Solved Ravi Shukla

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Investments

Solutions Manual

Ravi Shukla

Finance DepartmentSchool of Management

Syracuse University


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c 1995, Ravi Shukla.

Typeset in T EX and L A TEX 2 .


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Chapter 1

Securities Markets

1.1 Return Rebecca made a prot of $2,000 on her investment of $98,000 over three months. Her return, therefore,

was 2,000/ 98,000 = 0 .0204 or 2.04% per quarter.

1.2 Return Johns initial investment was 100 $24.50 = $2,450. The nal value of his investment was 100 $27.75 =$2,775. He made a prot of $2 ,775$2,450 = $325. His return, therefore, was 325 / 2,450 = 0 .1327 or 13.27%per year.

1.3 Round and Odd Lots The receipts from the trade will be the selling price minus the commissions. The selling price is 375 $2.75 = $1 ,031.25. The commission on the three round lots is 3 $12 = $36, and on the 75 shares sold in oddlot is 75$0.15 = $11 .25. The net proceeds to Mr. Williams, therefore, will be $1 ,031.25$36$11.25 = $984.

1.4 Transaction Costs and Return

(a) The cash outow was 200 13.25 (1 + 0 .02) = $2 ,703.(b) The cash inow was 200 15.50 (1 0.02) = $3 ,038.(c) The monthly rate of return was r = ($3 ,038 $2,703)/ $2,703 = 0 .1239 = 12.39%.1.5 Round Trip Transaction Cost

The shares were purchased at 47 1/ 2. Alice had to send in 100 $47.50 = $4 ,750.The shares were sold at 47 1/ 4. The proceeds were 100 47.25 = $4 ,725.The round trip transaction cost was $4 ,750$4,725 = $25. This cost arose from the bid-ask spread. Themoney went to the trader or the specialist who made the market in Kodak.With 2% commission, there would be the extra cost of 0 .02 $4,750 = $95 while buying the shares, and0.02$4,725 = $94 .50 while selling the shares. The round trip cost, therefore, would be $25+$95+$94 .50 =$214.50.

1.6 Long and Short

(a) The maximum prot you can make is because the share price can keep rising without any limit. Yourmaximum possible loss is $30 per share because it is possible for the share price to come down to zero.The maximum and the minimum returns corresponding to these prots and losses are and 100%.

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(b) Your friend will prot when the share price goes down and lose when the share price goes up. Therefore,her maximum and minimum prots and returns will be mirror images of yours, i.e., maximum loss of and maximum prot of $30. The corresponding returns are and +100%.

1.7 Short Interest

(a) Ramones short interest is zero because he hasnt sold any shares short.

(b) Sheilas net position is 100 shares long on Kodak. Therefore, her short interest is zero.

(c) The two positions taken by Andrew cancel each other out. His short interest in AT&T is zero.

(d) Brenda has a net short position of 300 shares on Sears. So her short interest is 300 shares of Sears.

(e) Glens short interest is 300 shares of McDonalds.

(f) Susan is short on McDonalds. Therefore, her short interest is 500 shares of McDonalds.

1.8 Shorting Against the Box The opening transaction on 3/15/88 resulted in a cash outow of $19 per share. On 8/30/88, Gorba

borrowed his friends shares and gave his own shares of Purex to his friend as collateral. Then Gorba soldhis friends shares. Therefore, there was a cash inow of $35 per share and a net prot of $16 per share.

On 1/1/89, there would be no cashow regardless of the price because Gorba would use his 500 shareskept with his friend as collateral to return the shares he borrowed.

1.9 Margin Accounts

(a) The market value of shares was 500 $40 = $20,000. Since she borrowed as much as she could for theopening transaction, her loan was (1 0.60) $20,000 = $8 ,000. The equity, therefore, was $12,000.Therefore, the account position was:Assets ($) Liabilities and Equity ($)Mkt. Val. of Shares 20 ,000 Loan 8,000

Equity 12 ,000Total 20 ,000 Total 20 ,000

(b) Suppose the stock price at which she receives the margin call is P . The market value of securities atthat price is 500 P . The loan has not been changed and it stays at $8,000, and therefore the equity is500P $8,000. For margin call:

Equity=

500P $8, 0000.40 =Mkt. Val. of securities 500 P

which gives P = $26 .67. Therefore, the margin call will go out at the price of $26.67.(c) At the price of $20, the market value of securities is $10,000. The loan is unchanged at $8,000 and

therefore the equity is $2,000. The account position is:

Assets ($) Liabilities and Equity ($)Mkt. Val. of Shares 10 ,000 Loan 8,000


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The margin, therefore, is 20%, which is below the maintenance level requirement. Suppose Ms. Millershould add C in cash to the account then the new value of equity would be $2 ,000 + C . To make theaccount current, C must be such that:

$2,000 + C 0.60 =$10,000

which gives C = 4 ,000. Therefore, $4,000 should be added to the account to make it current.

(d) Let us say the AE has to sell n shares, then 500 n shares remain in the account. The cash created byselling the shares will be kept in the account. The market value of shares is (500 n) $20. The loanis still $8,000 and the equity is $2,000. To restore the account to initial margin:2000

0.60 =(500 n) 20

which gives n = 333 .33. Since fractional number of shares cant be sold, the AE will sell 334 shares and

leave 166 shares in the account to keep the account above the initial margin level.

1.10 Margin Accounts

(a) The positions of his account after the initial sale and before liquidation are:

Loan Loan

After the Initial Sale Before LiquidationAssets ($) Liabilities and Equity ($) Assets ($) Liabilities and Equity (Mkt. Val. of Shares 9,000 3,600 Mkt. Val. of Shares 10,000 3,6

Equity 5,400 Equity 6,4Total 9,000 Total 9,000 Total 10,000 Total 10,0

(b) The gross prot is $1,000. The net prot must consider the interest payment on the loan of $3,600. Theinterest amount is 0 .11 $3,600 = $396. The net prot, therefore, is $1 ,000 $396 = $604. The rate of return is 604 / 5,400 = 0 .11185 or 11.185% per year.

(c) Suppose the margin call would have gone out at the price of P . The account would have looked as

500P Loan 3,600500P 3,600

500P 500P

follows:Assets ($) Liabilities and Equity ($)Mkt. Val. of Shares

EquityTotal Total

(d) The margin call price, therefore, can be determined as:

500P $3,6000.40 = P = $12500P

(e) The margin call, therefore, would have gone out at $12.

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1.11 Margin Accounts

(a) The shares were purchased at 24 1/ 2 or $24.50. The total value of the shares was $2,450. Kate neededequity worth 0 .75

$2,450 = $1,837.50 in the account. The remaining $612.50 was advanced as loan

to her. She also paid 0 .03 $2,450 = $73.50 in commissions. The total cash outow, therefore, was$1,837.50 + $73 .50 = $1 ,911.(b) The sell order was executed at $27. The gross proceeds were $2,700. The commission of 0 .03$2,700 =$81 were deducted from this amount leaving $2,619. The loan amount due was the principal of $612.50

plus the interest $612 .500.092 = $56.35, i.e., a total of $668.85. This means that Kate received a checkfor $2,619 $668.85 = $1,950.15.(c) Kate realized a net prot of $1 ,950.15 $1,911 = 39 .15. Her return, therefore, was $39 .15/ $1,911 =0.0205 or 2.05% per year.1.12 Margin Accounts

(a) The buy order was lled at 12 3/ 8. Since Dr. Lecter paid 90% cash, and borrowed only 10%, the accountposition was:

Assets ($) Liabilities and Equity ($)Mkt. Val. of Shares 6 ,187.50 Loan 618.75

Equity 5 ,568.75Total 6 ,187.50 Total 6 ,187.50

(b) Suppose the margin call will go out at the price of P per share, then the account position at the time of margin call will be:

Assets ($) Liabilities and Equity ($)

Mkt. Val. of Shares 500 P Loan 618.75Equity 500 P 618.75Total 500 P Total 500 P

For margin call:500P 618.750.40 =

500P which gives P = 2 .0625. Therefore, the margin call will go out when the price drops to $2.0625 per share(or below).

(c) The only way the shares can be converted to cash is by selling. Therefore, it is the bid price at whichthe shares are recorded. (Note that we should have done this in part a also.)

Assets ($) Liabilities and Equity ($)Mkt. Val. of Shares 7 ,125.00 Loan 618.75

Equity 6 ,506.25Total 7 ,125.00 Total 7 ,125.00

The margin, therefore, is 6,506.25/7,125=0.913 or 91.3%.

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1.13 Short Selling and Margin

(a) The $1,200 generated by short selling were kept as deposit in the account. Freddie was required todeposit cash equal to 70% of the market value of shares, i.e., 0 .70

$1,200 = $840. The account position

after this transaction was:

Assets ($)Cash 840Deposit 1 ,200Total 2 ,040

Liabilities and Equity ($)Mkt. Val. of Shares 1 ,200Equity 840Total 2 ,040

(b) Suppose the margin call will go out at P . The account position when margin call goes out is:

Assets ($)Cash 2 ,040 100P Deposit 100 P Total 2 ,040

Liabilities and Equity ($)Mkt. Val. of Shares 100 P

Equity 2 ,040 100P Total 2 ,040For margin call:

2040 100P 100P

0.40 =

which gives P = 14 .57. At this time, the cash balance is 2 ,040 100($14.57) = $583.

(c) The two choices available to Freddie are

Add some cash to the account.

Buy back and replace some shares.

