Growth-Optimality against Underperformance Greg Zitelli University - PowerPoint PPT Presentation

Growth-Optimality against Underperformance Greg Zitelli University of California, Irvine March 2016 Growth-Optimality against Underperformance University of California, Irvine

Large Deviation Theory (Markov’s Inequality) If X ≥ 0 is a random variable and x > 0 , then the Markov inequality is: x · P [ X ≥ x ] ≤ E [ X ] P [ X ≥ x ] ≤ E [ X ] x Now let X i be random variables, so that for any θ > 0 and x ∈ R we have � � e − θ � T i =1 X i � T � E 1 � e − θ � T i =1 X i ≥ e − θTx � � X i ≤ x = P P ≤ e − θTx T i =1 Growth-Optimality against Underperformance University of California, Irvine

Large Deviation Theory (Chernoff bounds) � � e − θ � T i =1 X i � T � E 1 � i =1 X i ≥ e − θTx � e − θ � T � P X i ≤ x = P ≤ T e − θTx i =1 If the X i are independent, we can split the expectation into the product, so it follows that � T � T 1 � e − θ � T � = e θTx + � T i =1 log E [ e − θXi ] � ≤ e θTx � i =1 X i X i ≤ x P E T i =1 i =1 Growth-Optimality against Underperformance University of California, Irvine

Large Deviation Theory (Chernoff bounds) � T � T 1 � e − θ � T � = e θTx + � T i =1 log E [ e − θXi ] � ≤ e θTx � i =1 X i X i ≤ x P E T i =1 i =1 The function ∞ κ n � � � e θX i n ! θ n λ i ( θ ) = log E = n =1 is called the cumulant generating function for X i , where κ 1 = E [ X i ] and κ 2 = var ( X i ) . For θ > 0 and X i independent our bound is now � T � 1 ≤ e θT ( x + 1 � T i =1 λ i ( − θ ) ) � P X i ≤ x T T i =1 Growth-Optimality against Underperformance University of California, Irvine

Large Deviation Theory (Chernoff bounds) Since this holds for all θ > 0 , we have T � � 1 � T θ> 0 e θT ( x + 1 i =1 λ i ( − θ ) ) � P X i ≤ x ≤ inf T T i =1 If the X i are i.i.d. then we have � � � T � 1 2 θ 2 − ... ] � T θT x + λ 1 ( − θ ) � e inf θ> 0 [ ( x − κ 1 ) θ + κ 2 � P X i ≤ x ≤ inf θ> 0 e = T i =1 If x − κ 1 = x − E [ X 1 ] < 0 and the power series for λ ( θ ) has nonzero radius then this bound is not trivial. In fact, if the X i are normal then κ n = 0 for all n ≥ 3 . Growth-Optimality against Underperformance University of California, Irvine

Gambling ◮ Let W 0 be our initial wealth. ◮ We choose to bet 0 ≤ p ≤ 1 fraction of our wealth on a gamble with odds π > 1 / 2 . ◮ After T rounds our wealth is � T T � � � W p,T = W 0 R p,i = W 0 exp log R p,i i =1 i =1 where P [ R p,i = 1 + p ] = π, P [ R p,i = 1 − p ] = 1 − π ◮ The Kelly criterion says to pick p so as to maximize (1 + p ) π (1 − p ) 1 − π � � 0 ≤ p ≤ 1 E [log R p,i ] = max max 0 ≤ p ≤ 1 log Growth-Optimality against Underperformance University of California, Irvine

Underperforming a benchmark Suppose we are now concerned about underperforming some benchmark rate a > 1 . � T � 1 W p,T ≤ W 0 a T � � � = P log R p,i ≤ log a P T i =1 Using large deviations we immediately have � e − θ log R p, 1 �� � � � W p,T ≤ W 0 a T � � ≤ inf θ> 0 exp θT log a + log E P � − θ �� T � �� R p, 1 = inf θ> 0 E a Growth-Optimality against Underperformance University of California, Irvine

Underperforming a benchmark � − θ �� T � �� R p, 1 W p,T ≤ W 0 a T � � inf P θ> 0 E ≤ a As T grows, suppose we want to minimize our chances of underperforming the benchmark. Our goal is to pick a 0 ≤ p ≤ 1 so as to minimize �� R p, 1 � − θ � min θ> 0 E inf a 0 ≤ p ≤ 1 Suppose π = 0 . 6 and the benchmark is 1% , then this becomes � � − θ � � − θ � 1 + p � 1 − p 0 ≤ p ≤ 1 inf min 0 . 6 + 0 . 4 1 . 01 1 . 01 θ> 0 Growth-Optimality against Underperformance University of California, Irvine

Kelly Criterion For π = 0 . 6 Kelly is 0 ≤ p ≤ 1 E [log R p, 1 ] = max max 0 ≤ p ≤ 1 0 . 6 log(1 + p ) + 0 . 4 log(1 − p ) � (1 + p ) 0 . 6 (1 − p ) 0 . 4 � = max 0 ≤ p ≤ 1 log which is realized when p = π − (1 − π ) = 2 π − 1 = 0 . 2 . Growth-Optimality against Underperformance University of California, Irvine

