Efficient
Diversification
CHAPTER 6
Diversification and Portfolio Risk

Suppose your portfolio has only 1 stock, how many sources of
risk can affect your portfolio?






Uncertainty at the market level
Uncertainty at the firm level
Market risk
 Systematic or Nondiversifiable
Firm-specific risk
 Diversifiable or nonsystematic
If your portfolio is not diversified, the total risk of portfolio
will have both market risk and specific risk
If it is diversified, the total risk has only market risk
Diversification and Portfolio Risk
Diversification and Portfolio Risk
Figure 6.1 Portfolio Risk as a
Function of the Number of Stocks
Covariance and Correlation



Why the std (total risk) decreases when more stocks are added to
the portfolio?
The std of a portfolio depends on both standard deviation of each
stock in the portfolio and the correlation between them
Example: return distribution of stock and bond, and a portfolio
consists of 60% stock and 40% bond
state
Recession
Normal
Boom
Prob.
0.3
0.4
0.3
stock (%)
-11
13
27
Bond (%)
16
6
-4
Portfolio
Covariance and Correlation

What is the E(rs) and σs?

What is the E(rb) and σb?

What is the E(rp) and σp?

Bond
Stock
Portfolio
E(r)
6
10
8.4
σ
7.75
14.92
5.92
Covariance and Correlation





When combining the stocks into the portfolio, you get the average
return but the std is less than the average of the std of the 2 stocks in the
portfolio
Why?
The risk of a portfolio also depends on the correlation between 2 stocks
How to measure the correlation between the 2 stocks
Covariance and correlation
n
Cov(rs , rb )   pi rs (i )  E (rs ) rb (i )  E (rb ) 
i 1
Corr (rs , rb ) 
Cov(rs , rb )
 s b
Covariance and Correlation

Prob
0.3
0.4
0.3
rs
-11
13
27
E(rs)
10
10
10
rb
16
6
-4
E(rb)
6
6
6
P(rs- E(rs))(rb- E(rb))
-63
0
-51

Cov (rs, rb) = -114
The covariance tells the direction of the relationship between the 2 assets,
but it does not tell the whether the relationship is weak or strong

Corr(rs, rb) = Cov (rs, rb)/ σs σb = -114/(14.92*7.75) = -0.99

Covariance
Cov(r1r2) = r1,212
r1,2 = Correlation coefficient of
returns
1 = Standard deviation of
returns for Security 1
2 = Standard deviation of
returns for Security 2
Correlation Coefficients: Possible Values
Range of values for r 1,2
-1.0 < r < 1.0
If r = 1.0, the securities would be
perfectly positively correlated
If r = - 1.0, the securities would be
perfectly negatively correlated
If ρ = 0, no correlation
Two Asset Portfolio St Dev – Stock and Bond
rp  wB rB  ws rs
E (rp )  wB E (rB )  ws E (rs )
  w   w   2w w   r
  Portfolio Variance
  Portfolio Standard Deviation
2
2
2
2
2
p
B
B
S
S
2
p
2
p
B
S
S
B
B ,S
Numerical Example: Bond and Stock
Returns
E(Bond) = 6%
E(Stock) = 10%
Standard Deviation
Bond = 12% Stock = 25%
Correlation Coefficient
(Bonds and Stock) = 0
Numerical Example: Bond and Stock
Case 1: Weights
Bond = .5
What is the E(rp) and σp
E(rp) = 8%
σp = 13.87%


Stock = .5
Average std = (25+12)/2 = 18.5
By combining stocks, get average return, but the risk is lower
than average
Numerical Example: Bond and Stock
Case 1: Weights
Bond = .75
What is the E(rp) and σp
E(rp) = 7%
σp = 10.96%
Stock = .25
• By combining you get higher return than bond but lower risk
than bond
• This is power of diversification
Two Asset Portfolio St Dev – Stock and Bond
wb b  ws s   p  wb b  ws s
• Std of the portfolio is always smaller than the
weighted average of the 2 std in the portfolio.
• Std of the portfolio is maximized when the correlation
= 1, and is minimized when correlation = -1
•No diversification benefit when the correlation = 1
Figure 6.3 Investment Opportunity
Set for Stock and Bonds
Figure 6.4 Investment Opportunity Set for Stock
and Bonds with Various Correlations
6.3 THE OPTIMAL RISKY PORTFOLIO
WITH A RISK-FREE ASSET
Extending to Include Riskless
Asset


