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WEEK 10 - CHAPTER 22 PROBLEM 3(a-d) Assuming that a one-year call option with an exercise price of $38 is available for the stock of the DEW Corp., consider the following price tree for DEW stock over the next year: Now S1 S2 One Year 46.31 44.10 42.00 40.00 42.34 40.32 38.40 38.71 36.86 35.39 a. If the sequence of stock prices that DEW stock follows over the year is $40.00, $42.00, $40.32, and $38.71, describe the composition of the initial riskless portfolio of stock and options you wuold form and all the subsequent adjustments you would have to make to keep this portfolio riskless. Assume the one-year risk-free rate is 6 percent. Ans I would want to have a riskless portfolio, which inevitably will be in the cash sort of position. Here as it could be watched that, the danger free rate is given as 6%. For every situation, the danger free rate to be added to the price of the stock and need to the guaranteed that whether the portfolio is riskless. It should be taken a gander at in a path that, at any rate the portfolio is in the method of no benefit and no misfortune. b. Given the initial DEW price of $40, what are the probabilities of observing each of the four terminal stock prices in one year? (Hint: In arriving at your answer, it will be useful to consider (1) the number of different ways that a particular terminal price could be achieved and (2) the probability of an up or down movement.) Ans Ending Price Number of Paths Path Probability Total Probability 46.31 1 0.2242016719 0.2242 42.34 3 0.1448545781 0.4346 38.71 3 0.0935891719 0.2808 35.39 1 0.0604670781 0.0604 c. Use the binominal option model to calculate the present value of this call option. Ans 6623 (8.31) + (3) (.6622) (1-.662) (4.34) + (3) (.662) (1-.662)2 (.71) 1.06 4.2452830189 4.24 d. Calculate the value of a one-year put option on DEW stock having an exercise price of $38; be sure your answer is consistent with the correct response to Part c. Ans P = (1- .662)3 (2.61) 1.06 0.1008/1.06 0.95 Check with put call parity. Does the following hold true C-P=S-PV (K) 4.24-.095=40-(38/1.06) 4.15 = 4.15 Put -call parity does hold exactly. GMBA 767-Security Analysis and Portfolio Management ARB Inc. Consider the following questions on the pricing of options on the stock of ARB Inc.: a. A share of ARB stock sells for $75 and has a standard deviation of returns equal to 20 percent per year. The current risk-free rate is 9 percent and the stock pays two dividends: (1) a $2 dividend just prior to the option's expiration day, which is 91 days from now (ie, exactly one-quarter of a year), and (2) a $2 dividend 182 days from now (ie., exactly one-half year). Calculate the Black-Shoules value for a European -style call option with an exercise price of $70. Ans: Current Security price = S1 = $ 75.00 Share value = S2 = $ $ 73.04 70.00 Exercise price = X = Time to expiration 1 = T1 S2 = Cum dividend value - present value of dividend 91 0.25 Time to expiration 2 = T2 0.50 182 0.5 Risk-free rate = RFR Security price volitility = = S2 = 75 - 2/en = 75 - 2/e 0.09*3/12 = 75 - 1.96 = 73.04 0.71 9% 0.20 Black-Shoules value for a European call option 1) d1 = ln(S2/D) + (r + 2/2)T T d1 = 0.20.25 d1 = ln(1.0434) + (0.09 + 0.2 /2)0.25 0.20.25 2 d1 = .0424 + 0.0275 0.10 N(d) = 0.5 + (d)(4.4-d)/10 V0 = SN(d1)-Ke-rT N(d2) N(d1) = 0.5 + (d1)(4.4-d1)/10 ln(73.04/70) + (0.09 + 0.22/2)0.25 V0 = 73.04*0.7587 - 70*0.98*0.7277 N(d1) = N(d1) = V0 = 5.495 N(d2) = 0.5 + (d2)(4.4-d2)/10 d1 = 0.699 N(d2) = N(d2) = d2 = 0.5 + (.699)(4.4 - 0.699)/10 0.7587 0.5 + (.599)(4.4 - 0.599)/10 0.7277 d1 - T d2 = 0.699 - 0.2*0.25 d2 = 0.599 b. What would be the price of a 91-day European-style put option on ARB stock having the same exercise price? Ans: S + P - C = PV of E 73.04 + P - 5.495 = PV of 