1
Interest Theory
A(t) is the amount function. a(t) is the accumulation function. a(t) =
k at time s ≡
vt =
1
,
a(t)
A(t)
.
A(0)
kA(t)
at time t.
A(s)
t ≥ 0, is called the discount function discount function.
k at time s ≡
kvt
at time t.
vs
Under compound interest vt = (1 + i)−t . i is the annual effective rate of interest. 1 + i
is the one year interest factor. ν = 1 − d is the one year discount factor. d is the
annual rate of discount.
i=
d
1−ν
i
1
=
, d=
= 1 − ν,
= 1 − d = ν, iν = d, 1 = (1 − d)(1 + i).
1−d
ν
i+1
1+i
i(m) is the nominal rate of interest compounded m times a year. d(m) is the nominal
rate of discount compounded m times a year.
µ
¶m
µ
¶−m
i(m)
d(m)
−1
1+i= 1+
= (1 − d) = 1 −
.
m
m
The force of interest is
δt = −
d
d
a0 (t)
d
A0 (t)
ln(vt ) =
ln a(t) =
=
ln A(t) =
.
dt
dt
a(t)
dt
A(t)
vt = e−
Rt
0
δs ds
, a(t) = e
Rt
0
δs ds
.
Annuities
The cashflow, present and future values of an annuity–due with level payments of one
are:
Contributions 1
Time
0
1
1
1
2
···
···
1
0
n−1 n
1 − νn
(1 + i)n − 1
and s̈n|i =
.
d
d
The cashflow, present and future values of an annuity–immediate with level payments
of one:
än|i =
Contributions 0
Time
0
1
1
1
1
2
···
···
1
n
1 − νn
(1 + i)n − 1
and sn|i =
.
i
i
The cashflow and present value of an perpetuity–due with level payments of one are:
an|i =
Contributions 1 1
Time
0 1
1
2
···
···
1
ä∞|i = .
d
The cashflow and present value of a perpetuity–immediate with level payments of one
are:
Contributions 0 1
Time
0 1
1
2
···
···
1
a∞|i = .
i
The cashflow and present value of a geometric annuity–due with first payment of one
are:
Payments
Time
1
0
(1 + r)2
2
1+r
1
(1 + r)n−1
n−1
···
···
and
(Gä)n|i,r = än| i−r .
1+r
The cashflow and present value of a geometric annuity–immediate with first payment
of one are:
Payments 1 1 + r
Time
1
2
(1 + r)2
3
···
···
(1 + r)n−1
and
n
1
a i−r .
1 + r n| 1+r
The cashflow and present value of a geometric perpetuity–due with first payment of
one are:
(Ga)n|i,r =
Payments
Time
1 1+r
0
1
(1 + r)2
2
and
(
(Gä)∞|i,r =
···
···
(1 + r)n−1
n
1+i
i−r
if i > r,
∞
if i ≤ r.
···
···
The cashflow and present value of a geometric perpetuity–immediate with first payment of one are:
2
Payments
Time
1 1+r
1
2
(1 + r)2
3
and
(
(Ga)∞−−|i,r =
(1 + r)n−1
n
···
···
1
i−r
if i > r
∞
if i ≤ r.
···
···
The cashflow, present and future values of a due increasing annuity with first payment
of one are:
Payments
Time
1 2
0 1
3
2
···
···
n
n−1
än|i − nν n
s̈n|i − n
and (I s̈)n|i =
.
d
d
The cashflow, present and future values of an immediate increasing annuity with first
payment of one are:
(Iä)n|i =
Payments
Time
1 2 3
1 2 3
···
···
n
n
än|i − nν n
s̈n|i − n
and (Is)n|i =
.
i
i
The cashflow and present value of an increasing due perpetuity with first payment of
one are:
(Ia)n|i =
Payments
Time
1 2
0 1
3
2
···
···
and (Iä)∞|i = d12 .
The cashflow and present value of an increasing immediate perpetuity with first
payment of one are:
Payments
Time
1 2
1 2
3
3
···
···
and (Ia)∞|i = id1 .
