hw1 - solutions

Finance 421

January - April 2014 Jason Chen

Homework 1: Suggested Answers

Note: unless otherwise specified, use $1,000 as the notional amount for treasury notes and bonds, use $10,000 for treasury bills.

Part I: Multiple Choice

1. The bid price of a treasury bill is __________. A) the price at which the dealer in treasury bills is willing to sell the bill B) the price at which the dealer in treasury bills is willing to buy the bill C) greater than the ask price of the treasury bill expressed in dollar terms D) the price at which the investor can buy the treasury bill

2. A 10 year Treasury bond with an 8% coupon rate should sell for ____ a 10 year

Treasury bond with a 12% coupon rate, all else equal. A) less than B) more than C) the same as D) indeterminate

3. Which of the following most closely approximates the performance of a buy and

hold portfolio strategy? A) an equally weighted index B) a price weighted index C) a value weighted index D) weights are not a factor in this situation

Note that a price weighted index does not follow a buy-and-hold portfolio when there are stock splits.

4. Underwriting is one of the services provided by ____. A) the SEC B) investment bankers C) publicly traded companies D) FDIC

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5. According to Loughran and Ritter, initial public offerings tend to exhibit

__________ performance initially, and __________ performance over the long term.

A) bad; good B) bad; bad C) good; good D) good; bad

6. The bid-ask spread exists because of ________________. A) market inefficiencies B) poor communication

C) the need for dealers to cover expenses and make a modest profit D) all of the above

7. You purchased 100 shares of ABC common stock on margin at $50 per share. Assume your initial margin is 50% and the maintenance margin requirement is 30%. Below the stock price of __________ you would get a margin call. Assume the stock pays no dividend and ignore interest on margin. A) $35.71 B) $42.86 C) $53.57 D) $57.14

8. Assume you purchased 200 shares of XYZ common stock on margin at $80 per share

from your broker. If the initial margin is 60%, the amount you borrowed from the broker is __________. A) $4000 B) $6400 C) $9600 D) $16000

9. You purchased 300 shares of common stock on margin for $50 per share. The initial margin is 60% and the stock pays no dividend. Your rate of return would be … if you sell the stock at $40 per share. Ignore interest on margin.

A) 33% B) -33% C) -44% D) -56%

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Part II: Detailed Questions

Question 1:

Consider the three stocks in the following table. Pt represents the price at time t, and Qt represents shares outstanding (in millions) at time t.

Name Symbol P01/02/14 Q01/02/14 P12/31/13 Q12/31/13 Citigroup C 57.25 500.22 55.94 500.22 Nortel Networks NT 3.18 4,330 3.06 4,330 British Petroleum BP 48.58 262.100 46.65 262.100

a. Compute the market capitalization of the three companies on December 31, 2013.

The market capitalization is the product of the number of shares outstanding and the price:

MCC= 500.22*55.94=$ 27,982.31M MCNT= 4,330*3.06=$ 13,249.80M MCBP= 262.1*46.65=$ 12,226.97M

b. Calculate the rate of return on an equally-weighted index of the three stocks on January 2nd, 2014

The return of the three stocks is: RC=57.25/55.94-1= 2.34%

RNT=3.18/3.06-1= 3.92% RBP=48.58/46.65-1= 4.14%;

The return of the equally-weighted index is:

EWR = (2.34+3.92+4.14)/3= 3.47%

c. Calculate the rate of return on a value-weighted index of the three stocks on January 2nd, 2014

The initial total market value of the three stocks is:

27,982.31 + 13,249.80 + 12,226.97 = $53,459.07 M

The relative market values of the three companies are: WC=27,982.31/53,459.07=0.5234 WNT=13,249.80/53,459.07=0.2479 WBP=12,226.97/53,459.07=0.2287

The return of the value-weighted index is:

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VWR=0.5234*2.34+0.2479*3.92+0.2287*4.14=3.14%.

d. Calculate the rate of return on a price-weighted index of the three stocks. (Assume that the initial divisor equals 3.)

The initial index is (55.94+3.06+46.65)/3=35.22. The next trading day, it is (57.25+3.18+48.58)/3=36.34. Thus the return is;

PWR=36.34/35.22-1 = 3.18%

Suppose for the remainder of this problem that BP splits its stock 2 for 1 after the market closes on December 31st, 2013.

e. What happens to the number of shares outstanding immediately after the split? At which price would BP have traded on January 2nd if markets are efficient?

