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# B 10year c 20year what would the results be in each

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(b) 10%/year? (c) 20%/year? What would the results be in each case if the loan period were 10 years? If the loan period is 5 years, then at the end of 5 years you will owe: (a) V 5 = 1000(1.05) 5 = 1000(1.27628) = \$1276.28; (b) V 5 = 1000(1.1) 5 = \$1610.51: (c) V 5 = 1000(1.2) 5 = \$2488.32. M9-4 MATH MODULE 9: GROWTH RATES, INTEREST RATES, AND INFLATION: THE ECONOMICS OF TIME

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If the loan period is 10 years, then at the end of 10 years you will owe: (a) V 10 = 1000(1.05) 10 = 1000(1.62889) = \$1628.89; (b) V 10 = 1000(1.1) 10 = \$2593.74: (c) V 10 = 1000(1.2) 10 = \$6191.74. [In solving Example M.9.5, you will need to use the y x (or x y ) button on your calculator. To calculate (1.05) 5 , on most calculators we enter “1.05” y x (or x y ) 5 “=.” Experiment with your own calculator to verify the correct sequence.] 9.3 CALCULATING THE PRESENT DISCOUNTED VALUE OF A FUTURE INCOME STREAM Suppose that someone guarantees to pay you an amount of \$1610.51, exactly 5 years 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 pro±ts 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
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

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b 10year c 20year What would the results be in each case if...

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