IE5203 (2021) soln-6-1

IE5203 Decision Analysis Solutions to Chapter 6 Exercises

P6.1 (a) u(w) = w – β w2

u’(w) = 1 – 2β w u’’(w) = –2β

Risk tolerance ββρ

221

)('')(')( w

wuwuw −

=−

=

Degree of absolute risk aversionw

wrββ21

2)(−

=

Note that for u(w) to be increasing and concave (i.e., risk averse), it is sufficient for coefficient β to be strictly positive.

(b) u(w) = ln w

u'(w) = 1/w u’’(w) = –1/w2

Risk tolerance wwuwuw =

−=

)('')(')(ρ

Degree of risk aversion = w

wr 1)( =

(c) u(w) = sgn(β) wβ

u’(w) = sgn(β) β wβ-1 u’’(w) = sgn(β) β (β – 1) wβ – 2

Risk tolerance β

ρ−

=−

=1)(''

)(')( wwuwuw

Degree of absolute risk aversion = w

wr β−=

1)(

IE5203 (2021) soln-6-2

P6.2 Given that John has the utility function u(x) = 1 – 3-x/50 over the range of x = -$50 to $5,000.

The utility function may be rewritten as 50/3ln50/ 131)( xx exu −− −=−= (a) The utility function is increasing and concave. Hence John is risk averse. (b) John’s risk tolerance = $ 50 / ln 3.

Hence John’s degree of risk aversion = 1 / risk tolerance = ln 3 / 50 = 0.0219722 $-1 (c) We want to find p such that certainty equivalent of the deal is zero.

u(0) = p u(50) + (1 – p) u(–50) 1 – 30 = p (1 – 3-1) + (1 – p) (1 – 31) 0 = p (2/3) + (1– p)( –2) p = 3/4

P6.3 Consider L2:

By the delta property, if we add the amount a to all its outcomes of L2, we get

Similarly, by the delta property, if we add the amount b to all the outcomes of L2, , we get

$50

-$50

p

1-p $ 0 ~

c + a

d + a

q

1- q CE(L2) + a ~

c + b

d + b

q

1- q CE(L2) + b ~

IE5203 (2021) soln-6-3

Consider L1: By the delta property, if we add the amount CE(L2) to all its outcomes, we get

Finally, by the subsitution rule, the above deal is equivalent to L3.

Hence CE(L3) = CE(L1) + CE(L2). P6.4 Since George has a constant risk tolerance of $1,000, it follows that he has constant absolute risk aversion or the delta property. Hence his utilty function is exponential in form. We let u(x) = – exp(– x/1,000).

Maximum Expected Utility at A = -0.6065307 Certainty Equivalent at A = u-1(-0.60653) = $500 The required preference probability is p, such that

Hence u(500) = p u(2,500) + (1 – p) u(–1,000)

⇒ ))(1())(( 000,1000,1

000,1500,2

000,1500

epepe −−+−=−−−

⇒ p = 0.80106

Note that we would have got the same answer if we had used the utility function u(x) = 1– exp(-x/1000), or if we had assumed u(x) = a – b exp(-x/1000) and fitted the constants a and b to the boundary conditions u(-$1,000) = 0 and u($2,500) = 1.

a + CE(L2)

b + CE(L2)

p

1- p CE(L1) + CE(L2) ~

$400 0.5

0.5 $300

$500

$900

$100

0.4

0.6

B

D

C

A

x u(x) -0.6065307

-0.6065307

-0.4065697

-0.9048374

-0.7055303

$2,500

-$1,000

p

1-p $500 ~

IE5203 (2021) soln-6-4

P6.5 (a) Susan satisfies delta property ⇒ utility function is of the form u(x) = a - b e –x/ρ where ρ is the

risk tolerance.

Given

u(0) = 0.8 u(2) + 0.2 u(-2) a – b = 0.8 (a – b e-2/ρ) + 0.2 (a – b e2/ρ) 1 = 0.8 e-2/ρ + 0.2 e2/ρ Let x = e2/ρ

x2 – 5x + 4 = 0 ⇒ (x – 1) (x – 4) = 0 ⇒ x = 1 or x = 4.

