On Tracking Portfolios with Certainty Equivalents on a Generalization ...
On Tracking Portfolios with Certainty Equivalents on a Generalization of Markowitz Model: the Fool, the Wise and the Adaptive Richard Nock Brice Magdalou Eric Briys Frank Nielsen
CEREGMIA - UAG, Martinique CEREGMIA - UAG, Martinique CEREGMIA - UAG & Cyberlibris, France Sony Computer Science Labs, Inc., Japan
Comp. Sc. Economics Finance Comp. Sc.
ICML’ 2011
Outline Generalization of Markowitz’ Mean-Variance How do individuals make decision under risk: Expected Utility Model
On-line Learning in the Mean-Divergence Model
More results: http://www1.univ-ag.fr/~rnock/Articles/ICML11/ Nock & al., 1/20
Key ingredients Market: d assets
d-dimensional
Investor’s portfolio: α ∈ Pd
probability simplex
Returns: w ∈ [−1, +∞)d
ei,current = (1 + wi ) × ei,previous assume w ∼ pψ
parameter
.
Investor’s wealth: ωinv = w � α Nock & al., 2
Key problem How does the investor builds preferences over portfolios, i.e. how does he/she ranks α� over α,
�
α �α
Nock & al., 3
Simple approach
) k r o w t o n s e o (d
Decision making under Risk
Chavas, 2004
People’s will to invest proportional to expected reward ? Coin flipping game: toss a fair coin, win 2n € iff first head on n th toss Limit expected reward: Hence...
�
n≥1
n
n
2 (1/2 ) →n ∞
Nock & al., 4
Decision making under Risk
Chavas, 2004
People’s will to invest proportional to expected reward ? Coin flipping game: toss a fair coin, win 2n € iff first head on n th toss Limit expected reward:
�
n≥1
n
n
2 (1/2 ) →n ∞
Hence... people would be infinitely willing to participate...
r o t s e v n i : x o d a r a p g r u b s r e t e St. P ω [ inv ] p E ∼ ψ e w z i m i x a m t o n does e sc) I I I V X , i l l u o n r e B ( Nock & al., 5
Normative approach
Expected utility setting for ωinv Make five assumptions about the way the investor builds preferences among portfolios. For example ( ∀α, α� , α�� ∈ Pd ) A1: (α � α� ) ∨ (α ≺ α� ) ∨ (α ∼ α� )
(α � α ) ∧ (α � α ) ⇒ (α � α )
A2:
�
�
��
��
(α � α� ) ⇔ (∀β ∈ (0, 1), βα + (1 − β)α�� � βα� + (1 − β)α�� )
Order Transitivity Independence
Then, under assumptions A1-A5,
u inv )] ≤ Ew∼pψ [u(ω u inv� )], α � α� ⇔ Ew∼pψ [u(ω 4 4 9 1 , n r e t s n e g Mor for some utility function u & n n a m u e N von Nock & al., 6
Certainty equivalent & Risk premium Theorem: the expected utility, over all w , equals the utility of a single situation (e.g. Chavas, 2004)
Ew∼pψ [u(ωinv )] = u(Ew∼pψ [ωinv ] − p(α; θ)) � �� � c(α;θ)
Certainty equivalent
Numerous cases
Risk premium
“Single” case
Nock & al., 7
Expressions of u,p(α; θ),c(α; θ)?
Finding u... Arrow-Pratt coefficient of absolute risk aversion for stock i: 2
∂ ri (ωinv ) = − 2 u(ωinv ) ∂wi .
�
∂ u(ωinv ) ∂wi
�−1
le o r r o j a m a s y Rationale: pla ) θ ; α ( p g n i t a m in approxi
Lemma: assume ri (ωinv ) = a, ∀i = 1, 2, ..., d , for some a ∈ R ;
ute l o s b A t n a t s n Co ARA) C ( n o i s r e v A Risk
then u(x) =
�
x − exp(−ax)
iff a = 0 . otherwise
(hereafter, investor risk averse: a
> 0)
Nock & al., 8
Finding p(α; θ) and c(α; θ) ... Theorem: assume pψ = N(µ, Σ) ; then p(α; θ) =
a � ; 2 α Σα
odel m e c n a i r a V n a Me 52) 9 1 , z t i w o k r a (M
�
hence c(α; θ) = α µ −
a � ; 2 α Σα
investor cares for average return and variance of returns ... ... but assumption known not to hold in practice; approximation by mean-variance always valid in the neighborhood of the riskless case; otherwise, can be devastating... Chavas, 2004
Nock & al., 9
Generalization of Mean-Variance nce Mean-Diverge model
Theorem: assume pψ in exponential families
�
�
pψ (w : θ) = exp w� θ − ψ(θ) b(w) then p(α; θ) =
1 a Dψ
hence c(α; θ) =
1 a
(θ − aα�θ);
Bregman divergence with generator ψ (strictly convex diff.)
