Management Decision Analysis

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

RNSengupta,IME Dept.,IIT Kanpur,INDIA
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. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis A B Outcome value(i) P[i] Outcome value(i) P[i] 15 1/ /3 10 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 and for B also it is 13.33 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 and for B it is still MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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

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. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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) Accordingly we have E[U(1)] = and E[U(2)] = So we can have a different decision depending on the form of utility function we are using. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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) Yes No No Yes No Yes Yes No Yes No 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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 Hence the first alternative is preferred. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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)? MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 Expected utility is 6.91 If U[P]= P0.25, then expected utility is 33.63 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Invest Prob Do not invest Prob 2 ½ 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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility curves MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Risk avoider MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Risk neutral MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Risk seeker MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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) MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 remains the same Increasing As wealth A(W) > 0 decreases MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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) MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 aversion assets remains the same Increasing As wealth increases R(W) > 0 aversion assets decreases MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

W W-b*W^2 A(W) A'(W) R(W) R'(W) 2.00 3.00 -0.25 0.06 -0.50 -0.13 5.25 -0.20 0.04 -0.60 -0.08 4.00 8.00 -0.17 0.03 -0.67 -0.06 5.00 11.25 -0.14 0.02 -0.71 -0.04 6.00 15.00 -0.75 -0.03 7.00 19.25 -0.11 0.01 -0.78 -0.02 24.00 -0.10 -0.80 9.00 29.25 -0.09 -0.82 10.00 35.00 -0.83 -0.01 11.00 41.25 -0.85 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

W ln(W) A(W) A'(W) R(W) R'(W) 1.00 0.00 -1.00 2.00 0.69 -0.50 -0.25 3.00 1.10 -0.33 -0.11 4.00 1.39 -0.06 5.00 1.61 -0.20 -0.04 6.00 1.79 -0.17 -0.03 7.00 1.95 -0.14 -0.02 8.00 2.08 -0.13 9.00 2.20 -0.01 10.00 2.30 -0.10 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

W U(W) A(W) A'(W) R(W) R'(W) 2.00 -1.65 -0.25 0.00 0.50 0.25 3.00 -2.12 0.75 4.00 -2.72 1.00 5.00 -3.49 1.25 6.00 -4.48 1.50 7.00 -5.75 1.75 8.00 -7.39 9.00 -9.49 2.25 10.00 -12.18 2.50 11.00 -15.64 2.75 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

W U(W) A(W) A'(W) R(W) R'(W) 2.00 0.30 0.38 -0.19 -0.75 0.00 3.00 0.33 0.25 -0.08 4.00 0.35 0.19 -0.05 5.00 0.37 0.15 -0.03 6.00 0.39 0.13 -0.02 7.00 0.41 0.11 8.00 0.42 0.09 -0.01 9.00 0.43 0.08 10.00 0.44 11.00 0.46 0.07 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

