [ i a gs =
ð ì ~î å =
( 0 .8 ) ( 2 ) =
1 .6 >ã>
>à î ü = ãð î üà î 7 =
( 0 .4 ) ( 1 ) = 0 .4 an d 4~à î ü =
4ð î üà î î =
( 0 .6 ) ( 1 .5 ) =
( 0 .2 ) ( 0 .5 ) =
0 .9 >3/àäü = Çð äüàî å
0 .1 . H en ce, t h e a n sw er is
5 .5 . A l t er n a t i v e ly , w e c a n ca l c u l a t e ex p ect ed p r ese n t v a l u es co n d i t i o n a l l y o n t h e
y ea r o f d e at h . T h er e a r e fi v e y ea r s o f i n t er est an d t h ey a r e eq u a l l y l i k el y . T h is
y i el d s ( 0 .5 + 2 .0 + 4 .5 + 8 .0 + 12 .5 ) / 5 = 5 .5 .
6 . U se f o r m u l a (4 .3 .9 ) o r i t s eq u i v al en t i n t er m s o f co m m u t a t i o n fu n ct i o n s .
| î >à ãü.1î ) = Ð ãü
( Ô çü — Ì 4ü ) = 4 .8 5 4 5 6 , ( Ð çü — 0 4ü ) / Ð ãü = 0 .24 35 5 , à ( 12 ) =
1 .00 0 2 0 , a n d 9 ( 12 ) = 0 .4 6 65 1 . Í åï ñå ,
l î l à.. ( | ã) 1
— a ( 12 ) | ù à ãü -ôî ] — p ( 12 ) ( i pp m v
— ãî ð ãüè
) = 4 .74 19 1 .
A P P E N D IX Ð . SO L UT I ON S
180
A not her sol u t io n is based on fo r m ula (4 .3 .2) an d (3.3 .10) , adj ust ed for t em p oãàãó r at her t h an w hole l i fe cont r act s:
À ( » ) 6]
1 ) Àç ù
.( è
—
+ l op35V1î
i
; (1ã) ( Ì 35 — ~Ñ~45) / Ð çü + Ð 4ü/ Ð çü = 0.6 18 14 .
Í
åï ñå ,
1ù à ã '. " ù
=
( Ð ç ü / Ð ãü )
1 -
À
~
/ à 1è 1 =
4
7 4 2 00
7 . U se for mu l a (4 .3 .9) t o der i ve à for mu l a an al ogou s t o for m u l a (4 .5.4) for
t em por ary an nu i t i es. T h en use t he for m u l as an alogous t o (À .3 .6) an d (À .3 .9)
t o w r i t e t h e r esu l t i n t er ms of com m ut at i on fu n ct ions:
(ëö.' „'
= c(m) ((è ). .-„ ,) —p(m) (à. ,-„„ S~ — S~q Ä — ï È ~ „
,
"„„ . )
, N ~ —Ì
» „ - ï Ð .~„
N ow cal cul at e t h e N an d D val ues by d i fferen ci ng t he su ccessi ve val ues of t he
gi ven val ues of S . W e need È ãî — Sro — Sy1 — 9597, Æâî = Sso — S51 = 2184,
Ð ðî = N ro — Ô ã1 = 9597 — 8477 = 1120 , an d D so = N so — N 51 = 368. W e get
( 1à ) 16~ — 29 .16 .
a*de ) + 4è~,—- ,1 (
~
1 *e*+a
a= o
òâ—1
d
à ~ Ðã(Ê > n ) + ~ à
w:= o
äÅ ( à .,„
Ðã(Ê = /ñ)
1)
~(àõ:%] = 1 — A * : ~)
9 . U se fo r mu l as (4 .2 .9 ) , (3 .2 .4 ) an d (3.2 .5) . V ar (Y ) =
N ow u se À = 1 — da t w i ce. À eval uat ed at 6 is
d iscount cor r esp on di n g Ñî 26 is 1 —V~ = 1 — (0 .96)~ =
«À eval u at ed at 26,» is 1 — (0.0784) (6 ) = 0 .5296 .