(d) Let us consider them one at a time:

Suppose Freddie adds C in cash. Then the account position would be:Assets ($)Cash 583 + C Deposit 1 ,457Total 2 ,040 + C

Liabilities and Equity ($)Mkt. Val. of Shares 1 ,457Equity 583 + C Total 2 ,040 + C

For the account to be current:583 + C

0.70 =1,457

which gives C = 436 .90. Therefore, Freddie would have to add $436.90 in cash. The accountposition after the cash has been added would be:

Assets ($)Cash 1,019.90Deposit 1,457Total 2,476.90

Liabilities and Equity ($)Mkt. Val. of Shares 1,457Equity 1,019.90Total 2,476.90

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Suppose Freddie buys back n shares and return them. Then the account position would be:Assets ($)Cash 583Deposit 14 .57(100 n)Total 583 + 14 .57(100 n)

Liabilities and Equity ($)Mkt. Val. of Shares 14 .57(100

n)

Equity 583Total 583 + 14 .57(100 n)

For the account to be current: 58314.57(100 n)

0.70 =

which gives n = 43. The account position after this transaction would be:

Assets ($)Cash 583Deposit 830.49Total 1,413.49

Liabilities and Equity ($)Mkt. Val. of Shares 830.49Equity 583Total 1,413.49

1.14 Long, Short, and Margin

(a) The market value of shares is 200 $30 = $6,000. Since the initial margin is 60%, he had to have0.60 $6,000 = $3,600 in equity. Therefore, Earl sent in $3,600 (plus some amount to cover thecommissions). The position of the account after this transaction was:Assets ($) Liabilities and Equity ($)Mkt. Val. of AT&T 6 ,000 Loan 2,400


(b) The position of his account just before the short sale of IBM was:

Assets ($) Liabilities and Equity ($)Mkt. Val. of AT&T 6 ,400 Loan 2,400


Short selling 100 shares of IBM generated 100 $104 = $10 ,400 which were kept as collateral. The marginin the account was $4,000/($6,400+$10,400)=0.238 or 23.8% which was much below the maintenancelevel. Therefore, Earl had to send some cash. Suppose he sent cash worth C , then his equity became4,000 + C . To meet the maintenance margin:

4,000 + C 0.35 =

6,400 + 10 ,400,

which gives C = $1 ,880. After the completion of the transactions, the account position was:

Assets ($)Mkt. Val. of AT&T 6 ,400Collateral 10 ,400Cash 1 ,880Total 18 ,680

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Liabilities and Equity ($)Mkt. Val. of IBM 10,400Loan 2,400Equity 5 ,880Total 18 ,680


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(c) The position of the account on April 1, was:

Assets ($)Mkt. Val. of AT&T 6 ,800Collateral 10 ,100Cash 2 ,180Total 19 ,080

Liabilities and Equity ($)Mkt. Val. of IBM 10,100Loan 2,400Equity 6 ,580Total 19 ,080

The margin was 6 ,580/ (6,800 + 10 ,100) = 0 .389 or 38.9%.

(d) If Earl decided to liquidate his account, he would buy back 100 shares of IBM using the collateral andreturn them. He would sell his shares of AT&T and generate cash. The total cash, therefore, would be$6,800 + $2 ,180 = $8 ,980. He would pay off the loan of $2,400 leaving him with $6,580. This wouldamount to a prot of $6 ,580 $3,600 $1,880 = $1 ,100.If he had to pay transaction costs and interest, they would have to be subtracted from his net proceeds.The transaction costs would be on both the buying and selling transactions. On AT&T, they would be

0.015 $6,000 = $90 for buying the shares, and 0 .015 $6,800 = $102 for selling the shares. On IBM,the transaction costs would be 0 .015 $10,400 = $156 for buying the shares and 0 .015 $10,100 =$151.50 for selling the shares. The interest on the loan would be 0 .021 $2,400 = $50.40. The totaltransactions and interest cost, therefore, would be: $90+$102+$156+$151.50+$50.40=$549.90. The netproceeds to Earl, therefore, would be $6 ,580 $549.90 = $6 ,030.10. Similarly, his net prot would be$1,100 $549.90 = $550 .10.

1.15 Shorting Against the Box and Margin Status of Ms. Forsythes account:

August 1, 1987Assets ($) Liabilities and Equity ($)Mkt. Val. of Shares 21,500 Loan 10,750

Equity 10,750Total 21,500 Total 21,500

October 1, 1987Assets ($) Liabilities and Equity ($)Mkt. Val. of Shares 45,750 Loan 10,750

Equity 35,000Total 45,750 Total 45,750

Interest on the loan: $10 ,750 0.01 2 = $215. Prot: $35 ,000 $10,750 $215 = $24 ,035.The position of her account after short selling:Assets ($)Mkt. Val. of UBM 45,750Collateral 45 ,750

Total 91 ,500

Liabilities and Equity ($)Mkt. Val. of UBM 45,750Loan 10,750Equity 35 ,000Total 91 ,500

The margin 35 ,000/ 91,500 = 0 .3825 or 38.25% is above the maintenance level. Therefore, Ms. Forsythe

would not have to add any cash to the account.January 1, 1988, Price=$50 : As the price rises, the collateral deposit would need to be increased.Thebroker would advance this as additional loan.

Assets ($)Mkt. Val. of UBM 50,000Collateral 50 ,000

Total 100 ,000

Liabilities and Equity ($)Mkt. Val. of UBM 50 ,000Loan 15,000Equity 35 ,000Total 100 ,000

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Interest on the loan between October 1, 1987 and January 1, 1988, assuming that the extra loan would bemade halfway between the two dates: ($10 ,750+$15 ,000)/ 20.013 = $386.25. The prot to Ms. Forsythe,therefore, would be $35 ,000 $10,750 $215 $386.25 = $23 ,648.75.January 1, 1988, Price=$20 : As the price declines, cash would be released from the collateral. Ms. Forsythewould use this cash to pay off the loan

,000 ,00020,000 Loan 0

,000 ,000,000 ,000

Assets ($) Liabilities and Equity ($)Mkt. Val. of UBM 20 Mkt. Val. of UBM 20CollateralCash 15 Equity 35Total 55 Total 55

Interest on the loan between October 1, 1987 and January 1, 1988, assuming that loan would be paid off halfway between the two dates: ($10 ,750+$0) / 20.013 = $161.25. The prot to Ms. Forsythe, therefore,would be $35,000 $10,750 $215 $161.25 = $23 ,873.75.

The brokers claims about shorting against the box locking in the prots are correct. Ms. Forsythesprots remain more or less unchanged regardless of the stock price. The small change comes from the extrainterest on the margin loan in the account.

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Chapter 2

Return and Risk

2.1 Return The rate of return was r = (31 .2530+0 .30)/ 30 = 0 .052 or 5.2% per quarter, or (1+0 .052)41 = 0 .2232or 22.32% per year.

2.2 Dividend Yield The dividend yield to Sylvia Potter would be 1.10/98=0.0112 or 1.12% per quarter or (1 .0112)4 1 =0.0457 or 4.57% per year.

2.3 Conversion of Units

(a) (1 + 0 .01)12 1 = 0 .1268 or 12.68% per year.(b) (1 + 0 .12)1/ 12 1 = 0 .0095 or 0.95% per month.(c) I am assuming 180 days in six months. (1 + 0 .0003)180

1 = 0 .0555 or 5.55% semiannually.

(d) (1 + 0 .005)52 1 = 0 .2961 or 29.61% per year.2.4 Conversion of Units

1% per month is equal to (1 + 0 .01)3 1 = 0 .030301 or 3.0301% per quarter. 3.0301% per quarter isequal to (1 + 0 .030301)4 1 = 0 .12682503 or 12.68% per year.1% per month is equal to (1 + 0 .01)12 1 = 0 .126802503 or 12.68% per year.2.5 Comparison of Returns

To compare these rates, we have to convert them to identical units. Let us convert 4% per quarter tothe annual rate: (1 + 0 .04)4 1 = 0 .16985856 or 16.99% per year. Therefore, my investment earned a higherreturn.2.6 Compounding and Effective Rates The effective rates for the banks can be calculated as:

A: (1 + 0 .07/ 4)4 1 = 0 .0719 or 7.19% per year.B: (1 + 0 .069/ 12)12 1 = 0 .0712 or 7.12% per year.C: e0.068 1 = 0 .0704 or 7.04% per year.

I will put my money in bank A, because it has the highest effective rate.

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2.7 Compounding and Effective Rates

(a) Suppose the compounding is being done m times a year. Then the basic equation is:

m0.101 + = 1 + 0 .1047m

Let us try the usual values of m that banks use: m = 2, for semiannual, m = 4 for quarterly, m = 12for monthly. One of those should t the equation.

Try m = 2 lhs = 1.102500 no!Try m = 4 lhs = 1.103812 no!Try m = 12 lhs = 1.104713 yes!Therefore, this bank is compounding monthly.

(b) Again, the basic equation is:

m

0.101 + = 1 + 0 .1052mFrom part a, we know that m has to be higher than 12.Try m = 365 lhs = 1.105155 maybe!Try continuous compounding ( m ) under which the basic equation transforms to e0.10 = 1+0 .1052,lhs = 1.1051709, maybe!Therefore, this bank is compounding either daily or continuously. To tell the continuous compoundingfrom daily compounding here, we need to know the effective rate with more precision.

2.8 Compounded Returns Let us say Mr. Fish started out with $100. The changes in his invested wealth are: His $100 got reduced

to $85 because of a 15% loss in the rst year. The second year he had a gain but only on his $85 leavinghim with $85(1+0.15)=$97.75. A position of net loss! So he is not back where he started.The nal result would be the same if the portfolio had gone up rst and then down: $100 up to $100(1+

0.15) = $115 and then down to $115(1 0.15) = $97 .75.2.9 Compounding

Mr. Fox is not giving the client interest on interest and therefore giving him less than he should receive.After the rst year, the deposit would have grown to $105 because of interest. During the second year, theinterest would have to be paid on the full $105. The interest amount, therefore, would be 0 .04105 = 4 .20.Therefore, the balance at the end of the second year would be $109.20. Similarly, the interest during thethird year would be paid on the full $109.20 and would equal 109 .20 0.06 = 6 .552 making the endingbalance equal 115.752 or approximately $115.75. The client, therefore, should receive $115.75.2.10 Leverage

Suppose the investor has E amount of equity. The investor borrows D amount at an interest rate of i. The combined funds, E + D are invested and they earn a rate equal ROA. The total income, therefore,would be (E + D )(ROA). Of this income, Di will have to be paid as interest on the loan. The net income,therefore, would be ( E + D )(ROA) Di = E (ROA) + D (ROA i). The return on equity is calculated bydividing the net prot by the equity investment to get:

DROE = ROA + (ROA i)E

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2.11 Statistical Analysis The following table presents the calculation details:

t1234...

181920

r H 0.01000.0200

0.01000.0200...

