CHAPTER 13: Asset Demand and Supply Under Uncertainty: the Risk-Return Relationship

CHAPTER 13: Asset Demand and Supply under Uncertainty: The Risk-Return Relationship

FOCUS OF THE CHAPTER

This chapter introduces various concepts of risk, and explains how risk and the expected returns on individual assets and portfolios can be measured using statistical techniques. The concepts and measurement of mean return (expected return), variance, covariance, and correlation coefficient are discussed. An introductory analysis of the optimalportfolio choice is presented. The capital asset pricing model (CAPM) and value-at-risk (VaR) approach are also discussed.

Learning Objectives:

Define risk and uncertainty and explain the difference between them

List some of the more common forms of risk

Describe how to measure and evaluate the return and risk of an asset portfolio

Determine how attitudes toward risk and risk aversion can be expressed

Identify how to select a portfolio and the implications of mixing risky and riskless assets

Describe what the capitalasset pricing model tells us about stock selection and portfolio risk

Analyze how the value-at-risk approach assists in business decision-making

Understand why the capital asset pricing model (CAPM) is a useful tool in finance

Determine how the matching principle aids in managing assets and debts and the role of risk

SECTION SUMMARIES

Risk and Attitudes toward It: Definitions

Risk is the likelihood that an investment may prove to be unprofitable. Risk is not the same as uncertainty (ignorance about the future).

Sources of Risk: Risk originates from a variety of sources, and is often named according to them. The following is a list of such various types of risk: 1) default risk; 2) liquidity risk; 3) market risk; 4) systematic risk; 5) foreign exchange risk; 6) income risk;

7) inflation risk; and 8) tax risk.

The Measurement and Consequences of Risk: Risk can be quantified with the help of statistical methods.

Variations in Asset Return: The rate of return on an investment can vary according to the various states of the world. (An asset may yield 6%, 9%, 12%, 14%, or 15%, depending on three different states of the world.) One can assign a probability (the chance of occurring) to each of the states (i.e., to each of the possible rates of return). A common

way of measuring the risk of an asset is to evaluate the 1) mean return (expected return), and 2) the variance of the returns (or the square root of the variance, called the standard deviation).

The expected return is the weighted average of possible returns, where the probabilities of occurrence are the weights. Given the possible rates of return (R) and their probability (Pr) of occurrence, the expected return (Re ) on an asset can be calculated as follows:

Re = Prs Rs,

where s = 1, 2,...n (s refers to the states of the world)

s = 1

=Pr1 R1 + Pr2 R2 +...+ Prn Rn

The variance (V) of the returns on an asset can be calculated as follows:

V = Prs (Rs-Re)2

s = 1

=Pr1 (R1 -Re)2 + Pr2( R2 -Re)2 +...+ Prn( Rn -Re)2

This measure of the variability of returns may be used as a measure of risk. The variance of a riskless asset is zero.

Correlation of Returns: A portfolio includes more than one asset. The return on one asset may vary systematically with another. The covariance and correlation coefficient are two statistical measures which show the degree of the relationship between the returns of two assets. The covariance (COV) indicates the degree to which the returns of two assets move together in the same or opposite directions.

COVij =  Prs (Ris -Re)(Rjs -Re)

s = 1

where i and j are the two assets.

The correlation coefficient indicates the strength of the linear relationship between two variables.

CORRij = COVij /(Vi1/2 Vj1/2)

where Vi1/2 and Vj1/2 are standard deviations (the square root of the variance) of the assets i and j respectively.

Risk in a Portfolio: The variance of a portfolio provides a measure of portfolio risk.

For a portfolio consisting of only two assets (i and j), the portfolio expected return (Rep) and portfolio variance (Vp) can be calculated as follows:

Rep= ki Rie + kj Rje

Vp= k2i Vi+ k2 j Vj + 2 ki k j COVij

where ki and k j are the proportions invested in i and j, respectively.

By examining the risk and return combinations of all possible portfolios, the efficient frontier, consisting of the portfolios yielding the highest returns at various levels of risk, can be derived.

