1. 5AC007 Business Finance Lecture Week 9
Risk and Return
5AC007 Business Finance
Lecture Week 9

2. Session Outcomes Calculate risk and return for individual assets
Calculate portfolio risk and return
Assess the benefits of a diversified portfolio

3. Return on an equity (share)
p1 - p0 p0 d0 p0
Return, r =
x 100%
x 100% +
Capital Gain/Loss
Dividend Yield

4. Vodafone Group Return = 169.35 – 175.50 175.50 x 100% 5.30% +
Share price as at opening 1st March 2011 = p
Share price as at closing 29th February 2012 = p
Dividend yield = 5.30%
Return = –
175.50
x 100%
5.30% +
Return = % % = 1.796%

5. Risk
Long-run stock returns best described using the normal distribution:
Probability
E(xi) = ∑ piXi
ø = √ ∑ p[xi - E(xi)]2 øa øb
E(xi)
Returns
E(x) = expected value (or mean)
ø = standard deviation (measure of risk)

6. Vodafone plc., in February 2012 and February 2011
Average daily returns = % (-0.026%)
Standard deviation of daily returns = (0.955)

7. Share price and daily returns for Vodafone plc., in February 2012
Date
Share price (in p)
Daily return (In %)
Daily return (in %)
31/01/12
170.8
15/02/12
173.55
01/02/12
170.55
16/02/12
173.85
02/02/12
17/02/12
174.6
03/02/12
175.1
20/02/12
175.4
06/02/12
177.85
21/02/12
175.8
07/02/12
174.95
22/02/12
173.9
08/02/12
173.6
23/02/12
173.5
09/02/12
174.5
24/02/12
171.75
10/02/12
172.65
27/02/12
172
13/02/12
174.4
28/02/12
172.35
14/02/12
29/02/12
169.35

8. Calculation of beta (measure of risk) for Vodafone in February 2012
Vodafone share price
FTSE All Share Index
Date
Price
Daily return on Vodafone shares
% Daily Return
Market daily return
% market daily return
31/01/2012
170.8
01/02/2012
170.55
02/02/2012
03/02/2012
175.1
06/02/2012
177.85
07/02/2012
174.95
08/02/2012
173.6
09/02/2012
174.5
10/02/2012
172.65
13/02/2012
174.4
14/02/2012
173.9
15/02/2012
173.55
16/02/2012
173.85
17/02/2012
174.6
20/02/2012
175.4
21/02/2012
175.8
22/02/2012
23/02/2012
173.5
24/02/2012
171.75
E-05
27/02/2012
172
28/02/2012
172.35
29/02/2012
169.35
Covariance with the market
E-05
Market Variance
E-05
Beta

9. Vodafone share price
file://localhost/Users/stuartfarquhar/Documents/5AC007 Business Finance/Lectures 2012/Lecture Week 9/Vodafoneand FTSE all share data Feb 2012.xls

10. Calculation of Beta Cov (ri, rm) Beta, ß = Var (rm)
Cov (ri, rm) = covariance of returns on the share to returns on the market
Var (rm) = variance of returns on the market

11. Vodafone’s beta for February 2012 & February 2011
Cov (ri, rm)
Var (rm)
Beta, ß =
Beta, ß =
Beta, ß =
0.847 (0.693)
Vodafone’s current beta = 0.69 (source: Reuters)

12. Fundamental Rule of Finance
Investors are risk averse therefore:
Maximise return for any given level of risk OR
Minimise risk for any given level of return
Higher returns compensate investors for higher risk

13. Portfolio of Assets
Most investors do not invest in a single asset, but a portfolio of assets.
A portfolio can be composed of both real assets (e.g. property) and financial assets (e.g., company shares, corporate bonds, government bonds)

14. The mean-variance approach to portfolios (Markowitz, 1952)
Any investment in financial assets (securities) is associated with a fundamental uncertainty about its outcome
As investors are assumed to be risk averse, they will prefer more return to less, and less variance (measure of risk) to more
Thus, investors utility function is an increasing function of the expected return and a decreasing function of the variance of their investment

15. Assumed objective of investors
To maximise the return on their portfolio of investments for a given level of risk OR
To minimise the risk of their portfolio for a given level of return

16. So how do we measure portfolio return and portfolio risk?

17. Expected return on a portfolio of assets
On the basis of the probability distribution of returns, investors can compute the mean return on the securities, which is a measure of the centre of the distribution.

