# Risk And Rates Of Return, Lecture Notes - Financial Management, Study notes for Financial Management. University of Michigan (MI)

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CHAPTER 5 Risk and Rates of Return

8-1

CHAPTER 8: Risk and Rates of Return Updated: September 20, 2011

All Financial Assets Produce CFs

Risk of Asset Depends on Risk of CFs

Stand-alone Risk of Asset’s CFs

Portfolio Risk of CFs

Diversifiable and Market Risk

Risk & return: CAPM / SML

8-2

Investment returns

The rate of return on an investment can be calculated as follows:

(Amount received – Amount invested) Return = ________________________

Amount invested

For example, if \$1,000 is invested and \$1,100 is returned after one year, the rate of return for this investment is:

(\$1,100 - \$1,000) / \$1,000 = 10%.

8-3

What is 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.

 Risk = Dispersion of Returns around mean, or expected mean: variance or standard deviation

8-4

Probability distributions

 A listing of all possible outcomes, and the probability of each occurrence.

 Can be shown graphically.

Expected Rate of Return

Rate of

Return (%) 100 15 0 -70

Firm X

Firm Y

8-5

Selected Realized Returns, 1926 – 2004

Average Standard

Return Deviation

Small-company stocks 17.5% 33.1%

Large-company stocks 12.4 20.3

L-T corporate bonds 6.2 8.6

L-T government bonds 5.8 9.3

U.S. Treasury bills 3.8 3.1

Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation

Edition) 2005 Yearbook (Chicago: Ibbotson Associates, 2005), p28.

8-6

Investment alternatives

Economy Prob. T-Bill HT Coll USR MKT.

Recession 0.1 5.5% -27.0% 27.0% 6.0% -17.0%

Below avg 0.2 5.5% -7.0% 13.0% -14.0% -3.0%

Average 0.4 5.5% 15.0% 0.0% 3.0% 10.0%

Above avg 0.2 5.5% 30.0% -11.0% 41.0% 25.0%

Boom 0.1 5.5% 45.0% -21.0% 26.0% 38.0%

8-7

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 5.5%, regardless of the economy.

 No, T-bills do not provide a completely risk-free return, as they are still exposed to inflation. Although, very little unexpected inflation is likely to occur over such a short period of time.

 T-bills are also risky in terms of reinvestment rate risk.

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

8-8

How do the returns of HT and Coll. behave in relation to the market?

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

 Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.

8-9

Calculating the expected return

12.4% (0.1) (45%)

(0.2) (30%) (0.4) (15%)

(0.2) (-7%) (0.1) (-27%) r

P r r

return of rate expected r

HT

^

N

1i ii

^

^







 

8-10

Summary of expected returns Expected return HT 12.4% Market 10.5% USR 9.8% T-bill 5.5% Coll. 1.0%

HT has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?

8-11

Calculating standard deviation

deviation Standard

2Variance 

i 2

N

1i i P)r(rσ 

 ˆ

8-12

Standard deviation for each investment

15.2%

18.8% 20.0%

13.2% 0.0%

(0.1)5.5) - (5.5

(0.2)5.5) - (5.5 (0.4)5.5) - (5.5

(0.2)5.5) - (5.5 (0.1)5.5) - (5.5

P )r (r

M

USRHT

CollbillsT

2

22

22

billsT

N

1i i

2 ^

i





  

  









2 1

8-13

Comparing standard deviations

USR

Prob. T - bill

HT

0 5.5 9.8 12.4 Rate of Return (%)

8-14

Comments on standard deviation as a measure of risk

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

 The larger σi is, the lower the probability that actual returns will be closer to expected returns.

 Larger σi is associated with a wider probability distribution of returns.

8-15

Comparing risk and return

Security Expected return, r

Risk, σ

T-bills 5.5% 0.0%

HT 12.4% 20.0%

Coll* 1.0% 13.2%

USR* 9.8% 18.8%

Market 10.5% 15.2%

* Seem out of place.

^

8-16

Coefficient of Variation (CV)

A standardized measure of dispersion about the expected value, that shows the risk per unit of return.

r

return Expected

deviation Standard CV

ˆ

 

8-17

Risk rankings, by coefficient of variation

CV T-bill 0.0 HT 1.6 Coll. 13.2 USR 1.9 Market 1.4

 Collections has the highest degree of risk per unit of return.

