EF8904Financial Theory

Fall 2013Ryerson University

Solutions to Problem Set 2

1.(i) [3 points] If he already owns the lottery, Ps must satisfy

U(Y + Ps) = U(Y +G) + (1 )U(Y +B) Ps = U1 (U(Y +G) + (1 )U(Y +B)) Y.

1.(ii) [3 points] If he does not own the lottery, the maximum he would be willing topay, Pb, must satisfy:

U(Y ) = U(Y Pb +G) + (1 )U(Y Pb +B)

1.(iii) [6 points] Assume now that = 1/2, G = 26, B = 6, Y = 10. Pb satisfies:

U(10) =1

2U(10 Pb + 26) +

1

2U(10 Pb + 6)

10 =

1

2(36 Pb)1/2 +

1

2(16 Pb)1/2

Pb 13.5

Ps satisfies:

(10 + Ps)1/2 =

1

2(10 + 26)1/2 +

1

2(10 + 6)1/2

Ps = 15

Clearly, Pb < Ps. If the agent already owned the asset his minimum wealth is 10+6 =16. If he is considering buying, its wealth is 10. In the former case, he is less riskaverse and the lottery is worth more. This is because risk tolerance generally varieswith wealth.

If the agent exhibits constant absolute risk aversion, we should get Ps = Pb. To verifythis, try using the CARA utility function, U(x) = 1 ex.

2.(i) [3 points] Notice that FB(3) = 1/3 > 1/4 = FA(3), but FB(5) = 1/3 < 3/4 =FA(5), violates first order stochastic dominance. Therefore, neither investment dom-inates the other in the first order stochastic dominance sense.

2.(ii) [3 points] The calculations for second order stochastic dominance are as follows:

1

EF8904Financial Theory

Fall 2013Ryerson University

x x0fB(t)df

x0FB(t)dt

x0fA(t)dt

x0FA(t)dt

x0

[FB(t) FA(t)]dt0 0 0 0 0 01 1/3 0 0 0 02 1/3 1/3 1/4 0 1/33 1/3 2/3 1/4 1/4 5/124 1/3 1 3/4 1/2 1/25 1/3 4/3 3/4 5/4 1/126 2/3 5/3 3/4 2 -1/3

There is no second order stochastic dominance as the sign of x0

[FB(t) FA(t)] dt isnot monotonic.

2.(iii) [3 points] See Figure 1.

0 1 2 3 4 5 6 7 8 9 100

1/41/3

1/2

2/3

3/4

1

FB

FA

Figure 1: The agents preferences.

2.(iv) [3 points] For investment A, the mean and variance are A = 4.75 and 2A =

6.6875. For investment B, they are B = 5 and 2B = 8.6667. Since A < B and

2A < 2B, neither investment dominates the other under the mean-variance criterion.

2

EF8904Financial Theory

Fall 2013Ryerson University

2.(v) [3 points] Bernies expected utilities from investments A and B are

EAU(xA) =1

4log(2) +

1

2log(4) +

1

4log(9) = 1.4157

EBU(xB) =1

3log(1) +

1

3log(6) +

1

3log(8) = 1.2904

Therefore, Bernie prefers investment opportunity A.

3. [10 points] Given , the minimum possible variance is

minw1

w2121 + (1 w1)222 + 2w1(1 w1)12

Imposing the condition that 1 = 2 = , the problem becomes

minw1

w212 + (1 w1)22 + 2w1(1 w1)2

= minw1

[w21 + (1 w1)2 + 2w1(1 w1)

]2

= minw1

[w21 + (1 w1)2 + 2w1(1 w1) 2w1(1 w1)(1 )

]2

= minw1

[1 2w1(1 w1)(1 )]2

Note that any w1 achieves the minimum variance if = 1. For any [1, 1), theobjective function is convex. Ignoring the trivial case, the first order condition istherefore necessary and sufficient. Taking the first order condition and solving yieldsw1 = 1/2, the desired result.

4.(i) [3 points] Notice that

EU(R) =

U(x)dF (x)

=

U(x)f(x)dx

With Normally distributed returns,

EU(R) =

U(x)1

2exp

[1

2

(x

)2]dx = (, 2)

4.(ii) [5 points] Differentiating with respect to , we have

EU(R)

=

1

2

U(x)(x )f(x)dx > 0

3

EF8904Financial Theory

Fall 2013Ryerson University

since utility is increasing. That is, the utility associated with each x < is less thanthe utility associated with the x > .

Alternatively, you could define z = (R )/. Using the method of integration bysubstitution, the expected utility can be written

EU(R) =

U(+ z)(z)dz

where is the probability density function for the standard normal distribution.Taking the derivative with respect to yields

EU(R)

=

U (+ z)(z)dz > 0

4.(iii) [5 points] Differentiating with respect to 2, we have

EU(R)

2=

1

24

U(x)[(x )2 2

]f(x)dx

By the definition of concavity, U(x) < U() + U ()(x ). Therefore,

EU(R)

2=

1

24

U(x)[(x )2 2

]f(x)dx