The cashflow, present and future values of a decreasing due annuity with first payment
of one are:
Payments
Time
n n − 1 n − 2 ···
0
1
2
···
1
n−1
n − an|i
n(1 + i)n − sn|i
and (Ds̈)n|i =
.
d
d
The cashflow, present and future values of a decreasing immediate annuity with first
payment of one are:
(Dä)n|i =
3
Payments
Time
n n − 1 n − 2 ···
1
2
3
···
1
n
n − an|i
n(1 + i)n − sn|i
and (Ds)n|i =
.
i
i
The cashflow, present and future values of a due annuity paid m times a year are
(Da)n|i =
Contributions
Time (in years)
1
m
0
1
m
1
m
1
m
2
m
1
m
m
m
···
···
1
m
m+1
m
···
···
···
···
1
m
nm−1
m
0
nm
m
1 − νn
(1 + i)n − 1
(m)
and
s̈
=
.
n|i
d(m)
d(m)
The cashflow, present and future values of an immediate annuity paid m times a year
(m)
än|i =
are
Contributions
Time (in years)
0
0
1
m
1
m
1
m
2
m
···
···
1
m
m
m
1
m
m+1
m
···
···
···
···
1
m
nm
m
1 − νn
(1 + i)n − 1
(m)
= (m) and sn|i =
.
i
i(m)
The present value of a continuous annuity with rate C(t) is
Z t
C(s)v s ds.
(m)
an|i
0
The present value of a continuous annuity with constant unit rate is
Z n
1 − vn
an|i =
v t dt =
.
δ
0
The present value of an annually increasing continuous annuity is
Z n
än|i − nv n
[t + 1]v t dt =
(Iā)n|i =
.
δ
0
The present value of a continuously increasing annuity is
Z n
¡ ¢
ān|i − nv n
t
¯
.
Iā
=
tv
dt
=
n|i
δ
0
The present value of an annually decreasing continuous annuity is
Z n
n − an|i
(Da)n|i =
[n + 1 − t]v t dt =
.
δ
0
The present value of a continuously decreasing continuous annuity with is
Z n
¡ ¢
n − an|i
(n − t)v t dt =
.
Da n|i =
δ
0
4
2
Survival models.
The cumulative distribution function of the r.v. X is FX (x) = P {X ≤ x}, x ∈ R.
The survival function of the nonnegative r.v. X is Sx (x) = s(x) = Pr{X > x}, x ≥ 0.
Rx
R∞
If h ≥ 0 and H(x) = 0 h(t) dt, x ≥ 0, then E[H(X)] = 0 s(t)h(t) dt. In particular,
Z ∞
Z ∞
Z a
p
p−1
E[X] =
s(t) dt, E[X ] =
s(t)pt dt, E[min(X, a)] =
s(t) dt.
0
0
0
If X is a discrete r.v.
E[H(X)] =
∞
X
Pr{X ≥ k}(H(k) − H(k − 1)).
k=1
In particular, for a positive integer a,
∞
∞
a
X
X
X
2
E[X] =
Pr{X ≥ k}, E[X ] =
Pr{X ≥ k}(2k − 1), E[min(X, a)] =
Pr{X ≥ k}.
k=1
k=1
k=1
(x) is called a life–age–x. T (x) = Tx = X − x is the future lifetime of (x).
The survival function of T (x) is t px = s(x+t)
, t ≥ 0. The c.d.f. of T (x) is t qx =
s(x)
t ≥ 0. We have that
t qx
= 1 − t px , px = 1 px , qx = 1 qx , s |t qx = Pr{s < T (x) ≤ s + t} = s px − s+t px = s px · t qx+s ,
m+n px
Pk
s(x)−s(x+t)
,
s(x)
j=1
= m px · n px+m , n px = px px+1 . . . px+n−1 ,
nj px
= n1 px · n2 px+n1 · n3 px+n1 +n2 · · · nk px+Pk−1 nj .
j=1
d
The force of mortality is µ(x) = µx = − dx
ln SX (x) = SfXX (x)
.
(x)
µ Z x
¶
R x+t
SX (x) = exp −
µ(t) dt , t px = e− x µ(y)dy , fT (x) (t) = t px µ(x + t).
0
Z
◦
∞
e0 = E[X] =
◦
ex:n|
Z
◦
∞
t p0 dt, ex = E[T (x)] =
t px dt,
0
0
Z n
◦
◦
◦
= E[min(T (x), n)] =
t px dt, ex = ex:n| + n px ex+n
0
dte is the least integer greater than or equal to t, dte = k if k − 1 < t ≤ k. Kx is the time
interval of death of a life age x. K(x) is the curtate duration of death of a life aged x,
i.e. the number of complete years lived by this life.
