ACC 471 Practice Problem Set # 4 Fall 2002 Suggested Solutions 1. Text Problems: 17-3 a. From put-call parity, C P S 0 X 1 r T f 4 50 50 1 10 1 4 $5 18. b. Sell a straddle, i.e. sell a call and a put to realize premium income of $4 $5 18 $9 18. If te stock ends up at $50, bot options will be wortless and your profit will be $9.18. Tis is your maximum possible profit since at any oter stock price, you will need to pay off on eiter te call or te put. Te stock price can move by $9.18 in eiter direction before your profits become negative. c. uy te call, sell te put, and lend 50 1 10 1 4 $48 82. Te payoff is as follows: Cas Flow in 3 Monts Position Cas Flow Now S T X S T X Call (long) C $5 18 0 S T $50 Put (sort) P $4 00 $50 S T 0 Loan 48 82 $50 $50 Total -$50 S T S T y put-call parity, te initial outlay equals te stock price of $50. In eiter scenario, you end up wit te same payoff as you would if you bougt te stock itself. 17-5 a. utterfly spread (wit call options): Position S T X 1 X 1 S T X 2 X 2 S T X 3 X 3 S T Long call (X 1 ) 0 S T X 1 S T X 1 S T X 1 Sort 2 calls (X 2 ) 0 0 2 S T X 2 2 S T X 2 Long call (X 3 ) 0 0 0 S T X 3 Total 0 S T X 1 2X 2 X 1 S T X 2 X 1 X 3 X 2 0 $ X 2 X 1 X 1 X 2 X 3 S T b. Vertical combination (using calls) Position S T X 1 X 1 S T X 2 X 2 S T Long call (X 2 ) 0 0 S T X 2 Long put (X 1 ) X 1 S T 0 0 Total X 1 S T 0 S T X 2 $ X 1 X 1 X 2 S T 1
17-7 a. y writing covered call options, Jones takes in premium income of $30,000. If te stock price in January is less tan or equal to $45, e will ave is stock plus te premium income. ut te most e can ave is $450,000 plus $30,000 because te stock will be called if its price exceeds $45. (Note tat we are ignoring interest earned on premium income from writing te option over tis sort time period.) Te payoff structure is: Stock price Portfolio value less tan $45 $30,000 plus 10,000 times stock price more tan $45 $30,000 plus $450,000, or $480,000 Tis strategy offers some extra premium income but leaves substantial downside risk. At an extreme, if te stock price fell to zero, Jones would be left wit only $30,000. Te strategy also puts a cap on te final value at $480,000, but tis is more tan sufficient to purcase te ouse. b. y buying put options wit a $35 exercise price, Jones will be paying $30,000 in premiums to insure a minimum level for te final value of is position. Tat minimum value is $35 10 000 $30 000 $320 000. Tis strategy allows for upside gain, but exposes Jones to te possibility of a moderate loss equal to te cost of te puts. Te payoff structure is: Stock price Portfolio value less tan $35 $350,000 less $30,000, or $320,000 more tan $35 10,000 times stock price less $30,000 c. Te net cost of te collar is zero. Te value of te portfolio will be as follows: Stock price Portfolio value less tan $35 $350,000 between $35 and $45 10,000 times stock price more tan $45 $450,000 If te stock price is below $35, te collar preserves te $350,000 in principal. If te price is above $45, te value is capped at $450,000. In between, is proceeds are 10,000 times te stock price. Te best strategy is te zero cost collar since it satisfies te two requirements of preserving te principal of $350,000 wile offering a cance at $450,000. Te worst strategy would be te covered call, since it leaves im exposed to risk of substantial loss of principal. 17-9 Te farmer as te option to sell te crop to te government for a guaranteed minimum price if te market price is too low. If te support price is denoted P S and te market price P M, ten te farmer as a put option to sell te crop (te asset) at an exercise price of P S, even if te price of te underlying asset, P M is less tan P S. 17-14 uy te X 62 put (wic sould cost more but does not) and write te X 60 put. Your net outlay is zero, since te options ave te same price. Your proceeds at maturity cannot be negative, and in some cases will be positive. Tis is an arbitrage opportunity since you ave te possibility of a gain at zero cost. Your final payoffs are: Position S T 60 60 S T 62 S T 62 Long put (X 62) 62 S T 62 S T 0 Sort put (X 60) 60 S T 0 0 Total 2 62 S T 0 Te profit diagram is: $ 2 0 60 S T 62 2
