LN08Brooks671956_02_LN08

7/28/2019 LN08Brooks671956_02_LN08

1/80

Chapter 8

Risk and Return

7/28/2019 LN08Brooks671956_02_LN08

2/80

2013 Pearson Education, Inc. All rights reserved.8-2

Learning Objectives

1. Calculate profits and returns on an investmentand convert holding period returns to annualreturns.

2. Define risk and explain how uncertainty

relates to risk.3. Appreciate the historical returns of various

investment choices.

4. Calculate standard deviations and variances

with historical data.5. Calculate expected returns and variances with

conditional returns and probabilities.

7/28/2019 LN08Brooks671956_02_LN08

3/80


Learning Objectives

6. Interpret the trade-off between risk andreturn.

7. Understand when and why diversificationworks at minimizing risk, and understand the

difference between systematic andunsystematic risk.

8. Explain beta as a measure of risk in a well-diversified portfolio.

9. Illustrate how the security market line and thecapital asset pricing model represent the two-parameter world of risk and return.

7/28/2019 LN08Brooks671956_02_LN08

4/80


8.1 Returns

Performance analysis of an investmentrequires investors to measure returns overtime.

Return and risk being intricately related,return measurement helps in theunderstanding of investment risk.

7/28/2019 LN08Brooks671956_02_LN08

5/80


8.1 (A) Dollar Profits andPercentage Returns

Dollar Profit or Loss = Ending value+ Distributions Original Cost

Rate of return = Dollar Profit or LossOriginal Cost

7/28/2019 LN08Brooks671956_02_LN08

6/80


8.1 (A) Dollar Profits andPercentage Returns (continued)

HPR = ProfitCost

HPR =Ending price + Distributions - Beginning price

Beginning price

HPR= Ending price + Distributions - 1Beginning price

7/28/2019 LN08Brooks671956_02_LN08

7/80 2013 Pearson Education, Inc. All rights reserved.8-7


Example 1: Calculating dollar andpercentage returns. Joe bought some gold coins for $1000 and sold

those 4 months later for $1200. Jane on the other hand bought 100 shares of a

stock for $10 and sold those 2 years later for $12per share after receiving $0.50 per share asdividends for the year.

Calculate the dollar profit and percent return

earned by each investor over their respectiveholding periods.

7/28/2019 LN08Brooks671956_02_LN08



Example 1 AnswerJoes Dollar Profit = Ending value Original cost

= $1200 - $1000 = $200Joes HPR = Dollar profit/Original cost

= $200/$1000 = 20%

Janes Dollar Profit = Ending value+Distributions- Original Cost= $12*100 + $0.50*100 - $10*100= $1200 + $50 - $1000=$250

Janes HPR = $250/$1000 = 25%

7/28/2019 LN08Brooks671956_02_LN08


8.1 (B) Converting HoldingPeriod Returns to Annual Returns

With varying holding periods, holding period returns

not good for comparison.

Necessary to state an investments performance in terms of

an annual percentage rate (APR) or an effective annual rate of

return (EAR) by using the following conversion formulas:

Simple annual return or APR = HPRn

EAR = (1 + HPR)

1/n

1 Where n is the number of years or proportion of a year that

the holding period consists of.

7/28/2019 LN08Brooks671956_02_LN08

10/80


8.1 (B) Converting Holding PeriodReturns to Annual Returns (continued)

Example 2: Comparing HPRs.

Given Joes HPR of 20% over 4 months andJanes HPR of 25% over 2 years, is it correct

to conclude that Janes investmentperformance was better than that of Joe?

7/28/2019 LN08Brooks671956_02_LN08

11/80


8.1 (B) Converting HoldingPeriod Returns to Annual Returns

Example 2 Answer

Compute each investorsAPR and EAR and then make thecomparison.

Joes holding period (n) = 4/12 = .333 years

JoesAPR = HPR/n = 20%/.333 = 60%Joes EAR = (1 + HPR)1/n 1 =(1.20)1/.33 1= 72.89%

Janes holding period = 2 yearsJanesAPR = HPR/n = 25%/2 = 12.5%

Janes EAR = (1 + HPR)1/n 1 = (1 .25)1/2 1=11.8%

Clearly, on an annual basis, Joes investment far outperformedJanes investment.

