LECTURE 3: RISK AND RETURN

3.0 INTRODUCTIONIn this lecture we explain how risk and returns of an asset are measured and the trade-off between risk and return. We also introduce the portfolio theory, the diversification effect, and the concept of market efficiency.

ObjectivesAt the end of this lecture you should be able to:

1. Explain the fundamentals of risk, return, and risk preferences.

2. Describe the procedures for measuring risk and return of single assets and portfolio of assets.

3. Show the effect of correlation and diversification on risk and return.

4. Discuss the meaning of beta and explain ther basics of the capital market pricing model (CAPM) and the security market line (SML).

Valuation and an understanding of the trade-off between risk and return form the

foundation of shareholders wealth maximization. Each financial decision presents certain

risk and return characteristics, and the unique combination of these characteristics

impacts on the value of the firm. We shall consider risk and return as they relate to both

single assets (arising share) and to a portfolio of assets.

3.1 FUNDAMENTALS OF RISK, RETURN AND

PREFERENCES

3.1.1 Risk

The term risk is used interchangeably with the term uncertainty to refer to the variability

of actual returns from those expected from a given asset. It is the chance of an

unexpected financial loss (or gain). The greater the variability the higher risk. Different

assets will have varying risk levels. For example, a government bond that guarantees its

holder Sh 100 interest after 30 days has no risk – it is risk free, because the return is

certain. On the hand, a Sh.1000 investment in a certain company’s shares which over the

1

same period could earn from Sh.0 to Sh.200 is very risky due to the high variability of

returns.

3.2.2 Return

The return on an asset is the total gain or loss experienced on an investment over a given

period of time. It is commonly measured as the change in value plus any cash

distribution during the period, expressed as a percentage of the beginning of the period

investment value.

The following equation captures the essence of this value.

kt = (Ct + [Pt – Pt-1])/ Pt-1 (3.1)

Where kt = actual, expected, or required rate of return during period t

Pt = Price (value) of asset at end of time period t

Pt-1 = Price value of asset atend of time period t-1

Ct = Cash (flow) received from the asset investment in the time period t.

t may be one day, one month, one year or 10 years. When it is one year kt represents an

annual rate of return. The return could be positive or negative in the event of a loss.

3.2.3 Risk Preferences

The three basic risk preference behaviors among managers are – risk-aversion, risk-

indifference and risk-seeking.

Risk-indifference, is the attitude toward risk in which no change in return would be

required for an increment risk

Risk-aversion is the attitude toward risk in which an increased return would be required

for an increase in risk.

Risk seeking is the attitude toward risk in which a decreased return would be accepted

for an increase in risk.

Figure 3.1 graphically illustrates the three risk preferences.

2

Figure 3.1 shows the three risk preferences of risk aversion, risk seeking and risk

indifference

Most managers and investors are risk-averse; for an increase in risk they require an,

increase in returns. Consequently, managers and investors tend to be conservative rather

than aggressive in accepting risk. Accordingly, unless specified otherwise, a risk averse

financial behavior will be assumed in this lecture.

3.3 RISK AND RETURN OF A SINGLE ASSET

For risky securities the actual rate of return can be viewed as a random variable subject to

a probability distribution (a set of possible values that a random variable can assume and

their associated probabilities of occurrence). This probability distribution for normal

populations) can be summarized in terms of two parameters: (1) the expected return

(mean), and (2) the standard deviation

Risk

Risk seeking

Risk indifference

Risk averse

Return

3

3.3.1 Expected Return

The expected return is the weighted average of possible returns, with the weight being the

probabilities of occurrences. The expected value of a return, k, is

(3.2)

Where ki = return for the ith outcome

Pri = Probability of occurrence of the ith outcome

n = Number of outcomes considered.

Example

Asset A’s return distribution is given in column 1 and 2 in the table below. Using the data

in the two columns, the expected return and the standard deviations (risk) of the asset are

computed in columns 3 and 4 in the same table.

(1) (2) (3) (4)

Possible returns Probability of

occurrence

Expected return Variance

= *

0.10 0.05 0.005 0.0000018

0.02 0.10 0.002 0.0005476

0.04 0.20 0.008 0.0005832

0.09 0.30 0.027 0.0000048

0.14 0.20 0.028 0.0004232

0.20 0.10 0.020 0.0011236

0.28 0.05 0.014 0.0017298

The expected return, , is 9.4% as computed in column 3 above.

