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Management Decision Analysis

Management Decision Analysis. Raghu Nandan Sengupta Industrial & Management Department Indian Institute of Technology Kanpur. Utility Analysis. Utility analysis

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Management Decision Analysis

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  1. Management Decision Analysis Raghu Nandan Sengupta Industrial & Management Department Indian Institute of Technology Kanpur RNSengupta,IME Dept.,IIT Kanpur,INDIA

  2. Utility Analysis Utility analysis Consider the same type of construction project is being undertaken by more than one company, who we will consider are the investors. Now different investors (considering they are investing their money, time, energy, skill, etc.) have different attributes and risk perception for the same project That is to say, each investor has with him/her an opportunity set. This opportunity set is specific to that person only. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  3. Utility Analysis Consider a shop floor manager has two different machines, A and B, (both doing the same operation) with him/her. The outcomes for the two different machines are given RNSengupta,IME Dept.,IIT Kanpur,INDIA

  4. Utility Analysis A B Outcome value(i) P[i] Outcome value(i) P[i] 15 1/3 20 1/3 10 1/3 12 1/3 15 1/3 8 1/3 In reality what would a person do if he or she has two outcome sets in front of him/her. For A we have the expected value of outcome as 13.33 and for B also it is 13.33 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  5. Utility Analysis A B Outcome value(i) P[i] Outcome value(i) P[i] 15 ½ 20 1/3 10 ¼ 12 1/3 15 ¼ 8 1/3 Now for A we have the expected value of outcome as 13.75 and for B it is still 13.33. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  6. Utility Analysis Outcome Team X Team Y Wins 40 45 Draws 20 5 Losses 10 20 Case I Case II Outcome Points Outcome Points Win 2 Win 5 Draw 1 Draw 1 Lose 0 Lose 0 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  7. Utility Analysis Case I Team A = 100; Team B = 95, which means A > B, i.e., A is ranked higher than B. Case II Team A = 220; Team B = 230, which means B > A, i.e., B is ranked higher than A. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  8. Utility Analysis On a general nomenclature we should have the expected value or utility given by here U(W) is the utility function which is a function of the wealth, W, while N(W) is the number of outcomes with respect to a certain level of income W. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  9. Utility Analysis Remember in general utility values cannot be negative, but many function may give negative values. For analysis to make the problem simple we may consider the value to be zero even though in actuality it is negative. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  10. Utility Analysis Consider an example where a single individual is facing the same set of outcomes at any instant of time but we try to analyze his/her expected value addition or utility separately based on two different utility functions 1) U[W(1)] = W(1) +1 2) U[W(2)] = W(2)2 + W(2) Outcome W(1) U[W(1)] P(W(1) W(2) U[W(2)] P(W(2) 15 1.5 2.5 0.15 1.5 3.75 0.15 20 2.0 3.0 0.20 2.0 6.00 0.20 25 2.5 3.5 0.25 2.5 8.75 0.25 10 3.0 4.0 0.10 3.0 12.00 0.10 5 0.5 1.5 0.05 0.5 0.75 0.05 25 5.0 6.0 0.25 5.0 30.00 0.25 Accordingly we have E[U(1)] = 3.825 and E[U(2)] = 12.69. So we can have a different decision depending on the form of utility function we are using. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  11. Utility Analysis Now we have two different utility functions used one at a time for two different decisions 1) U[W(1)] = W(1) - 5 and 2) U[W(2)] = 2*W(2)-W(2)1.25 Outcome W U[W(1)] U[W(2)] Decision (A) Decision (B) 8 4 0 2.34 Yes No 3 5 0 2.52 No Yes 4 6 1 2.60 No Yes 6 7 2 2.61 Yes No 9 8 3 2.54 Yes No 5 9 4 2.41 No Yes For utility function U[W(1)] U(A,1)=0*8/(8+6+9)+2*6/(8+6+9)+3*9/(8+6+9)=1.69 U(B,1)=0*3/(3+4+5)+1*4/(3+4+5)+4*5/(3+4+5)=2.00 For utility function U[W(2)] U(A,2)=2.34*8/(8+6+9)+2.61*6/(8+6+9)+2.54*9/(8+6+9)2.50 U(B,2)=2.52*3/(3+4+5)+2.60*4/(3+4+5)+2.41*5/(3+4+5) 2.50 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  12. Utility Analysis Example # 01: A venture capitalist is considering two possibilities of investment. The first alternative is buying government treasury bills which cost Rs. 6,00,000. While the second alternative has three possible outcomes, the cost of which are Rs.10,00,000, Rs. 5,00,000 and Rs. 1,00,000 respectively. The corresponding probabilities are 0.2, 0.4 and 0.4 respectively. If we consider the power utility function U(W)=W1/2, then the first alternative has a utility value of Rs.776 while the second has an expected utility value of Rs. 609. Hence the first alternative is preferred. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  13. Utility Analysis Would the above problem give a different answer if we used an utility function of the form U(W) = W1/2 + c (where c is a positive o a negative constant)? RNSengupta,IME Dept.,IIT Kanpur,INDIA

