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On the Optimality of Kelly Strategies


Mark Davis1 and Sebastien Lleo2

Workshop on Stochastic Models and ControlBad Herrenalb, April 1, 2011

1Department of Mathematics, Imperial College London, London SW7 2AZ,England, Email: [email protected]

2Finance Department, Reims Management School, 59 rue Pierre Taittinger,51100 Reims, France, Email: [email protected]


Outline

Outline

Where to invest?

Kelly Strategies

Back to Basics: Kelly Strategies in the Merton Model

Insights from the Merton Model: Change of Measure and Duality

Further Insights from the Merton Model...

Getting Kelly Strategies to Work for Factor Models

Beyond Factor Models: Random Coefficients

Conclusion and Next Steps


Where to invest?

Where to invest?

The central question for investment management practitioners is:where should I put my money?

Ideas such as the utility of wealth, and even expected portfolioreturns (and risks) take a backseat.

Yet, in finance theory, the optimal asset allocation appears as lessimportant than the utility of wealth. This is particularly true in theduality/martingale approach, where the optimal asset allocation isoften obtained last.


Kelly Strategies

Kelly Strategies

As a result of the differences between investment managementtheories and practice, a number of techniques and tools have beencreated to speed up the derivation of an optimal or near-optimalasset allocation.

Fractional Kelly strategies are one such technique.


Kelly Strategies

The Kelly (criterion) portfolio invest in the growth maximizingportfolio.

I In continuous time, this represents an investment in theoptimal portfolio under the logarithmic utility function.

I The Kelly criterion comes from the signalprocessing/gambling literature (see Kelly [7]).

I See MacLean, Thorpe and Ziemba [11] for a thorough view ofthe Kelly criterion.

A number of “great” investors from Keynes to Buffet can beviewed as Kelly investors. Others, such as Bill Gross probably are.

The Kelly criterion portfolio has a number of interesting propertiesbut it is inherently risky.


Kelly Strategies

To reduce the risk while keeping some of the nice properties,MacLean and Ziemba proposed the concept of fractional Kellyinvestment:

I Invest a fraction k of the wealth in the Kelly portfolio;

I invest a fraction 1− k in the risk-free asset.

Fractional Kelly strategies are

I optimal in the Merton world of lognormal asset prices andwith a globally risk-free asset;

I not optimal when the lognormality assumption is removed.


Kelly Strategies

Three questions:

I Why are fractional Kelly strategies not optimal outside of theMerton model?

I How is an optimal strategy constructed?

I Can we improve the definition of fractional Kelly strategies toguarantee optimality?


Kelly Strategies

In this talk, we will use the term Kelly strategies to refer to:

I The Kelly portfolio.

I Fractional Kelly investment.



Back to Basics: Kelly Strategies in the Merton ModelKey ingredients:

I Probability space (Ω, Ft ,F ,P);

I Rm-valued (Ft)-Brownian motion W (t);

I Price at time t of the ith security is Si (t), i = 0, . . . ,m;

The dynamics of the money market account and of the m riskysecurities are respectively given by:

dS0(t)

S0(t)= rdt, S0(0) = s0 (1)

dSi (t)

Si (t)= µidt +

N∑k=1

σikdWk(t), Si (0) = si , i = 1, . . . ,m

(2)where r , µ ∈ Rm and Σ := [σij ] is a m ×m matrix.



Assumption

The matrix Σ is positive definite.

Let Gt := σ(S(s), 0 ≤ s ≤ t) be the sigma-field generated by thesecurity process up to time t.

An investment strategy or control process is an Rm-valued processwith the interpretation that hi (t) is the fraction of current portfoliovalue invested in the ith asset, i = 1, . . . ,m.