70 P = 68.51 + 5.495 - 73.04 P= 0.965 c. Calculate the change in a call option's value that would occur if ARB's management suddenly decided to suspend dividend payments and this action had no effect on the price of the company's stock. Ans: d1 = ln(S1/D) + (r + 2/2)T V0 = SN(d1)-Ke-rT N(d2) N(d) = 0.5 + (d)(4.4-d)/10 N(d1) = 0.5 + (d1)(4.4-d1)/10 d1 = ln(75/70) + (.09 + .022/2)0.25 0.20.25 V0 = 75.00*0.8314 - 70*0.98*0.8057 N(d1) = N(d1) = T V0 = 7.084 0.5 + (.9647)(4.4 - 0.9647)/10 0.8314 d1 = ln(1.0714) + (0.09 + 0.22/2)0.25 d1 = d1 = Vo with dividends = 5.495 N(d2) = N(d2) = 0.6897 + 0.0275 0.10 7.084 N(d2) = 0.5 + (d2)(4.4-d2)/10 0.20.25 Vo without dividends = Change in value of option = 1.589 0.5 + (.8647)(4.4 - 0.8647)/10 0.8057 0.9647 d2 = d1 - T d2 = 0.9647 - 0.2*0.25 d2 = 0.8647 d. Briefly describe (without calculations) how your answer in Part a would differ under the following separate circumstances: (1) volitility of ARB stock increases to 30 percent, (2) the riskfree rate decreases to 8 percent. Ans: 1) Security price volitility = = d1 = 0.30 ln(S2/D) + (r + 2/2)T N(d) = 0.5 + (d)(4.4-d)/10 N(d1) = 0.5 + (d1)(4.4-d1)/10 T d1 = ln(73.04/70) + (0.09 + 0.3 /2)0.25 0.300.25 d1 = ln(1.0434) + (0.09 + 0.32/2)0.25 0.300.25 2 d1 = ln(1.0434 )+ 0.3375 0.15 d1 = 0.0424 + 0.03375 V0 = S2N(d1)-Ke-rT N(d2) V0 = 73.04*0.6976 - 70*0.98*0.6446 N(d1) = N(d1) = V0 = 6.7331 0.5 + (.5077)(4.4 - 0.5077)/10 0.6976 N(d2) = 0.5 + (d2)(4.4-d2)/10 N(d2) = N(d2) = 0.5 + (.3577)(4.4 - 0.3577)/10 0.6446 0.15 d1 = 0.5076666667 d2 = d1 - T d2 = 0.5077 - 0.3*0.25 d2 = 0.3576666667 2) Change in RFR = 8% 0.2 d1 = ln(S2/D) + (r + 2/2)T T N(d) = 0.5 + (d)(4.4-d)/10 V0 = S2N(d1)-Ke-rT N(d2) N(d1) = 0.5 + (d1)(4.4-d1)/10 V0 = 73.04*0.7511 - 70*0.98*0.7196 N(d1) = N(d1) = V0 = 5.4958 d1 = ln(73.04/70) + (0.08 + 0.32/2)0.25 0.200.25 d1 = ln(1.0434) + (0.08 + 0.32/2)0.25 0.200.25 N(d2) = 0.5 + (d2)(4.4-d2)/10 d1 = ln(1.0434 )+ 0.025 0.1 N(d2) = N(d2) = d1 = 0.0424 + 0.025 0.1 d1 = 0.674 d2 = d1 - T d2 = 0.674 - 0.2*0.25 d2 = 0.574 0.5 + (.674)(4.4 - 0.674)/10 0.7511 0.5 + (.574)(4.4 - 0.574)/10 0.7196 7 Suppose the current value of a popular stock index is 653.50 and the dividend yield on the index is 2.8%. Also, the yield curve is flat at a continuously compounded rate of 5.5%. a. If you estimate the volatility factor for the index to be 16%, calculate the value of an index call option with an exercise price of 670 and an expiration date in exactly three months. Ans: current value without dividend = 653.5 x 100/97.2 672.33 672.33 Compounded at rate 5.5% gives 670 One standard move = 670 x 16% x (90/360) = $26.8 Volatility lies between 643.2 and $696.8 b. If the actual market price of this option is $17.40, calculate its implied volatility coefficient. Ans: One standard move = $ 17.5 x 16% x (90/360) = $0.7 Volatility will be in the range of $16.8 to 18.2 c. Besides volatility estimation error, explain why your valuation and the option's traded price might differ from one another. Ans: The distinction is as a consequence of changes in the time that might influence the one standard move of the given price of choices. Solution: (a) Buying Call option @ $9 and Put option @ $8, both with strike price of $100 Share Price after 1 year 157 152 147 142 137 132 127 122 117 110 105 100 95 90 83 78 73 68 63 58 53 48 43 Gain / (Loss) 40 35 30 25 20 15 10 5 0 Break-even Point (7) (12) (17) Maximum Loss (12) (7) 0 Break-even Point 5 10 15 20 25 30 35 40 Solution (a) 50 40 40 40 35 35 30 30 30 25 25 20 20 20 15 15 10 10 10 Gain / (Loss) 5 5 0 15; 0 20 40 60 Bre ak-e ve n point at $ 83 and $ 117 9; 0 80 100 120 140 160 180 (7) (10) (7) (12) 12; (17) (12) (20) Share Price After 1 year Solution: (b) Buying Call option @ $6 with strike price of $110 and Put option @ $5 with strike price of $90 Share Price