454 Investments McGraw-Hill Q&A Ch-Q#-letter

b. Which stock has greater dispersion around the mean return? 18-1-b

XYZ

Consider the rate of return of stocks ABC and XYZ. Year rABC rXYZ 1 22% 36% 2 10 10 3 19 17 4 3 0 5 1 −8 a. Calculate the arithmetic average return on these stocks over the sample period. 18-1-a

Arithmetic average: r¯r¯ ABC = 11.00% r¯r¯ XYZ = 11.00%

What is the dollar-weighted rate of return? (Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2017, to January 1, 2020. If your calculator cannot calculate IRR, you will have to use trial and error or a spreadsheet program.) 18-2-b

Date Cash Flow Explanation 1/1/2017 −$570 Purchase of three shares at $190 each 1/1/2018 −$385 Purchase of two shares at $200 less dividend income on three shares held 1/1/2019 $205 Dividends on five shares plus sale of one share at $180 1/1/2020 $780 Dividends on four shares plus sale of four shares at $190 each ________________________________ Dollar-weighted return = Internal rate of return = −1.303% (CF 0 = −$570; CF 1 = −$385; CF 2 = $205; CF 3 = $780; Solve for IRR = −1.303%.)

a. Calculate the dollar-weighted average return on this portfolio. 18-3-c

Dollar-weighted average rate of return = IRR = 2.68%[Using a financial calculator, enter: n = 3, PV = -555, FV = 0, PMT = 195. Then compute the interest rate, or use the CF0 = -585, CF1 = 195, F1 = 3, then compute IRR]. The IRR exceeds the other averages because the investment fund was the largest when the highest return occurred.

What must be the beta of a portfolio with E(rP) = 12.70%, if rf = 5% and E(rM) = 12%? 7-3-a

E(rP) = rf + β[E(rM) - rf] Given rf = 5% and E(rM) = 12%, we can calculate β: 12.70% = 5% + β(12% - 5%) ⇒⇒ β = 1.10

The following price quotations are for exchange-listed options on Primo Corporation common stock. Company Strike Expiration Call Put Primo 61.12 54 Feb 7.36 0.47 With transaction costs ignored, how much would a buyer have to pay for one call option contract. Assume each contract is for 100 shares.

Each contract is for 100 shares: $7.36 × 100 = $736

We will derive a two-state call option value in this problem. Data: S0 = 260; X = 270; 1 + r = 1.1. The two possibilities for ST are 300 and 180. a. The range of S is 120 while that of C is 30 across the two states. What is the hedge ratio of the call? b. Calculate the value of a call option on the stock with an exercise price of 270. (Do not use continuous compounding to calculate the present value of X in this example, because the interest rate is quoted as an effective per-period rate.) 16-10-a&b

Explanation a) The hedge ratio for the call is: H = Cu - Cd = 30 - 0 = 0.25 uS0 - dS0 300 -180 b) Riskless Portfolio S = 180 S = 300 1 shares 180 300 Short 4 calls 0 -120 Total 180 180 Present value = $180/1.10 = $163.636Portfolio cost = 1S - 4C = 260 - 4C = $163.636 ⇒⇒ C = $24.09Put-call parity relationship: P = C - S0 + PV(X)$24.09 = $24.09 - $260 + ($270/1.1) = $24.09

a. If, in a two-state model, a stock can take a price of 184 or 138, what would be the hedge ratio for each of the following prices: (a) $184, (b) $180, (c) $170, (d) $138? 16-4-a

H = Cu − Cd / uS0 − dS0 uS0 − dS0 = 184 − 138 = 46 X Cu − Cd Hedge Ratio $184 0 − 0 0/46 = 0.00 $180 4 − 0 4/46 = 0.09 $170 14 − 0 14/46 = 0.30 $138 46 − 0 46/46 = 1.00 Note that, as the option becomes progressively more in the money, its hedge ratio increases to a maximum of 1.0.

