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Problem set 1 Q1. Consider the three stocks in the following table. Pt represents price at time t, and Qt represents shares outstanding at time t. Stock C splits two for one in the last period. P0 A 90 B 45 C 80

Q0 425 450 650

P1 95 40 90

Q1 425 450 650

P2 95 40 45

Q2 425 450 1,300

a. Calculate the rate of return on a price-weighted index of the three stocks for the first period (t = 0 to t = 1). (Round your answer to 2 decimal places.) b. Calculate the new divisor for the price-weighted index in year 2. (Round your answer to 2 decimal places.) c. Calculate the rate of return for the second period (t = 1 to t = 2). d. Calculate the first-period rates of return on the following indexes of the three stocks (t = 0 to t = 1): (Do not round intermediate calculations. Round your answers to 2 decimal places.) i. A market-value-weighted index. ii. An equally weighted index.

Solution: a. At t = 0, the value of the index is: (90 + 45 + 80)/3 = 71.667 At t = 1, the value of the index is: (95 + 40 + 90)/3 = 75 The rate of return is: (75/71.667) – 1 = 4.65% b. In the absence of a split, Stock C would sell for 90, so the value of the index would be: (95 + 40 + 45 + 45)/3 =225/3 = 75 with a divisor of 3. After the split, stock C sells for 45. Therefore, we need to find the divisor (d) such that: 75 = (95 + 40 + 45)/d⇒⇒d = 2.400. The divisor fell, which is always the case after one of the firms in an index splits its shares. c. The return is zero. The index remains unchanged because the return for each stock separately equals zero. d. i. Total market value at t = 0 is: ($90*425 + $45*450 + $80*650) = $110,500 Total market value at t = 1 is: ($95*425 + $40*450 + $90*650) = $116,875 Rate of return = ($116,875/$110,500) – 1 = 5.77%

d. ii. The return on each stock is as follows: rA = (95/90) – 1 = 0.0556 rB = (40/45) – 1 = –0.1111 rC = (90/80) – 1 = 0.1250 The equally weighted average is: [0.0556 + (−0.1111) + 0.125]/3 = 0.0231 = 2.31%.

Q2. Consider the following three stocks:

a. What is the price-weighted index constructed with the three stocks? b. If the market prices of each of the 3 stocks in the price-weighted index in a) all change by the same percentage amount during a given day, which stock will have the greatest impact on the index? c. What is the value-weighted index constructed with the three stocks using a divisor of 100? d. Assume at these prices that the value-weighted index constructed with the three stocks is 490. What would the index be if stock B is split 2 for 1 and stock C 4 for 1?

Solution: a. ($40 + $70 + $10)/3 = $40. b. Higher-priced stocks affect the DJIA more than lower-priced stocks. c. The sum of the value of the three stocks divided by 100 is 490: [($40 × 200) + ($70 × 500) + ($10 × 600)]/100 = 490. d. Value-weighted indexes are not affected by stock splits.

Q3. Assume you sell short 1,000 shares of common stock at $35 per share, with initial margin at 50%. What would be your rate of return if you repurchase the stock at $25 per share? The stock paid no dividends during the period, and you did not remove any money from the account before making the offsetting transaction.

Solution: Profit on stock = ($35 – $25)(1,000) = $10,000; initial investment = ($35)(1,000)(0.5) = $17,500; return = $10,000/$17,500 = 57.14%.

Q4. You sold short 100 shares of common stock at $45 per share. The initial margin is 50%. At what stock price would you receive a margin call if the maintenance margin is 35%?

Solution: Equity = 100($45) × 1.5 = $6,750; 0.35 = ($6,750 – 100P)/100P; 35P = $6,750 – 100P; 135P = $6,750; P = $50.00

Q5. Calculate the rate of return for investors in the following funds. Do not consider taxes and transactions costs. a. The Profitability Fund had NAV per share of $17.50 on January 1, 2016. On December 31 of the same year, the fund's NAV was $19.47. Income distributions were $0.75, and the fund had capital gain distributions of $1.00. b. The Yachtsman Fund had NAV per share of $36.12 on January 1, 2016. On December 31 of the same year, the fund's NAV was $39.71. Income distributions were $0.64, and the fund had capital gain distributions of $1.13.

Solution a. R = ($19.47 – 17.50 + 0.75 + 1.00)/$17.50 = 21.26%. b. R = ($39.71 – 36.12 + 0.64 + 1.13)/$36.12 = 14.84%.

Week 1 BKM Solutions Chapter 2 Question 5. Corp. Bonds Voting rights (typically) contractual obligation Perpetual payments Accumulated dividends Fixed payments (typically) Payment preference

Preferred Stock

Common Stock Yes

Yes Yes Yes Second

Yes

Yes Yes First

Third

Question 10. a. You could buy: $5,000/$142.97 = 34.97 shares. Since it is not possible to trade in fractions of shares, you could buy 34 shares of GD. b.