(ψ(θ) − ψ(θ − aα)) .
Nock & al., 10
Properties of the risk premium Convergence and monotonicity: lima→0 p(α; θ) = 0, limα→0 p(α; θ) = 0 ; p(α; θ) is strictly increasing in a. Toy upperbounds for particular cases: p(α; θ)
≤
� � � 1 � 1 dD � kl a 1−exp(−a) aλ 1 1 − λ� λ� +a
if if if
pψ = d-dim. multinomial pψ = Poisson(λ) pψ = exponential(λ� )
latter bounds proportional to (square root of) variance.
Nock & al., 11
Consequence of generalization
Duality allocations / returns Pops out from dual coordinates in exponential families
Allocations
α
∇ψ
θ
∇ψ �
Returns
Ew∼pψ [w]
w
θ =“natural market allocation”, optimal (information-theory) When pψ = N(µ, Σ), optimal allocation ∝ Σ−1 µ Markowitz, 1952 Nock & al., 12
Tracking portfolios
Setting & algorithm Shifting portfolios: NMA θt , our portfolio αt reference portfolio rt , that we wish to track.
and a
Algorithm: initialize: α0 = (1/d)1 , learning parameter η > 0 , strictly convex differentiable φ ; Nock & Nielsen, 2009 repeat for t = 0, 1, ..., T − 1:
αt+1 ←
−1 ∇φ (∇φ
Premium gradient
(αt ) − η∇p (αt ; θt ) − zt 1) Computed from wt
So that αt+1
∈ Pd
Nock & al., 13
Property Lowerbound on the certainty equivalent there exists some risk-aversion parameter a such that: T −1 �
cψ (αt ; θt )
t=0
� � T� −1 1 1 ≥ �rt+1 − rt �p cψ (rt ; θt ) − d q ln α t=0 t=0 � � 1 1−α − ln d ln +|T|υ − T ς − η α(1 − α) T −1 �
Nock & al., 14
Property Lowerbound on the certainty equivalent there exists some risk-aversion parameter a such that: T −1 �
cψ (αt ; θt )
t=0
≥
T −1 � t=0
cψ (rt ; θt ) − O(drift, sparsity of r. )
−O(T × max. scope of premium gradient)
Nock & al., 15
Experiments
Setting Four markets: DJIA, NYSE, TSE, S&P500, with daily or weekly returns, covering overall from 1962 to 2009. Tests with various values for a, η, ψ, φ + tests computing θt based on moving averages of returns (see paper + supplementary material)
Nock & al., 16
(some) Results for OMDkl,ψ
tur Cumulated re
nyse 100 80
returns
6
OMD (median) OMD (min) OMD (max) BEST UCRP
120
4
60 40
2
20
0
0
-2
-20
tse 35
OMD (median) OMD (min) OMD (max) BEST UCRP
14 12 10
returns
8
s&p500 25
8 6 4
5
0
300
400
500
OMD (median) OMD (min) OMD (max) BEST UCRP
14 12
140 120
0
100
200
300 T
400
500
600
0
60
OMD (median) OMD (min) OMD (max) BEST UCRP
10 8
6
returns
8
80
40
60 40
4
20
2
0
0
-20 0
100
200
300 T
400
500
0
1000 2000 3000 4000 5000 T
6 4
200 400 600 800 1000 1200 T OMD (median) OMD (min) OMD (max) BEST UCRP
50
100
returns
returns
1000 2000 3000 4000 5000 T OMD (median) OMD (min) OMD (max) BEST UCRP
160
10
is
0 0
T 16
Itakura-Saito
200
15 10
returns
m
100
20
2
-40 0
OMD (median) OMD (min) OMD (max) BEST UCRP
30
returns
OMD (median) OMD (min) OMD (max) BEST UCRP
10
returns
Markowitz
djia
ns
30 20
2
10
0
0 0
100
200
300 T
400
500
600
0
200 400 600 800 1000 1200 T
Nock & al., 17
(some) Results for OMDkl,ψ
s
emium r p d e t a l u m u C
djia Markowitz
100000
nyse 1e+06
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
10000 1000 100
100000 10000 1000
10000 1000 100
1000 100
10
1
1
1
0.1
0.1
0.1
0.01
0.01
0.01
0.01
0.001
0.001
0.001
0.001
0.0001
0.0001
0.1
0.0001 0
m
100
200
300
400
500
1e-05 0
T 1e+14
1e+16
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
1e+12 1e+10
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
1e+14 1e+12 1e+10
premiums
1e+08 1e+06 10000
1000 2000 3000 4000 5000 T
1e-05 0
100
1e+16
200
300 T
400
500
600
1e+14 1e+12 1e+10
1e+10 1e+08 1e+06 10000
10000
10000
100
100
100
100
1
1
1
1
0.01
0.01
0.01
0
100
200
300 T
400
500
0.01
0
1000 2000 3000 4000 5000 T
0.0001
200 400 600 800 1000 1200 T OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
1e+12
1e+06
1e+06
0
1e+14
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
1e+08
1e+08
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
10000
10
1
premiums
100000
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
10
0.0001
Itakura-Saito
100000
tse
100
10
is
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
s&p500
0
100
200
300 T
400
500
600
0.0001
0
200 400 600 800 1000 1200 T
Nock & al., 18
(some) Results for OMDkl,is (a = 100.0, η = 0.01) 0.9
0.35
INTEL CORP.
