4-riskandreturn

8/9/2019 4-RiskandReturn

1/30

Chapter 4 Risk and Return

Investment Returns

An individual or business spends moneytoday with the expectation of earningmore money in the future.

Returns can be expressed in dollar termsor percentage terms.


2/30

Exxon Stock Example

Lets say that in August 2005 you bought 20shares of XOM stock @ $60/share. Ignoringtransaction costs and dividends, lets say you soldthe stock one year later for $67/share.

In dollar terms, the return was:the amount received the amount invested Amount received: 20 * 67 = $1340

Amount invested: 20 * 60 = $1200= 1340 1200 = $140

Exxon Stock Example Continued

In order to make a meaningful judgment aboutthe return, you need to know:

scale (size) of the investment

timing of the return

Solution: rates of return (percentage returns):

Rate of return = amount received amount invested

amount invested


3/30

Rate of Return on Exxon Stock

Rate of return = amount received amount invested

amount invested

= 1340 1200

1200

= 11.67%

The rate of return calculation standardizes the return by

considering the return per unit of investment (perdollar).


4/30

Arithmetic versus Geometric Means

Geometric (compound) means are betterestimates of investment returns, particularlywhen you are referring to long-term and/orvolatile investments.

Arithmetic mean = annual rates of return

number of years

Geometric mean = e^ [ (ln annual rates of return)

number of years]

Choose an Investment

Stock BStock A

Return at endof year 4




-4%10%

52%9%

-22%11%

22%10%


5/30

To determine what your initial investment is worth:

Initial Investment * (1 + Geometric Mean)n


6/30

Figuring in Dividends

Lets say you bought 100 shares of GM in January 2002for $50/share. The stock prices over the following 4years are as follows:

In January of 2006 (exactly 4 years later) you sold the

shares for $25/share.

Each year a $2.00 dividend was paid.

Jan-02 50.00$

Jan-03 38.00$Jan-04 55.00$Jan-05 40.00$

Jan-06 25.00$

Calculate the Rate of Return (geometric mean)

Determine the annual returns by taking the endprice minus the beginning price. Add in thedividend. Then divide this capital gain/loss bythe beginning price.


7/30

Risk: Basics

Assumption: People are risk averse:people will invest in riskier assets only ifthey expect to receive higher returns.

In other words, no investment should beundertaken unless the expected rate ofreturn is high enough to compensateinvestors for the perceived level of risk ofthe investment.


8/30

Investment Risk

Risk can be considered on a stand-alone basis orin a portfolio context.

If assets always produced their expectedreturns, they wouldnt be risky.

Investment risk is related to the probability ofactually earning a low or negative return thegreater the chance of earning a low (or

negative) return, the riskier the investment.

Statistics Refresher

Probability is defined as the chance that anevent will occur.

Probability distribution is a list of all possible

outcomes with a probability assigned to eachevent.

The weighted average is the sum of theoutcomes multiplied by their probabilities. Theweighted average is the expected rate of return.


9/30

Demand for company'sproducts

Probability of Demandoccuring Overstock.com Proctor & Gamble

Strong 0.3 100% 20%

Normal 0.4 15% 15%Weak 0.3 -70% 10%

1

Rate of Return on Stock if Demand Occurs

Expected Rate of Return

Expected Rate of Return = r = P1r1 + P2r2 + + Pnrn

Overstock.coms Expected Return = 0.3(100%) + 0.4(15%) + 0.3(-70%)

= 15%

Proctor & Gambles Expected Return = 0.3(20%) + 0.4(15%) + 0.3(10%)

= 15%

^.

n

1=i

iiPr=r

Which stock would you choose?

Both stocks have expected returns of 15%.

Probability distribution

Rate of

return (%)50150-20

Stock X

Stock Y


10/30

Variance & Standard Deviation

To measure the tightness of a probability distribution wecan use the standard deviation.

.

Variance

deviationStandard

1

2

2

=

=

==

=

n

i

iiPrr

Measuring Risk

(r-E[r]) (r-E[r]) 2 {(r-E[r])^2}P85% 0.7225 0.216750% 0 0

-85% 0.7225 0.216750.4335 Variance

65.841% Standard Deviation

(r-E[r]) (r-E[r]) 2 {(r-E[r])^2}P5% 0.0025 0.000750% 0 0

-5% 0.0025 0.000750.0015 Variance

3.873% Standard Deviation

Overstock.com

Proctor & Gamble


11/30

Ok, so if a choice has to be made between twoinvestments with the same expected return butdifferent standard deviations, we would choosethe one with the lower standard deviation (andtherefore, lower risk).

How do we choose between two investments ifone has a higher expected return but the other

has a lower standard deviation?

Coefficient of Variation

Coefficient of Variation = CV =

r

This shows the risk per unit of return when

expected returns are not the same.

^


12/30

Coefficient of Variation Example

Project X has a 60% expected rate ofreturn and a 15% standard deviation.

Project Y has an 8% expected return anda 3% standard deviation.

Which project would you choose?