Hence = e2/ρ = 4 ⇒ 44.1$2ln

1==ρ

Susan risk tolerance = $1.44

(b) Susan’s risk attitude is risk averse since her risk tolerance is positive. (c) Susan’s utility function such that u(L= 0) = 0 and u(H =5) = 1 is

)21( 1.0322581)1( 1.0322581

11)(

2ln

/)(

/)(

-x

-x

LH

Lx

- - e

eexu

=

=−−

= −−

−−

ρ

ρ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Utili

ty

$X

$2

-$2

0.8 $ 0 ~

0.2

IE5203 (2021) soln-6-5

P6.6 Current wealth = $200.

Wealth Utility function 2000

)(2wwu = , w ≥ 0.

(a) Let s = personal indifferent selling price.

u(200 + s) = 0.25 u(200+25) + 0.5 u(200 + 50 ) + 0.25 u(200 – 50)

22.5562 $ )150( 25.0)250( 5.0)225( 25.0)200(

2000150 25.0

2000250 5.0

2000225 25.0

2000)200(

2222

2222

=++=+

+

+

=

+

ss

s

Hence Susan’s PISP = $22.56

(b) Let b = personal indifferent buying price.

u(200) = 0.25 u(200+25 – b) + 0.5 u(200 + 50 – b) + 0.25 u(200 – 50 – b)

2222

2222

)150()250(2)225()200(4

2000)150(25.0

2000)250(5.0

2000)225(25.0

2000200

bbb

bbb

−+−+−=

−+

−+

−=

03812517504 2 =+− bb

Solving: b = $22.99425 (okay) or $414.5057 (rejected)

Hence Susan’s PIBP = $22.99

$ 200 + 25 0.25

0.25 $ 200 - 50

$ 200 + 50 0.5 $ 200 + s ~

$ 200 + 25 - b 0.25

0.25 $ 200 – 50 - b

$ 200 + 50 - b 0.5 $ 200 ~

IE5203 (2021) soln-6-6

P6.7 Note that the risk neutral case for this problem was covered in Chapter 4 Exercises.

(a) Rex with 50001)(x

exu−

−= , satisfies the delta property. i. Base Decision model:

• Expected Utility at A = 0.048771 • Certainty Equivalent = u-1 (0.048771) = $250.00

ii. Decision model with clairvoyance on D before making decision A.

• Expected Utility of free clairvoyance on D before A = 0.057702 • Certainty Equivalent = u-1 (0.057702) = $297.17 • Therefore, Value of clairvoyance on D before A = $ 297.17 – 250.00 = $47.17

0.5 $375

0.5 D

$100

$107 C

$275 0.3

0.7

$250

B

A

x u(x) 0.053515

0.072257

0.019801

0.021173

0.048771

0.046029

0.021173

0.053515

0.0048275

0.048771

0.048771

0.5

$375

0 D

$100

$107

C

$275 0.3

0.7

$250

B

A

x

0 $375

1 D

$100

$107

C

$275 0.3

0.7

$250

B

A

0.030875 “D”

0.5

1

0.057702

u(x) 0.053515

0.072257

0.019801

0.021173

0.048771

0.053515

0.072257

0.019801

0.021173

0.048771

0.053515

0.072257

0.021173 0.072257

0.066634

0.048771

0.066634

0.053515

0.019801

0.021173

0.021173

0.048771

0.048771

IE5203 (2021) soln-6-7

Simplified decision model without 0-1 probability chance node.

iii. Decision model with clairvoyance on D before decision C but after decision A.