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 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Let us find absolute risk aversion and relative risk aversion properties of this particular utility function. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Now from the two equations we easily see that: 1. We have decreasing absolute risk aversion property, i.e., as the amount of wealth (W) increases the amount held in risky assets also increases. 2. We have constant relative risk aversion property, i.e., as the amount of wealth (W) increases the % held in risky assets remains the same. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Now the U(W) looks like as shown in the graph. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis The actual value of expected utility is of no use, except when comparing with other alternatives. Hence we use an important concept of certainty equivalent, which is the amount of certain wealth (risk free) that has the utility level exactly equal to this expected utility value. We define U(C) = E[U(W)], where C is the certainty value MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis How is this value of C useful Suppose that we have a decision process with a set of outcomes, their probabilities and the corresponding utility values. In case we want to compare this decision process we can find the certainty equivalent so that comparison is easier. To find the exact form of the utility function for a person who is not clear about the form of utility function he/she uses. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis Example # 03: Suppose you face two options. Under option # 1 you toss a coin and if head comes you win Rs. 10, while if tail appears you win Rs. 0. Under option # 2 you get an amount of Rs. M. Also assume that your utility function is of the form U(W) = W – 0.04*W2. It means that after you win any amount the utility you get from the amount you won. For the first option the expected utility value would be Rs. 3, while the second option has an expected utility of Rs. M – 0.04*M2. To find the certainty equivalent we should have U(M) = M – 0.04*M2 = 3. Thus M = 3.49, i.e., C = 3.49, as U(3.49) = E[U(W)] MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis The above example illustrates that you would be indifferent between option # 1 and option # 2. Now suppose if you face a different situation where you have option # 1 as before but a different option # 2 where you get Rs. 5. Then obviously you would choose option # 2 here, as U(5) = ( *52) = 4 > 3.49. For the venture capital problem the certainty value for the option # 2 is Rs , as U(370881) = (370881)0.5 = 609 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis A risk averse person will select a equivalent certain event rather than the gamble A risk neutral person will be indifferent between the equivalent certain event and the gamble A risk seeking person will select the gamble rather than the equivalent certain event MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis A and B are wealth values, i.e., values of W. Also for ease of our analysis we consider that U(W)=W. Form a lottery such that it has an outcome of A with probability p and the other outcome is B with a probability (1-p). Change the values of p and ask the investor how much certain wealth (C) he/she will have in place of the lottery. Thus C varies with p. Now the expected value of lottery is {p*A+(1-p)*B}. A risk averse person will have C < {p*A+(1-p)*B}. Plot the values of C and you already have the expected values of the lottery. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis How would you find the explicit form of the utility function of a person. Suppose you know that it is of the form U(W) = - e –aW. You ask the person that given a lottery which has a chance of winning Rs. 1,000,000 or Rs. 4,00,000. In order to buy this lottery what was he/she willing to pay. Suppose the answer is Rs. 5,00,000 (say for example), then it means that the person is indifferent between a certain equivalent amount of Rs. 5,00,000 and the lottery (which is a fair gamble). Hence: 1*(- e *a)= 0.5*(-e *a) + 0.5*(-e *a). Solving through iteration process we can find a. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Utility Analysis (Axioms)
Axioms of utility functions An investor can always say whether A = B, A> B or A < B If A > B and B > C, then A > C Consider X = Y. Then assume we combine with X with another decision Z, such that X is with P(X) = p and Z is with P(Z) =1-p. On the same lines we have the same decision Z with Y, such that Y is with P(Y) = p and Z is with P(Z) = 1-p. The X+Z = Y+Z For every gamble there is a certainty equivalent such that a person is indifferent between the gamble and the certainty equivalent MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Comparison of MV and Utility Analysis
Comparison between mean-variance and utility function The utility function used is (U(W)=W-bW2), which is quadratic Consider we have three assets and the prices are as follows No A B C R(A) R(B) R(C) P(i) /5 /5 /5 /5 /5 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Then: If risk less interest (in terms of total return) is 0.5, then using mean-variance analysis we rank the assets as Using quadratic utility function U(W) = W – b*W2, with b = we rank the assets as B [U(B) = 90.68] > A [U(A) = 88.01] > C [U(C) = 80.00] MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Consider the following example with two different sets of outcomes. The utility function is U[W] = W2 + W Outcome Outcome W U[W] P(W) Scenario 1 Scenario 2 (15+20)/212 (20+12)/212 (25+25)/212 (10+17)212 (5+8)/212 (25+30)/212 Accordingly we have to calculate the expected utility value MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Decisions and Utility Analysis
Deterministic vs Probabilistic MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

People have other criteria for investment/project/portfolio solutions and they are Geometric mean return Safety first criteria Stochastic dominance Analysis in terms of characteristics of the return distribution MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Geometric mean return For the selection process we consider the maximum GM has: The highest probability of reaching or exceeding any given wealth level in the shortest possible time. The highest probability of exceeding any given wealth level over any given period of time MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Ri,j = ith possible return on the jth portfolio. pi,j = probability of ith outcome for jth portfolio. Then choose the maximum of the GM values MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Example # 04: Consider we have the following combinations of assets A, B and C in the following ratios (weights) to form a portfolio P. The returns are 10, 20, 30 respectively. A B C 2 1/3 1/3 1/3 RP,1 = (1+0.10)0.20*(1+0.20)0.20*(1+0.30)0.60 – 1 = 0.237 RP,2 = (1+0.10)1/3*(1+0.20)1/3*(1+0.30)1/3 – 1 = 0.197 RP,3 = (1+0.10)0.25*(1+0.20)0.25*(1+0.30)0.50 – 1 = 0.222 Note: Hence choose scenario # 1 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Maximizing GM return is equivalent to maximizing the expected value of log utility function Projects/Investment/Portfolios that maximize the GM return are also mean-variance efficient if returns are log-normally distributed MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Safety first principle Under safety first principle the basic tenet is that the decision maker is unable or unwilling to consider the utility theorem for making his/her decision process. Under this methodology people make their decision placing more importance to bad outcomes MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Decisions and Utility Analysis (Concepts)
Safety first principles (rules) Min Pr[RP<RL] Max RL MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

If returns are normally distributed then the optimal portfolio would be the one where RL was the maximum number of SD away from the mean Let us consider an example for Min P[Rp < RL]. Remember we consider the returns are normally distributed and the suffix P denotes the portfolio while RL means a fixed level of return (5). A B C RP P Diff from 5% -1*A * B -1.5*C MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

In order to determine how many SDs, RL lies below the mean we calculate RL minus the mean return divided by the SD. Thus we have This is equivalent to MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Even though for our example we have simplified our assumption by considering only normal distribution, but this would hold for any distributions having first and second moments. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

According to Tchebychev (Chebyshev) inequality for any random variable X, such that E(X) and V(X) exists, then MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

As we are interested in lower limit hence we simply it and have MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

The right hand side of the inequality is is exactly equal to the decision process # 1 under safety first principle we have considered previously MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

For the second criterion we have max RL s.t.: Pr(RP < RL)   We are given  (say 0.05), then we should have MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

` MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

The criterion is such that Pr(RP  RL) = , here  is predertermined depending on the investors own constraints. Thus with the condition we have MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Safety First Principle (Example)
Example # 05: Considering we have projects A, B, C and D and we need to rank them using the concept of safety first principle. The information is as follows MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

As per safety first principle we have: min {RP,jRL} where: (i) i = 1, 2,….., m, (number of projects) and (ii) j = 1, 2,….., n (number of jobs/activities/financial decisions in each project). Thus: MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Loss Functions In the course of a statistical estimation problem, observations (collectively, the sample) pertaining to a data set are regarded as realizations of a random element (X) with which is associated a probability law P Usually, P is specified by the cumulative distribution function (d.f.) of X, namely, F(x) = P(X x), xR=(-, ) Generally, F is not completely known, and we are usually interested in drawing statistical conclusion about the parameters,  = (F) which are functionals of the d.f. F MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Loss Functions In a parametric model, the assumed functional form of F may involve some unknown algebraic constant(s), which are interpreted as the parameters, e.g., in the normal d.f., the algebraic constants are themselves the mean () and the variance (2) The objective in point estimation is to utilize the information in the given set (X1,….., Xn) of sample observations (random variables) to choose a suitable statistic Tn = T(X1, X2,..…, Xn) such that Tn estimates  (parameter) in a meaningful way MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Loss Functions Imposing consistency and unbiasedness does not always lead to a unique estimator of  A good idea is to locate an optimal estimator within the class of consistent (and possibly, unbiased) estimators One idea is to choose a nonnegative metric L(Tn , ) defined for all , where  varies over (the parameter space) while Tn varies over (the sample space, which is usually a subset of Rn, the n-dimensional Euclidean space) MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Loss Functions (SEL) MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Loss Functions (Mod Loss Function)

Loss Functions (Asymmentric Linear Loss Function)

Loss Functions (LINear EXponential Loss Function) (Varian, Zellner)

LINear EXponential Loss Function (Example # 01)
Consider a company plans to launch a new product, say a refrigerator in the consumer market. Also suppose that similar products from different manufacturers already exist in the market. Then the company is expected to give some warranty for the particular product, i.e., the refrigerator, to its customers in order to sell the product. Now if the value of this warranty is more than the average time of failure for the product, then the aforesaid mentioned company needs to replace the damaged products it sells, or face litigation charges. On the other if the warranty period is less than the average failure time of similar products available in the market, then the company losses the market share to its rivals, as naturally, customers are willing to buy the refrigerator from the competitors who assure a higher warranty period. Under such a situation it is definitely advisable to estimate the warranty life time using an asymmetric loss. What values of a one should use would then depend on the level of importance our company places on overestimation versus underestimation, i.e., the cost of litigation versus the cost of a loss in the market share of the company MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

As a second example, assume a civil engineer is building a dam and he/she is interested in finding the height of the dam which is being built. If due to some error the height is estimated to be greater than the actual value then the cost the engineer incurs are mainly due to material and labour. On the other hand, if the estimated height is less than what it should be, then the consequences can be disastrous in terms of an environmental impact, which in monetary terms can be very high. So it is logical to use a value of a < 0 in such situations such that underestimation is penalized more than over estimation MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Finally to illustrate the significance of over estimation when compared with underestimation let us consider a different real life example. Consider an electrical company manufactures vacuum circuit breakers/interrupters, which are used as a fuse in high voltage system. As for any product these circuit breakers have a working life and it is of utmost importance that this value is estimated as accurately as possible. In case they are underestimated than what its value is in reality, then the consequences is just labour and man hour loss in terms of production stoppage time. On the other hand if the working life of the circuit breaker is over estimated than the actual figure, then it would definitely signify an exponential form of loss in monetary terms due to an accident or major break down of machineries. So, for these categories of practical estimation problems we always consider a > 0 MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Balanced Loss Function (BLF)
A balanced loss function (BLF) is of the form, with w having a given value such that 0  w  1. The BLF, (Zellner (1994)), reflects both goodness of fit (lack of bias) and precision of estimation Note: The first term represents the goodness of fit while the second represents the precision of estimation, which is also, termed as accuracy MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Stochastic Dominance First-order stochastic dominance Second-order stochastic dominance Third-order stochastic dominance MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

First-order stochastic dominance A lottery F dominates G then the decision maker prefers F to G regardless of what U(W) is, as long as it is Weakly increasing (i.e., U(Wi)  U(Wj), where Wi  Wj) MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

First-order stochastic dominance F dominates G iff Simply stated it means that every individual with increasing utility function prefers FW to GW regardless of his/her risk preferences. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

First-order stochastic dominance F dominates G iff FW(W)GW(W) This definition requires that FW gives more wealth than GW realization by realization. MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Second-order stochastic dominance A lottery F dominates G if the decision maker prefers F to G as long as he/she is Risk averse U(W) is weakly increasing (i.e., U(Wi)  U(Wj), where Wi  Wj) MDA RNSengupta,IME Dept.,IIT Kanpur,INDIA

Management Decision Analysis

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Management Decision Analysis

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