PORTFOLIO THEORY
Portfolio theory provides a way to create efficient portfolios of assets, such as stocks and bonds.
A portfolio is efficient if it provides the highest expected return for a given level of risk.
In the context of portfolio theory, risk is measured by the standard deviation of assets’ returns.
Suppose you can invest in the following two stocks:
Stock A
15%
18,6%
Expected return
Standard deviation of returns
Correlation coefficient of returns
Stock B
21%
28%
0,5
If xA and xB are the weights assigned to each stock we can create the following portfolios:
xA
xB
E(rP)
σP
100%
0%
15%
18,6%
80%
20%
16,2%
18,3%
60%
40%
17,4%
19,4%
40%
60%
18,6%
21,5%
20%
80%
19,8%
24,5%
0%
100%
21%
28%
So, using more than one stock, we can increase our expected return and lower our risk.
This is the benefit of diversification.
Expected r
B
A
0
Standard Deviation σ
The risk-return combinations you can obtain are on the line AB.
For two assets A and B, the portfolio expected return is given by:
E(rP) = xArA + xBrB
The portfolio’s risk is:
σP2 = xA2 σΑ2 + xΒ2 σΒ2 + 2 xA xΒ σΑ σΒ ρAB
where ρAB is the correlation coefficient of the assets’ returns.
The product σΑ σΒ ρAB is the covariance of assets A and B.
For more than two stocks, the possible risk-return combinations are in the area ABC.
Expected r
B
C
A
Standard Deviation σ
0
An efficient portfolio is one which provides the highest expected return for a certain level of risk, an
investor should choose a portfolio on the segment CB
For n assets, the portfolio expected return is given by:
n
E(r) =
x r
i 1
i i
The portfolio’s risk is:
σP2 =
n
n
j 1
i 1
 x x  
i
j
i
j
 ij
Portfolio theory and borrowing
Suppose we can borrow or lend at the risk free rate.
Expected r
B
Borrow
S
Lend
rf
A
0
Standard Deviation σ
If we can lend or borrow at the risk free rate, we can obtain any risk-return combination on the
straight line starting at rf.
The straight line starting at rf is called the capital allocation line (CAL) and is given by:
 rS  r f

E(rP) = rf + 
 P 
 S

Where E(rP) is the expected return of the portfolio, rf is the risk free rate, rS the expected return of
portfolio S, σS the standard deviation of portfolio S and σP the standard deviation of our portfolio.
From the above, we obtain the separation property which separates the decision between portfolio
choice and risk-return choice.
If portfolio S is the market portfolio consisting of all assets in the economy, then the capital
allocation line is called the capital market line (CML).
Example:
Suppose that for a portfolio M, Ε(rM) = 15%, σM = 16% and rf = 5%. If you invest 50% of your funds
in portfolio M and 50% in the risk free asset, then:
E(rP) = (0,5 x 15%) + (0,5 x 5%) = 10%
Variance σ2 = (0,52 x 16%2) = 0,0064
Standard deviation σ = 0,0064 = 8%
If you borrow an amount equal to the amount you have and invest all the money in portfolio M:
E(rP) = (2 x 15%) - 5% = 25%
σ = (2 x σP) = 2 x 16% =32%
CAPITAL ASSET PRICING MODEL (CAPM)
Market risk
Idiosyncratic or non-systematic risk is the type of risk which affects only one or few assets.
Systematic risk is the type of risk which affects all assets.
Increasing the number of stocks in a portfolio, non-systematic risk is minimized.
Systematic risk is unavoidable.
The CAPM proposes that the risk of a well-diversified portfolio j is given by:
β = σj σmρjm/ σm2 = covjm / σm2
which is known as beta.
The expected return of an asset which is part of a well diversified portfolio is given by:
E(rj) = rf + [β (E(rm) - rf)]
Where E(rm) is the expected return on the market portfolio.
The market portfolio is a portfolio of all the assets of an economy.
The CAPM states that: the risk of a well-diversified portfolio is measured only by the market risk of
the assets included in the portfolio.
E(rP)
Security Market Line
rf
0
beta
Examples
1) You invest half your funds in a stock with β = 0,9 and the other half in a stock with β = 1,4. What
is the beta of your portfolio?
(0,5 x 0,9) + (0,5 x 1,4) = 1,15
2) Suppose a stock has σ = 12% and correlation coefficient with the market portfolio of 0,48. If the
market portfolio has σ = 20%, what is the stock’s beta?
β = (0,12 x 0,48) / 0,2 = 0,288
If E(rm) = 10% and rf = 4%, what is the expected return of the stock?
r = 0,04 + [0,288 x (0,1 - 0,04)] = 5,73%
3) You own a portfolio of stocks worth €100.000 with β = 1,05. If you invest an additional €5.000 in
the stock of the previous example, what is the beta of the new portfolio?
β = [(100/105) x 1,05] + [(5/105) x 0,288] = 1,014
What is the expected return of the portfolio if E(rm) = 10% and rf = 4%?
r = 0,04 + [1,014 x (0,1 - 0,04)] = 10,01%
Limitations of the CAPM
The market portfolio has to include all the assets of an economy so, we cannot measure it.
The results of studies on the validity of the CAPM may reject the model because they use the wrong
market portfolio.
Estimating beta has statistical problems.
Historical data do not confirm the predictions of the model.
Research has established the existence of other systematic factors.
Exercises
1) Portfolios Α and Β have expected returns Ε(rA) = 10% and Ε(rΒ) = 12%. You also estimate that σΑ
= 12% and σΒ = 16%. The correlation coefficient (ρ) between the two portfolios is 50%. The
expected return and risk of the market portfolio is Ε(rΜ) = 11% and σΜ = 14%, while the risk free
rate is 4%.
a) Show how you can build a portfolio with σP = 15% and calculate its expected return.
b) Is this portfolio efficient;
2) Suppose that a well-diversified portfolio A has beta = 0,9. Stock S has σS = 14% and correlation
coefficient with the market portfolio 90%. For the market portfolio, you estimate that Ε(rΜ) = 11%
and σΜ = 14%. The risk free rate is 4%.
a) What is the expected return of portfolio A and stock S based on the CAPM?
b) Would you rather buy portfolio A or stock S?
3) You are offered the following portfolio investments:
Portfolio A
Portfolio B
Expected return
9%
11%
Standard Deviation
14%
16%
The market portfolio has an expected return of 8% and standard deviation of 10%. The risk free rate
is 4%. If you can lend or borrow at the risk free rate, would you invest in one of those portfolios? If
yes, in which one and why?
4) “The benefit of diversification is that if you can find stocks with negative correlation you can
lower your risk without hurting your return”. Evaluate that statement.
5) You own a portfolio of stocks and you consider also investing into a stock for which you have the
following information: σ = 14% and beta = 0,9. The market portfolio has expected return of 9% and
standard deviation of 10%. The risk free rate is 3% and you can borrow or lend at that rate. Your
personal assessment if that the stock will generate a return of 10% in the next period. Calculate the
expected return of the stock according to the capital market line and according to the CAPM. Would
you buy that stock?