0.0300

0.02000.02000.2400

r M 0.01000.02000.0100

0.0200...0.01000.05000.0300

0.7000

2 2

r rH M 0.00010.00040.00010.0004

...0.00090.00040.0004

0.00010.00040.00010.0004

...0.00010.00250.0009

0.0140 0.0642

2

(a) From the table, we get the following information: T = 20, r H = 0 .2400, r M = 0 .7000, r H =0.0140 and r 2 = 0 .0642. Now the statistics can be calculated as:M

r H 0.2400H = = = 0 .0120T 20r M 0.7000M = = = 0 .0350T 20

2 ( r H )2

0.0140 0.24002

H =r H T = 20 = 0 .0242

T 1 192r M

( r M ) 2 0.0642 0.70002

M = T = 20 = 0 .0457T 1 19

(b) For the 90% condence interval, the z value is approximately 1.65. Therefore, the intervals are calculatedas [ 1.65, + 1 .65]. The intervals are [ 0.0279, 0.0519] for Huge and [0.0404, 0.1104] for Micro.

(c) The probability of losing money is the probability of getting a return less than zero. The z scores for thispossibility are (0 0.0120)/ 0.0242 = 0.496 for Huge and (0 0.035)/ 0.0457 = 0.766 for Micro. Thecorresponding probabilities from the normal distribution table are 0.31 and 0.22. So the probabilities of losing money on Huge and Micro are 31% and 22%, respectively.

(d) The z score for making more than 3% on Huge is (0 .03 0.0120)/ 0.0242 = 0 .744. The probability,therefore, is 0.23 or 23%.(e) To double the money, the return has to be 100% or 1. Therefore, the z score for this condition is

(1 0.035)/ 0.0457 = 21.11. Without even looking up the normal distribution table, we know that theprobability of earning 100% or more is zero for all practical purposes.2.12 Statistical Calculations

Before we can calculate the statistics, we have to convert the prices into returns as r t = ( pt pt 1)/p t 1 .

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The following table shows the basic calculations:

t

0123456789

10

10

p

1011121415141314141415

r

0.10000.09090.16670.0714

0.06670.07140.0769

0.00000.00000.0714

0.4393

2r

0.01000.00830.02780.00510.00440.00510.00590.00000.00000.0051

0.0717

0.4393= 0 .0439

10

From the table we get T = 10, r = 0 .4393, r 2 = 0 .0717, So that,

r = =

T

r 2 ( r ) 2

0.0717 0.43932

10 = T = = 0 .0763T 1 9

Therefore, the expected return is 0.0439 or 4.39% and the standard deviation of returns is 0.0763 or7.63%.

2.13 Security Statistics

From the information given in the problem we can make the following table:

Wk P ABC0123456789

10

39.5040.0041.0041.5041.0041.5042.0041.5042.0041.0041.00

P XYZ10.7511.0010.5011.2511.7511.5011.7512.0011.7512.0012.50

T

r ABC

0.012660.025000.01220

0.012050.012200.01205

0.011900.012050.023810.00000

100.0038380.015200

rXYZ

0.02326

0.045450.071430.04444

0.021280.021740.02128

0.020830.021280.04167

100.0157520.035283

The entries for T , , and were calculated using the Excel functions count , average , and stdev . Thetable answers parts (a) and (b) of the problem.

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(c) There is no dominant choice. ABC has a lower expected return with a lower risk and XYZ has a higherexpected return with a higher risk. The stock one chooses to invest in will be a function of ones tasteabout risk. For example, I would choose XYZ, because I believe that the extra weekly expected return

of 0.015752 0.003838 0.011 is a good compensation for the extra risk of 0 .035283 0.015200 = 0 .020in XYZ.(d) The 95% condence interval is calculated as [ 1.96]. For ABC it is [0.026, 0.034] while for XYZit is [0.053, 0.085].(e) The prices are related to the returns through the formula r t +1 = ( pt +1 pt )/p t . Therefore, pt +1 = pt (1 + r t +1 ). pt for XYZ is 12.50. Therefore, pt +1 can be calculated corresponding to the lowest,

expected, and the highest returns for each stock. The values turn out to be $11.83, $12.70, $13.56.

(f) If the price doubles, the return will be 100% or 1. Therefore, the z score for this event would be(1 0.003838)/ 0.0152 = 65.54. There is no need to look up the normal distribution table for this. Weknow that the probability of a z score equal to or higher than 65.54 is zero up to many decimal places.So the chance of the price of ABC doubling is practically zero.

2.14 Statistical Analysis

(a)

r 0.58 = = = 0 .0121 or 1.21% per month,

T 48

r 2 ( r ) 2

0.063 0.582

= T = 48 = 0 .0345 or 3.45% per month.T 1 47

(b) The 95% condence range is [ 1.96] or [

0.0556, 0.0797].

(c) For losing money, the return should be less than zero. The z score for this possibility is (0 .0 0.01208)/ 0.033 = 0.35. The probability, from the normal distribution table, is (0.5-0.1368)= 0.3632 or36.32%.2.15 Risk, Return, and Wealth

The table below shows the monthly returns for A and B and the growth of $100 invested in each stock.The assumption in calculating the growth is that all the wealth at the end of a month ( W ) is reinvested inthe same stock. Wealth at the end of a month is calculated as the wealth at the end of the previous month

(1 + r ). The bottom part of the table shows the statistics.

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A Bt

123456789

101112

T

r

0.010.03

0.020.040.03

0.040.050.02

0.020.040.020.06

120.015

0.0326

W

101.00104.03101.95106.03109.21104.84110.08112.28110.04114.44112.15118.88

r

0.07

0.030.050.050.06

0.04

0.050.060.08

0.070.050.07

120.015

0.0589

W

107.00103.79108.98103.53109.74114.13108.43114.93124.13115.44109.66117.34

Clearly, stock A dominates stock B because it offers the same average return as B for a lower risk. Ais superior in terms of accumulated wealth also. Starting with $100, A results in more money at the endof the year than B does, although both have equal average returns. This difference is attributable to theuctuation in returns measured by variance.

2.16 Returns and Prices We are given = 0 .03 and = 0 .04. We can calculate the 95% condence range as [ 1.96] =[0.03 1.96(0.04)] = [0.0484, 0.1084]. The highest and lowest possible returns with 95% condence,therefore, are 0 .1084 and

0.0484.

The formula for return is r = ( P 1 P 0)/P 0 which gives us P 1 = P 0(1 + r ). We know that P 0 = 12.Therefore, we can calculate the value of P 1 . We will get the expected value of P 1 if we use the expectedvalue of r , the highest value of P 1 if we use the highest value of r , and the lowest value of P 1 if we use thelowest value of r . On doing the calculation we nd that the expected value of the price next period is $12.36,the highest value (with 95% condence) is $13.30, and the lowest value (with 95% condence) is $11.42.

2.17 Risk Premium Denote risk of A by riskA . Then risk of B , riskB , is equal to 2 riskA . The equation for average returnsand risk is:

average return = risk-free rate + risk premium per unit of riskApply this equation to security A:

0.08 = 0 .06 + risk A

premium per unit of risk,

so that0.02 = risk A premium per unit of risk.

Now, for B :

average return = 0 .06 + 2 riskA premium per unit of risk= 0 .06 + 2 0.02 = 0 .10.Therefore, the average return on B should be 10% per year.

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2.18 Effect of Ination Mr. Johnsons $100 will become $100(1+ 0 .13) = $113 in one year. However, due to ination, they would

lose 5% in purchasing power. Therefore, the purchasing power of the amount received at the end of the year

would be $113/ 1.05 = $107 .62. Therefore, the real rate of return expected by Mr. Johnson is 7.62%.2.19 Term Premium

Let us say you lend $100 today for one year. At the end of one year you will receive $100(1+ 0 .12) = $112.You will lend the whole amount again during the second year, and at the end of the second year you willreceive $112(1+0 .14) = 127 .68. If you lend for two years at some rate r , then at the end of one year your $100would grow to 100(1+ r ). At the end of the second year, the amount would be 100(1+ r )(1+ r ) = 100(1+ r )2 .For you to be indifferent between lending for two years at rate r or lending one year at a time, the endingbalances should be the same, i.e., 100(1 + r )2 = 127 .68 which gives r = 0 .1299. Therefore, for a two yearloan, you should charge 12.99% per year.

2.20 Tax Premium What matters to you is the net income after all taxes because the two bonds are identical as far as risk

is concerned. The Treasury bonds will yield 0 .09

(1

0.28) = 0 .0648 or 6.48% per year, after taxes. Since

the income from the Thruway bonds will not be taxed at all, you will be interested in the Thruway bondsas long as they yield 6.48% or more per year.

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Chapter 3

Portfolio Analysis

3.1 Portfolio of Portfolios Henrys overall portfolio composition is 0 .3 {0.2, 0.4, 0.3, 0.1}+ 0 .7 {0.1, 0.2, 0.5, 0.2}= {0.3 0.2 +0.7 0.1, 0.3 0.4 + 0 .7 0.2, 0.3 0.3 + 0 .7 0.5, 0.3 0.1 + 0 .7 0.2}= {0.13, 0.26, 0.44, 0.17}

3.2 Security and Portfolio Statistics

(a) The following table was created using Lotus 1-2-3 to calculate the statistics:

t

1...

10

T

r 10.15

...0.17

10.0.1560.021

r20.12

...0.20

10.0.1350.038

21The variances are calculated by squaring the standard deviations. So,

The correlation between the returns of two securities was calculated by running a regression between thereturns of the stocks. The regression was run by invoking /Data Regression commands. The returnson one of the stocks were assigned to the X-range and the other one to the Y-range . An empty areaof the worksheet was used for Output range and then the regression was run by choosing Go. Theregression output is reproduced below:

Regression Output:

Constant 0.161094Std Err of Y Est 0.022419R Squared 0.004670No. of Observations 10Degrees of Freedom 8

X Coefficient(s) -0.03773Std Err of Coef. 0.194769

16

= 0 .000449 and 22 = 0 .001472.


21/54

From this output, the correlation is determined as the square root of R Squared . Negative square root istaken because the X Coefficient is negative. Therefore, 12 = 0.004670 = 0.0683. The covarianceis calculated as 12 = 1212 = (0 .021)(0 .038)(0.0683) = 0.000055555 = 0.000056.The information is presented below:

Covariance CorrelationStock 2

1 2 1 21 0.156 0.021 0.000449 0.000449 0.000056 1.0000 0.06832 0.135 0.038 0.001472 0.000056 0.001472 0.0683 1.0000

(b) The following formulae were used for the portfolio calculations:

P = x11 + x222P = x22 + x22 + 2 x1x212121 1 2

The answers are shown in the following table.

