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 o The greater the chance of low o

r negative returns, the riskier t r negative returns, the riskier t he investment. he 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 1.0

Average 0.4 8.0 20.0 0.0 7.0 15.0

Above avg. 0.2 8.0 35.0 -10.0 45.0 29.0

Boom 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. Nega tive 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?

k̂

A 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) Pi

2

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 2

0 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

returns 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 Return

Economy 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%

Returns Distribution for Two Returns Distribution for Two Perfectly Negatively Correlat Perfectly Negatively Correlat

- ed Stocks (r = 1.0) and for Po - ed Stocks (r = 1.0) and for Po rtfolio WM rtfolio WM

25

15

0

-10 -10 -10

0 0

15 15

25 25

Stock W Stock M Portfolio WM

.

. .

. .

.

.

..

.. . . . .

Returns Distributions for Two Returns Distributions for Two Perfectly Positively Correlate Perfectly Positively Correlate

d Stocks (r = +1.0) and for Po d Stocks (r = +1.0) and for Po rtfolio MM’ rtfolio MM’

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 a

dded stocks would not be perfect ly correlated but kp would remain

relatively 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 (%)

As more stocks are added, eac As more stocks are added, eac -h new stock has a smaller risk -h new stock has a smaller risk

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

40 stocks are included. The lo 40 stocks are included. The lo wer limit for wer 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= +

aaaaa aaaa aaa aaaaaaaaa aa aaaaaaaaaa aaaaaa aaaa aaa ,(35%. aaaaa aaaa aaa aaaaaaaaa aa aaaaaaaaaa aaaaaa aaaa aaa ,(35%.18%).18%).

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 measure- Stand alone risk as measure

d by a stock’s d by a stock’s or CV is not i or CV is not i -mportant to a well diversifie -mportant to a well diversifie

d investor. d investor. Rational, risk averse investo Rational, risk averse investo

rs are concerned with portfol rs are concerned with portfol io risk, and here the relevant io risk, and here the relevant

risk of an individual stock is i risk of an individual stock is i ts contribution to the riskine ts contribution to the riskine ss of a portfolio. ss of a portfolio.

There can only be one price, There can only be one price, hence market return, for a g hence market return, for a g

iven security. Therefore, n iven security. Therefore, n o compensation can be earn o compensation can be earn

ed for the additional risk of ed for the additional risk of - a one stock portfolio. - 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-divis its contribution to the riskiness of well-diversified portfolio.ersified 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 stock

’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 diff

erence in ki , the “run” is the di fference in kM . For example, ho

wmuch does ki increase or dec rease when kM increases from 0

10% to %?

Calculator. Calculator. Enter data points, and calculat Enter data points, and calculat or does least squares regression or does least squares regression

: k : kii = a + bk = a + bkMM - = 2.59 + 1.44k - = 2.59 + 1.44kMM . r . r = corr. coefficient = 0.997. = corr. coefficient = 0.997.

In the real world, we would use In the real world, we would use weekly or monthly returns weekly or monthly returns , with , with

at least a year of data, and woul at least a year of data, and woul d always use a computer or calcu d always use a computer or calcu

lator.lator.

10If beta = . , average risk. aaaaa aaaaaaa aaaa aaaaaa>1. 0,.

aaaaa aaaa aaaaa aaaa aaaa<1. 0,.

aa aaa aa aaaa0.51.5.

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%. (From . )

RPM = k M - kRF - =1 5 %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 50% A and 50 folio with 50% A and 50

%B %B

bp = 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 inf If investors raise inflationlation

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

1815

11 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 ver sts have problems which make ver

ification almost impossible. ification almost impossible.

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

re, the SMLmay not produce a correct re, the SMLmay not produce a correct estimate of k estimate of kii::

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