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15. Risk and Information

Learn about describing risky outcomes, evaluating probabilities, bearing and eliminating risk, and analyzing risky decisions. Explore objective and subjective probabilities, probability density functions, expected values, variances, and standard deviations.

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15. Risk and Information

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  1. 15. Risk and Information 15.1 Describing Risky Outcomes 15.2 Evaluating Risky Outcomes 15.3 Bearing and Eliminating Risk 15.4 Analyzing Risky Decisions

  2. 15.1 Probability Terminology • When there are multiple outcomes, probabilities can be assigned to the outcomes Terminology: Sample Space – set of all possible outcomes from a random experiment -ie S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} -ie E = {Pass exam, Fail exam, Fail horribly} Event – a subset of the sample space -ie B = {3, 6, 9, 12} ε S -ie F = {Fail exam, Fail horribly} ε E

  3. 15.1 Probability Probability = the likelihood of an event occurring (between 0 and 1) P(a) = Prob(a) = probability that event a will occur P(Y=y) = probability that the random variable Y will take on value y P(ylow < Y < yhigh) = probability that the rvariable Y takes on any value between ylow and yhigh

  4. 15.1 Probability Extremes If Prob(a) = 0, the event will never occur ie: Canada moves to Europe ie: the price of cars drops below zero ie: your instructor turns into a giant llama If Prob(b) = 1, the event will always occur ie: you will get a mark on your final exam ie: you will either marry your true love or not ie: the sun will rise tomorrow

  5. 15.1 Probability Types • There are two categories of probabilities: Objective Probabilities: Probabilities that are (mathematically) certain ie: rolling a dice, drawing a card Subjective Probabilities: Probabilities based on beliefs and expectations ie: gambling, stocks, many investments

  6. 15.1 Objective Probability –Card Example Sample space = {A, 1, 2…J, Q, K} of each suit -or [Ax,Kx] where x ε {hearts, diamonds, spades, clubs} Events: -drawing red card -drawing even card -drawing face card -drawing an ace -drawing a “one eyed jack” -drawing two cards of total value 15

  7. 15.1 Objective Probability Examples • Probability of drawing a heart = ¼ • Probability of drawing less than 3 = 2/13 • Probability of drawing a King or a heart = 13(hearts)+3(non-heart kings)/52 = 16/52 4) Probability of throwing a 13 = 0 5) Probability of tossing 6 heads in a row = 1/64 6) Probability of drawing a red or black card =1 7) Probability of passing the course = ?

  8. 15.1 Subjective Probability –Investment Example You decide to invest in Risktek Inc. Sample space = {-$1000, -$500, +$3000} Events: -losing $1000 -losing $500 -losing money -gaining $3000

  9. 15.1 Subjective Probability Examples Based on your subjective knowledge, probabilities are: • P {-$1000}=0.3 • P {-$500}=0.5 • P {$3000}=0.2

  10. 15.1 Probability Density Functions • The probability density function (pdf) summarizes probabilities associated with possible outcomes f(y) = Prob (Y=y) 0≤ f(y) ≤1 Σf(y) = 1 -the sum of the probabilities of all possible outcomes is one

  11. 15.1 Objective Dice Example • The probabilities of rolling a number with the sum of two six-sided die • Each number has different die combinations: 7={1+6, 2+5, 3+4, 4+3, 5+2, 6+1} • Exercise: Construct a table with 1 4-sided and 1 8-sided die

  12. 15.1 Expected Values Expected Value – measure of central tendency; center of the distribution; population mean - average outcome

  13. 15.1 Objective Example What is the expected value from a dice roll? E(W) = Σwf(w) =2(1/36)+3(2/36)+…+11(2/36)+12(1/36) =7 Exercise: What is the expected value of rolling a 4-sided and an 8-sided die? A 6-sided and a 10-sided die?

  14. 15.1 Subjective Example What is the expected value from investing in Risktek? Recall: P {-$1000}=0.3, P {-$500}=0.5 P {$3000}=0.2 E($) = Σ$f($) = -$1000(0.3)-$500(0.5)+$3000(0.2) = $50

  15. 15.1 Properties of Expected Values • Constant Property E(a) = a if a is a constant or non-random variable ie: E($100)=$100 b) Constants and random variables E(a+bW) = a+bE(W) If a and b are non-random and W is random ie: E[$100+2(investment)] =$100+2E(investment)

  16. 15.1 Variance Consider the following 3 midterm exams: • Average = 70%; everyone gets 70% • Average = 70%; the class is equally distributed between 50% and 90% • Average = 70%; most of the class gets 70%, with a few 100%’s and a few 40%’s who became sociologists

  17. 15.1 Variance Variance – a measure of dispersion (how far a distribution is spread out) Variance is a way of measuring risk σY2= Var(Y)= Σ(y-E(Y))2f(y)

  18. 15.1 Variances Example 1: E(Y)=70 Yi =70 for all i Var(Y) = Σ(y-E(Y))2f(y) = Σ(70-70)2 (1) = Σ(0)(1) =0 If all outcomes are the same, there is no variance.