You establish a straddle on Walmart using September call and put options with a strike price of $81. The call premium is $7.05 and the put premium is $7.80. a. What is the most you can lose on this position? b. What will be your profit or loss if Walmart is selling for $90 in September? c. At what stock prices will you break even on the straddle? 15-10-a,b,c

Explanation a) Maximum loss happens when the stock price is the same to the strike price upon expiration. Both the call and the put expire worthless, and the investor's outlay for the purchase of both options is lost: $7.05 + $7.80 = $14.85 b) Loss: Final value - Original investment = (ST − X) − (C + P) = $9 − $14.85 = −$5.85 c) There are two break even prices: ST > X(ST − X) − (C + P) = (ST − 81) − $14.85 = $0 ⇒⇒ ST = $95.85 ST < X(X - ST) − (C + P) = (81 − ST) − $14.85 = $0 ⇒⇒ ST = $66.15

The market price of a security is $52. Its expected rate of return is 9%. The risk-free rate is 5%, and the market risk premium is 7%. What will the market price of the security be if its beta doubles (and all other variables remain unchanged)? Assume the stock is expected to pay a constant dividend in perpetuity. 7-4-a

If the beta of the security doubles, then so will its risk premium. The current risk premium for the stock is: (9% - 5%) = 4%, so the new risk premium would be 8%, and the new discount rate for the security would be: 8% + 5% = 13% If the stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity: Price = Dividend/Discount rate 52 = D/0.09 ⇒⇒ D = 52 × 0.09 = $4.68 At the new discount rate of 13%, the stock would be worth: $4.68/0.13 = $36.00 The increase in stock risk has lowered the value of the stock by 30.77%.

You purchase one Microsoft June 74 put contract for a premium of $2.37. What is your maximum possible profit?

If the stock price drops to zero, you will make $74 − $2.37 per stock, or $71.63. Given 100 units per contract, the total potential profit is $7,163.

A call option with a strike price of $86 on a stock selling at $109 costs $25.4. What are the call option's intrinsic and time values? 16-1-a

Intrinsic value = S0 − X = $109 − $86 = $23.00Time value = C − Intrinsic value = $25.4 − $23.00 = $2.40

Given that the stock currently is selling at 130, calculate the put value. 16-6-d

Portfolio cost = 3S + 5P = $390+ 5P = $463.636Therefore 5P = $73.636 ⇒⇒ P = $73.636/5 = $14.73

What is the present value of the portfolio? 16-6-c

Present value = $510/1.10 = $463.636

b. Form a portfolio of 3 shares of stock and 5 puts. What is the (nonrandom) payoff to this portfolio? 16-6-b

Riskless port. St = 95 3 shares 285 5 puts 225 Total 510

You are attempting to value a call option with an exercise price of $120 and one year to expiration. The underlying stock pays no dividends, its current price is $120, and you believe it has a 50% chance of increasing to $150 and a 50% chance of decreasing to $90. The risk-free rate of interest is 6%. Based upon your assumptions, calculate your estimate of the the call option's value using the two-state stock price model. 16-9-a

Step 1: Calculate the option value at expiration based upon your assumption of a 50% chance of increasing to $150 and a 50% chance of decreasing to $90. The two possible stock prices are: S+ = $150 and S- = $90. Therefore, since the exercise price is $120, the corresponding two possible call values are: Cu = $30 and Cd = $0. Step 2: Calculate the hedge ratio: (Cu - Cd)/(uS0 - dS0) = (30 - 0)/(150 - 90) = 0.50 Step 3: Form a riskless portfolio made up of one share of stock and two written calls. The cost of the riskless portfolio is: (S0 - 2C0) = 120 - 2C0 and the certain end-of-year value is $90. Step 4: Calculate the present value of $90 with a one-year interest rate of 6%: $90/1.06 = $84.91 Step 5: Set the value of the hedged position equal to the present value of the certain payoff: $120 - 2C0 = $84.91 Step 6: Solve for the value of the call: C0 = $17.55 Notice that we never use the probabilities of a stock price increase or decrease. These are not needed to value the call option.