 HT, despite having the highest standard deviation of returns, has a relatively average CV.

8-18

Illustrating the CV as a measure of relative risk

σA = σB , but A is riskier because of a larger probability of losses. In other words, the same amount of risk (as measured by σ) for smaller returns.

0

A B

Rate of Return (%)

Prob.

8-19

Investor attitude towards risk

Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.

Risk premium – the difference between the return on a risky asset and a riskless asset, which serves as compensation for investors to hold riskier securities.

8-20

Portfolio construction: Risk and return

 Assume a two-stock portfolio is created with \$50,000 invested in both HT and Collections.

 A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets.

 Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.

8-21

Calculating portfolio expected return

6.7% (1.0%) 0.5 (12.4%) 0.5 r

rw r

:average weighted a is r

p

^

N

1i

i

^

ip

^

p

^



 

8-22

An alternative method for determining portfolio expected return

Economy Prob. HT Coll Port.

Recession 0.1 -27.0% 27.0% 0.0%

Below avg 0.2 -7.0% 13.0% 3.0%

Average 0.4 15.0% 0.0% 7.5%

Above avg 0.2 30.0% -11.0% 9.5%

Boom 0.1 45.0% -21.0% 12.0%

6.7% (12.0%) 0.10 (9.5%) 0.20

(7.5%) 0.40 (3.0%) 0.20 (0.0%) 0.10 rp ^





8-23

Calculating portfolio standard deviation and CV

0.51 6.7%

3.4% CV

3.4%

6.7) - (12.0 0.10

6.7) - (9.5 0.20

6.7) - (7.5 0.40

6.7) - (3.0 0.20

6.7) - (0.0 0.10

p

2 1

2

2

2

2

2

p



      

      



8-24

 σp = 3.4% is much lower than the σi of either stock (σHT = 20.0%; σColl. = 13.2%).

 σp = 3.4% is lower than the weighted average of HT and Coll.’s σ (16.6%).

 Therefore, the portfolio provides the average return of component stocks, but lower than the average risk.

 Why? Negative correlation between stocks.

8-25

 σ  35% for an average stock.

 Most stocks are positively (though not perfectly) correlated with the market (i.e., ρ between 0 and 1).

 Combining stocks in a portfolio generally lowers risk.

8-26

Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0)

See p. 260

-10

15 15

25 25 25

15

0

-10

Stock W

0

Stock M

-10

0

Portfolio WM

8-27

Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)

See p. 261

Stock M

0

15

25

-10

Stock M’

0

15

25

-10

Portfolio MM’

0

15

25

-10

8-28

Creating a portfolio: Beginning with one stock and adding randomly selected stocks to portfolio

 σp decreases as stocks added, because they would not be perfectly correlated with the existing portfolio.

 Expected return of the portfolio would remain relatively constant.

 Eventually the diversification benefits of adding more stocks dissipates (after about 40 stocks), and for large stock portfolios, σp tends to converge to  20%.

8-29

Illustrating diversification effects of a stock portfolio

# Stocks in Portfolio 10 20 30 40 2,000+

Diversifiable Risk

Market Risk

20

0

Stand-Alone Risk, p

p (%)

35

8-30

Breaking down sources of risk

Stand-alone risk = Market risk + Diversifiable risk

Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta.

Diversifiable risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.

8-31

Failure to diversify  If an investor chooses to hold a one-stock

portfolio (doesn’t diversify), would the investor be compensated for the extra risk they bear?  NO!  Stand-alone risk is not important to a well-

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

with σp, 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

holding unnecessary, diversifiable risk.

8-32

Capital Asset Pricing Model (CAPM)

 Model linking risk and required returns. CAPM suggests that there is a Security Market Line (SML) that states that a stock’s required return equals the risk-free return plus a risk premium that reflects the stock’s risk after diversification.

ri = rRF + (rM – rRF) bi

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

8-33

Beta

 Measures a stock’s market risk, and shows a stock’s volatility relative to the market.

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

8-34

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

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

 If beta < 1.0, the security is less risky than average.

 Most stocks have betas in the range of 0.5 to 1.5.

Beta = ΔY/ΔX or ΔKi /ΔKm

8-35

Can the beta of a security be negative?

