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## Please show all work. What should be the amounts of these payments if money is worth 4% compounded quarterly?

Steve Roper owes Phil Corrigan the following: $750 due in 3 years without interest,$800 due in 3.5 years with interest at 5% compounded quarterly, $500 due in 5 years with at 3.5% compounded semiannually. They agree to settle the debt by having Steve pay Phil$600 in 1 year and the remainder in two equal payments - one in 3 years and the other in 8 years. What should be the amounts of these payments if the money is worth 4% compounded quarterly?

## Answers (1)

• Answered by Anonymous 8 hours later
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I hope this might help you out.

EXAMPLE 1 Computing the Future Value of an Ordinary Annuity

Suppose \$1000 is invested in a savings plan at the end of each year and that 8% interest is paid, compounded annually. How much will be in the account after 4 years?

Solution To find the value of the annuity after 4 years, we compute the value of each of the four payments after 4 years and then find the sum of these payments. First, $1000 deposited at the end of the first year will be earning interest for 3 years. Thus the$1000 will be worth

Similarly, \$1000 deposited at the end of the second year will be earning interest for 2 years. Thus it will be worth

Continuing, we see that, after the fourth year, the \$1000 invested at the end of the third year will be worth

Finally, the $1000 invested at the end of the fourth year will not earn any interest and so will be worth$1000. Summing, we have

We illustrate this in Figure 1.

WARNING

In an ordinary annuity, payments are made at the end of each payment interval. This is a common practice as, for example, the first car or mortgage payment is usually due one month (or more) after the car or house purchase. However, in other cases, payments are made at the beginning of a period. This type of annuity is called an annuity due and is discussed on page 582. We stress that formulas (1), (2), (3), and (6), which we are about to develop, are valid only for ordinary annuities.

As Example 1 suggests, computing the future value of an annuity by calculating the future value of each payment separately can be a tedious affair. Imagine trying to compute the future value of an annuity consisting of 360 payments (as in a 30-year mortgage). Fortunately, there is a much easier way to do it

Suppose that B dollars is deposited or received at the end of each payment period. An interest rate of i is paid at the same time. Let us compute the amount in the account or future value (FV) after n payment periods:

After 1 period We deposit B dollars.

After 2 periods The first B dollars is now worth dollars, and we
deposit another B dollars to obtain

After 3 periods The dollars is now worth

and we deposit another B dollars, so

Continuing, we find that

The expression in brackets can be written more succinctly. As we show at the end of this section,

Thus we have the following.

FUTURE VALUE OF AN ANNUITY: COMPOUNDING
ONCE EACH PAYMENT PERIOD

The future value. FV, of an ordinary annuity with interest compounded at the end of each payment interval, is given by

where
B = amount deposited at the end of each payment period
i = interest rate per payment period
n = number of payment periods

EXAMPLE 1 (continued) Using Formula (1)

We can solve Example 1 much more quickly if we use equation (1). We have

If interest is compounded m times a year, a formula similar to (1) holds.

FUTURE VALUE OF AN ANNUITY: COMPOUNDING
m TIMES EACH PAYMENT PERIOD

The future value FV of an annuity, with interest compounded m times each payment period, is

where B, i, and n are as before.

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