Practice problems solutions

Practice problems solutions - Callahan 1 Solutions for...

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Unformatted text preview: Callahan 1. Solutions for practice problems Certainty equivalent method — In class we discussed the pricing of Boomers, Busters and Bogglers, but didn’t calculate the actual prices. Do so using the certainty equivalent method. Assume that each year the market return is either 20% or 4%, with equal probability. A Boomer is a perpetuity that pays $200 each year rm = 20% and 0 otherwise. A Buster is a perpetuity that pays $200 each year rm = 4% and 0 otherwise. A Boggler is a perpetuity that pays $200 each year rm = 4% and pays -$200 otherwise. That is, the Boggler has the same payoffs as a portfolio that is long one Buster and short one Boomer. Assume rf = 5%. These securities are all perpetairies, therefore the CEM valuation formula to be applied is: PL _ CE[CF] _ E[CF]— risk adj _ E[CF]— b- -MRP_ E[CF] —°i§§..%ffit’(E[rm ]— rf) ’3' ’"r ’3' ’"f The needed inputs are: 23. = 0.05 E04,) = 0.5(020) + 0.50004) = 0.12 var(rm) = 0.5(020 — 0.12)2 + 0.5(0.04 — 0.12)2 = 0.0064 E(CF,,W) = 0.5(200) + 0.5(0) = 100 E(CF.......) = 0.5(0) + 0.5(200) = 100 swamp) = 0.5(—200) + 0. 5(200) = 0 macaw” rm): 0. 5000- 100)(0 20— 0.12) + 0 5(0—100)(0.04 0.12)— s comm“, , rm) _ 0.5(0—100)(0.20 — 0.12) + 0.5(200 —100)(0.04 — 0.12) = —8 cov(CP;,gg,,,rm) = 0.5(—200 — 0)(0.20 — 0.12) + 0.5(200 — 0)(0.04 — 0.12) = —16 The values are: 100 — —— 0.12 0.05 PKQQMEP‘: "0.08064—‘(ft);—5v—.)fl= 250 100 — 0.12 0.05 Farmer = 06:640(05 —)—‘ —3750 46 (0.12— 0.05) PVIJoggIer= b29%.-—= 3500 Thus it is the case that PV(boggler) =PV(busier)—P Vwoomer) as it should be since CF (boggler) =CF (baster)—CF (boomer). In class we discussed put-call parity. 1’ d like you to check whether option prices conform to put- -call parity. Pick a stock (other than Google) with traded options. (In general, technology stocks and stocks with more volatile returns tend to have more active options markets ) Use YahooiFinance, or some other data source, to get the prices of a put and call with the same strike price and maturity date as each other. You want to pick options that are somewhat actively traded, therefore choose “at-the-money” options (with strike prices close to the current stock price). Determine the risk-free rate that corresponds with the maturity of the options. Check if put-call parity holds. Do this for two pairs of options: one pair with a relatively short maturity, and one pair with a relatively long maturity. Clearly present your data and results. Callahan Answer for GOOG, 09/23/08: Short maturity = 28 days. K=440 s = 437.60 C = 25.90 P = 27.47 rf = 0.76% T = 1 month S + P = C + PV(K) 437.60 + 27 .47 5 25.90 + —flm 1.0076 465 .07 r—: 465.62 which is reasonably close, given transaction cost bounds and asynchronous trade or delayed price update issues. The share price was bouncing all over as I copied down the numbers. Long maturity = 6 months. K=440 S = 437.30 C = 52.00 P =51.53 rf = 1.76% T = 6 month S + P = C + PV(K) 437.30+51.53 E 52.00+;4i‘% 1.0176 48883548818 which also is pretty decent. 3. Payoff diagrams — Suppose a stock is currently trading at $50. You are able to buy and sell risk~free bonds, buy and sell the stock, and buy and sell puts or calls with strike prices of $40, $50, or $60 and maturing in one month. Can you create a portfolio that will give you a positive payoff if the stock price in one month is between $40 and $60, and payoff nothing otherwise? Below is a picture of one payoff profile you can achieve. What is the portfolio with this payoff profile? This is a “Butterfly Spread ". Hopefllly, you can convince yourself that you can achieve this payofl diagram if you buy one call with K=40, sell two calls with K=50. and buy one call with K =60. You can also do it with puts. Buy on put with K260. sell two puts with K=50, and buy one put with K:40. 