  14. Utility Analysis In a span of 6 days the price of a security fluctuates and a person makes his/her transactions only at the following prices. We assume U[P] = ln(P) Day P U[P] Number of Outcomes Probability 1 1000 6.91 35 0.35 2 975 6.88 20 0.20 3 950 6.86 10 0.10 4 1050 6.96 15 0.15 5 925 6.83 05 0.05 6 1025 6.93 15 0.15 Expected utility is 6.91 If U[P]= P0.25, then expected utility is 33.63 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  15. Utility Analysis (Important properties) General properties of utility functions • Non-satiation: The first restriction placed on utility function is that it is consistent with more being preferred to less. This means that between two certain investments we always take the one with the largest outcome, i.e., U(W+1) > U(W) for all values of W. Thus dU(W)/dW > 0 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  16. Utility Analysis (Important properties) • If we consider the investors or the decision makers perception of absolute risk, then we have the concept/property of (i) risk aversion, (ii) risk neutrality and (iii) risk seeking. Let us consider an example now RNSengupta,IME Dept.,IIT Kanpur,INDIA

  17. Utility Analysis (Important properties) Invest Prob Do not invest Prob 2 ½ 1 1 0 ½ Price for investing is 1 and it is a fair gamble, in the sense its value is exactly equal to the decision of not investing RNSengupta,IME Dept.,IIT Kanpur,INDIA

  18. Utility Analysis (Important properties) Thus • U(I1)*P(I1) + U(I2)*P(I2) < U(DI)*1  risk averse • U(I1)*P(I1) + U(I2)*P(I2) = U(DI)*1  risk neutral • U(I1)*P(I1) + U(I2)*P(I2) > U(DI)*1  risk seeker RNSengupta,IME Dept.,IIT Kanpur,INDIA

  19. Utility Analysis (Important properties) Another characteristic by which to classify a risk averse, risk neutral and risk seeker person is • d2U(W)/dW2 = U(W) < 0  risk averse • d2U(W)/dW2 = U(W) = 0  risk neutral • d2U(W)/dW2 = U(W) > 0  risk seeker RNSengupta,IME Dept.,IIT Kanpur,INDIA

  20. Utility Analysis (Important properties) Utility curves RNSengupta,IME Dept.,IIT Kanpur,INDIA

  21. Utility Analysis and Marginal Utility Marginal Utility Function • Marginal utility function looks like a concave function  risk averse • Marginal utility function looks neither like a concave nor like a convex function  risk neutral • Marginal utility function looks like a convex function  risk seeker RNSengupta,IME Dept.,IIT Kanpur,INDIA

  22. Utility Analysis and Marginal Utility Marginal Utility Rate • Marginal utility rate is increasing at a decreasing rate  risk averse • Marginal utility rate is increasing at a constant rate  risk neutral • Marginal utility rate is increasing at a increasing rate  risk seeker RNSengupta,IME Dept.,IIT Kanpur,INDIA

  23. Utility Analysis and Marginal Utility Risk avoider RNSengupta,IME Dept.,IIT Kanpur,INDIA

  24. Utility Analysis and Marginal Utility Risk neutral RNSengupta,IME Dept.,IIT Kanpur,INDIA

  25. Utility Analysis and Marginal Utility Risk seeker RNSengupta,IME Dept.,IIT Kanpur,INDIA

  26. Utility Analysis and Marginal Utility Few other important concepts Condition Definition Implication Risk aversion Reject a U(W) < 0 fair gamble Risk neutrality Indifference to U(W) = 0 a fair gamble Risk seeking Select a U(W) > 0 fair gamble RNSengupta,IME Dept.,IIT Kanpur,INDIA

  27. Utility Analysis(Other concepts, i.e., A(W)) • Absolute risk aversion property of utility function where by absolute risk aversion we mean A(W) = - [d2U(W)/dW2]/[dU(W)/dW] = - U(W)/U(W) RNSengupta,IME Dept.,IIT Kanpur,INDIA

  28. Utility Analysis(Other concepts, i.e., A(W)) • Assume an investor has wealth of amount W and a security with an outcome represented by Z, which is a random variable. • Assume Z is a fair gamble, such that E[Z] = 0 and V[Z] = 2Z and the utility function is U(W). • If WC is the wealth such that we can write this as a decision process having two chooses, i.e., Choice A Choice B W+Z WC RNSengupta,IME Dept.,IIT Kanpur,INDIA

  29. Utility Analysis(Other concepts, i.e., A(W)) • Now if the person is indifferent between decision/choice A and decision/choice B, then we must have E[A] = E[B], i.e., E[U(W+Z)] = E[U(WC)] = U(WC)*1 • The person is willing to give maximum of (W – WC) to avoid risk, i.e., the absolute risk (say ) = (W- WC). • Expanding U(W+Z) in a Taylors series around W and we would get the answer. Assignment # 01: This is an assignment and for the proof check any good book in economics or game theory which has utility as a part of it RNSengupta,IME Dept.,IIT Kanpur,INDIA

