math 2040 introduction to mathematical finance

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MATH 2040 Introduction to Mathematical Finance

Instructor: Miss Liu Youmei

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Chapter 4 More General Annuities

Introduction

Annuities payable less frequently than interest is convertible

Annuities with payment period different from interest conversion period

Annuities payable more frequently than interest is convertible

Continuous annuities

Payments varying in arithmetic progression

Payments varying in geometric progression

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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.

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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.

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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

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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 semi-annually.

• Let i be the annual interest rate. • Then• Future value 10 years from now is

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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

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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

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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

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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 .

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Annuities-immediate payable more frequently than interest is convertible

)(

1

11

121

1

]1

[1

]...[1

m

n

m

mn

m

nmn

mm

iv

v

vvm

vvvvm

The accumulated value of this annuity immediately after the last payment

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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

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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.

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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

• Thus,

• Which gives

5.15.15.

11...110v

vvv

9.1)1(9. 5.5. iv

2346.,1)9.1( 2 ii

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• 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

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nn

tnt v

vvdtv

1|

ln 0

0• It could be obtained as follows:

• or

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• The accumulated value of a continuous annuity at the end of the term of the annuity is denoted by .

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• 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

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• Similarly, we have

• Finally, we can express the value of continuous annuities strictly in terms of the force of interest .

• and

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Example 4.6

• Find the force of interest at which

• Using the above formula,

0693.10

2ln,2

0)1)(2(

023

131

10

1010

1020

1020

e

ee

ee

oree

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• We now consider annuities with varying payments. In particularly, there are two types of commonly-encountered 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;

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• 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.

• 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.

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• Let A be the present value of the annuity

• Similarly,

nn vQnPvQnP

vQPvQPPvA

])1([])2([

...)2()(1

32

12

2

])1([])2([

...)2()()1(

nn vQnPvQnP

vQPvQPPAi

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• Subtract the first equation from the second one:

• The accumulated value is given by:

• Subtract the first equation from the second one:

• The accumulated value is given by:

Thus)...()1(

)1()...(12

12

nnnn

nnn

QnvvvvvQvP

QvnPvvvvQPiA

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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

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Increasing Annuity

• The alternative solution is to consider it to be the summation of a series of level deferred annuities.

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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

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Decreasing Annuity

• The alternative solution is to consider it to be the summation of a series of level annuities.

• The alternative solution is to consider it to be the summation of a series of level annuities.

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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

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• 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

2 1(1 ) (1 )n nv v k v k

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• This is a geometric progression whose sum is

• If k=i, then the above formula is undefined.• However, then the present value is just nv.

1 11 11 1111

n nk ki ivk i ki


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• This is a geometric progression whose sum is

• If k=i, then the above formula is undefined.• However, then the present value is just nv.

1 11 11 1111

n nk ki ivk i ki

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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%.

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Example

• The present value of the first 10 payments is

• Note that the 10th annual payment is made at time t=9.• The present value of the rest of the payments is

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91.061

1.06 1.06 1.08100 1 100 920.651.061.08 1.08 11.08

910 11 2

9

1 1 1.06 1 1100(1.06) 100(1.08) (1.08) 1.08 1.08 (1.08)

1.06 1 100 1056.451.08 .08

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Example

• Thus, the total present value equals

• 920.65+1056.45=$1977.10

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