Foundations of Financial Management: An Analysis (part 30)

“Money is only a tool. It will take you wherever you wish, but it will not replace you as the driver.” 

– Ayn Rand

 The Time Value of Money (part E)

 by

 Charles Lamson

Special Considerations in Time Value Analysis

In the previous posts covering the time value of money, we assumed interest was compounded or discounted on an annual basis. This assumption will now be relaxed. Contractual arrangements, such as an installment purchase agreement or a corporate bond contract, may call for semiannual, quarterly, or monthly compounding periods. The adjustment to the normal formula is quite simple. To determine n (number of periods), simply multiply the number of years by the number of compounding periods during the year. The factor for i (interest rate) is then determined by dividing the quoted annual interest rate by the number of compounding periods.

Case 1---Find the future value of a $1,000 investment after 5 years at 8% annual interest, compounded semi-annually.

n = 5 X 2 = 10      i = 8%/2 = 4 %

*Again, the IF in the subscript indicates we are using the interest factor from the table.

Case 2---Find the present value of 20 quarterly payments of $2,000 each to be received over the next five years. The stated interest rate is 8% per annum. The problem calls for the present value of an annuity. We again follow the same procedure as in Case 1 in regard to n and i.

*Again, the A in the subscript indicates we are talking about annuities rather than a single amount.

Patterns of Payment

Time value of money problems may evolve around a number of different payment or receipt patterns. Not every situation will involve a single amount or an annuity. For example a contract may call for the payment of a different amount each year over a three-year period. To determine present value, each payment is discounted (Table 2 from Part 26 of this analysis) to the present and then summed.

(Assume 8% discount rate)

A more involved problem might include a combination of single amounts and an annuity. If the annuity will be paid at some time in the future, it is referred to as a deferred annuity and it requires a special treatment. Assume the same problem as above, but with an annuity of $1,000 that will be paid at the end of each year from the fourth through the eighth year. With a discount rate of 8%, what is the present value of the cash flows?

We know the present value of the first three payments is $5,022, from our calculation above, but what about the annuity? Let's diagram the 5 annuity payments.

The information source is Table 4, the present value of an annuity of $1 in Part 27 of this analysis. For n = 5, i = 8 percent, the discount factor is 3.993---leaving a "present value" of the annuity of $3,993. However, tabular values only discount to the beginning of the first stated period of an annuity---in this case the beginning of the fourth year as diagrammed below.

The $3,993 must finally be discounted back to the present. Since this single amount falls at the beginning of the fourth period---in effect, the equivalent of the end of the third period---we discount back for 3 periods at the stated 8 percent interest rate. Using Table 2, we have:

A second method for finding the present value of a deferred annuity is to:

$3,170 is the same answer for the present value of the annuity as that reached by the first method. The present value of the five-year annuity may now be added to the present value of the inflows over the first three years to arrive at the total value. 

*MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 256-258*

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