7/28/2019 LN08Brooks671956_02_LN08

12/80


8.1 (C) Extrapolating HoldingPeriod Returns

Extrapolating short-term HPRs into APRsand EARs is mathematically correct, butoften unrealistic and infeasible.

Implies earning the same periodic rate overand over again in 1 year.

A short holding period with fairly high HPRwould lead to huge numbers if return is

extrapolated.

7/28/2019 LN08Brooks671956_02_LN08

13/80


8.1 (C) Extrapolating HoldingPeriod Returns (continued)

Example 3: Unrealistic nature of APR and EARLets say you buy a share of stock for $2 and sell it a weeklater for $2.50. Calculate your HPR,APR, and EAR. Howrealistic are the numbers?

N = 1/52 or 0.01923 of 1 year.

Profit = $2.50 - $2.00 = $0.50

HPR = $0.5/$2.00 = 25%

APR = 25%/0.01923=1300% or

=25%*52 weeks = 1300%

EAR = (1 + HPR)52

1=(1.25)52 1= 109,526.27%

Answer: Highly Improbable!

7/28/2019 LN08Brooks671956_02_LN08

14/80


8.2 Risk (Certainty andUncertainty)

Future performance of most investments isuncertain.

Risky Potential for loss exists

Risk can be defined as a measure of theuncertainty in a set of potential outcomes foran event in which there is a chance of someloss.

It is important to measure and analyze the riskpotential of an investment, so as to make aninformed decision.

7/28/2019 LN08Brooks671956_02_LN08

15/80


8.3 Historical Returns

Figure 8.1 Histograms of (A) U.S. Treasury bills from 1950 to1999, (B) long-term government bonds from 1950 to 1999, (C)large company stocks from 1950 to 1999, and (D) smallcompany stocks from 1950 to 1999.

7/28/2019 LN08Brooks671956_02_LN08

16/80


8.3 Historical Returns(continued)

Small company stocks earned the highestaverage return (17.10%) over the 5 decades,but also had the greatest variability 29.04%,widest range

(103.39% - (-40.54%)) = 143.93%), and weremost spread out.

3-month treasury bills earned the lowestaverage return, 5.23%, but their returns hadvery low variability (2.98%), a very small range(14.95%-0.86% = 15.91%) and were muchclosely clustered around the mean.

Returns and risk are positively related.

7/28/2019 LN08Brooks671956_02_LN08

17/80


8.4 Variance and StandardDeviation as a Measure of Risk

Variance and standard deviation are measures ofdispersion

Helps researchers determine how spread out orclustered together a set of numbers or outcomes is

around their mean or average value. The larger the variance, the greater is the variability

and hence the riskiness of a set of values.

7/28/2019 LN08Brooks671956_02_LN08

18/80


8.4 Variance and Standard Deviationas a Measure of Risk (continued)

Example 4: Calculating the variance ofreturns for large-company stocks

Listed below are the annual returnsassociated with the large-company stockportfolio from 1990 - 1999. Calculate thevariance and standard deviation of the

returns.

7/28/2019 LN08Brooks671956_02_LN08

19/80



Year Return (R - Mean) (R-Mean)2

1990 -3.20% -22.19% 0.0492396

1991 30.66% 11.67% 0.0136189

1992 7.71% -11.28% 0.0127238

1993 9.87% -9.12% 0.0083174

1994 1.29% -17.70% 0.031329

1995 37.71% 18.72% 0.0350438

1996 23.07% 4.08% 0.0016646

1997 33.17% 14.18% 0.0201072

1998 28.58% 9.59% 0.0091968

1999 21.04% 2.05% 0.0004203

Total 189.90% .18166156

Average 18.99%

Variance 0.020184618

Std. Dev 14.207%.

7/28/2019 LN08Brooks671956_02_LN08

20/80



Example 4 AnswerVariance = (R-Mean)2

N 1

= 0.1816615610-1

= 0.020184618

Std. Dev. = Variance

= .020184618 = 14.207%

7/28/2019 LN08Brooks671956_02_LN08

21/80


8.4 (A) Normal Distributions

Normal distribution with Mean = 0 and Std. Dev. = 1

About 68% of the area lies within 1 Std. Dev. from the mean.About 95% of the observations lie within 2 Std. Dev. from the mean.About 99% of the observations lie within 3 Std. Dev. from the mean.Smaller variances = less risky = less uncertainty about their futureperformance.

Figure 8.2Standardnormaldistribution.

