Models of Risk
2003,3,24
What is risk?
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Risk: to expose to hazard or danger
Risk = riscare (dare to do something)
Risk = 危機 (danger and opportunity)
Goal of finance (investment)
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To ensure that when an investor is exposed
to risk, there will be an “appropriate” reward
for taking the risk.
Characteristics of risk and reward
models
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Provide a measure of risk that applies to all assets
and not be asset-specific
Delineate those type of risk that are rewarded and
those that are not, and should give a rationale for the
delineation
Based on standardized risk measures ( to conclude
the asset involve above-average or below-average
risk)
Translate the measure of risk into a rate of return
that the investor can expect or demand as
compensation for bearing the risk
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It should work well not only as an explanation
of past returns, but also as a prediction of
future expected returns.
How risk is viewed in finance
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Defined the risk in terms of the distribution of
actual returns around an expected return
differentiate between risk that is specific to
one investment or a few investments, and
risk that affects a much wider cross section
of investment.
Measure risk
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Riskless investment: default-free bond, the
actual return is the expected return.
Risk investment: stock, the actual return will
almost certainly NOT be equal to the
expected return.
Symmetric?
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The spread of actual returns around the
expected return is captured by the variances
or standard deviation of the distribution.
The bias toward positive or negative returns
is captured by the skewness of the
distribution.
The shape of the tails of the distribution is
measured by the kurtosis of the distribution.
(fatter tails lead to higher kurtosis)
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Symmetric and normal: expected return and
variance
Symmetric : same standard deviation, chose
higher expected return
Neither symmetric nor normal : expected
return, variance, positive skewness, and then
lower kurtosis (lower likelihood of jumps)
than higher kurtosis (higher likelihood of
jumps).
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Investors tend to trade off good moment
(higher expected returns and more positive
skewnwess) than bad moment (higher
variance and higher kurtosis)
Capital asset pricing model (CAPM) explicitly
requires that choices be made only in terms
of expected returns and variances.
Utility function =?
Disney (Jan 1992 – Dec 1996)
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SD in monthly returns = 6.14%
Variance in monthly returns = 37.66%
(?) annualized SD = 6.14  12  21.26
(?) annualized variance = 37.66 12  452
Rewarded and Unrewarded Risk
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Firm-specific risk
Market risk
The components of Risk
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Project-specific risk
competitive risk
Industry-specific risk: technology risk, legal
risk, commodity risk
International risk: currency
Market risk: macroeconomic factors
Diversification
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Reduces or eliminates firm-specific risk
diversification
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Why marginal investor is assumed to be
diversified?
Risk increase if undiversified.
Diversified investors are willing to pay higher
price to reduce risk.
The asset will, over time, will end up being
held by diversified investors.
Measuring Market Risk
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Risk comes from the distribution of actual
returns around expected return.
Risk should be measured from the
perspective of marginal investor who is well
diversified. (spot price)
Methods for measuring risk
(Asset Pricing)
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Capital asset pricing theorem (CAPM)
Arbitrage pricing theory (APT)
Multi-factor models
Regression models
CAPM
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Assumptions: diversification is attractive
Real world: tractions cost and monitoring
cost, reduced risk is smaller when the
number of stock is larger. We believe we can
find under-valued stock.
Market portfolio: A portfolio that contains all
traded assets in the market.
Beta
CAPM
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Implications for investors: If all investors hold the
identical market portfolio, how do investors r?
Answer: Asset Allocation on riskless and risky
asset.
Measuring the market risk of an individual Asset:
Beta of an asset i = Covariance of asset I with
market portfolio/variance of portfolio
Beta of market portfolio=1
Expected return
E ( Ri )  R f  i [ E ( Rm )  R f ]
The CAPM in Practice (three inputs)
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Riskless asset: may dependent on time
period
risk premium: difference of expected return
(historical return) of risky asset and riskless
return
Beta: Covariance of risky asset and market
portfolio divided by the variance of market
portfolio (estimated by historical data)
The Arbitrage Pricing Model (APM)
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Assumption: If there are two portfolio has the
same exposure of risk but offer different
expected returns, the investor will buy the
portfolio with higher return. This action will
adjust the expected returns to equilibrium.
Risk=Market +Firm-specific
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m is the marketwise component of
unanticipated risk
is the firm-specific component
Return model:
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r  E ( R)  m  
Market Risk
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In CAPM, the market risk is capture by
market portfolio.
In APM, the market risk is coming from
economical fundamentals, gross national
product, interest rates, and inflation, and it
measures the sensitivity of investments to
these changes with factor betas.
Market Risk
m  1F1   2 F2 
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  n Fn
Where  j the sensitivity of the investment
to unanticipated changes in factor j
F unanticipated changes in factor j
j
The Effects of Diversification
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The returns of portfolio will not have firmspecific component of unanticipated returns.
Rp  ( w1R1   wn Rn )  ( w11,1   wn 1,n ) F1
  ( w1n,1   wn n,n ) Fn
Expected Returns and Betas
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The beta of a portfolio is the weighted
average of the betas of the assets in the
portfolio, in conjunction with absence of
arbitrage, leads to the conclusion that
expected returns should be linearly related to
betas.
Expected returns and Betas
Portfolio
Expected return
Beta
A
20%
2.0
B
12%
1.0
C
14%
1.5
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E(Rj )  Rf
of factor j
can be viewed as risk premium
E ( R)  R f  1[ E ( R1 )  R f ]
   n [ E ( Rn )  R f ]
APM in practice
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Estimates of each of the factors betas and
factor risk premium.