I Fraction invested in the money market account ish0(t) = 1−

∑mi=1 hi (t).;



DefinitionAn Rm-valued control process h(t) is in class A(T ) if the followingconditions are satisfied:

1. h(t) is progressively measurable with respect toB([0, t])⊗ Gtt≥0 and is cadlag;

2. P(∫ T

0 |h(s)|2 ds < +∞)

= 1, ∀T > 0;

3. the Doleans exponential χht , given by

χht := exp

γ

∫ t

0h(s)′ΣdWs −

1

2γ2

∫ t

0h(s)′ΣΣ′h(s)ds

(3)

is an exponential martingale, i.e. E[χhT

]= 1

We say that a control process h(t) is admissible if h(t) ∈ A(T ).



Taking the budget equation into consideration, the wealth, V (t),of the asset in response to an investment strategy h ∈ H followsthe dynamics

dV (t)

V (t)= rdt + h′(t) (µ− r1) dt + h′(t)ΣdWt

(4)

with initial endowment V (0) = v and where 1 ∈ Rm is them-element unit column vector.



The objective of an investor with a fixed time horizon T , noconsumption and a power utility function is to maximize theexpected utility of terminal wealth:

J(t, h; T , γ) = E [U(VT )] = E

[V γT

γ

]= E

[eγ lnVT

γ

]with risk aversion coefficient γ ∈ (−∞, 0) ∪ (0, 1).



Define the value function Φ corresponding to the maximization ofthe auxiliary criterion function J(t, x , h; θ; T ; v) as

Φ(t) = suph∈A

J(t, h; T , γ) (5)

By Ito’s lemma,

eγ lnV (t) = vγ exp

γ

∫ t

0g(h(s); γ)ds

χht (6)

where

g(h; γ) = −1

2(1− γ) h′ΣΣ′h + h′(µ− r1) + r

and the Doleans exponential χht is defined in (3)



We can solve the stochastic control problem associated with (5) bya change of measure argument (see exercise 8.18 in [8]).

Define a measure Ph on (Ω,FT ) via the Radon-Nikodym derivative

dPh

dP:= χh

T (7)

For h ∈ A(T ),

W ht = Wt − γ

∫ t

0Σ′h(s)ds

is a standard Ph-Brownian motion.

The control criterion under this new measure is

I (t, h; T , γ) =vγ

γEht

[exp

γ

∫ T

tg(h(s); θ)ds

](8)

where Eht [·] denotes the expectation taken with respect to the

measure Ph at an initial time t.



Under the measure Ph, the control problem can be solved througha pointwise maximisation of the auxiliary criterion functionI (v , x ; h; t,T ).

The optimal control h∗ is simply the maximizer of the functiong(x ; h; t,T ) given by

h∗ =1

1− γ(ΣΣ′)−1 (µ− r1)

which represents a position of 11−γ in the Kelly criterion portfolio.



The value function Φ(t), or optimal utility of wealth, equals

Φ(t) =vγ

γexp

γ

[r +

1

2(1− γ)(µ− r1) (ΣΣ′)−1 (µ− r1)

](T − t)

Substituting (9) into (3), we obtain an exact form for the Doleansexponential χ∗t associated with the control h∗:

χ∗t := exp

γ

1− γ(µ− r1)′Σ−1W (t)

−1

2

(γ

1− γ

)2

(µ− r1)′ (ΣΣ′)−1 (µ− r1) t

(9)

We can easily check that χ∗t is indeed an exponential martingale.Therefore h∗ is an admissible control.



In the Merton model, fractional Kelly strategies appear naturally asa result of a classical Fund Separation Theorem:

Theorem (Fund Separation Theorem)

Any portfolio can be expressed as a linear combination ofinvestments in the Kelly (log-utility) portfolio

hK (t) = (ΣΣ′)−1 (µ− r) (10)

and the risk-free rate. Moreover, if an investor has a risk aversionγ, then the proportion of the Kelly portfolio will equal 1

1−γ .


Insights from the Merton Model: Change of Measure and Duality

Insights from the Merton Model: Change of Measure andDuality

The measure Ph defined in (7) is also used in themartingale/duality approach to dynamic portfolio selection (see forexample [14] and references therein).

In complete market, the change of measure technique is equivalentto the martingale approach: the change of measure approach relieson the optimal asset allocation to identify the equivalent martingalemeasure, the martingale approach works in the opposite direction.




