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Only Time Will Tell Risk Optimization from a Dynamics Perspective Only time will tell Only time will Risk optimization from a dynamics perspective tell O. Peters Ole Peters Ole Peters Positioning Innocuous game Fellow External Professor Leverage optimization London Mathematical Laboratory Santa Fe Institute How deep the rabbit hole goes 12 May 2015 Global Quantitative Investment Strategies Conference Nomura, NYC Only time will tell O. Peters Ole Peters Name dropping: Positioning Many thanks to Alex Adamou, Bill Klein, Reuben Hersh, Innocuous game Murray Gell-Mann, Ken Arrow. Leverage optimization How deep the rabbit hole goes Only time will tell 1 Positioning O. Peters Ole Peters Positioning 2 Innocuous game Innocuous game Leverage optimization 3 Leverage optimization How deep the rabbit hole goes 4 How deep the rabbit hole goes • 17th century: mainstream economics went down a dead-end. • 19th { 21st centuries: relevant mathematics developed. Program Re-derive formal economics from modern starting point. Only time will tell My perspective O. Peters Ole Peters Positioning Innocuous game Leverage optimization How deep the rabbit hole goes • 19th { 21st centuries: relevant mathematics developed. Program Re-derive formal economics from modern starting point. Only time will tell My perspective O. Peters Ole Peters • 17th century: mainstream economics went down a Positioning dead-end. Innocuous game Leverage optimization How deep the rabbit hole goes Program Re-derive formal economics from modern starting point. Only time will tell My perspective O. Peters Ole Peters • 17th century: mainstream economics went down a Positioning dead-end. Innocuous game • 19th { 21st centuries: relevant mathematics developed. Leverage optimization How deep the rabbit hole goes Only time will tell My perspective O. Peters Ole Peters • 17th century: mainstream economics went down a Positioning dead-end. Innocuous game • 19th { 21st centuries: relevant mathematics developed. Leverage optimization How deep the Program rabbit hole goes Re-derive formal economics from modern starting point. Only time will tell O. Peters Thesis: Ole Peters Problem with randomness, i.e. risk. Positioning th Innocuous 17 -century key concept ! parallel worlds. game st Leverage 21 -century mathematics ! time. optimization How deep the rabbit hole goes Innocuous game Only time will tell O. Peters Ole Peters Positioning Innocuous game Leverage optimization How deep the rabbit hole goes Heads: win 50%. Tails: lose 40%. Innocuous game Toss coin once a minute Only time will tell O. Peters Ole Peters 120 Positioning 100 Innocuous game 80 Leverage optimization 60 How deep the rabbit hole goes 40 money in $ 20 0 1 2 3 4 5 Time in minutes One sequence Only time will tell O. Peters 10000 Ole Peters 1000 Positioning Innocuous game 100 Leverage optimization money in $ 10 How deep the rabbit hole goes 1 10 20 30 40 50 60 Time in minutes 10 sequences Only time will tell O. Peters 10000 Ole Peters 1000 Positioning Innocuous game 100 Leverage optimization money in $ 10 How deep the rabbit hole goes 1 10 20 30 40 50 60 Time in minutes 20 sequences Only time will tell O. Peters 10000 Ole Peters 1000 Positioning Innocuous game 100 Leverage optimization money in $ 10 How deep the rabbit hole goes 1 10 20 30 40 50 60 Time in minutes Average of 20 sequences Only time will tell O. Peters 10000 Ole Peters 1000 Positioning Innocuous game 100 Leverage optimization money in $ 10 How deep the rabbit hole goes 1 10 20 30 40 50 60 Time in minutes Average of 1000 sequences Only time will tell O. Peters 10000 Ole Peters 1000 Positioning Innocuous game 100 Leverage optimization money in $ 10 How deep the rabbit hole goes 1 10 20 30 40 50 60 Time in minutes Average of 1,000,000 sequences Only time will tell O. Peters 10000 Ole Peters 1000 Positioning Innocuous game 100 Leverage optimization money in $ 10 How deep the rabbit hole goes 1 10 20 30 40 50 60 Time in minutes Only time will tell O. Peters Ole Peters Positioning Good game? Innocuous game Leverage optimization How deep the rabbit hole goes Play for one hour... Only time will tell 10000 O. Peters Ole Peters 1000 Positioning Innocuous game 100 Leverage optimization money in $ 10 How deep the rabbit hole goes 1 10 20 30 40 50 60 Time in minutes ..continue one day (note scales)... Only time will tell O. Peters 1040 10000 Ole Peters 1000 "" 1020 " 100 Positioning " money in $ 10 " ((( 1 Innocuous "(( 10 20 30 40 50 60 game 100 Time in minutes Leverage optimization 10−20 money in $ How deep the rabbit hole −40 goes 10 6 12 18 24 Time in hours ..continue one week (note scales)... Only time will tell O. Peters 