7/28/2019 LN08Brooks671956_02_LN08

22/80


8.4 (A) Normal Distributions(continued)

If Mean =10% and Standard deviation = 12%and data are normally distributed:

68% probability that the return in theforthcoming period will lie between 10% + 12%

and 10% - 12% i.e. between -2% and 22%.

95% probability that the return will lie between10% + 24% and 10% - 24% i.e. between -14% and 34%

99% probability that the return will lie between10% + 36% and 10% - 36% i.e. between -26%and 46%.

7/28/2019 LN08Brooks671956_02_LN08

23/80



TABLE 8.2 Returns, Variances, and StandardDeviations of Investment Choices, 19501999

( ) l i ib i

7/28/2019 LN08Brooks671956_02_LN08

24/80



Over the past 5 decades (1950-1999), riskier investment groups haveearned higher returns and vice-versa.

History shows that the higher the return one expects the greaterwould be the risk (variability of return) that one would have totolerate.

Figure 8.3 Historicalreturns and standarddeviations of bonds andstocks.T = Treasury bills,B = government bonds,

L = large-companystocks, andS = small-companystocks.

7/28/2019 LN08Brooks671956_02_LN08

25/80


8.5 Returns in an Uncertain World(Expectations and Probabilities)

For future investments we need expected orex-anterather than ex-postreturn and risk measures.

For ex-ante measures we use probability distributions,

and then the expected return and risk measures areestimated using the following equations:

7/28/2019 LN08Brooks671956_02_LN08

26/80


8.5 (A) Determining the Probabilitiesof All Potential Outcomes

When setting up probability distributions thefollowing 2 rules must be followed:

1. The sum of the probabilities must always add up to1.0 or 100%.

2. Each individual probability estimate must bepositive.

7/28/2019 LN08Brooks671956_02_LN08

27/80


8.5 (A) Determining the Probabilitiesof All Potential Outcomes (continued)

Example 5: Expected return and riskmeasurement.

Using the probability distribution shown below,

calculate Stock XYZs expected return, E(r),andstandard deviation (r).

State of

theEconomy

Probability

of

EconomicState

Return in

EconomicState

Recession 45% -10%

Steady 35% 12%

Boom 20% 20%

7/28/2019 LN08Brooks671956_02_LN08

28/80


8.5 (A) Determining the Probabilitiesof All Potential Outcomes (continued)

Example 5 AnswerE(r) = Probability of Economic State * Return in Economic State

= 45%*(-10%) + 35%*(12%) + 20%*(20%)

= -4.5% + 4.2% + 4% = 3.7%

2 (r) = [Return in Statei E(r)]2 * Probability of Statei

= (-10%-3.7%)2*45% + (12%-3.7%)2*35%+(20%-3.7%)2*20%= 84.4605 +24.1115+53.138 = 161.71

(r) = 161.71 = 12.72%

8 6 Th Ri k d R t T d

7/28/2019 LN08Brooks671956_02_LN08

29/80


8.6 The Risk-and-Return Trade-off

Investments must be analyzed in terms of,both, their return potential as well as theirriskiness or variability.

Historically, its been proven that higherreturns are accompanied by higher risks.

7/28/2019 LN08Brooks671956_02_LN08

30/80


8.6 (A) Investment Rules

Investment rule number 1: If faced with 2 investment choiceshaving the same expected returns, select the one with the lower

expected risk.

Investment rule number 2: If two investment choices have similar

risk profiles, select the one with the higher expected return.

To maximize return and minimize risk, it would be ideal to select an

investment that has a higher expected return and a lower expected

risk than the other alternatives.

Realistically, higher expected returns are accompanied by greater

variances and the choice is not that clear cut. The investorstolerance for and attitude towards risk matters.

In a world fraught with uncertainty and risk, diversification is the

key!

7/28/2019 LN08Brooks671956_02_LN08

31/80

7/28/2019 LN08Brooks671956_02_LN08

32/80

8 7 Di ifi ti Mi i i i Ri k

7/28/2019 LN08Brooks671956_02_LN08

33/80


8.7 Diversification: Minimizing Riskor Uncertainty (continued)

Table 8.4 presents a probability distribution of theconditional returns of two firms, Zig and Zag, alongwith those of a 50-50 portfolio of the twocompanies.