The inclusion of more than one asset in a portfolio in order to reduce portfolio risk is called portfolio diversification. The inclusion of assets whose returns are uncorrelated reduces portfolio risk.

Preferences and the Equilibrium Portfolio

The equilibrium portfolio is the portfolio which yields the maximum utility subject to the efficient frontier (the optimal portfolio). The preferences of the individual play an important part in the choice of the optimal portfolio. Attitudes toward risk are generally of three types: 1) risk aversion (willing to give up return to reduce risk); 2) risk neutrality (concerned only with maximizing returns); and 3) risk-seeking (willing to give up returns to increase risk). It is argued that an individual’s risk aversion behaviour may change from time to time or under different circumstances.

Choosing among Risky Assets: Individuals maximize the following utility function (U), subject to the efficient frontier.

U = U(Rep, V p)

The indifference curve of a risk-averse individual is upward-sloping. An indifference curve to the left of another indicates a relatively higher level of utility. The optimal portfolio choice (or the optimal combination of expected returns and risk) is determined at a point where the efficient frontier is tangent to an indifference curve (see figure 13.2 on page 231 in the text).

The Case of a Risk-Free Asset: Consider a portfolio consisting of money (a risk-free asset with zero returns) and a corporate bond (a risky asset whose return is the opportunity cost of holding money). The portfolio expected return and variance are as follows:

Rep = km Rme + kb Rbe = kb Rbe, because Rme = 0.

Vp= k2m Vm+ k2 bVb + 2 kb k b COVmb = k2 b Vb

because Vm is zero and therefore,

COVmb is also zero.

The standard deviation of a portfolio (SDp ) is given below.

SDp = ( k2 b Vb)½ = k b SDb

whereSDb is the standard deviation of the bond returns.

The efficient frontier is an upward-sloping straight line in this case, because the expected return and variance of the portfolio depend only on the expected return and variance of the risky asset. A decrease in the returns on the bonds decreases the slope of the efficient frontier. As a result, it can be shown that the risk-averse individual increases the quantity (and proportion) of money held in the portfolio (see figure 13.3 on page 234 in the text). This shows that the quantity of money demanded is negatively related to the interest rate.

Value-at-Risk

The value-at-risk (VaR) approach refers to a set of techniques used to estimate the dollar cost of a portfolio in the worst case scenario, given the probability distribution of the portfolio returns. For example, if there is 5% probability of losing $100,000 over a month, then the VaR at 5% is $100,000. The VaR can be estimated at any probability level. The VaR varies depending on the underlying probability distribution of the gains and losses assumed.

Estimating VaR: Consider a portfolio consisting of assets i and j with a total investment of $50 million.

ki = 0.4 or 40% Rie = 0.10

kj = 0.6 0r 60% Rje = 0.06

SDi= 0.14 SDj = 0.09, CORRij = 0.45(note COVij = SDi SDj CORRij)

Rep= ki Rie + kj Rje = 0.4(0.10) + 0.6(0.06) = 0.076 or 7.6%.

Vp= k2i Vi+ k2 j Vj + 2 ki k j COVij

= 0.42(0.14)2 + 0.62(0.09)2 + 2 (0.4)(0.6)(0.14)(0.09)(0.45) = 0.0088

SDp = (Vp)1/2 = (0.0088)1/2 = 0.094

The Rep and SDp calculated above are the annual returns and their standard deviation. To calculate VaR on a monthly basis these values must be converted to a monthly basis.

Monthly Rep = Rep/ 12 = 0.076/12 = 0.0063

Monthly SDp= SDp/12 1/2 = 0.0271

Assuming that the underlying probability distribution of returns is normal, the VaR at the 5% level can be calculated as follows. Note that 1.65 is the critical value at the 5% level using the normal distribution.

VaR= [Monthly Rep - 1.65(Monthly SDp )] x Total investment

= [(0.0063 - 1.65(0.0271)] x 50

= (-0.0384) x 50 = 1.92 million dollars

The Capital Asset Pricing Model (CAPM)

Total portfolio risk is the sum of systematic risk and unsystematic risk. This implies that the return on an asset depends not only on its own characteristics but also on the returns of the other assets. This is the idea behind the characteristic equation or Sharpe’s model (named after William Sharpe).