Consider the two (excess return) index-model regression results for stocks A and B. The risk-free rate over the period was 7%, and the market's average return was 13%. Performance is measured using an index model regression on excess returns. Stock A Stock B Index model regression est.: 1% + 1.2(rM − rf) 2% + 0.8(rM − rf) R-square 0.629 0.463 Residual st.dev., σ(e) 11.2% 20% St.dev. of excess returns 22.5% 26.7% a. Calculate the following statistics for each stock: 18-4-a

Stock A Stock B (i) Alpha = regression intercept 1.0000% 2.0000% (ii)Information ratio = 0.0893 0.1000 (iii)*Sharpe measure = 0.3644 0.2547 rp −rfσprp/ −rfσp (iv)†Treynor measure = 6.8333 8.5000 rp −rfβprp/ −rfβp * To compute the Sharpe measure, note that for each stock, (rP − rf ) can be computed from the right-hand side of the regression equation, using the assumed parameters rM = 13% and rf = 7%. The standard deviation of each stock's returns is given in the problem. † The beta to use for the Treynor measure is the slope coefficient of the regression equation presented in the problem.

Assume both portfolios A and B are well diversified, that E(rA) = 14.8% and E(rB) = 15.8%. If the economy has only one factor, and βA = 1 while βB = 1.1, what must be the risk-free rate? 7-7-a

Substituting the portfolio returns and betas in the mean-beta relationship, we obtain two equations in the unknowns, the risk-free rate (rf) and the factor return (F): 14.8% = rf + 1 × (F - rf)15.8% = rf + 1.1 × (F - rf) From the first equation we find that F = 14.8%. Substituting this value for F into the second equation, we get: 15.8% = rf + 1.1 × (14.8% - rf) ⇒⇒ rf = 4.8%

A call option on Jupiter Motors stock with an exercise price of $95 and one-year expiration is selling at $5. A put option on Jupiter stock with an exercise price of $95 and one-year expiration is selling at $3.5. If the risk-free rate is 7% and Jupiter pays no dividends, what should the stock price be? 16-3-a

Using put-call parity: Put = C − S0 + PV(X) +PV(Dividends) $3.5 = $5 − S0 + $95/(1 + 0.07) + 0 ⇒⇒ S0 = $90.29

A collar is established by buying a share of stock for $54, buying a six-month put option with exercise price $50, and writing a six-month call option with exercise price $60. Based on the volatility of the stock, you calculate that for an exercise price of $50 and maturity of six months, N(d1) = 0.7001, whereas for the exercise price of $60, N(d1) = 0.6340. What will be the gain or loss on the collar if the stock price increases by $1? 16-8-a

The delta of the collar is calculated as follows: Delta Stock 1.00 Short call -N(d1) = -0.634 Long put N(d1) - 1 = -0.300 Total 0.066 If the stock price increases by $1, the value of the collar increases by $0.066. The stock will be worth $1 more, the loss on the short put is $0.300, and the call written is a liability that increases by $0.6340.

The hedge ratio (delta) of an at-the-money call option on IBM is 0.31. The hedge ratio of an at-the-money put option is −0.40. What is the hedge ratio of an at-the-money straddle position on IBM? 16-7-a

The hedge ratio of the straddle is the sum of the hedge ratios for the two options: 0.31 + −(0.40) = −0.09

An investor buys a call at a price of $6.50 with an exercise price of $60. At what stock price will the investor break even on the purchase of the call?