TABLE 8.4 Returns of Zig, Zag, and a 50/50 Portfolio ofZig and Zag

8 7 Diversification: Minimizing

7/28/2019 LN08Brooks671956_02_LN08

34/80


8.7 Diversification: MinimizingRisk or Uncertainty (continued)

The Portfolios expected return, E(rp), return can bemeasured in 2 ways:

1) Weighted average of each stocks expected return;

E(rp) = Weight in Zig * E(rZIG) + Weight in Zag*E(rZAG)

OR2) Expected return of the portfolios conditional returns.

E(rp) = Probability of Economic State * Portfolio Returnin Economic

State


7/28/2019 LN08Brooks671956_02_LN08

35/80



E(rp) = Weight in Zig * E(rZIG) + Weight in Zag*E(rZAG)= 0.50 * 15% + 0.50 * 15% = 15%

OR

(a) First calculate the state-dependent returns for the portfolio (Rps) as

follows:

Rps = Weight in Zig* R ZIG,S + Weight in Zag* R ZAG,S

Portfolio return in Boom economy = .5*25% + .5*5% = 15%

Portfolio return in Steady economy = .5*17%+.5*13% = 15%

Portfolio return in Recession economy = .5*5% + .5*25% = 15%

(b) Then, calculate the Portfolios expected return as follows:E(rp) = Probability of Economic State * Portfolio Return in

Economic State

= .2*(15%) + .5*(15%) + ,3*(15%)

= 3% + 7.5% + 4.5% = 15%


7/28/2019 LN08Brooks671956_02_LN08

36/80



The portfolios expected variance and standard deviation can bemeasured by using the following equations:

2 (rp) = [(Return in Statei E(rp)) 2 * Probability of Statei]

= [(15% 15%)2*.20 + (15%-15%)2*50%+(15%-

15%)2

*30%= 0 + 0 + 0 = 0

(rp) = 0 = 0%

Note: The squared differences are multiplied by the probability of

the economic state and then added across all economicstates.

8 7 (A) When Diversification

7/28/2019 LN08Brooks671956_02_LN08

37/80


8.7 (A) When DiversificationWorks

Must combine stocks that are not perfectlypositively correlated with each other.

the negative correlation between 2 stocksthe reduction in risk achieved by adding itto the portfolio.


7/28/2019 LN08Brooks671956_02_LN08

38/80


8.7 (A) When DiversificationWorks (continued)

Figure 8.7 Perfectly positive correlation of two assets'returns.


7/28/2019 LN08Brooks671956_02_LN08

39/80



Figure 8.8 Perfectly negative correlation of two assets'returns.


7/28/2019 LN08Brooks671956_02_LN08

40/80



Figure 8.9 Positive correlation of two assets' returns.

7/28/2019 LN08Brooks671956_02_LN08

41/80

8.7 (B) Adding More Stocks to the

7/28/2019 LN08Brooks671956_02_LN08

42/80


8.7 (B) Adding More Stocks to thePortfolio: Systematic and UnsystematicRisk

Total risk is made up of two parts:1. Unsystematic or Diversifiable risk and

2. Systematic or Non-diversifiable risk.

Unsystematic risk, Co-specific, Diversifiable Risk

product or labor problems.

Systematic risk, Market, Non-diversifiable Risk

recession or inflation

Well-diversified portfolio -- one whose unsystematic

risk has been completely eliminated. Large mutual fund companies.

8.7 (B) Adding More Stocks to the

7/28/2019 LN08Brooks671956_02_LN08

43/80


8.7 (B) Adding More Stocks to thePortfolio: Systematic and UnsystematicRisk

As the number of stocks in a portfolio approaches around 25, almostall of the unsystematic risk is eliminated, leaving behind onlysystematic risk.

Figure 8.11 Portfolio diversification and the eliminationof unsystematic risk.

8 8 Beta: The Measure of Risk in

7/28/2019 LN08Brooks671956_02_LN08

44/80


8.8 Beta: The Measure of Risk ina Well-Diversified Portfolio

Beta measures volatility of an individual security against themarket as a whole.

Average beta = 1.0 Market beta

Beta < 1.0 less risky than the market e.g. utility stocks

Beta > 1.0 more risky than the market e.g. high-tech stocks

Beta = 0 independent of the market e.g. T-billBetas are estimated by running a regression of stock returns

against market returns(independent variable). The slope ofthe regression line (coefficient of the independent variable)measures beta or the systematic risk estimate of the stock.

