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Chapter 5. Measuring Risk Defining and measuring Risk aversion & implications Diversification Defining and measuring Risk aversion & implications Diversification

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What is risk? Risk is about uncertainty In financial markets: Uncertainty about receiving promised cash flows Relative to other assets Over a certain time horizon Risk is about uncertainty In financial markets: Uncertainty about receiving promised cash flows Relative to other assets Over a certain time horizon

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Risk affects value So quantification is important! Examples: FICO score, beta Risk affects value So quantification is important! Examples: FICO score, beta

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Measuring risk Elements Distribution/probability Expected value Variance & standard deviation Elements Distribution/probability Expected value Variance & standard deviation

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ProbabilityProbability Likelihood of an event Between 0 and 1 Probabilities of all possible outcomes must add to 1 Probabilities distribution All outcomes and their associated probability Likelihood of an event Between 0 and 1 Probabilities of all possible outcomes must add to 1 Probabilities distribution All outcomes and their associated probability

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Example: coin flip Possible outcomes? 2: heads, tails Likelihood? 50% or.5 heads; 50% or.5 tails .5+.5 =1 Possible outcomes? 2: heads, tails Likelihood? 50% or.5 heads; 50% or.5 tails .5+.5 =1

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Expected value i.e. mean Need probability distribution Center of distribution i.e. mean Need probability distribution Center of distribution

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EVEV = sum of (outcome)(prob of outcome) Or if n outcomes, X 1, X 2,...,X n = sum of (outcome)(prob of outcome) Or if n outcomes, X 1, X 2,...,X n

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For a financial asset Outcomes = possible payoffs Or Possible returns on original investment Outcomes = possible payoffs Or Possible returns on original investment

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Example: two investments Initial investment: $1000

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Investment 1 Payoff (gross)ReturnProbability $500-50%0.2 $1,0000%0.4 $1,50050%0.4 EV = $500(.2) + $1000(.4) + $1500(.4) = $1100 or 10% return = -50%(.2) + 0%(.4) + 50%(.4) = 10%

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EV = $800(.25) + $1000(.35) + $1375(.4) = $1100 or 10% return = -20%(.25) + 0%(.35) + 37.5%(.4) = 10% Investment 2 PayoffReturnProbability $800-20%0.25 $1,0000%0.35 $1,37537.5%0.4

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Same EV—should we be indifferent? Differ in spread of payoffs How likely each payoff is Need another measure! Same EV—should we be indifferent? Differ in spread of payoffs How likely each payoff is Need another measure!

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Variance (σ 2 ) Deviation of outcome from EV Square it Wt. it by probability of outcome Sum up all outcomes standard deviation (σ) is sq. rt. of the variance Deviation of outcome from EV Square it Wt. it by probability of outcome Sum up all outcomes standard deviation (σ) is sq. rt. of the variance

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Investment 1 (500 -1100) 2 (.2) + (1000-1100) 2 (.4) + (1500-1100) 2 (.4) = 116,000 dollars 2 = variance Standard deviation = $341 (500 -1100) 2 (.2) + (1000-1100) 2 (.4) + (1500-1100) 2 (.4) = 116,000 dollars 2 = variance Standard deviation = $341

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Investment 2 (800 -1100) 2 (.25) + (1000-1100) 2 (.35) + (1375-1100) 2 (.4) = 56,250 dollars 2 = variance Standard deviation = $237 (800 -1100) 2 (.25) + (1000-1100) 2 (.35) + (1375-1100) 2 (.4) = 56,250 dollars 2 = variance Standard deviation = $237

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Lower std. dev Small range of likely outcomes Less risk Lower std. dev Small range of likely outcomes Less risk

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Alternative measures Skewness/kurtosis Value at risk (VaR) Value of the worst case scenario over a give horizon, at a given probability Import in mgmt. of financial institutions Skewness/kurtosis Value at risk (VaR) Value of the worst case scenario over a give horizon, at a given probability Import in mgmt. of financial institutions

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Risk aversion We assume people are risk averse. People do not like risk, ALL ELSE EQUAL investment 2 preferred people will take risk if the reward is there i.e. higher EV Risk requires compensation We assume people are risk averse. People do not like risk, ALL ELSE EQUAL investment 2 preferred people will take risk if the reward is there i.e. higher EV Risk requires compensation

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Risk premium = higher EV given to compensate the buyer of a risky asset Subprime mortgage rate vs. conforming mortgage rate = higher EV given to compensate the buyer of a risky asset Subprime mortgage rate vs. conforming mortgage rate

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Sources of Risk Idiosyncratic risk aka nonsytematic risk specific to a firm can be eliminated through diversification examples: -- Safeway and a strike -- Microsoft and antitrust cases Idiosyncratic risk aka nonsytematic risk specific to a firm can be eliminated through diversification examples: -- Safeway and a strike -- Microsoft and antitrust cases

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Systematic risk aka. Market risk cannot be eliminated through diversification due to factors affecting all assets -- energy prices, interest rates, inflation, business cycles Systematic risk aka. Market risk cannot be eliminated through diversification due to factors affecting all assets -- energy prices, interest rates, inflation, business cycles

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DiversificationDiversification Risk is unavoidable, but can be minimized Multiple assets, with different risks Combined, portfolio has smaller fluctuations Accomplished through Hedging Risk spreading Risk is unavoidable, but can be minimized Multiple assets, with different risks Combined, portfolio has smaller fluctuations Accomplished through Hedging Risk spreading

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HedgingHedging Combine investments with opposing risks Negative correlation in returns Combined payoff is stable Derivatives markets are a hedging tool Reality: a perfect hedge is hard to achieve Combine investments with opposing risks Negative correlation in returns Combined payoff is stable Derivatives markets are a hedging tool Reality: a perfect hedge is hard to achieve

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Spreading risk Portfolio of assets with low correlation Minimize idiosyncratic risk Pooling risk to minimize is key to insurance Portfolio of assets with low correlation Minimize idiosyncratic risk Pooling risk to minimize is key to insurance

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exampleexample choose stocks from NYSE listings go from 1 stock to 20 stocks reduce risk by 40-50% choose stocks from NYSE listings go from 1 stock to 20 stocks reduce risk by 40-50%

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# assets systematic risk idiosyncratic risk total risk

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