The number of shares would double to 524.2 million and the price of BP would need to halve to $24.29 if markets are efficient.

f. What must happen to the divisor for the price-weighted index after the close on December 31st, 2013? When a stock splits, the price-weighted index would change the divisor such that the index level does not change immediately before and after the stock split. See Example 2.3 (Splits and Price-Weighted Averages) in Chapter 2 of the textbook.

We need to set the divisor d such that the index does not change by the split: 35.22= (55.94+3.06+23.325)/d,

which implies a divisor of d=2.34.

g. Calculate the return of the equally-weighted, the value-weighted, and the price-weighted indices.

The equally-weighted and the value-weighted index returns are not affected by the split. On the other hand, the price-weighted index on January 2nd 2014 is (57.25+3.18+24.29)/2.34=36.21. Thus the return is;

PWR = 36.21/35.22-1 = 2.80%

Question 2:

Calculate the bank discount yield on each bill.

(1) A three-month bill selling at $9,764. (2) A six-month bill selling at $9,539.

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10,000 – 9764360

i. rBD = ? 91 = .0934 = 9.34% 10,000

ii.

10,000 – 9539360

rBD = 10,000 ? 182 = .0912 = 9.12%

Question 3:

Microsoft is currently selling at $26.84 per share. You buy 1000 shares by using $15,000 of your own money and borrowing the remainder of the purchase price from your broker. The rate on the margin loan is 8%. The account has an initial margin requirement of 50 percent and a maintenance margin of 25 percent.

a) What is the percentage increase in the net worth of your brokerage account if the price of MSFT immediately changes to: (i) $30; (ii) $25?

The purchase costs you $26,840. You borrow $11,840 from the broker and invest $15,000 of your own funds. The initial net value of the account is $26,840-$11,840=$15,000. Note that returns are always measured relative to the value you first invest in your account.

(i) The net account value increases to $30,000-$11,840=$18,160. Thus,

the return is 18,160/15,000-1=21.07%

(ii) The net account value falls to $25,000-$11,840=$13,160. Thus the

return is 13,160/15,000-1= -12.27%

b) If the maintenance margin is 25%, how low can the price of MSFT fall before you get a margin call? (You can ignore the interest expenses in the margin account.) The value of the 1,000 shares is 1000*P. Equity is 1000*P – 11,840. You will

receive a margin call when

1000P?11,840?0.25?P?$15.79.

1000P

c) How much money do you need to send your broker to satisfy the margin call if the stock price drops below the level computed in b?

You will need to add cash until your margin equals the initial margin of 50

percent. The value of the securities is $15,790 (15.79*1000) and you can borrow at most 50 percent of this (15790/2=7895). Thus, you need to pay back $3,945 (11840-7895) of the original margin loan of $11,840.

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d) What is the rate of return on your margined position if MSFT is selling after one year at: (i) $30; (ii) $25? (You can ignore the dividend payments by MSFT).

The margin loan with accumulated interest will be $11,840*1.08=$12,787.20. The equity in your account is 1000P-$12,787.2. Therefore your rates of return are:

1000?30?$12,787.2 (i) ?1?14.75%

$15,0001000?25?$12,787.2(ii) ?1??18.58%

15,000

Question 4:

\spreadsheet.

You have $20,000 to invest in the shares of Gucci Group (GUC). Let the price of its stock be $80 per share. You estimate that the stock will be selling at a price of $110 in one year. Assume that no cash dividends will be paid over the next year and that the rate on margin loans is currently 0%.

1. What would be the expected return on the investment assuming that you used the maximum allowable margin of 50%?

2. At what price would you get a margin call assuming the maintenance margin was 30%?

3. Construct two data tables that compare the return on investment for a margin trade and a trade with no margin for ending stock prices that range from $20 to $140 in increments of $10. How would you interpret the difference in returns? 4. What would be the expected return on investment if you were to use an initial margin of 80% rather than the maximum allowable margin of 50%?

5. How far could the stock price fall with an initial margin of 80% assuming the maintenance margin remains at 30%?

6. Construct two data tables that compare the return on investment for the margin trade and a trade with no margin for ending stock prices that range from $20 to $140 in increments of $10. Compare the results to the ranges in point 3.

