Risk and Return-F

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Risk and Rates of Return

Financial assets are expected to generate cash flows and hence theriskiness of a financial asset is measured in terms of the riskiness of itscash flows.

The riskiness of an asset is measured on a stand-alone basis or in a portfolio context . An asset may be very risky if held by itself but may bemuch less risky when it is part of a large portfolio.

In the context of portfolio, risk is divided into two parts: diversifiable riskand market risk. Diversifiable risk arises from compay-specific factors andhence can be washed away through diversification. Market risk stemsfrom general market movements and hence can not be diversifies away.

Investors are risk-averse. So they want to be compensated for bearing

the risk. In well oriented market, there is linear relationship betweenmarket risk and expected return.

This chapter focuses on risk and return from financial asset for anindividual investor, the concepts discussed here can be extended rophysical assets and other classes of investors (such as corporates).



Areas:

Risk and return of a single asset

Risk and return of a portfolio

Measurement of market risk Relationship between risk and return

Arbitrage pricing policy



Risk and Return of a Single Asset:

The rate of return on an asset for a given period

(usually 1 yr.) is defined as follows:

Annual Income + (Ending Price ± Beginning price)Rate of Return =

Beginning Price

Example:

Price at the beginning of the yr. = Rs. 60.00

Dividend paid at the end of the yr.= Rs. 2.40Price at the end of the yr. = Rs. 69.00

The rate of return on this stock is calculated as follows:

2.40 + (69.00 ± 60.00) = 0.19 or 19%

60.00



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.



Probability distributions

A listing of all possible outcomes, and theprobability of each occurrence.

Can be shown graphically.

Expected Rate of Return

Rate of

Return (%)100150-70

Firm X

Firm Y



Selected Realized Returns,

1971 ± 2001 Average Standard

Return Deviation

Small-company stocks 17.3% 33.2%Large-company stocks 12.7 20.2

L-T corporate bonds 6.1 8.6

L-T government bonds 5.7 9.4

Treasury bills 3.9 3.2

Source: Based on S tocks, Bonds, Bills, and Inflation: (ValuationEdition) 2002 Yearbook



Why is the T-bill return independent of the economy? Do T-bills promise a

completely risk-free return?

T-bills will return the promised 8%, regardless of the economy.

No, T-bills do not provide a risk-free return, asthey are still exposed to inflation. Although, verylittle unexpected inflation is likely to occur oversuch a short period of time.

T-bills are also risky in terms of reinvestment raterisk.

T-bills are risk-free in the default sense of theword.



Expected rate of return

It is the weighted average of all possible returnsmultiplied by their respective probabilities

n

E (R) = pi Ri

i = 1

Where E (R ) = expected return

Ri = return for the ith possible outcomepi = probability associated with Ri

n = no. of possible outcomes



Risk: Calculating the standard deviationfor each alternative

deviationStandard!W

2

Variance W!!W

i

2n

1i i

P)kk(§!

!W



Comments on standard deviation

as a measure of risk Standard deviation (i) measures total, or stand-alone, risk.

The larger i is, the lower the probabilitythat actual returns will be closer to expectedreturns.

Larger i is associated with a wider

probability distribution of returns. Difficult to compare standard deviations,because return has not been accounted for.



Coefficient of Variation (CV)

A standardized measure of dispersionabout the expected value, that shows therisk per unit of return.

^

k

Mean

devStd C V

W!!



Illustrating the CV as a measure of

relative risk

A = B , but A is riskier because of a larger probability of losses. In other words, the sameamount of risk (as measured by ) for less returns.

0

A B

Rate of Return (%)

Prob.



Investor attitude towards risk

Risk aversion ± assumes investors dislikerisk and require higher rates of return toencourage them to hold riskier securities.

Risk premium ± the difference betweenthe return on a risky asset and less riskyasset, which serves as compensation for investors to hold riskier securities.



Portfolio construction:

Risk and return Assume a two-stock portfolio is createdwith $50,000 invested in both HT andCollections.

Expected return of a portfolio is a weightedaverage of each of the component assetsof the portfolio.

Standard deviation is a little more trickyand requires that a new probabilitydistribution for the portfolio returns bedevised.



Calculating portfolio expected return

9.6%(1.7%)0.5(17.4%)0.5k

kwk

:averageweightedaisk

p

^

n

1i

i

^

ip

^

p

^

!!

!§!



Breaking down sources of risk

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

Market risk ± portion of a security¶s stand-alonerisk that cannot be eliminated throughdiversification. Measured by beta.

Firm-specific risk ± portion of a security¶sstand-alone risk that can be eliminated throughproper diversification.



Failure to diversify

If an investor chooses to hold a one-stockportfolio (exposed to more risk than adiversified investor), would the investor becompensated for the risk they bear?

± NO! ± Stand-alone risk is not important to a well-

diversified investor. ± Rational, risk-averse investors are concerned

with p, which is based upon market risk. ± There can be only one price (the market return)for a given security.

± No compensation should be earned for holdingunnecessary, diversifiable risk.



Capital Asset Pricing Model

(CAPM)

Model based upon concept that a stock¶srequired rate of return is equal to the risk-freerate of return plus a risk premium that reflectsthe riskiness of the stock after diversification.