after 1 year Gain / (Loss) 19 60 39 40 59 20 69 10 74 5 79 0 Break-even Point 85 (6) 90 (11) Maximum Loss 100 (11) Maximum Loss 110 (11) Maximum Loss 115 (6) 121 0 Break-even Point 126 5 131 10 141 20 161 40 181 60 Solution (b) 70 60 60 50 40 40 40 30 20 20 Gain / (Loss) 10 10 0 (10) 20 10 5 5 6; 0 0 20 40 60 80 12; 0 (6) 8; (11) 100 9; (11) (6) 120 10; (11) (20) Share Price After 1 year 140 160 60 Break- even point at $ 7 9 and $ 12 1 180 200 WEEK 10 - CHAPTER 22 PROBLEM 12 In developing the butterfly spread position, we showed that it could be broken down into two call option money spreads. Using the price data for SAS sock options from Exhibit 22.17, demonstrate how a butterfly profit structure similar to that shown in Exhibit 22.30 could be created using put options. Be specific as to the contract positions involved in the trade and show the expiration date net payoffs for the combined transation. Exhibit 22.17 Hypothetical SAS Corporation Stock and Option Prices Exercise Price ($) Instrument Stock: ------ Market Price ($) $ Intrinsic Value ($) ------ 40.00 Call: #1 #2 #3 $ $ $ 35.00 40.00 45.00 $ $ $ 8.07 5.24 3.24 Put: #1 #2 #3 $ $ $ 35.00 40.00 45.00 $ $ $ 1.70 3.67 6.47 $ $ Time Premium ($) ------ 5.00 $0.00 $0.00 $ $ $ 3.07 5.24 3.24 $0.00 $0.00 5.00 $ $ $ 1.70 3.67 1.47 Exhibit 22.30 Comparing the Butterfly Spread and Short Straddle Positions Net Profit 8.91 4.17 - -0.83 35 45 Short Straddle SAS Stock Price Butterat Expiration Spread A butterfly spread can be made by utilizing put choices. Bear spread is a spread in which we purchase a put choice with a higher exercise price and offer a put choice with a lower exercise price. Offering a bear spread would comprise of offering a put choice with a higher exercise price and purchasing a put choice with a lower exercise price. In the above case, to develop a butterfly spread utilizing puts, we would be purchasing a bear spread with puts comprising of purchasing the put with exercise price of $45 and offering the put with exercise price of $40 furthermore selling so as to offer a bear spread the put with exercise price of $40 and purchasing the put with exercise price of $35 Value at the beginning (V0) = P3 -2P2 + P1 = 6.47 - 2*3.67 + 1.70 = 0.83 P3 = Put premium paid for the $45 put option purchased. P1 = Put premium paid for the $35 put option purchased. P2 = Put premium received for the $40 put option sold. Value at Expiration = max (0, X1-ST) - 2 max (0, X2-ST) + max (0, X3-ST) Where, ST = Stock price at expiration. X1 = $35 X2 = $40 X3 = $45 Profit = Value at expiration - Value at the beginning Maximum profit = 45 - 40 - P3 - P1 + 2P2 = 45 - 40 - 6.47 - 1.70 + (3.67*2) = $4.17 Maximum profit occurs if the underlying ends up precisely at the middle exercise price of $40. Maximum loss = P1 + P3 -2*P2 0.83 Maximum loss would occur if the stock price ends above the Strike price of $45. There are two break-even points: Break-even point (a) = X1 + P1 - 2P2 + P3 35 + 1.70 -2*3.67 + 6.47 105.83 Break-even point (b) = 2X2 - X1 -P1 + 2P2 - P3 2*40 - 35 - 1.70 + 2*3.67 - 6.47 80-1.70+7.34-6.47 $79.17 se ercise of $35. Chapter 24 3 Consider the recent performance of the Closed Fund, a closed-end fund devoted to finding undervalued, thinly traded stocks: Period NAV 0 1 2 3 4 Premium/Discount $10.00 11.25 9.85 10.5 12.3 0 -5 2.3 -3.2 -7 Here, price premiums and discounts are indicated by pluses and minuses, respectively, and Period 0 represents Closed Fund's initiation date. a. Calculate the average return per period for an investor who bought 100 shares of the Closed Fund at the initiation and then sold her position at the end of Period 4. 