(Î 5296
(Î 6)ã)/ (Î 04) ã
106
10 . Use for m u l as (4 .2.13) an d (3 .2.12) .
d ã (Å [å ãü(~ + 1)] — A ~) .
1 — (0 .04 ) 10 = 0 .6 . T he
0 .0784 so Å [å ãü(~ã+ ' )] =
T her efor e and V ar ( Y ) =
 .4. L I F E A N N UI T I E S
18 1
11. Use formula (À .4.7). Ôãâ = S~ — Sgg = 97, Ngg = Sgg —ßçî = 93, and
è „ = N ss —N~
4. Hence, M ~ = 4 —(3/ 103)97 = 1.1748 where we used t he
commut at ion funct ion version of formula (4.2.8) : D = dN + Ì .
12. Use t he recursion relat ion ci = 1 + àð à » 1. 7.73 = 1 + (1.03) ' ðòç(7.43)
so ðòç = ( 1.03)(6 73)/ 7.43 = 0 93.
13. T he values of t he present value random variable Y are 2, 2 + Çv = 4.7273,
and 2 + Çå + 4vs = 8.0331. Hence, Pr (Y > 4) = Ðã(Ê > 0) = 1 —0.20 = 0.80
14 . Use formul a (4.4.8) wi t h r (t ) = 1 i f 0 ( t ( 1 and r (t ) = 2 for t >
1. Int egrat ion by par ts appl ied Ñî E(Y ) = Ä r (t )(1 — 0.05t )dt wit h w (t ) =
Ä r (s)ds yields E (Y ) = 0.05 Ä ø (Ô)é = 19.025. Al t ernat ively, t he annuity
can be viewed as t he sum of t wo annuit ies each having const ant ãàÑå of payment
of 1 per year . T he fi rst begins paying at age 80, t he second at age 81. Usi ng
t his approach we have E (Y ) = asia+ vpspasi = Å(Ò(80) ) + 0.95Å(Ò(81)) where
Ò(80) and Ò(81) are uni formly dist ri but ed on (0, 20) and (Î , 19), respect i vely.
Again we get E(Y ) = 10 + 0.95(9.5) = 19.025.
15 . Consi der t he sum of two annuit ies approach, as i n exercise 14: E (Y ) =
~80:ß + iipsOGsi ,g
p = E [min(T (80), 5)] + 0.95E[min(T (81),4)] = (2.5)(0.25) +
(5) (0.75) + (0.95) [(2) (4/ 19) + 4(15/ 19)] = 7.775.
16 . Since 6 = Î , 6 = Var (T ) = E(T ã) — (Å(Ò)) . Also E (T ã) = ]
Ñãä(é)é =
2 f p é(1 —G(t ))dt = 2ä by parts int egrat ion. Hence, Å(Ò) = ~/ Ãä~
17. (Ð à)òå,~i~~ = Ð ä ( 10Ôòî —ßò~ + Ssi ) = 42.09.
18 .
à 1] S — à ~] S + i + à ò] ß ».ã
èß —(1 + ý)S ».i + ß ».ã
D
~ ~~~è
N z+ 1
T he formula (À .4.6) Ñ„ = èÐö — D Ä».i , summed over y running from x t o t he
end of t he t able, gi ves Ì
= èÔ —N ».i , from which we see t he simpli fi cat ion
Ñî À .
19 . Let Z = å ~+ and Y = à ~] = á- ' ( 1—Z ) . From t he given dat a, we fi nd t hat
E (Z ) = 1 —106 E(Z ÿ) = 1 — 14.756, and 50 = Var (Y ) = 6 ã (E( Z ã) —E (Z )ã) .
First sol ve for á = 3.5%. T hen A = 1 —áà = 0.65.
20. Apply formula (4.8.9) to obt ain Ass.ò~ — 0.17509. Apply formula (3.3.5) Ñî
obt ai n A ss,òç = 0.18046. T hen àç5 zs = (1 — Ass.zs)/ 6 = 16.79725.