Ri = i + i Rm + ei

where Ri is the return on asset i, and Rm is the return on a market portfolio (or all the other assets). i and i are constants. The term ei accounts for the factors other than Rm that influence Ri. The beta of the asset ( i), which can be negative, positive, or zero, measures the strength of the relationship between Ri and Rm. The beta of the asset is a measure of systematic risk. By including the return on a risk-free asset (Ri), the following equation for the risk premium (Ri-RF) can be derived:

Ri - RF = i (Rm-RF) + vi , or Ri = RF + i (Rm-RF) + vi

where vi represents the other factors that can influence the risk premium.

The well-known capital asset pricing model (CAPM) which states that, on average, factors other than the beta of an asset should have no effect on the risk premium, is given below:

Ri - RF = i (Rm-RF)

Note that if I = 0 the risk premium is also zero. It can be shown that i = COVim /Vm

The arbitrage pricing theory, an alternative view to the CAPM, suggests that many other factors ignored by CAPM may influence the beta of an asset.

Matching Assets and Debts

Economic agents typically have assets as well as liabilities (debts). The difference is their net wealth. If asset yields are perfectly correlated with debt costs, the return on net wealth will be constant. Such a perfect correlation is unlikely. Therefore, to minimize the risk to their net wealth, economic agents try to match asset returns with debt costs so that the two are positively correlated.

MULTIPLE-CHOICE QUESTIONS

1. Bond rating schemes help investors identify the bonds associated with various levels of

a) default risk.

b) liquidity risk.

c) market risk.

d) foreign exchange risk.

2. The higher the marketability of an asset

a) the higher the liquidity risk.

b) the lower the liquidity risk.

c) the higher the income risk.

d) the higher the market risk.

3. The expected return on an asset

a) is the actual rate of return on the asset under any possible state of the world.

b) is the minimum possible rate of return on the asset.

c) is the maximum possible rate of return on the asset.

d) is a weighted average of possible rates of return.

4. The variance of returns

a) is a measure of the profitability of an investment.

b) is a measure of the average returns on a portfolio.

c) is a measure of the risk of an asset or a portfolio.

d) is the square root of the standard deviation.

5. If the variance for security A is higher than the variance for security B,

a) security A is a better investment than B.

b) security B is a better investment than A.

c) the possible outcomes from an investment in security A deviate relatively more from the expected return for security A than they do for security B.

d) security B is riskier than security A.

6. Consider two assets, A and B. The covariance of the two assets is -3.0. The variances of the assets A and B are 4.0 and 9.0, respectively. The correlation coefficient between the two assets is

a) - 4.33.

b) + 3.44

c) - 0.50

d) -0.08

7. Consider two assets, A and B, in a portfolio of $100 million. The expected returns on the assets A and B are 5.0 and 8.0, respectively. The investor has invested $80 million in A. The expected return on this portfolio is

a) 9%.

b) 4.5%.

c) 5.6%.

d) 4.2%.

8. By including assets whose returns are uncorrelated in a portfolio,

a) the portfolio can be made riskless.

b) the portfolio risk can be reduced.

c) the portfolio risk cannot be changed.

d) the portfolio risk can only be increased.

9. The set of portfolios that yields the highest expected returns at given levels of risk is given by

a) the feasible set of portfolios.

b) the optimal portfolio.

c) the indifference curve.

d) the efficient frontier.

10. The dollar exposure of a portfolio under the worst possible outcome is measured by

a) the beta () of the asset.

b) the value-at-risk.

c) the portfolio variance.

d) the expected returns on the portfolio.

11. The capital asset pricing model (CAPM)

a) relates the return on a particular asset to the return on a market portfolio.

b) relates the return on a particular asset to the covariance between the returns on a security with the market security.

c) relates the return on a particular asset to the corelation between the returns on a security with the market security.

d) relates the stock of capital to the liabilities of the company.