The level of risk aversion γ dictates the choice of measure Ph.Two special cases are worth mentioning.

Case 1: The Physical Measure

The measures Ph is the physical measure P in the limit as γ → 0,that is in the log utility or Kelly criterion case. This observationforms the basis for the ‘benchmark approach to finance’ (see [13]).



Case 2: The Dangers of Overbetting

A well established and somewhat surprising folk theorem holds that“when an agent overbets by investing in twice the Kelly portfolio,his/her expected return will be equal to the risk-free rate.” (see forinstance [15] and [11]).

Why is that???



The risk aversion of an agent who invest into twice the Kellyportfolio must be γ = 1

2 .

I Hence, the measures Ph coincides with the equivalentmartingale measure Q!

I Under this measure, the portfolio value discounted at therisk-free rate is a martingale.



In the setting of the Merton model and its lognormally distributedasset prices, the definition of Fractional Kelly allocationsguarantees optimality of the strategy.

However, Fractional Kelly strategies are no longer optimal as soonas the lognormality assumption is removed (see Thorpe in [11]).

This situation suggests that the definition of Fractional Kellystrategies could be broadened in order to guarantee optimality. Wecan take a first step in this direction by revisiting the ICAPM (seeMerton[12]) in which the drift rate of the asset prices depend on anumber of Normally-distributed factors.




Key ingredients:

I Probability space (Ω, Ft ,F ,P);

I RN -valued (Ft)-Brownian motion W (t), N := n + m;

I Si (t) denotes the price at time t of the ith security, withi = 0, . . . ,m;

I Xj(t) denotes the level at time t of the jth factor, withj = 1, . . . , n.

For the time being, we assume that the factors are observable.



The dynamics of the money market account is given by:

dS0(t)

S0(t)=(a0 + A′0X (t)

)dt, S0(0) = s0 (11)

The dynamics of the m risky securities and n factors are

dS(t) = D (S(t)) (a + AX (t))dt + D (S(t)) ΣdW (t),

Si (0) = si , i = 1, . . . ,m (12)

and

dX (t) = (b + BX (t))dt + ΛdW (t), X (0) = x (13)

where X (t) is the Rn-valued factor process with components Xj(t)and the market parameters a, A, b, B, Σ := [σij ], Λ := [Λij ] arevectors and matrices of appropriate dimensions.



Assumption

The matrices ΣΣ′ and ΛΛ′ are positive definite.

I The objective of an investor remains the maximization ofcriterion (15) at a fixed time horizon T ;

I The wealth V (t) of the portfolio in response to an investmentstrategy h ∈ A(T ) is now factor-dependent with dynamics:

dV (t)

V (t)=(a0 + A′0X (t)

)dt + h′(t)(a + AX (t))dt + h′(t)ΣdWt

(14)

with a := a− a01, A := A− 1A′0, and initial endowment V (0) = v .



The expected utility of terminal wealth J(t, x , h; T , γ) si factordependent:

J(t, x , h; T , γ) = E [U(VT )] = E

[V γT

γ

]= E

[eγ lnVT

γ

]By Ito’s lemma,

eγ lnV (t) = vγ exp

γ

∫ t

0g(Xs , h(s); θ)ds

χht (15)

where

g(x , h; γ) = −1

2(1− γ) h′ΣΣ′h + h′(a + Ax) + a0 + A′0x(16)

and the exponential martingale χht is still given by (3).



Applying the change of measure argument, we obtain the controlcriterion under the measure Ph

I (t, x , h; T , γ) =vγ

γEht,x

[exp

γ

∫ T

tg(Xs , h(s); θ)ds

](17)

where Eht,x [·] denotes the expectation taken with respect to the

measure Ph and with initial conditions (t, x).