Portfolio x1 x21 0.2 1.22 0.2 0.83 0.6 0.44 0.3 0.75 0.5 0.5

P 0.13080.13920.14760.14130.1455

P 0.04650.03070.01920.02720.0213

3.3 Optimal Portfolio Selection This portfolio calculations for this problem will be done using the formulae:

p = xA A + xB B + xC C

p = x2A 2A + x2B 2B + x2C 2C + 2 xA xB AB + 2 xA xC AC + 2 xB xC BC For Mr. Vestors portfolio:

P = (1 / 3)(0 .13) + (1 / 3)(0 .14) + (1 / 3)(0 .18) = 0 .15

P = (1 / 3)2(0.05) + (1 / 3)2(0.08) + (1 / 3)2(0.11) + 2(1 / 3)2(0.02) + 2(1 / 3)2(0.03) = 0 .1944

For the alternative portfolio:

P = (0 .40)(0 .13) + (0 .25)(0 .14) + (0 .35)(0 .18) = 0 .15

P = (0 .40)2(0.05) + (0 .25)2(0.08) + (0 .35)2(0.11) + 2(0 .40)(0 .25)(0 .02)+ 2(0 .25)(0 .35)(0 .03) = 0 .1890

Even though Mr. Vestor has diversied among the three stocks, his distribution does not give him as gooda combination as {0.4, 0.25, 0.35}. His portfolio has the same amount of expected return as the alternativeportfolio but a higher risk (standard deviation). Ill tell Mr. Vestor to not act too naive and try to nd anefficient portfolio.

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3.4 Portfolio Calculations Bob wants a portfolio such that p = 0 .14. He will accept the lowest return to be 8%. Obviously,

nobody can guarantee that. So Larry assumed a 95% condence range. In other words, Larry decided

that as long as there is only a 2.5% chance that the return will be less than 8%, it is ok. So he computedthe acceptable standard deviation as: Lowest return = 1.96. Therefore, 8 = 14 1.96 which gave = 3 .06%. Therefore, Bob would be satised a portfolio that has expected return of 14% per year andstandard deviation of 3.06% per year. If Larry can nd a better portfolio that will be an added bonus!

Now we go to do the calculations. The security statistics are given to us. So we calculate the portfoliostatistics using the following formulae:

p = x11 + x22 + x33

p = x21+1 x2222 + x2323 + 2 x1x21212 + 2 x1x31313 + 2 x2x32323The following table shows the results:

x10.30.2

0.20.10.5

x20.30.40.60.80.3

x30.40.40.60.10.2

P P 0.140 0.1310.144 0.1410.156 0.2090.154 0.2080.136 0.127

The rst portfolio has the expected return that Bob desires but the standard deviation is much too high.Other portfolios show similar magnitudes for standard deviations. So, a portfolio to meet Bobs specicationscannot be designed. The reason is that Bob has a mistaken notion about the level of risk in the market.Bob is being too naive if he expects to be able to make 14% per year with a standard deviation of only 3%.

3.5 Portfolios Involving a Risk-free Security We are given that r 1 = 0 .06, r2 = 0 .13, 2 = 0 .04. Since the rst security is risk-free, 1 = 0 .00 and

12 = 0 .00Various portfolios of the two securities were constructed and the statistics for these portfolios were

calculated using:

Port

1

2345678

x11.20

1.000.800.600.400.200.00

0.20

x2

0.200.000.200.400.600.801.001.20

P =

P =

p0.0460

0.06000.07400.08800.10200.11600.13000.1440

x11 + x222x22 + x22 + 2 x1x212121 1 2

p0.0080

0.00000.00800.01600.02400.03200.04000.0480

0.16

0.12..........

......................

.....................

......................

.....................

.....................

....

p 0.08

0.04..........

......................

................................

.....................

......................

.....................

...........................

.....................

......................

.....................

...

0.000.00 0.02 0.04 0.06

p

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The expected return-standard deviation plot is shown in the gure above. The plot shows that therisk-return tradeoff for the portfolios of a risky and a risk-free security indeed falls on straight line segments.Note: You may want to think about the segment that reverts back after hitting the y-axis.

3.6 Portfolio Calculations Two of the three correlations are equal to zero. This makes calculations a little easier. The portfolio

formulae are:

p = xA A + xB B + xC C = xA (0.16) + xB (0.20) + xC (0.10)2 2 22 = xA

2B + xC

2 + 2 xB xC B C BC p A + xB 2

C

Note that the other two cross-product terms are not there in 2 formula because AB = 0 and AC = 0.P Substitute the numbers in 2 equation to get:P

2 2 22 = xA (0.08)2 + xB (0.10)

2 + xC (0.06)2 + 2 xB xC (0.08)(0 .10)(0 .02) p

The calculations are done using these formulas and the following table results. The p and p for the rstthree portfolios are lled in just from the information matrix for securities given in the problem statement.

xA1.00.00.00.10.20.4

xB0.01.00.00.20.70.4

xC 0.00.01.00.70.10.2

P 0.160.200.100.130.180.16

P 0.080.100.060.050.070.05

0.24

0.18 T............

..........................

...

.........................................

...................................

......................... ............ ...

.... ...... ..... .... .... ....... ...... ...

..........................

........................

.........................

..........................

........................

........................

p 0.12.........

.................

..............

.......

..........................

........................

..........................

.....................

0.06

0.000.00 0.03 0.06 0.09 0.12

p

The graph between p and p is also shown above. The mean-standard deviation frontier has been drawnapproximately. Using 9% as the risk-free rate we draw the tangency line and nd that the tangency portfoliois approximately portfolio number 6! So we do not have to do any further guessing.

We know that Liz wants to get an expected return of 15%. The horizontal line shows her desired level of expected return. To nd how she should divide her money between the risk-free and the tangency portfoliouse the familiar equation:

L = x r f + (1 x) T L is the expected return on Lizs overall portfolio which is made up of investing x fraction of her money in therisk-free rate and (1 x) in the tangency portfolio. Substitute for L = 0 .15, r f = 0 .09 and T = 6 = 0 .16and solve for x to get x = 0 .14 so that 1

x = 0 .86.

Therefore, Liz should put 14% of her 250,000, i.e., $35,000 in the bank CD and the remaining $215,000 inthe tangency portfolio. She should divide the $215,000 into the three stocks in the proportions for portfolio#6. In other words, she should put 0 .4 $215, 000 = $86 ,000 in ALM, 0.4 $215, 000 = $86,000 in BPI,and 0 .2 $215, 000 = $43 ,000 in CRN.The standard deviation of her overall portfolio return can be calculated using a similar procedure aswe used for the expected return. This approach will work here because the correlation between the CD(risk-free) and the tangency portfolio is zero. So we get:

L = x 0.00 + (1 x) 0.05 = 0 .86 0.05 = 0 .04

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This number is conrmed by the graph also.The highest and the lowest returns would be based on some reasonable condence estimates. I am going

to use a 95% level of condence. Then the highest and the lowest returns are given by L 1.96L whichgives us that the highest and lowest possible return on her $250,000 investment would be 0.23 and 0.07,respectively.

3.7 Portfolio Analysis Suppose x fraction should be invested in security C and (1 x) in B , then to meet the objective:

x(0.13) + (1 x)(0 .19) = 0 .15Which gives, x = 2/ 3 and (1 x) = 1/ 3. Now, the standard deviation of this portfolio can be calculated as:

p = x22C + (1 x)22 + 2 x(1 x)BC B= ( 2/ 3)2(0.26) + ( 1/ 3)2(0.38) + 2( 2/ 3)( 1/ 3)(0 .16)

= 0 .478423

The standard deviation of A is A = 0 .32 = 0 .565685. Since the portfolio offers the same expected returnas security A, but has a lower standard deviation, it dominates security A.

3.8 Portfolio Calculations The covariance form of the portfolio variance formula should be used in this problem, i.e.:

p = x11 + x22 + x332 2 p = x22 + x22 + x32 + 2 x1x212 + 2 x1x313 + 2 x2x3231 1 2 3

(a) For Mr. Thomass portfolio:

p = (0 .33)(0 .15) + (0 .33)(0 .12) + (0 .34)(0 .17)= 0 .1469

p = (0 .33)2(0.0484) + (0 .33)2(0.0625) + (0 .34)2(0.0906)

+2(0 .33)(0 .33)(0 .033) + 2(0 .33)(0 .34)(0 .026) + 2(0 .33)(0 .34)(0 .038)]0.5

= 0 .2098

(b) For Ms. Walkers portfolio:

p = (0 .50)(0 .15) + (0 .15)(0 .12) + (0 .35)(0 .17)= 0 .1525

p = (0 .50)2(0.0484) + (0 .15)2(0.0625) + (0 .35)2(0.0906)+2(0 .50)(0 .15)(0 .033) + 2(0 .50)(0 .35)(0 .026) + 2(0 .15)(0 .35)(0 .038)]0.5

= 0 .2064

(c) Ms. Walker has the better portfolio because she is getting a higher expected return for a lower risk.

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3.9 Tangency Line The expected return and the risk for portfolios of a risk-free security and the tangency portfolio are given

by:

p = xr f + (1 x)T p = x22 + (1 x)22 + 2 x(1 x)f T f T f T

= x2 (0) + (1 x)22 + 2 x(1 x)(0) T f T T = (1 x)T

We are given r f = 0 .07, T = 0 .17, T = 0 .214.

(a) Directly apply the equations:

p = (0 .25)(0 .07) + (0 .75)(0 .17) = 0 .145 = 14.5% per year .

p= (0 .75)(0 .214) = 0 .1605 = 16.05% per year .

(b) Apply the equation for expected return:

p = 0 .15 = ( x)(0 .07) + (1 x)(0 .17)0.15 = 0 .17 0.10x0.02

x = = 0 .200.10

So, the investor should invest 20% of her wealth in the risk-free security and the remaining 80% in thetangency portfolio. For risk, apply the standard deviation equation:

p = (0 .80)(0 .214) = 0 .1712 = 17.12% per year

3.10 Hedge Portfolios For the hedge portfolio, we set the standard deviation of the portfolio equal to zero:

2 p = x22 + x22 + 2 x1x212 (+1) = 0 .1 1 2

Since x1 + x2 = 1, substitute x2 = 1 x1 in the equation, and square both sides to get:2x1

21 + (1 x1 )22 + 2 x1(1 x1)12 = 0 ,2

which can be factored and written as:

2

x11 + (1 x1 )2 = 0 ,so that,

x11 + (1 x1)2 = 0 .Now we can solve for x1 and then for x2 = 1 x1 to get:

x1 = 2 and x2 = 1 x1 =1 .