0.8
PHILIP MORRIS
djia::
0.3
None: 16.01% INTEL CORP.: 7.91% (�) AT&T CORP.: 7.31% HP: 6.32% JP MORGAN: 4.94% PHILIP MORRIS: 4.35% HONEYWELL: 4.35%
0.7 0.25
0.6
0.2
0.5 0.4
0.15
0.3 0.1
0.2 0.1
0.05
0
0 0
6
100
200
300
400
500
PURE GOLD MINERALS INC.
0
1.4
100
200
300
400
500
INTL FOREST PROD. LTD.
1.2
5
None:: 17.33% (�) PURE GOLD MIN.: 9.70% BREAKWATER RES.: 8.27% REPAP ENT. INC.: 5.72% GENTRA INC.: 3.50% COTT CORP.: 3.34% MIRAMAR MIN.: 3.18%
1 4 0.8 3
0.6
2
0.4
1
0.2 0
0 0
200
400
600
800 1000 1200
tse:
0
200
400
600
800 1000 1200
Nock & al., 19
Conclusion Interesting questions: Lift CARA to other models of Absolute Risk Aversion Can we efficiently learn / track the investor’s risk aversion parameter a ? Transaction costs to be included
Nock & al., 20
CEREGMIA - UAG, Martinique CEREGMIA - UAG, Martinique CEREGMIA - UAG & Cyberlibris, France Sony Computer Science Labs, Inc., Japan
Comp. Sc. Economics Finance Comp. Sc.
ICML’ 2011
Outline Generalization of Markowitz’ Mean-Variance How do individuals make decision under risk: Expected Utility Model
On-line Learning in the Mean-Divergence Model
More results: http://www1.univ-ag.fr/~rnock/Articles/ICML11/ Nock & al., 1/20
Key ingredients Market: d assets
d-dimensional
Investor’s portfolio: α ∈ Pd
probability simplex
Returns: w ∈ [−1, +∞)d
ei,current = (1 + wi ) × ei,previous assume w ∼ pψ
parameter
.
Investor’s wealth: ωinv = w � α Nock & al., 2
Key problem How does the investor builds preferences over portfolios, i.e. how does he/she ranks α� over α,
�
α �α
Nock & al., 3
Simple approach
) k r o w t o n s e o (d
Decision making under Risk
Chavas, 2004
People’s will to invest proportional to expected reward ? Coin flipping game: toss a fair coin, win 2n € iff first head on n th toss Limit expected reward: Hence...
�
n≥1
n
n
2 (1/2 ) →n ∞
Nock & al., 4
Decision making under Risk
Chavas, 2004
People’s will to invest proportional to expected reward ? Coin flipping game: toss a fair coin, win 2n € iff first head on n th toss Limit expected reward:
�
n≥1
n
n
2 (1/2 ) →n ∞
Hence... people would be infinitely willing to participate...
