CP

Annuities: Solving for Interest Rate, Number of Periods, or Payment 77

Switch to the beginning-of-period mode, and then enter N 3, I 5, PMT 
100,

and FV 0, and then press PV to get the answer, $285.94. Again, since most problems deal

with end-of-period cash flows, don’t forget to switch your calculator back to the “END” mode.

3. SPREADSHEET SOLUTION

For an annuity due, use the PV function just as for a regular annuity except enter 1

rather than 0 for Type to indicate that we now have an annuity due.

Which annuity has the greater present value: an ordinary annuity or an annuity

due? Why?

Explain how financial calculators can be used to find the present value of

annuities.

Annuities: Solving for Interest Rate,

Number of Periods, or Payment

Sometimes it is useful to calculate the interest rate, payment, or number of periods for

a given annuity. For example, suppose you can lease a computer from its manufacturer

for $78 per month. The lease runs for 36 months, with payments due at the end of the

month. As an alternative, you can buy it for $1,988.13. In either case, at the end of the

36 months the computer will be worth zero. You would like to know the “interest rate”

the manufacturer has built into the lease; if that rate is too high, you should buy the

computer rather than lease it.

Or suppose you are thinking ahead to retirement. If you save $4,000 per year at an

interest rate of 8 percent, how long will it take for you to accumulate $1 million? Or,

viewing the problem another way, if you earn an interest rate of 8 percent, how much

must you save for each of the next 20 years to accumulate the $1 million?

To solve problems such as these, we can use an equation that is built into financial

calculators and spreadsheets:

(2-6)

Note that some value must be negative. There are five variables: n, i, PV, PMT,

and FV.^6 In each of the three problems above, you know four of the variables. For ex-

ample, in the computer leasing problem, you know that n 36, PV 1,988.13 (this

is positive, since you get to keep this amount if you choose to lease rather than pur-

chase), PMT 
78 (this is negative since it is what you must pay each month), and

FV 0. Therefore, the equation is:

(2-6a)

Unless you use a financial calculator or a spreadsheet, the only way to solve for i is by

trial-and-error. However, with a financial calculator, you simply enter the values for

the four known variables (N 36, PV 1988.13, PMT 
78, and FV 0), and

then hit the key for the unknown fifth variable, in this case, I 2. Since this is an

(1,988.13)(1i)^36 (
78)a

(1i)^36 
 1

i

b 0 0.

PV(1i)nPMTa

(1i)n
 1

i

bFV0.

(^6) This is the equation for an ordinary annuity. Calculators and spreadsheets have a slightly different equation
for an annuity due.

Time Value of Money 75