The Ph-dynamics of the state variable X (t) is

dX (t) =(b + BX (t) + γΛΣ′h(t)

)dt + ΛdW h

t , t ∈ [0,T ]

(18)



Then value function Φ associated with the auxiliary criterionfunction I (t, x ; h; T , γ) is defined as

Φ(t, x) = suph∈A(T )

I (t, x ; h; T , γ) (19)

We use the Feynman-Kac formula to write down the HJB PDE:

∂Φ

∂t(t, x) +

1

2tr(ΛΛ′D2Φ(t, x)

)+ H(t, x ,Φ,DΦ) = 0 (20)

subject to terminal condition

Φ(T , x) =vγ

γ(21)

and where

H(s, x , r , p) = suph∈R

(b + Bx + γΛΣ′h

)′p − γg(x , h; γ)r

(22)

for r ∈ R and p ∈ Rn.



To obtain the optimal control in a more convenient format andderive the value function Φ(t, x) more easily, we find it convenientto consider the logarithmically transformed value functionΦ(t, x) := 1

γ ln γΦ(t, x) with associated HJB PDE

∂Φ

∂t(t, x) + inf

h∈RmLht (t, x ,DΦ,D2Φ) = 0 (23)

where

Lht (s, x , p,M) =

(b + Bx + γΛΣ′h(s)

)′p +

1

2tr(ΛΛ′M

)+γ

2p′ΛΛ′p − g(x , h; γ) (24)

for r ∈ R and p ∈ Rn and subject to terminal condition

Φ(T , x) = ln v (25)

This is in fact a risk-sensitive asset management problem(see [1], [9], [6]).



Solving the optimization problem in (24) gives the optimalinvestment policy h∗(t)

h∗(t) =1

1− γ(ΣΣ′

)−1[a + AX (t) + γΣΛ′DΦ(t,X (t))

](26)

The solution to HJB PDE (23) is

Φ(t, x) =1

2x ′Q(t)x + x ′q(t) + k(t)

where Q(t) satisfies a matrix Riccati equation, q(t) satisfies avector linear ODE and k(t) is an integral. As a result,

h∗(t) =1

1− γ(ΣΣ′

)−1[a + AX (t) + γΣΛ′ (Q(t)X (t) + q(t))

](27)



Substituting (26) into (3), we obtain an exact form for the Doleansexponential χ∗t associated with the control h∗:

χ∗t := exp

γ

1− γ

∫ t

0

[a + AX (s) + γΣΛ′ (Q(s)X (s) + q(s))

]′×(ΣΣ′)−1ΣdW (s)

−1

2

(γ

1− γ

)2 ∫ t

0

(a + AX (s) + γΣΛ′ (Q(s)X (s) + q(s))

)′×(ΣΣ′)−1

(a + AX (s) + γΣΛ′ (Q(s)X (s) + q(s))

)′ds

(28)



We can interpret the Girsanov kernel

γ

1− γΣ′(ΣΣ′)−1

[a + AX (s) + γΣΛ′ (Q(s)X (s) + q(s))

]as the projection of the factor-dependent returns

a + AX (s) + γΣΛ′ (Q(s)X (s) + q(s))

on the subspace spanned by the asset volatilities Σ, scaled by γ1−γ .

This observation also implies that the appropriate Girsanov kernelin a complete (factor-free) setting is

γ

1− γΣ′(ΣΣ′)−1 (µ− r1)

with a similar interpretation.



In the ICAPM, a new view of Fractional Kelly investing emerges:

Theorem (ICAPM Fund Separation Theorem)

Any portfolio can be expressed as a linear combination ofinvestments into two funds’ with respective risky asset allocations:

hK (t) = (ΣΣ′)−1(

a + AX (t))

hI (t) = (ΣΣ′)−1ΣΛ′ (Q(t)X (t) + q(t)) (29)

and respective allocation to the money market account given by

hK0 (t) = 1− 1′(ΣΣ′)−1

(a + AX (t)

)hI

0(t) = 1− 1′(ΣΣ′)−1ΣΛ′ (Q(t)X (t) + q(t))

Moreover, if an investor has a risk aversion γ, then the respectiveweights of each mutual fund in the investor’s portfolio equal 1

1−γand γ

1−γ , respectively.



