CHAPTER 5

Mathematics of Individual Finance

Present Value

The present value of a future sum is the amount that must be invested today at compound interest to reach a desired sum in the future. To solve for present value, we merely begin with Equation 5-5 and divide both sides of the equation by (1+r)n.

FVn = PV0(1+r)n

(Equation 5-5) PV0 = FVn/(1+r)n (Equation 5-6)

Example 5-5

Ginnie wishes to have $10,000 in her bank account to buy a car for her 18th birthday which is two years away. She can invest her money in a 2 year CD which pays an annually compounded rate of 6%. How much must she put into the CD now to achieve her goal?

Solution 5-5

PV0 = FV2/(1+r)n

= $10,000/(1.06)2

= $10,000/1.1236

= $8,899.96.

In Table A-1, 1/(1+k)n is the present value interest factor (remember that k is used instead of n in the tables). For 2 periods at 6% the present value interest factor is .8900. Multiplying this by $10,000 gives $8,900.00, a 4 cent rounding error.

Using the financial calculator, [2 N 6 %i 10000 FV CPT PV] gives $8,899.96. On LOTUS, the @PV function cannot be used since we only want the present value of a future sum, not an annuity. The formula would be 10000/(1.06^2).

Often in finance, you know only that you will be given a sum of money in the future and you want to know its present value. You do not have a stated interest rate and must assign a discount rate which reflects the rate on an similar investment instrument of equivalent risk and maturity.

For example, if you are to receive a sum of money a year from today, the discount rate that you would use would be equal to the rate that you would charge someone who wanted to borrow money from you for a year. However, the creditworthiness (reliability) of the two parties should be equivalent. If the U. S. Government has promised you a payment in a year, the discount rate should be equal to the rate on a one year Treasury bill, since it is free of default risk. If, however, the payment is from a friend of dubious character, a risk premium must be added to the risk free (Treasury bill) rate to arrive at the appropriate discount rate.

In the previous chapter we reviewed the yield curve which shows how market rates on Treasury instruments vary with maturity. Your discount rate should vary with maturity to reflect the yield curve but will probably be at a higher rate than the comparable Treasury to reflect the added risk. Also note that a Treasury bond pays interest semi-annually and the principal at maturity while a lump sum due in the future pays everything at the end. This means that the maturity of a Treasury bond understates the average period that you must wait to receive your cash flows from a lump sum. Later in this chapter we will study duration, which allows us to compare different types of instruments in terms of the average time to receipt of the discounted cash flow. For the moment, it is important only to understand that a lump sum paid in the future has an effective maturity (duration) which is longer than a Treasury bond whose maturity matches the time to payment of the lump sum.

Example 5-6

Under the terms of her grandmother's will, Karen will receive $50,000 on her 25th birthday, which is 3 years away. Since the funds are kept for her by the trust department of a bank, Karen has decided to use the 3 year CD bank rate of 5.7% for her discount rate. What is the present value of Karen's inheritance?

Solution 5-6

PV0 = $50,000/(1.057)3

= $50,000/1.1809322

= $42,339.43

[Annuities]

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