What is the pre sent discounted value v of that

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from today, and the interest rate over the period is a constant 10%/year. What is the pre- sent discounted value ( V 0 ) of that amount? It is the amount which (if you loaned it out today at an interest rate of 10%/year) would yield $1610.51 on that date. We already know the answer from Example 9.5: the present discounted value is $1000. Note, how- ever, that we could also use equation M.9.5 to calculate V 0 . We have V t V 5 = 1610.51 = V 0 (1 + i ) t = V 0 (1.1) 5 . Hence V 0 = V 5 /(1.1) 5 = V 5 (1.1) –5 = 1610.51/1.61051 = $1000. In general, we can write the formula for the present discounted value of a future payment as V 0 = V t (1 + i ) –t . (M.9.6) If there is a stream of future payments (for example, bond coupons that are clipped and cashed annually, or the annual profits of a business in which we have invested), then if we denote by R t the return (in dollars) in year t , the present discounted value of this future income stream is simply the sum of the discounted values of the returns for all years: T V 0 = R t (1 + i ) –t , (M.9.7) t = 0 where T is the last period in which there is a return. E XAMPLE M.9.6: You have the opportunity to purchase a machine from which you expect to receive the following future income stream: $1100 at the end of one year, $1221 at the end of 2 years, and $1331 at the end of 3 years. At that point, the machine self-destruc- ts in a puff of smoke. What is the maximum amount you would pay for the machine if the interest rate were 10%/year? What is the maximum amount you would pay if the interest rate were 5%/year? With the interest rate at 10%/year, the present discounted value of the income stream is V 0 = 1100/(1.1) + 1221/(1.1) 2 + 1331/(1.1) 3 = $3000. MATH MODULE 9: GROWTH RATES, INTEREST RATES, AND INFLATION: THE ECONOMICS OF TIME M9-5
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With the interest rate at only 5%/year, the present discounted value of the income stream is V 0 = 1100/(1.05) + 1221/(1.05) 2 + 1331/(1.05) 3 = $3294.90. When the interest rate is lower, the future earnings are discounted by a smaller factor, and hence their present discounted value is higher. 9.3 DISCRETE-TIME AND CONTINUOUS-TIME GROWTH RATES As Equation M.9.5 suggests, if a variable X grows from an initial value X 0 , at a constant growth rate g (in percent/year), compounded annually , its value at time t (in years) is given by the familiar “compound interest” formula, X t = X 0 (1 + g ) t . (M.9.8) Suppose that we set g =100%/year, and t =1 year. Then we have X t = X 1 = X 0 (1 + 1) 1 = 2 X 0 . That is, after one year, the value of X will double. Suppose now instead that we compound semi -annually, at the same annual rate. In this case, we halve the growth rate per subperiod, to 50% per 6 months, but now we compound twice per year. At the end of the year, X 1 = X 0 (1 + 1/2) 2 = 2.25 X 0 . More fre- quent compounding at the same annual rate increases the value of X at the end of the year. If we compound quarterly at the same annual rate, we have X 1 = X 0 (1 + 1/4) 4 = 2.44140625 X 0 . As we shorten the length of the subperiods and correspondingly increase the number of subperiods per year, we are moving towards continuous compounding.
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