r o t s e v n i : x o d a r a p g r u b s r e t e St. P ω [ inv ] p E ∼ ψ e w z i m i x a m t o n does e sc) I I I V X , i l l u o n r e B ( Nock & al., 5
Normative approach
Expected utility setting for ωinv Make five assumptions about the way the investor builds preferences among portfolios. For example ( ∀α, α� , α�� ∈ Pd ) A1: (α � α� ) ∨ (α ≺ α� ) ∨ (α ∼ α� )
(α � α ) ∧ (α � α ) ⇒ (α � α )
A2:
�
�
��
��
(α � α� ) ⇔ (∀β ∈ (0, 1), βα + (1 − β)α�� � βα� + (1 − β)α�� )
Order Transitivity Independence
Then, under assumptions A1-A5,
u inv )] ≤ Ew∼pψ [u(ω u inv� )], α � α� ⇔ Ew∼pψ [u(ω 4 4 9 1 , n r e t s n e g Mor for some utility function u & n n a m u e N von Nock & al., 6
Certainty equivalent & Risk premium Theorem: the expected utility, over all w , equals the utility of a single situation (e.g. Chavas, 2004)
Ew∼pψ [u(ωinv )] = u(Ew∼pψ [ωinv ] − p(α; θ)) � �� � c(α;θ)
Certainty equivalent
Numerous cases
Risk premium
“Single” case
Nock & al., 7
Expressions of u,p(α; θ),c(α; θ)?
Finding u... Arrow-Pratt coefficient of absolute risk aversion for stock i: 2
∂ ri (ωinv ) = − 2 u(ωinv ) ∂wi .
�
∂ u(ωinv ) ∂wi
�−1
le o r r o j a m a s y Rationale: pla ) θ ; α ( p g n i t a m in approxi
Lemma: assume ri (ωinv ) = a, ∀i = 1, 2, ..., d , for some a ∈ R ;
ute l o s b A t n a t s n Co ARA) C ( n o i s r e v A Risk
then u(x) =
�
x − exp(−ax)
iff a = 0 . otherwise
(hereafter, investor risk averse: a
> 0)
Nock & al., 8
Finding p(α; θ) and c(α; θ) ... Theorem: assume pψ = N(µ, Σ) ; then p(α; θ) =
a � ; 2 α Σα
odel m e c n a i r a V n a Me 52) 9 1 , z t i w o k r a (M
�
hence c(α; θ) = α µ −
a � ; 2 α Σα
investor cares for average return and variance of returns ... ... but assumption known not to hold in practice; approximation by mean-variance always valid in the neighborhood of the riskless case; otherwise, can be devastating... Chavas, 2004
Nock & al., 9
Generalization of Mean-Variance nce Mean-Diverge model
Theorem: assume pψ in exponential families
�
�
pψ (w : θ) = exp w� θ − ψ(θ) b(w) then p(α; θ) =
1 a Dψ
hence c(α; θ) =
1 a
(θ − aα�θ);
Bregman divergence with generator ψ (strictly convex diff.)
(ψ(θ) − ψ(θ − aα)) .
Nock & al., 10
Properties of the risk premium Convergence and monotonicity: lima→0 p(α; θ) = 0, limα→0 p(α; θ) = 0 ; p(α; θ) is strictly increasing in a. Toy upperbounds for particular cases: p(α; θ)
≤
� � � 1 � 1 dD � kl a 1−exp(−a) aλ 1 1 − λ� λ� +a
if if if
pψ = d-dim. multinomial pψ = Poisson(λ) pψ = exponential(λ� )
latter bounds proportional to (square root of) variance.
Nock & al., 11
Consequence of generalization
Duality allocations / returns Pops out from dual coordinates in exponential families
Allocations
α
∇ψ
θ
∇ψ �
Returns
Ew∼pψ [w]
w
θ =“natural market allocation”, optimal (information-theory) When pψ = N(µ, Σ), optimal allocation ∝ Σ−1 µ Markowitz, 1952 Nock & al., 12
Tracking portfolios
Setting & algorithm Shifting portfolios: NMA θt , our portfolio αt reference portfolio rt , that we wish to track.
