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- 1. Subjective Measures of Risk Eduardo Zambrano Department of Economics Cal Poly May 20, 2008
- 2. Example <ul><li>You are offered an investment g where you win $120 with probability ½ and you lose $100 with probability ½. </li></ul><ul><ul><li>g is a favorable investment: it’s expected value is $10. </li></ul></ul><ul><ul><li>Should you accept it? </li></ul></ul><ul><li>Better yet, what is the risk in accepting g ? </li></ul>
- 3. The purpose of this talk <ul><li>The problem: How to measure financial risk? </li></ul><ul><ul><li>Traditional measures have shortcomings </li></ul></ul><ul><li>Why is the problem important: </li></ul><ul><ul><li>Misrepresentation of the risk embedded in an investment can lead to serious, even catastrophic mistakes in decision making </li></ul></ul>
- 4. New solutions <ul><li>Aumann and Serrano (AS, 2007) </li></ul><ul><ul><li>Measure the risk of g as the number R that solves E e - g /R =1 . </li></ul></ul><ul><li>Foster and Hart (FH, 2007) </li></ul><ul><ul><li>Measure the risk of g as the number R that solves E log(1+ g /R) =0 . </li></ul></ul><ul><li>The FH measure has a clear operational interpretation, the AS does not. </li></ul>
- 5. My contribution <ul><li>To provide a clear operational interpretation of R AS </li></ul><ul><li>To understand further the relationship between R AS and R FH </li></ul><ul><li>To accomplish this it turned out to be useful to generalize their approach towards the development of a full family of subjective measures of risk . </li></ul><ul><li>To measure the riskiness of some known investments according to these measures to get a “feel” for their potential usefulness. </li></ul>
- 6. Example <ul><li>You are offered an investment g where you win $120 with probability ½ and you lose $100 with probability ½. </li></ul><ul><ul><li>g is a favorable investment: it’s expected value is $10. </li></ul></ul><ul><ul><li>Should you accept it? </li></ul></ul><ul><li>Better yet, what is the risk in accepting g ? </li></ul>
- 7. Traditional approach <ul><li>Use a statistical measure of dispersion to measure risk </li></ul><ul><ul><li>Standard deviation </li></ul></ul><ul><ul><li>Variance </li></ul></ul><ul><ul><li>Mean absolute deviation (E |g-Eg|) </li></ul></ul><ul><ul><li>Interquartile range </li></ul></ul><ul><li>These indices measure only dispersion, taking little account of the gamble’s actual values </li></ul>
- 8. <ul><li>Investment g 1 </li></ul><ul><li>Investment h 1 </li></ul>All the measures of dispersion would rate h as risky as g in spite of the fact that h is sure to yield more than g . <ul><ul><li>-Standard deviation </li></ul></ul><ul><ul><li>-Variance </li></ul></ul><ul><ul><li>-Mean absolute deviation </li></ul></ul><ul><ul><li>-Interquartile range </li></ul></ul>1/2 1/2 $120 $-100 g 1 1/2 1/2 $170 $-50 h 1
- 9. “Sharpe ratios” <ul><li>Investment g 1 </li></ul><ul><li>Investment h 1 </li></ul>1/2 1/2 $120 $-100 1/2 1/2 $170 $-50 μ =60 σ =110 σ / μ =1.83 σ 2 / μ =201.67 μ =10 σ =110 σ / μ =11 σ 2 / μ =1210 g 1 h 1
- 10. However… <ul><li>Investment g 2 </li></ul><ul><li>Investment h 2 </li></ul>.98 .02 $100 $-100 g 2 μ =96 σ =28 σ / μ = 0.29 σ 2 / μ = 8.2 .49 .02 $200 $-100 h 2 $100 .49 μ =145 σ =60.6 σ / μ = 0.42 σ 2 / μ = 25.3 σ / μ and σ 2 / μ rank h as more risky than g even though h never yields less than g and yields more with probability almost half.
