Chapter 5. Measuring Risk
• 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 affects value
 So quantification is important!
Measuring risk
• Elements
 Distribution/probability
 Expected value
 Variance & standard deviation
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
Expected value
• i.e. mean
• Need probability distribution
• Center of distribution
EV
= sum of (outcome)(prob of outcome)
Or if n outcomes, X1, X2, . . .,Xn
n
EV   X i Pr( X i )
i 1
For a financial asset
• Outcomes = possible payoffs
• Or
• Possible returns on original
investment
Example: two investments
•
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%
Investment 2
Payoff
Return
$800
-20%
$1,000
0%
$1,375
37.5%
Probability
0.25
0.35
0.4
EV = $800(.25) + $1000(.35) + $1375(.4)
= $1100 or 10% return
= -20%(.25) + 0%(.35) + 37.5%(.4)
= 10%
• Same EV—should we be
indifferent?
 Differ
• in spread of payoffs
• How likely each payoff is
 Need another measure!
Variance (σ2)
•
•
•
•
•
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
• (500 -1100)2(.2) +
•
(1000-1100)2(.4) +
(1500-1100)2(.4)
= 116,000 dollars2 = variance
Standard deviation = $341
Investment 2
• (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
Alternative measures
• Skewness
• Value at risk (VaR)
 Value of the worst case scenario
over a give horizon, at a given
probability
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
Risk premium
• = higher EV given to compensate
the buyer of a risky asset
 Subprime mortgage rate vs.
conforming mortgage rate
Sources of Risk
•
Idiosyncratic risk
 aka nonsystematic 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
Diversification
•
•
•
Risk is unavoidable, but can be
minimized
Multiple assets, with different risks
 Combined, portfolio has smaller
fluctuations
Accomplished through
 Hedging
 Risk spreading
Hedging
•
•
•
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
example
• choose stocks from NYSE listings
• go from 1 stock to 20 stocks
 reduce risk by 40-50%
s
idiosyncratic
risk
total
risk
systematic
risk
# assets