• Expected Utility of free clairvoyance on D before decision C but after A = 0.048771 • Certainty Equivalent = u-1 (0.048771) = $250.00 • Therefore, Value of clairvoyance on D before C but after decision A = $250 - $250 = $ 0

0.5

$375

$107

C

$275 0.3

0.7

$250

B

A

x

$100

$107

C

$275 0.3

0.7

$250

B

A

0.030875 D

0.5

0.057702

u(x)

0.053515

0.072257

0.021173

0.048771

0.053515

0.019801

0.021173

0.048771

0.072257

0.072257

0.066634

0.048771

0.066634

0.019801

0.021173

0.021173 0.048771

1 $375

0 D

$100

$107

C

$275

0.3

0.7

$250

B

A 0 $375

1 D

$100

$107

C

“D”

0.5

0.5

x u(x)

0.053515

0.072257

0.019801

0.021173

0.072257

0.019801

0.021173

0.048771

0.053515

0.072257

0.021173

0.072257

0.019801

0.021173

0.021173

0.046715

0.048755

0.048771

0.048771

IE5203 (2021) soln-6-8

Value of clairvoyance is zero because Rex will make exactly the same decisions here as he did without any information, i.e., the optimal decision policy is the same, with and without information. Hence the value of information is ZERO.

Simplified decision model without 0-1 probability chance node.

iv. Decision model with free clairvoyance on B before decision A.

• Expected Utility of free clairvoyance on B before making decision A = 0.050194 • Certainty Equivalent = u-1 (0.050194) = $257.49 • Hence, the most Jeanne would pay for clairvoyance on B before A is $257.49 – $250.00 =

$7.49.

$375

$107

C

$275

0.3

0.7

$250

B

A $100

$107

C

D

0.5

0.5

x u(x)

0.053515

0.072257

0.021173

0.019801

0.021173

0.048771

0.072257

0.021173

0.046715

0.048755

0.048771

0.5 $375

0.5 D

$100

$107

C

0

1

$250

B

A

0.046029

$250

A

B

$275

“B”

1

0

0.3

0.7

0.050194

x u(x) 0.053515

0.048771

0.072257

0.019801

0.021173

0.048771

0.053515

0.048771

0.053515

0.021173

0.046029

0.046029

0.048771

0.048771

IE5203 (2021) soln-6-9

Simplified decision model without 0-1 probability nodes:

(b) Paulina utility function is 10000

)(2wwu = , and her current wealth is $1,000.

Note that Paulina does not follow the delta property as her utility function is neither exponential nor linear. Hence we need to consider Paulina’s wealth in the analysis. i. Base Decision Model:

• Maximum Expected Wealth Utility = 157.29. • Wealth Certainty Equivalent = u-1 (157.29) = $ 1,254.31. • Certainty Equivalent for decision A is $ 1,254.31 – $1,000 = $ 254.31.

0.5 $375

0.5 D

$100

$107

C

$250

A

0.046029

$250 A

$275

B

0.3

0.7

0.050194

x u(x) 0.053515

0.048771

0.072257

0.019801

0.021173

0.048771

0.053515

0.046029

0.048771

0.5 $375

0.5 D

$100

$107 C

$275 0.3

0.7

$250

B

A

x u(w0+x) 162.56

189.06

121.06

122.54

156.25

155.03

122.54

162.56

157.29

156.25

w0+x

1375

1100

1107

1275

1250

157.29 155.03

IE5203 (2021) soln-6-10

ii. Decision model if Paulina pays $15 for clairvoyance on D before making decision A:

• Expected utility when Paulina pays $15 for clairvoyance on D before making decision A =

164.81 • Expected utility without information = 157.29 < 164.81 • Hence, Paulina should pay $15 for clairvoyance on D.

Note that Paulina does not satisfy the delta property. Therefore, we cannot use the “CE with free clairvoyance – CE no clairvoyance” method to compute her value of clairvoyance for any of the uncertain variables.