- 11. μ g = μ h <ul><li>Investment g 3 </li></ul><ul><li>Investment h 3 </li></ul>μ =120 μ =120 $100 $-300 h 3 $0 $300 $500 $256 $-423 g 3 $256 $256 $256
- 12. σ g = σ h <ul><li>Investment g 3 </li></ul><ul><li>Investment h 3 </li></ul>$100 $-300 h 3 μ =120 σ =303 μ =120 σ =303 $0 $300 $500 $256 $-423 g 3 $256 $256 $256
- 13. σ , σ 2 , E |g-Eg|, Q 3 -Q 1 , σ / μ , σ 2 / μ <ul><li>All these are good measures of dispersion and normalized dispersion, but… </li></ul><ul><li>Are they valid measures of risk for the purpose of decision making ? </li></ul>
- 14. Stochastic Dominance (I) <ul><li>Investment h (first order) stochastically dominates g when </li></ul><ul><ul><ul><li>h ≥ g for sure, and </li></ul></ul></ul><ul><ul><ul><li>h > g with positive probability </li></ul></ul></ul>1 x H(x) G(x)
- 15. Stochastic Dominance (II) <ul><li>Investment h (second order) stochastically dominates g when some values of g are replaced in h by their expectation </li></ul>1 x H(x) G(x)
- 16. <ul><li>If we could always compare investments in terms of their stochastic dominance, we would know which investment is more risky, for the purpose of decision making: </li></ul>If h stochastically dominates g then it will be preferred by any * risk averse expected utility decision maker
- 17. Problem (I) <ul><li>Stochastic dominance is not a complete order </li></ul><ul><ul><li>One would expect any reasonable notion of riskiness to extend the stochastic dominance orders </li></ul></ul>
- 18. Problem (II) <ul><li>The traditional measures all violate Stochastic Dominance </li></ul><ul><ul><li>Value at Risk also violates stochastic dominance </li></ul></ul><ul><ul><li>-Standard deviation </li></ul></ul><ul><ul><li>-Variance </li></ul></ul><ul><ul><li>-Mean absolute deviation </li></ul></ul><ul><ul><li>-Interquartile range </li></ul></ul><ul><ul><li>-Standard deviation/Mean </li></ul></ul><ul><ul><li>-Variance/Mean </li></ul></ul>
- 19. What to do?
- 20. Preliminaries <ul><li>An investment g is a random variable with real values some of which are negative, and that has a positive expectation </li></ul><ul><li>Burkhard accepts g at w if </li></ul><ul><ul><ul><li>Eu Burkhard (w+ g )>u Burkhard (w) </li></ul></ul></ul>
- 21. <ul><li>Klaus is more risk averse than Burkhard if [for all possible wealth levels for Klaus and Burkhard] Burkhard accepts all the investments than Klaus accepts, but not the other way around. </li></ul><ul><li>Consider investments g and h such that </li></ul><ul><ul><li>whenever Burkhard rejects g , Klaus rejects h . </li></ul></ul><ul><ul><li>Call investment h more risky than investment g . </li></ul></ul><ul><li>An index of riskiness Q( g ) is homogeneous of degree one if Q( g )=tQ(t g ) </li></ul>
- 22. Aumann, Serrano <ul><li>Theorem (AS, 2007):For each investment g there is a unique positive number R ( g ) with E e - g /R( g ) =1 . Then, </li></ul><ul><ul><ul><li>The index R thus defined satisfies the riskiness order and is homogeneous of degree one . </li></ul></ul></ul><ul><ul><ul><li>Any index satisfying these two principles is a positive multiple of R. </li></ul></ul></ul>Call R( g ) the riskiness of g .
- 23. Properties of R AS ( g ) <ul><li>It is measured in the same units as g is measured </li></ul><ul><li>It is monotone with respect to first and second order stochastic dominance </li></ul><ul><li>Property C : Is the reciprocal of the coefficient of absolute risk aversion of a CARA decision maker who is indifferent between taking and not taking the investment </li></ul>
- 24. Example ( μ g ≠ μ h ) <ul><li>Investment h 1 </li></ul><ul><li>Investment g 1 </li></ul>1/2 1/2 $120 $-100 g 1 σ =110 R AS ( g ) = 601.66 1/2 1/2 $170 $-50 h 1 σ =110 R AS ( h ) = 78.95
- 25. Example ( μ g = μ h ) <ul><li>Investment g 3 </li></ul><ul><li>Investment h 3 </li></ul>$100 $-300 h 3 σ =303 R AS ( h ) = 298.61 $0 $300 $500 Ok, so g is more risky than h , but what do those numbers mean? σ =303 R AS ( g ) = 396.94 $256 $-423 g 3 $256 $256 $256
- 26. Another approach <ul><li>How risky g is to you depends on how much wealth you have. </li></ul><ul><ul><li>If all you have is $100 then g is extremely risky: you risk bankruptcy </li></ul></ul><ul><ul><li>If your wealth is, say, $1,000,000, then g is not risky at all; moreover, you would love to be exposed to g repeatedly </li></ul></ul><ul><li>Investment g </li></ul>1/2 1/2 $120 $-100 g E g =10 R AS ( g ) = 601.66
- 27. Foster, Hart <ul><li>Theorem (FH, 2007): For each investment g there is a unique positive number R ( g ) with E log(1+ g /R( g )) =0 such that: </li></ul><ul><li>To guarantee no-bankruptcy, when one’s wealth is w, one must reject all investments g for which R( g )>w. </li></ul>Call R( g ) the riskiness of g .
