U6-1 UNIT 6 Risk and Return and Stock Valuation Risk return tradeoff Diversifiable risk vs. market risk Risk and return: CAPM/SML Stock valuation: constant,

U6-1

UNIT 6Risk and Return and Stock Valuation

Risk return tradeoff Diversifiable risk vs. market risk Risk and return: CAPM/SML Stock valuation: constant,

nonconstant, and zero growth

U6-2

Investment alternatives

Economy Prob. T-Bill HT Coll USR MP

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%

U6-3

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

1iii

^

^

U6-4

Summary of expected returns

Expected returnHT 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?

U6-5

Calculating standard deviation

deviation Standard

2Variance

i2

N

1ii P)r(rσ

ˆ

U6-6

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

1ii

2^

i

21

U6-7

Comparing standard deviations

USR

Prob.T - bill

HT

0 5.5 9.8 12.4 Rate of Return (%)

U6-8

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.

U6-9

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.

^

U6-10

Coefficient of Variation (CV)

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

r

return Expecteddeviation Standard

CV ˆ

U6-11

Risk rankings, by coefficient of variation

CVT-bill 0.0HT 1.6Coll. 13.2USR 1.9Market 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.

U6-12

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.

U6-13

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.

U6-14

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.

U6-15

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

^

U6-16

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

21

2

2

2

2

2

p

U6-17

Comments on portfolio risk measures

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.

U6-18

General comments about risk

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.

U6-19

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 10 stocks), and for large stock portfolios, σp tends to converge to 20%.

U6-20

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.

U6-21

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.

U6-22

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.

U6-23

Comments on beta 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.

U6-24

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.

U6-25

Comparing expected returns and beta coefficients

Security Expected Return Beta HT 12.4% 1.32Market 10.5 1.00USR 9.8 0.88T-Bills 5.5 0.00Coll. 1.0 -0.87

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

U6-26

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%.

U6-27

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%

U6-28

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

^

^

^

^

^

^

U6-29

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

U6-30

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

U6-31

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%

U6-32

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

13.510.5

8.5 5.5

I = 3%

Risk, bi

U6-33

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

13.510.5

5.5

RPM = 3%

Risk, bi

U6-34

Intrinsic Value and Stock Price

Outside investors, corporate insiders, and analysts use a variety of approaches to estimate a stock’s intrinsic value (P0).

In equilibrium we assume that a stock’s price equals its intrinsic value. Outsiders estimate intrinsic value to

help determine which stocks are attractive to buy and/or sell.

Stocks with a price below (above) its intrinsic value are undervalued (overvalued).

U6-35

Dividend growth model Value of a stock is the present value of

the future dividends expected to be generated by the stock.

)r(1D

... )r(1

D

)r(1D

)r(1

D P

s3

s

32

s

21

s

10

^

U6-36

Constant growth stock A stock whose dividends are expected

to grow forever at a constant rate, g.

D1 = D0 (1+g)1

D2 = D0 (1+g)2

Dt = D0 (1+g)t

If g is constant, the dividend growth formula converges to:

g -rD

g -rg)(1D

Ps

1

s

00

^

U6-37

Future dividends and their present values

t0t ) g 1 ( DD

tt

t )r 1 (D

PVD

t0 PVDP

$

0.25

Years (t)0

U6-38

What happens if g > rs?

If g > rs, the constant growth formula leads to a negative stock price, which does not make sense.

The constant growth model can only be used if: rs > g g is expected to be constant forever

U6-39

If rRF = 7%, rM = 12%, and b = 1.2, what is the required rate of return on the firm’s stock?

Use the SML to calculate the required rate of return (rs):

rs = rRF + (rM – rRF)b

= 7% + (12% - 7%)1.2= 13%

U6-40

If D0 = $2 and g is a constant 6%, find the expected dividend stream for the next 3 years, and their PVs.

1.8761

1.7599

D0 = 2.00

1.6509

rs = 13%

g = 6%0 1

2.247

2

2.382

3

2.12

U6-41

What is the stock’s intrinsic value?

Using the constant growth model:

$30.29

0.07$2.12

0.06 - 0.13$2.12

g - r

D P

s0

1ˆ

U6-42

What is the expected market price of the stock, one year from now?

D1 will have been paid out already. So, P1 is the present value (as of year 1) of D2, D3, D4, etc.

Could also find expected P1 as:

$32.10

0.06 - 0.13$2.247

g - r

D P

s

2^

1

$32.10 (1.06) P P 0

^

1

U6-43

What are the expected dividend yield, capital gains yield, and total return during the first year?

Dividend yield= D1 / P0 = $2.12 / $30.29 = 7.0%

Capital gains yield= (P1 – P0) / P0 = ($32.10 - $30.29) / $30.29 = 6.0%

Total return (rs)= Dividend Yield + Capital Gains Yield= 7.0% + 6.0% = 13.0%

U6-44

What would the expected price today be, if g = 0?

The dividend stream would be a perpetuity.

2.00 2.002.00

0 1 2 3rs = 13% ...

$15.38 0.13$2.00

r

PMT P

^

0

U6-45

Supernormal growth:What if g = 30% for 3 years before achieving long-run growth of 6%?

Can no longer use just the constant growth model to find stock value.

However, the growth does become constant after 3 years.

U6-46

Valuing common stock with nonconstant growth

rs = 13%

g = 30% g = 30% g = 30% g = 6%

P 0.06

$66.543

4.658

0.13

2.301

2.647

3.045

46.114

54.107 = P0

^

0 1 2 3 4

D0 = 2.00 2.600 3.380 4.394

...

4.658

U6-47

If the stock was expected to have negative growth (g = -6%), would anyone buy the stock, and what is its value?

The firm still has earnings and pays dividends, even though they may be declining, they still have value.

$9.89 0.19$1.88

(-0.06) - 0.13(0.94) $2.00

g - r) g 1 (D

g - r

D P

s

0

s

1^

0

U6-48

What is market equilibrium?

In equilibrium, stock prices are stable and there is no general tendency for people to buy versus to sell.

In equilibrium, two conditions hold: The current market stock price equals its

intrinsic value (P0 = P0).

Expected returns must equal required returns.

)br(r r r g PD

r RFMRFs0

1^

s

^