Primary conclusion: The relevant riskiness of a stock is its contribution to the riskiness of awell-diversified portfolio.



Beta

Measures a stock¶s market risk, andshows a stock¶s volatility relative to themarket.

Indicates how risky a stock is if the stock isheld in a well-diversified portfolio.



Calculating betas

Run a regression of past returns of asecurity against past returns on the market.

The slope of the regression line (sometimescalled the security¶s characteristic line) isdefined as the beta coefficient for thesecurity.



Illustrating the calculation of beta

.

.

.

ki

_

kM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Regression line:

ki = -2.59 + 1.44 kM^ ^

Year kM

ki

1 15% 18%

2 -5 -10

3 12 16



Comments on beta

If beta = 1.0, the security is just as risky as theaverage stock.

If beta > 1.0, the security is riskier than average.

If beta < 1.0, the security is less risky thanaverage.

Most stocks have betas in the range of 0.5 to

1.5.



Can the beta of a security benegative?

Yes, if the correlation between Stock i andthe market is negative (i.e., i,m < 0).

If the correlation is negative, the regressionline would slope downward, and the betawould be negative.

However, a negative beta is highly unlikely.



Can the beta of a security benegative?

Yes, if the correlation between Stock i andthe market is negative (i.e., i,m < 0).

If the correlation is negative, the regressionline would slope downward, and the betawould be negative.

However, a negative beta is highly unlikely.



Beta coefficients for HT, Coll, and T-Bills

ki

_

kM

_

-20 0 20 40

40

20

-20

HT: = 1.30

T-bills: = 0

Coll: = -0.87



Comparing expected return andbeta coefficients

Security Exp. Ret. BetaHT 17.4% 1.30

Market 15.0 1.00USR 13.8 0.89T-Bills 8.0 0.00Coll. 1.7 -0.87

Riskier securities have higher returns, so therank order is OK.



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

SML: ki = kRF + (kM ± kRF) i

Assume kRF = 8% and kM = 15%.

The market (or equity) risk premium is

RPM = kM ± kRF = 15% ± 8% = 7%.



What is the market risk premium?

Additional return over the risk-free rateneeded to compensate investors for assuming an average amount of risk.

Its size depends on the perceived risk of thestock market and investors¶ degree of riskaversion.

Varies from year to year, but most estimatessuggest that it ranges between 4% and 8%per year.



Calculating required rates of return

kHT = 8.0% + (15.0% - 8.0%)(1.30)

= 8.0% + (7.0%)(1.30)

= 8.0% + 9.1% = 17.10%

kM = 8.0% + (7.0%)(1.00) = 15.00%

kUSR = 8.0% + (7.0%)(0.89) = 14.23%

kT-bill = 8.0% + (7.0%)(0.00) = 8.00% kColl = 8.0% + (7.0%)(-0.87) = 1.91%



Expected vs. Required returns

k)k(Overvalued 1.9 1.7 Coll.

k)k(uedFairly val 8.0 8.0 bills-T

k)k(Overvalued 14.2 13.8 USR

k)k(uedFairly val 15.0 15.0Market k)k(dUndervalue 17.1% 17.4% HT

k k

^

^

^

^

^

^

!

!

"



Illustrating theSecurity Market Line

..Coll.

.HT

T-bills

.USR

SML

kM = 15

kRF = 8

-1 0 1 2

.

SML: ki = 8% + (15% ± 8%) i

ki (%)

Risk, i



An example:Equally-weighted two-stock portfolio

Create a portfolio with 50% invested in HTand 50% invested in Collections.

The beta of a portfolio is the weighted

average of each of the stock¶s betas.

P = wHT HT + wColl Coll

P = 0.5 (1.30) + 0.5 (-0.87)P = 0.215



Calculating portfolio required returns

The required return of a portfolio is the weightedaverage 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 usedto solve for expected return.

kP = kRF + (kM ± kRF) P

kP = 8.0% + (15.0% ± 8.0%) (0.215)

kP

= 9.5%



Factors that change the SML

What if investors raise inflation expectations by3%, what would happen to the SML?

SML1

ki (%)SML2

0 0.5 1.0 1.5

18

15

11

8

( I = 3%

Risk, i



Factors that change the SML

What if investors¶ risk aversion increased,causing the market risk premium to increaseby 3%, what would happen to the SML?

SML1

ki (%) SML2

0 0.5 1.0 1.5

18

1511

8

( RPM = 3%

Risk, i



Verifying the CAPM empirically

The CAPM has not been verifiedcompletely.

Statistical tests have problems that makeverification almost impossible.

Some argue that there are additional riskfactors, other than the market riskpremium, that must be considered.



More thoughts on the CAPM

Investors seem to be concerned with bothmarket risk and total risk. Therefore, the SMLmay not produce a correct estimate of ki.

ki = kRF + (kM ± kRF) i + ???

CAPM/SML concepts are based uponexpectations, but betas are calculated usinghistorical data. A company¶s historical data maynot reflect investors¶ expectations about futureriskiness.

Risk and Return-F

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