risk and return: the basics

Risk and Return: The Ba Risk and Return: The Basicssics - Stand alone risk Portfolio risk Risk and return: CAPM/SML

What is investment risk?

Investment risk Investment risk pertains to the pertains to the probability of earning less than probability of earning less than

the expected return. the expected return. The greater the chance of low or The greater the chance of low or

negative returns, the riskier the negative returns, the riskier theinvestment.investment.

Probability distribution

Expected Rate of Return

Rate ofreturn (%)100150-70

Firm X

Firm Y

Investment Alternatives

Economy Prob. T-Bill A B C Mkt Port.

Recession 0.1 8.0% -22.0% 28.0% 10.0% -13.0%Below avg. 0.2 8.0 -2.0 14.7 -10.0 1.0Average 0.4 8.0 20.0 0.0 7.0 15.0Above avg. 0.2 8.0 35.0 -10.0 45.0 29.0Boom 0.1 8.0 50.0 -20.0 30.0 43.0

1.0

- Why is the T bill return i - Why is the T bill return i ndependent ndependent

of the economy? of the economy?

Will return the promised 8%regardless of the state of the economy.

- Do T bills promise a - Do T bills promise a - completely risk free ret - completely risk free ret

urn?urn?

No, T-bills are still exposed to the risk of inflation.

However, not much unexpected inflation is likely to occur over a relatively short period.

Do the returns of A and Do the returns of A and Bmove with or counter Bmove with or counter to the economy? to the economy?

A : With. Positive correlation . Typical.

B : Countercyclical. Negat ive correlation . Unusual.

k = kiPi.

Calculate the expected r ate of

return on each alternative k = Expected rate of return

kA = (-22%)0.10 + (-2%)0.20 + (20%)0.40 + (35%)0.20 + (50%)0.10 = 17.4%.

^

^

^

i = 1

n

A appears to be the best, but is it really?

k̂A 17.4%Market 15.0C 13.8T-bill 8.0B. 1.7

What’s the standard de What’s the standard deviationviation

of returns for each alter of returns for each alternative?native?

= Variance = 2

= (k k) Pi2

ii=1

n

= Standard deviation

.

= (k k) Pi2

ii=1

n

T-bills = 0.0%.A = 20.0%.

B = 13.4%.C = 18.8%.M = 15.3%.

.

1/2

T-bills = 8.0- 8.0 + 8.0 - 8.0 8.0 - 8.0 + 8.0 - 8.0

2 2

2 2

2

01 0 20 4 0 2

8 0 - 8 0 01

. .. .

. . .

Prob.

Rate of Return (%)

T-bill

C

A

0 8 13.8 17.4

Standard deviation (i ) mea sures -stand alone risk.

The larger the i , the higher the probability that actual r

eturns will be far below the expected return.

Coefficient of Variation ( Coefficient of Variation (CV)CV)

Standardized measure of dispersionabout the expected value:

Shows risk per unit of return.

CV = = . Std dev k̂Mean

0

A B

A = B , but A is riskier because largerprobability of losses.

= CVA > CVB.k̂

Portfolio Risk and Retur Portfolio Risk and Returnn

Assume a two-stock portfolio with $50,000 in A and $50,000 in B.

Calculate kp and p.^

Portfolio Return, k Portfolio Return, kpp

kp is a weighted average:

kp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.

kp is between kA and kB.

^

^

^

^

^ ^

^ ^

kp = wikwn

i = 1

Alternative Method Alternative Method

kp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40 + (12.5%)0.20 + (15.0%)0.10 = 9.6%.

^

Estimated ReturnEconomy Prob. A B Port.Recession 0.10 -22.0% 28.0% 3.0%Below avg. 0.20 -2.0 14.7 6.4Average 0.40 20.0 0.0 10.0Above avg. 0.20 35.0 -10.0 12.5Boom 0.10 50.0 -20.0 15.0

= 3.3%.

p =

3.0 - 9.6 2

2

2

2

2

1 20 10

6 4 - 9 6 0 20

10 0 - 9 6 0 40

12 5 - 9 6 0 20

15 0 - 9 6 0 10

.

. . .

. . .

. . .

. . .