1 2 1 2

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3.11 Hedge Portfolios With perfectly negative correlation, we use equation (3.16):

2 0.20x1 = = = 0 .52631 + 2 0.18 + 0 .20x2 = 1 x1 = 1 0.5263 = 0 .4737

The statistics for the portfolio are:

p = (0 .5263)(0 .10) + (0 .4737)(0 .12) = 0 .1095

p = (0 .5263)2(0.18)2 + (0 .4737)2(0.20)2 + 2(0 .5263)(0 .4737)(0 .18)(0 .20)(1) = 0 .0000With perfectly positive correlation, we use the result from the previous exercise:

2 0.20x1 = = = 101 2 0.18 0.20x2 = 1

x1 = 1

10 =

9

The statistics for the portfolio are:

p = (10)(0 .10) + ( 9)(0 .12) = 0.08 p = (10) 2(0.18)2 + ( 9)2(0.20)2 + 2(10)( 9)(0 .18)(0 .20)(1) = 0 .0000

3.12 Rate of Return on a Mutual Fund To calculate the rate of return, we should use the total wealth here because the number of shares have

grown due to reinvestment. The initial wealth was 98 10.27 = 1006.46. The value of my investments todayis 130 9.54 = 1240 .20. The three year rate of return is (1240 .20 1006.46)/ 1006.46 = 0 .2322. Therefore,the annual rate is (1 + 0 .2322)1/ 3 1 = 0 .07209 or approximately 7.21%.3.13 Risk-free and Market Betas

From the denition of :i i = im m

Therefore,

f f = fm = (0)(0)

= 0m mm m = mm = (1)(1) = 1m

3.14 CAPM Calculations The of Canon stock is:

C = (0 .49)(0.30)

C = Cm = 0 .668.m (0.22)Therefore, the proper expected return is:

kC = r f + C (m r f ) = 0 .07 + 0 .668(0.12 0.07) = 0 .103,or 10.3% per year. The proper expected rate of return from the stock based on its risk is 0.103 or 10.3% peryear.

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3.15 and CAPM

(a) Since i = im / 2 , we need im , the covariance between security and market returns, and 2 , them mvariance of market returns. Let us calculate those rst:

r i r m

0.019455 (0.420)(0 .337)

im = r i r m T = 12 = 0 .000696363T 1 11( r m )

2

2r 0.016319 (0 .337) 2

m 122 = T = = 0 .000623174m T 1 11Now, we can calculate the :

im 0.000696363 = = = 1 .1172 0.000623174m

(b) Before doing the calculations, we need to convert the annual risk-free rate to monthly rate: r f =(1 + 0 .07)1/ 12

1 = 0 .00565 per month. The market return is expected to be equal to the historical

average, therefore, m = r m /T = 0 .337/ 12 = 0 .028 per month. Therefore, the proper expected returnon the stock ( k) is:

k = r f + (m r f ) = 0 .00565 + 1 .117(0.028 0.00565) = 0 .0306or 3.06% per month.

(c) Based on the historical returns, the stock has expected return of 0 .420/ 12 = 0 .035 or 3.5% per month.Since the stocks expected return is higher than its proper expected return, it is underpriced and shouldbe bought.

3.16 CAPM and Equilibrium

(a) The proper ( a la CAPM) expected return for Elwin is calculated as:ki = r f + i (m r f ) = 0 .08 + 1 .7(0.14 0.08) = 0 .182 = 18 .2%

(b) The proper stock price ( P ), therefore, is calculated using the rate of return equation:

18 + 2 P 0.182 =P

which gives us P = $16 .92.

(c) The stock is currently underpriced because the market price ($16.25) is less than its proper value ($16.92).Investors should buy the stock.

(d) The expected return, if the stock were bought, would be calculated as:18 + 2 16.25r = = 0 .2308 = 23 .08%

16.25

Since the stock is offering a higher return (23.08%) than what it should (18.2%), investors should buythe stock.

(e) Since investors will buy the stock, the demand pressure will push the stock price up till the stock isproperly priced.

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Chapter 4

Security Selection and PerformanceEvaluation

4.1 Present Value Calculations The PV expressions and the nal answers are shown below:

a.

b.

c.

d.

PV = 10( af 0.104 )PV = 10( af 0.104 )(df

0.101 )

PV = 10( af 0.105 )PV = 10( af 0.105 ) 10(df 0.103 )

=

=

=

=

31.70

28.82

37.91

30.39

e.

f.

PV = 10 / 0.1

PV = (10 / 0.1)df 0.10

3

=

=

100.00

75.13g.

h.

PV = 10 / (0.1 0.05)PV = [10 / (0.1 0.05)]df 0.102

=

=

200.00

165.29

4.2 Present Value

(a) This is a growing perpetuity. Let us rst calculate the growth rates: g1 2 = (33 / 30) 1 = 0 .1,g2 3 = (36 .3/ 33) 1 = 0 .1, g3 4 = (37 .03/ 36.3) 1 = 0 .02, g4 5 = (37 .77/ 37.03) 1 = 0 .02,g5 6 = (38 .52/ 37.77) 1 = 0 .02. So the growing perpetuity begins at t = 3 and has a growth rate of 0.02. The discount rate is 9% per period or 0.09. Therefore:36.3

df 0.09PV = 30 df 0.09 + 33 df 0.09 +21 2 0.09 0.0230 33 36.3 1

= + +1.09 (1.09)2 0.09 0.02 (1.09)2

= 491 .77

(b) This one has an annuity and a growing perpetuity. Just to make sure that there is no change in growthrate, calculate the growth rates: g5 6 = (315 / 300) 1 = 0 .05, g6 7 = (330 .75/ 315) 1 = 0 .05, g7 8 =(347.29/ 330.75)1 = 0 .05, g8 9 = (364 .65/ 347.29)1 = 0 .05, g9 10 = (382 .82/ 364.65)1 = 0 .05. The

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. . . . . .

growth rate, therefore, is 0.05 per period. The discount rate is 11.5% per period or 0.115. Therefore,

300df 0.115300af 0.115PV = 3 + 0.115

0.05 4

11 300 1= 300 1 +0.115 (1.115)3 0.115 0.05 (1.115)4= 3712 .91

4.3 Annuity Formula I will derive the formula for a growing annuity and then set the growth rate equal to zero to get the

formula for a simple annuity. A growing annuity, with a growth rate of g, is shown in the time line below:

c c(1 + g) c(1 + g)2 c(1 + g)n 2 c(1 + g)n 1

. . .0 1 2 3 n

1 n

This growing annuity can be written as a difference between the following two growing perpetuities:

c c(1 + g) c(1 + g)2 c(1 + g)n 2 c(1 + g)n 1 c(1 + g)n c(1 + g)n +1

. . . . . . 0 1 2 3 n 1 n n + 1 n + 2c(1 + g)n c(1 + g)n +1

0 1 2 3 n 1 n n + 1 n + 2Therefore, the present value of the growing annuity can be written as:

c c(1 + g)n 1PV = k g k g (1 + k)nc (1 + g)n

= 1 k g (1 + k)nNote that with lim n , the term in square brackets will go to 1 because with k > g , (1+ k)n approaches faster than (1 + g)n . The formula for an annuity is obtained by setting g = 0 in the above expression to get:

c 1PV = 1 k (1 + k)n

4.4 Rate of Return Calculations

In the equations in problem 4.1, we replace PV by the initial cash outow, i.e., $30, and instead of usinga known discount rate of 0.10, use the unknown r . Then we solve for r using algebraic or trial-and-error and

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interpolation methods. The nal answers are given below:

a.

b.

30 = 10( af r430 = 10( af r4

) r = 0 .126

)(df r1) r = 0 .087c.

d.

30 = 10( af r530 = 10( af r5

) r = 0 .199

) 10(df r 3) r = 0 .105e. 30 = 10/r r = 0 .333

f. 30 = (10 )df r/r 3 r = 0 .196

g. 30 = 10/ (r 0.05) r = 0 .383h. 30 = [10/ (r 0.05)]df r2 r = 0 .260

4.5 Rule of 72 Suppose it take n years for c to double at rate r , then c must be the PV of 2c to be received in n years

from now:2c

c =(1 + r )n

.

This equation can be solved for n as:log(2)

n =log(1 + r )

The Rule of 72 n is determined using 72 /r where r is expressed as a % rather than a decimal fraction. Letus see how well the rule of 72 holds up:

r

0.04000.05000.06000.07000.07850.08000.09000.10000.11000.12000.13000.14000.1500

0.16000.17000.1800

Exact n

17.673014.206711.895710.24489.17449.00658.04327.27256.64196.11635.67145.29014.9595

4.67024.41484.1878

Rule of 72 n

18.000014.400012.000010.28579.17439.00008.00007.20006.54556.00005.53855.14294.8000

4.50004.23534.0000

Error

0.32700.19330.10430.0409

0.00000.00650.04320.07250.09640.11630.13300.14720.15950.17020.17960.1878

The rule of 72 is exact for r = 7 .85% per year. The rule is in error for other rates. The farther we movefrom 7.85% the more the error.

4.6 Continuous Compounding

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The Treasury bill is maturing in 2/12 years and the compounding is continuous. The equation for annualrate, r , therefore, is:

100009823 = 2r 12

ewhich gives:2 10000

12 r =e9823

or, by taking logs:2 10000

r = ln12 9823

which gives r = 0 .10715 or 10.72% per year, compounded continuously.

4.7 Present Value and IRR

(a) The following time line shows the cashows on a per share basis.

45.00 0.30 0.30 0.35 44.301/1/88 3/31/88 6/30/88 9/30/88 12/31/88

(b) The equation for present value is:

0.30 0.30 0.35 44.30PV = + + +

(1 + k) (1 + k)2 (1 + k)3 (1 + k)4

The discount rate k is given to us as 14% per year. For our calculation we need it in the units of perquarter. Let us convert it to a quarterly rate as: (1 + 0 .14)1/ 4 1 = 0 .033299. Using this value as k inthe equation above we get PV = 39 .74819 or approx $39.75.Rather than converting the rate from annual to quarterly, we can write the equation in terms of annualrates, but we will have to measure distances in years, rather than quarters. With annual rate, the presentvalue equation would be:

0.30 0.30 0.35 44.30PV = + + +

(1 + k)0.25 (1 + k)0.5 (1 + k)0.75 (1 + k)1

In this equation, k would be 0.14. The nal answer given by this equation is exactly the same as above.