Once individual stock betas are determined, the portfolio beta iseasily calculated as the weighted average:

8 8 Beta: The Measure of Risk in a

7/28/2019 LN08Brooks671956_02_LN08

45/80


8.8 Beta: The Measure of Risk in aWell-Diversified Portfolio (continued)

Example 6. Calculating a portfolio beta.

Jonathan has invested $25,000 in Stock X,

$30,000 in stock Y, $45,000 in Stock Z, and$50,000 in stock K. Stock Xs beta is 1.5,Stock Ys beta is 1.3, Stock Zs beta is 0.8,and stock Ks beta is -0.6. Calculate

Jonathans portfolio beta.

8.8 Beta: The Measure of Risk in a

7/28/2019 LN08Brooks671956_02_LN08

46/80


8.8 Beta: The Measure of Risk in aWell-Diversified Portfolio (continued)Example 6 Answer

Stock Investment Weight Beta

X $25,000 0.1667 1.5Y $30,000 0.2000 1.3Z $45,000 0.3000 0.8K $50,000 0.3333 - 0.6

$150,000

Portfolio Beta = 0.1667*1.5 + 0.20*1.3 + 0.30*0.8 + 0.3333*-0.6=0.25005 + 0.26 + 0.24 + -0.19998= 0.55007

8 8 Beta: The Measure of Risk in a

7/28/2019 LN08Brooks671956_02_LN08

47/80


8.8 Beta: The Measure of Risk in aWell-Diversified Portfolio (continued)

2 different measures of risk related to financial assets;standard deviation (or variance) and beta.

Standard deviation -- measure of the total risk of anasset, both its systematic and unsystematic risk.

Beta -- measure of an assets systematic risk.

If an asset is part of a well-diversified portfolio use betaas the measure of risk.

If we do not have a well-diversified portfolio, it is moreprudent to use standard deviation as the measure of riskfor our asset.

8 9 The Capital Asset Pricing Model

7/28/2019 LN08Brooks671956_02_LN08

48/80


8.9 The Capital Asset Pricing Modeland the Security Market Line

The Security Market Line (SML) shows the relationship betweenan assets required rate of return and its systematic riskmeasure, i.e. beta. It is based on 3 assumptions:

1. There is a basic reward for waiting: the risk-free rate.consumption.

2. The greater the risk, the greater the expected reward.Investors expect to be proportionately compensated forbearing risk.

3. There is a consistent trade-off between risk and reward atall levels of risk. As risk doubles, so does the required rateof return, and vice-versa.

These three assumptions imply that the SML is upward sloping,has a constant slope (linear), and has the risk-free rate as its Y-intercept.

8.9 The Capital Asset Pricing Model

7/28/2019 LN08Brooks671956_02_LN08

49/80


p gand the Security Market Line(continued)

Figure 8.12 Security market line.

8 9 (A) The Capital Asset Pricing

7/28/2019 LN08Brooks671956_02_LN08

50/80


8.9 (A) The Capital Asset PricingModel (CAPM)

The CAPM equation form of the SMLUsed to quantify the relationship betweenexpected rate of return and systematicrisk.

It states that the expected return of aninvestment is a function of

1. The time value of money (the reward for

waiting)2. A reward for taking on risk

3. The amount of risk

8.9 (A) The Capital Asset Pricing

7/28/2019 LN08Brooks671956_02_LN08

51/80


8.9 (A) The Capital Asset PricingModel (CAPM) (continued)

The equation is in effect a straight line equation of the form:y= a + bx

Where, a intercept of the function;b the slope of the line,xthe value of the random variable on thex-axis.

Substituting E(ri)yvariable,

rf intercept a,(E(rm)-rf)the slope b,

random variable on thex-axis,we have the formal equation for the SML:

E(ri) = + rf + (E(rm)-rf)

Note: the slope of the SML is the market risk premium i.e.(E(rm)-rf) and not beta.

8.9 (A) The Capital Asset Pricing

7/28/2019 LN08Brooks671956_02_LN08

52/80


8.9 (A) The Capital Asset PricingModel (CAPM) (continued)

Example 7. Finding expected returns for a companywith known beta.

The New Ideas Corporations recent strategic moves haveresulted in its beta going from 0.8 to 1.2. If the risk-free rateis currently at 4% and the market risk premium is beingestimated at 7%, calculate its expected rate of return.

