Consumer s behavior under uncertainty

Transcription

1 Consumer s behavior under uncertainty Microéconomie, Chap 5 1

2 Plan of the talk What is a risk? Preferences under uncertainty Demand of risky assets Reducing risks 2

3 Introduction How does the consumer choose his consumption with uncertain prices and income (i.e. when he faces some risk)? 3

4 What is a risk? A risk consists of: 1. A collection of possible outcomes 2. A probability for each of the posible outcomes 4

5 What is a risk? Meaning of probabilities 1. Objective interpretation based on the relative frequency of past realizations of the given outcome 2. Subjective interpretation based on the belief of the future realization of the given outcome 5

6 Meaning of probabilities Subjective probabilities Different informations or abilities to analyse the same information can lead to different subjective probabilities It is based on personal experience and assessments 6

7 How to characterize a risk? In order to describe and compare two risks we need two statistics of the associated probability distributions 1. the expected value 2. the variance 7

8 How to characterize a risk? By the expected value It is the average of all possible outcomes weighted by their probabilities 8

9 Expected value example Investing in oil-drilling: Two possible outcomes success the price of the stock increases from 30 to 40 failure the price of the stock decreases from 30 to 20 9

10 Expected value example Objective probabilities Out of 100 explorations, 25 were a success and 75 a failure The probability (Pr) of success is 1/4 and the probability of failure 3/4 10

11 Expected value example EV = Pr(success)(value of succes) +Pr(failure)(value of failure) EV = 1 4 (40 euros per stock) + 3 (20 euros per stock) 4 EV = 25 euros per stock 11

12 Expected value If there are n possible outcomes: possible gains X 1, X 2,, X n associated probabilities Pr 1, Pr 2,, Pr n E(X) = Pr 1 X 1 + Pr 2 X Pr n X n 12

13 How to characterize a risk? By its variance It measures the variablity of all the different possible outcomes 13

14 Variance example Suppose you must choose between two jobs The first is remunerated by commissions on realized sales The second is remunerated by a fixed wage 14

15 Variance example By commission the gain can be either 2000 or 1000 with equal probabilities The fixed wage is 1510 but there is a probability 0,01 of the business going bankrupt and being fired with a compensation of

16 Variance example outcome 1 outcome 2 Prob. income Prob. income Job 1: commission 0, , Job 2: wage 0, ,

17 Variance example Expected income from job 1 Expected income from job 2 17

18 Variance example The two jobs have the same expected income, but they differ in the variance of this income A higher variance means a higher risk The variance take into account the possible deviations from the expected income i.e. the realized differences between actual income and expected income 18

19 Variance example Deviations from the expected income 1500 outcome 1 deviation outcome 2 deviation job with Pr=0, with Pr=0,5-500 job with Pr=0, with Pr=0,

20 Variance The expected deviation is always, by construction, zero It is necessary to get rid of the opposite signs of the different deviations This is achieved by the standard deviation The square root of the weighted average of the square of the deviations 20

21 Variance The standard deviation is: σ = Pr [ X E(X) ] 2 + Pr [ X E(X) ]

22 Variance The standard deviation measures risk It measures the variability of gains More variability means more risk The typical consumer prefers a small variability, i.e. low risks 22

23 Variance example Deviations from the expected income 1500 outcome 1 deviation outcome 2 deviation job with Pr=0, with Pr=0,5-500 job with Pr=0, with Pr=0,

24 Variance example The standard deviation in the two previous examples is: σ = Pr [ 1 X 1 E(X) ] 2 + Pr [ 2 X 2 E(X) ] 2 σ 1 = 0, , = = 500 σ 2 = 0, , = 9900 = 99,50 24

25 Variance example Job 1 has a bigger standard deviation and thus it is riskier than job 2 The standard deviation can equally be computed for any number of outcomes 25

26 Variance example 2 Income from job 1 is randomly between 1000 and 2000 (by multiples of 100) with uniform probability 0,09 everywhere Imcome from job 2 is randomly between 1300 and 1700 (by multiples of 100) with uniform probability 0,2 everywhere 26

27 Variance example 2 Probability The distribution of job 1 income is more dispersed, i.e. it has a bigger standard deviation and a higher risk 0,2 job 2 0,09 job income 27

28 Variance example 2 Probability The distribution of job 1 income is more dispersed, i.e. it has a bigger standard deviation and a higher risk 0,2 job 2 0,1 job income 28

29 Choice between risks if: job 1 and job 2 have the same expected income job 2 has a smaller standard deviation of income then the individual chooses job 2 29

30 Choice between risks Assume income from job 1 is increased by an additional 100 in every outcome, then job 1: the new expected income is 1600 and the standard deviation stays at 500 job 2: the expected income is still 1500 and the standard deviation 99,50 30