Kelly Criterion � (1 + p ) 0 . 6 (1 − p ) 0 . 4 � 0 ≤ p ≤ 1 log max Growth-Optimality against Underperformance University of California, Irvine

Underperforming a benchmark � − θ �� T � �� R p, 1 W p,T ≤ W 0 a T � � inf P θ> 0 E ≤ a Now suppose our goal is minimizing the probability of underperforming the benchmark 1% . We want to minimize �� R p, 1 � − θ � 0 ≤ p ≤ 1 ,θ> 0 E min a � � − θ � � − θ � 1 + p � 1 − p = min 0 . 6 + 0 . 4 1 . 01 1 . 01 0 ≤ p ≤ 1 ,θ> 0 Growth-Optimality against Underperformance University of California, Irvine

Underperforming a benchmark of 1% � − θ � − θ � 1 + p � 1 − p D p,θ = 0 . 6 + 0 . 4 1 . 01 1 . 01 Growth-Optimality against Underperformance University of California, Irvine

Underperforming a (smaller) benchmark of 0.1% � − θ � − θ � 1 + p � 1 − p D p,θ 0 . 6 + 0 . 4 1 . 001 1 . 001 Growth-Optimality against Underperformance University of California, Irvine

Underperforming a benchmark ◮ Our goal of minimizing the asymptotic probability � − θ �� T � �� R p, 1 W p,T ≤ W 0 a T � � inf P ≤ θ> 0 E a leads us to consider a dual optimization in terms of p and θ . �� R p, 1 � − θ � 0 ≤ p ≤ 1 min min θ> 0 E a ◮ 1 + θ plays the role of the bettor’s risk aversion. It is not exogenous, but rather determined by the inner maximization. For instance, a bettor who is concerned with outperforming returns of 1% exhibits risk aversion of 1 + θ = 1 . 43 . Growth-Optimality against Underperformance University of California, Irvine

Isoelastic Utility ◮ Note that our problem �� R p, 1 � − θ � 0 ≤ p ≤ 1 min min θ> 0 E a can be rephrased to appear similar to maximizing the isoelastic utility of our returns: � � 1 − γ � � R p, 1 0 ≤ p ≤ 1 max max γ> 1 E − a where γ is risk aversion. Growth-Optimality against Underperformance University of California, Irvine

Risk Aversion ◮ Consider the expected utility for the Blackjack game with π = 0 . 6 and varying risk aversion 1 + θ = γ . � � � − 0 . 6(1 + p ) − θ − 0 . 4(1 − p ) − θ � − R − θ 0 ≤ p ≤ 1 E max = max p, 1 0 ≤ p ≤ 1 Growth-Optimality against Underperformance University of California, Irvine

Risk Aversion ◮ If θ > 2 . 76 is a bettor’s exogenous risk aversion then a bettor considers a bet p = 10% to be unfavorable to a bet of p = 0% , regardless of their initial wealth W 0 or the number of trials T . Growth-Optimality against Underperformance University of California, Irvine

Measuring Risk Aversion ◮ Barsky et al. (1997) designed a questionnaire given to thousands of individuals in person by Federal interviewers, and about 2/3 of them had relative risk aversion higher than 3 . 76 = 1 + θ . ◮ Suppose I offer you the chance to play the blackjack π = 0 . 6 game 10,000 times instantly on a computer, but if you agree you must use the strategy p = 10% . ◮ Using the large deviation bound derived above, the long term behavior hinges on � − θ �� T � �� R p, 1 � W p,T ≤ W 0 a T � P θ> 0 E inf ≤ a Growth-Optimality against Underperformance University of California, Irvine

Measuring Risk Aversion ◮ The chances of underperforming an 0 . 6% benchmark are quite bad: � − θ �� 10 4 � �� R 10% , 1 � W 10% , 10 4 ≤ W 0 1 . 006 10 4 � inf P θ> 0 E ≤ 1 . 006 ≤ 0 . 998 10 4 < 10 − 8 so it is quite likely you will end up with more than W 0 × 10 24 , and all you stand to lose is W 0 . Growth-Optimality against Underperformance University of California, Irvine

Measuring Risk Aversion ◮ An individual with exogenous risk of 1 + θ = 3 . 76 or greater would not want to take this bet because they are principally interested in maximizing � � � − 0 . 6(1 + p ) − θ − 0 . 4(1 − p ) − θ � − R − θ max = max 0 ≤ p ≤ 1 E p, 1 0 ≤ p ≤ 1 and the choice p = 10% is worse (according to their expected utility) than a choice of p = 0% . ◮ On the other hand, an individual hoping to beat a modest benchmark of 0 . 6% is hoping to minimize � � − θ � − θ � � 1 + p � 1 − p 0 ≤ p ≤ 1 min min 0 . 6 + 0 . 4 1 . 006 1 . 006 θ> 0 Growth-Optimality against Underperformance University of California, Irvine

Measuring Risk Aversion ◮ Such an individual would be willing to take the bet. � − θ � − θ � 1 + p � 1 − p D p,θ = 0 . 6 + 0 . 4 1 . 006 1 . 006 Growth-Optimality against Underperformance University of California, Irvine