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Risk … … uncertainty about the future payoff of an investment measured over some time horizon and relative to a benchmark. Measuring Risk requires: List of all possible outcomes Chance of each one occurring.

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Measuring RiskCase 1 An Investment can rise or fall in value. Assume that an asset purchased for $1000 is equally likely to fall to $700 or rise to $1400

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Variance of Payoff Standard Deviation = Risk Variance of payoff = Expected squared deviation of return from its expected value = Expected squared deviation of return from its expected value =½($1400-$1050) 2 + ½($700-$1050) 2 =½($1400-$1050) 2 + ½($700-$1050) 2 = ½ ($350) 2 + ½ ($350) 2 = 122,500 $ 2 = ½ ($350) 2 + ½ ($350) 2 = 122,500 $ 2 Standard Deviation of Payoff = SQRT(Variance) = (122,500 $ 2 ) 1/2 = $350

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Measuring Risk: A second investment with same expected payoff but broader probability distribution

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Variance of Payoff Standard Deviation = Risk Variance of payoff = Expected squared deviation of return from its expected value = Expected squared deviation of return from its expected value =.1($100-$1050) 2 +.4($700-$1050) 2 =.1($100-$1050) 2 +.4($700-$1050) 2 +.4($1400-$1050) 2 +.1($2000-$1050) 2 +.4($1400-$1050) 2 +.1($2000-$1050) 2 = 278,500 $ 2 Standard Deviation of Payoff = SQRT(Variance) = (278,500 $ 2 ) 1/2 = $528

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A risk-free asset is an investment whose future value of known with certainty, and whose return is the risk- free rate of return. A risk-averse investor will always prefer an investment with a certain return to one with the same expected return but some risk. –The riskier an investment, the higher the compensation that investors require for holding it the higher the risk premium.

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Sources of Risk Idiosyncratic – Unique Risk Systematic – Economy-wide Risk

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Reducing Risk through Diversification Hedging Risk Reduce overall risk by making two investments with opposing risks. – –When one does poorly, the other does well, and vice versa. – –While the payoff from each investment is volatile, together their payoffs are stable.

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Reducing Risk through Diversification

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Compare three strategies for investing $100 1.Invest $100 in GE 2.Invest $100 in Texaco 3.Invest half in each company $50 in GE and $50 in Texaco

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Reducing Risk through Diversification To eliminate risk, find investments whose payoffs are negatively correlated: One does better than expected, the other does worse To spread risk, find investments whose payoffs are completely unrelated. To spread risk, find investments whose payoffs are completely unrelated. But perfectly negative correlation and even lack of correlation in payoffs is rarely possible systematic risk But perfectly negative correlation and even complete lack of correlation in payoffs is rarely possible systematic risk Diversification can still reduce risk (if not eliminate risk)

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: P Reducing Risk Through Diversification: Positively Correlated Payoffs

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Consider three investment strategies: (1) GE only, (2) Microsoft only, and (3) half in GE and half in Microsoft. The expected payoff on each of these strategies is the same: $110. For the first two strategies, $100 in either company, the standard deviation is still 10, just as it was before. But for the third strategy, the analysis is more complicated. – –There are four possible outcomes, two for each stock

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Variance of Payoff Standard Deviation = Risk Variance of payoff = Expected squared deviation of return from its expected value = Expected squared deviation of return from its expected value = ¼ ($120-$110) 2 + ½ ($110-$110) 2 = ¼ ($120-$110) 2 + ½ ($110-$110) 2 + ¼ ($100-$110) 2 + ¼ ($100-$110) 2 = 50 $ 2 Standard Deviation of Payoff = SQRT(Variance) = (50 $ 2 ) 1/2 = $ 7.07

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