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CHAPTER 9Risk and Rates of Return

Stand-alone risk (statistics review)

Portfolio risk (investor view) --

diversification important

Risk & return: CAPM/SML (market

equilibrium)

2Risk is viewed primarily from the stockholder perspective

Management cares about risk because stockholders care about risk.

If stockholders like or dislike something about a company (like risk) it affects the stock price.

Risk affects the discount rate for future returns -- directly affecting the multiple (P/E ratio)

Thus, the concern is still about the stock price. Stockholders have portfolios of investments –

they have stock in more than just one company and a great deal of flexibility in which stocks they buy.

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What is investment risk?

Investment risk pertains to the uncertainty regarding the rate of return.

Especially when it is less than the expected (mean) return.

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

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Return = dividend + capital gain or loss

Dividends are relatively stable Stock price changes (capital gains/losses)

are the major uncertain component There is a range of possible outcomes and

a likelihood of each -- a probability distribution.

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Expected Rate of Return

The mean value of the probability distribution of possible returns

It is a weighted average of the outcomes, where the weights are the probabilities

6Expected Rate of Return

(k hat)

n

1iii

nn2211

k

k...kkk̂

p

ppp

7

Investment Alternatives

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 1.0

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

8Why is the T-bill

return independent of the economy?

Will return be 8%regardless of the economy?

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Do T-bills really promise acompletely risk-free return?

No, T-bills are still exposed tothe risk of inflation.However, not much unexpectedinflation 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?

High Tech: With. Positive correlation.

Typical.

Collections: Countercyclical.

Negative correlation. Unusual.

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i

n

1=iipk = k

Calculate the expected rate ofreturn for each alternative:

k = 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?

^

Calculate others on your own

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What’s the standard deviationof returns for each alternative?

= Variance = 2

n

1=ii

2i p)k̂(k =

= standard deviation

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Normal Distribution

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One often assumes that data are from an approximately normally distributed population. then

about 68.26% of the values are at within 1 standard deviation away from the mean,

95.46% of the values are within two standard deviations and

99.73% lie within 3 standard deviations.

In a sample of observations

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= (k k) Pi2

ii=1

n

T-bills = 0.0%.HT = 20.0%. Coll = 13.4%.

USR = 18.8%. M = 15.3%.

^

0748599.20]403[]1.017.450

20.017.4-35 + 4.017.4-20 2.017.4-2- + 1.017.4-22-[ = 5.05.02

2222HT

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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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Security Expectedreturn

Risk,

High Tech 17.4% 20.0Market 15.0 15.3US Rubber 13.8* 18.8*T-bills 8.0 0.0Collections 1.7* 13.4*

Expected Returns vs. Risk:

*Return looks low relative to

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Coefficient of variation (CV):

Standardized measure of dispersionabout the expected value:

CV = Std devMean

=

k

Shows risk per unit of return.

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Portfolio Risk & Return

Assume a two-stock portfoliowith $50,000 in HighTech and $50,000 in Collections.

Calculate kp and p.

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Portfolio Expected Return, kp

n

1 = i.iip k̂w = k̂

kp is a weighted average:

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

kp is between kHT and kCOLL.

^

^

^

^

^ ^

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Alternative Method:

Estimated ReturnEconomy 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

kp = (3.0%)0.1 + (6.4%)0.20 + (10.0%)0.4 + (12.5%)0.20 + (15.0%)0.1 = 9.6%

^

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2/1

2

22

22

1.06.90.15

20.06.95.124.06.90.10

20.06.94.61.09.6-3.0

=

P

= 3.3%

CV = 3.3%9.6%

= 0.34.P

Note: you can work the variance calculation in either decimal or percentage

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p = 3.3% is much lower than that of either stock (20% and 13.4%).

p = 3.3% is also lower than avg. of HT and Coll, which is 16.7%.

Portfolio provides avg. return but lower risk.

Reason: diversification. Negative correlation is present

between HT and Coll but is not required to have a diversification effect

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General Statements about risk:

Most stocks are positively correlated. rk,m 0.65.

You still get a lot of diversification effect at .65 correlation

35% for an average stock. Combining stocks generally

lowers risk.

26What would happen to theriskiness of a 1-stock

portfolio as more randomlyselected stocks were added?

p would decrease because the added stocks would not be perfectly correlated

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# stocks in portfolio10 20 30 40 ...... 1500+

Company Specific risk

Market Risk

p %

35

20

0

Total Risk, P

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

p falls very slowly after about 40 stocks are included. The lower limit for p is about 20% = M .

