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52. a) (1) Show that . = -v(/a)

(2) Find J. - evaluated at i = 0.

b) (1) Show diat = -v(/a)jj,

(2) Find evaluated at / = 0.

53. d) Show diat -

1-i 2m

= l+i. 2m



57. Show tiiat

58. There are two perpetuities. The first has level payments of p at die end of each year. The second is increasing such that the payments are q, 2q, 3q,. . . Find the rate of interest which will make the difference in present value between these perpetuities:

a) Zero.

b) A maximum.

59. Fence posts set in soil last 9 years and cost $2. Posts set in concrete last 15 years and cost $2 + X. The posts will be needed for 35 years. Show that the break-even value of X, i.e. the value at which a buyer would be indifferent between the two types of posts is


60. If al ~ " and = b, express (/a) in terms of a and

b) Derive the following approximate equaUty:

(m) .

+ £Lli(i-v«).

54. For a given n, it is known that = n - 4 and 5 = 10%. Find dt.

55. Find expressions for:

a) E («)71

b) (o«)7i

4.8 Continuous varying annuities

46. Find the ratio of the total payments made under {1 ) during the second half of the term of the annuity to those made during the first half.

47. Evaluate (1 ) if 5 = .08.

48. Payments under a continuous peipetuity are made at the periodic rate of (1 + k) at time /. The annual effective rate of interest is /, where Q < < i. Find the present value of the perpetuity.

49. a) Find an integral expression for { > ) .

b) Find an expression not involving integrals for ( ) .

50. A perpetuity is payable continuously at the annual rate of 1 + time r. If 5 = .05, find the present value of the perpetuity.

51. A one-year deferred continuous varying annuity is payable for 13 years. The rate of payment at time is f. j per annum, and the force of interest at time t is (1+0 Find the present value of the annuity.

Miscellaneous problems

56. A family wishes to provide an annuity of $100 at the end of each month to their daughter now entering college. The annuity will be paid for only nine months each year for four years. Prove that the present value one month before the first payment

Yield rates


In Chapter 5 some important extensions of results covered in the earlier chapters are developed. These extensions involve concepts and techniques that are widely used in financial calculations in practice.

The technique of discounted cash flow analysis and the concept of yield rate are developed. The uniqueness of the yield rate is then considered. The use of these devices as tools for making financial decisions is stressed throughout.

Different techniques for measuring the interest return on an investment fund are developed and compared. The ramifications involved in the situation in which interest rates at the time of reinvestment differ from those prevailing at the time of initial investment are explored. Also developed is a generalization in which varying interest rates are a fiinction of both die date of original investment and the time since investment. Finally, more general borrowing/lending models are considered for certain complex financial transactions.

The reader will find that this chapter presents a number of important results for using the theory of interest in more complex "real-world" contexts than considered in the earlier chapters. Also, Chapter 5 extends the applications of the theory of interest beyond borrowing and lending transactions to a broader range of business and financial transactions.

It is important to note that the effect of taxes is largely ignored throughout this book. The tax implications involved in business and financial transactions of the type considered herein vary considerably depending upon the nature of the transaction and the political jurisdictions involved. Nevertheless, tax considerations are quite important in practical applications of the theory of interest. It is often important to know whether a particular financial calculation

involving interest is determined on a "before-tax" or an "after-tax" basis. For example, if interest income is subject to taxation, "after-tax" rates will be significantiy lower than "before-tax" rates. However, the principles contained in this book can be applied for calculations on either basis.

Another factor which may have a significant impact on financial calculations of the type discussed in this book is the treatment of expenses. Some rates of interest are quoted net of expenses and others are not. In the latter case, the net return to an investor will be reduced by the amount of expenses incurred. An example would be commissions and other fees involved in buying and selling securities.


In Chapters 3 and 4 we analyzed the present value of certain types of annuities consisting of a regular series of payments. This approach can be generalized to any pattern of payments and is termed discounted cash flow analysis.

Consider a situation in which an investor makes deposits or contributions into an investment of Cq, C, Cj, . . . ,C at times 0, 1, 2,. . . , n. For convenience, we assume that these times are equally spaced. If C, > 0, then there is a net cash flow into the investment at time /; while if C, < 0 , there is a net cash flow out of the investment.

