chapter 6 risk and rates of return 2 chapter 6 objectives inflation and rates of return how to...

Chapter 6

Risk and Rates of Return

2

Chapter 6 Objectives

Inflation and rates of returnHow to measure risk

(variance, standard deviation, beta)How to reduce risk

(diversification)How to price risk

(security market line, CAPM)

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Historical Risk and ReturnAnnual From 1926 to 1999

Avg. Return Std Dev.Small Stocks 17.6% 33.6%Large Co. Stocks 13.3% 20.1%L-T Corp Bonds 5.9% 8.7%L-T Govt. Bonds 5.5% 9.3%T-Bills 3.8% 3.2%Inflation 3.2% 4.6%

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Why are these rates different?

90-day Treasury Bill 1.7%90-day Commercial Paper 1.8%2-year US Treasury Note 3.0%10-year US Treasury Note 5.0%10-year Corporate Bond 6.9%

Inflation, Rates of Return, and the Fisher Effect

InterestRates

Interest RatesConceptually:

Interest RatesConceptually:

Nominalrisk-freeInterest

Rate

krf

=

Interest RatesConceptually:

Nominalrisk-freeInterest

Rate

krf

=

Realrisk-freeInterest

Rate

k*

Interest RatesConceptually:

Nominalrisk-freeInterest

Rate

krf

=

Realrisk-freeInterest

Rate

k*

+

Inflation-risk

premium

IRP

Conceptually:

Nominalrisk-freeInterest

Rate

krf

=

Realrisk-freeInterest

Rate

k*

+

Inflation-risk

premium

IRP

Mathematically:

(1 + krf) = (1 + k*) (1 + IRP)

This is known as the “Fisher Effect”

Interest Rates

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Suppose the real rate is 3%, and the nominal rate is 8%. What is the inflation rate premium?

(1 + krf) = (1 + k*) (1 + IRP)(1.08) = (1.03) (1 + IRP)(1 + IRP) = (1.0485), so

IRP = 4.85%

Interest Rates

Term Structure of Interest Rates

The pattern of rates of return for debt securities that differ only in the length of time to maturity.

Term Structure of Interest Rates

The pattern of rates of return for debt securities that differ only in the length of time to maturity.

yieldto

maturity

time to maturity (years)

Term Structure of Interest Rates

The pattern of rates of return for debt securities that differ only in the length of time to maturity.

yieldto

maturity

time to maturity (years)

Term Structure of Interest Rates

yieldto

maturity

time to maturity (years)

The yield curve may be downward sloping or “inverted” if rates are expected to fall.

Term Structure of Interest Rates

yieldto

maturity

time to maturity (years)

The yield curve may be downward sloping or “inverted” if rates are expected to fall.

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Recent US Treasury Yield Curve

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

0 5 10 15 20 25 30 35

Time to Maturity

Yie

ld

Last Semester Last Week

For a Treasury security, what is the required rate of return?

Since Treasuries are essentially free of default risk, the rate of return on a Treasury security is considered the

“risk-free” rate of return.

RequiredRequired

rate of rate of

returnreturn==

Risk-freeRisk-free

rate of rate of

returnreturn

For a corporate stock or bond, what is the required rate of return?

RequiredRequired

rate of rate of

returnreturn==

For a corporate stock or bond, what is the required rate of return?

RequiredRequired

rate of rate of

returnreturn==

Risk-freeRisk-free

rate of rate of

returnreturn

For a corporate stock or bond, what is the required rate of return?

How large of a risk premium should we require to buy a corporate security?

RequiredRequired

rate of rate of

returnreturn== + +

Risk-freeRisk-free

rate of rate of

returnreturn

RiskRisk

premiumpremium

ReturnsExpected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc.

Required Return - the return that an investor requires on an asset given its risk and market interest rates.

