Download Quantitative Investment Analysis by Richard A. DeFusco/ CFA

Topic 4:
Portfolio Concepts
Mean-Variance Analysis
• Mean–variance portfolio theory is based
on the idea that the value of investment
opportunities can be meaningfully
measured in terms of mean return and
variance of return.
Assumptions of the Model
1. All investors are risk averse; they prefer less risk
to more for the same level of expected return.
2. Expected returns for all assets are known.
3.The variances and covariances of all asset
returns are known.
4. Investors need only know the expected returns,
variances, and covariances of returns to
determine optimal portfolios.
5. There are no transaction costs or taxes.
Minimum Variance Frontier
• An investor’s objective in using a mean–
variance approach to portfolio selection is
to choose an efficient portfolio.
– An efficient portfolio is one offering the highest
expected return for a given level of risk as
measured by variance or standard deviation
of return.
– Portfolios that have the smallest variance for
each given level of expected return are called
minimum-variance portfolios.
Portfolio Expected Return
• The expected return for a portfolio is the
weighted average of the expected returns
of the securities in the portfolio.
E(R P )  w1E(R1 )  w 2 E(R 2 )    w n E(R n )
Portfolio Variance
• Although it might seem reasonable for the variance of a
portfolio to be the weighted average of the variances of
the securities in the portfolio, this is incorrect.
• Portfolio variance consists of the variances of the
individual securities, but must also consist of a factor that
measures the interaction of each pair of securities.
• Intuitively, if two risky securities are held in a portfolio,
but Security A tends to do well when Security B does
poorly, and vice versa, a portfolio of the two securities
will have less risk.
– we can account for the relationship between each pair of
securities by using the covariance or the correlation.
– Even though both assets are risky, a combination of the two will
create a portfolio that is less risky than each of its components.
• If we plot two assets in risk/expected return space we
get :
Negative Correlation
Return
A
B
Time
Minimum Variance Frontier: Large Cap
Stocks & Government Bonds
Minimum Variance Frontier for Varied
Correlations
Portfolio Risk for a Two-Asset Case
 2P  w12 12  w 22  22  2w1w 21, 2 1 2

 P  w12 12  w 22  22  2w1w 21, 21 2

1
2
Portfolio Risk Three-Asset Case
 2P  w1212  w 22  22  w 32 32  2w1w 21, 21 2  2w1w 31,313  2w 2 w 32,3 2 3

 P  w   w   w   2w1w 21, 21 2  2w1w 31,313  2w 2 w 32,3 23
2
1
2
1
2
2
2
2
2
3
2
3

1
2
Portfolio Risk and Return n Asset Case
n
E(R P )   w jE(R j )
j1
n
n
 2P   w i w jCov(R i , R j )
i 1 j1
n
w
j1
j
1
Example
• Given the information in Table 11-1, find
the expected return and variance for a
portfolio consisting of 40% in large-cap
stocks and 60% in government bonds.
Example
E ( R P )  w 1E ( R 1 )  w 2 E ( R 2 )
 .40(15%)  .60(5%)
 9%
 2P  w 12 12  w 22  22  2 w 1w 21, 2 1 2
 
 
 .4 2 152  .6 2 10 2  2(.4)(. 6)(0.5)(15)(10)
 108.0
Example
• We can find the expected return and
variance for portfolios with different
combinations of our two assets. Table 11-2
shows the different expected returns and
risk for various portfolios.
Example
Capital Market Line
E(R M  R F )
E(R P )  R F 
P
M
E(R P )  the expected return of portfolio p lying on the CML
R F  the risk free rate
E(R M )  the expected return on the market portfolio
 M  the standard deviation of return on the market portfolio
 P  the standard deviation of return on portfolio p
Capital Asset Pricing Model
• Assumptions of the CAPM
– Investors need only know the expected returns, the
variances, and the covariances of returns to
determine which portfolios are optimal for them.
– Investors have identical views about risky assets’
mean returns, variances of returns, and correlations.
– Investors can buy and sell assets in any quantity
without affecting price, and all assets are marketable
(can be traded).
– Investors can borrow and lend at the risk-free rate
without limit, and they can sell short any asset in any
quantity.
– Investors pay no taxes on returns and pay no
transaction costs on trades.
Capital Asset Pricing Model
E ( R i )  R F  i [ E ( R M )  R F ]
E(R i )  the expected return on asset i
R F  the risk free rate of return
E(R M )  the expected return on the market portfolio
Cov(R i , R M )
i 
Var (R M )
Mean Variance Portfolio Choice Rules
• The Markowitz decision rule provides the
principle by which a mean–variance investor
facing the choice of putting all her money in
Asset A or all her money in Asset B can
sometimes reach a decision.
– This investor prefers A to B if either
• the mean return on A is equal to or larger than that
on B, but A has a smaller standard deviation of
return than B
• the mean return on A is strictly larger than that on
B, but A and B have the same standard deviation
of return.
Decision to Add an Investment to
an Existing Portfolio
• Adding the new asset to your portfolio is
optimal if the following condition is met:
E(R new )  R F  E(R p )  R F 