In the factor-based ICAPM,(ΣΣ′

)−1[a + AX (t)

](30)

represents the Kelly (log utility) portfolio and(ΣΣ′

)−1ΣΛ′ (Q(t)X (t) + q(t)) (31)

is the ‘intertemporal hedging porfolio’ identified by Merton.

This mutual fund theorem raises some questions as to thepracticality of the intertemporal hedging portfolio as an investmentoption.




The change of measure approach still works well in a partialobservation case where we would need to estimate the coefficientsof the factor process through a Kalman filter.

However, this presupposes that we know the form of the factorprocess.

A more general approach would be to assume that the coefficientsof the asset price dynamics are random (See Bjørk, Davis andLanden [14] and references therein).



Key ingredients:I The “full” underlying probability space (Ω, Ft ,F ,P);I The filtration FW

t := σ(W (s)), 0 ≤ s ≤ t) generated by anm-dimensional Brownian motions driving the asset returns(augmented by the P-null sets);

I The filtration FSt := σ(S(s)), 0 ≤ s ≤ t) generated by the m

asset price (also augmented by the P-null sets);

The dynamics of the money market account is given by:

dS0(t)

S0(t)= r(t)dt, S0(0) = s0 (32)

where r ∈ R+ is a bounded FSt -adapted process. We will denote

by Z (t,T ) the discount factor:

Z (t,T ) = exp

−∫ T

tr(s)ds

(33)



The dynamics of the m risky securities and n factors can beexpressed as:

dSi (t)

Si (t)= µi (t)dt+

m∑k=1

σik(t)dWk(t), Si (0) = si , i = 1, . . . ,m

(34)where µ(t) = (µ1(t), . . . , µm(t))′ is an Ft-adapted process and thevolatility Σ(t) := [σij(t)] , i = 1, . . . ,m, j = 1, . . . ,m is anFSt -adapted process. More synthetically,

dS(t) = D(S(t))µ(t)dt + D(S(t))Σ(t)dW (t) (35)

Note that no Markovian structure is either assumed or required.

Assumption

We assume that the matrix Σ(t) is positive definite ∀t.



The critical step in this approach is to establish a projection ontothe observable filtration.

Once this is done, the partial observation problem related to thefiltration F can be rewritten as an equivalent complete observationproblem with respect to the filtration FS .

This effectively changes a utility maximizaton problem set in anincomplete market into a standard utility maximization problem setin a complete market.



Define the process Y (t) by:

dY (t) = Σ−1t D(St)

−1dSt = Σ−1t µtdt + ΣtdW (t) (36)

For any F-adapted process X , define the filter estimate process Xas the optional projection of X onto the filtration FS :

Yt = EP[Yt |FS

t

]



Next, define the innovation process U by

dU(t) = dZ (t)− (Σ−1µ(t))dt = dZ (t)− Σ−1µ(t)dt (37)

where the second equality follows from the observability of Σt .

Note that the innovation process U(t) is a standard FS -Brownianmotion (see for example Lipster and Shiryaev [10]).

As a result, we can rewrite (35) using the filter estimate for α andthe innovation process U obtained with respect to the filtrationFS :

dS(t) = D(S(t))µ(t)dt + D(S(t))Σ(t)dU(t) (38)

The remainder follows from a standard martingale argument.



DefinitionThe Girsanov kernel is the vector process ϕt given by

ϕ(t) = Σ−1(t)(µ(t)− r1) (39)

for all t.

Note that he Girsanov kernel ϕ(t) and the market price of riskvector λ(t) are related by λ(t) = −ϕ(t).



Assumption

We assume that the Girsanov kernel satisfies the Novikov condition

E[e

12

∫ T0 ‖ϕ(t)‖2dt

]<∞ (40)

and the integrability condition

E

[e

12

∫ T0

(γ

1−γ

)2‖ϕ(t)‖2dt

]<∞ (41)



DefinitionThe equivalent martingale measure Q is defined by theRadon-Nikodym derivative

dQdP

= Lt := exp

−∫ t

0ϕ′(s)dU(s)− 1

2

∫ t

0ϕ′(s)ϕ(s)ds

,

on FSt .