100*(12.30-10) 230 b. What was the average periodic growth rate in NAV over that same period? Average growth rate: (12.30-10)/5 0.46 c. Calculate the periodic return for another investor who bought 100 shares of Closed Fund at the end of Period 1 and sold his position at the end of Period 2. (9.85-11.25)*100 140 d. What was the periodic growth rate in NAV between Periods 1 and 2? 12.5% from 10 to 11.25 6 Suppose that at the start of the year, a no-load mutual fund has a net asset value of $27.15 per share. During the year it pays its shareholders a dividend distribution of $1.12 per share and finishes the year with an NAV of $30.34. a. What is the return to an investor who holds 257.876 shares of this fund in his (nontaxable) retirement account? Beginning value = $27.15 x 257.876 = $7,001.33 Dividends = $1.12 x 257.876 288.82 Ending value = $30.34 x 257.876 = 7,823.96 Return b. = [($7,823.96 - $7,001.33) + 288.82]/$7,001.33 = 15.87% What is the after-tax return for the same investor if these shares were held in an ordinary savings account? Assume that the investor is in the 30 percent tax bracket. Assuming that the tax rate of 30% is applied to all cash flows: ($7,823.96 - $7,001.33) + 288.82 = $1,111.45 $1, 111, 45(1 - .30) = $778.02 Return c. = $778.02/$7,001.33 11.11% If the investment company allowed the investor to automatically reinvest his cash distribution in additional fund shares, how many additional shares could the investor acquire? Assume that the distribution occurred at year end and that the proceeds from the distribution can be reinvested at the year-end NAV. Then again, the $1,111.45 could be reinvested at the year-end NAV of $30.34. The speculator could buy $1,111.45/$30.34 = 36.63 shares 8 Mutual funds can effectively charge sales fee in one of three ways: front-end load fees, 12b-1 (i.e., annual) fees, or deferred (i.e., back-end) load fee. Assume that the SAS Fund offers its investors the choice of the following sales fee arrangement: (1) a 3% front-end load, (2) a 0.50% annual deduction, (3) a 2% back-end load, paid at the liquidation of the investor's position. Also, assume that SAS Fund averages NAV growth of 12% per year. a. If you start with $100,000 in investment capital, calculate what an investment in SAS would be worth in three years under each of the proposed sales fee schemes. Which scheme would you choose? 3 percent front-end load = $100,000 (1 - .03) = $97,000 $97,000 (1 + .12)3 $97,000(1.4049) = $136,278 (2) a 0.50 percent annual deduction Year 1: $100,000(1 + .12) = $112,000 (1 - .005 = $111,440 Year 2: $111,440(1 + .12) = $124,812.80(1 - .005) = $124,188.74 Year 3: $124,188.74(1 + .12) = $139,091.38(1 - .005) = $138,395.93 3) a 2 percent back-end load $100,000(1 + .12)3 = $100,000(1.4049) = $140,492.80 $140,492.80(1 - .02) = $137,682.94 Choice (2) with ending wealth of $138,395.93 b. If your investment horizon were 10 years, would your answer in Part a change? Demonstrate. On the off chance that my speculation skyline is higher than I will take a 3% front end load and complete the circumstance of. This will guarantee that I don't need to pay whatever else. c. Explain the relationship between the timing of the sales charge and your investment horizon. In general In the event that my venture skyline is short of what I should utilize yearly charges as they will have less effect, and if my speculation skyline is long then I will contribute with front load and take one time charge. 