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 perspective Time perspective
Ole Peters 10000 1010,000 Positioning 1000 Innocuous game 100 100 Leverage
optimization money in $ money in $ 10 −10,000 How deep the 10 rabbit hole 1 goes 10 20 30 40 50 60 3 6 9 12 Time in minutes Time in months Only time will Non-ergodic tell
O. Peters Ensemble perspective 6= Time perspective
Ole Peters 10000 1010,000 Positioning 1000 Innocuous game 100 100 Leverage
optimization money in $ money in $ 10 −10,000 How deep the 10 rabbit hole 1 goes 10 20 30 40 50 60 3 6 9 12 Time in minutes Time in months Only time will Non-ergodic tell
O. Peters Ensemble perspective 6= Time perspective
Ole Peters 10000 1010,000 Positioning 1000 Innocuous game 100 100 Leverage
optimization money in $ money in $ 10 −10,000 How deep the 10 rabbit hole 1 goes 10 20 30 40 50 60 3 6 9 12 Time in minutes Time in months
Non-commuting limits
limT →∞ limN→∞ gest 6= limN→∞ limT →∞ 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 →∞ 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. O. Peters Ole Peters • Utility: 18th century (long before ergodicity debate).
Positioning • Innocuous Modern interpretation game Utility theory aims for ergodic observable, e.g. for GBM, Leverage optimization rate of change in log utility.
How deep the rabbit hole • Different questions answered goes Utility theory: u lopt corresponds to greatest expected happiness.
Time perspective: t lopt implies greatest growth rate. √ GBM parameters µr = 5.2% p.a., µe = 2.4% p.a., σ = 15.9% p. a.
3 10 Ergodicity economics Only time will Square root utility tell
O. Peters
2 Ole Peters 10
Positioning
Innocuous game
1 Leverage 10 optimization x(t)
How deep the rabbit hole goes
0 10
−1 10 0 10 20 30 40 50 60 70 80 90 100 t √ GBM parameters µr = 5.2% p.a., µe = 2.4% p.a., σ = 15.9% p. a.
3 10 Ergodicity economics Only time will u(x)=x3/4 tell
2 O. Peters 10
Ole Peters
1 Positioning 10
Innocuous game
0 Leverage 10 optimization x(t)
How deep the rabbit hole −1 goes 10
−2 10
−3 10 0 10 20 30 40 50 60 70 80 90 100 t √ GBM parameters µr = 5.2% p.a., µe = 2.4% p.a., σ = 15.9% p. a.
4 10 Ergodicity economics Only time will u(x)=x8/10 tell 2 10 O. Peters
Ole Peters 0 10 Positioning
Innocuous −2 game 10
Leverage optimization x(t) −4 10 How deep the rabbit hole goes −6 10
−8 10
−10 10 0 10 20 30 40 50 60 70 80 90 100 t Only time will tell
O. Peters
Ole Peters
Positioning Cruel world doesn’t care about my risk preferences. Innocuous game
Leverage optimization
How deep the rabbit hole goes How deep the rabbit hole goes
Only time will tell
O. Peters
Ole Peters
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes 384 – 322 BC Aristotle’s cosmology.
Only time will tell
O. Peters
Ole Peters
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell
O. Peters
Ole Peters
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell ??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
O. Peters
Ole Peters
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell ??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
O. Peters
Ole Peters 200 BC–1500 CE No challenge (Hypatia?).
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell ??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
O. Peters
Ole Peters 200 BC–1500 CE No challenge (Hypatia?).
Positioning 1473–1543 CE Copernicus’ model: perfect circles but heliocentric. Innocuous game
Leverage optimization
How deep the rabbit hole goes 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell ??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
O. Peters
Ole Peters 200 BC–1500 CE No challenge (Hypatia?).
Positioning 1473–1543 CE Copernicus’ model: perfect circles but heliocentric. Innocuous game 1546–1601 CE Tycho Brahe – geocentric but observed comet 1577. Leverage optimization
How deep the rabbit hole goes 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell ??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
O. Peters
Ole Peters 200 BC–1500 CE No challenge (Hypatia?).
Positioning 1473–1543 CE Copernicus’ model: perfect circles but heliocentric. Innocuous game 1546–1601 CE Tycho Brahe – geocentric but observed comet 1577. Leverage optimization 1571–1630 CE Kepler – Mars’ orbit elliptic! Heliocentric.
How deep the rabbit hole goes 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell ??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
O. Peters
Ole Peters 200 BC–1500 CE No challenge (Hypatia?).
Positioning 1473–1543 CE Copernicus’ model: perfect circles but heliocentric. Innocuous game 1546–1601 CE Tycho Brahe – geocentric but observed comet 1577. Leverage optimization 1571–1630 CE Kepler – Mars’ orbit elliptic! Heliocentric.
