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Understanding Risk-Aversion through Utility TheoryAshwin RaoICME, Stanford UniversityFebruary 3, 2020Ashwin Rao (Stanford)Utility TheoryFebruary 3, 20201 / 14
Intuition on Risk-Aversion and Risk-PremiumLet’s play a game where your payoff is based on outcome of a fair coinYou get 100 for HEAD and 0 for TAILHow much would you pay to play this game?You immediately say: “Of course, 50”Then you think a bit, and say: “A little less than 50”Less because you want to “be compensated for taking the risk”The word Risk refers to the degree of variation of the outcomeWe call this risk-compensation as Risk-PremiumOur personality-based degree of risk fear is known as Risk-AversionSo, we end up paying 50 minus Risk-Premium to play the gameRisk-Premium grows with Outcome-Variance & Risk-AversionAshwin Rao (Stanford)Utility TheoryFebruary 3, 20202 / 14
Specifying Risk-Aversion through a Utility functionWe seek a “valuation formula” for the amount we’d pay that:Increases one-to-one with the Mean of the outcomeDecreases as the Variance of the outcome (i.e. Risk) increasesDecreases as our Personal Risk-Aversion increasesThe last two properties above define the Risk-PremiumBut fundamentally why are we Risk-Averse?Why don’t we just pay the mean of the random outcome?Reason: Our satisfaction to better outcomes grows non-linearlyWe express this satisfaction non-linearity as a mathematical functionBased on a core economic concept called Utility of ConsumptionWe will illustrate this concept with a real-life exampleAshwin Rao (Stanford)Utility TheoryFebruary 3, 20203 / 14
Law of Diminishing Marginal UtilityAshwin Rao (Stanford)Utility TheoryFebruary 3, 20204 / 14
Utility of Consumption and Certainty-Equivalent ValueMarginal Satisfaction of eating cookies is a diminishing functionHence, Accumulated Satisfaction is a concave functionAccumulated Satisfaction represents Utility of Consumption U(x)Where x represents the uncertain outcome being consumedDegree of concavity represents extent of our Risk-AversionConcave U(·) function E[U(x)] U(E[x])We define Certainty-Equivalent Value xCE U 1 (E[U(x)])Denotes certain amount we’d pay to consume an uncertain outcomeAbsolute Risk-Premium πA E[x] xCERelative Risk-Premium πR Ashwin Rao (Stanford)πAE[x] Utility TheoryE[x] xCEE[x] 1 xCEE[x]February 3, 20205 / 14
Certainty-Equivalent ValueAshwin Rao (Stanford)Utility TheoryFebruary 3, 20206 / 14
Calculating the Risk-PremiumWe develop mathematical formalism to calculate Risk-Premia πA , πRTo lighten notation, we refer to E[x] as x̄ and Variance of x as σx2Taylor-expand U(x) around x̄, ignoring terms beyond quadratic1U(x) U(x̄) U 0 (x̄) · (x x̄) U 00 (x̄) · (x x̄)22Taylor-expand U(xCE ) around x̄, ignoring terms beyond linearU(xCE ) U(x̄) U 0 (x̄) · (xCE x̄)Taking the expectation of the U(x) expansion, we get:1· U 00 (x̄) · σx22Since E[U(x)] U(xCE ), the above two expressions are . Hence,E[U(x)] U(x̄) U 0 (x̄) · (xCE x̄) Ashwin Rao (Stanford)Utility Theory1· U 00 (x̄) · σx22February 3, 20207 / 14
Absolute & Relative Risk-AversionFrom the last equation on the previous slide, Absolute Risk-Premium1 U 00 (x̄) 2· σxπA x̄ xCE · 02 U (x̄)00(x)as the Absolute Risk-AversionWe refer to function A(x) UU 0 (x)1· A(x̄) · σx22In multiplicative uncertainty settings, we focus on variance σ 2x of xx̄x̄In multiplicative settings, we also focus on Relative Risk-Premium πRπA πR πA1 U 00 (x̄) · x̄ σx21 U 00 (x̄) · x̄ ·· ·· σ 2xx̄x̄2U 0 (x̄)x̄ 22U 0 (x̄)00We refer to function R(x) UU 0(x)·x(x) as the Relative Risk-AversionπR Ashwin Rao (Stanford)1· R(x̄) · σ 2xx̄2Utility TheoryFebruary 3, 20208 / 14