3.3.2 Standard Deviation

4

The risk of a return can be measured by the returns’ standard deviation or variance. The

standard deviation is a statistical measure of the dispersion of a distribution around the

expected value. It is the square root of the variance (the sum of squared deviations) of the

distribution. The standard deviation is the most common statistical measure of an asset’s

risk. The equation for determining the standard deviation (s.d.) is as follows

(3.3)

Using above preceding example we determine variance as in column 4 of the table.

= 0.0664

= 6.64%.

The greater the standard deviation, the riskier the asset.

3.3.3 Coefficient of Variation

The standard deviation can sometimes be misleading in comparing the risk of investment

alternatives if the alternatives differ in size. The coefficient of variation, CV, is a measure

of relative dispersion that is useful in comparing the risk of assets with differing expected

returns. The CV is a measure of risk that neutralizes the influence of size of the

investment.

The coefficient of variation (CV) is given by the formula,

(3.3)

The CV , thus, represents the amount of variation in the returns per unit of return.

The higher the CV, the greater the risk. The real utility of the CV comes in comparing

the risk of assets that have different expected returns.

5

Example

Consider investment opportunities A and B whose normal probability distributions of

one-year return are as follows.

Characteristics Investment A Investment B

Expected return 0.08 0.24

Standard deviation 0.06 0.08

CV 0.75 0.33

Based solely on the standard deviation, Investment B would be riskier than A (‘A’ would

be preferred). This would not be a rational investment decision because if one considers

the coefficient of variation, which measures the relative dispersion of risk – risk per unit

of expected return – it is evident that Investment B has the lower risk. B should therefore

be preferred over A.

3.4 RISK AND RETURN IN A PORTFOLIO CONTEXT

A portfolio is a combination of two or more assets. The risk of any single proposed asset

investment should not be viewed independently of other assets. New investment must be

considered in light of their impact on the risk and return of the portfolio of asset held by

an investor. The goal should be to create an efficient portfolio – one that minimizes risk

for a given level of return (or that maximizes returns for a given level of risk). We need

to extend our analysis of risk and return to portfolios of assets. Indeed this is only

necessary given that investors usually hold assets not singly but in combinations

.

3.4.1 Portfolio Return

The expected return on a portfolio is the weighted average of the expected returns of the

assets (securities) comprising that portfolio. The weights are equal to the proportions of

total funds investor in each security (weight must sum up to 100%) The general formula

for the expected return of a portfolio, kp, is

(3.5)

6

Where , is the proportion in weight of total funds invested in security j; is the

expected return for security j; and n is the total number of different securities in the

portfolio.

3.4.2 Standard Deviation of a portfolio

The standard deviation of a portfolio returns is found by applying the formula for the

standard deviation of a single asset.. Specifically equation 3.2 would be applied, i.e.

(3.6)

Portfolio risk and co-variation

While the portfolio expected return is a weighted average of return on the individual

assets, the portfolio s.d is not the simple weighed average of the standard deviations of

the individual assets making up the portfolio. The difference is due to the covariance

relationships between the returns on different assets comprising the portfolio, which

affects risk without affecting returns.

Covariance (correlation)

Covariance (correlation) is a statistical measure of the degree to which two variables (i.e.

securities return) move together over time. Positive correlation means that, on average,

the returns of the two assets move in the same direction (i.e. when the returns of one asset

increase (decrease) those of the other asset also increase (decrease)). Negatively

correlation suggests the returns of the two assets move in opposite directions (i.e. when

the returns of one asset increase (decrease) the returns of the other asset decrease

(increase)). Zero correlation would imply that the two variables show no tendency to vary

together.

7

The degree of correlation (co variation) is measured by a correlation coefficient, which

ranges from +1 for perfectly positively correlated series to -1 for perfectly negatively

correlated series. Uncorrelated series will have a coefficient of zero.

The two figures below, Figure 3.2 and Figure 3.3 show the effect of correlation on risk.

N

M

Return

Return

Time

Figure 3.2 A perfectly positively correlated returns of two assets M and N (no reduction in risk)

8

Covariance (correlation) between the returns of assets provides for the possibility of

eliminating some risk without reducing potential return. The result could be that a

combination of individual risky assets could deliver a low risk portfolio as long as the

individual assets’ returns do not move in lock step ( i.e. perfectly positively correlated).