- 28. Example <ul><li>If your wealth is more than $600 repeated exposure to g would (almost </li></ul><ul><li>surely) make you arbitrarily </li></ul><ul><li>wealthy. </li></ul><ul><li>If your wealth is less than $600 repeated exposure to g would bankrupt you with probability one. </li></ul>1/2 1/2 $120 $-100 g E g =10 R AS ( g ) = 601.66 R FH ( g )=600 <ul><li>Investment g </li></ul>
- 29. Properties of R FH ( g ) <ul><li>It is measured in the same units as g is measured </li></ul><ul><li>It is monotone with respect to first and second order stochastic dominance </li></ul><ul><li>Property C : Is the reciprocal of the coefficient of absolute risk aversion of a CRRA decision maker with relative risk aversion coefficient of one and who is indifferent between taking and not taking the investment </li></ul><ul><li>It has a clear operational interpretation </li></ul>
- 30. Question <ul><li>Can we come up with an operational interpretation of R AS ? </li></ul>
- 31. I pondered about this as I walked the shores of the State Park near my house… me My house
- 32. … I started thinking about subjective measures of riskiness
- 33. Objective vs. Subjective <ul><li>Both the AS and the FH approach are meant to be “objective” measures of riskiness </li></ul><ul><li>Yet they relate, respectively, to CARA and CRRA preferences in a particular way </li></ul><ul><li>… Use Property C to develop a definition of the riskiness of an investment for a specific decision maker </li></ul>
- 34. My approach <ul><li>risk tolerance= (absolute risk aversion) -1 </li></ul><ul><li>R i (w)=-u i ’(w)/u i ’’(w) </li></ul><ul><li>This paper: use R to define the riskiness of g </li></ul>Example (CARA) P[ g <CE-2R] < e -2 ≈14% <ul><li>Zambrano (2008, ET): given CE, R contains information about the riskiness of g </li></ul><ul><li>“ CE is to µ as R is to σ ” </li></ul>
- 35. <ul><li>Definition : For any investment g “find” the wealth w( g ) that makes the decision maker with utility function u i indifferent between accepting and not accepting g : </li></ul><ul><li> Eu i (w( g )+ g ) ≡ u i (w( g )) </li></ul>Call R i (w( g )) the riskiness of g for i. The “Riwi” of g
- 36. Properties of “Riwi” (I) <ul><li>It is measured in the same units as g is measured </li></ul><ul><li>It is monotone with respect to first and second order stochastic dominance </li></ul><ul><li>“ Property C” </li></ul>
- 37. Properties of Riwi (II) <ul><li>It functions as a criterion for decision making: given u, w and g compute </li></ul><ul><li> R i (w) and R i (w( g )) </li></ul><ul><li>If R i (w)< R i (w( g )) reject the investment </li></ul>R i (w( g )): how high the risk tolerance of an decision maker must be for that decision maker to want to hold g .