Coefficient of Variation = CV = r^

Risk in a Portfolio Context

Up to now, we have considered the risk ofassets held in isolation. Now we want toanalyze the risk of assets held in a

portfolio.


13/30

Expected Return on a Portfolio

Expected Return on a portfolio is simply the weightedaverage of the expected returns on the individual assets inthe portfolio with the weights being the fraction of the totalportfolio invested in each asset.

rp is a weighted average:^

^ ^rp = wiri.ni = 1

Calculate this Portfolios Expected Return

Form a $100,000 portfolio, investing$25,000 in each stock.

Stock Expected Return

Microsoft 12.0%General Electric 11.5%Pfizer 10.0%Coca-Cola 9.5%


14/30

Portfolio Risk

Unlike returns, the risk of a portfolio is not foundby taking the weighted average of the standarddeviations of the individual assets in theportfolio. The portfolios risk will almost alwaysbe smaller than the weighted average of theassets standard deviations. It is theoreticallypossible to combine stocks that are individuallyquite risky, forming a portfolio that is completelyriskless.

The tendency of two variables to move togetheris called correlation and is measured by thecorrelation coefficient ().

Two-Stock Portfolios

Two stocks can be combined to form ariskless portfolio if = -1.0.

Risk is not reduced at all if the two

stocks have = +1.0. In general, two randomly selected stocks

have 0.60, so risk is lowered but noteliminated.


15/30

What would happen to therisk of a 1-stock

portfolio as more randomlyselected stocks were added?

p would decrease because the added stockswould not be perfectly correlated, but rp would

remain relatively constant.

^

Large

0 15

Prob.

2

1

1 35% ; Large 20%.Return


16/30

Diversifiable Risk versus Market Risk

Diversifiable can be eliminated throughdiversification. Diversifiable risk is caused byrandom, company-specific, events such aslawsuits, strikes, winning or losing a majorcontract, etc..

Market (systematic) risk stems from factors thataffect most firms: war, recession, inflation,

interest rates, etc.. Market risk cannot beeliminated through diversification.

# Stocks in Portfolio

10 20 30 40 2,000+

Company Specific

(Diversifiable) Risk

Market Risk

20

0

Stand-Alone Risk, p

p (%)35


17/30

If investors are concerned with the risk oftheir portfolios, rather than the risk of the

individual securities, how should the risk ofan individual stock be measured?

Answer: by looking only at the relevantrisk: the contribution of a security to theoverall riskiness of the portfolio.

Beta Coefficient

The Capital Asset Pricing Model (CAPM) uses themarket portfolio (portfolio containing all stocks) asthe benchmark. Relevant risk is therefore definedas the amount of risk that the stock contributes tothe market portfolio. The relevant risk is calledthe stocks beta coefficient.

bi = (iM i) / M


18/30

How are betas calculated?

In addition to measuring a stockscontribution of risk to a portfolio, beta alsowhich measures the stocks volatilityrelative to the market.

Using a Regression to Estimate Beta

The Return on our Stock is dependent upon the Return on the Market.So, the return on our stock is referred to as the dependent variable (Y)and the return on the market is referred to as the independentvariable (X).

Run a regression with returns on the stock in question plotted on the Yaxis and returns on the market portfolio plotted on the X axis.

Our model looks like this:Return on our Stockt = + (Return on Markett)

Where is the intercept, is the slope. The slope measures relativevolatility, so it is defined as the stocks beta coefficient, or b.


19/30

Use the historical stock returns tocalculate the beta for AMR.

25.0%-13.1%10-25.0%-10.8%9 -10.0%10.0%8

42.0%40.0%730.0%13.7%610.0%32.5%535.0%15.0%4

-15.0%-11.0%3-15.0%8.0%240.0%25.7%1

AMRMarketYear

Data Analysis Add-In

The regression line, and hence beta, can be foundusing a calculator with a regression function or a

spreadsheet program.

In order to perform regression analysis in Excel, youmust have Excels Data Analysis feature enabled. DataAnalysis may not be automatically listed as an optionunder Tools. You may need to add the Data Analysisfunction by choosing Add-ins from the Tools menu andthen choosing Data Analysis Add-in.


20/30

Running the Regression

Adjusted R Square tells us the proportion of the variation in

the dependent variable that can be explained by the

independent variable. In this case, the markets return is

able to explain 27.4% of the variation in AMRs return.

1 the p-value tells us with how much certainty we can say

that there is a relationship between our variables. If our p-

value was close to 0, it would mean that we are close to

100% certain that the variation in the dependent variable

can be explained by the independent variable.

The intercept is the value

for in our regression

equation.

The return on the

markets coefficient is

the stocks beta.


21/30

Interpreting Regression Results

The R2 measures the percent of a stocksvariance that is explained by the market.The typical R2 is:

0.3 for an individual stock

over 0.9 for a well diversified portfolio

Calculating Beta for AMR

rAMR 0.83rM= 0.03 +

R2

= 27.4%-40%

-20%

0%

20%

40%

-40% -20% 0% 20% 40%

rM

rAMR


22/30

Calculating Beta in Practice

Many analysts use the S&P 500 tofind the market return.