Diversification

Diversification is the combining of assets (securities) in away that reduces risk (it

depends on how the returns of the assets co-vary not on the number of assets in the

basket). Diversification reduces risk because some of each individual security’s

variability is offset by the variability in the opposite direction of other securities Benefits

of diversification, in the form of risk reduction, occur as long as the security are not

perfectively positively correlated. Combining assets with perfect positive correlation does

not diversify risk. Combining assets with perfect negative correlation in returns confers

the greatest diversification impact as it reduces risk to the minimum. Combining assets

with correlation coefficients between +1 to -1 will result in diversification benefits,

Return

Time

N

M

Figure 3.3 Perfectly Negatively Correlated returns of two assets , M and N (variability (risk) reduced to nil).

9

whose magnitude depends on how far away the returns are from being perfectly

positively correlated. To reduce overall risk, it is best to combine or add to the portfolio

assets that have a negative or a low positive correlation.

component assets x and y.

Correlation, Diversification, Risk and return

The calculation of portfolios sad can be found using the following formula:

Portfolio std. deviation = √

Where n is the total number of different securities in the portfolio, and are the

proportions of total funds invested in securities k and j, and COV ( is the

covariance between possible return for securities j and k.

For a portfolio of two assets, X and Y, the portfolio’s standard deviation can be directly

calculated from the standard deviations of both assets using the following formula.

. Or, alternatively,

Where and are the proportion of funds invested in assets x and y, and

are the standard deviations of the returns of assets x and y,

10

is the covariance of the returns of assets x and y, and is the

correlation coefficient between the returns of component assets x and y. Note that =

.

The correlation coefficient takes values between -1, for perfectly negatively correlated

returns, through 0 for uncorrelated returns, to +1 for perfectly positively correlated

returns.

Example

Consider the following terms returns for three different assets A, B and C. Two

portfolios are to be formed from the assets: (i) AB and (2) AC.

The portfolios are formed by combining equal proportions of the component assets.

Asset A Asset B Asset C

8% 16 8%

10% 14% 10%

12% 12% 12%

14% 10% 14%

10% 8% 16%

Find expected returns of the two portfolios. Determine the Std. deviations of the two

portfolios. Comment on the diversification impacts of the portfolios formed.

The expected returns on the assets are as below.

A B C

% % %

0.2 8 1.6 16 3.2 8 1.6

0.2 10 2.0 14 2.8 10 2.0

0.2 12 2.4 12 2.4 12 2.4

11

0.2 14 2.8 10 2.0 14 2.8

0.2 16 3.2 8 1.6 16 3.2

Expected return 12 12 12

The standard deviations of the returns of the assets are similar as follows:

Asset A =

= (0.00032 + 0.00008+ 0+0.00008+0.00032)

=2.83%

Likewise the standard deviations of Asset B and C are 2.83%

Next we need to determine the covariance of returns of the assets being combined.

We begin with portfolio AB.

COV ( =

0.2(0.08-0.12) (0.16-0.12) = -0.00032

0.2(0.10-0.12)(0.14-0.12) = -0.00008

0.2(0,12-0.12)(0.12-0.12) = -0.0000

0.2(0.14-0.12)(0.10-0.12) = -0.00008

0.2(0.16-0.12)(0.08-0.12) -0.00032

-0.0008

12

Covariance of returns of portfolio AC =

0.2 0.08-0.12)(0.08-0.12) = 0.00032

0.2(0.10-0.12)(0.10-0.12) = 0.00008

0.2(0.12-0.12)(0.12-0.12) 0

0.2(0.14-0.12)(0.14-0.12) 0.000008

0.2(0.16x0.12)(0.16-0.12) 0.00032

0.0008

Expected returns of portfolio AB = 0.5x12+0.5x12=12%

Standard deviation of returns of portfolio AB

= = 0

Assets A and B are perfectly negatively correlated and combining them in a portfolio

completely eliminates any variability (risk) in returns.

The expected returns of portfolio AC= 0.5x12+0.5x12=12%

Standard deviation of returns of portfolio AC =

13

=√ (0.0008)

= 0.0283

=2.83%

Assets A and C are perfectly positively correlated (correlation coefficient of

+1).Combining the two assets in a portfolio has no diversification effect as shown by the

unchanged portfolio standard deviation of 2.83%.

Most assets are positively correlated but with a correlation coefficient of less than +1.