10 Peter and Andrew Mueller have built up their $600 000 investment portfolio over many years through regular purchases of mutual funds holding only U.S. securities. Each purchase was based on personal research but without consideration of their other holdings. They would now like advice on their total portfolio, which follows: Andrea's company Stock Blue-chip growth fund Super-beta fund Conservative fund Index fund No-dividend fund long-term zero-coupon fund Type Stock Stock Stock Stock Stock Stock Stock Market Sector Small-cap growth Large-cap growth Small-cap growth Large-cap growth Large-cap growth Large-cap growth Government Beta Percent of Total 1.4 1.2 1.6 1.05 1 1.25 0 35 20 10 2 3 25 5 Evaluate the Mueller's portfolio in terms of the following criteria: a. Preference for \"minimal volatility\" The unpredictability of the Mueller's' portfolio is liable to be much more noteworthy than insignificant. The advantage portion of 95 percent stocks 5 percent bonds demonstrates that generous changes in resource worth will probably happen after some time. The advantage distribution's unpredictability is exacerbated by the way that the beta coefficient of 90 percent of the portfolio (i.e., the four development stock allotments) is significantly more prominent than 1.0. In this way the designation to stocks ought to be diminished, as ought to the extent of development stocks or higher beta issues. Moreover, the 5 percent assignment to securities is in a long haul zero coupon security support that will be profoundly unstable in light of long haul loan cost changes; this bond portion ought to be traded for one with lower unpredictability (maybe shorter development, higher evaluation issues.) b. Equity diversification The most evident value enhancement issue is the convergence of 35% of the portfolio in the high beta little top stock of Andrea's organization, an organization with a very questionable future. A significant part of the stock can and ought to be sold, which should be without possible or to a great extent free of expense risk as a result of the accessible duty misfortune convey forward. Another issue is the 90 percent focus in high beta development stocks, which repudiates the Mueller's' inclination for negligible instability ventures. The same is valid for the portfolio's 45 percent distribution to higher instability little top stocks. At long last, the whole portfolio is gathered in the household market. Broadening far from Andrea's organization's stock, into more esteem stocks, into bigger top stocks, and into at any rate some global stocks is justified. c. Asset allocation (including Cash flow needs) According to the Muller's money related circumstance, it appears that the portfolio has more value weighting which appears to be more forceful towards the goal of Muller. Their goals of underneath danger resilience normal with restricted development desire educates that an extremely adjusted and a moderate assignment would be more proper. The Muller's dared to contribute whatever other resources but stocks and the little security store holding. Decrease in value interest if there should arise an occurrence of little tops and development values and obligation venture build alerts to produce more predictable as far as sought results that finish the short market cycle. Also, the Mueller's have no money store or possessions of fleeting high review obligation resources. In the precise not so distant future, the Mueller's will require $50,000 (forthright installment) and at any rate $40,000 (first year educational cost and everyday costs) for their little girl's school instruction, and additionally a store against typical costs. Also, they hope to have negative income every year their little girl is in school, which ought to lead them to expand their money hold. The present portfolio is likely to deliver a low level of pay in light of the vast weighting in development stocks and in light of the fact that the main bond holding is a long haul zero coupon security reserve. Likewise, the attractiveness of Andrea's organization stock obscure and could introduce a liquidity issue on the off chance that it should be sold rapidly. After their quick money needs are met, the Mueller's will require a humble, continuous distribution to money counterparts. Security Modified Duration Portfolio U.S. Treasury bond futures contract Basis Point Value 10 Years 8 Years Conversion Factor for Cheapest to Deliver Bond Portfolio Value/ Future Contact Price $100,000.00 Not Applicable $100,000,000.00 $75.32 1 94-05 a. Discuss two reasons for using futures rather than selling bonds to hedge a bond portfolio. No calculations required. 