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11

a) On the actual/actual basis?

b) On the 30/360 basis?

6. A sum of $10,000 is invested for the months of July and August at 6% simple interest. Find the amount of interest earned:

a) Assuming exact simple interest.

b) Assuming ordinary simple interest.

c) Assuming the Bankers Rule.

7. a) Show that the Bankers Rule is always more favorable to the lender than is

exact simple interest.

b) Show that the Bankers Rule is usually more favorable to the lender than is ordinary simple interest.

c) Find a counterexample in ) for which the opposite relationship holds.

2.5 Equations of value

In return for payments of $2000 at the end of four years and $5000 at the end of ten n investor agrees to pay $3000 immediately and to make an additional at the enH nf thiw. vpflrs Find the amount of the additional payment

years, an ,

payment at the end of three years, if = .06.

9. At a certain interest rate the present value of the following two payment patterns are equal:

(0 $200 at the end of 5 years plus $500 at the end of 10 years.

(h) $400.94 at the end of 5 years. At the same iiiterest rate $100 invested now plus $120 invested at the end of 5 years will accumulate to P at the end of 10 years. Calculate P.

10. An investor makes three deposits into a fiind, at the end of 1, 3, and 5 years. The amount of the deposit at time t is 100(1.025). Find the size of the ftind at the end of 7 years, if the nominal rate of discount convertible quarterly is 4/41.

11. Whereas the choice of a comparison date has no effect on the answer obtained with compound interest, the same cannot be said of simple interest. Find the amount to be paid at the end of 10 years which is equivalent to two payments of $100 each, the first to be paid immediately and the second to be paid at the end of 5 years. Assume 5% simple interest is earned from the date each payment is made and use a comparison date of:

a) The end of 10 years.

b) The end of 15 years.

2.6 Unknown time

12. Solve formula (2.8) exacdy.

13 Find how long $1000 should be left to accumulate at 6% effective in order that it will amount to twice the accumulated value of another $1000 deposited at the same time at 4% effective.

14 The present value of two payments of $100 each to be made at the end of n years " and In years is $100. If / = .08, find n.

15. A payment of n is made at the end of n years, In at the end of In years.....nat

the end of /i years. Find the value of t by the method of equated time.

16. You are asked to develop a rule of n to approximate how long it takes money to triple. Find n.

17. A loan is negotiated with the lender agreeing to accept $1000 after 10 years, $2000 t; after 20 years, and $3000 after 30 years in fiiU repayment. The borrower wishes J.. to liquidate the loan with a single $6000 payment. Let , represent the time of the

$6000 payment calculated by an equation of value. Let 2 represent the time determined by the method of equated time. If / = .01, find T - Tj to the nearest .10. Use linear interpolation in the tables.

18. Fund A accumulates at a rate of 12% convertible monthly. Fund accumulates - with a force of interest 5, = r/6. At time t = 0 equal deposits are made in each fl- fund. Find the next time that the two fiinds are equal.

2.7 Unknown rate of interest

19. Find the nominal late of interest convertible seminannually at which the accumulated value of $1000 at the end of 15 years is $3000.

21.

Ifh.

Find an expression for the exact effective rate of interest at which payments of $300 at the present, $200 at the end of one year, and $100 at the end of two years will accumulate to $700 at the end of two years.

It is known that an investment of $1000 will accumulate to $1825 at the end of 10 years. If it is assumed that the investment earns simple interest at rate i during the 1st year, li during the 2nd year,. . . , lOi during the 10th year, find i.

It is known that an amoimt of money will double itself in 10 years at a varying force of interest 5, = . Find an expression for k.

The sum of the accumulated value of 1 at the end of three years at a certain effective rate of interest, and the present value of 1 to be paid at the end of three years at an effective rate of discount numerically equal to / is 2.0096. Find the rate.

Rework Examples 2.9 and 2.10 using the following formula to generate successive values:

/(/,)

+1= -



2.8 Practical examples

A bill for $100 is purchased for $96 three months before it is due. Find:

a) The nominal rate of discount convertible quarterly earned by the purchaser.

b) The annual effective rate of interest earned by the purchaser.

A two-year certificate of deposit pays an annual effective rate of 9%. The purchaser is offered two options for prepayment penalties in the event of early withdrawal:

A - a reduction in the rate of interest to 7%.

- loss of three months interest. In order to assist the purchaser in deciding which option to select, compute the ratio of the proceeds under Option A to those under Option if the certificate of deposit is surrendered:

a) At the end of 6 months.

b) At the end of 18 months.

