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Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Only time will tellRisk optimization from a dynamics perspective
Ole Peters
Fellow External ProfessorLondon Mathematical Laboratory Santa Fe Institute
12 May 2015Global Quantitative Investment Strategies Conference
Nomura, NYC
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Name dropping:Many thanks to Alex Adamou, Bill Klein, Reuben Hersh,Murray Gell-Mann, Ken Arrow.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
1 Positioning
2 Innocuous game
3 Leverage optimization
4 How deep the rabbit hole goes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
My perspective
• 17th century: mainstream economics went down adead-end.
• 19th – 21st centuries: relevant mathematics developed.
ProgramRe-derive formal economics from modern starting point.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
My perspective
• 17th century: mainstream economics went down adead-end.
• 19th – 21st centuries: relevant mathematics developed.
ProgramRe-derive formal economics from modern starting point.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
My perspective
• 17th century: mainstream economics went down adead-end.
• 19th – 21st centuries: relevant mathematics developed.
ProgramRe-derive formal economics from modern starting point.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
My perspective
• 17th century: mainstream economics went down adead-end.
• 19th – 21st centuries: relevant mathematics developed.
ProgramRe-derive formal economics from modern starting point.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Thesis:
Problem with randomness, i.e. risk.
17th-century key concept → parallel worlds.
21st-century mathematics → time.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Innocuous game
Heads: win 50%. Tails: lose 40%.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Innocuous game
Toss coin once a minute
0
20
40
60
80
100
120
1 2 3 4 5
mon
ey in
$
Time in minutes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
One sequence
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
10 sequences
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
20 sequences
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Average of 20 sequences
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Average of 1000 sequences
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Average of 1,000,000 sequences
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Good game?
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Play for one hour...
mon
ey in
$
Time in minutes
10000
1000
100
10
110 20 30 40 50 60
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
..continue one day (note scales)...
mon
ey in
$
Time in hours
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
1040
1020
100
10−20
10−40
6 12 18 24
""""""
(((((
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
..continue one week (note scales)...
mon
ey in
$
Time in days
10-4010-20
10010201040
0 6 12 18 24
mon
ey in
$
Time in hours
10300
10150
100
10−150
10−300
1 2 3 4 5 6 7
""""""
((((
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
..continue one year (note scales)...
mon
ey in
$
Time in months
10-30010-150
1001015010300
0 1 2 3 4 5 6 7
mon
ey in
$
Time in days
1010,000
100
10−10,000
3 6 9 12
""""""
((((((
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Ensemble perspective
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
1010,000
100
10−10,000
3 6 9 12
Time perspective
mon
ey in
$
Time in months
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Ensemble perspective
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
Non-ergodic
6=
1010,000
100
10−10,000
3 6 9 12
Time perspective
mon
ey in
$
Time in months
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Ensemble perspective
1
10
100
1000
10000
10 20 30 40 50 60
mon
ey in
$
Time in minutes
Non-ergodic
6=
1010,000
100
10−10,000
3 6 9 12
Time perspective
mon
ey in
$
Time in months
Non-commuting limits
limT→∞ limN→∞ gest 6= limN→∞ limT→∞ gest
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
No magic.
Ensemble perspective
$150
$100
$50
0 1 2
heads universe
average(probabilities = weights)
tails universe
Result: $110.25
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
No magic.
Ensemble perspective
$150
$100
$50
0 1 2
heads universe
average(probabilities = weights)
tails universe
Result: $110.25
Time perspective
$150
$100
$50
0 1 2
one trajectory(probabilities = frequencies)
Result: $90
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Message:
Expectation value meaningful only if
• observable is ergodic
• a physical ensemble exists
Otherwise meaningless.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Leverage optimization
Problem: find proportion of wealth to invest in some venture.
Neoclassicaleconomics
Timeperspective
Model Random variable ∆xto represent changesin wealth.
Stochastic processx(t) to representwealth over time.
