Chapter+1

Chapter 1: Decision analysis
1. This exercise is about London Underground's courtcase. My tree is Therefore the decision is go for settlement and offer 400

2.This exercise is about Rickie's option to set up a business. His gross revenues are 9,000 or 4,000 with a 50-50 chance. His initial wealth is zero. What is the largest value of the cost he'll accept?

a. U(x) = ax+b, where a>0 In this case he is clearly risk-neutral. Anyway, he'll just move if his utility for his current wealth exceeds the expected utility of the venture: U(0)=b E[U(x)]=p.U(9000-C) + (1-p).U(4000-C) > b => 0.5 (9000a - a.C + b) + 0.5 (4000a - a.C + b) > b => 4500 a - 0.5 a C + 0.5 b + 2000a - 0.5ac + 0.5b > b => 6500 a - a C + b > b => aC < 6500a and, with a >0, C<6500

b.U(x)=x^1/2 for x>0 and U(x)=-(-x)^1/2 for x<0 In this case he is clearly risk averse. The same formulation applies: U(0)=0 E[U(x)]= 0.5 (9000-C)^1/2 - 0.5(C-4000)^1/2 (I'm assuming that C will turn out to be > 4000) 0.5 (9000-C)^1/2 - 0.5(C-4000)^1/2 > 0 (9000-C)^1/2 - (C-4000)^1/2 > 0 (9000-C)^1/2 > (C-4000)^1/2 9000 - C > C - 4000 2 C C < 6500 (again!)

c. U(x) = X^2 for x>0 and U(x) = -(x^2) for x<0 In this case he is risk lover Still U(0)=0 E[U(x)] = 0.5 (9000 - C)^2 - 0.5(4000-C)^2 (again I'm alreadu assuming that C>4000) 0.5 (9000 - C)^2 - 0.5(4000-C)^2 > 0 (9000-C)^2 > (4000-C)^2 9000-C > C-4000 (because 4000-C < 0) and C < 6500 again.
 * 9000-C| > |4000-C|

3. Find CARA of U(x)=a-b.exp(-cx) and CRRA of U(x)=a+b.ln(x) For the first function: U'(x)=b.c.exp(-cx) U''(x)=-c^2.b.exp(-cx) CARA=-U''(x)/U'(x) = - [-c^2.b.exp(-cx)]/[c.b.exp(-cx)] = c for c>0 individual is risk averse

For the second function: U'(x)=b/x U''(x)=-b/x^2 CRRA = -U''(x).x/U'(x) CRRA = - [-b/x^2].x/[b/x] = 1 so individual is risk averse

5. This exercise asks to draw the utility function of an utility maximiser who spends $10 on a lottery ticket with a chance of 1 in 1 million of $1 million and takes insurance at a premium of $100 with a 1% chance of claiming $1000. Using as benchmark a risk neutral individual, the fair value of the lottery ticket would be $1, while the fair value of the premium should be $10. So it is clear that the individual is risk lover for low values and risk averse for high(ish) values. His utility function therefore is convex-concave as in the figure below:

6. A decision-maker must choose betwwen (1) a sure payment of $200 (2) a gamble with prizes $0, $20, $450 and $1000 with resective probabilities 0.5, 0.3, 0.1 and 0.1; (3) a gamble with prizes $0, $100, $200, and $520 each with probability 0.25 a. Which choice will be made if the decision-maker is risk neutral? This is a straightforward tree: In this case wither gamble 1 or gamble 2 would have an expected value of 205 so the decision-maker would pick either

b. This item has a confusing question: "assume the decision-maker has a CARA (constant absolute risk aversion) utility of money function U(x)=-a.exp(-cx)+b and her certainty equivalent for a gamble with prizes $1000 and $0 equally likely is $470. Which choice will be made?" My doubt is whether U(x) is the utility function itself or the CARA. Because the text only mentions how to calculate CARA and never said how it should be used, I'll assume the function is indeed U(x). Th exercise tell us the equivalent for a gamble of $1000 and $0 so we can derive the coefficients a, b and c. Let's say that U(0), the worst outcome of the gamble equals 0 and U(1000) the best outcome of the gamble equals 1. We can find the coefficients of U(x): U(0)=-a+b=0 => b=a U(1000)=-a.exp(-1000c)+b=1 => U(1000) = a(1-exp(-1000c))=1 We also now that U(470)=0.5 U(0) + 0.5 U(1000) = 0.5 So U(470)=-a.exp(-470c)+b = a(1-exp(-470c) = 0.5 Joining the equations: a(1-exp(-1000c)) = 2.a.(1-exp(-470c)) 1-exp(-1000c)=2-2exp(-470c) 2exp(-470c)-exp(-1000c)=1 At this point my brain fails to solve algebraically (if someone know how to do it, please fix this!), so I'll use Excel's Goal Seek function to get c=0.000246 Therefore a=1/[1-exp(-1000c)]=4.58 So, U(X)=4.58(1-exp(-0.000246x)) Back to the tree, this time replacing for U(X): The option would be Gamble 2 with an Expected Utility of 0.2203

7. Henrika has U(x)=M^1/2 for M>0 and -(-M)^1/2 for M<0 over money payoffs M a. Given a lottery with outcomes $0 and $36 with probabilities 2/3 and 1/3, how much is she willing to pay to replace the lottery with its expected value? E[U(x)]=2/3.U(0) + 1/3.U(36) = 1/3.6 = 2 E(x)=2 => x=4, so Henrika will pay $4 for the lottery ticket, although its fair valye would be $12 => she is risk averse (we already knew that because of the shape of her utility function being concave).

b. Given the table of money payoffs below, which action maximises her expected utility? EU1=1/3 U(4) + 2/3 U(4) = 2 EU2 = 1/3 U(9) + 2/3 U(1) = 5/3
 * || S1 || S2 ||
 * A1 || 4 || 4 ||
 * A2 || 9 || 1 ||
 * || 1/3 || 2/3 ||

So action 1 maximises her expected utility

c. How much would Henrika be willing o pay for perfect information regarding the state of nature? Henrika's original tree is

With perfect information (for free), she can now make the optimal choices A2 if S1 and A1 if S2, so her Expected Value will become 2.333 (tree below) Her original Expected Utility was 2, so she increases her utility by 1/3. She would be willing to pay up to R for perfect information if she gets the same utility as before, considering the price to be paid, so: 2 = 1/3 U(9-R) + 2/3 U(4-R) 2 = 1/3 (9-R)^1/2 + 2/3 (4-R)^1/2 Solving the equation, we get R=$1.38