17-15 Te following payoff table sows tat te portfolio is risk free wit a payoff at T of $10. Tus te risk free interest rate must be 10 9 50 1 5 26%. Position S T 10 S T 10 uy stock S T S T Write call 0 S T 10 uy put 10 S T 0 Total 10 10 17-16 From put-call parity, C P S 0 X 1 r f T. If te options are at te money, ten S 0 wic must be positive. Terefore, C P. C P X X 1 r f 18-5 a. Wen S T 130, P 0. Wen S T 80, P 30. Te edge ratio is P P S S 0 30 130 80 T 3 5 X and tus b. We ave: Position S 80 S 130 3 sares $240 $390 5 puts $150 $0 Total $390 $390 Tis portfolio is riskless and as a present value of 390 1 1 $354 55. c. Te portfolio cost is 3S 5P 300 5P, and it is wort $354.55. Terefore P 54 55 5 $10 91. 18-6 Te edge ratio for te call is Terefore: C C S S 20 0 130 80 2 5 Position S 80 S 130 2 sares $160 $260 5 calls sold $0 -$100 Total $160 $160 Tis portfolio is riskless and as a present value of 160 1 1 $145 45. Te portfolio cost is 2S 5C 200 5C, so te call is wort 200 145 45 5 $10 91 today. Note tat put-call parity is satisfied: P C PV X S 10 91 10 91 110 1 10 100 18-10 Less. Te cange in te call price would be $1 only if (i) te probability of exercise was 100%; and (ii) te risk free interest rate was zero. 18-11 Holding firm-specific risk constant, iger β implies iger total stock volatility. Terefore, te value of te put option will increase as β increases. 18-12 Holding β constant, te ig firm-specific risk stock will ave iger total volatility. Te option on te stock wit iger firm-specific risk will be wort more. 18-25 Wen r 0, one sould never exercise a put early. Tere is no time value cost to waiting to exercise, but tere is a volatility benefit from waiting. To sow tis more rigorously, consider a portfolio were you lend $X and sort one sare of stock. Te cost to establis tis portfolio is X S 0. Te payoff at time T (wit zero interest earnings on te loan) is X S T. In contrast, a put option as a payoff at time T of X S T if tat value is positive, and zero oterwise. Te put s payoff is at least as large as te portfolio s, and terefore te put must cost at least as muc to purcase. Hence, P X S 0, and te put can be sold for more tan te proceeds from early exercise. We conclude tat it doesn t pay to exercise early (wen r 0). 3
18-27 Since te two possible call option values at expiration are $20 (if te stock is wort $120) and $0 (if te stock is wort $80), te edge ratio is: 20 0 0 5 120 80 Tis means tat we can form a riskless portfolio by purcasing one sare of stock and writing two calls. Te cost of te portfolio is S 2C 100 2C. At expiry, if te stock is wort $120, te portfolio is wort 120 2 20 $80. If te stock is wort $80, te portfolio is wort $80 (since te calls expire wortless). Terefore: 100 2C 80 1 10 C $13 64 Notice tat we never used te probabilities of a stock price increase or decrease tese are not needed to value te call option. 18-28 Te edge ratio is 30 0 130 70 0 5. Form te riskless portfolio by buying one sare of stock and writing two call options. Te portfolio costs S 2C 100 2C. Its payoff is $70 (if te stock drops, te calls are wortless and te sare is wort $70; if te stock rises, te stock is wort $130 but eac call is wort -$30, so te overall value is $70). Tus 100 2C 70 1 10 C $18 18, wic exceeds te value in te lower volatility scenario. 18-29 Te edge ratio for a put wit X 100 would be 0 20 120 80 0 5. Form te riskless portfolio by buying one sare and two put options. Te portfolio costs 100 2P. Its payoff in te down state is 80 2 20 $120, matcing its payoff in te up state (120 2 0 ). Terefore 100 2P 120 1 10 P $4 54. According to put-cal parity, S P C PV X. Our estimates of option value satisfy tis relationsip: 4 54 100 13 64 100 1 10. 