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

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

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

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

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

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

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

Example: two investments Initial investment: $1000

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%

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

Investment 1 ( ) 2 (.2) + ( ) 2 (.4) + ( ) 2 (.4) = 116,000 dollars 2 = variance Standard deviation = $341 ( ) 2 (.2) + ( ) 2 (.4) + ( ) 2 (.4) = 116,000 dollars 2 = variance Standard deviation = $341

Investment 2 ( ) 2 (.25) + ( ) 2 (.35) + ( ) 2 (.4) = 56,250 dollars 2 = variance Standard deviation = $237 ( ) 2 (.25) + ( ) 2 (.35) + ( ) 2 (.4) = 56,250 dollars 2 = 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 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 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 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 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

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

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

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

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%

 # assets systematic risk idiosyncratic risk total risk