AnswerUsing the CAPM equation we have:

Where;

Rf= 4%; E(rm)-rf=7%; and = 1.2Expected rate of return = 4% + 7%*1.2 = 4% + 8.4 =12.4%

8 9 ( ) li i f h S

7/28/2019 LN08Brooks671956_02_LN08

53/80


8.9 (B) Application of the SML

The SML has many practical applications suchas.

1. Determining the prevailing market oraverage risk premium

2. Determining the investment attractivenessof stocks.

3. Determining portfolio allocation weights

and expected return.


7/28/2019 LN08Brooks671956_02_LN08

54/80


8.9 (B) Application of the SML(continued)

Example 8: Determining the market riskpremium.

Stocks X and Y seem to be selling at theirequilibrium values as per the opinions of themajority of analysts.

Stock X has a beta of 1.5 and an expectedreturn of 14.5%, and

Stock Y has a beta of 0.8 and an expectedreturn of 9.6%

Calculate the prevailing market risk premiumand the risk-free rate.

7/28/2019 LN08Brooks671956_02_LN08

55/80


7/28/2019 LN08Brooks671956_02_LN08

56/80



Example 8 Answer (continued)To calculate the risk-free rate we use theSML equation by plugging in the expectedrate for any of the stocks along with its beta

and the market risk premium of 7% andsolve.

Using Stock Xs information we have:

14.5% = rf+ 7%*1.5 rf= 14.5- 10.5 = 4%


7/28/2019 LN08Brooks671956_02_LN08

57/80



Example 9: Assessing marketattractiveness.

Lets say that you are looking at investing in 2stocks A and B.

A has a beta of 1.3 and based on your bestestimates is expected to have a return of 15%,

B has a beta of 0.9 and is expected to earn 9%.

If the risk-free rate is currently 4% and the

expected return on the market is 11%,determine whether these stocks are worthinvesting in.


7/28/2019 LN08Brooks671956_02_LN08

58/80



Example 9 AnswerUsing the SML:

Stock As expected return = 4% + (11%-4%)*1.3

= 13.1%Stock Bs expected return = 4% + (11%-4%)*0.9

= 10.3%

So, Stock A would plot above the SML, since

15%>13.1% and would be considered undervalued,while stock B would plot below the SML (9%

7/28/2019 LN08Brooks671956_02_LN08

59/80



Example 10: Calculating portfolio expectedreturn and allocation using 2 stocks.

Andrew has decided that given the currenteconomic conditions he wants to have a portfolio

with a beta of 0.9, and is considering Stock R witha beta of 1.3 and Stock S with a beta of 0.7 as theonly 2 candidates for inclusion.

If the risk-free rate is 4% and the market riskpremium is 7%, what will his portfolios expectedreturn be and how should he allocate his moneyamong the two stocks?


7/28/2019 LN08Brooks671956_02_LN08

60/80


( ) pp(continued)

Example 10 AnswerDetermine portfolio expected return using the SML

= 4% + 7%*0.9 = 4%+6.3%=10.3%

Next, using the two stock betas and the desired portfolio beta, infer theallocation weights as follows:

Let Stock Rs weight = X%; Stock Ss weight = (1-X)%

Portfolio Beta = 0.9 = X%*1.3 + (1-X)%*0.7=1.3X+0.7-0.7X0.6X+0.7

0.9 =0.6X+0.70.2=0.6XX = 0.2/0.6 = 1/3 1-X = 2/3

To check: 1/3*1.3 + 2/3*0.7 = 0.4333+0.4667 = 0.9 = Portfolio Beta

Additional Problems with Answers

7/28/2019 LN08Brooks671956_02_LN08

61/80


Problem 1

Comparing HPRs, APRs and EARs: Two yearsago, Jim bought 100 shares of IBM stock at $50 pershare, and just sold them for $65 per share afterreceiving dividends worth $3 per share over thetwo year holding period.

Mary, bought 5 ounces of gold at $800 per ounce,three months ago, and just sold it for $1000 perounce.

Calculate each investors HPR, APR, and EAR andcomment on your findings.