Kx = dT (x)e, K(x) = Kx − 1, K(x) = dT (x)e − 1,
∞
∞
X
X
2
ex = E[K(x)] =
p
,
E[(K(x))
]
=
(2k − 1) · k px ,
k x
k=1
ex = px (1 + ex+1 ), ex:n| =
k=1
n
X
k px , e x
◦
◦
◦
= ex:n| + n px ex+n , ex:m+n| = ex:m| + m px ex+m:n| .
k=1
5
For de Moivre’s law:
1
ω−x
1
, SX (x) =
, µ(x) =
, for 0 ≤ x < ω,
ω
ω
ω−x
ω−x−t
t
, t qx =
, 0 ≤ t ≤ ω − x,
t px =
ω−x
ω−x
ω−x
(ω − x)2
ω−x−1
(ω − x)2 − 1
◦
ex =
, Var(T (x)) =
, ex =
, Var(K(x)) =
2
12
2
12
fX (x) =
Under constant force of mortality µ:
SX (x) = e−µx , FX (x) = 1 − e−µx , fX (x) = µe−µx , µ(x) = µ, for x > 0,
s(x + t)
= e−µt ,
t px = Pr{T (x) > t} =
s(x)
1 − e−µn
px (1 − pnx )
px
1 ◦
1
px
◦
ex = , ex:n| =
, Var(T (x)) = 2 , ex = , ex:n| =
, Var(K(x)) = 2 .
µ
µ
µ
qx
qx
qx
3
Life tables.
`x denote the number of individuals alive at age x. The number of individuals which died
between ages x and x + t is t dx = `x − `x+t . The number of individuals which died between
ages x and x + 1 is dx = `x − `x+1 . We have that
`x
`0 − `x
d
, FX (x) =
, µ(x) = − log(`x ),
`0
`0
dx
d
`x+t
`x − `x+t
`x+1
`x − `x+1
dx
`x+n − `x+n+m
t x
, t qx =
=
, px =
, qx =
= , n |m q x =
.
t px =
`x
`x
`x
`x
`x
`x
`x
s(x) =
◦
Z
∞
e0 =
0
`x
◦
dx, ex =
`0
Z
∞
0
`x+t
◦
dt, ex:n| =
`x
Z
n
0
∞
n
X `x+k
X `x+k
`x+t
dt, ex =
, ex:n| =
.
`x
`
`
x
x
k=1
k=1
The expected number of years lived between age x and age x + n by the `x survivors at
age x is n Lx .
Px+n−1
P∞
Z n
Lk
◦
◦
◦
k=x Lk ◦
`x+t dt, Lx = 1 Lx = `x ex:1| , ex =
, ex:n| = k=x
.
n Lx = `x ex:n| =
`x
`x
0
Interpolation
uniform distribution of deaths
`x+t
t px
`x + t(`x+1 − `x ) 1 − tqx
Lx
exponential interpolation
`x ptx
ptx
dx
− log px
Balducci assumption
1
(1−t) `1 +t `
px
t+(1−t)px
−`x+1 log px
qx
x
6
1
x+1
`x +`x+1
2
Under uniform distribution of deaths:
`x+t = `x + t(`x+1 − `x ), t px = 1 − tqx , fT (x) (t) = qx , µx+t =
Lx =
qx
, 0 ≤ t ≤ 1,
1 − tqx
1
`x + `x+1 ◦
, ex = ex + .
2
2
Under exponential interpolation:
`x+t = `x ptx , t px = ptx , fTx (t) = −ptx log px , µx+t = − log px , 0 ≤ t ≤ 1.
Under (Balducci assumption) harmonic interpolation:
1
`x+t
4
= (1 − t)
1
1
px
+t
, t px =
.
