Ch 13. Return, Risk and
Security Market Line (SML)
1. Expected Returns and Variance
• Until now, we mainly concerned historical returns and
risks. However, in this chapter, we will cover returns and
variance in future. Without estimating returns and
variance in future, we can not make decisions.
• 1) Expected return

E ( R) 
T
 P  E(R )
i 1
i
i
• Ex)
State of Economy
Recession
Boon
Sum
Probability
0.6
0.4
1
Stock L
-20%
70%
Stock U
30%
10%
• E(RL) = 0.6×(-0.2)+0.4×0.7 = 0.16
• E(Ru) = 0.6×(0.3)+0.4×0.1 = 0.22
• If a risk free rate is 8%, then risk premium
is 0.08 and 0.14, respectively.
• 2) (Expected) Variance
T

Var ( R)   Pi  [ E ( Ri )  E ( R)]
2
i 1
• Ex) in the previous example,
• Var(L) = 0.6*(-0.2-0.16)^2+0.4*(0.7-0.16)^2 =
0.1944 and STDEV(L) = 0.4409.
• Var(U) = 0.6*(0.3-0.22)^2+0.4*(0.1-0.22)^2 =
0.0096 and STDEV(U) = 0.09798.
• Which one do you want to buy?
• 3) Portfolios: group of stocks, bonds or
investments.
• Portfolio Weight: percentage of total
capital for individual investment or
security. Sum of all portfolio weights
should be 1.
• (1) Portfolio Expected Returns
E(Rp ) 
E(Rp ) 
T
W
i
i 1
 E ( Ri )
T
 P  E(R
i 1
i
pi
)
• Ex) Assuming weights for L and U are
50%
State of Economy
Recession
Boon
Sum
Weight
Expected Portfolio Return
Probability
0.6
0.4
1
Stock L
-20%
70%
Stock U
30%
10%
50%
50%
Portfolio
5%
40%
19%
• (2) Portfolio Variance
State of Economy
Recession
Boon
Sum
Weight
Expected Portfolio Return
Variance
Probability
0.6
0.4
1
Portfolio
5%
40%
Deivation
1.96%
4.41%
19%
2.94%
• 4) Total return
=Expected Return + Unexpected Return
= E(R) + U
• Total return (actual return) differs from the
expected return, because of surprises that
occur during the year. Surprise usually
relates to the unexpected return portion
and generate the risk of investment.
2. Diversification and Portfolio risk
As Table 13-7 and Figure 13-1 show, the
more assets in your portfolio, the lower the
risk (standard deviation). It means that by
including more assets, you can reduce the
risk –the principle of diversification.
• Also it shows that after a certain point, we
can not reduce the risk furthermore.
1) Total Risk = unsystematic (diversifiable) risk +
systematic (nondiversifiable) risk.
• Systematic risk: risk that has marketwise effects
or influences, called market risk. Ex) inflation or
interest rate
• Unsystematic risk: risk that affects a single asset
or a small group of assets. Ex) strike of union.
• Unsystematic risk is essentially eliminated by
diversification, so a portfolio with many assets
has almost no unsystematic risk.
• Risk decomposition implies that
• R=E(R)+systematic portion +
unsystematic portion
• =E(R) + m + ᵋ
• 2) Systematic risk.
• As discussed, systematic risk would not be
eliminated. The rewards (or risk premium) to
taking risks would directly relate to the amount of
systematic risk in individual asset or portfolio.
• The systematic risk is measured by beta
coefficient. It measures relative movements of
an asset or a portfolio to the market or
systematic risk relative to that in an average risk
asset.
Cov( M , i )
 
Var ( M )
• Portfolio beta is a weighted average of
systematic risks of assets in a portfolio.
n
 port   Wi   i
i 1
• 3. Security Market Line (SML)
• A line showing the relationship between
systematic risks and expected returns.
• The slop of the SML means a reward-to-risk
ratio.
E ( R)  R f

• If the market is in equilibrium, assets should be
on SML. Any asset on SML should have the
same reward to risk ratios (slope).
E ( Ra )  R f
a

E ( Rb )  R f
b
• Ex) Security A has an expected return of 10%
and a beta of 0.7. Security B has an expected
return of 8%. Risk free return is 3%. If the
market is in equilibrium, what would be the beta
of security B?
• 4. Capital Asset Pricing Model (CAPM):
• A model estimating an expected return of
an asset basing on the systematic risk.
• Assuming two assets. One is Security i
and the other one is a market portfolio
supposedly composed of all securities
available in the world. The systematic risk
of the market portfolio is always 1.
• The ratios should be the same.
E ( Ri )  R f
i

E ( Rm )  R f
m
 E ( Rm )  R f
E ( Ri )  R f   i  ( E ( Rm )  R f )
• This is called CAPM. If we know the estimate of
an security, market return and risk free return,
we can estimate the expected return of a
security. Practically, the market index is used as
a proxy for the market portfolio. T-Bill or Bond is
used as a proxy for the risk free security.
• Ex) 3 years T-bond generates a return of
3%. S&P 500 produce a return of 10%.
What is an expected return of a security A
if it has a beta of 0.8?