31 Choice between risks Deviations from expected income outcome 1 deviation outcome 2 deviation job with Pr=0, with Pr=0,5-500 job with Pr=0, with Pr=0,

32 Choice between risks What would the individual choose? It depends on his preferences Some will find the increase in expected income up to 1600 is worth the higher risk of job 1 Others will prefer the lower risk of job 2 even at the price of a smaller expected income of

33 Preferences over risks Risky options can be compared assigning a utility to each of them 33

34 Preferences over risks - example Assume: a job pays another job pays with probability 0, with probability 0,5 the utility from is 10 the utility from is 13,5 the utility from is 18 34

35 Preferences over risks - example To compare the two jobs we compute the expected utility of the risky job E(U) = (Prob. of utility 1) x (utility 1) + (Prob. of utility 2) x (utility 2) 35

36 Preference over risks - example The expected utility of the risky job is: E(U) = 0,5 x U( ) + 0,5 x U( ) = 0,5 x ,5 x 18 = 14 The expected utility E(U) of the risky job is 14, higher than 13,5 for the riskless job 36

37 Preferences over risks Individuals can have different attitudes towards risk They can be risk averse risk neutral risk lover 37

38 Preferences over risks Risk aversion The individual prefers a sure income to a risky income with the same expected value Income provides a decreasing marginal utility to the individual It is the most common attitude It is what makes possible the insurance market 38

39 Risk aversion - example An individual can get a salary of (with utility 16) with probability 100% He can also get at another job (utility 18) with probability 0,5 or (utility 10) with probability 0,5 39

40 Risk aversion - example Expected income from the risky job E(I) = 0,5 x ,5 x E(I) = Expected utility from the risky job E(U) = 0,5 x ,5 x 18 E(U) = 14 40

41 Risk aversion - example Since both jobs have the same expected income, a risk averse individual would choose the riskless job Expected utility from the riskless job is higher For a risk averse individual possible losses matter more than uncertain gains 41

42 Utility function with risk aversion Utility A C D F E the risky job has an expected income of and an expected utility of 14 (point F) the riskless job has an expected income of and an expected utility of 16 (point D) income ( 1000) 42

43 Preferences over risks An individual is risk neutral if he is indifferent between two outcomes with the same expected value Marginal utility of income is constant for him 43

44 Risk neutral Utility 18 E 12 6 A C The individual is indifferent between and with equal probabilities 1/2 (expected income and expected utility 12) or with probability 1 (expected utility 12) income ( 1000) 44

45 Preferences over risks An individual is risk lover if he prefers a risky outcome to a sure one with the same expected value Examples: gambling, some criminal activities Marginal utility from income is increasing 45

46 Risk lover Utility F C E Risky option: point F E(R) = 0,5 x ,5 x = E(U) = 0,5 x 3 + 0,5 x 18 = 10,5 Sure income with utility 8 point C 3 A income ( 1000) 46

47 Preferences over risks The risk premium is the maximum amount an individual is willing to pay to get the same expected income with no risk It is also the minimum amount that the individual would ask for bearing a risk with the same expected value It is the difference bewteen the expected income and its certainty equivalent, the riskless income that makes him indifferent It depends on the preferences of the agent and the specific risk 47

48 Risk premium example Utility A C Risk premium F E G For a risk of or with probabilities 1/2, the risk premium is 4000, since with probability 1 (the certainty equivalent) gives a utility 14 equal to the expected utility from the risky option income ( 1000) 48

49 Risk premium example Utility A C Risk premium E F F G For a risk of 0 or with probabilities 1/2, the risk premium is , since with probability 1 (the certainty equivalent) gives a utility 10 equal to the expected utility from the risky option income ( 1000) 49

50 Risk premium The higher the risk variance, the bigger the risk premium The higher the risk aversion, the bigger the risk premium 50

51 Preferences over risks and indifference curves The attitude towards risk can be described by means of indifference curves over the expected value and standard deviation of income For a risk averse individual, the riskier an option, the higher must be its expected gain to compensate for its risk Thus his indifferent curves are increasing 51

52 Preferences over risk and indifference curves Expected income U 3 U 2 U 1 Strong risk aversion: an increase in the standard deviation needs to be compensated by a big increase in expected income Standard deviation 52

53 Preferences over risks and indifference curves Expected income Weak risk aversion: an increase in the stadard deviation is compensated by a small increase in expected income U 3 U 2 U 1 Standard deviation 53

54 Demand for risky assets Future returns are not known with certainty, so investment decisions have to be made based on expected returns Actual returns can be higher or lower than expected 54

55 Demand for risky assets Risk and return ( ) 55

56 Demand for risky assets Higher returns are associated to higher risks Risk averse individuals seek to balance high returns and small risks 56

