MATH 2040 Introduction to Mathematical Finance Instructor: Miss Liu Youmei

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

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.

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.

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

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

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

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

Example 4.3 So the future value in 10 years is 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

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 .

Annuities-immediate payable more frequently than interest is convertible The accumulated value of this annuity immediately after the last payment

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

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.

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

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

Continuous annuities It could be obtained as follows: or

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

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

Continuous annuities Similarly, we have Finally, we can express the value of continuous annuities strictly in terms of the force of interest . and

Example 4.6 Find the force of interest at which Using the above formula,

Payments varying in arithmetic progression 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. Annuities with payments varying in arithmetic progression; Annuities with payments varying in geometric progression;

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. 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.

Payments varying in arithmetic progression Let A be the present value of the annuity Similarly,

Payments varying in arithmetic progression 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:

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

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

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

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.

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

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

Payments varying in geometric progression 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.

Payments varying in geometric progression 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.

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%.

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

Example Thus, the total present value equals 920.65+1056.45=$1977.10