LOGO
MATH 2040
Introduction to
Mathematical Finance
Instructor: Miss Liu Youmei
1
Chapter 4 More General Annuities
Introduction
Annuities with payment period
different from interest conversion period
Annuities payable less frequently than
interest is convertible
Annuities payable more frequently than
interest is convertible
Continuous annuities
Payments varying in arithmetic progression
Payments varying in geometric progression
2
Introduction
• In the previous Chapter, annuities were described as
a series of level payments made at the same
frequency as what the interest rate was being
converted at.
• In this chapter, non-level payments are examined as
well as the case where the interest conversion
period and the payment frequency no longer
coincide.
3
Annuities with payment period different from
interest conversion period
• Let the payments remain level for the time being.
• Suppose the interest conversion period does not
coincide with the payment frequency,
• Sometimes we can take the given rate of interest and
convert it to an interest rate that does coincide.
4
Example 4.1
• Find the accumulated value in 10 years if
monthly payments of 100, starting now, are
being made into a fund that credits a nominal rate
of interest at 10%, convertible semi-annually.
• Let j be the month interest rate.
• Then (1 + j)6 = 1.05, or j = 0.8165%
• Now future value 10 year from now is
5
Example 4.2
• Find the accumulated value in 10 years if
annual payments of 100, starting now, are
being made into a fund that credits a nominal
rate of interest at 10%, convertible semiannually.
• Let i be the annual interest rate.
• Then
• Future value 10 years from now is
6
Annuities payable less frequently than interest is
convertible
• Suppose an annuity pays 1 at the end of each k interest
conversion period for a total of n payments.
• The present value of these n payments are :
The accumulated value of this annuity immediately after the
last payment is
7
Annuities due payable less frequently than interest
is convertible
• Suppose an annuity pays 1 at the beginning of each k
interest conversion period for a total of n payments.
• The present value of these n payments are :
The accumulated value of this annuity k interest conversion periods
after the last payment is
8
Example 4.3
• Find the accumulated value in 10 years if annual
payments of 100, starting now, are being made into a
fund that credits a nominal rate of interest at 10%,
convertible semi-annually. Using the approach
developed in last two slides.
• It is an annuity pays 100 at the beginning of each 2
interest conversion periods for a total 10 payments.
• So the future value in 10 years is
9
Annuities-immediate payable more frequently
than interest is convertible
• Let m be the number of payment periods in one interest
conversion period, let n be the term of the annuity measured
in interest conversion periods. And let i be the interest rate
per interest conversion period.
• The number of annuity payments made is mn
• The present value of an annuity which pays 1/m at the end
of each mth of an interest conversion period for a total of n
interest conversion periods is denoted by
.
• For
, the total amount paid during each interest
conversion period is 1 .
10
Annuities-immediate payable more frequently
than interest is convertible
1
2
1
m
n
1
n
1 m
m
 [v  v  ...  v m  v n ]
m
1 v v
 [
1
m
1 v m
1 vn
 (m)
i
1
m
]
The accumulated value of this annuity immediately after the last payment
11
Annuities-due payable more frequently
than interest is convertible
• The present value of an annuity which pays 1/m at the
beginning of each mth of an interest conversion period for a
total of n interest conversion periods denoted by
The accumulated value of this annuity one mth of an interest
conversion period after the last payment is
12
Example 4.4
• Find the accumulated value in 10 years if
monthly payments of 100, starting now, are
being made into a fund that credits a nominal rate
of interest at 10%, convertible semi-annually.
Using the approach developed in last two slides.
13
Example 4.5
• At what annual effective rate of interest is the
present value of a series of payments of $1
every six months forever, with the first payment
made immediately, equal to $10?
• Solution
• The equation of value is
10  1  v.5  v  v1.5  ... 
• Thus,
1
1  v1.5
1
.9
1 2
i  ( )  1, i  .2346
.9
v.5  .9  (1  i ).5 
• Which gives
14
Continuous annuities
• A special case of annuities payable more frequently
than interest is convertible is one in which the
frequency of payment become infinite, i.e., payments
are made continuously.
• We will denote the present value of an annuity payable
continuously for n interest conversion periods, such that
the total amount paid during each interest conversion
period is 1, by the symbol
.
• An expression for
is
15
Continuous annuities
t
n
v
1