57 Trade-off between return and risk An investor must chose the composition of his portfolio of stocks and Treasury bills: 1. Treasury bills are riskless assets 2. Stocks are risky 57

58 Trade-off between return and risk R f = expected return of Treasury bills = actual return of Treasury bills R m = expected return of stocks r m = actual return of stocks typically R m > R f, otherwise no risk averse investor would hold stocks 58

59 Trade-off between return and risk What share of the portfolio must be invested in stocks and what in Treasury bills? let b be the share of stocks Let 1-b be the share of Treasury bills The portfolio s expected return R p is 59

60 Trade-of between return and risk Example: if R m = 12%, R f = 4%, and b = 1/2 60

61 Trade-off between return and risk Which is the portfolio s risk? It is measured by the standard deviation σ p of its return if σ m is the standard deviation of the return to stocks and b is the share of stocks in the portfolio, then 61

62 Trade-off between return and risk [( ) ( br m + (1 b)r )] 2 f σ p 2 = E br m + (1 b)r f σ 2 p = E[ b(r m R m )] 2 σ 2 p = b 2 E[ r m R ] 2 m 62

63 Trade-off between return and risk How much does the market allow for trading off return and risk? 63

64 Trade-off between return and risk The portfolio s expected return R p increases linearly with standard deviation σ p 64

65 Trade-off between return and risk Expected return R p budget line R m R f standard deviation of return, σ p 65

66 Trade-off between return and risk The slope of this «budget» line is the price of reducing the risk It represent the rate at which the return of the portfolio must decrease in order to decrease its risk slope = R m R f σ m 66

67 Choice between return and risk If the individual only invests in Treasure bills (b=0), then return is R f If the individual only invests in stocks (b=1), then the expected return is R m but with standard deviation σ m If the individual invests in both (0<b<1), then the expected return will be between R f and R m with a standard deviation between 0 and σ m 67

68 Choice between return and risk The investor seeks to reach the highest indifference curve between return and risk on the «budget» line 68

69 Choice between return and risk Expected return R p Expected return R * and standard deviation σ * is the optimal choice because it equalizes the price of risk and MRS between return and risk U 3 U 2 U 1 budget line R m R * R f standard deviation of return, σ p 69

70 Choice between return and risk A strongly risk averse investor chooses mostly Treasure bills and few stocks, with an expected return slightly above R f and little risk A weakly risk averse investor chooses few Treasure bills and a lot of stocks, with an expected return close to r m but a high standard deviation 70

71 Choice between return and risk Expected return R p R m R B R A U A U B budget line Investor A: strongly risk averse Investor B: weakly risk averse R f standard deviation of return σ p 71

72 Reducing risk exposure Most individuals have some degree of risk aversion and want to reduce their risk exposure three ways to do it: 1. obtaining additional information 2. diversifying risks 3. insuring risks 72

73 Reducing risk exposure Obtaining additional information A risk may originate in missing information about the consequences of a decision Information is valued by risk averse investors and is regularly traded E.g.: marketing 73

74 Reducing risk exposure Diversifying risks Risk exposure is reduced holding a portofolio of assets the risks of which offset each other 74

75 Reducing risk exposure Diversifying risks - example a firm can sell either air conditioning units, heating, or both whether is cold or warm with equal probability.5 what should the firm decide to sell? 75

76 Reducing risk exposure Diversifying risks - example warm cold AC units sales heating sales

77 Reducing risk exposure Diversifying risks - example the income from selling only AC units or only heating is either or expected income is: 1/2( ) + 1/2( ) =

78 Reducing risk exposure Diversifying risks - example Devoting resources equally to sell both AC and heating the firm obtains half of the sales of each If warm, income will be from AC sales and from heating sales, i.e If cold, income will be from AC sales and from heating, i.e

79 Reducing risk exposure Diversifying risks - example diversifying sales, the firm obtains the same expected income with no risk 79

80 Reducing risk exposure Diversifying risks example 2 Investing only in a single stock is very risky If the stock loses value, the entire investment may be lost Risk exposure can be reduced investing in several stocks of unrelated sectors e.g.: investment funds 80

81 Reducing risk exposure Insurance Risk averse individuals are willing to pay for getting rid of a risk If cost of insurance amounts to the expected loss, risk averse individuals would insure completely the risk 81

82 Reducing risk exposure Insurance insurance loss (Pr 0,1) no loss (Pr 0,9) expected wealth standard deviation no yes

83 Reducing risk exposure Insurance risk averse individuals prefer a sure amount to a risky option with the same expected value 83

84 Insurance contract fixed term Insurance problem: w wealth L possible loss π probability of loss Insurance contract : p insurance fee L compensation in case of loss requires p>πl 84