7/28/2019 LN08Brooks671956_02_LN08

62/80


Problem 1 (Answer)

Jims holding period (n) = 2 years

Jims HPR =( Selling Price + Distributions-Purchase Price)/Purchase Price

= [$65(100) +$3(100) - $50(100)]/$50(100)

= [$6500 + $300 - $5000]/$5000 = $1800/$5000 = 36%

Jims APR = HPR/n = 36%/2 = 18%

Jims EAR = (1 + HPR)1/n 1 =(1.36)1/2 1= 16.62%


7/28/2019 LN08Brooks671956_02_LN08

63/80


Problem 1 (Answer) (continued)

Marys holding period = 3/12 = 0.25 of a yearMarys HPR = (Selling Price Purchase Price)/ Purchase Price

= ($1000 * 5 - $800 * 5)/ $800*5

= ($5000 -$4000)/$4000

= $1000/$4000 = 25%

Marys APR = HPR/n = 25%/0.25 = 100%

Marys EAR = (1 + HPR)1/n 1 = (1 .25)1/.25 1=144.14%

Clearly, Mary had a higher HPR, APR, and EAR than Jim.

However, the APR and HPR seem unrealistic because of hershort holding period. It implies that Mary would make3additional trades of 25% profit over the next 3 quarters.


7/28/2019 LN08Brooks671956_02_LN08

64/80


Problem 2

Calculate ex-post risk measures: Listedbelow are the annual rates of return earnedon Stock X and Stock Y over the past 6years. Which stock was riskier and why?

YearStock

XStock

Y

2004 20% 16%2005 15% 17%2006 -10% 20%

2007 30% 24%2008 25% 23%2009 14% -10%


7/28/2019 LN08Brooks671956_02_LN08

65/80


Problem 2 (Answer)

Year

Stock

X

Stock

Y (X-Mean)2

(Y-

Mean)22004 20% 16% 0.001877778 0.00012005 15% 17% 4.44444E-05 0.00042006 -10% 20% 0.065877778 0.00252007 30% 24% 0.020544444 0.00812008 25% 23% 0.008711111 0.0064

2009 14% -10% 0.000277778 0.0625

Average 16% 15% 0.019466667 0.016 Variance13.95% 12.65% Std. dev.

We calculate each stocks average return, variance,and standard deviation over the past 6 years and

compare their risk per unit of return i.e. /Average


7/28/2019 LN08Brooks671956_02_LN08

66/80



Stock X Stock Y

Average return 16% 15%

Standard Deviat ion 13.95% 12.65%

Standard Deviat ion 0.87% 0.84%

Average Return

Stock X was riskier than Stock Y since it had the higher Standard

Deviation of the two, and its average return was not muchhigher than Stock Ys average return resulting in 0.87% risk perunit of return versus Stock Ys 0.84%risk per unit of return.


7/28/2019 LN08Brooks671956_02_LN08

67/80


Problem 3

Calculating ex-ante risk and return measures.Using the probability distribution shown below,calculate the expected risk and return estimates ofeach stock and of a portfolio comprised of 40% ofStock A and 60% of Stock B.

State ofEconomy

Probabilityof State

occurring

Stock A'sConditional

return

Stock B'sConditional

return

Recession 0.3 -12% 20%Normal 0.5 14% 12%Boom 0.2 25% -10%


7/28/2019 LN08Brooks671956_02_LN08

68/80


Problem 3 (Answer)

Stock As E(r)= 0.3*(-12%)+0.5*(14%)+0.2*(25%)=8.4%

Stock Bs E(r)= 0.3*(20%)+0.5*(12%)+0.2*(-10%)=10%

Stock As Exp. Var= 0.3*(-12-8.4)2+0.5*(14-8.4)2+0.2*(25-8.4)2

= 124.848 + 15.68 + 55.112

= 195.64

Stock As Exp. Std. dev. = 195.64 = 13.99%

Stock Bs Exp. Var. = 0.3*(20-10)2+0.5*(12-10)2+0.2*(-10-10)2= 30 + 2 + 80

= 112

Stock Bs Exp. Std. dev. = 112 = 10.58%

Portfolio ABs E(r)= Wt. in A * E(RA) + Wt. in B * E(RB)

= .4 * 8.4% + .6*10% = 9.36%


7/28/2019 LN08Brooks671956_02_LN08

69/80



ALTERNATIVE METHOD

Calculate the portfolios conditional returns and then computethe E(r)and standard deviation/variance.

Portfolio ABs recession return = .4*(-12) + .6*(20) = 7.2%

Portfolio ABs normal return = .4*(14) + .6*(12) = 12.8%

Portfolio ABs boom return = .4*(25) + .6*(-10) = 4%

Portfolio ABs E(r)= 0.3*7.2+0.5*12.8+0.2*4 = 9.36%

Portfolio ABs Exp. Var.