`x
`x+1
t + (1 − t)px
Life insurance.
type of insurance
whole life insurance
n–year term life insurance
n–year deferred life insurance
n–year pure endowment life insurance
n–year endowment life insurance
m–year deferred n–year term life insurance
payment
Zx = v Kx
1
= v Kx I(Kx ≤ n)
Zx:n|
Kx
I(n < Kx )
n |Zx = v
n
1
Zx:n| = v I(n < Kx )
Zx:n| = v min(Kx ,n)
Kx
I(m < Kx ≤ m + n)
m |n Z x = v
Whole life insurance paid at the end of the year:
Zx = v
Kx
, Ax = E[Zx ] =
∞
X
v
k
k−1 px
2
· qx+k−1 , Ax =
Var(Zx ) = Ax −
A2x ,
v 2k k−1 px · qx+k−1 ,
k=1
k=1
2
∞
X
Ax = vqx + vpx Ax+1 .
n–year term life insurance paid at the end of the year:
1
Zx:n|
=v
Kx
I(Kx ≤
n), A1x:n|
=
1
E[Zx:n|
]
=
n
X
k
v · k−1 |qx ,
2
A1x:n|
=
k=1
1
)
Var(Zx:n|
=
2
A1x:n|
−
2
A1x:n| ,
A1x:n|
= vqx +
n
X
v 2k · k−1 |qx ,
k=1
vpx A1x+1:n−1| .
n–year deferred life insurance paid at the end of the year:
n |Zx
=v
Kx
I(n < Kx ), n |Ax = E[n |Zx ] =
∞
X
k
v · k−1 |qx ,
2
n |Ax
k=n+1
2
Var(n |Zx ) = 2 n |Ax − n |Ax , n |Ax = v n n−1 px · qx+n−1 + n+1 |Ax .
7
=
∞
X
k=n+1
v 2k · k−1 |qx ,
n–year pure endowment life insurance paid at the end of the year:
1
1
1
Zx:n|
= v n I(n < Kx ), Ax:n|
= E[Zx:n|
] = n Ex = v n · n px ,
2
2
1
1
1
1
.
Ax:n|
= v 2n · n px , Var(Zx:n|
) = 2 Ax:n|
− Ax:n|
n–year endowment life insurance paid at the end of the year:
Zx:n| = v
min(Kx ,n)
, Ax:n| = n Ex = E[Zx:n| ] =
n
X
v k · k−1 |qx + v n n px ,
k=1
2
Ax:n| =
n
X
v 2k · k−1 |qx + v 2n n px , Var(Zx:n| ) = 2 Ax:n| − Ax:n| 2 .
k=1
n |Ax
= n Ex Ax+n , Ax = A1x:n| + n |Ax = A1x:n| + n Ex Ax+n , 2 Ax = 2 A1x:n| + 2 n |Ax ,
1
1
Ax:n| = A1x:n| + Ax:n|
, 2 Ax:n| = 2 A1x:n| + 2 Ax:n|
,
Increasing/decreasing life insurance paid at the end of the year:
(IA)x =
∞
X
k=1
k
kv ·
1
k−1 |qx , (IA)x:n|
=
n
X
k
kv · k−1 |qx ,
k=1
(DA)1x:n|
=
n
X
(n + 1 − k)v k · k−1 |qx .
k=1
Under de Moivre’s model with terminal age ω, if ω, x, n are a positive integers,
aω−x|i
aω−x−n|i
an|
ω−x−n
1
Ax =
, A1x:n| =
, Ax:n|
= vn
, n |Ax = v n
.
ω−x
ω−x
ω−x
ω−x
Under constant force of mortality:
qx
qx
qx
1
Ax =
, Ax:n|
= e−n(µ+δ) , n |Ax = e−n(µ+δ)
, A1x:n| = (1 − e−n(µ+δ) )
.
qx + i
qx + i
qx + i
type of insurance
whole life insurance
n–year term life insurance
n–year deferred life insurance
n–year pure endowment life insurance
n–year endowment life insurance
m–year deferred n-year term life insurance
payment
Z x = v Kx
1
Z x:n| = v Tx I(Tx ≤ n)
Tx
n |Z x = v I(n < Tx )
1
Z x:n| = v n I(n < Tx )
Z x:n| = v min(Tx ,n)
Tx
m |n Z x = v I(m ≤ Tx ≤ m + n)
Whole life insurance paid at the time of death:
Z ∞
Tx
Z x = v Ax = E[Z x ] =
v t fTx (t) dt,
Z ∞ 0
2
2
Ax = E[(Z x )2 ] =
v 2t fTx (t) dt, Var(Z x ) = 2 Ax − Ax .
0
8
n–year term life insurance paid at the time of death:
Z n
1
1
1
Tx
Z x:n| = v I(Tx ≤ n), Ax:n| = E[Z x:n| ] =
v t fTx (t) dt,
0
Z n
2
1
1
1
2
2
Ax:n| = E[Z x:n| ] =
v 2t fTx (t) dt, Var(Z x:n| ) = 2 A1x:n| − A1x:n| .