and a
Algorithm: initialize: α0 = (1/d)1 , learning parameter η > 0 , strictly convex differentiable φ ; Nock & Nielsen, 2009 repeat for t = 0, 1, ..., T − 1:
αt+1 ←
−1 ∇φ (∇φ
Premium gradient
(αt ) − η∇p (αt ; θt ) − zt 1) Computed from wt
So that αt+1
∈ Pd
Nock & al., 13
Property Lowerbound on the certainty equivalent there exists some risk-aversion parameter a such that: T −1 �
cψ (αt ; θt )
t=0
� � T� −1 1 1 ≥ �rt+1 − rt �p cψ (rt ; θt ) − d q ln α t=0 t=0 � � 1 1−α − ln d ln +|T|υ − T ς − η α(1 − α) T −1 �
Nock & al., 14
Property Lowerbound on the certainty equivalent there exists some risk-aversion parameter a such that: T −1 �
cψ (αt ; θt )
t=0
≥
T −1 � t=0
cψ (rt ; θt ) − O(drift, sparsity of r. )
−O(T × max. scope of premium gradient)
Nock & al., 15
Experiments
Setting Four markets: DJIA, NYSE, TSE, S&P500, with daily or weekly returns, covering overall from 1962 to 2009. Tests with various values for a, η, ψ, φ + tests computing θt based on moving averages of returns (see paper + supplementary material)
Nock & al., 16
(some) Results for OMDkl,ψ
tur Cumulated re
nyse 100 80
returns
6
OMD (median) OMD (min) OMD (max) BEST UCRP
120
4
60 40
2
20
0
0
-2
-20
tse 35
OMD (median) OMD (min) OMD (max) BEST UCRP
14 12 10
returns
8
s&p500 25
8 6 4
5
0
300
400
500
OMD (median) OMD (min) OMD (max) BEST UCRP
14 12
140 120
0
100
200
300 T
400
500
600
0
60
OMD (median) OMD (min) OMD (max) BEST UCRP
10 8
6
returns
8
80
40
60 40
4
20
2
0
0
-20 0
100
200
300 T
400
500
0
1000 2000 3000 4000 5000 T
6 4
200 400 600 800 1000 1200 T OMD (median) OMD (min) OMD (max) BEST UCRP
50
100
returns
returns
1000 2000 3000 4000 5000 T OMD (median) OMD (min) OMD (max) BEST UCRP
160
10
is
0 0
T 16
Itakura-Saito
200
15 10
returns
m
100
20
2
-40 0
OMD (median) OMD (min) OMD (max) BEST UCRP
30
returns
OMD (median) OMD (min) OMD (max) BEST UCRP
10
returns
Markowitz
djia
ns
30 20
2
10
0
0 0
100
200
300 T
400
500
600
0
200 400 600 800 1000 1200 T
Nock & al., 17
(some) Results for OMDkl,ψ
s
emium r p d e t a l u m u C
djia Markowitz
100000
nyse 1e+06
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
10000 1000 100
100000 10000 1000
10000 1000 100
1000 100
10
1
1
1
0.1
0.1
0.1
0.01
0.01
0.01
0.01
0.001
0.001
0.001
0.001
0.0001
0.0001
0.1
0.0001 0
m
100
200
300
400
500
1e-05 0
T 1e+14
1e+16
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
1e+12 1e+10
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
1e+14 1e+12 1e+10
premiums
1e+08 1e+06 10000
1000 2000 3000 4000 5000 T
1e-05 0
100
1e+16
200
300 T
400
500
600
1e+14 1e+12 1e+10
1e+10 1e+08 1e+06 10000
10000
10000
100
100
100
100
1
1
1
1
0.01
0.01
0.01
0
100
200
300 T
400
500
0.01
0
1000 2000 3000 4000 5000 T
0.0001
200 400 600 800 1000 1200 T OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
1e+12
1e+06
1e+06
0
1e+14
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
1e+08
1e+08
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
10000
10
1
premiums
100000
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
10
0.0001
Itakura-Saito
100000
tse
100
10
is
OMD (median) OMD (min) OMD (max) BEST (median) UCRP (median)
s&p500
0
100
200
300 T
400
500
600
0.0001
0
200 400 600 800 1000 1200 T
Nock & al., 18
(some) Results for OMDkl,is (a = 100.0, η = 0.01) 0.9
0.35
INTEL CORP.
0.8
PHILIP MORRIS
djia::
0.3
None: 16.01% INTEL CORP.: 7.91% (�) AT&T CORP.: 7.31% HP: 6.32% JP MORGAN: 4.94% PHILIP MORRIS: 4.35% HONEYWELL: 4.35%
0.7 0.25
0.6
0.2
0.5 0.4
0.15
0.3 0.1
0.2 0.1
0.05
0
0 0
6
100
200
300
400
500
PURE GOLD MINERALS INC.
0
1.4
100
200
300
400
500
INTL FOREST PROD. LTD.
1.2
5
None:: 17.33% (�) PURE GOLD MIN.: 9.70% BREAKWATER RES.: 8.27% REPAP ENT. INC.: 5.72% GENTRA INC.: 3.50% COTT CORP.: 3.34% MIRAMAR MIN.: 3.18%
1 4 0.8 3
0.6
2
0.4
1
0.2 0
0 0
200
400
600
800 1000 1200
tse:
0
200
400
600
800 1000 1200
Nock & al., 19
Conclusion Interesting questions: Lift CARA to other models of Absolute Risk Aversion Can we efficiently learn / track the investor’s risk aversion parameter a ? Transaction costs to be included
Nock & al., 20