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Holding Period (Actual) Returns

The realized return over a period of time (HPR).HPR=(Ending Price - Beginning Price + Distributions Received)/Beginning PriceExample: What is your HPR if you buy a stock for $20, receive $1 in dividends, and then sell it for $25.HPR = ($25-$20+$1)/$20 = 0.3 = 30%

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Calculation of Expected ReturnsExpected Rate of Return (Expected Value) given a probability distribution of possible returns(ki): E(k) or k

_ n

E(k)=k = ki P(ki)

i=1

Realized or Average Return on Historical Data: - n

k = 1/n k i

i=1

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Expected Return and Standard Deviation Example

MAD E(r) = .25(80%) + .60(30%) + .15(-30%) = 33.5%CON E(r) = .25(5%) + .60(10%) + .15(15%) = 9.5%

St at e of Cont raryEconom y Probabilit y MAD I nc. Co. ( CON)Boom 0.25 80% 5%Normal 0.60 30% 10%Recession 0.15 - 30% 15%

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Definition of Risk

Risk is an uncertain outcome or chance of an adverse outcome.Concerned with the riskiness of cash flows from financial assets.Namely, the chance that actual cash flows will be different from forecasted cash flows.Standard Deviation can measure this type of risk.

How do we Measure Risk?

A more scientific approach is to examine the stock’s standard deviation of returns.Standard deviation is a measure of the dispersion of possible outcomes. The greater the standard deviation, the greater the uncertainty, and therefore , the greater the risk.

Standard Deviation

= (ki - k)2 P(ki) n

i=1

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Expected Return and Standard Deviation Example

MAD E(r) = .25(80%) + .60(30%) + .15(-30%) = 33.5%CON E(r) = .25(5%) + .60(10%) + .15(15%) = 9.5%

St at e of Cont raryEconom y Probabilit y MAD I nc. Co. ( CON)Boom 0.25 80% 5%Normal 0.60 30% 10%Recession 0.15 - 30% 15%

MAD, Inc. ( 80% - 33.5%)2 (.25) = 540.56(30% - 33.5%)2 (.6) = 7.35(-30% - 33.5%)2 (.15) = 604.84

Variance = 1152.75%

Stand. dev. = 1152.75 = 34.0%

MAD, Inc. ( 80% - 33.5%)2 (.25) = 540.56(30% - 33.5%)2 (.6) = 7.35(-30% - 33.5%)2 (.15) = 604.84

Variance = 1152.75%

Stand. dev. = 1152.75 = 34.0%

= (ki - k)2 P(ki) n

i=1

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Expected Return and Standard Deviation Example

St at e of Cont raryEconom y Probabilit y MAD I nc. Co. ( CON)Boom 0.25 80% 5%Normal 0.60 30% 10%Recession 0.15 - 30% 15%

MAD E(r) = .25(80%) + .60(30%) + .15(-30%) = 33.5%CON E(r) = .25(5%) + .60(10%) + .15(15%) = 9.5%

Contrary Co. (5% - 9.5%)2 (.25) = 5.06(10% - 9.5%)2 (.6) = 0.15(15% - 9.5%)2 (.15) = 4.54Variance = 9.75%

Stand. dev. = 9.75 = 3.1%

= (ki - k)2 P(ki) n

i=1

Which stock would you prefer?How would you decide?

Which stock would you prefer?How would you decide?

It depends on your tolerance for risk!

Remember, there’s a tradeoff between risk and return.

Return

Risk

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Coefficient of Variation

A relative measure of risk. Whereas, is an absolute measure of risk. Relates risk to expected return.CV = /E(k)MAD’s CV = 34%/33.5% = 1.01CON’s CV = 3.1%/9.5% = 0.33CONtrary is the less risky of the two investments. Would choose CON if risk averse.

Portfolios

Expected Portfolio Return is weighted average of the expected returns of the individual stocks = Σwjkj.However, portfolio risk (standard deviation) is NOT the weighted average of the standard deviations of the individual stocks.Combining several securities in a portfolio can actually reduce overall risk.How does this work?

Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated).

rateof

return

time

kA

kB

rateof

return

time

kpkA

kB

What has happened to the variability of returns for the

portfolio?

Diversification

Investing in more than one security to reduce risk.If two stocks are perfectly positively correlated, diversification has no effect on risk.If two stocks are perfectly negatively correlated, the portfolio is perfectly diversified.

If you owned a share of every stock traded on the NYSE and NASDAQ, would you be diversified?