Analysts typically use four or fiveyears of monthly returns to establishthe regression line.

Some analysts use 52 weeks ofweekly returns.

If b = 1.0, stock has average risk.

If b > 1.0, stock is riskier than average.

If b < 1.0, stock is less risky thanaverage.

Most stocks have betas in the range of0.5 to 1.5.

Can a stock have a negative beta?

How is beta interpreted?


23/30

Finding Beta Estimates on the Web

Go to www.msn.com

Then click on the link for Money.

Enter the ticker symbol for aStock Quote, such as IBM orDell, then click GO.

When the quote detail comes upyou will see the stocks Beta.

Use the SML to calculate eachalternatives required return

The Security Market Line (SML) is part of the Capital Asset Pricing Model(CAPM). The Capital Asset Pricing Model (CAPM) provides us with a modelfor determining required returns.

SML: Required Ratei = Rf+ i [ E(Rm) - Rf]

Where Rf is the risk-free rate of interest, i is a measure of the riskiness

of security i relative to the riskiness of the market portfolio, and E(Rm) isthe expected rate of return on the market portfolio.

In CAPM, Rfserves as the base rate of interest. It is defined as the rateof return on a security with no risk. We know this risk-free rate inadvance. Ordinarily, the risk-free rate is assumed to be the rate on a U.S.Treasury (a 91-day T-bill), because it is a short term rate and it isassumed to be free of default risk. The risk-free rate is also referred toas the pure time value of money; the rate of return that is earned fordelaying consumption but not accepting any risk.


24/30

CAPM in Action

As a financial analyst, you are asked to prepare areport detailing your firms expectations regarding twostocks for the coming year.

Given:S&P 500 is expected to earn 11% in the year ahead.The risk-free (T-bill) rate of return is 5%.According to msn.com, the betas for stocks X and Yare 0.5 and 1.5 respectively.

What are the required returns for X and Y?

CAPM in Action

Given:

S&P 500 is expected to earn 11% in the year ahead.

The risk-free (T-bill) rate of return is 5%.

According to Reuters, the betas for stocks X and Y are 0.5 and 1.5 respectively.


How do we know that beta of a risk-free asset is 0?

How do we know that beta of the market is 1?

Recall that beta is a measure of the riskiness of the asset relative to the market.

Required Return


25/30

CAPM in Action


Simply use the CAPM formula to get results:

Required Returni = Rf+ i [ E(Rm) - Rf]

Required Return

CAPM in Action

We can create an XY Scatter Plot of our results which will depict the SecurityMarket Line.

As risk increases, required return increases.

The Security Market Line

0.00%

5.00%

10.00%

15.00%

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

Beta

ExpectedR

eturn

RequiredReturn


26/30

Expected Return versus Market Risk

Which of the alternatives is best?

-0.861.7Repo Men

0.008.0T-bills

0.6813.8Am. Foam

1.0015.0Market

1.2917.4%Alta

Risk, breturnSecurity

Expected

Use the SML to calculate eachalternatives required return

Assume:

rRF = 8%; rM = rM = 15%.

RPM = (rM - rRF) = 15% - 8% = 7%.

^


27/30

Required Rates of Return

rAlta = 8.0% + (7%)(1.29)

= 8.0% + 9.0% = 17.0%.

rM = 8.0% + (7%)(1.00) = 15.0%.

rAm. F.= 8.0% + (7%)(0.68) = 12.8%.

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

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

Expected versus Required Returns

^

Overvalued2.01.7Repo

Fairly valued8.08.0T-bills

Undervalued12.813.8Am. F.

Fairly valued15.015.0Market

Undervalued17.0%17.4%Alta

rr


28/30

..Repo

.Alta

T-bills

.Am. Foam

rM = 15

rRF = 8

-1 0 1 2

.

SML: ri = rRF + (RPM) biri = 8% + (7%) bi

ri (%)

Risk, bi

SML and Investment Alternatives

Market

Calculate beta for a portfolio with 50%Alta and 50% Repo

bp = Weighted average

= 0.5(bAlta) + 0.5(bRepo)

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


29/30

What is the required rate of returnon the Alta/Repo portfolio?

rp = Weighted average r

= 0.5(17%) + 0.5(2%) = 9.5%.

Or use SML:

rp

= rRF

+ (RPM

) bp= 8.0% + 7%(0.22) = 9.5%.

SML1

Original situation

Required Rate

of Return r (%)

SML2

0 0.5 1.0 1.5 2.0

1815

11

8

New SML

I = 3%

Impact of Inflation Change on SML


30/30

rM = 18%

rM = 15%

SML1

Original situation

Required Rateof Return (%)SML2

After increasein risk aversion

Risk, bi

18

15

8

1.0

RPM = 3%

Impact of Risk Aversion Change

Has the CAPM been completely confirmed orrefuted through empirical tests?

No. The statistical tests have problemsthat make empirical verification orrejection virtually impossible. Investors required returns are based on

future risk, but betas are calculated withhistorical data. Investors may be concerned about both

stand-alone and market risk.

4-riskandreturn

Documents