A savings and loan association pays 7% effective on deposits at the end of each year. At the end of every three years a 2% bonus is paid on the balance at that time. Find the effective rate of interest earned by an investor if the money is left on deposit:

a) Two years.

b) Three years.

c) Four years.

A bank offers the following certificates of deposit:

Term in years

Nominal annual interest rate ("convertible semiannually)

2 6%

3 7%

4 8%

The bank does not permit early withdrawal. The certificates manire at die end of die term. During the next six years die bank will continue to offer these certificates of deposit. An investor deposits $1000 in the bank. Calculate the maximum amount that can be withdrawn at the end of six years.

Miscellaneous problems

A manufacturer sells a product to a retailer who has the option of paying 30% below the retail price immediately, or 25% below the retail price m six months. Find the annual effective rate of interest at which the retailer would be indifferent between the two options.

Show that the answer obtained in Example 2.10 is obtained as j\ after only one iteration using the answer obtained in Example 2.9 as the starting value Jq.

If an invesUnent will be doubled in 8 years at a force of interest 6, in how many years will an invesUnent be tripled at a nominal rate of interest numerically equal to 5 and convertible once every three years?

Fund A accumulates at 6% effective and Fund accumulates at 8% effective. At the end of 20 years the total of the two funds is $2000. At the end of 10 years the amount in Fund A is half that in Fund B. What is tlie total of the two funds at the end of 5 years?

An investor deposits $10,000 in a bank. During the first year, the bank credits an annual effective rate of interest /. During the second year, the bank credits an annual effective rate of interest / - .05. At the end of two years the account balance is $12,093.75. What would the account balance have been at the end of three years, if the annual effective rate of interest were ; + .09 for each of the three years?

A signs a one-year note for $1000 and receives $920 from the bank. At the end of sbt months A makes a payment of $288. Assuming simple discount, to what amount does this reduce the face amount of the note?



Basic annuities

3.1 INTRODUCTION

An annuity may be defined as a series of payments made at equal intervals of time. Annuities are common in our economic life. House rents, mortgage payments, installment payments on automobiles, and interest payments on money invested are all examples of annuities. Originally the meaning of the word "annuity" was restricted to annual payments, but it has been extended to include payments made at other regular intervals as well.

Consider an annuity such that payments are certain to be made for a fixed period of time. An annuity with these properties is called an annuity-certain. The fixed period of time for which payments are made is called the term of the annuity. For example, mortgage payments on a home or business constitute an annuity-certain.

Not all annuities are annuities-certain. An annuity under which the payments are not certain to be made is called a contingent annuity. A common type of contingent annuity is one in which payments are made only if a person is alive. Such an annuity is called a life annuity. For example, monthly retirement benefits from a pension plan, which continue for the life of a retiree, constitute a life annuity.

In this book, we will largely restrict our attention to annuities-certain. However, the risk of default or non-payment is addressed in Section 9.5. It will often be convenient to drop the word "certain" and use the term "annuity" to refer to an annuity-certain.

The interval between annuity payments is called the payment period. In this chapter, we consider annuities for which the payment period and the interest conversion period are equal and coincide, and for which the payments are of

Basic annuities 59

level amount. Since the payment period and the interest conversion period are equal and coincide, we will just use the term "period" for both. In Chapter 4, annuities for which payments are made more or less frequently than interest is converted and armuities with varying payments are examined.

3.2 ANNUITY-IMMEDIATE

Consider an aimuity under which payments of 1 are made at the end of each period for n periods. Such an annuity is called an annuity-immediate. Figure 3.1 is a time diagram for such an annuity. Arrow 1 appears one period before the first payment is made. We assume that the rate of interest is i per period. The present value of the annuity at this point in time is denoted by a-. Arrow 2 appears n periods after arrow 1, just after the last payment is made. The accumulated value of the annuity at this point in time is denoted by s.

Figure 3.1 Time diagram for an annuity-immediate

We can derive an expression for as an equation of value at the beginning of the first period. The present value of a payment of 1 made at the end of the first period is v. The present value of a payment of 1 made at the end of the second period is v. This process is continued until the present "value of a payment of 1 made at the end of the rath period is v". The total present value must equal the sum of the present values of each payment,

This formula could be used to evaluate , for large n. It is possible to derive a more compact expression by recognizing Jhat formula (3.1) is a geometric progression

-hv". (3.1)

but it would become inefficient

= v + v

+ v"-! -I- v"

= V.

= V.

1 -v" 1 -V 1 -v«

1 -v" i

(3.2)



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