  19. 15.1 Variances Example 2: E(Y)=70 Y= 50, 60, 70, 80 ,90 Var(Y) = Σ(y-E(Y))2f(y) = (50-70)2(1/5)+ (60-70)2(1/5)+ (70-70)2(1/5)+ (80-70)2(1/5)+ (90-70)2(1/5)+ =400/5+100/5+0/5+100/5+400/5 =1000/5 =200

  20. 15.1 Variances Example 3: E(Y)=70 Y= 40, 70, 70, 70 ,100 Var(Y) = Σ(y-E(Y))2f(y) = (40-70)2(1/5)+ (70-70)2(1/5)+ (70-70)2(1/5)+ (70-70)2(1/5)+ (100-70)2(1/5)+ =900/5+0/5+0/5+0/5+900/5 =1800/5 =360

  21. 15.1 Standard Deviation Standard Deviation is more useful for a visual view of dispersion: Standard Deviation = Variance1/2 sd(W)=[var(W)]1/2 σ= (σ2)1/2

  22. 15.1 SD Examples In our first example, σ =01/2=0 No dispersion exists In our second example, σ =2001/2≈14.1 In our third example, σ =3601/2=19.0 If you could choose an exam to take, the third exam would be the riskiest.

  23. 15.1 Constant Property of Variance Constant Property Var(a) = 0 if a is a constant Ie: Var($100)=0, the risk of having $100 (and not gambling) is zero.

  24. 15.2 Risk and Utility Option 1 – Government job. Wage = $50,000 Option 2 – Start-Up Company. Wage = $10,000 Plus: $100,000 if successful (0.4) $0 otherwise (0.6) E($) = Σ$f($) = $10,000(0.6)+$110,000(0.4) = $50,000 Which should you choose?

  25. 15.2 Expected Utility Expected Utility – probability-weighted average of the utility from each outcome E(U) = ΣUf(U) If U=($)1/2, Option 1: E(U) = (50,000)1/2 (1) E(U) = 224

  26. 15.2 Expected Utility If U=($)1/2, Option 2: E(U) = ΣUf(U) E(U) = (10,000)1/2 (0.6)+($110,000)1/2(0.4) E(U) = 60 + 133 E(U) = 193 Option 1 has a higher expected utility, (224>193) so you would choose option 1.

  27. 15.2 Risk Characteristics Different people would make different decisions given the above choices. Your choice depends on your RISK CHARACTERISTIC: • Risk Neutral • Risk Averse • Risk Loving

  28. 15.2a Risk Neutral Someone is RISK NEUTRAL if they will always choose the highest expected income. A RISK NEUTRAL agent has CONSTANT MARGINAL UTILITY:

  29. 15.2a Risk Neutral Example Ned’s Utility is U(I) = 5I. Ned could: a) Work for Sony for $60,000 a year b) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%) Ned would choose option a.

  30. 15.2a Risk Neutral Example Ned’s Utility is U(I) = 5I. Ned could: a) Work for Sony for $60,000 a year b) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%) Ned would choose option a.

  31. U U=5(I) Ned has a constant marginal utility. Choosing the highest expected value give him the highest utility. 300K 230K Income 40K 60K 100K E(I)= 46K

  32. 15.2b Risk Averse Someone is RISK AVERSE if they prefer a certain income to a risky income with the same expected value A RISK AVERSE agent has DECREASING MARGINAL UTILITY:

  33. 15.2b Risk Averse Example Averly’s Utility is U(I) = √I. She could: a) Work for Sony for $46,000 a year b) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%) Here both expected incomes are equal.

  34. 15.2b Risk Averse Example Averly’s Utility is U(I) = √I. She could: a) Work for Sony for $46,000 a year b) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%) Averly would choose option a.

  35. Averly has a decreasing marginal utility. She prefers the certain income. U U= √I 214 212 Income 40K 100K E(I)= 46K

  36. 15.2c Risk Loving Someone is RISK LOVING if they prefer a risky income to a certain income with the same expected value A RISK LOVING agent has INCREASING MARGINAL UTILITY:

  37. 15.2c Risk Loving Example Lana’s Utility is U(I) = (I/1,000)2. She could: a) Work for Sony for $46,000 a year b) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%) Here both expected incomes are equal.

  38. 15.2c Risk Loving Example Lana’s Utility is U(I) = (I/1,000)2. She could: a) Work for Sony for $46,000 a year b) Work for Risky for $100,000 a year (10%) or $40,000 a year (90%) Lana would choose option b.

  39. Lana has an increasing marginal utility. She prefers the risky income. U (U= I/1000)2 2440 2116 Income 40K 100K E(I)= 46K

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