Combining them in a portfolio will result in diversification gains depending on how far

from +1 the correlation coefficient is.

3.5 THE CAPITAL ASSET PRICING MODEL (CAPM)

One of the basic theories that links together risk and return for all marketable assets is the

capital asset pricing model (CAPM) initially developed by Sharpe (1964) and Lintner

(1965). A number of other economists subsequently tested, advanced, refined and

extended its applicability (Black (`972), Merton (1973)).

3.5.1 Systematic Vs. Unsystematic Risk

The total risk of an asset can be decomposed into two basic components:

Unsystematic (Diversifiable) Risk

14

This is that part of total risk that can be diversified away by holding the investment in a

suitably wide portfolio. Research has shown that on average, most of the reduction

benefits of diversification can be gained by forming portfolios containing 15 -20

randomly selected securities. Diversifiable risk is the portion of total risk that is

associated with random (idiosyncratic causes which can be eliminated through

diversification. At the limit the market portfolio, comprising an appropriate portion of

each asset in the market has no undiversifiable risk The causes are firm-specific and

include labour unrests, law suits, regulatory action, competition, loss of a key customer

etc.

Non-diversifiable (Systematic) Risk

This is the risk inherent in the market as a whole and is attributable to market wide

factors. This risk component is not diversifiable and must thus be accepted by any

investor who chooses to hold the asset. Factors such as war, inflation, international

incidents, government macroeconomic policies and political events account for non-

diversifiable risk.

Because any investor can costlessly create a portfolio of assets that will eliminate

virtually all diversifiable risk, the only risk relevant in determination of the prices and

returns of an asset is its non-diversifiable risk.

Interpretation

The CAPM links together non-diversifiable risk and the return for all assets. The model is

concerned with: (1) how systematic risk is measured , and (2) how systematic risk affects

required returns and share values. The CAPM theory includes the following propositions:

a. Investors require a return in excess of the risk-free rate to compensate them for

systematic risk.

b. Investors require no premium for bearing unsystematic risk because it can be

diversified away.

15

c. Because systematic risk varies between companies, investors will require a higher

return from investments where systematic risk is greater.

The Formula

The CAPM can be stated as follows.

(3.7)

Where: is the expected return from asset i.

is the risk-free rate of return (return on the 91-day treasury bill

is the return from the market as a whole: The market portfolio will ,

by definition be fully diversified as it comprises all marketable assets.

is the beta factor of asset i..

is the market premium

The Beta Coefficient and the Market Premium

The beta coefficient, , measures the non-diversifiable risk. It is an index of the degree

of volatility of asset i’s returns in terms of the volatility of the returns of the market

portfolio (market’s risk). The beta factor for the market portfolio is 1.0: the risk free asset

will have a beta of 0. Assets that are riskier than the market will have betas > 1.0 while

those which are less risky will have betas less than 1.0.

Example

16

ABC Ltd. wishes to determine the required return on asset Z which has a beta of 1.5. The

risk-free rate of return is found to be 7%; the return on the market portfolio is 11%. Find

the required rate of return on asset Z.

Using the CAPM formula,

= 7% + 1.5(11% - 7%) =7% + 6% = 13%

The markets risk premium of 4% (11% - 7%), when adjusted for asset Z’s index of risk

(beta) of 1.5 results in the asset’s risk premium of 6% (1.5 * 4%). That risk premium

when added to 7% risk-free rate, results in a 13% required rate.

Security Market Line (SML)

When the CAPM is depicted graphically it is called the security market line (SML). In

the graph, risk, as measured by beta, is plotted on the X-axis and the required return are

shown on the Y-axis. Two points to note in graphing the SML are:

(i) The risk-free asset has a beta of 0.

(ii) The market portfolio has a beta of 1.The risk-return trade-off is clearly shown

by the SML. For the preceding example ,the SML will appear as below

return

E(ri) SML

17

m

rf 7%

0 1 beta

(i) Figure 3.1 The figure show the Security Market Line with beta on the x-axis and expected return on the y-axis. Note that the market portfolio has a beta of 1 and the risk free asset of 0.

REVIEW QUESTIONS

1. Define the terms return and risk as they relate to financial decision making.

2. What is the coefficient of variation? When is it preferred over the standard

deviation when company risk?

3. What is an efficient portfolio? Why is the correlation between asset return

important.

4. What is the relationship of total risk, non diversifiable risk and diversifiable

risk? Why is non diversifiable risk the only relevant risk in asset pricing?