How deep the 1564–1642 CE Galileo – earthly motion in perfect shapes, parabolas! rabbit hole goes 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell ??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
O. Peters
Ole Peters 200 BC–1500 CE No challenge (Hypatia?).
Positioning 1473–1543 CE Copernicus’ model: perfect circles but heliocentric. Innocuous game 1546–1601 CE Tycho Brahe – geocentric but observed comet 1577. Leverage optimization 1571–1630 CE Kepler – Mars’ orbit elliptic! Heliocentric.
How deep the 1564–1642 CE Galileo – earthly motion in perfect shapes, parabolas! rabbit hole goes 1643–1727 CE Newton – laws on earth and in heaven identical. 384 – 322 BC Aristotle’s cosmology. 310 – 230 BC Aristarchus model: heliocentric – dismissed. Only time will tell ??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
O. Peters
Ole Peters 200 BC–1500 CE No challenge (Hypatia?).
Positioning 1473–1543 CE Copernicus’ model: perfect circles but heliocentric. Innocuous game 1546–1601 CE Tycho Brahe – geocentric but observed comet 1577. Leverage optimization 1571–1630 CE Kepler – Mars’ orbit elliptic! Heliocentric.
How deep the 1564–1642 CE Galileo – earthly motion in perfect shapes, parabolas! rabbit hole goes 1643–1727 CE Newton – laws on earth and in heaven identical.
Mid-17th century: Crisis! All or nothing time-bound? 1654 Probability theory
Only time will tell
O. Peters
Ole Peters
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes 1654 Probability theory
Only time will tell
O. Peters
Ole Peters
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes 1654 Probability theory
Only time will tell
O. Peters
Ole Peters
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes Evolution of structure
Only time will Two entities follow GBM. tell dx1 = x1(µdt + σdW1) and dx2 = x2(µdt + σdW2) O. Peters Ole Peters If entities pool resources, they will follow
Positioning 1 1 dx12 = x12 µdt + σ 2 dW1 + 2 dW2 Innocuous game Cooperation conundrum Leverage optimization Lucky partner gives, unlucky partner receives. How deep the rabbit hole goes Expectation value grows at µ, irrespective of cooperation. Lucky partner: why cooperate? But: cooperation exists (multicellularity, firms, states etc).
→ Expectation value model fails. Cooperators do better over time (though not in expectation).
von Neumann (1944): “We need inventions on the scale of a new calculus to make progress on dynamics.”
Correct, and we have that since 1944 (Itoˆ).
Usual story: very complicated.
Only time will tell Our story: non-ergodic system → compute expectation value of ergodic observable under specified dynamics (time-average O. Peters
Ole Peters growth rate).
Positioning dhln(x1)i dhln(x2)i 2 Innocuous 1 No cooperation: dt = dt = µ − σ /2 game dhln(x )i Leverage 2 12 2 optimization Cooperation: dt = µ − σ /4 How deep the rabbit hole goes Usual story: very complicated.
Only time will tell Our story: non-ergodic system → compute expectation value of ergodic observable under specified dynamics (time-average O. Peters
Ole Peters growth rate).
Positioning dhln(x1)i dhln(x2)i 2 Innocuous 1 No cooperation: dt = dt = µ − σ /2 game dhln(x )i Leverage 2 12 2 optimization Cooperation: dt = µ − σ /4 How deep the rabbit hole Cooperators do better over time goes (though not in expectation).
von Neumann (1944): “We need inventions on the scale of a new calculus to make progress on dynamics.”
Correct, and we have that since 1944 (Itoˆ).
Individual 1 Individual 2 Cooperating unit Expectation value Only time will $1e+40 tell
O. Peters
Ole Peters Wealth $1e+20 Positioning
Innocuous game
Leverage optimization $1
How deep the 0 2000 4000 6000 8000 10000 rabbit hole Time goes
• Good risk management = faster growth. • Evolutionary advantage of structure (multicellularity, tribes, firms, nations) over no structure. Only time will Problems we can address (solve) tell
O. Peters • optimize leverage (for any dynamic) Ole Peters • map utility theory → dynamics
Positioning • 300-year old St. Petersburg paradox Innocuous game • dynamics of wealth distribution Leverage • better economic measures than GDP optimization
How deep the • price insurance contracts (derivatives) rabbit hole goes • explain emergence of structure • ... Conclusion
Only time will tell
O. Peters
Ole Peters
Positioning
Innocuous game Correcting one deep conceptual flaw in economics Leverage enables powerful quantitative theory. optimization
How deep the rabbit hole goes Only time will tell Thank you. O. Peters
Ole Peters
Positioning
Innocuous game
Leverage optimization
How deep the rabbit hole goes