0
n–year deferred life insurance paid at the time of death:
Z ∞
Tx
v t fTx (t) dt,
n |Z x = v I(n < Tx ), n |Ax = E[n |Z x ] =
n
Z ∞
2
2
2
v 2t fTx (t) dt, Var(n |Z x ) = 2 n |Ax − n |Ax .
n |Ax = E[n |Z x ] =
n
n–year endowment life insurance:
Z n
min(Tx ,n)
Z x:n| = v
, Ax:n| = E[Z x:n| ] =
v t fTx (t) dt + v n Pr{Tx > n},
0
Z n
2
2
Ax:n| = E[(Z x:n| )2 ] =
v 2t fTx (t) dt + v 2n Pr{Tx > n}, Var(Z x:n| ) = 2 Ax:n| − Ax:n| .
0
1
1
1
Z x = Z x:n| + n |Z x , Ax = Ax:n| + n |Ax , 2 Ax = 2 Ax:n| + 2 n |Ax ,
1
1
1
1
1
1
Z x:n| = Z x:n| + Ex:n|
, Ax:n| = Ax:n| + Ax:n|
, 2 Ax:n| = 2 Ax:n| + 2 Ax:n|
,
n |Ax
= n Ex Ax+n .
Under de Moivre’s model with terminal age ω,
Ax =
aω−x|i
aω−x−n|i
an|i
ω−x−n
1
1
, Ax:n| =
, Ax:n| = e−nδ
, n |Ax = e−nδ
.
ω−x
ω−x
ω−x
ω−x
Under constant force of mortality:
Ax =
µ
µ
µ
1
1
, Ax:n| = e−n(µ+δ) , n |Ax = e−n(µ+δ)
, Ax:n| = (1 − e−n(µ+δ) )
.
µ+δ
µ+δ
µ+δ
Continuously increasing whole life insurance: bt = t, t ≥ 0,
Z ∞
¡
¢
I A x=
tv t · t px µx+t dt.
0
Annually increasing whole life insurance: bt = dte, t ≥ 0, present value is denoted
by
¡
∞
X
¢
I A x=
k=1
Z
k
k−1
9
kv t · t px µx+t dt.
n–year term continuously increasing whole life insurance: bt = t, 0 ≤ t ≤ n,
Z n
¢1
¡
I A x:n| =
tv t · t px µx+t dt.
0
n–year term annually increasing whole life insurance: bt = dte, 0 ≤ t ≤ n,
¡
n
X
¢1
I A x:n| =
k=1
Z
k
kv t · t px µx+t dt.
k−1
Continuously decreasing life insurance: bt = n − t, 0 ≤ t ≤ n,
Z n
¡
¢1
(n − t)v t · t px µx+t dt.
D A x:n| =
0
Annually decreasing life insurance: bt = dn − te, 0 ≤ t ≤ n,
n
X
¡
¢1
D A x:n| =
k=1
Z
k
(n + 1 − k)v t · t px µx+t dt.
k−1
Assuming a uniform distribution of deaths:
i
i
i
i
1
1
Ax = Ax , Ax:n| = A1x:n| , n |Ax = · n |Ax , Ax:n| = A1x:n| + Ax:n|
,
δ
δ
δ
δ
(m)
i
i 1
i
i
(m)
(m)
1
1
A(m)
=
A
,
A
= (m) · n |Ax , Ax:n| = (m) A1x:n| + Ax:n|
.
x
x
x:n| = (m) Ax:n| , n |Ax
(m)
i
i
i
i
5
Life annuities.
due annuities
whole life
n–year deferred life insurance
n–year term
present value
x
Ÿx = äKx | = 1−Z
d
n
n |Ÿx = v äKx −n| I(Kx > n)
1−Z
Ÿx:n| = ämin(Kn ,n)| = dx:n|
APV
x
äx = 1−A
d
n |äx = n Ex äx+n
1−A
äx:n| = dx:n|
immediate annuities
whole life
n–year deferred life insurance
present value
x
Yx = aKx −1| = v−Z
d
n
n |Yx = v aKx −n−1| I(Kx > n + 1)
APV
x
ax = v−A
d
n |ax = n Ex · ax+n
n–year term
Yx:n| = amin(Kx −1,n)| =
ax:n| =
v−Zx:n+1|
d
continuous annuities
whole life
n–year deferred life insurance
present value
x
Y x = aTx | = 1−Z
δ
n
n |Y x = v aTx −n| I(Tx > n)
n–year term
Y x:n| = amin(Tx ,n)| =
10
1−v min(Tx ,n)
δ
v−Ax:n+1|
d
APV
x
ax = 1−A
δ
n |ax = n Ex · ax+n
ax:n| =
1−Ax:n|
δ
Discrete whole life due annuity:
∞
Ÿx = äKx |
2
1 − Zx
1 − Ax X k
Ax − A2x
, äx =
=
v k px , Var(Ÿx ) =
, äx = 1 + vpx äx+1 .