19-3 Sort-selling results in an immediate cas inflow, wereas te sort futures position does not: Position Initial Cas Flow Final Cas Flow Sort sale P 0 P T Sort futures 0 F 0 P T 19-4 a. Take a sort position in T-bond futures, to offset interest rate risk. If rates increase, te loss on te bond will be offset to some extent by gains on te futures. b. Again, a sort position in T-bond futures will offset te interest rate risk. c. You wis to protect your cas outlay wen te bond is purcased. If bond prices increase, you will need extra cas to purcase te bond. Tus, you want to take a long futures position tat will generate a profit if prices increase. 19-5 a. We ave: Cas Flows Position Now T 1 T 2 Long futures (T 1 ) 0 P 1 F T 1 0 Sort futures (T 2 ) 0 0 F T 2 P 2 uy asset at T 1, sell at T 2 0 P 1 P 2 At T 1, borrow F T 1 0 F T 1 F T 1 1 r f Total 0 0 F T 2 F T 1 1 r f b. Since te T 2 cas flow is risk free and no net investment was made, any profits would represent an arbitrage opportunity. c. Te zero profit no-arbitrage restriction implies tat: 19-6 Te put-call parity relationsip states tat: F T 2 F T 1 1 r f T 2 X P C S 0 1 r T f T 1 T 2 T 1 T 2 T 1 4
If F X, ten P C S 0 F 1 r f T. ut spot-futures parity tells us tat F S 0 1 r f T. Substituting, we find tat: S 0 1 r T f P C S 0 1 C r T S 0 S 0 C f i.e. P C. 19-11 According to te parity relation, te proper price for December futures is: F Dec F June 1 r 1 2 f 346 30 1 05 1 2 354 85 Te actual futures price for December is too ig relative to te June price. You sould sort te December contract and take a long position in te June contract. 19-15 Te treasurer would like to buy bonds today, but cannot. As a proxy for tis purcase, T-bond futures contracts can be purcased. If rates do in fact fall, te treasurer will ave to buy back te bonds for te sinking fund at prices iger tan te prices at wic tey could be purcased today. However, te gains on te futures contract will offset tis iger cost to some extent. 19-16 a. S 0 1 r f D 950 1 06 10 997. b. S 0 1 r f D 950 1 03 10 968. c. Te futures price is too low. uy futures, sort te index, and invest te proceeds of te sort sale in T-bills: Position Cas Flows Now Cas Flows in 6 Monts uy futures 0 S T 948 Sort index 950 S T 10 uy bills -950 978.50 Total 0 20.50 19-18 a. From parity, F 0 400 1 03 5 407. Actual F 0 406, so te futures price is 1 below te proper level. b. uy te relatively ceap futures and sell te relatively expensive stock: Position Cas Flows Now Cas Flows in 6 Monts Sell sares 400 S T 5 Sort index 0 S T 406 Lend 400 412 Total 0 1 c. If you do not receive interest on te proceeds of te sort sales, ten te $400 you receive will not be invested but will simply be returned to you. Te proceeds from te strategy in (b) are now negative. An arbitrage opportunity no longer exists: Position Cas Flows Now Cas Flows in 6 Monts Sell sares 400 S T 5 Sort index 0 S T 406 Place 400 in margin account 400 400 Total 0 11 20-3 a. i. Te αs are te regression intercepts: α A 01, α 02. ii. Te appraisal ratio is α p σ e p. For stock A, we ave 01 103 0971, wile for is is 02 191 1047. iii. Te Sarpe measure S p r p r f σ p. We are told tat, over te sample period, r M 14 and r f 06. Terefore: Stock A: r p r f 01 106 1 2 08 106 S A 216 Stock : r p r f 02 084 0 8 08 084 S 249 4907 3373 5
iv. Te Treynor measure T p r p r f β p. For stock A, T A 106 1 2 0883, and for stock, T 084 8 1050. b. i. If tis is te only risky asset, ten Sarpe s measure is te one to use. Stock A s is iger, so it is preferred. ii. If te stock is mixed wit te index fund, te contribution to te overall Sarpe measure is determined by te appraisal ratio; terefore is preferred. iii. If it is one of many stocks, ten Treynor s measure is te correct criterion, and is preferred. 