Chapter 6

Risk and Rates of Return

2

Chapter 6 Objectives

Inflation and rates of return

How to measure risk

(variance, standard deviation, beta)

How to reduce risk

(diversification)

How to price risk

(security market line, CAPM)

3

Historical Risk and Return

Annual 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%

4

Why are these rates different?

90-day Treasury Bill 1.7%

90-day Commercial Paper 1.8%

2-year US Treasury Note 1.9%

10-year US Treasury Note 3.8%

10-year AAA Corporate Bond 5.0%

10-year BBB Corporate Bond 6.1%

Inflation, Rates of Return, and the Fisher Effect

InterestRates

Interest Rates

Conceptually:

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

8

Proof of Fisher Effect Equation

You have a $100 to buy items costing a $1 each. You can buy 100 items now.

Instead of spending the $100 now, you decide to invest the money at 7% (nominal risk-free rate) for a year giving you $107 at the end of the year so you can hopefully buy more than 100 items at the end of the year.

9

Proof of Fisher Effect Equation

At the end of the year, the items now cost $1.04 each (4% inflation).You can buy $107/$1.04 = 102.88 of these items at the end of the year.This represents a 2.88% increase in your real purchasing power (real interest rate).We used (1+krf) = (1+k*)(1+IRP) (1.07) = (1+k*)(1.04): 1+k* = 1.07/1.04 = 1.0288; k* = .0288 = 2.88%

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.

12

Recent US Treasury Yield Curves

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 Week Last Semester

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?

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

Returns

Expected 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%

17

Calculation of Expected Returns

Expected 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%

19

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.For a stock, we can examine the standard deviation of the stock’s returns.

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

23

Expected Return and Standard Deviation ExampleSt at e of Cont rary

Econom 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?

It depends on your tolerance for risk!

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

Return

Risk

27

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.

32

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.

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

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. We use beta as a measure of market risk.

36

The Concept of Beta

Beta() 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

37

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.

38

CAPM Equation

krp= 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%

39

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

Required vs. Expected Return

15%18.30%

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

40

Portfolio Beta and CAPM

The 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) = 1.28

kp = 6% + 1.28(16% - 6%) = 18.8%

41

More CAPM/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.)

42

Changes to SML

Security Market Line

0%

5%

10%

15%

20%

25%

30%

0 0.5 1 1.5 2 2.5

Beta

Re

turn

Original SML

43

Changes to SML

Security Market Line

0%

5%

10%

15%

20%

25%

30%

0 1 2 3

Beta

Re

turn Original SML

Increased Inflation

44

Changes to SML

Security Market Line

0%

5%

10%

15%

20%

25%

30%

0 1 2 3

Beta

Re

turn Original SML

Less Risk Aversion

45

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.