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**Unformatted text preview: **Geometric Average Versus Arithmetic Average 1 Suppose you invest $435 in a zero coupon bond for one year and earn a return of 8%. You then reinvest the proceeds at 12% for a second year. How do you describe your average annual return over the two-year period? Arithemtic Average One possibility is to calculate the simple arithmetic average of the annual returns, (8% + 12%)/2 = 10%. We suspect that it might be more complicated than that because we know that compounding is involved whenever the proceeds of year 1 are reinvested in year 2 . We check using our notion of holding period returns. Geometric Average and HPR We know how to calculate the average annual return, assuming annual compound- ing, of an initial sum V that grows to V t over t years: HPR = V t V ¶ 1 /t- 1 We can also calculate exactly what V 2 will be after 2 years in our case since we know that V = 435. In particular, V 2 = 435(1 . 08)(1 . 12) = 526 . 176 Therefore, we know that in this case, HPR = 526 . 176 435 ¶ 1 / 2- 1 = . 0998 Thus, we see that the arithmetic mean is bigger than the true annual average return because we know that .0998 is correct since we calculated it from first principles, that is, we calculated it using the proper definition of annual returns.is, we calculated it using the proper definition of annual returns....

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