=
d
d
d2
k=0
Whole life immediate annuity:
∞
v − Zx
v − Ax X k
= Ÿx − 1 =
, ax =
=
v k px ,
d
d
k=1
Yx = aKx −1|
2
Ax − A2x
, ax = vpx äx+1 = vpx (1 + ax+1 ).
d2
Whole life continuous annuity:
Z ∞
2
2
1 − Zx
1 − Ax
Ax − Ax
t
Y x = aTx | =
, ax =
=
.
v · t px dt, Var(Y x ) =
δ
δ
δ2
0
Var(Yx ) =
n–year deferred discrete due annuity:
n
n |Ÿx = v äKx −n| I(Kx > n), n |äx =
∞
X
v k · k px = n Ex äx+n .
k=n
n–year deferred discrete immediate annuity:
n |Yx
= n+1 |Ÿx , n |ax = n+1 |äx = vpx · n−1 |ax+1 .
n–year deferred continuous annuity:
n |Y x
Z
∞
n
= v aTx −n| I(Tx > n), n |ax =
v t · t px dt = n Ex · ax+n .
n
n–year term due discrete annuity:
n−1
Ÿx:n| = ämin(Kn ,n)|
X
1 − Ax:n|
1 − Zx:n|
v k k px =
=
, äx:n| =
,
d
d
k=0
2
Ax:n| − (Ax:n| )2
, äx:n+m| = äx:n| + n Ex · äx+n:m| ,
d2
äx = äx:n| + n |äx = äx:n| + n Ex äx+n .
Var(Ÿx:n| ) =
n–year term discrete immediate annuity:
v − Zx:n+1|
,
d
n
X
v − Ax:n+1|
= äx:n+1| − 1 =
v k · k px =
d
k=1
Yx:n| = amin(Kx −1,n)| = Ÿx:n+1| − 1 =
ax:n|
2
Ax:n+1| − (Ax:n+1| )2
,
d2
ax = n |ax + ax:n| = n |ax + n Ex ax+n ,
Var(Yx:n| ) =
11
n–year term continuous annuity:
Z n
1 − Ax:n|
1 − Z x:n|
1 − v min(Tx ,n)
Y x:n| = amin(Tx ,n)| =
=
, ax:n| =
v s s px ds =
,
δ
δ
δ
0
2
Ax:n| − (Ax:n| )2
Var(Y x:n| ) =
, ax:n+m| = ax:n| + n Ex · ax+n:m| , ax = ax:n| + n |ax .
δ2
Under constant force of mortality:
1
1+i
vpx
1 − qx
1
1
e−(δ+µ)
äx =
=
, ax =
=
, ax =
=
=
.
−(δ+µ)
−(δ+µ)
1 − vpx
qx + i
1−e
1 − vpx
qx + i
1−e
µ+δ
Annuities paid m times a year.
For a whole life unity annuity–due to (x) paid m times a year:
(m)
Ÿx(m)
(m)
1 − Zx
1 − Ax
=
, ä(m)
=
x
(m)
d
d(m)
∞
1 X k
=
vm ·
m k=0
k
m
px , Var(Ÿx(m) ) =
2
(m)
Ax
(m)
− (Ax )2
.
(d(m) )2
For a whole life unity annuity–immediate to (x) paid m times a year:
(m)
Yx(m)
a(m)
x
1
v 1/m − Zx
=
−
=
,
m
d(m)
∞
(m)
1
v 1/m − Ax
1 X k
(m)
vm ·
= äx −
=
=
m
d(m)
m k=1
Ÿx(m)
Var(Yx(m) )
2
=
k
m
px ,
(m)
(m)
− (Ax )2
.