Technique 1656 –1738: computeexpectation value〈∆x〉.1738 onwards: findutility function u(x),optimize expectationvalue 〈∆u(x)〉.
Find ergodic observ-able f (x). Optimizetime-average perfor-mance by computingexpectation value ofergodic observable.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Example:geometric Brownian motion (GBM) with leverage l .
• Proportion l invested in GBM
• Proportion 1− l invested in risk-free asset
• constant rebalancing, self-financed portfolio.
Wealth follows: dx = x((µr + lµe)dt + lσdW )
Solution: x(t) = x0 exp((µr + lµe − l2σ2
2
)t + lσW (t)
)→ Find optimal leverage using
a) Utility theory.
b) Time perspective.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
a) Utility theory with power-law utility, u(x) = xα:
• Fix horizon ∆t, consider random variable x(∆t).
• Convert x(∆t) to utility u(x(∆t)) = x(∆t)α,u(x(∆t)) = xα0 exp
(α
(µr + lµe − l2σ2
2
)∆t + αlσW (∆t)
)
• Find expectation value of u(x(∆t)),〈u(x(∆t))〉 = xα0 exp
(α∆t
(µr + lµe − l2σ2
2+ αl2σ2
2
))
Implies expected change in utility 〈∆u〉 = 〈u(x(∆t))〉 − u(x0).
• Set derivative to zero,d〈∆u〉
dl= 0
= xα0 α∆t(µe − lσ2 + αlσ2
)exp
(α∆t
(µr + lµe − l2σ2
2+ αl2σ2
2
)).
• Solve for l
luopt = µe(1−α)σ2 .
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
b) Time perspective
• Given x(t), find ergodic observable, i.e. f (x) such thatTime average︷ ︸︸ ︷
limT→∞
1
T
∫ T
0f (x(t))dt =
Ensemble average︷ ︸︸ ︷1
N
N∑i
f (xi (t)) = 〈f (x(t))〉 .
Solution is a growth rate, determined by dynamics,f (x) = 1
∆tlog(x(t + ∆t)/x(t)).
• Find time average (or expectation value)
f = µr + lµe − l2σ2
2.
• Set derivative to zero, solve for l
l topt = µeσ2 .
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Comments:
• luopt = µe(1−α)σ2 depends on utility function (α).
l topt = µeσ2 set by dynamics.
• Utility: 18th century (long before ergodicity debate).
• Modern interpretationUtility theory aims for ergodic observable, e.g. for GBM,rate of change in log utility.
• Different questions answeredUtility theory:luopt corresponds to greatest expected happiness.
Time perspective:l topt implies greatest growth rate.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
GBM parameters µr = 5.2% p.a., µe = 2.4% p.a., σ = 15.9% p.√a.
0 10 20 30 40 50 60 70 80 90 10010
−1
100
101
102
103
t
x(t)
Ergodicity economicsSquare root utility
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
GBM parameters µr = 5.2% p.a., µe = 2.4% p.a., σ = 15.9% p.√a.
0 10 20 30 40 50 60 70 80 90 10010
−3
10−2
10−1
100
101
102
103
t
x(t)
Ergodicity economics
u(x)=x3/4
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
GBM parameters µr = 5.2% p.a., µe = 2.4% p.a., σ = 15.9% p.√a.
0 10 20 30 40 50 60 70 80 90 10010
−10
10−8
10−6
10−4
10−2
100
102
104
t
x(t)
Ergodicity economics
u(x)=x8/10
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Cruel world doesn’t care about my risk preferences.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
How deep the rabbit hole goes
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
200 BC–1500 CE No challenge (Hypatia?).
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
200 BC–1500 CE No challenge (Hypatia?).
1473–1543 CE Copernicus’ model: perfect circles but heliocentric.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
200 BC–1500 CE No challenge (Hypatia?).
1473–1543 CE Copernicus’ model: perfect circles but heliocentric.
1546–1601 CE Tycho Brahe – geocentric but observed comet 1577.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
200 BC–1500 CE No challenge (Hypatia?).
1473–1543 CE Copernicus’ model: perfect circles but heliocentric.
