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Orientation
Preassessment
Essential Concepts
Time Value of Money
Evaluating Cash Flows
Introduction
Challenge
Annuities
Perpetuities
NPV & IRR
Risk and Return
Postassessment


PreMBA Analytical Methods
Evaluating Cash Flows: Annuities

An annuity is a stream or series of identical cash flows occurring at regular intervals through time for a specified number of periods. In Challenge problems A and B, you were asked to solve problems involving annuity cash flows. Challenge A had two separate annuity flows for different time periods. As you will see in the following text, if an annuity cash flow begins after the first period, the present value annuity (PVA) formula discounts the cash flows only to the period when the annuity stream began. It cannot discount all the way back to the present, the end of time period zero, without another present value calculation. Two common examples of annuities are fully amortizing loans, such as mortgages and automobile loans, and the coupon payments on bonds. Can you think of other examples of annuities?

When do bank customers use annuities?
Low bandwith

High bandwith

The time line below depicts the cash flows for a $100 loan repaid in 12 equal installments of $8.79, based on a 10 percent annual rate of interest. The payments represent an annuity.

Annuities are either ordinary or due. Ordinary annuities, the most common form of annuity, have payments that occur at the end of a period. Most loan payments are ordinary annuities, where the first payment includes interest. An annuity due has payments at the beginning of the period. Most leases and rent agreements require payments due at the beginning of the lease or rental contract, and are examples of annuities due.

Future Value of Annuities

After you have properly characterized a set of cash flows as being either an ordinary annuity or an annuity due, you can find the future value or present value of all the cash flows at a single point in time. How would you find the future value of a whole stream of cash flows? You could use the future value formula for a single cash flow for each cash flow and then sum the results to calculate the future value. How would this look? Using the formula for future value of a single payment, let FVN be the future value at the end of period N of a payment, Ct, made at the end of time period t. Then

Annuities appeal to and reward the "slow and steady" investor. Read about how an annuity investment compares with a lump-sum investment.

Negative Compounding

That is, the future value of a cash flow (payment) at the end of period t is compounded N – t times into the future. Use the time line below to verify this. The cash flow at the end of period 1 has three periods to compound (N – t is 4 – 1). Similarly, cash received at the end of period 2 has only two periods to compound (4 – 2), and so with period 4. A cash flow 4 periods into the future is already at its future value.

By extension, for all N periods, the future value of N cash flows is

To compute the future value of monthly payments for 30 years would require you to calculate 360 exponentiations, multiplications, and additons using this formula. Happily, there is an easier way, the formula for the future value of an annuity. How is FVA derived?

Exponents

That's it. Now for any number of time periods, N, you can compute the future value of a set of equal cash flows, C, at a periodic compound rate of r.

Annuities
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Notice that with the formula for FVA, if you know the other variables, you can solve for the cash payment, C, or the term, N. Unfortunately, it is difficult to calculate r directly. How is r computed?

Now, consider the general form of the future value of an annuity due.

Notice that the FVADue equation is just like the FVA equation, except for the extra term (1 + r) on the far right. The term (1 + r) in the future value of an annuity due formula values the cash flow stream of an ordinary annuity one period later. What does this imply?

Present Value of Annuities

What if you wanted to find the present value of an annuity stream? As illustrated in the solution to Challenge A, this can be solved by using the PVA formula. The formula for finding the present value of an annuity will let you find the present value of a stream of cash flows simultaneously.

Start by considering the present value of an ordinary annuity, stated in general form.

As was done above in computing the future value, remember that the present value of a single cash flow t periods into the future at a discounting interest rate of r percent is

For a series of N cash flows, the present value is

Recall from Time Value of Money that the negative exponent in the PV equation is equivalent to the reciprocal of the number raised to the positive of the exponent. In other words,

The purpose of the present value formula is to calculate today's value of a future cash payment. Certainly you would pay less than a dollar today for a dollar in the future. To obtain a future value for a current amount, you multiply the current amount by a factor. Similarly, you can divide a future value by a factor to compute the present value. This is represented mathematically as

Using the same procedure used before to obtain the formula for the future value of an annuity, you derive the following formula for the present value of an annuity.

PVA = present value of an ordinary annuity

C = the fixed annuity payment

r = the periodic interest rate

N = the total number of annuity payments

Solving the PVA equation for C would give you the annual cash payment needed to pay a fully amortizing loan over N periods in the amount of PVA at a periodic interest rate of r.

Example: Ordinary annuity
Your company needs a substantial amount of money today, and your boss has asked you to ascertain how much the company can borrow. You learn that the company's operating cash flow will be sufficient to support quarterly debt service payments of $45,000 for 20 years. The company's bank is willing to offer a 20-year loan requiring quarterly payments and an annual interest rate of 13.5 percent. What is the maximum amount the company can borrow?

Notice this is an annuity, because you plan to make equal periodic debt service payments of $45,000 per quarter for 20 years. For this problem, C is the $45,000 quarterly payment; r is the 13.5 percent annual rate divided by the compounding frequency, m, which is 4, so r = 3.375 percent or .03375. N is equal to the payment frequency, 4, times the number of years, 20, or 80.

Construct a time line. Remember that each payment, a cash outflow, occurs at the end of a period. The first payment occurs at the end of period 1, and the last payment occurs at the end of period 80. The time line would show 80 end-of-period payments of $45,000, occurring one per quarter for 20 years. For reference, the end of the first period is at the mark for time period 1, and the end of the last period is at the mark for time period 79.