(d(m) )2
Ax
For a n–year unity annuity–due to (x) paid m times a year:
(m)
(m)
Ÿx:n|
=
1 − Zx:n|
d(m)
(m)
,
(m)
äx:n|
=
1 − Ax:n|
d(m)
=
nm−1
1 X 1
vm ·
m k=0
k
m
px ,
(m)
(m)
Var(Ÿx:n| )
=
Var(Zx:n| )
(d(m) )2
.
For a n–year unity annuity–due to (x) paid m times a year:
(m)
(m)
Ÿx:n|
=
1 − Zx:n|
d(m)
(m)
,
(m)
äx:n|
=
1 − Ax:n|
d(m)
nm−1
1 X 1
=
vm ·
m k=0
(m)
k
m
px ,
(m)
Var(Ÿx:n| )
=
Var(Zx:n| )
(d(m) )2
For a n–year unity annuity–immediate to (x) paid m times a year:
(m)
(m)
Yx:n| = Ÿx:n| −
1
1
1
1
(m)
(m)
1
+ Zx:n|
, ax:n| = äx:n| −
+
· n Ex .
m m
m m
12
.
For a n–year deferred unity annuity–due to (x) paid m times a year:
(m)
(m)
n |Ÿx
=
1
Zx:n|
− n |Zx
d(m)
(m)
(m)
,
(m)
n |äx
=
(m)
1
Ax:n|
− n |Ax
d(m)
∞
1 X k
=
vm ·
m k=nm
k
m
(m)
px = n Ex · äx+n ,
(m)
ä(m)
= äx:n| + n |ä(m)
= äx:n| + n Ex äx+n .
x
x
For a n–year deferred unity annuity–immediate to (x) paid m times a year:
1
1
(m)
1
− n Ex ,
= n Ex · ax+n = n |ä(m)
, n |a(m)
Zx:n|
x
x
m
m
(m)
(m)
(m)
(m)
= ax:n| + n |ax = ax:n| + n Ex ax+n .
= n |Ÿx(m) −
(m)
n |Yx
a(m)
x
Under an uniform distribution of deaths within each year:
ä(m)
x
6
i
i
1 − i(m)
Ax (m)
Ax
v 1/m − i(m)
1 − δi Ax
1
(m)
=
, ax = äx −
=
, ax =
.
d(m)
m
d(m)
δ
Benefit Premiums.
Fully discrete insurance
Whole life insurance:
µ
¶
P
− P äKx | = Zx − P Ÿx = Zx − P Ÿx = Zx
− ,
d
¶
µ
P
P
E[Lx ] = Ax − P äx = Ax 1 +
− ,
d
d
µ
¶2
µ
¶2
¡2
¢
P
P
Var(Lx ) = 1 +
Var(Zx ) = 1 +
Ax − Ax 2 .
d
d
Lx = v
Kx
P
1+
d
Under the equivalence principle:
Ax
dAx
1
=
=
− d,
äx
1 − Ax
äx
2
2
Ax
Ax − Ax 2
Ax − Ax 2
Var(Lx ) =
=
, t Px =
.
2
2
äx:t|
(1 − Ax )
(däx )
Px =
n–year term insurance:
1
1
− P Ÿx:n| = Zx:n|
−P
L1x:n| = Zx:n|
1
Px:n|
= P (A1x:n| ) =
A1x:n|
äx:n|
1 − Zx:n|
,
d
1
, t Px:n|
= P (t A1x:n| ) =
13
A1x:n|
äx:t|
.
n–year pure endowment:
1
1
1
= Zx:n|
− P Ÿx:n| = Zx:n|
−P
Lx:n|
1
Px:n|1 = P (Ax:n|
)=
1
Ax:n|
äx:n|
1 − Zx:n|
,
d
1
, t Px:n|1 = P (t Ax:n|
)=
1
Ax:n|
äx:t|
.
n–year endowment:
¶
µ
1 − Zx:n|
P
P
Zx:n| − ,
Lx:n| = v
− P ämin(Kx ,n)| = Zx:n| − P Ÿx:n| = Zx:n| − P
= 1+
d
d
d
µ
¶2
µ
¶2
¡2
¢
P
P
Var(Lx:n| ) = 1 +
Var(Zx:n| ) = 1 +
Ax:n| − (Ax:n| )2 ,
d
d
Ax:n|
Ax:n|
Px:n| = P (Ax:n| ) =
, t Px:n| = P (t Ax:n| ) =
,
äx:n|
äx:t|
µ
¶
2
¢ 2 Ax:n| − Ax:n| 2
Ax:n| − Ax:n| 2
Px:n| 2 ¡2
Var(Lx:n| ) = 1 +
Ax:n| − Ax:n| 2 = ¡
¢2 =
¡
¢2 .
d
1 − Ax:n|
däx:n|
min(n,Kx )
n–year deferred insurance:
n |Zx
n |Ax
− P Ÿx , P (n |Ax ) =
äx
, t P (n |Ax ) =
n |Ax
äx:t|
.