Sometimes it is more convenient to analyze a financial transaction in terms of withdrawals or returns from the investment rather than deposits or contributions into the investment. Thus, we can denote the returns as Rq, R, R2, . . . , Rji at times 0, 1, 2, . . . , n. It is obvious that contributions and returns are equivalent concepts viewed from opposite sides of the transaction. Thus, we have

-C, for r = 0,1,2.....n. (5.1)

Aldiough it is not necessary to define and Rj separately in this fashion, we will find it convenient to have both symbols available in developing certain formulas in this chapter.

It may happen that there is both a contribution and return at the same point in time. In this case, the two are offset against each other. For example, if we have a contribution of $5000 at time 5 and also a return of $1 0 at time 5, then C5 = $4000 and 5 = - $4000.

We have chosen the time periods such that the investment begins at time r = 0 and ends at time t = n. Thus, if the investment is positive during this interval, we have Cq > 0 {Rq < 0) and C < 0 ( > 0). However, = -R may be either positive, negative, or zero for r = 1, 2, . . . , n - 1.







- 5,000


- 1,000


- 1,000


- 1,000


- 1,000



















We now address the problem more generally. Assume that the rate of interest per period is /. Then the net present value at rate / of investment returns by the discounted cash flow technique is denoted by P{i) and is given by

Pit) = Ev%. t=0


The value of P(0 can be either positive or negative, depending on i. For the investment project illustrated in Table 5.1, will be positive for "low" values of i and negative for "high" values of i. The positive net returns during the last few years of the investment will dominate the negative net retiarns during the early years when present values are computed at a "low" rate of interest, while the opposite is true at a "high" rate of interest.

A very important special case of formula (5.2) is the one in whichP(0 = 0,


To illustrate these definitions, consider a ten-year investment project in which the investor contributes $10,000 at the beginning of the first year, $5000 at the beginning of the second year, and then incurs maintenance expenses of $1000 at the beginning of each remaining year thereafter. The project is expected to provide an investment return at the end of each year for the last five years of the project, starting at $8000 and increasing $1000 per year thereafter.

Table 5.1 summarizes the cash flows for this investment project. The last column contains values of J?, in order to display returns from the investment. However, the table could have been set up to contain values of C, instead.

Table 5.1 Cash Flows for Investment Project Dlustrated in Section 5.2

Pit) = vR, = 0. (5.3)


The rate of interest / which satisfies formula (5.3) is called the yield rate on the investment. Stated in words:

JJie yield rate is that rate of interest at which the present value of returns from the investment is equal to the present value of contributions into the

V investment.

In the business and finance literature the yield rate is often called the internal rate of return. The terms "yield rate" and "internal rate of return" can be used interchangeably.

Yield rates are not an entirely new concept; we have encountered them fore. The unknown rate of interest problems in Chapters 2, 3 and 4 can be characterized as yield rate problems. For example, we showed that the yield rate on an investment of $16,0( which returns $10( at the end of every jjuarter for five years is 2.22623% per quarter, or a nominal rate of 8.9049% convertible quarterly in Example 3.8.

v In this section so far we have adopted the vantage point of the investor, i.e. lender. However, if we are dealing with two-party transactions, then we could just as easily adopt the vantage point of the borrower. If this is done, then the values of C, and /?, change signs.

However, the value of the yield rate given by formula (5.3) remains unchanged. Thus, the yield rate on a transaction is totally determined by the ,cash flows defined in that transaction and their timing, and is the same from either the borrowers or lenders perspective.

Yield rates are frequentiy used as an index to measure how favorable or unfavorable a particular transaction may be. From the lenders perspective, the higher the yield rate the more favorable the transaction. From the borrowers prspective, the opposite is the case. Although these convenient rules will usually produce reasonable results. Section 5.3, 5.8, and 5.9 contain illustrations in which difficulties arise in using yield rates in this fashion. Sections 5.8 and 5.9 contain a more systematic discussion of methods of comparing different financial transactions.

Yield rates need not be positive. If the yield rate is zero, then the investor (lender) received no return on investment. If the yield rate is negative, then the investor (lender) lost money on the investment. We will assume that such negative yield rates satisfy -1 < i < 0. It is difficult to find any practical interpretation for a situation in which i < -1, i.e. 1 -f i < 0.

Not all transactions are two-party transactions. For example, consider the investment project summarized in Table 5.1. Quite conceivably, the cash

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