Ec 1723 Pset 1

Fanny Chen

October 1, 2007

1. (a) Let p =

(21

)and X =

(5 01 1

).

Choose w1 s.t.

wT1 X =

(1 0

)Thus,

wT1 =

(1/50

)

So we long 1/5 share of stock and 0 shares of the risk-free asset.To calculate the price, we take

q = wT p = $0.40

(b) Let p and X be defined as in part (a).Choose w2 s.t.

wX2 =

(0 1

)Thus,

w2 =

(−1/5

1

)

So wee short 1/5 share of stock and long 1 share of the risk-freeasset.To calculate the price, we take

q = wT p = $0.60.

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(c) The weight matrix of the payoffs to make Arrow-Debreu securitiesis:

W = X−1 =

(1/5 0−1/5 1

)

Thus, we can derive the Arrow-Debreu state price vector to be:

q = X−1p =

(0.40.6

)

The first row of each matrix denotes the payoffs and prices ofstates 1 and 2, respectively.

(d)p = Xq

Thus, we have

p =(

1 0.25)( 0.4

0.6

)

p =(

0.55)

You can conostruct a portfolio of identical payoffs to that of abond with a price of $0.55 using stocks and risk-free assets.

For example, suppose we want payoff to be(

1 0.25).

We take(1 0.25

)X−1

(21

)=(

1 0.25)( 0.4

0.6

)=(

0.55)

Now, to take advantage of this arbitrage opportunity, we shsould

short bonds and buy(

1 0.25)( 1/5 0

−1/5 1

)=(

3/20 1/4)

which translates into 0.15 shares of stock sand 0.25 shares of risk-free assets per bond we are shorting. In period 2, we then sellthe stock sand risk-free assets and buy back bonds.

(e) We don’t need to use probabilities because we have a completemarket which determines prices in each state, given payoffs inperiod 2.

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2. (a) X1 = asset1payoff =(

1 −0.5)

X2 = asset2payoff =(

3 −1.5)

X1 = 1/3 ∗X2 ⇒ p1 = 1/3p2 by the Law of One Price. However,since 0.5 6= 1/3∗2, asset 2 is clearly expensive. To take advantageof this arbitrage opportunity, we long asset 1 an dshort asset 2.

(b) 3X1 =(

1.5 3)

3p1 = 1.54

X2 =(

1 3)

p2 = 1.5By the law of one price, asset 2 is overvalued because asset 1 givesgreater payoffs for an equivalent price. To take advantage of thisarbitrage opportunity, long asset 1 and short asset 2.

(c) Construct the Arrow-Debreu complete market. Since the weightmatrix w = X( − 1) for an AD security, we must find X( − 1).(

2 00 1/2

)(1/2 00 2

)=

(1 00 1

)

We can tell that w =

(2 00 1/2

)To find the price matrix p = w ∗ q, we compute:(

2 00 1/2

)(0.21

)=

(0.40.5

)(

4 1.5)( 0.4

0.5

)= $2.35

The AD security is more expensive than asset 3.Thus, long asset 3 and short assets 1 and 2 in the ratio(

4 1.5)( 2 0

0 1/2

)=

(8

.75

)

or in other words, 8 shares of asset 1 and .75 shares of asset 2 per1 share of asset 3.

3. (a) The bank discount yield is inaccurate because

i. it uses 360 instead of 365 daysii. it does not continuously compound, and thus overestimates

the returns on the investment

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iii. it takes the rate as a fraction of par value, not the ask price.

(b)

R =10, 000

P

365/n

(c) Before computers, the bank discount yield was significantly easierto compute than the effective annual yield. For small periods oftime, the bank discount yield will not diverge significantly fromits compounded counterpart. However, for large transactions,such a difference will be evident and the actual returns on theinvestment will be smaller than the estimated returns of the bankdiscount yield.

