Ec 1723 Pset 1

Ec 1723 Pset 1 Fanny Chen October 1, 2007 !

2 1. (a) Let p = and X = 1 Choose w1 s.t.

!

5 0 1 1

w1T X =

.



1 0



Thus, w1T

1/5 0

=

!

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. w2X =



0 1



−1/5 1

!

Thus, w2 =

So wee short 1/5 share of stock and long 1 share of the risk-free asset. To calculate the price, we take q = wT p = $0.60. 1

(c) The weight matrix of the payoffs to make Arrow-Debreu securities is: −1

W =X

!

1/5 0 −1/5 1

=

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

−1

0.4 0.6

p=

!

The first row of each matrix denotes the payoffs and prices of states 1 and 2, respectively. (d) p = Xq Thus, we have p=



1 0.25 p=



0.4 0.6



0.55

!



You can conostruct a portfolio of identical payoffs to that of a bond 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

2 1

!

=



1 0.25



0.4 0.6

!

=



0.55



Now, to take advantage of this arbitrage opportunity, we shsould !     1/5 0 short bonds and buy 1 0.25 = 3/20 1/4 −1/5 1 which translates into 0.15 shares of stock sand 0.25 shares of riskfree assets per bond we are shorting. In period 2, we then sell the stock sand risk-free assets and buy back bonds. (e) We don’t need to use probabilities because we have a complete market which determines prices in each state, given payoffs in period 2. 2

2. (a) X1 = asset1payof f =



1 −0.5



X2 = asset2payof f =



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 advantage of this arbitrage opportunity, we long asset 1 an dshort asset 2. (b) 3X1 = X2 =





1.5 3

1 3



3p1 = 1.54



p2 = 1.5

By the law of one price, asset 2 is overvalued because asset 1 gives greater payoffs for an equivalent price. To take advantage of this arbitrage opportunity, long asset 1 and short asset 2. (c) Construct the Arrow-Debreu complete market. Since the weight matrix w = X ( − 1) for an AD security, we must find X ( − 1). 2 0 0 1/2

!

1/2 0 0 2

!

=

1 0 0 1

!

!

2 0 We can tell that w = 0 1/2 To find the price matrix p = w ∗ q, we compute: 2 0 0 1/2 

!

4 1.5

0.2 1



!

0.4 0.5

=

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 per 1 share of asset 3. 3. (a) The bank discount yield is inaccurate because i. it uses 360 instead of 365 days ii. it does not continuously compound, and thus overestimates the returns on the investment 3

iii. it takes the rate as a fraction of par value, not the ask price. (b) R=

10, 000 365/n P

(c) Before computers, the bank discount yield was significantly easier to compute than the effective annual yield. For small periods of time, the bank discount yield will not diverge significantly from its compounded counterpart. However, for large transactions, such a difference will be evident and the actual returns on the investment will be smaller than the estimated returns of the bank discount 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 pay 0.70. 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 are making a fair bet. (d) Three cases i. Case 1: Harvard wins the game EV(betting H wins by the end of the second round) = P(Harvard wins both rounds)*$1 + P(Harvard does not win both rounds)*0 = P(Harvard wins first round)*P(Harvard wins second round |Harvardwinsf irstround)∗$1 =0.7*0.5*$1 = $0.35 Intuitively, we should only have to invest the expected value to obtain $1 if Harvard wins the game. Let us check. 4

Suppose you invest $0.35 on a bet that Harvard wins the first round. If you win, you get $1 ∗ 0.35 0.7 = $0.5, which you would then reinvest in betting that Harvard will win the second game. If Harvard wins again, you will have captured 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 the last 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 alewinsf irstround)∗ $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, you get $1 ∗ 0.09 0.3 = $0.30, which you would then reinvest in betting that Yale will win the second game. If Yale wins again, you will have captured a profit of exactly $1. If Yale lost either game, you will have $0. iii. Case 3: H-Y Tie EV(betting they tie) = P(Yale wins round 1 and H wins Round 2)*$1 + P(H wins round 1 and Y wins round 2)*$1 = 0.3 ∗ 0.7 ∗ $1 + 0.7 ∗ 0.5 ∗ $1 = $0.21 + $0.35 = $0.56 In 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 from winning either bet in round 1 in a bet that the opposite team wins in round 2. 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 to hold when you have complete markets and no transaction costs. (f) This question is about a dynamically complete market, which is an extension of the static complete market idea discussed more fully in lecture. In a dynamically complete market, the ArrowDebreu securities involve a predetermined trading strategy in which 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 the static market, in which the cost of an asset is its expected value (assuming no transaction costs), except that you would have 4 AD 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 5

c = 425.69 (b) There are several conditions that we must consider, some more interesting than others. First, the mundane ones. First, we assume perfect markets, so companies will charge the equilibrium price cΛ such that cΛ > c but cΛ < cinsurance , the cost to the insurance company of providing such insurance. Since the problem does not stipulate zero transaction costs (meaning zero operating costs and zero profit for the insurance company), and it does not seem a reasonable assumption to make, we cannot determine specifically the price that the company will charge. Second, we consider the real world risks to insurance companies. In particular, consider the covariance of the events that neighboring houses burn down. If the covariance is high, then the potential risk of insurance for an insurance company covering houses in one geographic location is extremely high, and insurance companies may be less willing to insure this area.

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