(c) The following equation can be written for the quarterly rate of return r :

0.30 0.30 0.35 44.3045 = + + +

(1 + r ) (1 + r )2 (1 + r )3 (1 + r )4

Now solve for r using the trial-and-error and interpolation:

r

0.0200.0100.0050.0030.001

r

rhs

41.8386343.5022544.3652944.7165445.0712945

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4.13 Security Valuation I will use m to denote market and i to denote Murphy. i = im i m = 0 .8

0.240.16 = 1 .2. Therefore,

ki = r f + i (m r f ) = 0 .08 + 1 .2(0.14 0.08) = 0 .152. The discount rate, therefore, is 0.152 or 15.2% peryear. Since the problem involves quarterly dividend payments, we need the quarterly discount rate. Thequarterly discount rate is calculated as (1 + 0 .152)1/ 4 1 = 0 .036 or 3.6%. Now we can calculate the properprice (the present value) of the stock:

PV = 0 .40af 0.0368 + 15 df 0.0368 = 14 .042

Clearly, at the current market price of $10.50, the stock is underpriced. Ziggy should buy it.

4.14 Security Selection We know the following: O = 1 .08, r f = 0 .08 per year, or (1 .08)1/ 12 1 = 0 .006434 per month, andm = 0 .012 per month. Therefore, kO = r f + O (m r f ) = 0 .006434 + 1 .08(0.012 0.006434) = 0 .012445per month.Since the stock can be sold for $12 a month from now, the proper price (present value) of Orpheus is:

12/ (1+ 0 .012445) = 11 .85. Since the current market price is $10.20, the stock is underpriced and should be

bought.Since the stock can be bought today for $10.20 and be sold a month later for $12, the internal rate returnfrom the stock for the coming month would be (12 10.20)/ 10.20 = 0 .1765. Since this is higher than thediscount rate, the stock is underpriced.4.15 Security Selection

The following facts can be collected from the information given:

Expected dividend for the next quarter = d = 0 .20(1.015) = 0 .203Dividend growth rate = g = 0 .015 per quarter

Stocks = 1 .2Risk-free rate = r f = 0 .078 per year

Expected return on the market = m = 0 .135 per yearTherefore,

Stocks discount rate = k = r f + (m r f ) = 0 .078 + 1 .2(0.135 0.078)= 0 .1464 per year= (1 + 0 .1464)0.25 1 = 0 .034746 per quarter

Therefore,

Stocks proper price = PV =d

k g=

0.2030.034746 0.015

= 10 .28020

= $10 .28

Since the market price of 107/

8=$10.875 is higher than the proper price, the stock is overpriced. It shouldbe sold. If the stock is not currently owned, it may be sold short.

4.16 Security Selection

(a) The quarterly dividends expected by Carol are shown on the time line below.

0.50 0.50 0.50 0.505 0.51005

. . . . . . . . . 0 1 2 16 17 18 28 29 30

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(b) The following equation can be written equating the present value of the future dividends to the currentstock price. r in this equation is the unknown expected return.

8 = 0 .50af r12 df

r16 +

0.505r 0.01df

r28

To solve for the expected return, r , we use the trial-and-error and interpolation techniques. The detailsare shown below:

r af r12 df r16 df

r28 rhs

0.0200 10.575340 0.728445 0.574374 32.8576900.0400 9.385073 0.533908 0.333477 8.1189210.0450 9.118580 0.494469 0.291570 6.4613770.0410 9.330853 0.525760 0.324623 7.7411160.0405 9.357905 0.529817 0.329019 7.926692

r 8

r = 0 .0405 +0.0405 0.04

7.926692 8.118921(8 7.926692) = 0 .040309

The expected rate of return, therefore, is 0.040309 or 4.0309% per quarter.

(c) The risk-premium offered by the average stock is 0 .12 0.07 = 0 .05. Therefore, being twice as risky asthe average stock, Surya should offer twice the risk premium, i.e., 0.10. Therefore, the rate of return onSurya should be 0 .07 + 0 .10 = 0 .17 or 17% per year.

(d) The expected return on Surya, based on the dividends, is 4.0309% per quarter or (1 + 0 .040309)4 1 =0.17125 or 17.125% per year. From part c we know that for the amount of risk contained in Surya, itshould pay 17% per year. Since the stock is expected to pay a higher rate of return than it should basedon its risk, Carol should invest in the stock.

4.17 Market Efficiency

(a) High volatility does not imply market inefficiency. Market prices uctuate because of the release of newsand information. Continuous release of new information and its instantaneous availability to investorsimplies that in an efficient market, investors will react to the news and this reaction will be evident inthe price uctuations, which is the cause of volatility.

(b) An efficient market does not mean that investors cannot make any returns in the market. What it meansis that they cannot make any more returns than justied by the amount of risk they take. Therefore,this statement is misguided.

4.18 Performance Evaluation Realized return: (11 .60 10.00)/ 10.00 = 0 .16 per month. Based on the market conditions ( r f = (1 +0.09)1/ 12 1 = 0 .0072 per month, and r m = 0 .026 per month), and the risk of the stock ( O = 1 .08),its return should have been: 0 .0072 + 1 .08(0.026 0.0072) = 0 .0275. The abnormal return, therefore, is0.16 0.027 = 0 .133. Since the abnormal return is positive, her stock returned more than what it shouldhave and therefore it was a good investment.

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4.23 Performance Measurement The following facts can be collected from the information given:

Average fund return = r i = 0 .126 per yearAverage market return = rm = 0 .142 per year

Average risk-free rate = r f = 0 .074 per yearStd Dev of market = m = 0 .42 per year

Std Dev of fund = i = 0 .33 per yearCorrelation between fund and mkt. = im = 0 .60

Funds = i = imim

= 0 .600.330.42

= 0 .471428

Funds abnormal return = r i [r f + i (rm r f )]= 0 .126 [0.074 + 0 .471428(0.142 0.074)]= 0 .019942 per year

Since the abnormal return is positive, the fund did exhibit superior performance.

4.24 Performance Measurement

(a) Comparing fund returns with the S&P 500 is not appropriate if the funds risk is not the same as thatof the S&P 500.

(b) Statistical analysis of the data shows that the of the fund is 1.35559 and the average return on thefund and the S&P 500 during the 10 years were 19.5% and 17.7%, respectively. The average abnormalreturn for the fund, therefore, is:

ar = 19 .5 [8.2 + 1 .35559(17.7 8.2)] = 1 .578% per year.Since the abnormal return is positive, the fund performed well.

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The stock paid $2.55 in dividends during the past year. The dividend yield of the stock is 6.07% per year. The dividend yield is calculated by dividing theannual dividend by the closing price. Since the closing price for the day (as we see later) is $42,

the dividend yield is 2 .55/ 42 = 0 .0607, which agrees with the reported value.

The ratio of price per share to earnings per share is 7. 2000 shares of the stock were traded in the market. The high, low, and closing prices for the stock were all $42. The stock closed up by 3/ 8. Therefore, the closing price on the previous trading day must have been415/ 8.

(b) The dividend payout ratio is calculated by dividing the dividends per share by the earnings per share. Weknow that the dividends per share for the last year were $2.55. The earnings per share can be calculatedusing the PE ratio. Since, the PE ratio is dened as the price per share ( p) divided by earnings pershare ( e), 7 = p/e = 42 /e which gives us earnings per share of 6. The dividend payout ratio, therefore,

is 2.55/6=0.425 or 42.5%. I would classify this rm as a growing rm.(c) The cash ows expected from the stock are:

0.60 0.60 0.80 46.60

1 2 3 4

To calculate the proper price (present value) of the stock we need the discount rate. The for the stockis calculated as im i / m = 0 .800.22/ 0.14 = 1 .257. The discount rate for the stock is then calculatedas r f + (m r f ) = 0 .07+ 1 .257(0.12 0.07) = 0 .13285. This rate is on a per year basis. We estimatethe quarterly rate needed for computation as (1 + 0 .13285)1/ 4 1 = 0 .03168. Now we can calculate thepresent value as:

PV = 0.601+0 .03168 +0.60

(1+0 .03168) 2 +0.80

(1+0 .03168) 3 +46 .60

(1+0 .03168) 4

= 43 .01

The proper price, therefore, is $43.01. Since the stock is selling for $42, it is underpriced and should bepurchased.

5.4 Common Stock Valuation

(a) The entries in the listing are explained below:

30 and 18 are the highest and the lowest prices reached by the stock during the past 52 weeks.

Jumbo Peanuts is the name of the company and JP is the ticker symbol.

The stock paid $2.70 in dividends during the past year. The dividend yield of the stock is 12.3% per year. The dividend yield is calculated by dividing theannual dividend by the closing price. Since the closing price for the day (as we see later) is $22,

the dividend yield is 2 .70/ 22 = 0 .1227 0.123, which agrees with the reported value.

The ratio of price per share to earnings per share is 6. 2000 shares of the stock were traded in the market. The high, low, and closing prices for the stock were $23, $21, and $22, respectively.

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(a) The for Curio is calculated by running the market model regression. The result gives C = 0.017831.Since no growth or cyclical behavior is apparent in the S&P 500 returns, we can estimate the expectedreturn on the market by the average of past returns. Therefore, on calculation, we see m = 0 .010227.

The proper rate of return expected from Curio is calculated using CAPM as:C = r f + C (m r f ) = 0 .006 + (0.017831)(0 .0102270.006) = 0 .005924

.

(b) The current price of Curio is $19.20 and the next months price is expected to be $20. Therefore, basedon the private information, the rate of return is expected to be (20 19.20)/ 19.20 = 0 .041666.

(c) Since the rate of return expected based on private information is higher than that based on the risk, thestock is underpriced and should be purchased.

5.6 Price Earning RatioWith constant growth, the price is equal to the present value of growing perpetuity beginning with the

next periods dividend which equals d(1 + g) where d is the current periods dividend. Therefore:

p =d(1 + g)

k g=

e(1 + g)k g

pe

=(1 + g)

k gwhere the second step follows from the denition of the payout ratio. From this equation we can see that astock with high growth ( g) will have a high price earnings ratio. The ratio could also be high if the discountrate k is low.

