chapter 5. measuring risk defining and measuring risk aversion & implications diversification...
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Chapter 5. Measuring RiskChapter 5. Measuring RiskChapter 5. Measuring RiskChapter 5. Measuring Risk
• Defining and measuring
• Risk aversion & implications
• Diversification
• Defining and measuring
• Risk aversion & implications
• Diversification
What is risk?What is risk?What is risk?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
• Risk affects value So quantification is important! Examples: FICO score, beta
• Risk affects value So quantification is important! Examples: FICO score, beta
Measuring riskMeasuring riskMeasuring riskMeasuring risk
• Elements Distribution/probability Expected value Variance & standard deviation
• Elements Distribution/probability Expected value Variance & standard deviation
ProbabilityProbabilityProbabilityProbability
• 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
Example: coin flipExample: coin flipExample: coin flipExample: 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
Expected valueExpected valueExpected valueExpected value
• i.e. mean
• Need probability distribution
• Center of distribution
• i.e. mean
• Need probability distribution
• Center of distribution
EVEVEVEV
= sum of (outcome)(prob of outcome)
Or if n outcomes, X1, X2, . . .,Xn
= sum of (outcome)(prob of outcome)
Or if n outcomes, X1, X2, . . .,Xn
n
iii XXEV
1
)Pr(
For a financial assetFor a financial assetFor a financial assetFor a financial asset
• Outcomes = possible payoffs
• Or
• Possible returns on original investment
• Outcomes = possible payoffs
• Or
• Possible returns on original investment
Example: two investmentsExample: two investmentsExample: two investmentsExample: two investments
• Initial investment: $1000
• Initial investment: $1000
Investment 1
Payoff (gross) Return Probability
$500 -50% 0.2
$1,000 0% 0.4
$1,500 50% 0.4
EV = $500(.2) + $1000(.4) + $1500(.4)
= $1100 or 10% return
= -50%(.2) + 0%(.4) + 50%(.4) = 10%
EV = $800(.25) + $1000(.35) + $1375(.4)
= $1100 or 10% return
= -20%(.25) + 0%(.35) + 37.5%(.4) = 10%
Investment 2
Payoff Return Probability
$800 -20% 0.25
$1,000 0% 0.35
$1,375 37.5% 0.4
• 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!
Variance (Variance (σσ22))Variance (Variance (σσ22))
• 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
Investment 1Investment 1Investment 1Investment 1
• (500 -1100)2(.2) +
(1000-1100)2(.4) +
(1500-1100)2(.4)
= 116,000 dollars2 = variance
• Standard deviation = $341
• (500 -1100)2(.2) +
(1000-1100)2(.4) +
(1500-1100)2(.4)
= 116,000 dollars2 = variance
• Standard deviation = $341
Investment 2Investment 2Investment 2Investment 2
• (800 -1100)2(.25) +
(1000-1100)2(.35) +
(1375-1100)2(.4)
= 56,250 dollars2 = variance
• Standard deviation = $237
• (800 -1100)2(.25) +
(1000-1100)2(.35) +
(1375-1100)2(.4)
= 56,250 dollars2 = variance
• Standard deviation = $237
• Lower std. dev Small range of likely outcomes Less risk
• Lower std. dev Small range of likely outcomes Less risk
Alternative measuresAlternative measuresAlternative measuresAlternative 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
Risk aversionRisk aversionRisk aversionRisk 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
Risk premiumRisk premiumRisk premiumRisk 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
Sources of RiskSources of RiskSources of RiskSources 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
• 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
DiversificationDiversificationDiversificationDiversification
• 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
HedgingHedgingHedgingHedging
• 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
Spreading riskSpreading riskSpreading riskSpreading 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
exampleexampleexampleexample
• 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%
# assets
systematicrisk
idiosyncratic risk
totalrisk