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Solution Set 2

1. (10 points)Consider the following analysts expected return on two stocks for two particularmarket returns. The probability of each state 1/3 (the states are equally likely). The risk-free rateis 2%.

a. (4 points) There are four steps in calculating the betas.Step1: CalculateE(Rm), the expected return on the market

Step2: Calculate 2m , the variance of the market

Step 3: Calculate the expected returns (E(Ri)) for each stockStep 4: Calculate the covariance between each stock and the market (Cov(Rm,Ri))

Step1: %2*33.%10*33.%25*33.)( ++=mRE =11%

Step 2: 2222 %]11%2[*33.%]11%10[*33.%]11%25[*33. ++=m =122.0%

%0.11=m Step 3: %10*33.%15*33.%40*33.)( ++=StockAggressiveRE = 15.0%

%1*33.%8*33.%12*33.)( ++=tockDefensiveSRE = 7.00%Step 4:

)]()][()[Pr(

)]()][()[Pr(),(

m

Bull

mAgg

Bull

Agg

m

Bear

mAgg

Bear

AggmAgg

RERRERBullMarket

RERRERBearMarketRRCov

+

=

%]112%][15%10[*33.

%]11%10%][15%15[*33.%]11%25%][15%40[*33.),(

+

+=mAgg RRCov

%0.225),( = mAgg RRCov

%]11%2%][7%1[*33.%]11%10%][7%8[*33.%]11%25%][7%12[*33.),(

++=mDef RRCov

= ),( mDef RRCov 49.0%

Now we have everything we need to calculate the betas on both stocks.

Recall the equation for beta:2

),(

m

mi

i

RRCov

=

%0.122

%0.225),(2

==m

mAgg

AggStock

RRCov = 1.84

%0.122%0.49),(

2 ==

m

mDef

DefStock

RRCov = 0.40

Notice that in class we defined stocks with >1 as aggressivestocks and stocks with

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industries we talked about in class were luxury goods, consumer goods, wirelesscompanies, on-line retailers, and car manufacturers. Some of the Defensive industries

we talked about in class were utilities, beer manufacturers, gaming companies (casinos),and tobacco manufacturers.

c. (2 points) Now that we have the betas on both securities it is straight forward to find theexpected risk premia that CAPM predicts. CAPM predicts the following relationship:

])([)( fmStockIfStockI rRErRE += That is, the risk premium on a stock equals the stocks beta times the risk premium onthe market.

=+= ])([)( fmAggfAgg rRErRE 2.5%+ 1.84*(11%-2.5%)= 18.2%

=+= ])([)( fmDeffDef rRErRE 2.5%+ 0.40*(11%-2.5%)= 5.91%

d. (2 points)According to CAPM, what are the alphas on each stock and what do thesealpha values mean in terms of the stock being underpriced, overpriced or perfectlypriced? The actual expected returns on both stocks differ from those predicted byCAPM, which means that their prices are out of equilibrium. We calculate the alphas as

follows:

stockstockstock turnmExpectedEquilibriuturnctedActualExpe ReRe =

In the case of the aggressive stock %2.18%0.15 =aggressuve = -3.2%. The return islower than it should be, which means the stock is overpriced.

In the case of the defensive stock, %9.5%0.7 =defensive = 1.1%. Thereturn is higher than it should be, which means the stock is underpriced.

e. (1point) To profit from the mispricings I would do the following:Aggressive:Since Aggressive is overpriced, at its current price demand for thestock should fall, driving down the price, and driving the return up to 14.8%predicted by CAPM. I could profit from this mispricing by shorting aggressive stock.Defensive: Since Defensive is underpriced, at its current price demand for thestock will rise, driving up the price and pushing the return down to the 4.933%predicted by CAPM. I could profit from this mispricing by buying defensivestock on margin.

2. (4 points) Suppose that the return on short-term government securities (perceived to be risk-free)is 5%. Suppose also that the expected return required by the market for a portfolio with a betaof 1.6 is 18%. According to the capital asset pricing model

a. (1 point) What is the expected return on the market portfolio?%]5)([6.1%5%18])([)( +=+= mfmifi rErrErrE

%12.13)( =mrE b. (1 point) The expected return for a security with a zero beta is just the risk-free rate, inthis case, 5%.

c. (2 points) First we need to calculate the actual expected return on this stock at the currentprice of $40:

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iceCurrent

videndExpectedDiiceCurrenticeleExpectedSaturnActual cecurrentpri

Pr

PrPrRe

+=

40$

3$40$41$Re

+=cecurrentpriturnActual =10%

Now we need to use find the expected return predicted by CAPM:%]5%12.13[*5.0%5Re +=turnmEquilibriu =9.06%

We calculate the stock's alpha to find the degree of the mispricing:

%06.9%10 =stock =0.94% This stock has a positive alpha, meaning that it isyielding a higher return than it should yield according to CAPM. The stock is thereforeunderpriced at $40 a share.