The price has to be at least as much as the sum of the exercise price and the premium of the option to break even: $60 + $6.50 = $66.50

Suppose two factors are identified for the U.S. economy: the growth rate of industrial production, IP, and the inflation rate, IR. IP is expected to be 6% and IR 6%. A stock with a beta of 1 on IP and 0.5 on IR currently is expected to provide a rate of return of 10%. If industrial production actually grows by 7%, while the inflation rate turns out to be 8%, what is your best guess for the rate of return on the stock? 7-9-a

The revised estimate of the expected rate of return of the stock would be the old estimate plus the sum of the unexpected changes in the factors times the sensitivity coefficients, as follows: Revised estimate = 10% + [(1 × 1%) + (0.5 × 2%)] = 12.0%

A put option on a stock with a current price of $46 has an exercise price of $48. The price of the corresponding call option is $4.20. According to put-call parity, if the effective annual risk-free rate of interest is 4% and there are three months until expiration, what should be the price of the put? 16-2-a

Using put-call parity: Put =C − S0 + PV(X) + PV(Dividends) =$4.20 − $46 + $48/(1 + 0.04)3/12 + 0 = $5.73

2XYZ's stock price and dividend history are as follows: Year Beg$ Dividend Paid 2017 $190 $5 2018 200 5 2019 180 5 2020 190 5 An investor buys three shares of XYZ at the beginning of 2017, buys another two shares at the beginning of 2018, sells one share at the beginning of 2019, and sells all four remaining shares at the beginning of 2020. a. What are the arithmetic and geometric average time-weighted rates of return for the investor? 18-2-a

Time-weighted average returns are based on year-by-year rates of return: Year Return = (Capital gain + Dividend)/Price 2017 − 2018 [($200 - $190) + $5]/$190 = 7.89% 2018 − 2019 [($180 - $200) + $5]/$200 = -7.5% 2019 − 2020 [($190 - $180) + $5]/$180 = 8.33% Arithmetic mean: (7.89% - 7.5% + 8.33%)/3 = 2.91% Geometric mean: (1.0789 × 0.925 × 1.0833)1/3 - 1 = 0.0264 = 2.64%

A nine-year bond has a yield of 10% and a duration of 7.207 years. If the bond's yield increases by 40 basis points, what is the percentage change in the bond's price as predicted by the duration formula? 11-2-a

Using Equation 11.2, the percentage change in the bond price is: ΔP = -Duration × Δy = -7.207 × 0.0040 = -0.0262 or a 2.62% decl

Consider a 1-year option with exercise price $95 on a stock with annual standard deviation 10%. The T-bill rate is 3% per year. Find N(d1) for stock prices (a) $90, (b) $95, and (c) $100. 16-5-a

We first calculate d1 = ln(S0/X) + (r − δ + σ2/2)TσT√ln(S0/X) + (r⁢ − δ⁢ + σ2/2)TσT , and then find N(d1), which is the Black Scholes hedge ratio for the call. We can observe from the following that when the stock price increases, N(d1) increases as well. X 95 r 3% σ 10% T 1 S d1 N(d1) $90 −0.1907 0.4244 $95 0.3500 0.6368 $100 0.8629 0.8059 = normsdist(d1)

We will derive a two-state put option value in this problem. Data: S0 = 130; X = 140; 1 + r = 1.1. The two possibilities for ST are 170 and 95.a. The range of S is 75 while that of P is 45 across the two states. What is the hedge ratio of the put? a. The range of S is 75 while that of P is 45 across the two states. What is the hedge ratio of the put? 16-6-a

When ST = $170, then P = 0.When ST = $95, then P = 45. hedge ratio is: Pu - Pd / uS0 - dS0 = H 0 -45 / 170 -95 = -0.60

d. If you were equally likely to earn a return of 22%, 10%, 19%, 3%, or 1%, in each year (these are the five annual returns for stock ABC), what would be your expected rate of return? 18-1-d

Your expected rate of return would be the arithmetic average, or 11.00%.

c. Calculate the geometric average returns of each stock. What do you conclude? 18-1-c

rABC = (1.22 × 1.1 × 1.19 × 1.03 × 1.01)1/5 - 1 = 0.1069 = 10.69% rXYZ = (1.36 × 1.1 × 1.17 × 1 × 0.92)1/5 - 1 = 0.1 = 10% Despite the fact that the two stocks have the same arithmetic average, the geometric average for XYZ is less than the geometric average for ABC. The reason for this result is the fact that the greater variance of XYZ drives the geometric average further below the arithmetic average.