annual payments of $5,000 each. The payments will begin 5 years from today. Given an annual interest rate of 8%, what is the present value of the six payments?
Option 1: First, compute the present value of an ordinary annuity of $1 as if the payments had occurred for the entire period. We then subtract the present value of payments that were not received during the deferral period. We are left with the present value of the payments actually received subsequent to the deferral period. Use only Table 4, as shown, step by step, below.
Step 1:
Obtain the present value of an ordinary annuity of $1 for total periods (10) [number of payments (6) plus number of deferred periods (4)] at 8%
Step 2:
Get the present value of an ordinary annuity of $1 for the number of deferred periods (4) at 8%
Step 3:
Take the difference.
Step 4:
Multiply an annual payment by this difference to yield the present value of six annual payments of $5,000 deferred 4 periods
The subtraction of the present value of an annuity of $1 for the deferred periods eliminates the nonexistent payments during the deferral period. It converts the present value of an ordinary annuity of $1.00 for 10 periods to the present value of 6 annual payments of $1.00, deferred 4 periods. Symbolically,
Option 2: Use both Table 4 and Table 3 (in this sequence) to compute the present value of the 6 payments. You can first discount the annuity 6 periods. However, because the annuity is deferred 4 periods, you must treat the present value of the annuity as a future amount to be discounted another 4 periods. Calculation using formulas would be done in two steps, as follows.
Step 1:
Present value of an ordinary annuity = $5,000 T4(8%, 6 years) = $5,000 (4.623) = $23,115
Step 2:
Present value of a single sum = $23,115 T3(8%, 4 years) = $23,115 (.735) = $16,990
The present value of $16,990 computed by using both options should be the same, except for rounding errors.
What are the Applications of Future Values and Present Values?
Future and present values have numerous applications in financial and investment decisions. Each of these applications is presented below.
Deposits to Accumulate a Future Sum (or Sinking Fund)
A financial manager might wish to find the annual deposit (or payment) that is necessary to accumulate a future sum. To find this future amount (or sinking fund) we can use the formula for finding the future value of an annuity.
Solving for A, we obtain:
Example 19
You wish to determine the equal annual end-of-year deposits required to accumulate $5,000 at the end of 5 years in a fund. The interest rate is 10 percent. The annual deposit is:
In other words, if you deposit $819 at the end of each year for 5 years at 10 percent interest, you will have accumulated $5,000 at the end of the fifth year.
Example 20
You need a sinking fund for the retirement of a bond 30 years from now. The interest rate is 10 percent. The annual year-end contribution needed to accumulate $1,000,000 is
Example 21
A company needs to create a $15 million sinking fund at the end of 8 years at 10 percent interest to retire $15 million in outstanding bonds. The amount that should be deposited in the account at the end of each year is:
Thus, if the company deposits $1,311,647.42 at the end of each year for the next 8 years in an account earning 10 percent interest, it will accumulate the $15 million needed to retire the bonds.
Amounts of periodic withdrawals
You might want to determine how much you can withdraw with present savings earning interests over time.
Example 22
Jack and Jill Smiths have saved $40,000 to finance their daughter’s college education. They deposited the money in the Downey Savings and Loan Association, where it earns 4% interest compounded semiannually. What equal amounts can their daughter withdraw at the end of every 6 months during her 4 college years, without exhausting the fund?
Determining the answer by simply dividing $40,000 by 8 withdrawals is wrong, since this ignores the interest earned on the money remaining on deposit. She must consider that interest is compounded semiannually at 2% (4%/2) for 8 periods (4 years x 2). Thus, using the same present value of an ordinary annuity formula, she determines the amount of each withdrawal that she can make as follows.
Determining the Number of Periods Required
You might want to know how long it will take your money to double or how long it will take to reach your monetary goal.
Case 1: Single-Deposit Investment. The number of years it will take to reach a certain future sum can be computed as follows:
Solving for T1 we obtain:
where Fn = future value in period n, and P = a present sum of money or base-period value.
Example 23
At an interest rate of 12 percent, you want to know how long it will take for your money to double. The value is computed as follows:
Therefore,
From Table 1 in the Appendix, a T1 of 2 at 12 percent gives n = almost 6 years. It will take almost 6 years. Alternatively, from Table 3 in the Appendix, a T3 of 0.5 at 12 percent also yields n = almost 6 years.
Example 24
You want to have $250,000. You have $30,000 to invest. The interest rate is 12 percent. The number of years it will take to reach your goal is computed below: