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CHAPTER 5 Risk and Rates of Return

- Stand-alone risk
- Portfolio risk
- Risk & return: CAPM / SML

Investment returns

The rate of return on an investment can be

calculated as follows:

For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is:

($1,100 - $1,000) / $1,000 = 10%.

What is investment risk?

- Investment risk is related to the probability of earning a low or negative actual return. The greater the chance of lower than expected or negative returns, the riskier the investment.

Two types of investment risk

- Stand-alone risk: all our money is tied to a single asset
- Portfolio risk : Asset is held as one of many assets in the portfolio

Probability distributions

- A listing of all possible outcomes, and the probability of each occurrence.
- Can be shown graphically.

T-bills

- T-bills will return the promised return, regardless of the economy.
- T-bills do not provide a risk-free return, as they are still exposed to inflation. Although, very little unexpected inflation is likely to occur over such a short period of time.
- T-bills are also risky in terms of reinvestment rate risk.
- T-bills are risk-free in the default sense of the word.

How do the returns of HT and Coll. behave in relation to the market?

- HT – Moves with the economy, and has a positive correlation. This is typical.
- Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.

Expected Return

- Weighted average
- Weights are probabilities
- Weights add up to 1

Measuring Stand-alone Risk

Calculating the standard deviation for each alternative

standard deviation is the square root of variance. It has

same unit as expected return

Comments on standard deviation as a measure of risk

- Standard deviation (σi) measures total, or stand-alone, risk.
- The larger σi is, the lower the probability that actual returns will be closer to expected returns.
- Larger σi is associated with a wider probability distribution of returns.
- Difficult to compare standard deviations, because return has not been accounted for.

Investor attitude towards risk

- Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.
- Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.

Portfolio construction: Risk and return

- Assume a two-stock portfolio is created with $50,000 invested in HT and $50,000 invested in Collections.
- Expected return of a portfolio is a weighted average of each of the component assets of the portfolio.
- Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.

An alternative method for determining portfolio expected return

In recession, $ return from HT$50,000*(-22%)= -$11,000

$ return from Coll $50,000*(28%)= $14,000

Total $ return is $3,000 so % return is 3%

Comments on portfolio risk measures

- σp = 3.3% is much lower than the σi of either stock (σHT = 20.0%; σColl. = 13.4%).
- σp = 3.3% is lower than the weighted average of HT and Coll.’s σ (16.7%).
- Portfolio provides average return of component stocks, but lower than average risk.
- Why? Negative correlation between stocks.

Combination line

- The risk of a portfolio is very different from a simple average of the risk of individual assets in the portfolio. There is a risk reduction from holding a portfolio of assets if assets do not move in perfect unison.
- Combination line shows how the expected return and risk of a two-security portfolio changes as weights are changed.

General comments about risk

- Most stocks are positively correlated with the market (ρk,m 0.65).
- σ 35% for an average stock.
- Combining stocks in a portfolio generally lowers risk.

Illustrating diversification effects of a stock portfolio

To examine the relationship between portfolio size and portfolio

risk, consider average annual standard deviations for equally-

weighted portfolios that contain different numbers of randomly

selected NYSE securities.

diversification

Principle of diversification: Spreading an investment across many assets will eliminate some of the risk.

Breaking down sources of risk

Stand-alone risk = Market risk + Firm-specific risk

- Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta.
- Firm-specific risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.

In other words

- Firm-specific or diversifiable risk is caused by such random events as lawsuits, strikes and other events that are unique to a particular firm. Since these events are random, their effects on a portfolio can be eliminated by diversification-bad events in one firm will be offset by good events in another.
- Market or non-diversifiable risk stems from factors that systematically affect most firms: war, inflation, recessions, and high interest rates. Since most stocks are negatively affected by these factors, market risk cannot be eliminated by diversification.

single index model

Models return generating process. It offers significant new insights

into the nature of systematic vs. firm-specific risk

ri - rf = i + i(rM-rf) + i single index model

Single-index (factor) model assumes

Var(ri - rf )= Var(i + i(rM-rf) + i)= i2Var(rM-rf) + Var(i)

Total risk = market risk + firm-specific risk

Well-diversified portfolio

Well-diversified portfolio

rp= Σwiri = Σwi(rf +i + i(rM-rf) + i)

=rf + Σwii + (Σwii)(rM-rf) + Σwii

Last term becomes zero

rp-rf = Σwii + (Σwii)(rM-rf) = p + p(rM-rf)

So well-diversified portfolio does not have firm-specific component

Var(rp-rf )= p2Var(rM-rf)

Total risk of well-diversified portfolio is proportional to total risk of

market portfolio

For the pricing of risk what is the relevant measure?

- If investors are primarily concerned with the riskiness of their portfolios rather than the riskiness of the individual securities in the portfolio, how should the riskiness of an individual stock be measured?
- CAPM states that the relevant riskiness of an individual stock is its contribution to the riskiness of a well-diversified portfolio.
- The risk that remains after diversifying is market risk, or the risk that is inherent in the market, and it can be measured by the degree to which a given stock tends to move up or down with the market.

contribution of a stock to the riskiness of market portfolio

If there are N securities in the market, it can be shown that

M2=w1M1M+w2M2M+.. wNMNM

So covariance of a security is directly related to the total risk of the

market portfolio. The higher a securities covariance with market,

the higher is security’s contribution to total risk of market.

contribution of a single stock to the riskiness of a well-diversified portfolio.