1546–1601 CE Tycho Brahe – geocentric but observed comet 1577.
1571–1630 CE Kepler – Mars’ orbit elliptic! Heliocentric.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
200 BC–1500 CE No challenge (Hypatia?).
1473–1543 CE Copernicus’ model: perfect circles but heliocentric.
1546–1601 CE Tycho Brahe – geocentric but observed comet 1577.
1571–1630 CE Kepler – Mars’ orbit elliptic! Heliocentric.
1564–1642 CE Galileo – earthly motion in perfect shapes, parabolas!
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
200 BC–1500 CE No challenge (Hypatia?).
1473–1543 CE Copernicus’ model: perfect circles but heliocentric.
1546–1601 CE Tycho Brahe – geocentric but observed comet 1577.
1571–1630 CE Kepler – Mars’ orbit elliptic! Heliocentric.
1564–1642 CE Galileo – earthly motion in perfect shapes, parabolas!
1643–1727 CE Newton – laws on earth and in heaven identical.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
384 – 322 BC Aristotle’s cosmology.
310 – 230 BC Aristarchus model: heliocentric – dismissed.
??? – 178 BC Ptolemy’s model: geocentric, perfect circles.
200 BC–1500 CE No challenge (Hypatia?).
1473–1543 CE Copernicus’ model: perfect circles but heliocentric.
1546–1601 CE Tycho Brahe – geocentric but observed comet 1577.
1571–1630 CE Kepler – Mars’ orbit elliptic! Heliocentric.
1564–1642 CE Galileo – earthly motion in perfect shapes, parabolas!
1643–1727 CE Newton – laws on earth and in heaven identical.
Mid-17th century: Crisis! All or nothing time-bound?
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
1654 Probability theory
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
1654 Probability theoryProbability theory
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
1654 Probability theoryProbability theory
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Evolution of structure
Two entities follow GBM.
dx1 = x1(µdt + σdW1) and dx2 = x2(µdt + σdW2)
If entities pool resources, they will follow
dx12 = x12
[µdt + σ
(12dW1 + 1
2dW2
)]Cooperation conundrum
Lucky partner gives, unlucky partner receives.
Expectation value grows at µ, irrespective of cooperation.
Lucky partner: why cooperate?
But: cooperation exists (multicellularity, firms, states etc).
→ Expectation value model fails.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Usual story: very complicated.
Our story: non-ergodic system → compute expectation valueof ergodic observable under specified dynamics (time-averagegrowth rate).
1 No cooperation: d〈ln(x1)〉dt = d〈ln(x2)〉
dt = µ− σ2/2
2 Cooperation: d〈ln(x12)〉dt = µ− σ2/4
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 (Ito).
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Usual story: very complicated.
Our story: non-ergodic system → compute expectation valueof ergodic observable under specified dynamics (time-averagegrowth rate).
1 No cooperation: d〈ln(x1)〉dt = d〈ln(x2)〉
dt = µ− σ2/2
2 Cooperation: d〈ln(x12)〉dt = µ− σ2/4
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 (Ito).
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
0 2000 4000 6000 8000 10000$1
$1e+20
$1e+40
Time
Wea
lth
Individual 1Individual 2Cooperating unitExpectation value
• Good risk management = faster growth.
• Evolutionary advantage of structure (multicellularity,tribes, firms, nations) over no structure.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Problems we can address (solve)
• optimize leverage (for any dynamic)
• map utility theory → dynamics
• 300-year old St. Petersburg paradox
• dynamics of wealth distribution
• better economic measures than GDP
• price insurance contracts (derivatives)
• explain emergence of structure
• ...
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Conclusion
Correcting one deep conceptual flaw in economicsenables powerful quantitative theory.
Only time willtell
O. Peters
Ole Peters
Positioning
Innocuousgame
Leverageoptimization
How deep therabbit holegoes
Thank you.