12. The capital asset pricing model (CAPM) tells us that the expected return on a security will be higher if any of the following increase except

a) the risk free rate.

b) the market rate of return.

c) the standard deviation of returns on the security.

d) the beta of the security.

PROBLEMS

1. Consider the following information about an investment:

State of the WorldProbability of OccurrenceRate of Return

130%10%

240% 8%

320% 5%

410% 1%

a) Calculate the expected returns on this investment.

b) Calculate the variance and standard deviation of returns.

2. Consider the following information about a portfolio consisting of two financial assets, A and B:

Expected rates of return on A and B are 10% and 8%, respectively.

Standard deviations of returns on A and B are 4 and 5, respectively.

The correlation coefficient of returns on A and B is 0.5.

The proportion of A in the portfolio is 40%.

a) Calculate the expected portfolio return.

b) Calculate the covariance of asset returns.

c) Calculate the portfolio risk (variance).

3. Consider the following information about a financial asset A, and a market portfolio consisting of other financial assets(M):

Mean returns on A and M are 9% and 6%, respectively.

Standard deviations of returns on A and M are 10 and 5, respectively.

Covariance of returns on A and M is 40.

a) Calculate the beta of asset A.

b) Write the capital asset pricing model if the return on the risk-free asset is 4%.

c) Predict the return on asset A if the return on M is 16%.

4. What is portfolio diversification? Does it necessarily lead to a lower risk? Explain.

ANSWER SECTION

Answers to multiple-choice questions:

a (see page 237)
b (see page 237)
d (see pages 239-242)
c (see page 242)
c (see page 242)
c (see page 243)
c (see page 244)
b (see pages 244-245)
d (see pages 244-245)
b (see page 251)
c (see pages 253-256)
c (see pages 253-256)

Answers to problems:

1. a) Re =Pr1 R1 + Pr2 R2 + Pr3 R3 + Pr4 R4

=0.3(10) + 0.4(8) + 0.2(5) + 0.10(1)

=3.0 + 3.2 + 1.0 + 1.0

=8.2

b)V =Pr1 (R1-Re)2 + Pr2( R2-Re)2 + Pr3( R3-Re)2 + Pr4( R4-Re)2

=0.3(10-8.2)2 + 0.4(8-8.2)2 + 0.2(5-8.2)2 + 0.10(1-8.2)2

= 0.3(1.8)2 + 0.4(-0.2)2 + 0.2(-3.2)2 + 0.10(-7.2)2

=0.3(3.24) + 0.4(0.04) + 0.2(10.24) + 0.10(51.84)

= 0.972 + 0.016 + 2.048 + 5.184

=8.22

SD = V 1/2 (i.e., SD is the square root of V)

= (8.22)1/2 = 2.867

2. a) Rep = kA RAe + kBRBe = 0.4(10) + 0.6(8) = 8.8 (i.e., 8.8%)

b) COVAB = SDA SDB CORRAB = (4)(5)(0.5) = 10

c) Vp = k2A VA + k2 B VB + 2 kA k B COVAB

= 0.42(4)2 + 0.62(5)2 + 2(0.4)(0.6)10

= 2.56 + 9.00 + 4.8 = 16.36

3. a) A = COVAM / VM = 40/(5)2 = 40/25 = 1.6

b) RA -RF = i (RM - RF)

RA = RF + i (RM - RF)

RA = 4 + 1.6 (RM - RF)

RA = 4 + 1.6 (RM - 4) = (4-6.4) + 1.6 RM = -2.4 + 1.6 RM

c) RA = 4 + 1.6 (RM-RF) = 4 + 1.6 (16-4) = 4 + 19.2 = 23.2 (i.e., 23.2%)

4. In general, the inclusion of more than one asset in a portfolio is referred to as portfolio diversification. The inclusion of more than one asset in a portfolio does not necessarily lead to a lower risk. Risk is measured by the variance or standard deviation of portfolio returns. The variance or standard deviation of a portfolio decreases as more and more uncorrelated or negatively correlated assets are included in the portfolio. The inclusion of more and more positively or highly correlated assets does not lead to lower portfolio risk.

Chapter 13Page 1 of 9