Let

1

• Lets summarize where we are so far
• The optimal combinations result in lowest level
  of risk for a given return.
• The optimal trade-off is described as the
  efficient frontier.
• These portfolios are dominant (i.e., better).

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Including Riskless Investments

The optimal combination becomes linear A single
combination of risky and riskless assets will
dominate
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ALTERNATIVE CALS
E(r)
Efficient Frontier
P
F Risk Free
s
?A
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The Capital Market Line or CML
CAL (P) CML
E(r)
Efficient Frontier
E(rPF)

• The optimal CAL is called the Capital Market Line
  or CML
• The CML dominates the EF

P
E(rP)
E(rPF)
F Risk Free
s
?PF
?P
?PF
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Dominant CAL with a Risk-Free Investment (F)
CAL(P) Capital Market Line or CML dominates
other lines because it has the the largest
slope Slope (E(rp) - rf) / sp (CML maximizes
the slope or the return per unit of risk or it
equivalently maximizes the Sharpe ratio) We want
our CAL to be drawn tangent to the Efficient
Frontier and the risk-rate. This tells us the
optimal risky portfolio. Once weve drawn the
CAL, we can use the investors risk-aversion to
determine where he or she should be on the CAL.
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Capital Allocation Line and Investor Risk Aversion

• Assume that for the optimal risky portfolio
  E(rp) 15, sp 22, and rf 7.
• Each investors complete portfolio (well use
  subscript C to designate it) is determined by
  their risk aversion. Let y be the fraction of
  the dollars they invest in the optimal risky
  portfolio and 1-y equal the fraction in
  T-bills.
• Expected reward, risk, and Sharpe measure for
  each investors complete portfolio
• E(rc) (1- y) rf (y) E(rp)
• sc y sp
• S (E(rp) - rf ) / sp
• y 1-y E(rc) rf
  sc S
• Retiree Fred 0 1.0 0
  0 ?
• Mid-life Rose 0.5 0.5 (.5)(8) 4
  11 8/22
• The Just-Married Jones 1.0 0.0 8
  22 8/22
• Single and Loving it Sali 1.4 -
  0.4 (1.4)(8) 11.2 30.8 8/22
• Doubling the risk, doubles the expected reward.
  Sharpe ratio doesnt change.

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Capital Allocation Line and Investor Risk Aversion

• Here is the Capital Allocation Line for the
  example E(rp) 15, sp 22, and rf 7.

Sali (y1.4)
Optimal risky portfolio is point P
Fred (y0)
Just-Married Jones (y1.0)
Rose (y.5)
The Capital Allocation Line has an intercept of
rf and a slope (rise/run) equal to the Sharpe
ratio.
8

• 6.5 A Single Index Asset Market

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Individual securities
We have learned that investors should
diversify. Individual securities will be held in
a portfolio. What do we call the risk that
cannot be diversified away, i.e., the risk that
remains when the stock is put into a
portfolio? How do we measure a stocks
systematic risk?
Consequently, the relevant risk of an individual
security is the risk that remains when the
security is placed in a portfolio.
Systematic risk
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Systematic risk

• Systematic risk arises from events that effect
  the entire economy such as a change in interest
  rates or GDP or a financial crisis such as
  occurred in 2007and 2008.
• If a well diversified portfolio has no
  unsystematic risk then any risk that remains must
  be systematic.
• That is, the variation in returns of a well
  diversified portfolio must be due to changes in
  systematic factors.

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Individual securities

• How do we measure a stocks systematic risk?

Systematic Factors
Returns well diversifiedportfolio
?
? interest rates, ? GDP, ? consumer spending, etc.
Returns Stock A
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Risk Premium Format
Let Ri (ri - rf)
Risk premium format
Rm (rm - rf)
The Model
Ri ai ßi(Rm) ei
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Single Factor Model
Ri E(Ri) ßiM ei Ri Actual excess return
ri rf E(Ri) expected excess return Two
sources of Uncertainty M ßi ei
some systematic factor or proxy in this case
M is unanticipated movement in a well diversified
broad market index like the SP500 sensitivity
of a securities particular return to the
factor unanticipated firm specific events
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Single Index Model Parameter Estimation
Market Risk Prem
Risk Prem
or Index Risk Prem
ai
the stocks expected excess return if the
markets excess return is zero, i.e., (rm - rf)
0
ßi(rm - rf) the component of excess return due
to movements in the market index
ei firm specific component of excess return
that is not due to market movements
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Estimating the Index Model
Scatter Plot
Excess Returns (i)
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Excess returns on market index
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Ri a i ßiRm ei Slope of SCL
beta y-intercept alpha
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Components of Risk
ßiM ei

• Market or systematic risk
• Unsystematic or firm specific risk
• Total risk

risk related to the systematic or macro economic
factorin this case the market index
risk not related to the macro factor or market
index
Systematic Unsystematic
?i2 Systematic risk Unsystematic Risk
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Measuring Components of Risk
bi2 sm2 s2(ei)

• si2
• where

si2 total variance bi2 sm2
systematic variance s2(ei) unsystematic
variance
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Examining Percentage of Variance

• Total Risk
• Systematic Risk / Total Risk

Systematic Risk Unsystematic Risk
r2
ßi2 s m2 / si2 r2 bi2 sm2 / (bi2 sm2 s2(ei))
r2
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Advantages of the Single Index Model

Reduces the number of inputs needed to account
for diversification benefits
If you want to know the risk of a 25 stock
portfolio you would have to calculate 25
variances and (25x24) 600 covariance terms With
the index model you need only 25 betas
Easy reference point for understanding stock
risk. ßM 1, so if ßi gt 1 what do we know? If ßi
lt 1?
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Sharpe Ratios and alphas

When ranking portfolios and security
performance we must consider both return
risk Well performing diversified portfolios
provide high Sharpe ratios
Sharpe (rp rf) / ?p
You can also use the Sharpe ratio to evaluate an
individual stock if the investor does not
diversify
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Sharpe Ratios and alphas

Well performing individual stocks held in
diversified portfolios can be evaluated by the
stocks alpha in relation to the stocks
unsystematic risk.
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