/

CVp = = 0.34. 3.3% 9.6%

ReturnsDistribution forTwoPerf ReturnsDistribution forTwoPerf ectlyNegatively CorrelatedStoc ectlyNegatively CorrelatedStoc

- ks (r= 1 .0 ) and forPortfolioW - ks (r= 1 .0 ) and forPortfolioWMM

25

15

0

-10 -10 -10

0 0

15 15

25 25

Stock W Stock M Portfolio WM

.. .

. .

..

..

.. . . . .

ReturnsDistributions forTwoPer ReturnsDistributions forTwoPerffffff ffffffffff ffffffffff fffffffffff ffffffffff ffffffffff fffff s (r= +1 .0 ) and forPortfolioM s (r= +1 .0 ) and forPortfolioMM’M’

Stock M

0

15

25

-10

0

15

25

-10

Stock M’

0

15

25

-10

Portfolio MM’

What would happen to t What would happen to thehe

riskiness of an average riskiness of an average-1 stock-1 stock

portfolio as more rando portfolio as more randomlymly

selected stocks were ad selected stocks were added?ded?p would decrease because the ad

ded stocks would not be perfectly correlated but kp would remain rel

atively constant.

^

Large

0 15

Prob.

2

1

# Stocks in Portfolio10 20 30 40 2000+

Company Specific Risk

Market Risk

35

18

0

Stand-Alone Risk, p

p (%)

Asmore stocks are added, each Asmore stocks are added, each -new stock has a smaller risk re -new stock has a smaller risk re

ducing impact. ducing impact.pp falls very slowly after about falls very slowly after about

40 stocks are included. The low 40 stocks are included. The low er limit for er limit for pp is about is about MM 18= 18=

%.%.

- Stand alone Market -Firm specific

Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification.Firm-specific risk is that part of a security’s stand-alone risk which can be eliminated by proper diversification.

risk risk risk= +

By formingportfolios,we caneliminate By formingportfolios,we caneliminate abouthalf the riskinessof individual st abouthalf the riskinessof individual st

ocks (3 5 %vs. 1 8 %). ocks (3 5 %vs. 1 8 %).

If you chose to hold a one-stock portfolio and thus are exposed to more risk t

han diversified investors, wouldyou be compensated for all the risk yo

u bear?

NO!NO! - Stand alone risk as measured- Stand alone risk as measured

by a stock’s by a stock’s or CV is not imp or CV is not imp - ortant to a well diversified inv - ortant to a well diversified inv

estor.estor. Rational, risk averse investor Rational, risk averse investor

s are concerned with portfolio s are concerned with portfolio risk, and here the relevant ris risk, and here the relevant ris

k of an individual stock is its c k of an individual stock is its c ontribution to the riskiness of ontribution to the riskiness of

a portfolio. a portfolio.

There can only be one pric There can only be one pric e, hence market return, for e, hence market return, for

a given security. Therefor a given security. Therefor e, no compensation can be e, no compensation can be

earned for the additional ri earned for the additional ri - sk of a one stock portfolio. - sk of a one stock portfolio.

CAPM( Capital Asset P ricing Model)

Conclusion:Conclusion:The relevant riskiness of an individual stock The relevant riskiness of an individual stock

is its contribution to the riskiness of well-diveis its contribution to the riskiness of well-diversified portfolio.rsified portfolio.

CAPM links risk and required rate of returnCAPM links risk and required rate of return

Beta measures a stock’s m arket risk. It shows a stoc

k’s volatility relative to themarket. Beta shows how risky a sto

ck is if the stock is held in a- well diversified portfolio.

The concept of beta, “b”The concept of beta, “b”

Year kM ki 1 15% 18% 2 -5 -10 3 12 16

.

.

.ki

_

kM

_-5 0 5 10 15 20

20

15

10

5

-5

-10

Illustration of beta calculations:Regression line:ki = -2.59 + 1.44 kM^ ^

Find beta Find beta

““ By Eye.” By Eye.” Plot points, draw in r egression line, get slope as b = Rise/Run. The “rise” is the diffe

rence in ki , the “run” is the diffe rence in kM . For example, howm

uch does ki increase or decrease when kM increases from 0% to 1

0%?