  30. Utility Analysis(Other concepts, i.e., A(W)) For the three different types of persons • Decreasing absolute risk aversion  A(W) = dA(W)/d(W) < 0 • Constant absolute risk aversion  A(W) = dA(W)/d(W) = 0 • Increasing absolute risk aversion  A(W) = dA(W)/d(W) > 0 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  31. Utility Analysis(Other concepts, i.e., A(W)) Condition Definition Property • Decreasing As wealth A(W) < 0 absolute risk increases the amount aversion held in risk assets increases • Constant As wealth A(W) = 0 absolute risk increases the amount aversion held in risk assets remains the same • Increasing As wealth A(W) > 0 absolute risk increases the amount aversion held in risk assets decreases RNSengupta,IME Dept.,IIT Kanpur,INDIA

  32. Utility Analysis(Other concepts, i.e., R(W)) • Relative risk aversion property of utility function where by relative risk aversion we mean R(W) = - W * [d2U(W)/dW2]/[dU(W)/dW] = - W * U(W)/U(W) RNSengupta,IME Dept.,IIT Kanpur,INDIA

  33. Utility Analysis(Other concepts, i.e., R(W)) • Consider the same example as the previous prove but now with /= (W- WC)/W, which is the per cent of money the person will give up in order to avoid the gamble and E[Z]=1. • Z represented the outcome per rupee invested. • Therefore for W invested we obtain W*Z amount of money. On the other hand we have a sure investment of WC. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  34. Utility Analysis(Other concepts, i.e., R(W)) • For the investor to be indifferent between the two decision processes we must have:  E[U(W*Z)] = E[U(WC)] • Consider now E(U(W*Z)] and expanding it in a Taylors series around W and we would get our result Assignment # 02: This is an assignment and for the proof check any good book in economics or game theory which has utility as a part of it RNSengupta,IME Dept.,IIT Kanpur,INDIA

  35. Utility Analysis(Other concepts, i.e., R(W)) For the three different types of persons • Decreasing relative risk aversion  R(W) = dR(W)/dW < 0 • Constant relative risk aversion  R(W) = dR(W)/dW = 0 • Increasing relative risk aversion  R(W) = dR(W)/dW > 0 RNSengupta,IME Dept.,IIT Kanpur,INDIA

  36. Utility Analysis(Other concepts, i.e., R(W)) Condition Definition Property • Decreasing As wealth increases R(W) < 0 relative risk the % held in risky aversion assets increases • Constant As wealth increases R(W) = 0 relative risk the % held in risky aversion assets remains the same • Increasing As wealth increases R(W) > 0 relative risk the % held in risky aversion assets decreases RNSengupta,IME Dept.,IIT Kanpur,INDIA

  37. Examples of Utility Functions U(W) = W – b*W2 Then: • A(W)=4*b2/(1- 2*b*W)2 • R(W)=2*b/(1- 2*b*W)2 Hence we use this utility function for people with (i) increasing absolute risk aversion and (ii) increasing relative risk aversion. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  38. Examples of Utility Functions RNSengupta,IME Dept.,IIT Kanpur,INDIA

  39. Examples of Utility Functions RNSengupta,IME Dept.,IIT Kanpur,INDIA

  40. Examples of Utility Functions U(W) = ln(W) Then: • A(W) = - 1/W2 • R(W) = 0 We use this utility function for people with (i) decreasing absolute risk aversion and (ii) constant relative risk aversion RNSengupta,IME Dept.,IIT Kanpur,INDIA

  41. Examples of Utility Functions RNSengupta,IME Dept.,IIT Kanpur,INDIA

  42. Examples of Utility Functions RNSengupta,IME Dept.,IIT Kanpur,INDIA

  43. Examples of Utility Functions U(W) = - e-aW Then: • A(W) = 0 • R(W) = a We use this utility function for people with (i) constant absolute risk aversion and (ii) increasing relative risk aversion. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  44. Examples of Utility Functions RNSengupta,IME Dept.,IIT Kanpur,INDIA

  45. Examples of Utility Functions RNSengupta,IME Dept.,IIT Kanpur,INDIA

  46. Examples of Utility Functions U(W) = c*Wc Then: • A(W) = (c-1)/W2 • R(W) = 0. We use this utility function for people with (i) decreasing absolute risk aversion (ii) constant relative risk aversion. RNSengupta,IME Dept.,IIT Kanpur,INDIA

  47. Examples of Utility Functions RNSengupta,IME Dept.,IIT Kanpur,INDIA

  48. Examples of Utility Functions RNSengupta,IME Dept.,IIT Kanpur,INDIA

  49. Utility Function (An Example) Example # 02:Suppose U(W) = W1/4 and we are required to find the properties of this utility function and also draw the utility function graph. Now RNSengupta,IME Dept.,IIT Kanpur,INDIA

  50. Utility Function (An Example) Let us find absolute risk aversion and relative risk aversion properties of this particular utility function. RNSengupta,IME Dept.,IIT Kanpur,INDIA

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