3. (2 points) Two investment advisers, 1 and 2 are comparing performance. Investor 1 averaged a19% rate of return and investor 2 averaged a 16% rate of return. The beta of the first investor is1.5 while the beta of the second investor is 1.1.

a. If the risk free rate is 6% and the return on the market is 14%, which investor did abetter job?To answer this we first need to find the returns both investors should have according toCAPM.

Investor 1: %6%]6%14[*5.1)( 1 +=RE =18%%18%191 =Alpha =1%

Currently Investor 1 is yielding 1% more than he/she needs toaccording to CAPM.

Investor 2: %6%]6%14[*1.1)( 2 +=RE =14.8%%8.14%161 =Alpha =1.2%

Currently Investor 2 is yielding 1.2% more than he/she needs to

according to CAPM.

In this scenario both investors are outperforming other investments with similar risklevels, since both have returns that are higher than those predicted by CAPM.However, since Investor 2 has the higher "extra" return, he/she had the bestperformance. (Investor 2 is providing a return that is 1.2% higher than that requiredfor the risk level of the investment. According to CAPM if you found an investmentwith an equal level of risk, as measured by beta, it should yield 14.8%)

b. If the risk free rate is 3% and the return on the market 15%, which investor did a betterjob?

To answer this we first need to find the returns both investors should have according to

CAPM.Investor 1: %3%]3%15[*5.1)( 1 +=RE =21%%21%191 =Alpha =-2%

Currently Investor 1 is yielding 1% less than he/she needs toaccording to CAPM.

Investor 2: %3%]3%15[*1.1)( 2 +=RE =16.2%%2.16%161 =Alpha =-0.2%

Currently Investor 2 is yielding 0.2% less than he/she needs to

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according to CAPM.

In this scenario both investors are underperforming compared to otherinvestments with the similar risk levels, since both have returns that are lower than thosepredicted by CAPM. However, Investor 2's underperformance is less than that of Investor1, which means he/she did a better job.

4. (4 points)3 year, 7.4% coupon bonds of LuvFin Co. are currently selling for $975.a. What is the current yield on this bond?

975$

74$

Pr==

iceQuoted

onAnnualCoupldCurrentYie = 7.59%

Because the bond is selling below par, the current yield is greater than the coupon rate.

b. The current yield gives us an approximation of an investors return on this bond ifhe/she bought it today for a price of $975 and held it until maturity. However, a moreprecise measure of the return on this bond is the yield to maturity. The yield to maturity(YTM) is the return an investor would get by buying the bond today at $960 andholding it until maturity. The YTM solves the following equation:

6

6

1 )1(

1000$

)1(Pr

yy

couponiceQuoted

tt +

++

= =

665432 )1(

1000$

)1(

37$

)1(

37$

)1(

37$

)1(

37$

)1(

37$

)1(

37$975$

yyyyyyy ++

++

++

++

++

++

+= Red

c. Any event that would cause the perceived default risk of LuvFin, the issuer, to increasewould increase the yield on these bonds. You received full credit if you listed any eventthat would increase default risk.

5. (1 points)As we outlined in class, if a bond is selling at a discount, or less than par, then the yieldto maturity is greater than the current yield which is greater than the coupon rate. YTM>currentyield> coupon rate. This means that that C is true.

6. (2 points)a. (0.5 point) The risk premium on the 5 year General Motors bond equals

yearTBondGMbond YieldYield 5 = 7.9%- 6.7% = 1.2%. Expressing the risk premium inbasis points, the spread to Treasury on the GM bond is 120 basis points. This meansthat GM is paying 1.2% more than the U.S. government to borrow funds over a 5 yearperiod in the market.

The risk premium on the Exxon bond equals 7.2%-6.2% = 1%. Thus, the spread toTreasury on the Exxon bond is 100 basis points. This means that Exxon is paying 1%

more than the U.S. government to borrow funds over a 1 year period.

b. (1 point) If Exxon is selling above par, the yield is lower than the coupon rate on thebond. Since the coupon rate is approximately the yield on the Exxon bond when it wasissued, this means that the yield on Exxon has fallen. Given that Treasury yields on oneyear bonds have stayed the same, this means that the risk premium on the bond is lowertoday than when the bond was originally issued. The risk premium has declined.

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Note: comparing spreads across time can tell us about the evolution of the risk premium on a bond.One key point of comparison is the risk premium when the bond was issued, which we take as thedifference between the coupon rate and comparable Treasury yields.

c) (0.5 point) If you bought the Exxon bond today and hold it until maturity your annual returnis simply the yield to maturity, or 7.2%

7. (2 points) The price of this bond is simply the present discounted value of the stream of futurepayments, where the discount rate is just the yield to maturity.

