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# Notice that when the number of stocks increases by a

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Notice that, when the number of stocks increases by a factor of 5 (i.e., from 20 to 100), standard deviation decreases by a factor of 5 = 2.23607 (from \$134,164 to \$60,000). 7 a. ) e ( 2 2 M 2 2 σ + σ β = σ 881 25 ) 20 8 . 0 ( 2 2 2 2 A = + × = σ 500 10 ) 20 0 . 1 ( 2 2 2 2 B = + × = σ 976 20 ) 20 2 . 1 ( 2 2 2 2 C = + × = σ b. If there are an infinite number of assets with identical characteristics, then a well-diversified portfolio of each type will have only systematic risk since the non-systematic risk will approach zero with large n. The mean will equal that of the individual (identical) stocks. c. There is no arbitrage opportunity because the well-diversified portfolios all plot on the security market line (SML). Because they are fairly priced, there is no arbitrage. 8. a. A long position in a portfolio (P) comprised of Portfolios A and B will offer an expected return-beta tradeoff lying on a straight line between points A and B. Therefore, we can choose weights such that β P = β C but with expected return higher than that of Portfolio C. Hence, combining P with a short position in C will create an arbitrage portfolio with zero investment, zero beta, and positive rate of return. b. The argument in part (a) leads to the proposition that the coefficient of β 2 must be zero in order to preclude arbitrage opportunities. 10-3

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10-4 9. The APT factors must correlate with major sources of uncertainty, i.e., sources of uncertainty that are of concern to many investors. Researchers should investigate factors that correlate with uncertainty in consumption and investment opportunities. GDP, the inflation rate, and interest rates are among the factors that can be expected to determine risk premiums. In particular, industrial production (IP) is a good indicator of changes in the business cycle. Thus, IP is a candidate for a factor that is highly correlated with uncertainties that have to do with investment and consumption opportunities in the economy. 10. Any pattern of returns can be “explained” if we are free to choose an indefinitely large number of explanatory factors. If a theory of asset pricing is to have value, it must explain returns using a reasonably limited number of explanatory variables
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