Your annual dividend income would be: 34  $3.04 = $103.36

c. The price-to-earnings ratio is 15.39 and the price is $142.97. Therefore: $142.97/Earnings per share = 15.39  Earnings per share = $9.29 d. General Dynamics closed today at $142.97, which was $0.47 lower than yesterday’s price of $143.44.

Chapter 3 Question 1. Stop-loss order: allows a stock to be sold if the price falls below a predetermined level. Stoploss orders often accompany short sales. Limit sell order: sells stock when the price rises above a predetermined level. Market order: either a buy or sell order that is executed immediately at the current market price. Question 6. a.

The stock is purchased for: 300  $40 = $12,000 The amount borrowed is $4,000. Therefore, the investor put up equity, or margin, of $8,000.

b.

If the share price falls to $30, then the value of the stock falls to $9,000. By the end of the year, the amount of the loan owed to the broker grows to: $4,000  1.08 = $4,320 Therefore, the remaining margin in the investor’s account is: $9,000  $4,320 = $4,680 The percentage margin is now: $4,680/$9,000 = 0.52, or 52% Therefore, the investor will not receive a margin call.

c.

The rate of return on the investment over the year is: (Ending equity in the account  Initial equity)/Initial equity = ($4,680  $8,000)/$8,000 = 0.415, or 41.5% Alternatively, divide the initial equity investments into the change in value plus the interest payment: ($3,000 loss + $320 interest)/$8,000 = -0.415.

Question 12. a. The gain or loss on the short position is: (–1,000  ΔP) Invested funds = $15,000 Therefore: rate of return = (–1,000  ΔP)/15,000 The rate of return in each of the three scenarios is: (i) Rate of return = (–1,000  $2)/$15,000 = –0.1333, or –13.33% (ii) Rate of return = (–1,000  $0)/$15,000 = 0% (iii) Rate of return = [–1,000  (–$2)]/$15,000 = +0.1333, or +13.33% b. Total assets in the margin account equal: $20,000 (from the sale of the stock) + $15,000 (the initial margin) = $35,000 Liabilities are 500P. You will receive a margin call when: $35,000  1,000 P = 0.25  when P = $28 or higher 1,000 P

c. With a $1 dividend, the short position must now pay on the borrowed shares: ($1/share  1000 shares) = $1000. Rate of return is now: [(–1,000  ΔP) – 1,000]/15,000 (i)

Rate of return = [(–1,000  $2) – $1,000]/$15,000 = –0.2000, or –20.00%

(ii) (iii)

Rate of return = [(–1,000  $0) – $1,000]/$15,000 = –0.0667, or –6.67% Rate of return = [(–1,000)  (–$2) – $1,000]/$15,000 = +0.067, or +6.67%

Total assets are $35,000, and liabilities are (1,000P + 1,000). A margin call will be issued when: 35,000  1,000 P  1,000 = 0.25  when P = $27.2 or higher 1,000P

Chapter 4 Question 11. a.

b.

NAV 

$200,000,000  $3,000,000  $39.40 5,000,000

Premium (or discount) =

Pr ice  NAV $36  $39.40 = = –0.086, or -8.6% NAV $39.40

The fund sells at an 8.6% discount from NAV.

Question 13. a.

Start-of-year price: P0 = $12.00 × 1.02 = $12.24 End-of-year price: P1 = $12.10 × 0.93 = $11.25 Although NAV increased by $0.10, the price of the fund decreased by $0.99. Rate of return =

P1  P0  Distributions $11.25  $12.24  $1.50   0.042, or 4.2% $12.24 P0

b. An investor holding the same securities as the fund manager would have earned a rate of return based on the increase in the NAV of the portfolio:

NAV1  NAV0  Distributions $12.10  $12.00  $1.50   0.133, or 13.3% NAV0 $12.00

Chapter 26 Question 3.

There are a number of factors that make it harder to assess the performance of a hedge fund portfolio manager than a typical mutual fund manager. Some of these factors are:  Hedge funds tend to invest in more illiquid assets so that an apparent alpha may be in fact simply compensation for illiquidity.  Hedge funds’ valuation of less liquid assets is questionable.  Survivorship bias and backfill bias result in hedge fund databases that report performance only for more successful hedge funds.  Hedge funds typically have unstable risk characteristics making performance evaluation that depends on a consistent risk profile problematic.  Tail events skew the distribution of hedge fund outcomes, making it difficult to obtain a representative sample of returns over relatively short periods of time....