7. Show how your results for point 3 would change if the rate on margin loans was 7%, instead of 0%. This is a more realistic assumption, which you may face in reality. Hint: the rate on margin loans is the cost for the investor, so it should probably decrease his/her income from investment. 1) Since the initial margin is 0.5 we can calculate the amount of loan that can be used to lever up your investment.

Margin = (Value of Investment - Loan)/Value of Investment

Or 0.5 = 1 - Loan/Value of Investment

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Since Value of Investment = Loan + 20,000 we have the following:

0.5 = 1 - Loan/(Loan + 20,000) => Loan = $20,000

It means that an investor uses $20,000 from his account and $20,000 from the loan.

Since the total investment is $40,000, he can buy 40,000/80 = 500 of the stocks.

Therefore, the return on his investment is:

Return = (500*110-500*80)/20000 = 75%

2) In order to find the price you have to solve the following inequality:

(500*P-20000)/500*P <0.3 0.7*500*P < 20000

Hence, P < $57.14

3) The table below provides you with the returns calculated the way as in point a)

Return on

Ending Investment Ending Return with Stock Price with margin Stock Price no margin

20 -150.00% 20 -75.00% 30 -125.00% 30 -62.50% 40 -100.00% 40 -50.00% 50 -75.00% 50 -37.50% 60 -50.00% 60 -25.00% 70 -25.00% 70 -12.50% 80 0.00% 80 0.00% 90 25.00% 90 12.50% 100 50.00% 100 25.00% 110 75.00% 110 37.50% 120 100.00% 120 50.00% 130 125.00% 130 62.50% 140 150.00% 140 75.00%

4) To calculate the return with margin 0.8 you have to calculate the loan as in point a) and then appropriate return on your investment.

Margin = (Value of Investment - Loan)/Value of Investment

Or 0.8 = 1 - Loan/Value of Investment

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Since Value of Investment = Loan + 20,000 we have the following:

0.8 = 1 - Loan/(Loan + 20,000) => Loan = $5,000

It means that an investor uses $20,000 from his account and $5,000 from the loan.

Since the total investment is $25,000, he can buy 25,000/80 = 312.5 of the stocks.

Therefore, the return on his investment is:

Return = (312.5*110-312.5*80)/20000 = 46.88%

5) Similar to b) we can calculate the price below which we would get a margin call.

(312.5*P-5000)/312.5*P <0.3 0.7*312.5*P < 5000

Hence, P < $22.86

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6) The table with the returns is depicted below:

Ending Return on Ending Return with St Price Investment St Price No Margin

20 -93.75% 20 -75.00% 30 -78.13% 30 -62.50% 40 -62.50% 40 -50.00% 50 -46.88% 50 -37.50% 60 -31.25% 60 -25.00% 70 -15.63% 70 -12.50% 80 0.00% 80 0.00% 90 15.63% 90 12.50% 100 31.25% 100 25.00% 110 46.88% 110 37.50% 120 62.50% 120 50.00% 130 78.13% 130 62.50% 140 93.75% 140 75.00%

Since we are using much lower amounts of leverage with the initial margin at 80%, return on investment is much less volatile that when using 50% margin.

7) In order to calculate the returns on your position, which accounts for the interest on the loan we assume that after one year we return only the interest on the loan but not the entire principal.

Hence, given 7% interest rate the return on our investment can be calculated as follows: Return = (110*500-80*500-7%*20,000)/20,000 = (15,000-1400)/20,000 = 13,600/20,000 = 68%

Similar to b) we can calculate the price below which we would get a margin call. (500*P- 20,000) /500*P <0.3 0.7*500*P < 20,000

Hence, P < $57.14, which is exactly the same as the price derived in b), because the interest on loan is not included in the calculation of margin. Finally, the table with the returns is included below:

Ending Return on Ending Return with St Price Investment St Price No Margin

20 -157.00% 20 -75.00% 30 -132.00% 30 -62.50% 40 -107.00% 40 -50.00% 50 -82.00% 50 -37.50% 60 -57.00% 60 -25.00% 70 -32.00% 70 -12.50% 80 -7.00% 80 0.00% 90 18.00% 90 12.50% 100 43.00% 100 25.00% 110 68.00% 110 37.50% 120 93.00% 120 50.00% 130 118.00% 130 62.50% 140 143.00% 140 75.00%

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