The optimal combination becomes linear
A single combination of risky and riskless
assets will dominate
Figure 6.5 Opportunity Set Using Stocks and
Bonds and Two Capital Allocation Lines
Dominant CAL with a Risk-Free
Investment (F)
CAL(O) dominates other lines -- it has the
best risk/return or the largest slope
Slope = (E(R) - Rf) / 
[ E(RP) - Rf) /  P ] > [E(RA) - Rf) / A]
Regardless of risk preferences combinations
of O & F dominate
Figure 6.6 Optimal Capital Allocation
Line for Bonds, Stocks and T-Bills
Figure 6.7 The Complete
Portfolio
Figure 6.8 The Complete Portfolio –
Solution to the Asset Allocation
Problem
6.4 EFFICIENT DIVERSIFICATION WITH
MANY RISKY ASSETS
Extending Concepts to All
Securities
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

Figure 6.9 Portfolios Constructed from
Three Stocks A, B and C
Figure 6.10 The Efficient Frontier of Risky Assets
and Individual Assets
Portfolio Selection

Asset allocation
Security selection

These two are separable!

Asset Allocation



“Asset allocation accounts for 94% of the differences in
pension fund performance”
Identify investment opportunities (risk-return
combinations)
Choose the optimal combination according to investor’s
risk attitude
Optimal Portfolio Construction
Step 1: Using available risky securities (stocks) to construct
efficient frontier.
Step 2: Find the optimal risky portfolio using risk-free asset
Step 3: Now We have a risk-return tradeoff, choose your
most favorable asset allocation
Expected
Portfolio
Return, rp
Efficient Set
Feasible Set
Risk, p
Feasible and Efficient Portfolios

The feasible set of portfolios represents all
portfolios that can be constructed from a
given set of stocks.

An efficient portfolio is one that offers:

 the
most return for a given amount of risk, or
 the
least risk for a give amount of return.
The collection of efficient portfolios is called
the efficient set or efficient frontier.
Expected
Return, rp
IB2 I
B1
IA2
IA1
Optimal Portfolio
Investor B
Optimal Portfolio
Investor A
Optimal Portfolios
Risk p
What impact does rRF have on
the efficient frontier?

When a risk-free asset is added to the
feasible set, investors can create
portfolios that combine this asset with a
portfolio of risky assets.

The straight line connecting rRF with M,
the tangency point between the line and
the old efficient set, becomes the new
efficient frontier.
Efficient Set with a Risk-Free Asset
Expected
Return, rp
Z
.
B
M
^
rM
rRF
.
A
The Capital Market
Line (CML):
New Efficient Set
.
M
Risk, p
What is the Capital Market Line?
The Capital Market Line (CML) is all
linear combinations of the risk-free asset
and Portfolio M.
 Portfolios below the CML are inferior.

 The
CML defines the new efficient set.
 All investors will choose a portfolio on the
CML.
Expected
Return, rp
CML
I1
^
rM
^r
R
.
.
M
C
C = Optimal
Portfolio
rRF
c
M
Risk, p
Disadvantages of the efficient frontier approach


The efficient frontier was introduced by Markowitz (1952) and
later earned him a Nobel prize in 1990.
However, the approach involved too many inputs, calculations

If a portfolio includes only 2 stocks, to calculate the variance of the
portfolio, how many variance and covariance you need?

If a portfolio includes only 3 stocks, to calculate the variance of the
portfolio, how many variance and covariance you need?

If a portfolio includes only n stocks, to calculate the variance of the
portfolio, how many variance and covariance you need?


n variances
n(n-1)/2 covariances
Single index model
Ri   i   i Rm  ei
Ri  ri  rf
Rm  rm  rf
Ri : risk premium of stock i
Rm : risk premium of market
 i : intercept
 i : responsive ness of stock i to the market
 i Rm : component of return due to uncertaint y at the market level
ei : component of return due to uncertaint y at the firm level
Single index model
 i2   i2 m2   ei2
 i2 : Total risk
 i2 m2 : systematic risk component
 ei2 : specific risk
When we diversify, all the specific risk will go away, the only
risk left is systematic risk component
 p2  12 m2  ..........  n2 m2
Now, all we need is to estimate beta1, beta2, ...., beta n,
and the variance of the market. No need to calculate n
variance, n(n-1)/2 covariances as before
Estimate beta






Run a linear regression according to the index model, the slope
is the beta
For simplicity, we assume beta is the measure for market risk
Beta = 0
Beta = 1
Beta > 1
Beta < 1
Advantages of the Single Index Model
Reduces the number of inputs for
diversification
 Easier for security analysts to specialize