Corr (R new , R p


 new
p


Market Model
R i   i  i R M   i
R i  the return on asset i
R M  the return on the market portfolio
 i  average return on asset i unrelated to the market return
i  the sensitivit y of the return on asset i to the return on the market portfolio
 i  an error term
E ( R i )   i  i E ( R M )
Var ( R i )  i2  2i
Cov( R i R j )  i j 2
M
Multifactor Models
R i  a i  b i1F1  b i 2 F2  ...  b iK FK   i
R i  the return on asset i
a i  the expected return to asset i
Fk  the surprise in factor k, k  1, 2, . . ., K
b ik  the sensitivit y of the return on asset i to a surprise in factor k
 i  an error term with zero mean that represents the portion of
the portion of the return to asset i not explained by the
factor model
An Example
R i  a i  b i1FINT  b i 2 FGDP   i
R i  the return on stock i
a i  the expected return to stock i
FINT  the surprise in interest rates
b i1  the sensitivit y of the return to stock i to interest rate surprises
FGDP  the surprise in GDP growth
b i1  the sensitivit y of the return to stock i to GDP growth surprises
 i  an error term with zero mean that represents the portion of
the portion of the return to asset i not explained by the
factor model
Arbitrage Pricing Theory (APT) and the Factor Model
• APT relies on three assumptions:
1. A factor model describes asset returns.
2. There are many assets, so investors
can form welldiversified portfolios that eliminate asset-specific risk.
3. No arbitrage opportunities exist among well-diversified
portfolios.
• Arbitrage is a risk-free operation that earns an expected positive net
profit but requires no net investment of money.
• An arbitrage opportunity is an opportunity to earn an expected
positive net profit without risk and with no net investment of money.
Arbitrage Pricing Theory
E (R P )  R F  1 P ,1  ...  1 P ,K
E (R P )  the expected return on portfolio P
R F  the risk free rate
 j  the risk premium for factor j
 P , j  the sensitivit y of the portfolio to factor j
K  the number of factors
Analyzing Sources of Return
• Multifactor models can help us understand in detail the
sources of a manager’s returns relative to a benchmark.
• The return on a portfolio, Rp, can be viewed as the sum
of the benchmark’s return, RB, and the active return.
K
Active Return   [( Portfolio sensitivit y) j (Benchmark sensitivit y) j ]  (Factor return)
j1
 Asset selection
• Active risk is the standard deviation of active returns.
• Tracking error is the total return on a portfolio (gross of
fees) minus the total return on a benchmark.
• The information ratio (IR), is a tool for evaluating mean
active returns per unit of active risk.
j
Analyzing Sources of Return
• How can an analyst appraise the individual contributions
of a manager’s active factor exposures to active risk
squared?
• We can usea a factor’s marginal contribution to active
risk squared (FMCAR).
• With K factors, the marginal contribution to active risk
squared for a factor j, FMCARj is
K
FMCAR j 
b aj  bia Cov(Fj , Fi )
i 1
Active risk squared
Creating a Tracking Portfolio
• In a risk-controlled active or enhanced
index strategy, the portfolio manager may
attempt to earn a small incremental return
relative to the benchmark while controlling
risk by matching the factor sensitivities of
the portfolio to her benchmark.
• A tracking portfolio is a portfolio having
factor sensitivities that are matched to
those of a benchmark or other portfolio.