4. (a) Event tree:

(b) EV(betting H wins the first round) = (0.7)($1)+(0.3)(0) = $0.7.Since bookmakers do not make any money, you should have to pay0.70. Likewise, EV(betting Y wins the first round) = (0.3)($1) +(0.7)(0) = $0.30. Thus, you would have to pay $0.30.

(c) $0.50 for each option because both are equally likely, and you aremaking a fair bet.

(d) Three cases

i. Case 1: Harvard wins the game EV(betting H wins bythe end of the second round)

= P(Harvard wins both rounds)*$1 + P(Harvard does notwin both rounds)*0

= P(Harvard wins first round)*P(Harvard wins secondround |Harvardwinsfirstround)∗$1 =0.7*0.5*$1 = $0.35

Intuitively, we should only have to invest the expected value to obtain $1 ifHarvard wins the game. Let us check.

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Suppose you invest $0.35 on a bet that Harvard wins the first round. If youwin, you get $1 ∗ 0.35

0.7 = $0.5, which you would then reinvest in betting thatHarvard will win the second game. If Harvard wins again, you will havecaptured a profit of exactly $1. If Harvard lost either game, you will have$0.

ii. Case 2: Yale wins the game We perform the same operations as in thelast section.EV(betting Y wins by the end of the second round)

= P(Yale wins both rounds)*$1 + P(Yale does not win both rounds)*0= P(Yale wins first round)*P(Yale wins second round |Y alewinsfirstround)∗

$1= 0.3 ∗ 0.3 ∗ $1 = $0.09

IF you invest $0.09 on a bet that Yale wins the first round. If you win, youget $1∗ 0.09

0.3 = $0.30, which you would then reinvest in betting that Yale willwin the second game. If Yale wins again, you will have captured a profit ofexactly $1. If Yale lost either game, you will have $0.

iii. Case 3: H-Y TieEV(betting they tie)

= P(Yale wins round 1 and H wins Round 2)*$1 + P(H wins round 1 andY wins round 2)*$1 = 0.3∗0.7∗$1+0.7∗0.5∗$1 = $0.21+$0.35 = $0.56In this game, you would invest $0.21 on a bet that H wins round 1, and$0.35 on a bet that Yale wins in round 2. You reinvest your earnings fromwinning either bet in round 1 in a bet that the opposite team wins in round2. Thus, you will earn exactly $1 if there is a tie and $0 otherwise.

(e) These costs are identical to the probabilities of Harvard winning,Yale winning, or H-Y tying. You can expect this relationship tohold when you have complete markets and no transaction costs.

(f) This question is about a dynamically complete market, which isan extension of the static complete market idea discussed morefully in lecture. In a dynamically complete market, the Arrow-Debreu securities involve a predetermined trading strategy inwhich you reinvest whatever returns you get to finally realize$1 if you win your bet and $0 if you do not. It is similar to thestatic market, in which the cost of an asset is its expected value(assuming no transaction costs), except that you would have 4AD securities, one for each of the possible outcomes HH, HY,YH, YY.

5. (a) ln(300, 000 + (200, 000 − c) ∗ 1.06) = ln(300, 000 + (200, 000) ∗1.06) ∗ .999+ln(200, 000 ∗ 1.06) ∗ .001

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c = 425.69

(b) There are several conditions that we must consider, some moreinteresting than others. First, the mundane ones.

First, we assume perfect markets, so companies will chargethe equilibrium price cΛ such that cΛ > c but cΛ < cinsurance, thecost to the insurance company of providing such insurance. Sincethe problem does not stipulate zero transaction costs (meaningzero operating costs and zero profit for the insurance company),and it does not seem a reasonable assumption to make, we cannotdetermine specifically the price that the company will charge.

Second, we consider the real world risks to insurance com-panies. In particular, consider the covariance of the events thatneighboring houses burn down. If the covariance is high, thenthe potential risk of insurance for an insurance company coveringhouses in one geographic location is extremely high, and insur-ance companies may be less willing to insure this area.

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