This document discusses portfolio risk and return, including expected return, measures of risk like variance and standard deviation, and how diversification can reduce risk. It provides examples of calculating expected return, variance, and standard deviation for individual stocks and portfolios. It then introduces the Capital Asset Pricing Model (CAPM), which specifies the relationship between risk and required return of individual stocks based on the stock's beta. It provides examples of using the CAPM equation to calculate required return given beta and market factors, and calculating beta given expected return and market factors.

1. 1
PORTFOLIO RISK &
RETURN

2. 2
Expected Return
 The future is uncertain.
 Investors do not know with certainty whether the economy will be
growing rapidly or be in recession.
 Investors do not know what rate of return their investments will
yield.
 Therefore, they base their decisions on their expectations
concerning the future.
 The expected rate of return on a stock represents the mean of
a probability distribution of possible future returns on the stock.

3. 3
Expected Return
 The table below provides a probability distribution for the returns on stocks
A and B
State Probability Return On Return On
Stock A Stock B
1 20% 5% 50%
2 30% 10% 30%
3 30% 15% 10%
4 20% 20% -10%
 The state represents the state of the economy one period in the future i.e.
state 1 could represent a recession and state 2 a growth economy.
 The probability reflects how likely it is that the state will occur. The sum of
the probabilities must equal 100%.
 The last two columns present the returns or outcomes for stocks A and B
that will occur in each of the four states.

4. 4
Expected Return
 Given a probability distribution of returns, the expected
return can be calculated using the following equation:
N
E[R] = Σ (piRi)
i=1
 Where:
 E[R] = the expected return on the stock
 N = the number of states
 pi
= the probability of state i
 Ri
= the return on the stock in state i.

5. 5
Expected Return
 In this example, the expected return for stock A would
be calculated as follows:
E[R]A = .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%
 Now you try calculating the expected return for stock
B!

6. 6
Expected Return
 Did you get 20%? If so, you are correct.
 If not, here is how to get the correct answer:
E[R]B = .2(50%) + .3(30%) + .3(10%) + .2(-10%) = 20%
 So we see that Stock B offers a higher expected
return than Stock A.
 However, that is only part of the story; we haven't
considered risk.

7. 7
Measures of Risk
 Risk reflects the chance that the actual return on an investment
may be different than the expected return.
 One way to measure risk is to calculate the variance and
standard deviation of the distribution of returns.
 We will once again use a probability distribution in our
calculations.
 The distribution used earlier is provided again for ease of use.

8. 8
Measures of Risk
 Probability Distribution:
State Probability Return On Return On
Stock A Stock B
1 20% 5% 50%
2 30% 10% 30%
3 30% 15% 10%
4 20% 20% -10%
 E[R]A = 12.5%
 E[R]B = 20%

9. 9
Measures of Risk
 Given an asset's expected return, its variance can be
calculated using the following equation:
N
Var(R) = σ2
= Σ pi(Ri – E[R])2
i=1
 Where:
 N = the number of states
 pi
= the probability of state i
 Ri
= the return on the stock in state i
 E[R] = the expected return on the stock

10. 10
Measures of Risk
 The standard deviation is calculated as the positive square root
of the variance:
SD(R) = σ = σ2
= (σ2
)1/2
= (σ2
)0.5

11. 11
Measures of Risk
 The variance and standard deviation for stock A is calculated as
follows:
σ2
A = .2(.05 -.125)2
+ .3(.1 -.125)2
+ .3(.15 -.125)2
+ .2(.2 -.125)2
= .002625
σΑ = (.002625)0.5
= .0512 = 5.12%
 Now you try the variance and standard deviation for stock B!
 If you got .042 and 20.49% you are correct.

12. 12
Measures of Risk
 If you didn’t get the correct answer, here is how to get it:
σ2
B = .2(.50 -.20)2
+ .3(.30 -.20)2
+ .3(.10 -.20)2
+ .2(-.10 - .20)2
= .042
σΒ = (.042)0.5
= .2049 = 20.49%
 Although Stock B offers a higher expected return than Stock A, it
also is riskier since its variance and standard deviation are greater
than Stock A's.
 This, however, is still only part of the picture because most investors
choose to hold securities as part of a diversified portfolio.

13. 13
Portfolio Risk and Return
 Most investors do not hold stocks in isolation.
 Instead, they choose to hold a portfolio of several stocks.
 When this is the case, a portion of an individual stock's risk can
be eliminated, i.e., diversified away.
 From our previous calculations, we know that:
 the expected return on Stock A is 12.5%
 the expected return on Stock B is 20%
 the variance on Stock A is .00263
 the variance on Stock B is .04200
 the standard deviation on Stock A is 5.12%
 the standard deviation on Stock B is 20.49%

14. 14
Portfolio Risk and Return
 The Expected Return on a Portfolio is computed as the weighted
average of the expected returns on the stocks which comprise the
portfolio.
 The weights reflect the proportion of the portfolio invested in the
stocks.
 This can be expressed as follows:
N
E[Rp] = Σ wiE[Ri]
i=1
 Where:
 E[Rp
] = the expected return on the portfolio
 N = the number of stocks in the portfolio
 wi
= the proportion of the portfolio invested in stock i
 E[Ri
] = the expected return on stock i

