Laurence Booth Sean Cleary. LEARNING OBJECTIVES Risk, Return, and Portfolio Theory 8 8.1 Distinguish between ex post and ex ante returns and explain how

Laurence Booth

Sean Cleary

LEARNING OBJECTIVES

Risk, Return, and Portfolio Theory8

8.1 Distinguish between ex post and ex ante returns and explain how they are estimated.

8.2 Distinguish between arithmetic and geometric means.8.3 Explain how common risk measures are calculated and what they

mean.8.4 Describe what happens to risk and return when securities are

combined in a portfolio.8.5 Explain what is meant by the “efficient frontier.”.8.6 Define diversification and explain why it is important to investors.8.7 Construct two-security portfolio risk-efficient frontiers.

8.1 MEASURING RETURNS• Ex post returns are past or historical returns, while ex ante returns

are future or expected returns• The income yield is the return earned by investors as a periodic

cash flow• The capital gain (or loss) is the appreciation (depreciation) in the

price of an asset from some starting price, usually the purchase price or the price at the start of a period

• The total return is the capital gain yield plus the income yield• Equations 8-1, 8-2 and 8-3 can be summarized as:

Booth • Cleary – 3rd Edition 3© John Wiley & Sons Canada, Ltd.

0

011

0

1

0

01return total

yield income return (loss)gain capital return total

P

PPCF

P

CF

P

PP

• Figure 8-1 shows the earnings yield on the S&P/TSX Composite Index and the YTM on the long Canada bond since 1986

• Notice that usually yields on bonds exceed stocks, but this relationship actually reversed during the recent financial crisis


8.1 MEASURING RETURNS

• Table 8-1 indicates that the yield gap between common shares and long Canada bonds varies over time depending on bond yields and the perceived riskiness of equity investments

• The yield on the long Canada bond is a fixed return earned by buying and holding the bond until maturity

• Investing in common shares, however, should yield capital gains over the long-term



• Owning assets includes an attachment effect causing some investors to refuse to accept capital losses in the total return calculation until they are realized; paper losses, unlike realized losses, are perceived as not being completely real

• Day traders and investors who must mark to market (i.e., carry securities at the current market value regardless of whether they will be sold at that price), on the other hand, do not suffer from attachment effect

• Securities cannot be sold for the purchase price, but they can be sold for the current market value and the funds reinvested elsewhere.

• This is the basic opportunity cost argument (that is, what is the best alternative use for the funds tied up in the investment?)

• Marking securities to market reflects the economic value of past investment decisions



Measuring Average Returns• The arithmetic mean is commonly used in statistics and is appropriate if

the investment horizon is one year, but the geometric mean is useful for measuring the compound growth rate over multiple periods

• Standard deviation, the square root of variance, is a measure of risk (see next slide)



Measuring Average Returns• Equation 8-4 gives the arithmetic mean and equation 8-5 the geometric

mean:

• Equation 8-6 can be used to estimate expected returns as a probability-weighted average return:

• For short-term forecasts a scenario-based approach (Equation 8-6) makes more sense, but for longer-run forecasts, the historical approach (Equation 8-5) tends to be better because it reflects what actually happens even if it was not expected


n

rn

ii

1(AM)mean Arithmetic

1)r1)...(r1)(r1)(r1((GM)mean Geometric n1

n321

)Prob(1

i

n

iirER


8.2 MEASURING RISK• Risk is the probability of incurring harm, and for our purposes harm means

that the actual return from an investment is less than its expected return• Figure 8-2 shows the annual returns on common asset classes. Notice

that the asset classes with more variable returns are considered more risky


• Equation 8-7 gives the standard deviation (square root of the variance) for a series of historical or ex post returns:

• But this risk was not constant over the 1938 to 2008 period; Figure 8-3 (next slide) shows that the relative risk of equities versus bonds changed over the 1938 to 2008 period: the returns on stocks are less variable in recent years relative to bonds than they were in earlier years


n

ii rr

n 1

2)(1

1post Ex

Asset Class Standard Deviation of Annual Investment Returns (1938 to 2008)

Common shares 16.75%Bonds 9.13%

8.2 MEASURING RISK

• Equation 8-8 gives the standard deviation for a series of forecast or ex ante returns:


n

iiii ERr

1

2)()(Prob anteEx

8.2 MEASURING RISK

8.3 EXPECTED RETURN AND RISK FOR PORTFOLIOS

• A portfolio is a collection of securities, such as stocks and bonds, that are combined and considered a single investible asset