20-4 We need to distinguis between market timing and security selection abilities. Te intercept of te scatter diagram is a measure of stock selection ability. If te manager tends to ave a positive excess return even wen te market s performance is merely neutral (i.e. as zero excess return), ten we conclude tat te manager as on average made good stock picks. Stock selection must be te source of positive excess returns. Timing ability is indicated by te curvature of te plotted line. Lines tat become steeper as we move to te rigt side of te grap sow good timing ability. Te steeper slope sows tat te manager maintained iger portfolio sensitivity to market swings (i.e. a iger β) in periods wen te market performed well. Tis ability to coose more market-sensitive securities in anticipation of market upturns is te essence of good timing. In contrast, a declining slope toward te rigt and side of te grap means tat te portfolio was more sensitive to te market wen te market did poorly and less sensitive wen te market did well. Tis indicates poor timing. We can terefore classify performance for te four managers as follows: 20-8 a. Treynor measures: Sarpe measures: Manager Selection Ability Timing Ability A ad Good Good Good C Good ad D ad ad Portfolio X: Portfolio X: Market: Market: 10 06 6 10 06 18 12 06 1 0667 12 06 13 2222 0600 4615 Portfolio X outperforms te market based on te Treynor measure, but underperforms based on te Sarpe measure. b. Te two measures of performance are in conflict because tey use different measures of risk. Portfolio X as less systematic risk tan te market based on its lower β, but more total risk (standard deviation). Terefore, te portfolio outperforms te market based on te Treynor measure but underperforms based on te Sarpe measure. 20-9 Support: A manager could be a better performer in one type of circumstance. For example, a manager wo does no timing, but simply maintains a ig β, will do better in up markets and worse in down markets. Terefore, we sould observe performance over an entire cycle. Also, to te extent tat observing a manager over an entire cycle increases te number of observations, it would improve te reliability of te measurement. Contradict: If we adequately control for exposure to te market (i.e. adjust for β), ten market performance sould not affect te relative performance of individual managers. It is terefore not necessary to wait for a complete market cycle to pass before you evaluate a manager. 6
20-12 a. Treynor measures: Portfolio A: Portfolio : y tis criterion, te portfolios are ranked equally. Jensen measures: 24 12 1 0 30 12 1 5 12 12 Portfolio A: 24 12 1 0 21 12 03 Portfolio : 30 12 1 5 21 12 045 Tis indicates tat portfolio as superior performance. b. i. One year s data is too small a sample for reliable results. ii. It is possible tat te fund managers were trying to time te market, resulting in a non-linear security caracteristic line. iii. Tere is no indication about te diversification of te two funds, so tat systematic risk may not be te appropriate measure to use in evaluating te two funds. For example, portfolio may be very undiversified and significantly riskier. 2. True. From put-call parity: S 0 P C PV X C P S 0 PV X 0 (since S 0 X). 3. a. From te information given, te stock price will be eiter $118.32 or $84.52 after 6 monts. Te tree possible values at te end of te year are $140, $100, and $71.43. Te corresponding terminal payoffs of te European put are $0, $10, and $38.57. Te risk-neutral probability is: R d π u d 1 1 1 1 4 1 4 1 1 4 6024 If te stock price is $118.32 after 6 monts, ten te option value is πp uu 1 π P ud 6024 0 3976 10 P u R At tis node we also ave: P uu P ud 0 10 S uu S ud 140 100 up ud dp uu R u d 0 25 1 1832 10 8452 0 1 1832 8452 $3 79 33 37 so te replicating portfolio involves sorting 0.25 sares of te stock and lending $33.37. If te stock price is $84.52 after 6 monts, ten te option value is: and 6024 10 3976 38 57 P d 10 38 57 100 71 43 1 1 1832 38 57 8452 10 1 1832 8452 $20 37 104 88 7
so te replicating portfolio involves sorting 1 sare of te stock and lending $104.88. Working back to today, te option value is: and 6024 3 79 3976 20 37 P 0 3 79 20 37 118 32 84 52 49 1 1832 20 37 8452 3 79 1 1832 8452 $9 90 58 93 so te replicating portfolio involves sorting 0.49 sares of te stock and lending $58.93. If te option is American, it is not wort exercising if te stock price rises after 6 monts (te stock price is $118.32 but te strike is $110), so noting canges in tat case. If te stock price drops after 6 monts, it would be wort 110-84.52 = $25.48 if exercised but, from above, only $20.37 if eld. Tis means tat te option value