Let’s 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).
1
Including Riskless Investments
• The optimal combination becomes linear
• A single combination of risky and riskless
assets will dominate
6-2
ALTERNATIVE CALS
CAL (P)
E(r)
Efficient
Frontier
E(rP&F)
P
E(rP)
E(rA)
CAL (A)
CAL (Global
minimum variance)
A
G
F
Risk Free
sA sP sP&F
s
6-3
The Capital Market Line or CML
CAL (P) = CML
E(r)
Efficient
Frontier
E(rP&F)
P
E(rP)
o The optimal CAL is
called the Capital
Market Line or CML
o The CML dominates
the EF
E(rP&F)
F
Risk Free
sP&F
sP sP&F
s
6-4
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 we’ve drawn the CAL, we can use the investor’s riskaversion to determine where he or she should be on the CAL.
6-5
Capital Allocation Line and Investor Risk Aversion
Assume that for the “optimal risky portfolio”: E(rp) =15%, σp = 22%, and rf = 7%.
Each investor’s “complete portfolio” (we’ll 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 investor’s “complete portfolio”:
• E(rc) = (1- y) rf + (y) E(rp)
• σc = y σp
• S
= (E(rp) - rf ) / σp




Retiree Fred
Mid-life Rose
The Just-Married Jones
“Single and Loving it” Sali *
y
1-y
0
0.5
1.0
1.4
1.0
0.5
0.0
- 0.4
E(rc) – rf
sc
S
0
0
?
(.5)(8%) = 4%
11% 8/22
8%
22% 8/22
(1.4)(8%) =11.2% 30.8% 8/22
Doubling the risk, doubles the expected reward. Sharpe ratio doesn’t change.
Capital Allocation Line and Investor Risk Aversion
Here is the Capital Allocation Line for the example: E(rp) =15%, σp = 22%, and rf = 7%.
Optimal
risky
portfolio is
point “P”
Fred
(y=0)
Rose
(y=.5)
Sali (y=1.4)
JustMarried
Jones
(y=1.0)
The Capital Allocation Line has an intercept of rf and a slope (rise/run) equal to the Sharpe ratio.
6.5 A Single Index Asset Market
6-8
Individual securities
• We have learned that investors should diversify.
• Individual securities will be held in a portfolio.
Consequently, the relevant risk of an individual
security is the risk that remains when the security
is placed 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? Systematic risk
• How do we measure a stock’s systematic risk?
6-9
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.
6-10
Individual securities
How do we measure a stock’s systematic
risk?
Systematic Factors

Returns
Stock A
Returns
well
diversified
portfolio
Δ interest rates,
Δ GDP,
Δ consumer spending,
etc.
6-11
Risk Premium Format
Let: Ri = (ri - rf)
Rm = (rm - rf)
Risk premium
format
The Model:
Ri = ai + ßi(Rm) + ei
6-12
Single Factor Model
Ri = E(Ri) + ßiM + ei
Ri = Actual excess return = ri – rf
E(Ri) = expected excess return
Two sources of Uncertainty
M = some systematic factor or proxy; in this case
M is unanticipated movement in a well
diversified broad market index like the
S&P500
ßi
= sensitivity of a securities’ particular return to
the factor
ei = unanticipated firm specific events
6-13
Single Index Model Parameter Estimation
r  r  a   r  r  e
i
f
i
i
m
f
i
Market Risk Prem
or Index Risk Prem
αi = the stock’s expected excess return if the
market’s 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
Risk Prem
6-14
Estimating the Index Model
Scatter
Plot
Excess Returns (i)
. ..
. ..
.
.
.
.
.
. . ..
.. . .
Security
.
.
.
.
Characteristic
.
.
.
. .
Line
.
. .. . .
.
. . . Excess returns
. . . . on market index
.
.
.
.
.
.
. R =. a + ß R + e
i
i
i
m
i
Slope of SCL = beta
y-intercept = alpha
6-15
Components of Risk
• Market or systematic risk:
ßiM + ei
risk related to the systematic or macro economic factor
in this case the market index
• Unsystematic or firm specific risk:
risk not related to the macro factor or market index
• Total risk =
Systematic + Unsystematic
si2 = Systematic risk + Unsystematic Risk
6-16
Measuring Components of Risk
si2
=
i2 sm2 + s2(ei)
where;
si2
= total variance
i2 sm2 = systematic variance
s2(ei) = unsystematic variance
6-17
Examining Percentage of
Variance
Total Risk =
Systematic Risk + Unsystematic Risk
Systematic Risk / Total Risk
r2
=
ßi2 s m2 / si2 = r2
i2 sm2 / (i2 sm2 + s2(ei)) = r2
6-18
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.
βM = 1, so if βi > 1 what do we know?
If βi < 1?
6-19
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) / sp
• You can also use the Sharpe ratio to evaluate an
individual stock if the investor does not diversify
6-20
Sharpe Ratios and alphas
•
“Well performing” individual stocks held in
diversified portfolios can be evaluated by the
stock’s alpha in relation to the stock’s
unsystematic risk.
6-21