• The expected return on a portfolio is the weighted average of the expected returns on the individual securities in the portfolio, as in Equation 8-9:

• where:ERP = expected return on the portfolioERi = expected return on the security iwi = portfolio weight of security INote that portfolio weights must sum to 1.For a two-security portfolio, Equation 8-9 becomes Equation 8-10:


n

iiiP ERwER

1

)(

)()(1

BAB

n

iiiP ERERwERERwER

• Example: Security A earns 10% and security B earns 8%. Figure 8-4 shows the portfolio return varying the weights in each security between 0% and 100%



• The standard deviation of a two-security portfolio can be estimated using Equation 8-11:

• where:• σP = portfolio standard deviation• COVAB = covariance of the returns of securities A and B

• The covariance is calculated using Equation 8-12:


ABBABBAAP COVwwww 22222

n

iBiBAiAAB rrrrCOV

1,, ))((Prob


• Example: Security A earns 10% and security B earns 8%; Figure 8-5 shows the portfolio standard deviation varying the weights in each security between 0% and 100%



Correlation Coefficient• The correlation coefficient is a statistical measure that

identifies how security returns move in relation to one another and is given by equations 8-13 or 8-14:

• And, substituting into Equation 8-11 gives Equation 8-15:


BA

ABABBAABAB

COVCOV

BAABBABBAAP wwww 22222


Correlation Coefficient• The correlation coefficient will range between -1 and +1 • Securities that have -1 correlation are perfectly negatively

correlated (as one goes up, the other goes down)• Securities that have +1 correlation are perfectly positively

correlated (both go up together)• Securities that have zero correlation have no relationship• The closer the absolute value of the correlation is to 1, the

stronger the relationship between the securities



Correlation Coefficient• Figure 8-7a shows positive correlation between Canadian and

U.S. stock returns



Correlation Coefficient• Figure 8-7b shows no correlation between T-Bill returns and

Canadian stock returns



Correlation Coefficients and Portfolio Standard Deviation• Figure 8-8 shows a non-linear relationship between

correlation coefficient and standard deviation; notice that when perfect negative correlation exists, there is a set of weights for which risk is eliminated



Correlation Coefficients and Portfolio Standard Deviation• Figure 8-9 shows a non-linear relationship between correlation

coefficient and standard deviation; notice that when perfect negative correlation exists, there is a set of weights for which risk is eliminated



Special Case• Equation 8-16 gives the standard deviation of a portfolio

where the correlation between securities A and B is perfectly negative:

• The portfolio weight of security A for this portfolio is then given by Equation 8-17:

• Example: In Figure 8-8 we see the portfolio weights that give zero risk if securities A and B are perfectly negatively correlated are 27.76% in A and 72.24% in B.


BAP ww )1(

BA

Bw


8.4 THE EFFICIENT FRONTIER• For n securities, the expected portfolio return is still a

weighted average of individual security expected returns, while the portfolio standard deviation must take the correlation of each pair of securities into consideration to measure total risk

• Equation 8-18 calculates the risk of a three-security portfolio:

• The more securities in a portfolio, the greater the relative impact of the security co-movements on the overall portfolio’s risk and the lower the relative impact of the individual risks


CBBCCBCAACCABAABBACCBBAAP wwwwwwwww 222222222

• Harry Markowitz was awarded the Nobel Prize in Economics in 1990 for work on portfolio theory in the 1950s

• Markowitz assumed:1. Investors are rational decision makers2. Investors are risk averse, and so must be compensated for

assuming additional risk3. Investor preferences are based on portfolio expected return

and risk, as measured by variance or standard deviation• Based on these assumptions, efficient portfolios can be

constructed from a set of available securities• Efficient portfolios offer the highest expected return for a

given level of risk or offer the lowest risk for a given expected return


8.4 THE EFFICIENT FRONTIER

• Attainable portfolios can be constructed by combining the underlying securities; unattainable ones cannot

• In Figure 8-10, portfolio A is unattainable, portfolios B, D and E lie along the minimum variance frontier, and portfolio C is attainable only if some portion of investible wealth remains uninvested (i.e., C is inefficient)

• The blue line represents the minimum variance frontier, the risk-return combinations available to investors from a given set of securities by allowing portfolio weights to vary



• The minimum variance frontier can be divided into three sections:• Portfolio E is the minimum variance portfolio (MVP), because it has the

minimum amount of risk available for any possible combination of securities

• The segment of the frontier above E is the efficient frontier, while the segment of the frontier below E is the dominated frontier

• The efficient frontier is the set of portfolios that offer the highest expected return for their given level of risk; these are the only portfolios that rational, risk-averse investors would hold



