Subjective Measures of Risk

Subjective Measures of Risk Eduardo Zambrano Department of Economics Cal Poly May 20, 2008

Example ,[object Object],[object Object],[object Object],[object Object]

The purpose of this talk ,[object Object],[object Object],[object Object],[object Object]

New solutions ,[object Object],[object Object],[object Object],[object Object],[object Object]

My contribution ,[object Object],[object Object],[object Object],[object Object]

Traditional approach ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],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 . ,[object Object],[object Object],[object Object],[object Object],1/2 1/2 $120 $-100 g 1 1/2 1/2 $170 $-50 h 1

“Sharpe ratios” ,[object Object],[object Object],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

However… ,[object Object],[object Object],.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.

μ g = μ h ,[object Object],[object Object],μ =120 μ =120 $100 $-300 h 3 $0 $300 $500 $256 $-423 g 3 $256 $256 $256

σ g = σ h ,[object Object],[object Object],$100 $-300 h 3 μ =120 σ =303 μ =120 σ =303 $0 $300 $500 $256 $-423 g 3 $256 $256 $256

σ , σ 2 , E |g-Eg|, Q 3 -Q 1 , σ / μ , σ 2 / μ ,[object Object],[object Object]

Stochastic Dominance (I) ,[object Object],[object Object],[object Object],1 x H(x) G(x)

Stochastic Dominance (II) ,[object Object],1 x H(x) G(x)

[object Object],If h stochastically dominates g then it will be preferred by any * risk averse expected utility decision maker

Problem (I) ,[object Object],[object Object]

Problem (II) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Preliminaries ,[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object]

Aumann, Serrano ,[object Object],[object Object],[object Object],Call R( g ) the riskiness of g .

Properties of R AS ( g ) ,[object Object],[object Object],[object Object]

Example ( μ g ≠ μ h ) ,[object Object],[object Object],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

Example ( μ g = μ h ) ,[object Object],[object Object],$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

Another approach ,[object Object],[object Object],[object Object],[object Object],1/2 1/2 $120 $-100 g E g =10 R AS ( g ) = 601.66

Foster, Hart ,[object Object],[object Object],Call R( g ) the riskiness of g .

Example ,[object Object],[object Object],[object Object],[object Object],1/2 1/2 $120 $-100 g E g =10 R AS ( g ) = 601.66 R FH ( g )=600 ,[object Object]

Properties of R FH ( g ) ,[object Object],[object Object],[object Object],[object Object]

I pondered about this as I walked the shores of the State Park near my house… me My house

… I started thinking about subjective measures of riskiness

Objective vs. Subjective ,[object Object],[object Object],[object Object]

My approach ,[object Object],[object Object],[object Object],Example (CARA) P[ g <CE-2R] < e -2 ≈14% ,[object Object],[object Object]

[object Object],[object Object],Call R i (w( g )) the riskiness of g for i. The “Riwi” of g

Properties of “Riwi” (I) ,[object Object],[object Object],[object Object]

Properties of Riwi (II) ,[object Object],[object Object],[object Object],R i (w( g )): how high the risk tolerance of an decision maker must be for that decision maker to want to hold g .

Example ,[object Object],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 ,[object Object]

Notice all these measures of risk are similar for this investment ,[object Object],[object Object],[object Object],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 )

Properties of Riwi (III) ,[object Object],[object Object],[object Object],Max Eu(w+x( g / σ -p)) x

[object Object],[object Object],[object Object],dx/dp | x=0 = -R i (w)

[object Object],[object Object]

w 0 w 0 +g g is too risky for i

w 1 w 1 +g g is not too risky for i

w(g) w(g) +g g is “just right” for i

An operational interpretation of R i (w( g )) ,[object Object],The “reservation slope” of the demand for g

Decision making, again ,[object Object],[object Object],[object Object],Desired Marginal exposure Required marginal exposure

An investment 27.83 RiWi(3) 44.82 R FH 25.50 R AS 20.42 Stdev 8.47 E Rm -Rf

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

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

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

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

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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