15. 15
Portfolio Risk and Return
 For a portfolio consisting of two assets, the above equation can
be expressed as:
E[Rp] = w1E[R1] + w2E[R2]
 If we have an equally weighted portfolio of stock A and stock B
(50% in each stock), then the expected return of the portfolio is:
E[Rp] = .50(.125) + .50(.20) = 16.25%

16. 16
Portfolio Risk and Return
 The variance/standard deviation of a portfolio reflects not only the
variance/standard deviation of the stocks that make up the portfolio
but also how the returns on the stocks which comprise the portfolio
vary together.
 Two measures of how the returns on a pair of stocks vary together
are the covariance and the correlation coefficient.
 Covariance is a measure that combines the variance of a stock’s
returns with the tendency of those returns to move up or down at the
same time other stocks move up or down.
 Since it is difficult to interpret the magnitude of the covariance terms,
a related statistic, the correlation coefficient, is often used to measure
the degree of co-movement between two variables. The correlation
coefficient simply standardizes the covariance.

17. 17
Portfolio Risk and Return
 The Covariance between the returns on two stocks can be
calculated as follows:
N
Cov(RA,RB) = σA,B = Σ pi(RAi - E[RA])(RBi - E[RB])
i=1
 Where:
 σΑ,Β = the covariance between the returns on stocks A and B
 N = the number of states
 pi
= the probability of state i
 RAi
= the return on stock A in state i
 E[RA
] = the expected return on stock A
 RBi
= the return on stock B in state i
 E[RB
] = the expected return on stock B

18. 18
Portfolio Risk and Return
 The Correlation Coefficient between the returns on two stocks can
be calculated as follows:
σA,B Cov(RA,RB)
Corr(RA,RB) = ρA,B = σAσB = SD(RA)SD(RB)
 Where:
 ρA,B
=the correlation coefficient between the returns on stocks A and B
 σA,B
=the covariance between the returns on stocks A and B,
 σA
=the standard deviation on stock A, and
 σB
=the standard deviation on stock B

19. 19
Portfolio Risk and Return
 The covariance between stock A and stock B is as follows:
σA,B= .2(.05-.125)(.5-.2) + .3(.1-.125)(.3-.2) +
.3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105
 The correlation coefficient between stock A and stock B is as follows:
-.0105
ρA,B = (.0512)(.2049) = -1.00

20. 20
Portfolio Risk and Return
 Using either the correlation coefficient or the covariance, the
Variance on a Two-Asset Portfolio can be calculated as follows:
σ2
p = (wA)2
σ2
A + (wB)2
σ2
B + 2wAwBρA,B σAσB
OR
σ2
p = (wA)2
σ2
A + (wB)2
σ2
B + 2wAwB σA,B
 The Standard Deviation of the Portfolio equals the positive
square root of the the variance.

21. 21
Portfolio Risk and Return
 Let’s calculate the variance and standard deviation of a portfolio
comprised of 75% stock A and 25% stock B:
σ2
p =(.75)2
(.0512)2
+(.25)2
(.2049)2
+2(.75)(.25)(-1)(.0512)(.2049)= .00016
σp = .00016 = .0128 = 1.28%
 Notice that the portfolio formed by investing 75% in Stock A and 25%
in Stock B has a lower variance and standard deviation than either
Stocks A or B and the portfolio has a higher expected return than
Stock A.
 This is the purpose of diversification; by forming portfolios, some of
the risk inherent in the individual stocks can be eliminated.

22. 22
Capital Asset Pricing Model
(CAPM)
 If investors are mainly concerned with the risk of their portfolio rather than
the risk of the individual securities in the portfolio, how should the risk of
an individual stock be measured?
 In important tool is the CAPM.
 CAPM concludes that the relevant risk of an individual stock is its contribution
to the risk of a well-diversified portfolio.
 CAPM specifies a linear relationship between risk and required return.
 The equation used for CAPM is as follows:
Ki = Krf + βi(Km - Krf)
 Where:
 Ki = the required return for the individual security
 Krf = the risk-free rate of return
 βi= the beta of the individual security
 Km= the expected return on the market portfolio
 (Km - Krf) is called the market risk premium
 This equation can be used to find any of the variables listed above, given
the rest of the variables are known.

23. 23
CAPM Example
 Find the required return on a stock given that the risk-free rate is
8%, the expected return on the market portfolio is 12%, and the beta
of the stock is 2.
 Ki = Krf + βi(Km - Krf)
 Ki = 8%+ 2(12% - 8%)
 Ki = 16%
 Note that you can then compare the required rate of return to the
expected rate of return. You would only invest in stocks where the
expected rate of return exceeded the required rate of return.

24. 24
Another CAPM Example
 Find the beta on a stock given that its expected return is 12%, the risk-
free rate is 4%, and the expected return on the market portfolio is 10%.
 12%= 4%+ βi(10% - 4%)
 βi = 12% - 4%
10% - 4%
 βi = 1.33
 Note that beta measures the stock’s volatility (or risk) relative to the
market.

25. 25
THANKS TO ALL