1. Measuring Returns Converting Dollar Returns to Percentage Returns
PORTFOLIO RISK AND RETURN
Measuring Returns Converting Dollar Returns to Percentage Returns
An investor receives the following dollar returns a stock investment of $25:
$1.00 of dividends
Share price rise of $2.00
The capital gain (or loss) return component of total return is calculated: ending price – minus beginning price, divided by beginning price
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2. Portfolio’s Risk and 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. 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
% % %
% % %
% % %
% % %
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. Expected Return
Given a probability distribution of returns, the expected return can be calculated using the following equation: N
E[R] = S (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. 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. 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. 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. Measures of Risk Probability Distribution:
State Probability Return On Return On
Stock A Stock B
% % %
% % %
% % %
% % %
E[R]A = 12.5%
E[R]B = 20%

9. Measures of Risk
Given an asset's expected return, its variance can be calculated using the following equation: N
Var(R) = s2 = S 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. Measures of Risk
The standard deviation is calculated as the positive square root of the variance:
SD(R) = s = s2 = (s2)1/2 = (s2)0.5

11. Measures of Risk
The variance and standard deviation for stock A is calculated as follows:
s2A = 0.2( ) ( ) ( ) ( )2 =
sA = ( )0.5 = = 5.12%
Now you try the variance and standard deviation for stock B!
If you got .042 and 20.49% you are correct.

12. Measures of Risk
If you didn’t get the correct answer, here is how to get it:
s2B = .2( )2 + .3( )2 + .3( )2 + .2( )2 = .042
sB = (.042)0.5 = = 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. 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
the variance on Stock B is
the standard deviation on Stock A is 5.12%
the standard deviation on Stock B is 20.49%

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] = S 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. 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. Portfolio Risk and Return
Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows:
s2p = (wA)2s2A + (wB)2s2B + 2wAwBrA,B sAsB OR
s2p = (wA)2s2A + (wB)2s2B + 2wAwB sA,B
The Standard Deviation of the Portfolio equals the positive square root of the the variance.

17. Portfolio Risk and Return
The Covariance between the returns on two stocks can be calculated as follows: N
Cov(RA,RB) = sA,B = S pi(RAi - E[RA])(RBi - E[RB])
i=1
Where:
sA,B = 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. Portfolio Risk and Return
The Correlation Coefficient between the returns on two stocks can be calculated as follows:
Cov(RA,RB)/sA,B
Corr(RA,RB) = rA,B = sAsB = SD(RA)SD(RB)
Where:
rA,B=the correlation coefficient between the returns on stocks A and B
sA,B=the covariance between the returns on stocks A and B,
sA=the standard deviation on stock A, and
sB=the standard deviation on stock B

19. Portfolio Risk and Return
The covariance between stock A and stock B is as follows:
sA,B = .2( )(.5-.2) + .3( )(.3-.2) +
.3( )(.1-.2) +.2( )(-.1-.2) =
The correlation coefficient between stock A and stock B is as follows:
-.0105
rA,B = (.0512)(.2049) =

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:
s2p = (wA)2s2A + (wB)2s2B + 2wAwBrA,B sAsB OR
s2p = (wA)2s2A + (wB)2s2B + 2wAwB sA,B
The Standard Deviation of the Portfolio equals the positive square root of the the variance.

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:
s2p =(.75)2(.0512)2+(.25)2(.2049)2+2(.75)(.25)(-1)(.0512)(.2049)=
sp = = = 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 in the individual stocks can be eliminated.

22. RISK AND RETURN EXAMPLE 2

24. Measuring Risk Ex post Standard Deviation
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25. Measuring Risk Example Using the Ex post Standard Deviation
Problem
Estimate the standard deviation of the historical returns on investment A that were: 10%, 24%, -12%, 8% and 10%.
Step 1 – Calculate the Historical Average Return
Step 2 – Calculate the Standard Deviation

26. Scenario-based Estimate of Risk Example Using the Ex ante Standard Deviation – Raw Data
GIVEN INFORMATION INCLUDES:
Possible returns on the investment for different discrete states
Associated probabilities for those possible returns

27. Scenario-based Estimate of Risk First Step – Calculate the Expected Return
Determined by multiplying the probability times the possible return.
Expected return equals the sum of the weighted possible returns.

28. Scenario-based Estimate of Risk Second Step – Measure the Weighted and Squared Deviations
Now multiply the square deviations by their probability of occurrence.
First calculate the deviation of possible returns from the expected.
Second, square those deviations from the mean.
The sum of the weighted and square deviations is the variance in percent squared terms.
The standard deviation is the square root of the variance (in percent terms).

29. Scenario-based Estimate of Risk Example Using the Ex ante Standard Deviation Formula

30. Expected Return of a Portfolio Example
Portfolio value = $2,000 + $5,000 = $7,000
rA = 14%, rB = 6%,
wA = weight of security A = $2,000 / $7,000 = 28.6%
wB = weight of security B = $5,000 / $7,000 = (1-28.6%)= 71.4%

31. Expected Return and Risk For Portfolios Standard Deviation of a Two-Asset Portfolio using Covariance
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Risk of Asset A adjusted for weight in the portfolio
Risk of Asset B adjusted for weight in the portfolio
Factor to take into account comovement of returns. This factor can be negative.
CHAPTER 8 – Risk, Return and Portfolio Theory

32. Grouping Individual Assets into Portfolios
The riskiness of a portfolio that is made of different risky assets is a function of three different factors:
the riskiness of the individual assets that make up the portfolio
the relative weights of the assets in the portfolio
the degree of comovement of returns of the assets making up the portfolio
The standard deviation of a two-asset portfolio may be measured using the Markowitz model:

33. Range of Returns in a Two Asset Portfolio E(r)A= 14%, E(r)B= 6%
A graph of this relationship is found on the following slide.

34. Expected Portfolio Returns Example of a Three Asset Portfolio5

35. Correlation
The degree to which the returns of two stocks co-move is measured by the correlation coefficient (ρ).
The correlation coefficient (ρ) between the returns on two securities will lie in the range of +1 through - 1.
+1 is perfect positive correlation
-1 is perfect negative correlation
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36. 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 + bi(Km - Krf)
Where:
Ki = the required return for the individual security
Krf = the risk-free rate of return
bi = 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.

37. 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 + bi(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.

38. 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% + bi(10% - 4%)
bi = 12% - 4%
10% - 4%
bi = 1.33
Note that beta measures the stock’s volatility (or risk) relative to the market.