v
t
n
  v dt 
|0 
ln v

0
n
• It could be obtained as follows:
• or
16
Continuous annuities
• The accumulated value of a continuous annuity at
the end of the term of the annuity is denoted
by
.
17
Continuous annuities
• Alternatively,
• Consider an investment fund in which money is
continuously being deposited at the rate of 1 per interest
conversion period. The fund balance at time t is equal to
• The fund balance is changing instantaneously for two
reasons. First, new deposits are occurring at the constant
rate of 1 per interest conversion period and interest is
being earned at force  on the fund balance
18
Continuous annuities
• Similarly, we have
• Finally, we can express the value of continuous
annuities strictly in terms of the force of interest  .
•
and
19
Example 4.6
• Find the force of interest at which
• Using the above formula,
e 20  1 e10  1
3
or


e 20  3e10  2  0
(e10  2)(e10  1)  0
ln 2
10
 e  2,  
 .0693
10
20
Payments varying in arithmetic progression
• We now consider annuities with varying payments.
In particularly, there are two types of commonlyencountered varying annuities for which relatively
simple expressions can be developed.
①Annuities with payments varying in arithmetic
progression;
② Annuities with payments varying in geometric
progression;
21
Payments varying in arithmetic progression
• Consider a general annuity-immediate with a term of n
periods in which payments begin at P and increase by Q
per period thereafter. The interest rate is i per period.
• P is positive and Q can be either positive or negative
and P+(n-1)Q>0.
22
Payments varying in arithmetic progression
• Let A be the present value of the annuity
A  Pv  ( P  Q)v 2  ( P  2Q)v3  ...
 [ P  (n  2)Q]v n1  [ P  (n  1)Q]v n
• Similarly, (1  i) A  P  ( P  Q)v  ( P  2Q)v 2  ...
 [ P  (n  2)Q]v n2  [ P  (n  1)Q]v n1
23
Payments varying in arithmetic progression
• Subtract the first equation from the second one:
iA  P  Q(v  v 2  ...  v n 1 )  Pv n  (n  1)Qv n
 P(1  v )  Q(v  v  ...  v
Thus
n
2
n 1
 v )  Qnv
n
n
• The accumulated value is given by:
24
Two special cases
• The first of these is the increasing annuity in which
P=Q=1,denoted by
• The accumulated value of this annuity, denoted by
25
Increasing Annuity
• The alternative solution is to consider it to be the
summation of a series of level deferred annuities.
26
Decreasing Annuity
• The second of these is the decreasing annuity in which
P=n, Q=-1,denoted by
• The accumulated value of this annuity, denoted by
27
Decreasing Annuity
• The alternative solution is to consider it to be the
summation of a series of level annuities.
28
Annuity-Due
• The similar results could be obtained by replacing i into d.
• Also, it is possible to have varying perpetuities. We can
find the general form for a perpetuity-immediate by taking
the limit, obtaining
• since
29
Payments varying in geometric progression
• Consider an annuity-immediate with a term of n
periods in which the first payment is 1 and
successive payments increase in geometric
procession with common ratio 1+ k.
• The present value of this annuity is
v  v (1  k ) 
2
 v (1  k )
n
n 1
30
Payments varying in geometric progression
• This is a geometric progression whose sum is
n
  1  k n 
 1 k 
1  
  1 

1 i  
1 i 



v

  1 k  
ik
 1  1 i  

 
• If k=i, then the above formula is undefined.
• However, then the present value is just nv.
31
Payments varying in geometric progression
• This is a geometric progression whose sum is
n
  1  k n 
 1 k 
1

1

 
 


1

i
1

i
  

v 
  1 k  
ik
 1  1 i  

 
• If k=i, then the above formula is undefined.
• However, then the present value is just nv.
32
Example
• A perpetuity-due makes annual payments which
begin at $100 for the first year, then increase at 6%
per year through the 10th year, and then remain level
thereafter. Calculate the present value of this
perpetuity, if the annual effective rate of interest is
equal to 8%.
33
Example
• The present value of the first 10 payments is
 1.06
100 1 

 1.08
  1.06 10 
9
1  
 

1.08
 1.06 
   920.65
 

   100 
1.06 
 1.08  
1



1.08


• Note that the 10th annual payment is made at time t=9.
• The present value of the rest of the payments is
 1
1
100(1.06)9 


10
11
(1.08)
 (1.08)

1
 1.06   1

100





2
 1.08  1.08 (1.08)




9
 1.06  1
 100 
 1056.45

 1.08  .08
34
Example
• Thus, the total present value equals
• 920.65+1056.45=$1977.10
35