= 0.3*(7.2-9.36)2+0.5*(12.8-9.36)2+0.2*(4-9.36)2

= 1.39968 + 5.9168 + 5.74592

= 13.0624Portfolio ABs Exp. Std. dev. = 13.0624 = 3.61%


7/28/2019 LN08Brooks671956_02_LN08

70/80


Problem 4

Calculate a portfolios expected rate ofreturn using the CAPM.

Annie is curious to know what her portfoliosCAPM-based expected rate of return should be.

After doing some research she figures out themarket values and betas of each of her 5 stocks(shown below) and is told by her consultantthat the risk-free rate is 3% and the market

risk premium is 8%.Help Annie calculate her portfolios expectedrate of return.

7/28/2019 LN08Brooks671956_02_LN08

71/80


7/28/2019 LN08Brooks671956_02_LN08

72/80



Portfolio Beta = 0.14*1.6+.16*1.2+.18*1.0+.2*-0.8+.32*0.8

= 0.224 + 0.192 +0.18 + (-.16) + 0.256 =0.692

Next, using the CAPM equation and

rf= 3%, E(rm-rf) = 8%;

Calculate the portfolios expected rate.

E(rp) = 3% + 8%*(.692) = 8.54%


7/28/2019 LN08Brooks671956_02_LN08

73/80


Problem 5 (A)

Applying the CAPM to determine market attractiveness.

Annie is curious to know whether the following 5 stocks are

appropriately valued in the market. Accordingly, she creates a table

(shown below) listing the betas of each stock along with their ex-

ante expected return values that have been calculated using a

probability distribution. She also lists the current risk-free rate and

the expected rate of return on the broad market index. Help her outand state your steps.

Stock Expected Return Beta1 22% 1.82 8% 0.93 14% 1.24 10% 1.15 16% 1.4

Rf 3.5% ----Rm 15% 1.0


7/28/2019 LN08Brooks671956_02_LN08

74/80


Problem 5 (A) (Answer)

Step 1. Using the CAPM equation calculate therisk-based return of each stock

StockExpected

Return BetaCAPME(Ri) Comment

1 26% 1.8 24.20% Undervalued2 16% 0.9 13.85% Undervalued3 14% 1.2 17.30% Overvalued4 16.15% 1.1 16.15% Correct5 20% 1.4 19.60% UndervaluedRf 3.50% ----Rm 15% 1


7/28/2019 LN08Brooks671956_02_LN08

75/80


Problem 5 (B)

If Annie wants to form a 2-stock portfolio ofthe most undervalued stocks with a beta of1.3, how much will she have to weight eachof the stocks by?


7/28/2019 LN08Brooks671956_02_LN08

76/80


Problem 5 (B) (Answer)

Based on the results in (A), Stocks 1 and 2 are most undervalued and

would be chosen by Annie to form the 2-stock portfolio with a beta =1.3.

Stock 1s beta = 1.8; Stock 2s beta = 0.9; Desired Portfolio beta =1.3

Since the portfolio beta = weighted average of individual stock betas

Let Stock 1s Weight be X%; Thus Stock 2s Weight would be (1-X)%

1.8*X% + 0.9*(1-X)% = 1.3 1.8X + 0.9 -0.9X = 1.3 0.9X = 0.4

X = 0.4/0.9 = 0.4444 or 44.44% = Stock 1s Weight (1-X) = 1-0.4444 = .5556 or 55.56 = Stock 2s Weight

Check.0.4444*1.8 + 0.5556*0.9 = 0.79992+ 0.50004=1.3

7/28/2019 LN08Brooks671956_02_LN08

77/80


TABLE 8.1 Year-by-YearReturns and DecadeAverages, 19501999

7/28/2019 LN08Brooks671956_02_LN08

78/80


TABLE 8.1 Year-by-

Year Returns andDecade Averages,19501999(continued)

TABLE 8.3 Conditional Returns ofh i

7/28/2019 LN08Brooks671956_02_LN08

79/80


Investment Choices

FIGURE 8.13 Security market linei h i di id l

7/28/2019 LN08Brooks671956_02_LN08

80/80

with individual assets.

LN08Brooks671956_02_LN08

Documents

Transcript of LN08Brooks671956_02_LN08