1. Futures will help the portfolio being hedged against any parallel shift while selling the portfolio means there is no gain/loss in future. 2. 2. Selling means the manager will have liquid cash and there is a cost of holding cash while futures is an investment that would help them in gain in future. b. Formulate Klein's hedging strategy using only the futures contract shown. Calculate the number of futures contracts to implement the strategy. Show all calculations. No of futures = (10000000*10) / [(94+5/32)*75.32*8] 1762.5869418041 = $17,625.87 c. Determine how each of the following should change in value if interest rates increase by 10 basis points as anticipated. Show all calculations. Increase 0.001 Change in portfolio -0.001 x 10 x 10,000,000 = -1,000,000.00 Change in T bond position 17,625.87*94.15625*75.32*0.001 =1,000,000.00 Portfolio is equal to change in portfolio plus change in T bond position 0 d. State three reasons why Klein's hedging strategy might not fully protect the portfolio against interest rate risk. It will not hedge fully since the cheapest to deliver bond value will change and hence the number of futures need to be adjusted e. Describe a zero-duration hedging strategy using only the government bond portfolio and options on U.S. Treasury bond futures contracts. No calculations required. In this strategy the government bonds are hedged by using the options. The options are used to hedge downward risk in the portfolio while the upside potential is unlimited. Portfolio Bond 1A B 2C Market Value (Mil) $6.00 4 11.5 Coupon Rate Compounding Frequency 0.00% Annual 0.00% Annual 4.60% Annual Yield to Maturity Maturity 3 yrs. 14 yrs. 9 yrs. 7.31% 7.31% 7.31% a. Calculate the modified duration (expressed in years) for each of all yields increase by 60 basis points on an annual basis? Modified Duration = (1/1+10.355) = 8.81% b. Without performing the calculations, explain which of the portfolios will actual have its value impacted to the greatest extent (in absolute terms) by th (Hint: This explanation requires knowledge of the concept of bond convexity). Yield shift will lead to shift of around 8.81% in the value of the bond. That is what the modified duration tells us. c. Assuming the bond speculator wants to hedge her net bond position, what is the optimal number of futures contracts that must be bought or sold? Start by calculating the optimal hedge ratio between the futures contract and the two bond portfolios separately and then combine them. Based on the assumption that interest rates would increase sharply in the next year, the TD team recommended \"plain vanilla\" interest rate swaps as the mo mechanism. Interest rate swaps would effectively lock in a fixed rate on CRP's floating rate bank debt at interest rate levels that were attractive to the company. Andrew and Sara had certainly heard of interest rate swaps, but to this point CRP had never borrowed money at floating rates and had no real need for hedging pr Even the LIBOR floating rate borrowing option was a new concept to the company. by the shift yields. d? e most attractive hedging y. ng products. As a relationship officer for a money-center commercial bank, one of your corporate accounts has just approached you about a one-year loan for $1,000,000. The customer would pay a quarter interest expense based on the prevailing level of LIBOR at the beginning of each three-month period. As is the bank's convention on all such loans, the amount of the interest payment would then be paid at the end of the quarterly circle when the new rate for the next 90-day LIBOR 4.60% 180-day LIBOR 4.75% 270-day LIBOR 5.00% 360-day LIBOR 5.30% a.If 90-day LIBOR rises to the levels "predicted" by the implied forward rates, what will the dollar level of the bank's interest receipt be at the end of each quarter during the one-year loan period? 