1. CHAPTER 10
RISK AND RETURN 1

2. Chapter outline Introduction
Assessing the return and risk characteristics of a single security
Assessing the return and risk characteristics of a portfolio
The security market line
Conclusion

3. Learning outcomes By the end of this chapter, you should be able to:
calculate the holding period return of a single security
differentiate between arithmetic and geometric average returns
compute the expected returns of a single security
calculate the historic standard deviation of a single security
determine the expected risk of a single security
compute the coefficient of variation for a single security
calculate the expected return of a portfolio

4. Learning outcomes (cont.)
By the end of this chapter, you should be able to:
explain what is meant by covariance and correlation coefficient
compute the expected risk of a portfolio consisting of two securities
differentiate between systematic and non-systematic risk
calculate the beta coefficient of a single security and a portfolio
compute the required rate of return of a single security and portfolio using the security market line.

5. Introduction
Value-adding capital project yields a return in excess of its required rate of return
Investors demand higher required rates of return from riskier projects
Evaluate the return and risk characteristics of single (individual) securities
Portfolios

6. Assessing the return and risk characteristics of a single security
Return and risk characteristics of financial assets can be assessed from two perspectives:
Ex post (using historic data)
Ex ante (using expected future data)

7. Example 10.1
Suppose you bought shares in the Oasis Company one year ago at c. Today, exactly one year later, you receive a dividend of 400c per share. The current share price is
12 000c. If you choose to sell the shares today, what is the return you earned on this investment?

8. Example 10.1 Where: = historic holding period return
= price of the security at the end of the period
= price of the security at the beginning of the period
= distributions received at the end of the period

9. Calculating historic returns
If the security is held for more than one year we need to compute an average return
Two alternatives:
Arithmetic mean
Geometric mean

10. Example 10.2
Use the following data (in cents) for the Oasis Company to calculate its historic return using both the arithmetic and geometric averaging techniques:
Initial price (P0):
Price at the end of the first year (P1):
Price at the end of the second year (P2):
Dividend received at the end of the 1st year (D1): 400
Dividend received at the end of the 2nd year (D2): 600

11. Example 10.2
Historic holding period return (HPR) at the end of year 1 = 24% (as calculated earlier)

12. Evaluating expected returns
Making estimates about future returns involves uncertainty – probability theory for a solution
A probability refers to the chance or odds that a future event will occur
Probabilities always have to add up to 100%

13. Example 10.3
Assume that you are interested in buying ordinary shares in the Forest Beer Company. This company produces a premium beer sold in upmarket clubs across South Africa. Using your insight of the local economy and demand for beer, you attach the following probabilities to five possible states of the economy. You also determine a likely rate of return of the Forest Beer Company shares for each of the economic states.

14. Example 10.3 What is average expected return of this share?
State of the economy
Probability of this state occurring
Anticipated rate of return if this state occurs
Very deep recession
10%
−15%
Mild recession
20% 0%
Normal growth
40% 5%
Strong growth
Exceptionally strong growth
25%
What is average expected return of this share?
= (0,1)( −15) + (0,2)0 + (0,4)(5) + (0,2)(10) + (0,1)(25) = 5%

15. Evaluating historic risk
Historic risk can be calculated by means of the variance (2) and standard deviation
The bigger the standard deviation:
the more actual returns tend to differ from the average expected return
the more spread out the actual returns tend to be
the more risk there is when investing in a particular security.

16. Evaluating expected risk
Expected risk can also be calculated by means of the variance (2) and standard deviation, but now probabilities are incorporated
What is the expected  of the Forest Beer Company (based on information provided earlier)?
2 = (0.1)(-15-5)2 + (0.2)(0-5)2 + (0.4)(5-5) 2
+ (0.2)(10-5)2 + (0.1)(25-5)2 = 90
 = 9.49%
Interpretation?

17. Coefficient of variation
The CV standardises the risk and return characteristics of securities

18. Example 10.4
Compute the CV for the Forest Beer Company. A competitor the Dune Beer Company has a CV of 1,27. Interpret the CVs of the two shares and indicate which share you would select. Assume that you are a risk averse investor.

19. Example 10.4
CV for the Forest Beer Company = 9,49 ÷ 5 = 1,9
Interpretation: For every 1% return that this company offers, investors can expect returns to vary by 1,9%
Select the Dune Beer Company as it has a lower CV and hence lower risk-per-unit of return

20. Assessing expected portfolio returns
Expected return of a portfolio
Where:
= expected return of a portfolio
= number of securities included in the portfolio
= weight of security (i) in the overall portfolio
= average expected return of the ith security
Expected portfolio return is thus simply a weighted average of the expected returns of the individual securities

21. Average expected return
Example 10.5
You would like to invest in a portfolio consisting of two shares, M and N . You will invest R and R in the two shares respectively. Based on your economic forecasts, the average expected returns of the two shares are 10% and 25%. What is the return that you expect to earn on this portfolio?
Share
Investment
Weight
Average expected return M R
0,25(a)
10% N R
0,75
25%
Total R
1,00
(a) 100 ÷ 400 = 0,25 (or 25%)

22. Assessing expected portfolio risk
Unlike expected portfolio return, expected portfolio standard deviation is NOT a weighted average of the standard deviations of the individual securities
Why?
Because the returns of securities in a portfolio tend to move together
This can be measured through the covariance and correlation coefficient (also called rho)
Covariance is used to compute the standard deviation of the portfolio

