Foundations of Financial Management: An Analysis (part 28)

“Friendship is like money, easier made than kept.” 

– Samuel Butler

The Time Value of Money (Part C)

by

Charles Lamson

 

Determining the Annuity Value

In our prior discussion of annuities from last post, we assumed the unknown variable was the future value or the present value---with specific information available on the annuity value (A), the interest rate, and the number of periods of years. In certain cases our emphasis may shift to solving for one of these other values on the assumption that future value or present value is given. For now, we will concentrate on determining an unknown annuity value.

Annuity Equaling a Future Value

Assuming we wish to accumulate $4,641 after 4 years at a 10% interest rate, how much must be set aside at the end of each of the four periods? We take the previously developed statement from last post for the future value of an annuity and solve for A.

The solution is the exact reverse of that previously presented under the discussion of the future value of an annuity. As a second example, assume the director of the Women's Tennis Association must set aside an equal amount for each of the next 10 years to accumulate $1,000,000 in retirement funds and the return on deposited funds is 6%. Solve for the annual contribution, A, using Table 3 from last post.

Annuity Equaling a Present Value

In this instance, we assume you know the present value and you wish to determine what size annuity can be equated to that amount. Suppose your wealthy uncle presents you with $10,000 now to help you get through the next four years of college, or whatever. If you are able to earn 6 percent on deposited funds, how many equal payments can you withdraw at the end of each year for 4 years? We need to know the value of an annuity equal to a given present value. We take the previously developed statement for the present value of an annuity and we reverse it to solve for A.

 The appropriate table is Table 4 from last post (present value of an annuity). We determine an answer of $2,886.

The flow of funds follow the pattern in Table 5. Annual interest is based on the beginning balance for each year.

Table 5 Relationship of present value to annuity

The same process can be used to indicate necessary repayments on a loan. Suppose a homeowner signs a $40,000 mortgage to be repaid over 20 years at 8 percent interest. How much must he or she pay annually to eventually liquidate the loan? In other words, what annuity paid over 20 years is the equivalent of a $40,000 present value with an 8 percent interest rate?

Part of the payments to the mortgage company will go toward the payment of interest, with the remainder applied to debt reduction, as indicated in Table 6.

Table 6 Payoff table for loan (amortization table)

If this same process is followed over 20 years, the balance will be reduced to zero. You might note that the homeowner will pay over $41,000 of interest during the term of the loan, as indicated below.

*MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 252-254*

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