At rst glance, it appears that the price earnings could also be high if the payout ratio ( ) is high.However, a high payout ratio would lower the growth rate g so that which will bring the price earnings ratiodown. The net effect of a high payout ratio on the price earnings ratio is negative.

5.7 Price Earning Ratio

(a) PE ratio = price/earnings. Therefore, earnings = Price/PE ratio. So, earnings=31 5/ 8/52=0.6082, orapproximately $0.61 per share.

(b) The dividend payout ratio is dened as dividends per share divided by earnings per share which equals0.92/0.61=1.508. So the company paid 150% of its earnings as dividends. This situation may arise if the company has low earnings during the year but the company does not want to change its dividendpolicy.

(c) k = r f + (m r f ) = 0 .065 + 1 .2(0.13 0.065) = 0 .143 or 14.3% per year.(d) Suppose the annual growth rate is g, then:

p = 31 .625 =0.92(1 + g)0.143 g

which gives g = 0 .1104 or 11.04% per year.

5.8 PVGO

PVGO = p ek

= 14 1.200.10

= 14 12 = 2

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5.9 PVGO PVGO, for a constant growth stock, is dened as

PVGO = P V e

kThe PV for a growth company is calculated using growing perpetuity, so that:

PVGO =d

k g ek

Now substitute g = (1 )i from equation (5.4) and d = e from the denition of payout ratio and simplify:PVGO = ek (1 ) i ek

= ek ek + e(1 ) ik (k (1 ) i )= ek ek + ei eik (k (1 ) i )

= e ( i

k )

e ( i

k )k (k (1 ) i )

= (e e )( i k )k (k (1 ) i )= e (1 )( i k )k (k (1 ) i )

= (1 )k e( i k )

k (1 ) i

If i = k, the second term goes to zero, making PVGO=0.

5.10 Valuing a Growth Stock The growth rates in dividends for the previous three years are calculated as (0 .62/ 0.55) 1 = 0 .127,(0.70/ 0.62) 1 = 0 .129, (0.79/ 0.70) 1 = 0 .129. The average annualized growth rate therefore seems tobe 0.129 or 12.9%. The expected dividend next year, therefore, is 0 .79

(1 + 0 .129) = 0 .89. The discount

rate is 15% per year. Assuming that the company will be able to maintain this growth rate forever, we cancalculate the present value of the stock as: 0 .89/ (0.15 0.129) = $42 .38.5.11 Delayed Growth

The earnings per share in Qtr 1 of 1995 would be 0 .75(1 + 0 .02)21 = 1 .136749. Therefore, the dividendsper share would be 0 .60 1.136749 = 0 .682049. Thereafter, the dividends will grow at the rate of 1.4%or 0.014 per quarter. The discount rate for the stock can be calculated using the CAPM to be 0 .065 +1.4(0.125 0.065) = 0 .149 per year. For present value calculation, this has to be converted to per quarterbasis as (1 + 0 .149)0.25 1 = 0 .035333. Now we can calculate the present value as:

P V =0.682049

(0.035333 0.014)1

(1 + 0 .035333)20= 15 .96

5.12 Information and Price Reaction

(a) The proper expected return is calculated using CAPM as: k = r f + (m r f ). We need to know touse this equation. can be calculated as: i = im

im

= 0 .200.250.20

= 0 .25

Now, the discount rate is calculated as: k = 0 .06 + 0 .25(0.13 0.06) = 0 .0775, or 7.75% per year.

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(b) The dividend payments from Merit constitute a perpetuity. Therefore,

p =d

k20 =

d

0.0775which gives d = $1 .55 per share.

(c) With the new tax law, the dividends per share will become $1.62. This change will not have any effecton the discount rate. Therefore, the new price will be:

p =1.62

0.0775= 20 .90

So, the price of Merit company will go up by $0.90 after the announcement.

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Chapter 6

Fixed Income Securities

6.1 Bond Valuation The annual coupon rate is 5 5/ 8 or 5.625% per year. Therefore, the bond will pay annual coupon of $56.25

in two equal payments of $28.125. The promised cashows from the bond are shown below:

$28.125 $28.125 $28.125 $1028.125

. . .0 1 2 8 9

7/1/91 12/31/91 6/30/92 6/30/95 12/31/95

We can write the following equation for the semiannual yield to maturity:

910 = 28.125af r8 + 1028 .125df r9

To solve it, we resort to trial-and-error. Since the bond is selling at a discount rate, the yield to maturitymust be higher than the coupon rate. So we start with a value higher than 2.8125%.

r af r8 df r8 rhs

0.030 7.0197 0.7664 985.40100.035 6.8740 0.7337 947.69710.040 6.7327 0.7026 911.70540.041 6.7050 0.6995 904.7048

r 910

r = 0 .040 +0.041 0.040

904.7048 911.7054(910 904.7048) = 0 .040243

or 4.0243% per six month or 8.21% per year.

6.2 Yield of a Bond Selling at Par The semiannual coupon rate is 3.5%. So the cemiannual coupon amount is $35. The equation for

calculating the semiannual yield to maturity may be written as:

1000 = 35af r12 + 1000 df r12

To solve for r , we use the trial-and-error.

r af r12 df r12 rhs

0.035 9.66333 0.66178 1000.0000

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So the semiannual yield to maturity is equal to the semiannual coupon rate, i.e., 3.5%. The annual yieldto maturity is (1 + 0 .035)2 1 = 0 .071225 or 7.1225%.6.3 Convertible Bonds

Since the risk of the ABC9s92 and ABC8s91 are identical, the yield to maturity of the two bonds shouldbe identical. The semiannual yield to maturity of ABC8s91 is calculated as:

920 = 40df r1 + 1040 df r2 =

40(1 + r )

+1040

(1 + r )2

which can be solved using trial-and-error or an algebraic method. Let us use the algebraic method. Multiplythe whole equation by (1+r) and take all terms to the left hand side to get:

920(1 + r )2 40(1 + r ) 1040 = 0This is a quadratic equation for (1 + r ). The solutions for (1 + r ) are:

(1 + r ) = (40) (40)2 4(920)(1040)2(920)= 40 38288001840=

40 1956.73201840

= 1 .0852 or 1.0417so that

r = 0 .0852 or 2.0417

Since the yield to maturity cannot be negative, the proper yield to maturity is 0.0852 or 8.52% semiannually.Now, as we mentioned above, the yield to maturity of ABC9s92 should also be 8.52% semiannually. So

the value of the bond can be calculated as:

PV = 45 af 0.08523 + 1045 0.08524 = 868 .38

So ABC9s92 should be selling for $868.38. However, this bond also has a convertibility feature. Therefore,the value of the bond, if converted to shares would be 50 18 = $900. The bond is worth more if convertedto shares. Therefore, this will be the determining factor in bond price. The price of the bond, therefore, willbe $900 or more because of the possibility that the prices of shares may go up even more making the bondmore valuable.6.4 Pre-tax and After-tax Yields

The pre-tax semiannual yield to maturity is calculated using the equation:

920 = 50af r10 + 1000 df r10

which, when solved using trial-and-error and interpolation, gives r = 0 .060916.

The after-tax coupon income to the investor paying 28% on interest income and capital gains would be$50(1 0.28) = $36. The total income on maturity would be $1,000 but $1 , 000 $920 = $80 of this wouldbe capital gains. The investor will have to pay 0 .28 $80 = $22 .4 in capital gains taxes making his netcashow to be $1 , 000 22.40 = $977 .60. The semiannual yield to maturity based on the after-tax cashowswould be calculated as:920 = 36af r10 + 977 .60df

r10

which, when solved using trial-and-error and interpolation, gives r = 0 .044243.Using the intuitive formula, the after-tax rate would have been calculated to be 0 .060916(1 28) =0.0438592 which is a little less than the true after-tax yield of 0.044243.

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6.5 Basic Bond Calculations

(a) The bond is below its face value. So it is selling at a discount or below par.

(b) The current yield of the bond is 90 / 985 = 0 .0914 or 9.14%. It is an approximate measure of the nextyears rate of return the investor can expect from the interest income alone.

(c) The semiannual yield to maturity can be calculated using the following equation:

985 = 45af r34 + 1000 df r34

which, when solved using trial-and-error and interpolation, gives r = 0 .045879 semiannually or an annualrate of 9.3864%. This is the annual rate of return expected from the bond if the bond is held to maturityand the bond makes all the promised cashows.

(d) The yield to call is calculated assuming the bond will be called the rst chance the issuer has to call thebond. Since the bond is callable in 1997, let us assume that the bond will be called on January 1, 1997,which is essentially the same as calling it on December 31, 1997. The semiannual yield to maturity can

be calculated using the following equation:985 = 45af r16 + 1075 df

r16

which, when solved using trial-and-error and interpolation, gives r = 0 .049561 semiannually or an annualrate of 10.1578%. This is the annual rate of return expected from the bond if the bond is held till calland the bond makes all the promised cashows.

6.6 Bond Pricing There are two steps required to solve this problem. In the rst step, we determine the yield to maturity

of CRX and then, use it as the discount rate for Lauras bond.The cashows from the CRX bond are given below. In drawing this diagram I assumed that the bond

pays its coupon interests on June 30 and December 31, and that the bond will mature on December 31,

1995. Since today is May 12, 132 days have passed since the previous coupon payment date and 50 daysremain to the next coupon payment.

$41.25 $41.25 $41.25 $1041.25

. . .#1 6/30/88 12/31/88 6/30/95 12/31/95

We can write the following equation to solve for the yield to maturity, r :

952.50 +132182

41.25 =1

(1 + r )50 / 182[41.25 + 41 .25af r15 + 1000 df

r15 ]

982.42 =1

(1 + r )50 / 182[41.25 + 41 .25af r15 + 1000 df

r15 ]

We can solve for r using trial-and-error and interpolation:

r af r15 df r15 rhs

0.040 11.1184 0.5553 1043.83990.050 10.3797 0.4810 937.77360.045 10.7395 0.5167 988.94520.046 10.6661 0.5094 978.4224

y 982.4200

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Now we use interpolation:

r = 0 .046 +0.045 0.046

988.9452

978.4224

(982.4200 978.4224) = 0 .0456201Since Lauras bond has identical risk as the CRX, then under market equilibrium, the discount rate for

Lauras bond would also be 0.0456201. Assuming that the discount rate for Lauras bond on July 1 wouldalso be be 0.0456201 the price of Lauras bond on July 1 would be calculated as:

P = 45 .00af y23 + 1000 df y23

where y = 0 .0456201. Again, I have assumed that the coupon payments are made on June 30 and December31, and that the bond matures on December 31. On calculation we nd the price to be 991.28.

