CHAPTER 6 Risk and Rates of Return

Stand-alone risk Portfolio risk Risk & return: CAPM/SML

What is investment risk?

Investment risk pertains to the probability of actually earning a low or negative return.

The greater the chance of low or negative returns, the riskier the investment.

Probability distribution

Expected Rate of Return

Rate ofreturn (%)100150-70

Firm X

Firm Y

Annual Total Returns,1926-1998Average StandardReturn Deviation Distribution

Small-companystocks 17.4% 33.8%

Large-companystocks 13.2 20.3

Long-termcorporate bonds 6.1 8.6

Long-termgovernment 5.7 9.2

Intermediate-termgovernment 5.5 5.7

U.S. Treasurybills 3.8 3.2

Inflation 3.2 4.5

0 17.4%

0 13.2%

0 6.1%

0 5.7%

0 5.5%

0 3.8%

0 3.2%

Investment Alternatives(Given in the problem)

Economy Prob. T-Bill HT Coll USR MP

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 independent of the economy?

Will return the promised 8% regardless of the economy.

Do T-bills promise a completely risk-free return?

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 HT and Coll. move with or counter to the conomy?

HT: Moves with the economy, and has a positive correlation. This is typical.

Coll: Is countercyclical of the economy, and has a negative correlation. This is unusual.

Calculate the expected rate of return on each alternative:

.Pk = k̂n

1=i

iik = expected rate of return.

kHT = (-22%)0.1 + (-2%)0.20 + (20%)0.40 + (35%)0.20 + (50%)0.1 = 17.4%.

^

^

k

HT 17.4%

Market 15.0

USR 13.8

T-bill 8.0

Coll. 1.7

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

^

What’s the standard deviationof returns for each alternative?

= Standard deviation.

= =

=

Variance 2

.P)k̂k(n

1ii

2i

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

Coll = 13.4%.USR = 18.8%. M = 15.3%.

1/2

T-bills =

.P)k̂k(n

1ii

2

i

(8.0 – 8.0)20.1 + (8.0 – 8.0)20.2

+ (8.0 – 8.0)20.4 + (8.0 – 8.0)20.2

+ (8.0 – 8.0)20.1

Prob.

Rate of Return (%)

T-bill

USR

HT

0 8 13.8 17.4

Standard deviation (i) measures total, or stand-alone, risk.

The larger the i , the lower the probability that actual returns will be close to the expected return.

Expected Returns vs. Risk

SecurityExpected

return Risk,

HT 17.4% 20.0%Market 15.0 15.3USR 13.8* 18.8*T-bills 8.0 0.0Coll. 1.7* 13.4*

*Seems misplaced.

Coefficient of Variation (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 Return

Assume a two-stock portfolio with $50,000 in HT and $50,000 in Collections.

Calculate kp and p.^

Portfolio Return, kp

kp is a weighted average:

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

kp is between kHT and kCOLL.

^

^

^

^

^ ^

^ ^

kp = wikin

i = 1

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. HT Coll. 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

CVp = = 0.34. 3.3% 9.6%

p = = 3.3%.

1 2/

(3.0 – 9.6)20.10

+ (6.4 – 9.6)20.20

+ (10.0 – 9.6)20.40

+ (12.5 – 9.6)20.20

+ (15.0 – 9.6)20.10

p = 3.3% is much lower than that of

either stock (20% and 13.4%).

p = 3.3% is lower than average of HT

and Coll = 16.7%. Portfolio provides average k but

lower risk. Reason: negative correlation.

^

General statements about risk

Most stocks are positively correlated. rk,m 0.65.

35% for an average stock. Combining stocks generally lowers

risk.

Returns Distribution for Two Perfectly Negatively Correlated Stocks (r = -1.0) and for Portfolio WM

25

15

0

-10 -10 -10

0 0

15 15

25 25

Stock W Stock M Portfolio WM

.. .

. .

.

.

..

.. . . . .

Returns Distributions for Two Perfectly Positively Correlated Stocks (r = +1.0) and for Portfolio MM’

Stock M

0

15

25

-10

Stock M’

0

15

25

-10

Portfolio MM’

0

15

25

-10

What would happen to the riskiness of an average 1-stock portfolio as more randomly selected stocks were added?

p would decrease because the added

stocks would not be perfectly correlated but kp would remain

relatively constant.

^

Large

0 15

Prob.

2

1

Even with large N, p 20%

# Stocks in Portfolio10 20 30 40 2,000+

Company Specific Risk

Market Risk

20

0

Stand-Alone Risk, p

p (%)

35

As more stocks are added, each new stock has a smaller risk-reducing impact.

p falls very slowly after about 10

stocks are included, and after 40 stocks, there is little, if any, effect. The lower limit for p is

about 20% = M .

Stand-alone Market Firm-specific

Market risk is that part of a security’s stand-alone risk that cannot be eliminated by diversification, and is measured by beta.

Firm-specific risk is that part of a security’s stand-alone risk that can be eliminated by proper diversification.

risk risk risk= +

By forming portfolios, we can eliminate about half the riskiness of individual stocks (35% vs. 20%).

If you chose to hold a one-stock portfolio and thus are exposed to more risk than diversified investors, would you be compensated for all the risk you bear?

NO! Stand-alone risk as measured by

a stock’s or CV is not important to a well-diversified investor.

Rational, risk averse investors are concerned with p , which is based on market risk.

There can only be one price, hence market return, for a given security. Therefore, no compensation can be earned for the additional risk of a one-stock portfolio.

Beta measures a stock’s market risk. It shows a stock’s volatility relative to the market.

Beta shows how risky a stock is if the stock is held in a well-diversified portfolio.

How are betas calculated?

Run a regression of past returns on Stock i versus returns on the market. Returns = D/P + g.

The slope of the regression line is defined as the beta coefficient.

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 calculation:

Regression line:ki = -2.59 + 1.44 kM^ ^

If beta = 1.0, average stock. If beta > 1.0, stock riskier than

average. If beta < 1.0, stock less risky than

average. Most stocks have betas in the

range of 0.5 to 1.5.

List of Beta Coefficients

Stock BetaMerrill Lynch 2.00America Online 1.70General Electric 1.20Microsoft Corp. 1.10Coca-Cola 1.05IBM 1.05Procter & Gamble 0.85Heinz 0.80Energen Corp. 0.80Empire District Electric 0.45

Can a beta be negative?

Answer: Yes, if ri, m is negative. Then in a “beta graph” the regression line will slope downward. Though, a negative beta is highly unlikely.

HT

T-Bills

b = 0

ki

_

kM

_-20 0 20 40

40

20

-20

b = 1.29

Coll.b = -0.86

Riskier securities have higher returns, so the rank order is OK.

HT 17.4% 1.29Market 15.0 1.00USR 13.8 0.68T-bills 8.0 0.00Coll. 1.7 -0.86

Expected RiskSecurity Return (Beta)

Use the SML to calculate therequired returns.

Assume kRF = 8%. Note that kM = kM is 15%. (Equil.) RPM = kM – kRF = 15% – 8% = 7%.

SML: ki = kRF + (kM – kRF)bi .

^

Required Rates of Return

kHT = 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%.

kUSR = 8.0% + (7%)(0.68) = 12.8%.

kT-bill = 8.0% + (7%)(0.00) = 8.0%.

kColl = 8.0% + (7%)(-0.86) = 2.0%.

HT 17.4% 17.0% Undervalued: k > k

Market 15.0 15.0 Fairly valuedUSR 13.8 12.8 Undervalued:

k > kT-bills 8.0 8.0 Fairly valuedColl. 1.7 2.0 Overvalued:

k < k

Expected vs. Required Returns

^

^

^

^ k k

..Coll.

.HT

T-bills

.USR

SML

kM = 15

kRF = 8

-1 0 1 2

.

SML: ki = 8% + (15% – 8%) bi .

ki (%)

Risk, bi

Calculate beta for a portfolio with 50% HT and 50% Collections

bp= Weighted average = 0.5(bHT) + 0.5(bColl) = 0.5(1.29) + 0.5(-0.86) = 0.22.

The required return on the HT/Coll. 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 inflation expectations by 3%, what would happen to the SML?

SML1

Original situation

Required Rate of Return k (%)

SML2

0 0.5 1.0 1.5 Risk, bi

1815

11 8

New SML I = 3%

If inflation did not changebut risk aversion increasedenough to cause the marketrisk premium to increase by3 percentage points, whatwould happen to the SML?

kM = 18%

kM = 15%

SML1

Original situation

Required Rate of Return (%)

SML2

After increasein risk aversion

Risk, bi

18

15

8

1.0

RPM = 3%

Has the CAPM been verified through empirical tests?

Not completely. Those statistical tests have problems that make verification almost impossible.

Investors seem to be concerned with both market risk and total risk. Therefore, the SML may not produce a correct estimate of ki:

ki = kRF + (kM – kRF)b + ?

Also, CAPM/SML concepts are based on expectations, yet betas are calculated using historical data. A company’s historical data may not reflect investors’ expectations about future riskiness.