Custom Search

Solving Annuity Problems Using a Financial Calculator

The formulas for both the present value and future value of an annuity are also built into your financial calculator. Again, we can input all but one of the five financial keys, and let the calculator solve for the remaining variable. In these applications, the PMT key is used to either enter or solve for the value of an annuity.

Solving for an Annuity

In Example 3.12, we determined the savings stream that would provide a retirement goal of $500,000 after 50 years of saving at an interest rate of 10 percent. To find the required savings each year, enter n = 50, i = 10, FV = 500,000, and PV = 0 (because your savings account currently is empty). Compute PMT and find that it is $429.59. Again, your calculator is likely to display the solution as 429.59, since the positive $500,000 cash value in 50 years will require 50 cash payments (outflows) of $429.59.

The sequence of key strokes on three popular calculators necessary to solve this problem is as follows:

Your calculator displays a negative number, as the 50 cash outflows of $429.59 are necessary to provide for the $500,000 cash value at retirement.

Present Value of an Annuity

In Example 3.10 we considered a 30-year mortgage with monthly payments of $1,028.61 and an interest rate of 1 percent. Suppose we didn t know the amount of the mortgage loan. Enter n = 360 (months), i = 1, PMT = 1,028.61 (we enter the annuity level paid by the borrower to the lender as a negative number since it is a cash outflow), and FV = 0 (the mortgage is wholly paid off after 30 years; there are no final future payments beyond the normal monthly payment). Compute PV to find that the value of the loan is $100,000.

What about the balance left on the mortgage after 10 years have passed? This is easy: the monthly payment is still PMT = 1,028.61, and we continue to use i = 1 and FV = 0. The only change is that the number of monthly payments remaining has fallen from 360 to 240 (20 years are left on the loan). So enter n = 240 and compute PV as 93,417.76. This is the balance remaining on the mortgage.

Future Value of an Annuity

In Figure 3.12, we showed that a 4-year annuity of $3,000 invested at 8 percent would accumulate to a future value of $13,518. To solve this on your calculator, enter n = 4, I = 8, PMT = 3,000 (we enter the annuity paid by the investor to her savings account as a negative number since it is a cash outflow), and PV = 0 (the account starts with no funds). Compute FV to find that the future value of thesavings account after 3 years is $13,518.

Calculator Self-Test Review (answers follow)

1. Turn back to Kangaroo Autos in Example 3.8. Can you now solve for the present value of the three installment payments using your financial calculator? What key strokes must you use?

2. Now use your calculator to solve for the present value of the three installment payments if the first payment comes immediately, that is, as an annuity due.

3. Find the annual spending available to Bill Gates using the data in Example 3.11 and your financial calculator.

Solutions to Calculator Self-Test Review Questions

1. Inputs are n = 3, i = 10, FV = 0, and PMT = 4,000. Compute PV to find the present value of the cash flows as $9,947.41.

2. If you put your calculator in BEGIN mode and recalculate PV using the same inputs, you will find that PV has increased by 10 percent to $10,942.15. Alternatively, as depicted in Figure 3.10, you can calculate the value of the $4,000 immediate payment plus the value of a 2-year annuity of $4,000. Inputs for the 2-year annuity are n = 2, I = 10, FV = 0, and PMT = 4,000. Compute PV to find the present value of the cash flows as $6,942.15. This amount plus the immediate $4,000 payment results in the same total present value: $10,942.15.

3. Inputs are n = 40, i = 9, FV = 0, PV = 96,000 million. Compute PMT to find that the 40-year annuity with present value of $96 billion is $8,924 million.