# What is the pre sent discounted value v of that

• Notes
• 8

This preview shows pages 5–7. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern