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CHAPTER 6 Risk and Rates of Return

Stand-alone risk

Portfolio risk

Risk & return: CAPM/SML

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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.

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Probability distribution

Expected Rate of Return

Rate ofreturn (%)100150-70

Firm X

Firm Y

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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%

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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

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Why is the T-bill return independent of the economy?

Will return the promised 8% regardless of the economy.

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Do T-bills promise a completelyrisk-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.

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Do the returns of HT and Coll. move with or counter to the economy?

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.

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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%.

^

^

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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?

^

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What’s the standard deviationof returns for each alternative?

= Standard deviation.

= =

=

Variance 2

.P)k̂k(n

1ii

2i

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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

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Prob.

Rate of Return (%)

T-bill

USR

HT

0 8 13.8 17.4

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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.

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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.

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Coefficient of Variation (CV)

Standardized measure of dispersionabout the expected value:

Shows risk per unit of return.

CV = = . Std dev

k̂Mean

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0

A B

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

= CVA > CVB.k̂

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Portfolio Risk and Return

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

Calculate kp and p.^

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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

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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

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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

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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.

^

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General statements about risk

Most stocks are positively correlated. rk,m 0.65.

35% for an average stock.

Combining stocks generally lowers risk.

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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

.. .

. .

.

.

..

.. . . . .

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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

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What would happen to theriskiness of an average 1-stock

portfolio as more randomlyselected stocks were added?

p would decrease because the added

stocks would not be perfectly correlated but kp would remain relatively constant.^

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Large

0 15

Prob.

2

1

Even with large N, p 20%

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# Stocks in Portfolio10 20 30 40 2,000+

Company Specific Risk

Market Risk

20

0

Stand-Alone Risk, p

p (%)

35

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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 .

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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= +

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

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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?

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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.

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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.

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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.

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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.

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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^ ^

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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.

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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

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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.

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HT

T-Bills

b = 0

ki

_

kM

_-20 0 20 40

40

20

-20

b = 1.29

Coll.b = -0.86

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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)

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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 .

^

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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%.

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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

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..Coll.

.HT

T-bills

.USR

SML

kM = 15

kRF = 8

-1 0 1 2

.

SML: ki = 8% + (15% – 8%) bi .

ki (%)

Risk, bi

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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.

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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%.

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If investors raise inflation expectations by 3%, what would

happen to the SML?

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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%

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

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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%

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Has the CAPM been verified through empirical tests?

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

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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 + ?

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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.