0.5

$375

$107

C

$275 0.3

0.7

$250

B

A

x

$100

$107

C

$275 0.3

0.7

$250

B

A

131.10 D

0.5

164.81

158.76

184.96

119.25

152.52

158.76

117.73

119.25

152.52

184.96 177.10

w0 -15 + x u(w0 -15 + x)

1360

1092

1260

1235

1085

1092

1260

1235

158.76

184.96 177.10

152.52 158.76

117.73

119.25 119.25

152.52

152.52

IE5203 (2021) soln-6-11

P6.8 (a) Stan is risk neutral with initial wealth = $20.

i. Stan’s certainty equivalent for Deal A = (0.5)(15) + (0.5)( –9) = $3 Since the certainty equivalent > $0, Stan should accept Deal A. ii. Stan’s certainty equivalent for Deal B = (0.5)(25) + (0.5)( –10) = $7.50. Yes, Stan should accept Deal B also since its certainty equivalent > $0. iii. If Deals A and B are bundled, the situation appears as follows:

Stan’s certainty equivalent for the bundle = (0.25) (40 + 5+ 16 – 19) = $10.50

Note that this is exactly CE(A) + CE(B) = $3.00 + $7.50 = $10.50 (See P6.3 for the general case with delta property).

$15 0.5

0.5 -$9

$25 0.5

0.5 -$10

Deal A Deal B

0.5

0.5

15 + 25 = $40 0.5

0.5

Deal A+B

0.5

0.5

15 - 10 = $5

-9 + 25 = $16

-9 - 10 = -$19

IE5203 (2021) soln-6-12

(b) Burke: u(w)=ln(w), where w is total wealth. Initial wealth wo = $20.

i. Burke is offered Deal A:

Expected utility for Deal A = (0.5) ln(35) + (0.5) ln(11) = 2.9766 Wealth CE for A = u-1(2.9766) = exp(2.9766) = $19.62 CE for A = Wealth CE – Initial wealth = $19.62 – $20.00 = –$0.38 < 0 Since the Certainty Equivalent of A < 0, Burke should not accept Deal A. ii. Burke is offered Deal B:

Expected wealth utility for Deal B = (0.5) ln(45) + (0.5) ln(10) = 3.0546 Wealth CE for B = u-1(3.0546) = exp(3.0546) = $21.21 CE for A = Wealth CE - Initial wealth = $21.21 – $20.00 = $1.21 > 0 Since the Certainty Equivalent of B > 0, Burke should accept Deal B.

iii. If Deals A and B are bundled, the situation appears as follows:

Expected Wealth Utility for bundle A+B = (0.25)(ln(60) + ln(25) + ln(36) + ln(1)) = 2.724 Wealth CE for bundle = exp (2.724) = $15.24 CE for bundle = $15.24 – $20.00 = –$4.76 < 0 Therefore Burke should not accept the bundled deal.

(c) For Stan (risk neutral ⇒ delta property), CE(A) + CE(B) = CE(A+B). For Burke (non delta property), CE(A) + CE(B) ≠ CE(A+B).

20+15=$35 0.5

0.5 20 – 9 =$11

Deal A wo+x ln(wo+x)

3.55534

2.3979

20+25=$45 0.5

0.5 20 - 10=$10

Deal B wo+x ln(wo+x)

3.80666

2.30259

0.5

0.5

15+25+20 = $60 0.5

0.5

Deal A+B

0.5

0.5

15–10+20 = $25

-9+25+20 = $36

-9-10+20 = $1

wo+x

IE5203 (2021) soln-6-13

P6.9 Kay utility function is 100092)(x

xu−

−= ⇒ Delta property.

(a) Expected utility for “Invest” = 0.50591

Expected utility for “Do not invest” = 1 (b) Expected dollar value for “Invest”

= (0.4)(0.9 × 600 + 0.1 × -300) + (0.6)( 0.1 × 500 + 0.9 × –400) = $18.

Expected dollar value for “Do not invest” = $0. (c) Since EU(Do not invest) > EU(Invest), Kay’s best decision is “Do not invest” Certainty Equivalent = $0. (d) We use the answers found in part (a) and not part (b) because Kay is not risk neutral (even

though she satisfies the delta property).

We have to compare expected utilities and not expected dollar values. It would be okay to use expected dollar values if Kay was risk neutral. If we had used expected dollar values instead, we would have ignored Kay’s risk attitude and may get a wrong answer since expected dollar values are not the same as the certainty equivalents.