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

 If the correlation is negative, the regression line would slope downward, and the beta would be negative.

 However, a negative beta is highly unlikely.

8-36

Calculating betas

 Well-diversified investors are primarily concerned with how a stock is expected to move relative to the market in the future.

 Without a crystal ball to predict the future, analysts are forced to rely on historical data. A typical approach to estimate beta is to run a regression of the security’s past returns against the past returns of the market.

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

8-37

Illustrating the calculation of beta

.

.

.

ri

_

rM

_ -5 0 5 10 15 20

20

15

10 5

-5

-10

Regression line:

ri = -2.59 + 1.44 rM ^ ^

Year rM ri

1 15% 18%

2 -5 -10

3 12 16

8-38

Beta coefficients for HT, Coll, and T-Bills

ri

_

kM

_

-20 0 20 40

40 20

-20

HT: b = 1.30

T-bills: b = 0

Coll: b = -0.87

8-39

Comparing expected returns and beta coefficients

Security Expected Return Beta HT 12.4% 1.32 Market 10.5 1.00 USR 9.8 0.88 T-Bills 5.5 0.00 Coll. 1.0 -0.87 Riskier securities have higher returns, so the rank order is OK.

8-40

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

SML: ri = rRF + (rM – rRF) bi ri = rRF + (RPM) bi

 Assume the yield curve is flat and that rRF = 5.5% and RPM = 5.0%.  The market (or equity) risk premium is

RPM = (kM – kRF )= 10.5% – 5.5% = 5%.

8-41

What is the market risk premium?

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

 Its size depends on the perceived risk of the stock market and investors’ degree of risk aversion.

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

8-42

Calculating required rates of return

 rHT = 5.5% + (5.0%)(1.32)

= 5.5% + 6.6% = 12.10%

 rM = 5.5% + (5.0%)(1.00) = 10.50%  rUSR = 5.5% + (5.0%)(0.88) = 9.90%  rT-bill = 5.5% + (5.0%)(0.00) = 5.50%  rColl = 5.5% + (5.0%)(-0.87) = 1.15%

8-43

Expected vs. Required returns

r) r( Overvalued 1.2 1.0 Coll.

r) r( uedFairly val 5.5 5.5 bills-T

r) r( Overvalued 9.9 9.8 USR

r) r( uedFairly val 10.5 10.5 Market

r) r( dUndervalue 12.1% 12.4% HT

r r

^

^

^

^

^

^

8-44

Illustrating the Security Market Line

. . Coll.

. HT

T-bills

. USR

SML

rM = 10.5

rRF = 5.5

-1 0 1 2

.

SML: ri = 5.5% + (5.0%) bi

ri (%)

Risk, bi

8-45

An example: Equally-weighted two-stock portfolio

 Create a portfolio with 50% invested in HT and 50% invested in Collections.

 The beta of a portfolio is the weighted average of each of the stock’s betas.

bP = wHT bHT + wColl bColl

bP = 0.5 (1.32) + 0.5 (-0.87)

bP = 0.225

8-46

Calculating portfolio required returns

 The required return of a portfolio is the weighted average of each of the stock’s required returns.

rP = wHT rHT + wColl rColl

rP = 0.5 (12.10%) + 0.5 (1.15%)

rP = 6.63%

 Or, using the portfolio’s beta, CAPM can be used to solve for expected return.

rP = rRF + (RPM) bP

rP = 5.5% + (5.0%) (0.225)

rP = 6.63%

8-47

Factors that change the SML

 What if investors raise inflation expectations by 3%, what would happen to the SML?

SML1

ri (%) SML2

0 0.5 1.0 1.5

13.5

10.5

8.5

5.5

D I = 3%

Risk, bi

8-48

Factors that change the SML

 What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML?

SML1

ri (%) SML2

0 0.5 1.0 1.5

13.5

10.5

5.5

D RPM = 3%

Risk, bi

8-49

Verifying the CAPM empirically

 The CAPM has not been verified completely.

 Statistical tests have problems that make verification almost impossible.

 Some argue that there are additional risk factors, other than the market risk premium, that must be considered.

8-50

More thoughts on the CAPM

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

ri = rRF + (rM – rRF) bi + ???

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