18. Expected return on a portfolio of assets
Portfolio Rate
of Return for n assets
fraction of portfolio
in first asset
rate of return
on first asset = x
fraction of portfolio
in second asset
rate of return
on second asset + x
fraction of portfolio
in nth asset
rate of return
on nth asset
...+ x
i = n RP =
E (RP) =
∑ xi Ri
i = 1

19. Portfolio Return - An Example
Suppose you have a portfolio consisting of two equity assets. You invest 60% of your portfolio in shares of BP and 40% of your portfolio in shares of HSBC. The expected sterling return on your BP shares is 15% and the expected sterling return on your HSBC shares is 10%. Calculate the expected return on your portfolio.
E(Rp) =
[%PBP x E(RBP)]+ [%(RHSBC) x E(RHSBC)]
E(Rp) = (0.6 x 15) + (o.4 x 10) = 13%

20. Student Question
Suppose you have a portfolio consisting of two equity assets. You invest 70% of your portfolio in shares of Aggreko and 30% of your portfolio in shares of Experian . The expected sterling return on your Aggreko shares is 20% and the expected sterling return on your Experian shares is 15%. Calculate the expected return on your portfolio.

21. Expected portfolio return – a further example
You have a portfolio consisting of ten FTSE 100 companies. The proportion of your shares in each company and the expected return on each company is given in the table below.
Company
% of portfolio
Expected Return (%)
Burberry Group 12
Kingfisher 7 5
Compass Group 8
National Grid 15 14
Diageo 10
Next 4
GKN 20
Rolls-Royce Group 3
Johnson Matthey
Schroders 25

22. Portfolio Return - Solution
E(Rp) = (0.12x12) + (0.08x8) + (0.1x10) + (0.15x20) + (0.08x8) + (0.07x5) + (0.15x14) + (0.05x4) + (0.05x3) + (0.15x25)
E(Rp) = 13.27%
The expected return on the ten asset portfolio is 13.27%

23. Risk
Types of Risk
Market Risk - Risk factors affecting the economy as a whole - Also known as Systematic Risk
Specific (Unique) Risk - Risk factors affecting a specific company - Also known as Diversifiable Risk
Diversification - strategy designed to lower risk by spreading the portfolio across many investments

24. Diversifying Risk Portfolio Standard Deviation Specific Risk
Market Risk
Number of Assets

25. Measuring portfolio risk
Portfolio Variance - a measure of the spread/dispersion or variation around the mean of the portfolio returns
Portfolio Standard Deviation - the square root of variance

26. Calculating variance & standard deviation
State of Economy
Rate of Return of Almeria plc
(RA)
Deviation from Expected Return
(Expected return = 0.175)
[(RA - E(RA)]
Squared Value of Deviation
[(RA - E(RA)]2
Depression
-0.20
-0.375
Recession
0.10
-0.075
Normal
0.30
0.125
Boom
0.50
0.325
Variance = /4 = ; SD = √ = = 25.86%

27. Student Question
Calculate the variance & standard deviation for Barak plc
State of Economy
Rate of Return of Barak
(RB)
Deviation from Expected Return
(Expected return = 0.055)
[(RB - E(RB)]
Squared Value of Deviation
[(RB - E(RB)]2
Depression
0.05
Recession
0.20
Normal
-0.12
Boom
0.09

28. Measuring portfolio risk
When computing the variance, we also have to concern ourselves with how the asset returns vary together. We do this by calculating covariance of the returns on one asset to the returns on another asset (or alternatively by calculating the correlation coefficient between the returns on the two assets).
If the returns tend to move in opposite directions, then this produces the effect of reducing the overall variability of the portfolio.