1st Quar 2nd quar 3rd quar 4th quar Rate 4.60% 4.75% 5% 5.30% Interest ### ### ### ### a. If the bank wanted to hedge its exposure to falling LIBOR on this loan commitment, describe the sequence of transactions in the futures If bank want to hedge the transaction for the falling interest rate then in that case bank needs to deal with interest rate swaps. b. Assuming the yields inferred from the Eurodollar futures contract prices for the next three settlement periods are equal to the implied fo calculate the annuity value that would leave the bank indifferent between making the floating-rate loan and annuity value in both dollar and Rate Interest 1st Quar 4.60% ### 2nd quar 4.75% ### 3rd quart 5% ### 4th quar 5.30% ### Total Interest Earned ### Interest Rate % 6.55% Percent rate should be 6.55% then the choice would be indifferent. 0. r the next cycle is determined. e futures market it could undertake. mplied forward rates, dollar and annual (360-day) percentage terms. 9. Alex Andrew, who manages a $95 million large- capitalization U.S. equity portfolio, currently forecasts that equity markets will decline soon. Andrew prefers to avoid the transacti Because Andrew realizes that his portfolio will not track the S&P 500 Index exactly, he performs a regression analysis on his actual portfolio returns versus the S&P 500 futures retu Future Contract Data S&P 500 futures price S&P 500 index S&P 500 index multiplier 1,000 999 250 a. Calculate the number of future contracts required to hedge $15 million of Andrew's portfolio, using the data shown. State whether the he The number of future contracts required is = (value of the portfolio / value of the index futures) x beta of the portfolio = [15,000,000/ (1,000 * 250)] * 0.88 = (15,000,000/250,000)*0.88 = 52.8 or 53 contracts Selling (going short) 52 or 53 contracts will hedge $15,000,000 of equity exposure. b. Identify two alternative methods (other than selling securities from the portfolio or using futures) that replicate the strategy in Part a. Co Two alternative methods that replicate the future strategy in Part a. are the Shorting SPDRs and Shortening a forward contract on the S&P 500 index. Shortening the SPDRs would be more expensive than the futures to trade in terms of liquidity and transaction costs. Tracking error, in theory, would be higher for fu Shortening a forward contract on the S&P500 index. The price of this transaction is negotiable between the two parties. Forwards are not liquid, may prove difficult action costs of making sales but wants to hedge $15 million of the portfolio's current value using S&P 500 futures. returns over the past year. The regression analysis indicates a risk-minimizing beta of 0.88 with an R 2 of 0.92. e hedge is long or short. Show all calculations. Contract each of these methods with the futures strategy. or futures than SPDRs because S&P 500 futures can close under and over fair value. SPDRs do not incur the cost of rolling over, which a position in future s would cult to reverse once entered, and may involve counterparty risk. Unlike futures, which are standardized contracts with no customization possible, this alternative has would incur if held longer that one expiration date. e has the advantage of customization; negotiable terms including length of time of the contract, margin requirements, cost of closing the position early, and timing o ming of payments. 