8.5 DIVERSIFICATIONDiversification• Reduce a portfolio’s risk by

spreading investment funds across several assets, with the key being to choose assets whose returns are less than perfectly positively correlated

• Even with random selection or naïve diversification, the risk of a portfolio can be reduced

• Figure 8-11 shows that as the number of securities increases, the diversification benefit decreases; eventually, additional securities are superfluous. Also see Table 8-3 in the text


• Equation 8-19 states:• Total risk = Market (systematic) risk + Unique (non-

systematic) risk• As Figure 8-11 shows, it is not possible to eliminate total risk

through diversification because market risk remains after diversification

• Unique or non-systematic risk can, however, be eliminated with sufficient amounts of diversification

• Diversification therefore adds value to a portfolio by reducing risk without sacrificing return

• Most of the benefits of diversification can be achieved by investing in 40 to 50 different securities


8.5 DIVERSIFICATION

• Theoretically, international diversification could reduce more risk than investing only in domestic capital markets because international economies may not be well correlated

• But, evidence suggests the benefits of international diversification are declining as global securities markets become more integrated


8.5 DIVERSIFICATION

Appendix 8A TWO-SECURITY PORTFOLIO FRONTIERS

• Equation 8A-1 shows the portfolio weight as we change our expected return:

• Substituting equation 8A-1 into equation 8-15 gives equation 8A-2:


BA

BP

ERER

ERERw

BAABBA

BP

BA

BPB

BA

BPA

BA

BPP ERER

ERER

ERER

ERER

ERER

ERER

ERER

ERER

121 2

2

2

2

• Figure 8A-1 shows that:• The risk of the portfolio falls, unless securities A and B are perfectly

correlated, as we add risky security A to the portfolio• The decline in risk is much more dramatic if the securities are

negatively correlated• Eventually, the risk reduction from holding A falls and portfolio risk is

minimized



• Example: Returning to the earlier example, where the correlation coefficient was 0.379, the efficient frontier is given in Figure 8A-2 • Note that axes are flipped because it is customary to have expected

return on the vertical axis and risk on the horizontal axis• Also, security weights fall below zero (short) or above one (borrowed

to invest) to obtain the entire IOS graphed in Figure 8A-2



Appendix 8B VALUE AT RISK (VaR)• Value at Risk (VaR) is a risk-management technique that measures potential

loss (in money terms) that could be exceeded (minimum loss) at a given level of probability

• Example: a $1 million daily VaR at 5% means that there is a 5% chance of losing at least $1 million in one day or, equivalently, a 95% chance that daily losses will be less than $1 million

• There are three ways to estimate VaR:

1. The analytical (variance-covariance) method

2. The historical method

3. The Monte Carlo method• Advantages: quantifies potential loss in simple terms, is widely accepted by

regulators and is versatile• Disadvantages: can be difficult to estimate, can promote a false sense of

security for portfolio managers and may underestimate the severity of worst-case scenarios


The Analytical (Variance-Covariance) Method • The analytical method assumes that portfolio returns are normally

distributed, and requires estimates of portfolio expected returns and standard deviations

• Daily VaR = position dollar value × portfolio return volatility• Portfolio return volatility = portfolio standard deviation multiplied by the

Z-score appropriate for the required probability• The main advantage of the analytical method is its simplicity , but its

disadvantage is the assumption that returns are normally distributed and that estimates of return standard deviations are representative of the future

• Example: Estimate the 5% daily VaR of a $2 million position in the market which has a 2% daily standard deviation in price changes– Note: the Z-score for 5% VaR is 1.65– Daily VaR = $2 million × 1.65 × 0.02 = $66,000


Appendix 8B VALUE AT RISK (VaR)

The Historical and Monte Carlo Methods • Historical Method

– Use actual daily returns from a specified past period to estimate future results

– Implicit assumption: the future will be like the past– The key advantage is that this method is non-parametric: no

probability distribution needs to be assumed for any of the variables– The key disadvantage is that this method may suffer from a lack of

historical data• Monte Carlo Method

– Random outcomes are simulated using a computer to examine the effects of particular sets of risks using probability distributions for each variable of interest

– Historical variance and covariance estimate are often used as inputs– Non-normal probability can be used– For large portfolios, this method can be computationally intensive


Appendix 8B VALUE AT RISK (VaR)

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Laurence Booth Sean Cleary. LEARNING OBJECTIVES Risk, Return, and Portfolio Theory 8 8.1 Distinguish between ex post and ex ante returns and explain how