29. Calculating Covariance
State of Economy
Rate of return of Almeria
(RA)
Deviation from Expected Return
[RA - E(RA)]
(expected return = 0.175)
Rate of Return of Barak
(RB)
[RB - E(RB)]
(expected return = 0.055)
Product of Deviations
[RA - E(RA)] x [RB - E(RB)]
Depression
-0.20
-0.375
0.05
-0.005
Recession
0.10
-0.075
0.20
0.145
Normal
0.30
0.125
-0.12
-0.175
Boom
0.50
0.325
0.09
0.035
0.70
0.22
øAB = Cov (RA, RB) = /4 =

30. Covariance
øAB = Cov (RA, RB) = Expected value of [(RA - RA) x (RB - RB)
If the two returns are positively related, then covariance is positive
If they are negatively related, then covariance is negative
If they are unrelated, then covariance will be zero

31. Correlation Coefficient
The correlation coefficient measures the strength of association between two variables
Cov(RA,RB)
SD(RA) x SD(RB)
pAB = Corr (RA, RB) =
The correlation coefficient will always have the same sign as covariance. If covariance is -ve, correlation coefficient is -ve.

32. Correlation Coefficient – an example
Using the data for calculating covariance and standard deviations of Almeria plc, and Barak plc, to calculate the correlation coefficient between the two assets
pAB = Corr (RA, RB) =
Cov(RA,RB)
SD(RA) x SD(RB) = x =
Correlation coefficients vary between +1 and -1
+1 implies perfect positive correlation
-1 implies perfect negative correlation
0 implies no correlation

33. Variance of a Portfolio
Given by

34. The variance of a two-asset portfolio
Var(portfolio) = XA2σA2 + XB2σB2 + 2XAXBσA,B
XA2σA2 = contribution of the variance of asset A
XB2σB2 = contribution of the variance of asset B
2XAXBσA,B = contribution of the covariance between assets A and B

35. The return & variance of a two asset portfolio - example
Assume you have 60% of your portfolio in Almeria with an expected return of 17.5% and 40% in Barak with an expected return of 5.5%. The variance of Almeria is , standard deviation is The variance of Barak is , standard deviation is Covariance between Almeria and Barak is
Var(portfolio) = XA2σA2 + XB2σB2 + 2XAXBσA,B
Var (portfolio) = 0.62 x x (2 x 0.6 x 0.4 x )
Variance (portfolio) =
Standard deviation = √ = = 15.44%

36. The variance of a two asset portfolio – alternative approach
As Cov (RA, RB) = Corr (RA, RB) øAøB
We can express var(pf) in terms of correlation coefficient as follows:
Var(portfolio) = XA2σA2 + XB2σB2 + 2XAXBpA,BσAσB
XA2øA2 = contribution of the variance of asset A
XB2øB2 = contribution of the variance of asset B
pA,B = correlation coefficient between assets A and B
øA = standard deviation of asset A
øB = standard deviation of asset B

37. The variance of a two asset portfolio – alternative approach
The correlation coefficient between Almeria and Barak is
Var(portfolio) = XA2σA2 + XB2σB2 + 2XAXBpA,BσAσB
Var(portfolio) = 0.62 x x (2 x 0.6 x 0.4 x x x 0.115)
Variance (portfolio) =
Standard deviation = √ = = 15.44%

38. The Diversification Effect
Weighted average of standard deviations of Almeria and Barak
WA of SD = XAσA + XBσB
WA of SD = (0.6 x ) + (0.4 x 0.115) = 20.12%
SD (portfolio) = 15.44%
SD (portfolio) is lower than WA of SD due to correlation coefficient between Almeria and Barak being less than +1.

39. Summary
A diversified portfolio where the correlation coefficient is less than +1 will have lower standard deviation, than the weighted average of the standard deviations of the assets. This means we can lower risk through investing in a portfolio of assets.
Next week, we will examine risk, return and capital budgeting (investment decisions)

40. Reading
Arnold, G. (2007) Essentials of corporate financial management, Harlow, Essex: Pearson Education Ltd. - chapter 5
Block, S.B. & Hirt, G.A. (2008) Foundations of financial management, 12th ed., New York, New York: McGraw-Hill/Irwin - chapter 13
Brealey, S.C., Myers, S.C., & Marcus, A.J. (2009) Fundamentals of corporate finance, 6th ed., New York, New York: McGraw-Hill/Irwin - chapter 11
Marney, J-P., & Tarbert, H. (2011) Corporate finance for business, Oxford: Oxford University Press Chapter 5
Ross, S.A., Westerfield, R.W., & Jordan, B.D. (2008) Essentials of corporate finance, 6th ed., New York, New York: McGraw-Hill/Irwin - chapter 10