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CHAPTER 5Risk and Rates of Return

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

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Risk and Rates of Return

How do you determine the rate of return that an investment in a new, fixed asset should provide?

It will depend on the project’s risk. But how do you define “risk”? And how do you measure risk?

And once you’ve measured the risk, how do you determine the rate of return that is appropriate for that risk?

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Calculating Rates of Return for Stocks

A stock’s rate of return for a past or future year is calculated by:r = D/P0 + (P1 – P0)/P0

The expected rate of return (“expected return”) to be realized from an investment is the mean value of the probability distribution of possible returns.

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What is investment risk? Two types of investment risk

Stand-alone risk Portfolio 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.

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Probability Distributions

Probability Distribution – A list of all the possible outcomes of a future event together with the probability (chance of occurrence) for each outcome.

You can calculate the mean (expected value), the standard deviation, and the variance of a probability distribution.

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Calculating Expected Returns for Stocks

The “expected value of returns” or “expected return” for a stock is the weighted average of the possible outcomes (possible returns) where the weights are the probabilities associated with the outcomes.

If there are n possible outcomes for a given stock:

)(Pr

)(Pr)(Pr)(Prˆ

1

2211

i

n

ii

nn

k

kkkr

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Measuring the Risk of Stocks: The Variance of Returns The standard deviation, denoted by “sigma”(σ), is a

measure of the variability, tightness, or spread of a set of outcomes expressed in a probability distribution.

Variance (σ2) is the standard deviation squared. Variance is the expected value of the squared deviations. Deviationi=

i

N

ii kkVariance Pr)ˆ( 2

1

2

)ˆ( kki

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Selected Realized Returns, 1926 – 2001

Average Standard Return Deviation

Small-company stocks 17.3% 33.2%Large-company stocks 12.7 20.2L-T corporate bonds 6.1 8.6L-T government bonds 5.7 9.4U.S. Treasury bills 3.9 3.2

Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition) 2002 Yearbook (Chicago: Ibbotson Associates, 2002), 28.

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Coefficient of Variation (CV)

A standardized measure of dispersion about the expected value, that shows the risk per unit of return.

^

k

Meandev Std

CV

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How Investors View Risk and Return

Investors like return. They seek to maximize return.

But investors dislike risk. They seek to avoid or minimize risk. Why? Because human beings possess the

psychological trait of “risk aversion” which is a dislike for taking risks.

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Implications of Risk Aversion The “risk-return tradeoff” - Risk averse

investors require higher rates of return to induce them to invest in higher risk securities.

The higher a security’s risk, the higher the return investors demand. Thus, the less they are willing to pay for the investment, i.e. as risk increase, P0 decreases.

Risk averse investors will diversify their investments in order to reduce risk.

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Diversification

Definition - An investment strategy designed to reduce risk by spreading the funds invested across many securities.

It is holding a broad portfolio of securities so as “not to have all your eggs in one basket.”

Since people hold diversified portfolios of securities, they are not very concerned about the risk and return of a single security. They are more concerned about the risk and return of their entire portfolio.

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The Two Components of a Security’s Variance (Risk)

1. Unique Risk - Also called “diversifiable risk” and “unsystematic risk.” The part of a security’s risk associated with random outcomes generated by events specific to the firm. This risk can be eliminated by proper diversification.

2. Market Risk – Also called “systematic risk.” The part of a security’s risk that cannot be eliminated by diversification because it is associated with economic or market factors that systematically affect most firms.

Market risk reflects economy-wide sources of risk that affect most firms and, hence, the overall stock market.

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The Expected Return on a Portfolio of Stocks

Assume N stocks are held in the portfolio.

Stock i is held in the proportion, wi

i

N

iip

NNp

kwk

kwkwkwk

1

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ˆ

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N

iix

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The Variance of Returns for a Portfolio of Stocks

jiijj

N

i

N

jip

ijj

N

i

N

jip

ww

ww

1 1

2

1 1

2

Variance Portfolio

σij= the covariance between stocks i and j ρij= the correlation coefficient for stocks i and j

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Correlation Coefficient

The “Correlation Coefficient” is a measure of the extent that two variables move or vary together.

It ranges between –1.0 and +1.0 Positive correlation: a high value on one variable is

likely to be associated with a high value on the other.

Negative correlation: a high value on one variable is likely to be associated with a low value on the other.

No correlation: values of each are independent of the other

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Correlation Coefficient-Cont’d

It is denoted by the Greek letter, “rho”: ρ If ρ = +1.0, perfect positive correlation If ρ = -1.0, perfect negative correlation If ρ = 0, uncorrelated or independent

ρij = the correlation coefficient for returns of stock i and stock j

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The Variance of Returns for a Portfolio of Stocks

jiijj

N

i

N

jip

ijj

N

i

N

jip

ww

ww

1 1

2

1 1

2

Variance Portfolio

σij= the covariance between stocks i and j ρij= the correlation coefficient for stocks i and j

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How Diversification Reduces Risk

Combining stocks into a portfolio reduces the variability of possible returns as long as the returns on the individual stocks are not perfectly correlated, i.e. as long as their correlation coefficients are less than +1.0.