23. Assessing expected portfolio risk
Interpretation of the correlation coefficient

24. Example 10.6
You have inherited some money and are considering investing in the ordinary shares of two listed companies. The first company, Country Lodges, is listed in the Travel and Leisure sector, whereas the second company, Oil 4 Africa, is listed in the Oil and Gas sector. More details on the two companies are:
Based on your expectation, what will be the portfolio risk of this two-asset portfolio?
Country Lodges
Oil 4
Total investment
R2m
R3.5m
Expected return
15%
38%
Expected standard deviation 8%
25%
Correlation coefficient
0.40

25. Example 10.6 First find the weights: = 0.4 x 8 x 25 = 80
Investment
Weight
Country Lodges
40%
Oil 4
60%
Total
100%
= 0.4 x 8 x 25 = 80
= [(0.42x82) + (0.62 x 252) + (2x0.4x0.6x80)]1/2 = 16.54%

26. Portfolio risk: A closer look
By combining securities that are negatively correlated, the risk of a portfolio can be substantially lowered
Q: Can risk ever be completely eliminated?
A: No! The reason being that total portfolio risk consists of two elements
Diversifiable (company-specific) risk
Non-diversifiable (market) risk

27. Portfolio risk: A closer look
Total portfolio risk = diversifiable risk + non-diversifiable risk

28. Portfolio risk: A closer look
Diversifiable (company-specific) risk
Risk that is unique to one or a few companies
Examples: strikes, shortages of raw materials, pollution, failed marketing campaigns, lawsuits, fraud, etc
Can be diversified away in a large portfolio
Also called non-systematic risk

29. Portfolio risk: A closer look
Non-diversifiable (market) risk
Risk that affects large numbers of companies in a market
Examples: unexpected global market events (such as 9/11), and unexpected changes in economic conditions (such as interest-rate hikes and oil-price surprises)
Cannot be eliminated irrespective of the size of the portfolio 
Also called systematic risk

30. Beta coefficient
As company-specific risk can be diversified away, investors are only rewarded for bearing market risk
A security’s exposure to market risk can be measured by calculating its beta coefficient ()
It indicates the degree to which its returns tend to move with the overall market (FTSE/JSE All Share Index)
= beta coefficient of security i
= covariance of the returns of security i and the overall market
= variance of the returns of all securities contained in the market portfolio

31. Beta coefficient
 = 1: the security’s returns move in sync with those of the market; it has the same risk as the average share
 > 1: the security’s returns are more volatile than those of the average share; it has more market risk
 < 1: the security’s returns are less volatile than those of the average share; it has less market risk
 < 0: the security’s returns move in the opposite direction as compared to the rest of the market
Formula to calculate beta coefficient of a portfolio:

32. Example 10.7
Calculate and interpret the beta coefficient of the following portfolio of three shares:
p = (1÷3)(0,6) + (1÷3)(1) + (1÷3)(1,8) = 1,1
Interpretation: The return of this three-share portfolio is slightly riskier than the average risky portfolio
Beta is used to represent a security’s relevant risk in the Security Market Line (SML)
Share
Investment
Beta coefficient X R
0,6 Y
1,0 Z
1,8

33. The security market line (SML)
Investors should base their investment decisions on the following rules:
If expected return > required return buy the share, as it is undervalued
If expected return < required return do not buy the share as the share is overvalued
The SML provides investors with the required rate of return
Where:
= required rate of return
= risk-free return on a government security
= beta coefficient of the ith security
= return of the overall market portfolio

34. Example 10.8
You are considering whether to invest in the following portfolio, which consists of four shares. The invested amounts and beta coefficients of each share are as follows
Share
Investment (R)
Beta coefficient A
1,2 B
−0,4 C
1,5 D
0,8

35. Example 10.8
The expected rate of return on the FTSE/JSE ALSI is 12% and the risk-free rate on a T-Bill 6%. Based on your economic forecasts, the expected rate of return on the portfolio ( ) is 15%. Should you invest in this portfolio?

36. Example 10.8 Step 1: Calculate the portfolio’s beta Share
Investment (R)
Weight
(wi)
Beta
(i) A
0,10
1,2
0,12 B
0,15
−0,4
−0,06 C
0,25
1,5
0,38 D
0,50
0,8
0,40
Total
1,00

37. Example 10.8
Step 2: Use the SML to calculate the rate that investors should require on this investment to compensate them for market risk
= 6 + 0,835(12-6) = 11.01%

38. Example 10.8
Step 3: Compare the expected portfolio return ( ) with the required rate of return ( )
Expected return on this portfolio = 15%
Required return on this portfolio = 11.01%
Invest in the portfolio because its >
Expected return exceeds its required return

39. Conclusion
Evaluating the return and risk characteristics of financial securities and portfolios relies heavily on the use of statistical techniques.
Risk can never be completely eliminated, but it can be substantially reduced by creating diversified portfolios.
Take into consideration the correlation between securities and the level of market risk to which a portfolio is exposed.

40. Conclusions (cont.)
Investors should only invest in single securities and portfolios of which the expected returns exceed the required rate of returns as indicated by the SML.
Investors demand higher required rates of return from riskier projects.
The return and risk characteristics of securities and portfolios can be evaluated from an ex ante and ex post perspective.

41. Conclusions (cont.)
Portfolio risk can be reduced by investing in securities that are negatively correlated.
The SML uses beta coefficient in determining the rate of return that investors will require before investing.
Investors should only invest if the expected return of a security exceeds the required return in which case the intrinsic value will also exceed the market value.