$600

$0

0.9

0.1

A up

1.56845

0.1

0.9

B up

B down -$300

$500

-$400

A down

B up

B down

Do not invest

-0.20071

0.50591

Invest

0.4

0.6

x u(x)

1.7324

0.0668

1.6667

-0.4082

1

0.50591

1

IE5203 (2021) soln-6-14

(e) Since Kay satisfies the delta property, we can easily find her value of clairvoyance on A using the difference of CEs method.

Decision model with free clairvoyance on Stock A:

Expected Utility with free clairvoyance = 1.22634. To find the certainty equivalent: 22634.192 1000 =− − x

⇒ x = $116.8 Certainty equivalent with free clairvoyance on A = $116.80 Hence value of clairvoyance on A = $116.80 –$ 0 = $116.80 > $10. Therefore Kay should pay $10 for clairvoyance on A. (f) Decision model with free clairvoyance on Stock B:

Expected utility with free clairvoyance on B = 1.30366 30366.192 1000 =− − x

⇒ x = $164.71 Certainty equivalent with free clairvoyance on B = $164.71 Hence value of clairvoyance on B = $164.71 –$ 0 = $164.71 > $10.

Therefore Kay should pay $10 for clairvoyance on B.

$600

$0

0.9

0.1

Invest

1.56584

B up

B down -$300

Do not invest

1.22634 1

$0

0.1

0.9

Invest -0.2007

B up

B down -$400

Do not invest 1

$500

A up

A down

0.4

0.6 1

x u(x) 1.7324

0.0668

1

1.6667

-0.4082 1

1.56584

$600

$0

6/7

1/7

Invest

1.7230

A up

A down $500

Do not invest

1.30366 1

$0

2/29

27/29

Invest -0.3754

A up

A down -$400

Do not invest 1

-$300

B up

B down

0.42

0.58 1

x u(x) 1.7324

1.6667

1

0.0668

-0.4082 1

1.7230

IE5203 (2021) soln-6-15

(g) Decision model with free joint clairvoyance on Stock A & Stock B:

Expected utility with free clairvoyance on both A and B = 1.30366 Certainty Equivalent with free clairvoyance on both A and B = u-1 (1.30366) = $164.71

Kay’s value of clairvoyance on A and B together = $164.71 –$ 0 = $164.71.

$600

$0 0.9

0.1

B up

B down

1.30366

1

Invest

Do not invest A up

A down

0.4

0.6

-$300

$0

Invest

Do not invest

$500

$0 0.9

0.1

B up

B down 1

Invest

Do not invest

-$400

$0

Invest

Do not invest

x u(x) 1.7324

1

0.0668

1

1

1

1.6667

-0.4082

1.7324

1.6667

IE5203 (2021) soln-6-16

P6.10 Daniel: 00

,106,10

)(<≥

++

=ww

ww

wu , where w is total assets in dollars. wo = $20.

Mad Dog: 502)(

x

xu = ; wo = $75. Note that Mad Dog has the delta property.

(a) Daniel’s expected wealth utility = (0.7) u(80 + 20) + (0.3) u(–40 + 20)

= (0.7) u(100) + (0.3) u(–20) = (0.7) (110) + (0.3)( –110) = 44 Wealth Certainty Equivalent = 44 – 10 = $34.00 Daniel’s Certainty Equivalent for deal = $34.00 – $20.00 = $14.00

(b) Daniel personal indifferent selling price is $14.00

We need to compute Mad Dog personal indifferent buying price. Since Mad Dog has exponential utility curve, he satisfies the delta property. Furthermore, personal indifferent buying price = personal indifferent selling price = certainty equivalent. CE for Mad Dog = u-1 (0.7 u(80) + 0.3 u(–40) ) = u-1 (0.7 ( 2^(80/50)) + 0.3 ( 2^(–40/50) )) = u-1 (2.2943)

= $59.90 = Personal indifferent buying price for Mad Dog.

Therefore, we can make a maximum of $59.90 – $14.00 = $45.90 by buying the deal from Daniel and selling it to Mad Dog.

10

0 w

u(w)

$80 0.7

0.3 -$40