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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 putcall 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 “atthemoney” options (with strike prices close to the current
stock price). Determine the riskfree rate that corresponds with the maturity of the options.
Check if putcall 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, 137) = 6, max(0, 8.507) = 1.50, and max(0, 47) = 0
b. Pricing by replication: DN ET OFT: _ . 403': LSD _ a O
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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 riskfree 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
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 Fall '07
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