It is only the market risk of stock A that will affect the risk of the well-diversified portfolio.

Well-diversified

portfolio

Stock A

Has only market risk

But no firm-specific risk

Has both market and firm-specific risks

Capital Asset Pricing Model (CAPM)

Model based upon concept that a stock’s required rate of return is

equal to the risk-free rate of return plus a risk premium that

reflects the riskiness of the stock after diversification.

Assumptions:

- All investors employ Markowitz portfolio theory to find portfolios in the efficient set and then based on their individual risk preferences invest in one of the portfolios in the efficient set
- All investors have the same planning horizon and identical beliefs about the distributions of security returns
- No barriers to flow of capital or information. E.g. no transaction costs, no taxes etc.

Standard CAPMrisk-free lending and borrowing and short sales

Recall the efficient set

- The tangency portfolio and Rf span the efficient set.
- Since investors have homogenous expectations, everybody comes up with the same tangency portfolio.
- A risk-averse investor (investor A) will invest positive amounts in the risk-free asset (lends at Rf) and the tangency portfolio.
- A less risk-averse investor (investor B) will short risk-free asset (borrows at Rf) and buy the tangency portfolio.

In equilibrium, the prices for all assets must adjust so that aggregate amount of borrowing equals aggregate amount of lending.

CAPM

All efficient portfolios are combinations of market with risk free security

rp=wMP rM+wrfP rf

E(rP)=wMP E(rM)+(1-wMP) rf = rf + wMP [E(rM)-rf]

P2=wMP2M2 or P=wMPM

Efficient portfolios’ total risk depends on market risk

[E(rP)- rf]/[E(rM)-rf]=P/M

E(rP)= rf +P [E(rM)-rf]/M

The last relation is called Capital Market Line.

We need a relation between risk and return that holds for all

securities/portfolios. This is call Security Market Line in CAPM.

A measure of market risk: Beta

- The tendency of a stock to move up or down with the market is reflected in its beta coefficient.
- Indicates how risky a stock is if the stock is held in a well-diversified portfolio.

Calculating betas

- Run a regression of past returns of a security against past returns on the market.
- The slope of the regression line (sometimes called the security’s characteristic line) is defined as the beta coefficient for the security.

Illustrating the calculation of beta

Returns of asset i and market are excess returns. Practitioners often use total rather than excess returns. This practice is most common when daily data is used where total and excess returns are almost indistinguishable.

Comments on beta

- If beta = 1.0, the security is just as risky as the average stock.
- If beta > 1.0, the security is riskier than average.
- If beta < 1.0, the security is less risky than average.
- Most stocks have betas in the range of 0.5 to 1.5.

Can the beta of a security be negative?

- Yes, if the correlation between Stock i and the market is negative (i.e., ρi,m < 0).
- If the correlation is negative, the regression line would slope downward, and the beta would be negative.
- However, a negative beta is highly unlikely.

Beta coefficients for HT, Coll, and T-Bills

Slope of the regression line is given by the following formula:

Given our payoff matrix we can calculate:

Comparing expected return and beta coefficients

SecurityExp. Ret. Beta

HT 17.4% 1.30

Market 15.0 1.00

USR 13.8 0.89

T-Bills 8.0 0.00

Coll. 1.7 -0.87

Riskier securities have higher returns, so the rank order is OK.

?

The Security Market Line (SML): Calculating required rates of return

Recall that CAPM is based upon concept that a stock’s required

rate of return is equal to the risk-free rate of return plus a risk

premium that reflects the riskiness of the stock after diversification.

SML equation states that the risk premium is the product of risk and

extra compensation per unit of risk. Risk is measured by beta, and

extra compensation by excess return on market portfolio.

What is the market risk premium?

- Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.
- Its size depends on the perceived risk of the stock market and investors’ degree of risk aversion.
- Varies from year to year, but most estimates suggest that it ranges between 4% and 8% per year.

Expected vs. Required returns

Ceteris paribus as price rises expected return falls

Calculating portfolio beta

Example: Equally-weighted two-stock portfolio

- Create a portfolio with 50% invested in HT and 50% invested in Collections.
- The beta of a portfolio is the weighted average of each of the stock’s betas.

Calculating portfolio required returns

The required return of a portfolio is the weighted

average of each of the stock’s required returns.

kP = wHT kHT + wColl kColl

kP = 0.5 (17.1%) + 0.5 (1.9%)

kP = 9.5%

Or, using the portfolio’s beta, CAPM can be used to

solve for required return.

kP = kRF + (kM – kRF) βP

kP = 8.0% + (15.0% – 8.0%) (0.215)

kP = 9.5%

Factors that change the SML

What if investors raise inflation expectations by 3%,

what would happen to the SML?

recall that kRF =k*+IP

kRF will increase by 3%, RPM stays constant since kM also increases by the same amount

Increase in risk aversion

What if investors’ risk aversion increased, causing the market risk

premium to increase by 3%, what would happen to the SML?

Investors would require higher risk premium per unit of risk

how can we derive the formula for assuming that SML holds (by using single index model)

decomposition of total (stand-alone) risk

Again you may assume that kA and kM show excess returns.

Useful formulas

recall: cov(x,y)=E(xy) – E(x) E(y)

if cov(x,y)=0 then E(xy) = E(x) E(y)

therefore E(xy) = 0 if E(x)=0 or E(y)=0

so since cov(kM,A)=0 and E(A)=0 then E(kMA)=0

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