D .4 .2
S o l u t i o n s t o Sp r ea d sh eet E x er c i se s
1. Set up your spreadsheet t o calcul at e t he required annuity val ue wit h reference
t o à single age and int erest r ate. Use VL OOK UP() references to t he mor t al ity
182
A P P E N D I X Î . SOL UT I ON S
values, which may be on à separate sheet . T hen use t he Dat a Table feat ure t o
calculat e t he array of values for x running down à column and i accross à row.
Check values: à~~ = 18.058 at i = 5% and à~~ = 13.753 at i = 7.5%.
2 . T he expect ed market val ue is 309,153.
3 . Guide: Set up à t able t o cal cul at e A and E [Z s] wit h à reference t o à single
value of c. Use formula (3.2.4). T hen use t he Table feat ure t o allow for different
values of c.
4 . For i = 5%, as s — 18.3831 and A~~ s = 0.11255. For i = 6%, àäà ,~ —
15.37108 and À çî .à~ = 0 10369.
5 . Gui de: From t he Il lustrat ive Li fe Table set up à t able wit h t he cummul at ive distr ibut ion funct ion of Ê . Fill à column wit h 200 ran dom numbers from
t he int erval [Î , 1] using RAND (). Use t he VL OOK U P() funct ion t o fi nd t he
correspondi ng val ue of Ê . Evaluat e Ó for each value of Ê , t hen calculat e t he
sample mean and var iance using t he built -in funct ions. T he t heoret ical answers
are abc = 16.632 and Var (Y ) = 10.65022.
D
D.5..5 NETNPREM
et PI rUM
em
S:S0
ium
L UÒ10
s: Solut
ÈS
ions
D .5.1
18 3
T heor y Ex er ci ses
1. Use formula (5.3.15): Ðãü,.~ö — Ð '
rel at ion ãî Ðãü = Ð
~p + Ð
Ð
'
„ + Ð
> ~ — 0.064. Now use t he
~ Àùü = 0.046 Ñî obt ain
= (0.064 —0.046)/ (1 —0.64) = 0.05.
T herefore P '
= 0.064 —0.05 = 0.014.
ãü:Ù
2 . L' = è~ã+ ~ —Ga
= - G/ d + (1 + G/ d)v + + ~ and hence
Ê + 1[
Var [L ' ] = ( 1 1 ~ / ô ~Var [va +
Si m i l ar l y ,
an d
L = è~ +
—Ð à
= (ñ~.+ Ñ )ã(Ã ã× àã[è~ +| ].
+ , [ — —Ð / é + ( 1 + Ð / ô è
+
Var [L ] = (d + P ) dà Var [v + ] = 0.30.
Now use E [L ] = 0 and E[L ' ] = —0.20 t o fi nd t hat 0 = À —P a = 1 —(d+ P )a
and —0.2 = 1 — (d + G)a . Hence
Var [L ' ]
=
(d + Ñ )) é ~× àã[è~ + | ] = 0.30(d + G) / (d + P )~
0.30[(d + G)/ (d + P~)]~ = 0.432.
3 . L et Ð d enot e t he net annu al p r em iu m .
5 000( 10Àãü + ü]Àãü + ö)]À ãü + 1ü[À ãü + ãî [Àãü + ãü]Àãü
àãü . ÃÎ ]
5.000( 10Ì ãü + M sp + Ì çü + Ì 4î + Ì 4ü + Ì üî
N~s —Nss
1012.33.
L oss À
=
4v+ + —0.18à +
Ê + 1!
—0.18/ d + (4 + 0.18/ 4)è~ + ' = - 2.25 + 6.25è~ +
Using t he t able we fi nd t hat Var [Loss À] = 3.25 = (6.25) × àã[è~ ~ ~]. Similàãló, Loss  = 6èê + ' — 0.226 + ] — —2.75 + 8.75ÄK+ i and Var [Loss Â] =
(8.75)ã× àã[è~ + ' ] = (8.75)ã(3.25)/ (6.25)ã = 6.37.
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