Substitute the given values in the formula for present value of an ordinary annuity, and solve.

Therefore, assuming that the company can make quarterly payments of $45,000 each over 20 years, it can borrow $1,239,643 at 13.5 percent compounded quarterly. The first payment will occur one quarter after the company borrows $1,239,643.

As with the FVA problems, you can solve this problem with Excel.

Using either the PVA equation or Excel's built-in PV function, discount all 80 payments to t = 0.

Reconsider the above situation with each $45,000 payment being made at the beginning of each period. This is an example of the present value of an annuity due, denoted as PVADue. The general form of PVADue is

Similar to FVADue and FVA, the PVADue equation is just like the PVA equation except for the extra term (1 + r) on the far right. The term (1 + r) in the PVADue equation serves to value the cash flow stream one period forward, or into the future. Thus, to find the present value of an annuity due, you calculate the present value as if it were an ordinary annuity, then adjust the present value forward one period by the periodic interest rate.

Example: annuity due
Now find the present value of the $45,000 annuity due.

For this calculation, C is the $45,000 quarterly payment; r is the 13.5 percent annual rate divided by the compounding frequency, m, which is 4, or 3.375 percent; and N is equal to the payment frequency, 4, times the number of years, 20, or 80. All the terms are exactly the same as they were for the ordinary annuity calculation.

You may wish to make a time line representing the annuity's cash flows and their timing at the beginning of each period.

Note that the first payment is made at the beginning of the first period, which is the same as the end of time period 0. The last payment is made at the beginning of period 80, which is at the mark for the end of time period 79. The time values of the payments, denoted as cash outflows for the company, all need to be discounted to the beginning of the first period, time period 0.

Insert the given values into the formula for the present value of an annuity due, and solve.

Assuming that the company can make quarterly payments of $45,000 starting at the beginning of the first period, over 20 years, it can borrow $1,281,480 at 13.5 percent compounded quarterly over 20 years. Thus, the PVADue equation gives you a lump-sum present value on the date of the first payment. This implies that the first payment is not discounted at all, and that the last payment is discounted back 79 periods.

You can solve this problem with the help of an Excel worksheet.

Summary

Before leaving this section, you may want to review the following essential concepts.

When you compute the FV of an ordinary annuity, the result is the equivalent lump-sum payment on the date of the last annuity payment.

When you compute the FV of an annuity due, the result is the equivalent lump-sum payment one period after the last annuity payment.

When you compute the PV of an ordinary annuity, the result is the equivalent lump-sum payment one period before the first annuity payment.

When you compute the PV of an annuity due, the result is the equivalent lump-sum payment on the date of the first annuity payment.

Always use an interest rate appropriate for the length of the period between annuity payments. For example, if the payments are annual, use a one-year interest rate; if they are monthly, then use a monthly interest rate.

How many decimal places should you use?

Analytical Methods: Summation

1. You sell retirement plans. You offer customers a fixed rate of 6.76 percent, compounded monthly, on their monthly contributions. Mr. Smith, a potential new customer, has told you that he can make monthly contributions of $250, starting one month from today for 32 years. How much will he accumulate after 32 years? Assuming that Mr. Smith makes all of his contributions at the end of every month, what will be the future value of his retirement fund after 32 years?

Solution 1

You can use Excel to solve FV.
Download FV Excel Tutorial

2. Your company buys structured settlements or annuities from people who want their money now. If your company requires an annual rate of 28 percent compounded quarterly, how much would it pay for an inheritance that will generate 180 quarterly payments of $15,000 starting today?

Solution 2

3. You are an investment attorney, and a new client who just won $60 million in the lottery seeks your advice. Lottery officials have given her two payment options. Under the first option, the lottery would pay your client a one-time payment of $29,888,572.92 today. Under the second option, the lottery would pay your client $250,000 per month for the next 20 years, with the first payment to be made immediately.

If your client believes that she can earn a guaranteed 8 percent return, compounded monthly over the next 20 years, which option should she select?

Solution 3

4. Having landed a large signing bonus upon completing your MBA and taking a job, you have decided to lease a new car for three years. The dealer offers a 36-month lease that requires you to pay $550 per month and a $2,000 down payment due at lease signing. If you were able to earn 3 percent compounded monthly in a money market account, how much would you have to put in the account today in order to meet your lease obligations? (Include the amounts due today.)

Solution 4

5. Assume a government bond has a face value of $1,000 (it pays $1,000 at maturity), a coupon of 6 percent (it pays interest at an annual rate of 6 percent), semiannual payments of interest ($30 twice a year), and a five-year maturity. If the market interest rate for such a bond is 5 percent, how much is the bond worth? (Discount the cash flows at an annual rate of 5 percent, compounded semiannually.) How much is it worth at a market rate of 8 percent? (Discount the cash flows at an annual rate of 8 percent compounded semiannually.)

Solution 5

You can use Excel to solve PV.
Download PV Excel Tutorial

6. Given a mortgage of $100,000 at a 9 percent annual interest rate compounded monthly and 360 monthly payments, how much interest will be charged during the fifth year of the loan?

Solution 6

You can use Excel to solve NPER.
Download NPER Excel Tutorial

7. A $100,000 mortgage carries a 9 percent annual interest rate compounded monthly and a maximum term of 360 monthly payments. How many payments of $1,000 will pay off the loan?

Solution 7

Sources
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Analytical Methods: Logarithms



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