Calculator. Calculator. Enter data points, an Enter data points, an d calculator does least squares r d calculator does least squares r

egression: k egression: kii = a + bk = a + bkMM - = 2 .5 9 - = 2 .5 9 + 1 .4 4 k + 1 .4 4 kMM . r= corr. coefficient= . r= corr. coefficient=

0 .9 9 7 .0 .9 9 7 . Inthe real worl d,wewoul duse Inthe real worl d,wewoul duse weekweek

l y or monthl y returns l y or monthl y returns ,withatl east ayea ,withatl east ayea rofdata, andwouldalwaysusea co rofdata, andwouldalwaysusea co

mputeror calculator. mputeror calculator.

10If beta = . , average risk. 10Ifbeta> . , stockr i ski er t han aver ag. fffff ffff fffff ffff ffff<1. 0,. ff fff ff ffff0.51.5.

Can a beta be negative?

Yes, in theory, if a stock’s returns are negatively correlated with the market. Then in a “beta graph” the regression line will slope downward.

In the “real world,” negative beta stocks do not exist.

A

T-Bills

b = 0

ki

_

kM

_-20 0 20 40

40

20

-20

b = 1.29

Bb = -0.86

Use the SML to calculate t he required returns.

Assume kRF = 8%. Note that kM = k M is 15%. (Fromm

arket portfolio.)RPM = k M - kRF - 15 8 7= % % = %.

SML: ki = kRF + (kM - kRF)bi .

^

Required Rates of Retur Required Rates of Returnn

kA = 8.0% + (15.0% - 8.0%)(1.29)= 8.0% + (7%)(1.29)= 8.0% + 9.0% = 17.0%.

kM = 8.0% + (7%)(1.00) = 15.0%.kC = 8.0% + (7%)(0.68) = 12.8%.kT-bill = 8.0% + (7%)(0.00) = 8.0%.kB = 8.0% + (7%)(-0.86) = 2.0%.

Expected vs. Required Returns

^

^

^

^

A 17.4% 17.0% Undervalued: k > k

Market 15.0 15.0 Fairly valued C 13.8 12.8 Undervalued:

k > k T-bills 8.0 8.0 Fairly valued B 1.7 2.0 Overvalued:

k < k

k k

..B

.A

T-bills

.C

SML

kM = 15

kRF = 8

-1 0 1 2

.

SML: ki = 8% + (15% - 8%) bi .

ki (%)

Risk, bi

Calculate beta for a port Calculate beta for a port folio with 5 0 % A and folio with 5 0 % A and 50% B 50% Bbp = Weighted average

= 0.5(bA) + 0.5(bB)= 0.5(1.29) + 0.5(-0.86)= 0.22.

The required return on t The required return on t he A/B portfolio is: he A/B portfolio is:

kp =Weighted average k=0.5(17%) + 0.5(2%)=9.5%.

Or use SML:

kp=kRF + (kM - kRF) bp

=8.0% + (15.0% - 8.0%)(0.22) =8.0% + 7%(0.22)=9.5%.

If investors raise in If investors raise inflationflation

expectations by 3 p expectations by 3 p ercentage points, w ercentage points, w

hat would happen t hat would happen t o the SML? o the SML?

SML1

Original situation

Required Rate of Return k (%)

SML2

0 0.5 1.0 1.5 2.0

181511 8

New SML I = 3%

If inflation did not chan If inflation did not chan ge but risk aversion incr ge but risk aversion incr

eased enough to cause t eased enough to cause t hemarket risk premium hemarket risk premium to increase to increase by 3 percentage points, by 3 percentage points,

what would happen to t what would happen to t he SML? he SML?

kM = 18%kM = 15%

SML1

Original situation

Required Rate of Return (

%)SML2

After increasein risk aversion

Risk, bi

18

15

8

1.0

MRP = 3%

Has the CAPM been verifie Has the CAPM been verifie d through empirical tests? d through empirical tests?

Not completely. That statistical te Not completely. That statistical te sts have problems which make veri sts have problems which make veri

fication almost impossible. fication almost impossible.

Investors seem to be concerned with b Investors seem to be concerned with b oth market risk and total risk. Therefor oth market risk and total risk. Therefor

e, the SMLmay not produce a correct e e, the SMLmay not produce a correct e stimate of k stimate of kii::

ki = kRF + (kM - kRF)b + ?