Taking stock of what we’re learning hereWe’ve shown that Risk-Premium can be expressed as the product of:Extent of Risk-Aversion: either A(x̄) or R(x̄)Extent of uncertainty of outcome: either σx2 or σ 2xx̄We’ve expressed the extent of Risk-Aversion as the ratio of:Concavity of the Utility function (at x̄): U 00 (x̄)Slope of the Utility function (at x̄): U 0 (x̄)For optimization problems, we ought to maximize E[U(x)] (not E[x])Linear Utility function U(x) a b · x implies Risk-NeutralityNow we look at typically-used Utility functions U(·) with:Constant Absolute Risk-Aversion (CARA)Constant Relative Risk-Aversion (CRRA)Ashwin Rao (Stanford)Utility TheoryFebruary 3, 20209 / 14
Constant Absolute Risk-Aversion (CARA)1 e axa U 00 (x)U 0 (x) aConsider the Utility function U(x) for a 6 0Absolute Risk-Aversion A(x) a is called Coefficient of Constant Absolute Risk-Aversion (CARA)For a 0, U(x) x (meaning Risk-Neutral)If the random outcome x N (µ, σ 2 ), 1 e aµ a22σ2for a 6 0aE[U(x)] µfor a 0xCE µ aσ 22aσ 222For optimization problems where σ is a function of µ, we seek the2distribution that maximizes µ aσ2Absolute Risk Premium πA µ xCE Ashwin Rao (Stanford)Utility TheoryFebruary 3, 202010 / 14
A Portfolio Application of CARAWe are given 1 to invest and hold for a horizon of 1 yearInvestment choices are 1 risky asset and 1 riskless assetRisky Asset Annual Return N (µ, σ 2 )Riskless Asset Annual Return rDetermine unconstrained π to allocate to risky asset (1 π to riskless)Such that Portfolio has maximum Utility of Wealth in 1 year aWfor a 6 0With CARA Utility U(W ) 1 eaPortfolio Wealth W N (1 r π(µ r ), π 2 σ 2 )From the section on CARA Utility, we know we need to maximize:aπ 2 σ 22So optimal investment fraction in risky assetµ rπ aσ 21 r π(µ r ) Ashwin Rao (Stanford)Utility TheoryFebruary 3, 202011 / 14
Constant Relative Risk-Aversion (CRRA)x 1 γ 11 γ U 00 (x)·x γU 0 (x)Consider the Utility function U(x) Relative Risk-Aversion R(x) for γ 6 1γ is called Coefficient of Constant Relative Risk-Aversion (CRRA)For γ 1, U(x) log(x). For γ 0, U(x) x 1 (Risk-Neutral)If the random outcome x is lognormal, with log(x) N (µ, σ 2 ), e µ(1 γ) σ22 (1 γ)2 1for γ 6 11 γE[U(x)] µfor γ 1xCE e µ σ2(1 γ)2σ2 γxCE 1 e 2x̄2For optimization problems where σ is a function of µ, we seek the2distribution that maximizes µ σ2 (1 γ)Relative Risk Premium πR 1 Ashwin Rao (Stanford)Utility TheoryFebruary 3, 202012 / 14
A Portfolio Application of CRRA (Merton 1969)We work in the setting of Merton’s 1969 Portfolio problemWe only consider the single-period (static) problem with 1 risky assetRiskless asset: dRt r · Rt · dtRisky asset: dSt µ · St · dt σ · St · dzt (i.e. Geometric Brownian)We are given 1 to invest, with continuous rebalancing for 1 yearDetermine constant fraction π of Wt to allocate to risky assetTo maximize Expected Utility of Wealth W W1 (at time t 1)Constraint: Portfolio is continuously rebalanced to maintain fraction πSo, the process for wealth Wt is given by:dWt (r π(µ r )) · Wt · dt π · σ · Wt · dztAssume CRRA Utility U(W ) Ashwin Rao (Stanford)W 1 γ 11 γ , 0Utility Theory γ 6 1February 3, 202013 / 14
Recovering Merton’s solution (for this static case)Applying Ito’s Lemma on log Wt gives us:Z tZ tπ2σ2log Wt ) · du (r π(µ r ) π · σ · dzu200π2σ2 2 2,π σ )2From the section on CRRA Utility, we know we need to maximize: log W N (r π(µ r ) r π(µ r ) π 2 σ 2 π 2 σ 2 (1 γ) 22π2σ2γ2So optimal investment fraction in risky asset r π(µ r ) π Ashwin Rao (Stanford)µ rγσ 2Utility TheoryFebruary 3, 202014 / 14
The word Risk refers to the degree of variation of the outcome We call this risk-compensation as Risk-Premium Our personality-based degree of risk fear is known as Risk-Aversion So, we end up paying 50 minus Risk-Premium to play the game Risk-Premium grows with Outcome-Variance & Risk-Aversion Ashwin Rao (Stanford) Utility Theory February 3, 2020 2/14. Specifying Risk-Aversion through a .