Assume: Invest 50% in Stock A and 50% in Stock B Stock A: r = 13%; σ = 20%, σ2 = 400 Stock B: r = 13%; σ = 20%, σ2 = 400

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How Diversification Reduces Risk - Cont’d

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Portfolio Risk Falls As You Add Securities

05 10 15

Number of Securities

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You Can’t Eliminate “Market Risk”

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Number of Securities

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Market risk

Uniquerisk

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This Pattern Occurs Because of the Two Components of a Stock’s Variance (Risk):

1. Unique Risk 2. Market Risk

The unique risk is “diversified away” when individual stocks are combined in a portfolio.

Only market risk remains. The amount of the market risk is

determined by the market risk of the individual stocks in the portfolio.

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How Should We Measure Portfolio Risk Now?

Diversification eliminates unique risk and leaves market risk.

Therefore, the relevant measure of risk for a portfolio is the portfolio’s “beta”: a measure of the sensitivity of the portfolio’s returns to changes in the return on the “market portfolio” which is closely approximated by a portfolio consisting of the S&P 500 stocks.

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How Do Investors View the Risk of a Single Security Held in a Portfolio?

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Answer: “Beta” Measures a Stock’s Market Risk

2m

imiB

Covariance with the market

Variance of the market

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How to Interpret a Beta

If βi > 1, returns to stock i are amplified relative to the market.

If βi is between 0 and 1.0, returns to stock i tend to move in the same direction as the market but not as far.

If βi < 1(very rare), returns to stock i tend to move in the opposite direction as the market.

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How To Interpret a Beta-Cont’d

A stock with β = 1 has average market risk. A well-diversified portfolio of such stocks tends

to move by the same percentage as the overall market moves and has the same σ as the overall market.

A stock with β = +.5 has below average market risk. A well-diversified portfolio of these stocks tends

to move half as far as the overall market moves and has half the standard deviation

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Betas Are Calculated Using Regression Analysis

beta

Expected

return

Expectedmarketreturn

10%10%- +

-10%+10%

stock

Copyright 1996 by The McGraw-Hill Companies, Inc

-10%

1. Total risk = diversifiable risk + market risk2. Market risk is measured by beta, the sensitivity to market changes

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General comments about risk

Most stocks are positively correlated with the market (ρi,m 0.65).

σ 35% for an average stock. Combining stocks in a portfolio

generally lowers risk.

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A model that relates an asset’s risk to its rate of return

The “Capital Asset Pricing Model” won the Nobel Prize in economics.

Referred to as the “CAP-M”

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General Comments About CAPM

Concerned with “equilibrium conditions”

CAPM seeks to predict: In equilibrium, what will be the

relationship between expected return and risk for portfolios?

In equilibrium, what will be the relationship between expected return and risk for individual securities?

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Capital Asset Pricing Model

ki = kRF + Bi ( kM - kRF )

CAPM

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CAPM Graphically: The Security Market LineReturn

BETA

kRF

SML

SML Equation: ki = kRF + Bi( kM - kRF )

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Plotting the Security Market Line

Return

BETA

.Risk Free

Rate = kRF

Market Return = kM

Market Portfolio

1.0

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

SML: ki = kRF + (kM – kRF) βi

Assume kRF = 8% and kM = 15%. The market (or equity) risk

premium is RPM = kM – kRF = 15% – 8% = 7%.

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An 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.

βP = wHT βHT + wColl βColl

βP = 0.5 (1.30) + 0.5 (-0.87)

βP = 0.215

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Factors that change the SML What if investors raise inflation expectations

by 3%, what would happen to the SML?

SML1

ki (%)SML2

0 0.5 1.0 1.5

1815

11 8

I = 3%

Risk, βi

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Factors that change the SML What if investors’ risk aversion increased,

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

SML1

ki (%) SML2

0 0.5 1.0 1.5

1815

11 8

RPM = 3%

Risk, βi

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Verifying the CAPM empirically

The CAPM has not been verified completely.

Statistical tests have problems that make verification almost impossible.

Some argue that there are additional risk factors, other than the market risk premium, that must be considered.

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More thoughts on the CAPM Investors seem to be concerned with both

market risk and total risk. Therefore, the SML may not produce a correct estimate of ki.

ki = kRF + (kM – kRF) βi + ???

CAPM/SML concepts are based upon expectations, but betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.