5. If corporate managers are risk averse, does this mean that they will not take

risks? Explain.

PROBLEMS

18

3.1 Mbalamwezi Ltd must choose between two assets purchases: The annual rate of

return and related probabilities for the assets are shown below:

Project ABC Project XYZRate of return Probability Rate return Probability

-10%10%20%30%40%45%50%60%70%80%100%

0.010.040.050.100.150.300.150.100.050.040.01

10%1520253035404550

0.050.100.100.150.200.150.100.100.05

a) For each projects, compute

i) The range of possible rates of return

ii) The expected value of return

iii) The standard deviation of returns

iv) The coefficient of variations.

b) Construct a bar chart of each distributions of rates of return.

c) Which project would you consider less risky? Why?

3-2 The following data has been gathered in order to help in graphically estimating

the betas of two assets A and B.

Actual return

Year Market portfolio Asset A Asset B1996199719981999200020012002200320042005

6%2-13-4-8161015813

11%8-4391914181217

16%11-103-33022291926

19

a) On a set of market return (x-axis) – asset return (y-axis) axes, use the data above

to draw the characteristics line for asset A and for asset B (On the same set of

axes).

b) Use the characteristic lines from (a) to estimate the betas for assets A and B.

c) Use the betas in (b) to comment on the relative risks of assets A and B.

3.3 The risk free rate in the economy is currently 8%, with the market return at 12%.

Asset A has a beta of 1.10

a) Draw the security market line (SML)

b) Use CAPM to calculate the required return on asset A. Depict asset A’s

position on the SML in (a)

c) Assume as a result of recent economic events the risk free rate and the

market return have declined to 6% and 10% respectively. Draw the new

SML on same axes as used before and show new position of asset A.

d) Assume as a result of recent events, the market return has risen to 13%.

Ignoring the shift in.

c) Draw the new SML on same axes as before and show the new position for

asset A.

3.5 A company is considering developing and raising two apartment complexes, WA

and HA. The following estimate of cash flows has been generated for each

apartment.

WA HAProbability Annual cash flows Probability Annual cash

flows0.10.20.40.20.1

1,000.0001,500,0003,000.0004,500.0005,000.000

0.20.30.40.1

1,500.0002,500.0003,500.0004,500.000

20

a) Find the expected cash flows from each apartment complex.

b) What is the coefficient of variation for each apartment complex

c) Which apartment complex has more risk?

3-6 The company in the preceding question will hold the apartments for 10 years.

Either apartment would cost sh.10,000,000. The company uses risk adjusted

discount rate when considering investments with coefficient of variation (CV)

greater than 0.35. He estimates the cost of capital to be 12%. For projects with

CV between 0.35 and 0.40, he adds 2% to the cost of capital and for projects with

CV between 0.40 and 0.50 he adds 4%. The company would not consider an

investment with a CV more than 0.50.

a) Compute the risk adjusted net present values for WA and HA apartments.

(Use cash flows from previous problem).

b) Which investment should company accept if the investments are mutually

are exclusive?

c) If projects are not mutuality exclusive, and in the absence of capital

positioning, how would your decision in (b) be affected.

3.6 Tobacco Company of Kenya (TCK) is a stable company with sales growth of

about 5% per year in good or bad economic conditions. Because of this stability

(correlation coefficient with economy of +0.3 and standard deviation of sales of

about 5% from the mean) the management the company can absorb some small

risky outfits, which could add quite a bit of a return with affecting company’s

risk. Two alternative outfits are being considered for acquisition (i.e. ABC and

XYZ) TCK cost of capital is 10%.

Probability After tax cash flows for 10 years

Sh ‘000’

Probability After tax cash flows for 10 yearsSh ‘000’

0.30.30.20.2

600010,00016,00025,000

0.20.20.20.30.1

(1,000)300010,00025,00031,000

21

a) What is the expected cash flow from each outfit.

b) Which outfit has the lower coefficient of variation

c) Compute the net present value of each outfit

d) Which outfit would you pick based on NPV.

e) Would you change your mind if you added the risk dimensions into the problem?

Explain.

f) If ABC had a correlation coefficient with the economy of 0.5 and XYZ had one of

-0.1, which outfit would give you best portfolio effects for risk reduction? Which

would give the highest potential return?

g) What might be the effect of the acquisitions on the market value of TCK shares?

22