10. The treasurer of a middle market, import-export Company has approached you for advice on how to best invest some of the firm's short-term cash balances. Th U.S. Swiss Dollar per Francs per Swiss U.S. Dollar Franc (CHF) Spot 1 - year CHF futures 1.5035 0.6651 0.6586 a. Calculate the one-year bond equivalent yield for the Swiss government security that would support the interest rate parity condition. To get a return of 4.25% by converting to CHF it must be the case that a dollar converted today, at rate R, and converted back at the end of the year would be wort ($1/0.6651) * (1 + R) * 0.6586 = 1.0425 0.990 = 0.990 R = 1.0425 R= 5.28% b. Assuming the actual yield on a one-year Swiss government bond is 5.50 percent, which strategy would leave the treasurer with the grea If the actual rate is 5.5% for a one year Swiss government bond, then the return on CHF investment would be greater. This can be illustrated by plugg ($1 / 0.6651) * (1 + 0.55) * 0.6586 = 1.0447 c. Describe the transactions that an arbitrageur could use to take advantage of this apparent mispricing, and calculate what the profit wou An arbitrageur could borrow $250, 000.00 domestically at 4.25%, convert it into $375, 883.33 CHF, buy Swiss government bonds, and enter into a forward contract ances. The company, which has been a client of the bank that employs you for a few years, has $250, 000 that is able to commit for a one year holding period. The ition. d be worth $1.0425 the greatest return after one year? d by plugging in R= 0.55 into the equation ($1 / 0.6651) * (1 + R) * 0.6586 rofit would be for a $250, 000 transaction. d contract to reconvert the proceeds after a year. After a year invested at 5.5% the arbitrageur would have $396,556.91 which could then be converted back into dol he treasurer is currently considering two alternatives: (1) invest all the funds in one year U.S. Treasury bill offering a bond equivalent yield of 4.25 percent, and (2) in dollars at the forward rate of 0.6586 $ / CHF. This would result in $261,172.38 of which $260,625.00 would be needed to repay the loan and the interest. Therefore, ) invest all the funds in a Swiss government security over the same horizon, locking in the spot and forward currency exchanges in the FX market. A quick call to th re, the arbitrageur would have made a profit of $547.38 before commissions the bank's FX desk gives the following two-way currency exchange quotes. 11. Bonita Singer is a hedge fund manager specializing in futures arbitrage involving stock index contracts. She is investigating potential trading opportunities in S& a. Assume that the Treasury yield curve is flat at 3.2 percent and the annualized dividend yield on the S&P index is 1.8 percent. Using the c S= Spot Price= $1100 S= Storage Cost=0 R= Risk Free Interest Rate= 3.2% C= Convenience Yield= 1.8% T= Term of Contract=1/2 year F= $1107.727 (As per formula & Calculation) b. Described the set of transactions that Bonita would have to undertake to take advantage of an actual futures contract price and was (1) Bonita needs to take long positions in the higher contracts and hence use the arbitrage available to her in the future. c. Assuming that total round-trip arbitrage transaction costs are $20 for the set trades described in Part b, calculate the upper and lower b Arbitrage won't be profitable if S&P is between $1100 and $1127 S&P 500 index futures to see if there are any inefficiencies that she can exploit. She knows that the S&P 500 stock index is currently trading at $1100. e cost of carry model, demonstrate what the theoretical contract price should be for the futures position expiring six months from now. 1) substantially higher or (2)substantially lower than the theoretical value you established in Part a. r bounds for the theoretical contract price such that arbitrage trading would not be profitable