Properties:
1
1
Px:n| = Px:n|
+ Px:n|1 , n Px = Px:n|
+ Px:n|1 Ax+n .
Semicontinuous annual benefit premiums
Whole life insurance:
P x = P (Ax ) =
Ax
Ax
, t P x = t P (Ax ) =
.
ax
ax:t|
n–year term insurance:
1
P x:n|
=
A1x:n|
ax:n|
,
1
t P x:n|
=
1
t P (Ax:n| )
=
A1x:n|
ax:t|
n–year pure endowment:
1
P x:n|
=
1
P (Ax:n|
)
=
1
Ax:n|
ax:n|
,
1
t P x:n|
14
=
1
t P (Ax:n| )
=
1
Ax:n|
ax:t|
.
n–year endowment:
P x:n| = P (Ax:n| ) =
Ax:n|
Ax:n|
, t P x:n| = t P (Ax:n| ) =
.
ax:n|
ax:t|
n–year deferred insurance:
P (n |Ax ) =
n |Ax
ax:n|
, t P (n |Ax ) =
n |Ax
ax:n|
Fully continuous insurance
Whole life insurance:
¶
µ
1 − Zx
P
P
− ,
L(Ax ) = v − P aTx | = Z x − P Y x = Z x −
= Zx 1 +
δ
δ
δ
¶2
¶2
µ
µ
¡2
¢
P
P
Var(L(Ax )) = 1 +
Var(Z x ) = 1 +
Ax − Ax 2 ,
δ
δ
Ax
1
Ax
δAx
P (Ax ) =
=
=
− δ, t P (Ax ) =
.
ax
ax
ax:t|
1 − Ax
µ
¶2
2
¢ 2 Ax − Ax 2
P (Ax ) ¡2
Ax − Ax 2
=
Var(L(Ax )) = 1 +
Ax − Ax 2 =
.
δ
(δax )2
(1 − Ax )2
Tx
n–year term insurance:
1
L=
1
Z x:n|
− P Y x:n| ,
1
1
P (Ax:n| )
Ax:n|
Ax:n|
1
, t P (Ax:n| ) =
.
=
ax:n|
ax:t|
1
P (Ax:n| )
Ax:n|
Ax:n|
1
=
, t P (Ax:n| ) =
.
ax:n|
ax:t|
n–year pure endowment:
1
L=
1
Z x:n|
− P Y x:n| ,
1
n–year endowment:
µ
¶
1 − Z x:n|
P
P
L = Z x:n| − P Y x:n| = Z x:n| − P
= 1+
− Z x:n| ,
δ
δ
δ
µ
¶³
´
¡
¢2
P
2
Var(L) = 1 +
Ax:n| − Ax:n|
,
δ
2
Ax:n|
Ax:n| − Ax:n| 2
1 − δax:n|
δAx:n|
Ax:n|
P (Ax:n| ) =
, Var(L) = ¡
=
=
.
¢2 , t P (Ax:n| ) =
ax:n|
ax:n|
ax:t|
1 − Ax:n|
1 − Ax:n|
n–year deferred insurance:
L = n |Z x − P Y x:n| , P (n |Ax ) =
15
n |Ax
ax:n|
.
n–year deferred annuities
n–year deferred due annuity:
L = n |Ÿx − P Ÿx:n| , P (n |äx ) =
n |äx
äx:n|
.
n–year deferred immediate annuity:
L = n |Yx − P Ÿx:n| , P (n |ax ) =
n |ax
äx:n|
.
n–year deferred continuous annuity funded discretely:
L = n |Y x − P Ÿx:n| , P (n |ax ) =
n |ax
äx:n|
.
n–year deferred continuous annuity funded continuously:
L = n |Y x − P Y x:n| , P (n |ax ) =
16
n |ax
äx:n|
.