today is: 6024 3 79 3976 25 48 P 0 $11 84 and 3 79 25 48 118 32 84 52 64 1 1832 25 48 8452 3 79 1 1832 8452 76 01 so te replicating portfolio involves sorting 0.64 sares of te stock and lending $76.01. b. y te convergence property, te tree possible futures prices after a year are equal to te tree possible stock prices, i.e. $140, $100, and $71.43. y te martingale property: F u πf uu 1 π F ud 6024 140 F d πf ud 1 π F dd 6024 100 F 0 πf u 1 π F d 6024 124 10 3928 100 $124 10 3928 71 43 $88 64 3928 88 64 $110 (Note: you ave to carry more decimal places tan sown ere to avoid some rounding errors.) If te stock price rises after 6 monts, to replicate te option wit futures we ave to solve: 140 124 10 F 100 124 10 F 0 10 Te solution is F 25, 3 79. If te stock price falls after 6 monts: Te solution is F 1, 20 37. Similarly, as of today: 100 88 64 F 71 43 88 64 F 124 10 110 F 88 64 110 F 10 38 57 3 79 20 37 Te solution is F 47, 9 90. Recall tat te value of te option is tus $9.90, as it costs noting to establis te futures position. 8
c. Te terminal payoff of te powered call is $2,025 if te stock price is $140, $25 if te stock price is $100, and $0 if te stock price is $71.43. If te stock price is $118.32 after 6 monts, ten te option value is C u πc uu At tis node we also ave: 1 π C ud R C uu C ud 2025 25 C uu C ud 140 100 uc ud dc uu R u d 6024 2025 3976 25 50 1 1832 25 8452 2025 1 1832 8452 $1 172 60 4 743 48 so te replicating portfolio involves buying 50 sares of te stock and borrowing $4,743.48. If te stock price is $84.52 after 6 monts, ten te option value is: and 6024 25 3976 0 C d 25 0 100 71 43 875 1 1832 0 8452 25 1 1832 8452 $14 36 59 59 so te replicating portfolio involves buying.875 sares of te stock and borrowing $59.59. Working back to today, te option value is: and 6024 1172 60 3976 14 36 C 0 1172 60 14 36 118 32 84 52 34 26 1 1832 14 36 8452 1172 60 1 1832 8452 $678 97 2747 16 so te replicating portfolio involves buying 34.26 sares of te stock and borrowing $2,747.16 (note again tat te answers ere will be quite sensitive to te number of decimal places carried trougout). 4. A Canadian investor can invest $100 today and receive a risk free payoff of $105.81 in one year. Tis investor can also obtain a risk free return by converting te $100 into German marks, buying a one year German Treasury bill, and converting back into Canadian dollars at te one year forward rate. Tese two strategies must offer te same payoffs to avoid arbitrage. Terefore: 105 81 100 7131 1 r G f 7245 101 5986 1 r G f r G f 4 145% 5. Te PV of storage costs is: Y t 05 08 3 05e 12 08 6 05e 12 05e 08 9 12 0 194138 Terefore te futures price is: F t S t Y t e r T t 8 25 194138 e 08 9 12 $8 97 9
6. Te futures contract price F is 250 400 $100 000. To reduce te overall portfolio β to zero, you sould sort βv 0 1 25 100 000 000 N 1 250 contracts. F 100 000 To cange te overall portfolio β to 0.5, recall tat if you sort N ˆβV0 F contracts, your portfolio β will become approximately β Terefore you sould sort 0 75 100 000 000 N 100 000 ˆβ. Here β 1 25 and β 750 contracts. ˆβ 0 5 ˆβ 0 75. 7. a. We need to calculate Joanne s portfolio return, β, and standard deviation: r P 45 0 β P 45 6 σ P 85 2 20 2 35 30 35 1 2 45 2 30 2 20 10 1250 20 8 85 35 2 60 2 20 2 25 2 1 2 3061 Ten: 1250 05 Sarpe: S P 3061 1250 05 Treynor: T P 85 10 05 2450 S M 20 10 05 0882 T M 1 0 0 2500 0 0500 Jensen: α P 125 05 85 10 05 0325 α M 10 05 1 0 10 05 0 0000 b. Her portfolio outperformed te index based on te Jensen and Treynor measures, but it underperformed based on te Sarpe measure. Since tis portfolio is er entire investment in risky assets, se sould switc to te index portfolio. c. We need to recalculate er portfolio weigts, based on ow er portfolio performed last year. We know se earned a total return of 12.5%, so for every dollar se invested in te portfolio, se now as $1.125. Terefore te weigts are: Her portfolio β is now: β P 4000 6 45 1 0 w X 1 125 35 1 3 w Y 1 125 20 1 10 w Z 1 125 4044 1 2 4000 4044 1956 1956 8 8818 so er expected return is: E r P 05 8818 08 1205 10