6.7 Bond Yields using Real Data The bond is to be purchased on February 9, 1988: 40 days after the December 31, 1987 coupon payment.

The semiannual coupon rate is 4.3125%. The semiannual coupon payment, therefore, is $43.125 per bond.

The bond has a face value of $1,000. The accrued interest, therefore, is 43 .125 40

182 or $9.48. I used 182because the number of days between January 1, 1988 and June 30, 1988 is 182. This leads to the purchaseprice of $940+$9.48=$949.48. This is the price to be paid for the bond regardless of whether the investorhas to pay taxes or not. The total number of semiannual coupon payments are 42 because 21 years passbetween the beginning of 1988 and the end of 2008. The time between the day of purchase and the rstcoupon payment is 182 40 or 142 days. This translates into 0.78 semiannual periods.(a) The yield to maturity r is solved using the following equation:

949.48 = [43.125 + 43 .125af r41 + 1000 df r41 ]

1(1 + r )0.78

The value of r is determined by the trial and error method.

r af r41 df r41 (1 + r )0.78 1072.2830

0.0400 19.9930 0.200277 1.031073 1072.28300.0450 18.5661 0.164525 1.034939 974.27300.0460 18.3000 0.158197 1.035711 956.35880.0470 18.0400 0.152119 1.036484 938.96240.0465 18.1692 0.155128 1.036098 947.59700.0464 18.1953 0.155737 1.036021 949.3391

r 949.48

Now we interpolate using the last two iterations to get the semiannual yield as:

r = 0 .0464 +0.0465

0.0464

947.5970 949.3391 (949.48 949.3391) = 0 .04639.The annualized yield is (1 + 0 .04639)2 1 = 0 .09493 or 9.48% per year.

(b) The transaction cost is 0 .03 949.48 or $28.48. The total cash outow on February 9, therefore, wouldbe 949.48 + 28.48 or $977.96. Taxes would reduce the coupon amounts. The after-tax coupon amountwould be 43.125(1 0.28) or $31.05. On maturity, capital gain tax would have to be paid. The capitalgain is $1,000 $949.48 = $50 .52. The capital gain tax would be 50 .52 0.28 or $14.15. The net nalcashow, therefore, would be 1 ,000 14.15 or $985.85.

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I have made a simplifying assumption that the taxes are deducted at the source, i.e., at the time thecoupon income is received. In reality, the coupon incomes may be received in June and December butthe taxes may not be paid till the following April. A very careful analysis will take that into account

also.To solve for the yield to maturity r we set up the following equation:

977.96 = [31.05 + 31 .05af r41 + 985 .85df r41 ]

1(1 + r )0.78

Using trial-and-error and interpolation, we nd the yield to be 6.53% per year.

(c) With possibilities of default, the pricing and yield calculations should be based on the expected valuesrather that the promised values. Since in this particular problem default is possible only during thepayment of the face value we do not have to make too many changes from part b. The expected nalcashow is calculated as 0 .80 1000 + 0 .20 850 = 970. This means that the expected capital gainwould be 970 949.48 or $20.52 on which the capital gain tax would be $5.75 resulting in the net nalexpected payment of $964.25.The yield to maturity r is solved by trial and error using the following equation:

977.96 = [31.05 + 31 .05af r41 + 964 .25df r41 ]

1(1 + r )0.78

to give an annual rate of approximately 6.48% per year.

6.8 Treasury Note

(a) The annual coupon rate on the note is 13 7/ 8 or 13.875%. The note matures in June 1992. Its bid andasked prices were 105 13/ 32 and 105 17/ 32 . The bid price had changed by 2/ 32 since the previous trading.The yield to maturity is estimated to be 8.85% per year.

(b) The semiannual yield to maturity may be calculated using the the ask price of 105.53125 using following

equation:

105.53125 + 6 .9375 (123/ 182) =1

(1 + r )59 / 182[6.9375 + 6 .9375af r8 + 100 df

r8 ]

which, when solved using trial-and-error and interpolation gives r = 0 .060647. The annual yield, there-fore, is 12.4972% per year.

6.9 Comparing Bond Investments The annual yield to maturity on the 15 year Treasury strip is calculated as:

26.02 =100

(1 + r )15r =

10026.02

( 115 )

1 = 0 .0939

or 9.39% per year.The semiannual yield to maturity of the 15 year 12% Treasury bond is calculated as:

123.14 = 6af r30 + 100 df r30

which, when solved using trial-and-error and interpolation, gives r = 0 .045680 semiannually or an annualrate of 9.3446%.

The 12% coupon bond is yielding a little less than the Treasury strip. The interest rate risk of the 12%coupon bond is also a little bit higher because the coupon income has to be reinvested and if the rates godown the coupons will earn a lower rate.

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6.10 Treasury Bill The bid and ask prices can be calculated using the corresponding discounts using the equation:

p = 100 100nd

360

So that the bid and ask prices are 98.8065 and 98.81078, respectively. The 77-day ask yield is (100 98.81078)/ 98.81078 = 0 .012035. Therefore, the annual yield as reported in The Wall Street Journal is0.012035 (365/ 77) = 0 .0571 or 5.71%.The effective annual yield is (1 + 0 .012035)365 / 77 1 = 0 .058347 or 5.83% per year.Suppose the continuously compounded annual rate is r , then:

98.81078 = 100/e (77 / 365) r r = 0 .0567

or 5.67% per year.

6.11 Yield of a Bond Portfolio

(a) The semiannual yield to maturity for bonds are calculated as:

1050 = 80af r A6 + 1000 df r A6

940 = 60af r A4 + 1000 df r A4

which give rA = 0 .069525 and r B = 0 .078036.

(b) Cost of 3 A bonds is 31050 = 2100 and cost of 3 B bonds is 3 940 = 2820. The portfolio, therefore, willcost 2100 + 2820 = 4920. The fraction of money invested in bonds A and B are 2100 / 4920 = 0 .426829and 2820/ 4920 = 0 .573171. The weighted average yield of the portfolio, therefore, is 0 .426829rA +0.573171r B = 0 .426829 0.069525 + 0 .573171 0.078036 = 0 .074403 or 7.44% semiannually.

(c) The coupon income from a portfolio of 2 A bonds and 3 B bonds would be 2

80+3

60 = 340 for the

rst two years and 2 80 = $160 for the third year. The face value of A will be received 3 years fromnow and that of B, two years from now. The cashows are shown below on a semiannual time line:4920 340 340 340 3340 160 2160

0 1 2 3 4 5 6

The yield to maturity, r p, from these cashows is calculated using the following equation:

4920 = 340af r p3 + 3340 df r p4 + 160 df

r p5 + 2160 df

r p6

which, when solved, gives r p = 0 .073768 or 7.3768% semiannually.

6.12 Interest Rate Sensitivity

(a) The semiannual yield to maturity r can be solved using the following equation:

978 = 25af r16 + 1000 df r16

which gives r = 0 .026708. The annual yield to maturity, therefore, is (1 + 0 .026708)2 1 = 0 .054129 or5.4129% per year.

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(b) With a 1% increase, the annual yield will be 6.4129%. The semiannual yield will be (1+0 .064129)0.51 =0.031566. The price of the bond can now be calculated as:PV = 25 af r

16+ 1000 df r

16

where r = 0 .031566. Upon calculation, we get PV = 918 .5015. So the bond price drops from $978 to$918.5015 because of a 1% increase in yield. This is a 6.084% price in drop. Therefore, a 1% increase ininterest rate results in a 6.084% price drop for this bond.

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7.2 Option Pricing P s = 58 1/ 4, E = 55, P c = 3 7/ 8. Intrinsic value = Max[ P s E, 0] = Max[58 1/ 455, 0] = 3.25. Time value =P c Intrinsic value = 3 .875 3.25 = 0 .625If the stock price were $60, the options intrinsic value would be 60 55 = 5. The time value should stillbe about 0.625. It may be a little bit less, because the higher the price, the lesser the chance of price going

even higher. So the option price would be just slightly below $5.625.If the option were expiring on April 20, it would have zero or negligible time value. Therefore, the option

price would be $3.875.7.3 Payoff Diagrams

The payoff diagram is shown below:

100 110 120 P s

10

Payof f

............................................................................................................................................................................................................................... ...........................................................

...................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................

The cost of this combination is 7 3/ 8 + 1/ 8 2(1 3/ 8) = 4.75. The combination allows an investor to makeprot if the stock price does not move much.7.4 Put-call Parity

The put-call parity relationship is:

P s + P p P c =E

(1 + r f )T .

We are given P c = 6 .75, E = 40, r f = 0 .08 per year, T = 0 .5 years, P s = 45. So:

45 + P p

6.75 =

40

(1.08)0.5

which gives P p = 0 .24. The price of the put option should be $0.24.7.5 Binomial Option Pricing

Consider a portfolio that is long 1 share and short m puts with $45 exercise price. The payoff on thisportfolio, if the future stock price is $48, would be $48 because the option would be worthless. If the futurestock price is $34, the portfolio would be worth 34 11m. For the payoff to be free of risk, it should besame regardless of the stock price. So 48 = 34 11m which gives us m = 1.2727. So the portfolio shouldactually be long 1.2727 puts. The payoff on the portfolio will be $48, regardless of the stock price. Thepresent value of the portfolio, therefore, should be 48 / (1 + 0 .06)0.5 = 46 .622. The cost of the portfolio is40 + 1 .2727P p should be equal 46.622. From that condition we get P p = 5 .203.7.6 Black-Scholes Formula

First we calculate d1 and d2:

d1 =ln( P sE ) + r f T

T +12

T = ln(3020 ) + (0 .08)(0 .25)

0.10 0.25 + (0 .5)0.10 0.25 = 8 .51

d1 =ln( P sE ) + r f T

T 12

T = ln(3020 ) + (0 .08)(0 .25)

0.10 0.25 (0.5)0.10 0.25 = 8 .46So that, N (d1) = N (8.51) = 1 .00, N (d2) = N (8.46) = 1 .00. Finally:

P c = P s N (d1) Ee r f T N (d2) = (30)(1 .00) (20)e(

0.08)(0 .25) (1.00) = 10 .39

Solved Ravi Shukla

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