8180 8080 80

801 1

(6.75 / 4) 100Pr 1.69 100 1.69 100

(1 ) (1 ) 1

t

tt t

Quoted icey y

= =

= + = + = + + +

Where1

1 y=

+and y is the quarterly yield to maturity. So we need to calculate this quarterly

yield. Since we have the annual yield (6.95%), we just need to express it in quarterly terms asfollows:

( ) ( ) ( )4 1/41 0.0695 1 1 0.0695 1 0.0169y y+ = + = + =

Therefore:1 1

0.98341 1 0.0169y

= = =+ +

So:

[ ]81

800.9834 0.9834Pr 1.69 100(0.9834) 1.69 43.7157 100(0.2621) 100.091 0.9834

Quoted ice

= + = + =

8. Stocks with beta higher than 1, imply a higher risk than the market. Conversely, stocks with betasmaller than 1, imply a lower risk than the market

9. (5 points)The premium on a Microsoft July call option is $2.05 and the premium on aMicrosoft July put option is $2.50. Assuming that the stock price at expiration (ST) is

$25. Note that we have three scenarios: one where STX. The payoffs and profits for the calls and puts vary accordingly.

a. Call option X=$22.50: ST>XExercise the call1. Payoff for call= max[ST-X, 0] recall, if you exercise the call, your

payoff is the difference between what you sell the shares for, ST, and

what you buy them for, X. If ST

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2. Profit=$0-$2.50=-$2.50c. Call option X=$25:ST=XIndifferent between exercising and not exercising

1. Payoff = $25-$25= $0.2. Profit=$0-$2.05= -$2.05

d. Put option, X=$25: ST=XIndifferent between exercising and not exercising1. Payoff = $25-$25= $0.2. Profit=$0-$2.50= -$2.50

e. Call option, X=$27.50: STX and the other when ST

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the minimum amount that Joseph needs for his down payment is $350,000 and that

$450,000 is more that enough to cover his costs and a cash cushion.

a. Advantages of writing the call options:1. Joseph earns $30,000 from the premium of the call options. Where

did this number come from? To cover all 10,000 shares he will have

to sell 100 options contracts (each contract is for 100 shares). Eachoption contract will cost $300 ($3 per share * 100 shares), translating

into a total of $30,000.

2. He also sets an upper bound of $45 for the sale price of his shares.This eliminates some uncertainty.

b. Disadvantages of writing the call options:1. Joseph is giving up the chance to benefit if the stock price goes

above $45. For the price of the option he is essentially selling his

claim to increased earnings if the stock price goes above $45.

2. This written call does not cover Joseph if the stock price falls,leaving him unprotected in a downside scenario. If the price falls

below $32, Joseph will not have enough money to make his downpayment. (If the price moves to $32, Joseph sells and makes

$32,000. Combined with the $30,000 from the premiums just covers

the down payment. If the price moves lower than $32, however, he

will have insufficient funds).

c. Advantages of buying the put options with strike price $35:1. He puts a lower bound on the sale price of the stock, thus fully

protecting himself in the case that the stock price falls below $35.

He is now guaranteed to be able to sell his shares for $35.

d. Disadvantages of buying the put options with strike price $351. Establishing this position is going to cost Joseph $30,000

(premium=$3), which is not a small amount. While he has protectedthe downside risk, this protection is not cheap. Indeed, if we assume

that the $30,000 is coming out of funds for the down payment, then

Joseph will need to sell for $38 a share to cover the down payment

and the cost of the options position.

12.(6points) Suppose you purchase one IBM May 100 call contract at $5 and write oneIBM May 105 call contract at $2. This is a money spread on calls.

a. You benefit most if the price on the expiration date is 105. In this case you canexercise the call option you purchased, earning a payoff of $5 per share, and

the one that you wrote will not be exercised. A standard contract has 100

options, which means your payoff will be $500. From this we must add or

subtract the difference in the premiums to find the profit. This difference is

$2-$5=-$3that is, you pay $3 per option to obtain this position. Total profit= $2 per option =$200 for a standard contract. (Note, that if the price at

expiration is above $107, the total profit is the same as you are in the same

situation of buying the stock at $100 and selling it at $105)

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b. If at expiration, the price of a share of IBM stock is $103, you will exerciseyour call option but the holder of the call you wrote will not exercise. By

exercising your option you obtain a payoff of $3 per option. However, this

exactly offsets the cost of obtaining the position, so the profit=$0.

c. The maximum loss you can suffer with this spread is the difference betweenthe two premiums, or -$3 per option =-$300for a standard contract. This willhappen if at the expiration date the price is at or below $100. In this case both

calls go unexercised.

d. By writing an uncovered call contract, called writing a call naked, yourmaximum loss is unbounded. If the price of IBM rises dramatically, the

holder will exercise the call, and you will be forced to buy IBM for a very high

price and sell it for a significantly lower price. Specifically, the loss can be

written as X-ST, where X is the strike price and STis IBM's current price.

Technically, STcould reach infinity, which means your losses are also infinite.

e. The need for covering the written call position is obvious. You can cover thisposition bygoing long IBM stock: you will buy IBM shares when you write

the call. Now, if the call is exercised, you have shares on hand which you can

sell to the call owner, freeing you from having to buy the shares in the market

at the high current price. This reduces your losses.

The offsetting stock position is one that hedges your call position. Recall that

a good hedge is an asset whose returns move in the opposite direction as those

of the original security. So we want to find a stock position that is up when

the call position is down and vice versa. The call position is down when the

stock price rises; but this is exactly when the stock position is up.Alternatively, the call position is up when the stock price falls; exactly when

the stock position is down. By combining these two positions, we have

removed much of the risk of our portfolio.