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VALUING REAL CASH PAYMENTS

Think again about how to value future cash payments. Earlier you learned how to value payments in current dollars by discounting at the nominal interest rate. For example, suppose that the nominal interest rate is 10 percent. How much do you need to invest now to produce $100 in a year`s time? Easy! Calculate the present value of $100 by discounting by 10 percent:

PV = $100 = $90.91 1.10

You get exactly the same result if you discount the real payment by the real interest rate. For example, assume that you expect inflation of 7 percent over the next year. The real value of that $100 is therefore only $100/1.07 = $93.46. In one year`s time your $100 will buy only as much as $93.46 today. Also with a 7 percent inflation rate the real rate of interest is only about 3 percent. We can calculate it exactly from the formula

(1 + real interest rate) = 1 + nominal interest rate 1 + inflation rate = 1.10 = 1.028 1.07

real interest rate = .028, or 2.8%

If we now discount the $93.46 real payment by the 2.8 percent real interest rate, we have a present value of $90.91, just as before:

PV = $93.46 = $90.91 1.028

The two methods should always give the same answer.8 8 If they don`t there must be an error in your calculations. All we have done in the second calculation is to divide both the numerator (the cash payment) and the denominator (one plus the nominal interest rate) by the same number (one plus the inflation rate):

PV = payment in current dollars 1 + nominal interest rate = (payment in current dollars)/(1 + inflation rate) (1 + nominal interest rate)/(1 + inflation rate) = payment in constant dollars 1 + real interest rate

Remember:

Current dollar cash flows must be discounted by the nominal interest rate; real cash flows must be discounted by the real interest rate.

Mixing up nominal cash flows and real discount rates (or real rates and nominal flows) is an unforgivable sin. It is surprising how many sinners one finds.

How Inflation Might Affect Bill Gates

We showed earlier (Example 11) that at an interest rate of 9 percent Bill Gates could, if he wished, turn his $96 billion wealth into a 40-year annuity of $8.9 billion per year of luxury and excitement (L&E). Unfortunately L&E expenses inflate just like gasoline and groceries. Thus Mr. Gates would find the purchasing power of that $8.9 billion steadily declining. If he wants the same luxuries in 2040 as in 2000, he`ll have to spend less in 2000, and then increase expenditures in line with inflation. How much should he spend in 2000? Assume the long-run inflation rate is 5 percent. Mr. Gates needs to calculate a 40-year real annuity. The real interest rate is a little less than 4 percent:

1 + real interest rate =

1 + nominal interest rate 1 + inflation rate 1.09 = 1.038 1.05

so the real rate is 3.8 percent. The 40-year annuity factor at 3.8 percent is 20.4. Therefore, annual spending (in 2000 dollars) should be chosen so that $96,000,000,000 = annual spending Р РЈР Р· 20.4 annual spending = $4,706,000,000

Mr. Gates could spend that amount on L&E in 2000 and 5 percent more (in line with inflation) in each subsequent year. This is only about half the value we calculated when we ignored inflation. Life has many disappointments, even for tycoons.

You are owed $5,000 by a relative who will pay back in 1 year. The nominal interest rate is 8 percent and the inflation rate is 5 percent. What is the present value of your relative`s IOU? Show that you get the same answer (a) discounting the nominal payment at the nominal rate and (b) discounting the real payment at the real rate.

You have reached age 60 with a modest fortune of $3 million and are considering early retirement. How much can you spend each year for the next 30 years? Assume that spending is stable in real terms. The nominal interest rate is 10 percent and the inflation rate is 5 percent.