- 38. Example <ul><li>A CRRA decision maker with relative risk aversion of 3 needs to have an (absolute) tolerance for risk of at least 600.37 for this decision maker to want to hold g </li></ul>1/2 1/2 $120 $-100 g E g =10 R AS ( g ) = 601.66 R FH ( g )=600 R i (w( g ))=600.37 <ul><li>Investment g </li></ul>
- 39. Notice all these measures of risk are similar for this investment <ul><li>This is so because they all satisfy the identity </li></ul><ul><li>R( g ) ≡ ½ [ E g 2 /E g ] + </li></ul><ul><li>When those terms are small, R( g ) ≈ R 0 ( g ) </li></ul>R AS ( g ) = 601.66 R FH ( g )=600 R i (w( g ))=600.37 R 0 ( g ) =610 { third order terms in a Taylor series expansion of Eu i (w( g )+ g ) } 1/2 1/2 $120 $-100 g R 0 ( g )
- 40. Properties of Riwi (III) <ul><li>Imagine that σ ( g ) shares on the investment g are offered at some price p </li></ul><ul><li>How many such shares of g will a risk averse decision maker want? </li></ul><ul><li>The answer to this problem is x(p), the demand function for shares of g . </li></ul>Max Eu(w+x( g / σ -p)) x
- 41. <ul><li>We know than when those shares are priced at expected value the decision maker will want zero shares of g . </li></ul><ul><li>As price drops slightly from expected value, how many shares will the decision maker want? </li></ul><ul><li>He (or she) will want R i (w) shares. </li></ul>dx/dp | x=0 = -R i (w)
- 42. <ul><li>R i (w) = the slope (at the origin) of the normalized demand function for shares of g for individual i and wealth w. </li></ul><ul><li>R i (w) = how much exposure to g you would want at the margin if g wasn’t priced exactly at fair value </li></ul>
- 43. w 0 w 0 +g g is too risky for i
- 44. w 1 w 1 +g g is not too risky for i
- 45. w(g) w(g) +g g is “just right” for i
- 46. An operational interpretation of R i (w( g )) <ul><li>Given investment g , R i (w( g )) is a measure of how sensitive the demand for the shares of g must be to a small drop in their price [from expected value] for the individual to want to hold the entire issue of g [at the current price]. </li></ul>The “reservation slope” of the demand for g
- 47. Decision making, again <ul><li>R i (w( g )) = the marginal exposure to g at the origin that would make you want to own all of g . </li></ul><ul><li>R i (w) = how much marginal exposure to g at the origin you actually want </li></ul><ul><li>If R i (w)< R i (w( g )) reject the investment </li></ul>Desired Marginal exposure Required marginal exposure
- 48. An investment 27.83 RiWi(3) 44.82 R FH 25.50 R AS 20.42 Stdev 8.47 E Rm -Rf
- 49. An investment A CRRA(3) decision maker with wealth $100,000 has R i (w)= 33,333 27.83 RiWi(3) 44.82 R FH 25.50 R AS 20.42 Stdev 8.47 E Rm -Rf
- 50. Scaling up A CRRA(3) decision maker with wealth $100,000 has R i (w)= 33,333 27833 27.83 RiWi(3) 44820 44.82 R FH 25500 25.50 R AS 20424 20.42 Stdev 8471 8.47 E 1000*( Rm –Rf) Rm -Rf
- 51. Two investments 24.21 27.83 RiWi(3) 29.71 44.82 R FH 23.82 25.50 R AS 14.47 20.42 Stdev 3.82 8.47 E SMB Rm -Rf
- 52. Three investments 25.39 24.21 27.83 RiWi(3) 39.56 29.71 44.82 R FH 23.16 23.82 25.50 R AS 14.02 14.47 20.42 Stdev 4.61 3.82 8.47 E HML SMB Rm -Rf
- 53. My contribution <ul><li>To provide a clear operational interpretation of R AS </li></ul><ul><li>To understand further the relationship between R AS and R FH </li></ul><ul><li>To accomplish this it turned out to be useful to generalize their approach towards the development of a full family of subjective measures of risk . </li></ul><ul><li>To measure the riskiness of some known investments according to these measures to get a “feel” for their potential usefulness. </li></ul>
- 54. Thank you for coming!
- 56. Take your pick 25.39 24.21 27.83 RiWi(3) 39.56 29.71 44.82 R FH 23.16 23.82 25.50 R AS 23.64 29.34 28.86 R 0 11.16 10.79 16.22 E|g-Eg| 18.97 17.04 28.27 Q3-Q1 15.32 14.01 24.99 VaR(5%) -39.40 - 28.68 -44.80 Min 42.66 54.86 49.24 Var/E 3.04 3.79 2.41 Stdev/E 196.69 209.42 417.15 Var 14.02 14.47 20.42 Stdev 4.61 3.82 8.47 E HML SMB Rm -Rf

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