YES!Would you have eliminated all of your risk?

NO! Common stock portfolios still have risk.

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Some risk can be diversified away and some cannot.

Market risk (systematic risk) is nondiversifiable. This type of risk cannot be diversified away.Company-unique risk (unsystematic risk) is diversifiable. This type of risk can be reduced through diversification.

Market Risk

Unexpected changes in interest rates.Unexpected changes in cash flows due to tax rate changes, foreign competition, and the overall business cycle.

Company-unique Risk

A company’s labor force goes on strike.A company’s top management dies in a plane crash.A huge oil tank bursts and floods a company’s production area.

As you add stocks to your portfolio, company-unique risk is reduced.

portfoliorisk

number of stocks

Market risk

As you add stocks to your portfolio, company-unique risk is reduced.

portfoliorisk

number of stocks

Market risk

company-unique

risk

Do some firms have more market risk than others?

Yes. For example:Interest rate changes affect

all firms, but which would be more affected:

a) Retail food chainb) Commercial bank

Note:

The market compensates investors for accepting risk - but only for market risk. Company-unique risk can and should be diversified away.

So - we need to be able to measure market risk.

This is why we have Beta.Beta: a measure of market risk.

Specifically, beta is a measure of how an individual stock’s returns vary with market returns.

It’s a measure of the “sensitivity” of an individual stock’s returns to changes in the market.

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The Concept of BetaBeta() measures how the return of an individual asset (or even a portfolio) varies with the market portfolio. = 1.0 : same risk as the market < 1.0 : less risky than the market > 1.0 : more risky than the marketBeta is the slope of the regression line (y = a + x) between a stock’s return(y) and the market return(x) over time, from simple linear regression.i = Covariancei,m/Mkt. Var. =imim/m

2

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Relating Market Risk and Required Return: the CAPM

Here’s the word story: a stock’s required rate of return = risk-free rate + the stock’s risk premium.The main assumption is investors hold well diversified portfolios = only concerned with market risk. A stock’s risk premium = measure of systematic risk X market risk premium.

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CAPM Equationkrp= market risk premium = km - krf

stock risk premium = j(krp)

kj = krf + j(km - krf )

= krf + j (krp)Example: What is Yahoo’s required return if

its b = 1.75, the current 3-mo. T-bill rate is 1.7%, and the historical market risk premium of 9.5% is demanded?

Yahoo k = 1.7% + 1.75(9.5%) = 18.3%

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Question: If Yahoo’s exp. Return = 15%, what to do?

Required vs. Expected Return

15%19.10%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

0 0.5 1 1.5 2 2.5

Beta

Retu

rn

Req. Return Exp Return

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Portfolio Beta and CAPMThe for a portfolio of stocks is the weighted average of the individual stock s.

p = wjj

Example: The risk-free rate is 6%, the market return is 16%. What is the required return for a portfolio consisting of 40% AOL with b = 1.7, 30% Exxon with b = 0.85, and 30% Fox Corp. with b = 1.15.Bp = .4(1.7)+.3(0.85)+.3(1.15) = 1.28kp = 6% + 1.28(16% - 6%) = 18.8%

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More SML Fun!

According to the CAPM and SML equation with k = 6% + (16% - 6%)How would a change in inflation affect required returns? (Say inflation increases 2% points)How would a change in risk aversion (market risk premium) affect required returns? (Say market risk premium decreases 2% points.)

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Changes to SMLSecurity Market Line

0%

5%

10%

15%

20%

25%

30%

0 1 2 3

Beta

Retu

rn

Original SML

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Changes to SMLSecurity Market Line

0%

5%

10%

15%

20%

25%

30%

0 1 2 3

Beta

Retu

rn

Original SML

IncreasedInflation

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Changes to SMLSecurity Market Line

0%

5%

10%

15%

20%

25%

30%

0 1 2 3

Beta

Retu

rn

Original SML

Less RiskAversion

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Limitations of CAPM/SML

Don’t really know what the market portfolio is, which makes it hard to estimate market expected or required return.Beta estimates can be unstable and might not reflect the future.Maturity debate over proper risk-free estimate.Most investors focus on more than systematic risk.