29Decomposing Risk—Systematic (Market) and

Unsystematic (Business-Specific) Risk

Fundamental truth of the investment world– The returns on securities tend to move up and

down together• Not exactly together or proportionately

Events and Conditions Causing Movement in Returns– Some things influence all stocks (market risk)

• Political news, inflation, interest rates, war, etc.– Some things influence only particular firms

(business-specific risk)• Earnings reports, unexpected death of key

executive, etc.– Some things affect all companies within an

industry• A labor dispute, shortage of a raw material

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Total = Market + Firm specificrisk risk risk

Market risk is that part of a security’s risk that cannot beeliminated by diversification.Firm-specific risk is that partof a security’s risk which canbe eliminated withdiversification.

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

(actually35% vs. 20% is a 43%reduction)

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

CAPM -- Capital Asset Pricing Model

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NO! Stand-alone risk as

measured by a stock’s

or CV is not important to well-diversified investors.

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

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

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

The higher beta, the higher the expected rate of return.

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How are betas calculated? Run a regression of past

returns on Stock i versus returns on the market.

The slope coefficient is the beta coefficient.

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-5 0 5 10 15 20

20

15

10

5

-5

-10

Year kM ki 1 15% 18% 2 -5 -10 3 12 16

.

.

.

ki = -2.59 + 1.44 kM

Illustration of beta = slope:

Regression line

ik

Mk

37Find beta:

Statistics program or spreadsheet regression

Find someone’s estimate of beta for a given stock on the web

Generally use weekly or monthly returns, with at least a year of data

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If beta = 1.0, average risk stock. (The ‘market’ portfolio has a beta of 1.0.)

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.

Some ag. related companies have betas less than 0.5

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=1, get the market expected return

<1, earn less than the market expected return

>1, get an expected return greater than the market

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Answer: Yes, if the correlation between ki and kM is negative.

Then in a “beta graph” the regression line will slope downward. Negative beta -- rare

Can a beta be negative?

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HT

T-bills

b=0

ki

kM

-20 0 20 40

40

20

-20

b = 1.29

Collb = -0.86

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SecurityExpected

ReturnRisk

(Beta)HighTech 17.4% 1.29“Market” 15.0 1.00US Rubber 13.8 0.68T-bills 8.0 0.00Collections 1.7 -0.86

Riskier securities have higherreturns, so the rank order is O.K.

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Given the beta of a stock, a theoretical required rate of return can be

calculated. The Security Market Line (SML) is used. SML: ki = kRF + (kM - kRF)bi

MRP

MRP= market risk premium

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ki = kRF + (kM - kRF)bi

45For Term Projects (2013)

Use KRF = 3.0%; this is a bit more than the current 10 year treasury rate of 2.75%. Sometimes analysts use a shorter term rate and short term treasuries are still extremely low, but we are going to use 3.0%.

Use MRP = 5%. This is MRP, not KM. The historical average MRP is about 5%. Find your own beta from the web On Yahoo Finance look up your company and

then the “key statistics” tab on the left will give you their beta

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The Bottom Line on Riskfree Rates  Using a long term government rate (even on a coupon bond) as the riskfree rate on all of the cash flows in a long term analysis will yield a close approximation of the true value. For short term analysis, it is entirely appropriate to use a short term government security rate as the riskfree rate.

  The riskfree rate that you use in an analysis should be in the same currency that your cashflows are estimated in. •  In other words, if your cashflows are in U.S. dollars, your riskfree rate has to be in U.S. dollars as well.•  If your cash flows are in Euros, your riskfree rate should be a Euro riskfree rate.  The conventional practice of estimating riskfree rates is to use the government bond rate, with the government being the one that is in control of issuing that currency. In US dollars, this has translated into using the US treasury rate as the riskfree rate. In May 2009, for instance, the ten-year US treasury bond rate was 3.5%.

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Use the SML to calculatethe required returns (for the example)

Assume kRF = 8%. Note that kM = kM is 15%. MRP = 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%kTbill = 8.0% + (7%)0.00 = 8.0%kColl = 8.0% + (7%)(-0.86) = 2.0%

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

Portfolio Beta

bP = weighted average of the betas of the stocks in the portfolio

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

Weights are the proportions invested in each stock.

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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 for the portfolio:

kP = kRF + (kM - kRF) bP

= 8.0% + (15.0% - 8.0%) (0.22)= 8.0% + 7%(0.22) = 9.5% .

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Using Beta—The Capital Asset Pricing Model (CAPM)

The CAPM helps us determine how stock prices are set in the marketDeveloped in 1950s and 1960s by Harry

Markowitz and William Sharpe The CAPM's Approach

People won't invest unless a stock's expected return is at least equal to their required return

The CAPM attempts to explain how investors' required returns are determined

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Has the CAPM been verified through empirical tests?

Not completely. Because statistical tests have problems which 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.