Example 8
You deposit $30,000 semiannually into a fund for ten years. The annual interest rate is 8 percent. The amount accumulated at the end of the tenth year is calculated as follows:
where A = $30,000
Therefore,
An excerpt from Table 2 is given below.
Table 2: The Future Value of an ordinary Annuity of $1.00 (Compounded Amount of an ordinary Annuity of $1.00) = FVF-OA (i,n) = T2(i,n)
Example 9
Michael receives an annual annuity of $2,000 that is invested at 10 percent immediately upon receipt of the annual payment. At the end of the 5-year annuity, Michael will have accumulated the following sums.
Michael will have $12,210 at the end of the annual annuity if it is invested at 10 percent.
Annuity Due
The formula for computing the future value of an annuity due must take into consideration one additional year of compounding, since the payment comes at the beginning of the year. Therefore, the future-value formula must be modified to take this into consideration by compounding it for one more year:
Note: The future value of an annuity due of $1is found by subtracting 1 (one extra payment) from the future value of an ordinary annuity of $1 for one more period. That is:
Example 10
Using the same investment as Example 8, the future-value computations for the annuity due would be as follows:
Future-Value Difference between Ordinary Annuity and Annuity Due
The future-value difference between the ordinary annuity and an annuity due in our examples is $13,431 - $12,210 = $1,221. Generally speaking, an annuity due is preferable over an ordinary annuity, since an amount equivalent to an additional year of compounding is received.
Can a Computer Help?
Financial calculators marketed by several manufacturers (e.g., Hewlett-Packard, Sharp, Texas Instruments) have a “future (compound) value” function. Future value is also incorporated as a built-in function in spreadsheet programs. For example, Excel has a routine FV(rate,nper,pmt,pv,type), which calculates the future value of an investment based on periodic, constant payments and a constant interest rate. To calculate the future value of an annuity due, use the formula FV × (1 + interest).
What is Present Value - How Much Money is Worth Now?
Present value is the present worth of future sums of money. The process of calculating present values, or discounting, is actually the opposite of finding the compounded future value, compounding. In connection with present value calculations, the interest rate i is called the discount rate. The discount rate we use is more commonly called the cost of capital, which is the minimum rate of return required by the investor (to be discussed later in the book).
Recall that Fn = P (1+i)n
Therefore,
Where PVF(i,n) = T3(i,n) represents the present value of $1 and is given in Table 3 in the Appendix.
Example 11
You have been given an opportunity to receive $20,000 6 years from now. If you can earn 10 percent on your investments, what is the most you should pay for this opportunity? To answer this question, you must compute the present value of $20,000 to be received 6 years from now at a 10 percent rate of discount. F6 is $20,000, i is 10 percent, and n is 6 years. PVF(10%,6) = T3 (10%,6) from Table 3 is 0.565.
An excerpt from Table 3 is given below.
Table 3: The Present Value of $1.00 (Discounted Amount of $1.00) = PVF (i,n) = T3(i,n)
This means that you can earn 10 percent on your investment, and you would be indifferent to receiving $12,800 now or $20,000 6 years from today since the amounts are time equivalent. In other words, you could invest $12,800 today at 10 percent and have $20,000 in 6 years.
An Application
Suppose you purchase a building with a noninterest-bearing as a consideration. There is no established exchange price for the building, and the note had no ready market. The noninterest-bearing note should be recorded at its fair market value, which is the present value of the future cash flows discounted at the prevailing rate of interest.
Present Value of Mixed Streams of Cash Flows
The present value of a series of mixed payments (or receipts) is the sum of the present value of each individual payment. We know that the present value of each individual payment is the payment times the appropriate T3 value.
Example 12
You are thinking of starting a new product line that initially costs $32,000. Your annual projected cash inflows are:
| Year 1 | $10,000 |
| Year 2 | $20,000 |
| Year 3 | $5,000 |
If you must earn a minimum of 10 percent on your investment, should you undertake this new product line?
The present value of this series of mixed streams of cash inflows is calculated as follows:
Since the present value of your projected cash inflows is less than the initial investment, you should not undertake this project.
Present Value of an Annuity
Interest received from bonds, pension funds, and insurance obligations all involve annuities. To compare these financial instruments, we need to know the present value of each. The present value of an annuity is a method of discounting an annuity to determine its worth in present-day dollars. It shows the amount of the lump-sum payment that would have to be received today to equal the annuity.
This analysis accommodates two types of annuities – ordinary annuities in which the equal payment comes at the end of the year, and annuities due