4. This begins similar to number 1 above. For (a) we start with the same CEM pricing equation. The inputs are: Callahan rf = 0.04 E03”) = 05(012) + 0.5(002) = 0.07 var(rm) = 0.5(0.12—0.07)2 + 0.5(002 —0.07)2 = 0.0025 E(CF) = 0.5(200) + 0.5(100) = 150 cov(CF,rm) = 0.5000 — 150)(0.12 — 0.07) + 0.5(200 4150x002 — 0.07) = —2.5 150 + 2.5(.07 — .04) r’ .0025 .04 b. Once we know the fair value and the cash flows, we can calculate the returns. Up market return is 100/4500 = 2.22%. Down market return is 200/4500 = 4.44%. Expected return is 3.33% c. We can calculate the CAPM beta using the expected return and the CAPM equation, or directly using the formula for beta: E[rl=rf +l3(E[rml—r3«) => 3.33=4+fi(7—4) 2 PV = = 4500 => [3 = —O.22 _ or _ cov(r,r ) 0.5(0222 — .0333)(.12 — .07) + 0.5(.0444 — .0333)( .02 — .07) ,3 = __,L = —————.—__—_ = —0 .22 var(rm) .0025 5. There is no number 5. 6. This will be an upward sloping line. It will have a slope of 1 from 0 to 40 and from 50 onward. It will have a slope of 2 between 40 and 50. ‘1'” 5° >1” 7. The price goes from 8 to 10.50 or 6. From 10.50 to 13 or 8.50. And from 6 to 8.50 or 4. Callahan a. For K=7, the final payoffs are max(0, 13-7) = 6, max(0, 8.50-7) = 1.50, and max(0, 4-7) = 0 b. Pricing by replication: DN ET OFT: _ . 403': LSD _ a O (War.- A‘tg'fi- 3402.: 4, WT. A $.Sh*‘3_‘o_s=o Amuse-+8 W5: 3'1 Au Sits-- tL-m: LS‘ '3‘ ' Li. _________+g a, 6 +g-to3-a .7\ _. r 2' Ir ____________——-———‘ 4.9; ~43 45$ 4%”.41 '7 bf:— '13 B="‘—i '5 A=iz$='f;5 r gag IA=Z/‘3 c'i" C 'l-Im —- ,L __ ”rim .. 0'“ g: — gfi'zflfl‘ = \ I'D-SD)" T51;— '= 3.70 :3 E ”—3“ - i M} c. Use put-call parity: 7_ $13 __ l l: ”T c: 1533'— 103 — Z 4' l): C+Q+£f"9=Z-ll1+fgt—2=O. ‘1 (1. Less — option value is less with shorter maturity. g §L_\oo+l?_m 0k) NPV —= -* \.'Lh'+ I006 = b7§looo (3)1? CF=1u=a N?\I: -—I 2n + gig-‘5‘ ‘- 230mm (é) ISM: FFV 9:; WT = ‘01 : [37%|C. ‘) M5501”; —'—’9 afim'vl- ctli Vajcvi' ‘iDGQAfi , 05) (fl CF 2 it“) (TV eq— [“me : LEO-[L :- |29OIQ 4'. [.3aoIQ. J}? 3; g, (1001 wouiok Q4“ 1;» W13 CREE (:2) vase?” g, woman PMM 5. 0 ZS‘D‘OA-Zxro ('33 {515 [#0 5°K- 12.573 +wo my A 2.700 + B w0§ -— O A1251, +3 '°§:5—° A IZBS‘D ’3'” A‘— 9" 'ms—b I 32‘0“ \oS‘ V4\[email protected]+=u = -5“? [L34 .-: gjflc. + 13'1L 'hu — qu 9. You own a piece of property on which you can build a strip mall or an office building. At today’s prices the strip mall will cost $300,000 and has a PV of $400,000, while the office building will cost $500,000 and has a PV of $550,000. Next year the market/economy will either do well (and appreciate by 25%) or do poorly (and depreciate by 10%). Therefore, if the market is valued at $100 today, next year it will be valued at either $125 or $90. The risk-free rate is 4%. If the market does well, the PV of the strip mall will increase by 15% while the PV of the office building will increase by 30%. If the market does poorly, both PVs will decrease by 10%. Costs are expected to increase by 5% next year whether the market does well or poorly. a. What would you choose to build if you Were to build today? b. What would you choose to build if you were to build next period in a good economy? In a bad economy? 0. Should you build today or wait on year and what is the value of the property? 00 MAM— : NW: Hoo— 3c... -_- tut-AL e some hAbL'TbDAL‘ U’FGFLEZ Mfg ; 5'31) -— SSE-a = $21k (B) CD501) itemsan 7 Maw—2 taco (MS) ~ 3w (ms) = M? m “Hing—C 3.30 (1.30) -— S’do {L533 .3 \qok é Bo?» PM EQNOM‘j‘. MkLL.‘ l400 (6‘) " Eat (\ot') : Mglg é.— Efllb) W: m (st -— WU“) : ”3““ (C) ‘2‘ W1 1‘1 {00 [0,9 / > / 0 Ta \ [0.1 I \ ‘13 A- '30 + B-luw = big” a; gust. V - Mm” W1 : 331;: HS“ 0 5.194: Baguio: Bitmap E'U’w 31” 10K "Ba/[email protected] ND «1L. ...
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