10300 Ole Peters " 1040 " 1020 150 " 100 Positioning 10 -20 " money in $ 10 " 10-40 Innocuous " (( 0 6 12 18 24 game 100 (( Time in hours Leverage optimization 10−150 money in $ How deep the rabbit hole −300 goes 10 1 2 3 4 5 6 7 Time in days ..continue one year (note scales)... Only time will tell O. Peters 10;000 Ole Peters 10 " 10300 " 10150 " 100 Positioning 10-150 " money in $ " (((( 10-300 Innocuous "(( 0 1 2 3 4 5 6 7 game 100 Time in days Leverage optimization money in $ How deep the −10;000 rabbit hole 10 goes 3 6 9 12 Time in months Only time will tell O. Peters Ensemble 10000 perspective Time perspective Ole Peters 1000 1010;000 Positioning Innocuous game 100 100 Leverage optimization money in $ money in $ 10 −10;000 How deep the 10 rabbit hole goes 1 3 6 9 12 10 20 30 40 50Time 60 in months Time in minutes Only time will Non-ergodic tell O. Peters Ensemble 10000 perspective 6= Time perspective Ole Peters 1000 1010;000 Positioning Innocuous game 100 100 Leverage optimization money in $ money in $ 10 −10;000 How deep the 10 rabbit hole goes 1 3 6 9 12 10 20 30 40 50Time 60 in months Time in minutes Only time will Non-ergodic tell O. Peters Ensemble 10000 perspective 6= Time perspective Ole Peters 1000 1010;000 Positioning Innocuous game 100 100 Leverage optimization money in $ money in $ 10 −10;000 How deep the 10 rabbit hole goes 1 3 6 9 12 10 20 30 40 50Time 60 in months Non-commutingTime in minutes limits limT !1 limN!1 gest 6= limN!1 limT !1 gest No magic. Only time will tell O. Peters Ensemble perspective Ole Peters Positioning Innocuous $150 game heads universe Leverage optimization $100 average (probabilities = weights) How deep the rabbit hole $50 goes tails universe 0 1 2 Result: $110.25 No magic. Only time will tell O. Peters Ensemble perspective Time perspective Ole Peters Positioning Innocuous $150 $150 game heads universe Leverage one trajectory $100 average $100 optimization (probabilities = frequencies) (probabilities = weights) How deep the rabbit hole $50 $50 goes tails universe 0 1 2 0 1 2 Result: $110.25 Result: $90 Only time will tell O. Peters Message: Ole Peters Expectation value meaningful only if Positioning • observable is ergodic Innocuous game • a physical ensemble exists Leverage optimization How deep the Otherwise meaningless. rabbit hole goes Leverage optimization Only time will tell Problem: find proportion of wealth to invest in some venture. O. Peters Neoclassical Time Ole Peters economics perspective Positioning Model Random variable ∆x Stochastic process Innocuous game to represent changes x(t) to represent Leverage in wealth. wealth over time. optimization Technique 1656 {1738: compute Find ergodic observ- How deep the rabbit hole expectation value able f (x). Optimize goes h∆xi. time-average perfor- 1738 onwards: find mance by computing utility function u(x), expectation value of optimize expectation ergodic observable. value h∆u(x)i. Example: Only time will geometric Brownian motion (GBM) with leverage l. tell O. Peters • Proportion l invested in GBM Ole Peters • Proportion 1 − l invested in risk-free asset Positioning • constant rebalancing, self-financed portfolio. Innocuous game Wealth follows: dx = x((µr + lµe )dt + lσdW ) Leverage optimization l 2σ2 Solution: x(t) = x0 exp µr + lµe − 2 t + lσW (t) How deep the rabbit hole goes ! Find optimal leverage using a) Utility theory. b) Time perspective. a) Utility theory with power-law utility, u(x) = xα: • Fix horizon ∆t, consider random variable x(∆t). Only time will α tell • Convert x(∆t) to utility u(x(∆t)) = x(∆t) , α l2σ2 u(x(∆t)) = x exp α µr + lµe − ∆t + αlσW (∆t) O. Peters 0 2 Ole Peters • Find expectation value of u(x(∆t)), Positioning α l2σ2 αl2σ2 hu(x(∆t))i = x0 exp α∆t µr + lµe − 2 + 2 Innocuous game Implies expected change in utility h∆ui = hu(x(∆t))i − u(x0). Leverage optimization • Set derivative to zero, How deep the dh∆ui = 0 rabbit hole dl goes α 2 2 l2σ2 αl2σ2 = x0 α∆t µe − lσ + αlσ exp α∆t µr + lµe − 2 + 2 : • Solve for l u µe lopt = (1−α)σ2 . b) Time perspective Only time will • Given x(t), find ergodic observable, i.e. f (x) such that tell Ensemble average Time average z }| { O. Peters z }| { N 1 Z T 1 X lim f (x(t))dt = f (xi (t)) = hf (x(t))i : Ole Peters T !1 T 0 N i Positioning Solution is a growth rate, determined by dynamics, Innocuous f (x) = 1 log(x(t + ∆t)=x(t)): game ∆t Leverage • Find time average (or expectation value) optimization l2σ2 How deep the f = µr + lµe − 2 : rabbit hole goes • Set derivative to zero, solve for l t µe lopt = σ2 : Comments: • u µe lopt = (1−α)σ2 depends on utility function (α). Only time will tell t µe lopt = σ2 set by dynamics.
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