85 Insurance contract fixed term loss without insurance w-l with insurance w-p-l+l = w-p no loss w w-p π(w-l)+(1-π)w = expected value w-πl will buy the contract if π(w-p)+(1-π) (w-p) = w-p πu(w-l)+(1-π)u(w) < or = U(w-p) 85

86 Insurance contract fixed term Utility Maximum insurance fee Minimum insurance fee U(w) U(w-p M ) U(w-L) A C F E The higher the risk aversion, the more willing to pay a risk premium over the expected loss 0 w-l w-p M w-πl w income ( 1000) 86

87 Insurance contract fixed term risk neutral individuals are indifferent between a fixed term insurance for a fee p=πl and no insurance risk averse individuals prefer insurance for a fee p=πl and even p>πl up to a maximum p M a risk averse individual prefers a sure wealth smaller than its expected value: if p M >p>πl, he prefers a sure wealth w-p =[initial wealth insurance fee] to the lottery : w with probability 1-π and w-l with probabilty π w-p M is the certainty equivalent of the lottery without insurance the risk premium is p M -πl 87

88 Insurance contract fixed term loss no loss : probability 1-π loss: probability π A,E=(w,w-L) wealth without insurance C=(w-p M,w-p M ) wealth with insurance C determines the maxmimum insurance fee B determines the minimum insurance fee 45 o w-πl w-p M w-l O w-p M 1- π π w-πl C B w U(w- πl) A,E (1-π)U(w)+ πu(w-l) =U(w- p M ) 1- π π no loss 88

89 Insurance contract flexible terms Insurance problem: w wealth L possible loss π probability of loss Insurance contract : p insurance fee per euro reimbursed thus p<1 R reimbursement in case of loss necessarily p > π, since total fee = pr > πr = expected reimbursment the individual chooses whether to buy insurance or not, and by how much 89

90 Insurance contract flexible terms If insured the risk becomes: If no loss, paid fee pr and gets w-pr loss, paid fee pr, receives R and gets w-pr-l+r The individual cannot choose the insurance fee p but can choose the amount of insurance R What is the optimal choice? 90

91 Insurance contract flexible terms loss without insurance w-l with insurance w-pr-l+r = w+(1-p)r-l no loss w w-pr π(w-l)+(1-π)w = expected value w-πl will buy insurance R if π[w+(1-p)r-l] +(1-π)[w-pR] = w-(p-π)r-πl πu(w-l)+(1-π)u(w) < or = πu(w+(1-p)r-l)+(1- π)u(w-pr) 91

92 Insurance contract flexible terms loss no loss : probability 1-π loss: probability π A,E=(w,w-L) wealth without insurance B=(w-pR,w-pR-L+R) wealth with insurance the agents chooses not to insure completely 45 o W-pR-L+R w-l O 1- π π w-pr B w 1- p p A,E (1-π)U(w)+ πu(w-l) no loss 92

93 Insurance contract flexible terms Utility U(w) B E U(w-L) A B 0 w-l w-pr-l+r w-pr w income ( 1000) 93

94 Insurance contract flexible terms if p = π a risk neutral agent is indifference between buying insurance an not buying a risk averse agent buys full insurance, i.e. R=L if p > π a risk neutral agent does not buy insurance a risk averse agent buys partial insurance, i.e. R<L 94

95 Insurance contract To be purchased, an insurance contract must increase expected utility The agent s decision depends on his attitude towards risk and on the insurance fee 95

96 problems adverse selection if a common fee p = π (average probability loss) is charged to all individuals, but some have a low probability π l of suffering a loss, and some have a high probability π h then for low risk individuals π l <p : they buy partial insurance for high risk individuals π h >p : the insurer loses money on them Therefore, the insurance company would not maximize profits Either no insurer would be willing to cover the risk, or he charges a fee p = π h so that low risk individuals buy too little or no insurance (inefficient) 96

97 problems adverse selection to address the adverse selection problem : reduce asymmetric information between insurer and insured: forecast good and bad risks: time series (life insurance), socio-economic data (loan insurance), objective criteria (e.g. home insurance, theft insurance) signalling : make reimbursement contingent to past behavior (bonus-malus) 97

98 problems moral hazard an agent s behavior can change, as a consequence of obtaining insurance, in such way that the probability of loss increases there is a need to introduce incentives for prudent behavior, e.g. making the insured to bear some of the risk: deductibles co-insurance and limited insurance 98

99 Insurance Insurers can know the average probability of a loss across large populations, even if the probability of a loss for a specific individual cannot be known Thus with large population insurers face little risk 99

100 Insurance Big losses (earthquakes, floods, etc.) are difficult to insure in the market Probabilities and expected losses are difficult to estimate Governments tend to act as insurers of last resort 100