I It follows from Assumption 4 that Lt is a true martingale.

I Moreover, Q is unique: this is a direct consequence of the useof filtered estimates in (38).



The objective is to maximize the expected utility of terminalwealth:

J(t, h; T , γ) = EP[

V γT

γ

]subject to the budget constraint

EQ [K0,TVT ] = v = EP [K0,TLTVT ] (42)

Next, form the Lagrangian

L(h, λ; T ) =1

γEP [V γ

T

]− λ

(EP [K0,TLTVT ]− v

)=

∫Ω

1

γV γT − λ (K0,T (ω)LT (ω)VT (ω)− v) dP(ω)



This separable problem can be maximized for each ω. The firstorder condition leads to

V ∗T = [λK0,TLT ]−1

1−γ

Substituting in the budget equation (42), we get

λ−1

1−γEP[(K0,TLT )

−γ1−γ

]= v

and as a result,

V ∗T =v

H0K

−11−γ

0,T L−1

1−γ

T (43)

with

H0 = EP[(K0,TLT )

−γ1−γ

]



Therefore, the optimal expected utility is equal to:

U0 = EP[

(V ∗T )γ

γ

]= H1−γ

0

v

γ(44)

Observe that H0 can be expressed as

H0 = EP[

L0T exp

∫ T

0

(rt +

1

2(1− γ)‖ϕ(t)‖2

)dt

]where

L0t := exp

γ

1− γ

∫ t

0ϕ′(s)dU(s)− 1

2

(γ

1− γ

)2 ∫ t

0ϕ′(s)ϕ(s)ds

(45)

is a true martingale. This leads us to the following definition:



Definition

(i). The measure Q0 is defined by the Radon-Nikodym derivative

dQ0

dP= L0

t , on FSt

(ii). The process H is defined by

Ht = E0

[exp

γ

1− γ

∫ T

t

(rs +

1

2(1− γ)‖ϕ(s)‖2

)ds

∣∣∣∣Ft

](46)

where the expectation E0 [·] is taken with respect to the Q0

measure.



Finally, we recall Proposition 4.1 in Bjørk, Davis and Landen [14]:

Proposition

The following hold:

(i). The optimal wealth process V ∗(t) is given by

V ∗(t) =H(t)

H(0)

(K0,t Lt

) −11−γ x

(ii). The optimal weight vector h∗ is given by

h∗(t) =1

1− γ(ΣtΣ

′t)−1 (µt − r1) + σH(t)

(Σ−1t

)′(47)

where σH is the volatility term of H, i.e. H has dynamics ofthe form

H(t) = µH(t)dt + σH(t)dU(t)



In the limit as γ → 0, the optimal wealth process V ∗(t) is given by

V ∗(t) =(K0,t Lt

) −11−γ x

and the optimal investment is the Kelly portfolio,

h∗(t) = (ΣtΣ′t)−1 (µt − r1) (48)



With the choice γ = 12 , and

h∗(t) = 2(ΣtΣ′t)−1 (µt − r1) + σH(t)

(Σ−1t

)′(49)

As a result,

V ∗(t) =H(t)

H(0)

(K0,t Lt

)2x = x exp

∫ T

tr(s)ds

as expected.



However, there is still something unsatisfactory about this result:the optimal strategy depends on the volatility of the process H(t):

Ht = E0

[exp

γ

1− γ

∫ T

t

(rs +

1

2(1− γ)‖ϕ(s)‖2

)ds

∣∣∣∣Ft

]As a result, getting some intuition on the intertemporal hedgingterm is ‘difficult’.




I Get stronger statements of equivalence between themartingale/duality approach and the change of measuretechnique in incomplete markets;

I Study the practicality of intertemporal hedging portfolio as aninvestment option;

I Add jumps.



Thank you!

3

Any question?3Copyright: W. Krawcewicz, University of Alberta



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