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LEARNING ABOUT PREDICTABILITY: THE EFFECTS OFPARAMETER UNCERTAINTY ON DYNAMIC ASSET

ALLOCATION

Yihong Xia, Journal of Finance 2001

Roine Vestman, NYU, September 11 2007

PORTFOLIO CHOICE IN THE PRESENCE OF DYNAMIC LEARNING ABOUTPREDICTABILITY

I There are evidence of return predictability and predictable variation in theequity premium in the finance literature (Campbell (1987), Campbell andShiller (1988,1989), among others)

I Consequently, people have investigated the effect of return predictabilityon portfolio choice (BSL: Brennan, Schwartz and Lagnado (1997), CV:Campbell and Viceira (1999), B: Barberis (2000)).

I This paper, Xia (2001), revises this effect by properly taking into accountthe fact that investors are not certain about the exact nature of returnpredictability, or even whether it exists.

I BSL (1997), CV (1999): Constant and known return predictabilityrelationship.

I B (2000): Unknown return predictability relationship

I This paper: Unknown return predictability relationship that the investortries to learn about over time (parameter uncertainty)

RESEARCH QUESTIONS

I Process for the excess return

µ(t) = α + β′S(t)where (α, β) are unknown to the investor

I What is the optimal allocation in the presence of parameter uncertaintyand learning?

I What is the role of the investment horizon in the presence of parameteruncertainty and learning?

I What is effect on market timing in the presence of parameter uncertaintyand learning?

RESEARCH QUESTIONS

MODEL (1)

I Price of the risk-free asset

dB = rBdt

I Stock price with dividend reinvested ([dz ]k×1)dPP = µ(t)dt + σPdz

I DGP of expected return (dividend reinvested)

I Generally, β may not be a constant

dβ = (α0(P,S , t) + α1(P,S , t)β)dt + η(P,S , t)dz

I The predictive variables S (dt/pt) follow

dS = (A0(P,S , t) + A1(P,S , t)β)dt + σS (P,S , t)dz

MODEL (1)

dB = rBdt

MODEL (1)

dB = rBdt

MODEL (1)

dB = rBdt

MODEL (1)

dB = rBdt

MODEL (1)

dB = rBdt

I Stock price with dividend reinvested ([dz ]k×1)dPP = µ(t)dt + σPdz (Measurement 1)

I The predictive variables S (d/pt) follow

dS = (A0(P,S , t) + A1(P,S , t)β)dt + σS (P,S , t)dz (Measurement 2)

MODEL (2)

I The investor’s problem

max{x(t),c(t)}

E{∫ T

0 U(c(t), t)dt + B(W (T ),T )|F I0

}s.t. dW = (rW + xW (µ + β(S − S)− r)− c)dt + xW σPdz

U(c(t), t) = e−ρt[c1−γt /(1− γ)

I Separation of the investor problem

1. Inference problem for β

I Priors β ∼ N(b0, ν0)I I (t) = (P(t),S(t))I E

[β|F I

]∼ N(bt , νt )

2. Optimization conditional on (bt , νt ) (two additional state variables)

MODEL (2)

I The investor’s problem

max{x(t),c(t)}

E{∫ T

0 U(c(t), t)dt + B(W (T ),T )|F I0

}s.t. dW = (rW + xW (µ + β(S − S)− r)− c)dt + xW σPdz

U(c(t), t) = e−ρt[c1−γt /(1− γ)

]I Separation of the investor problem

1. Inference problem for β

I Priors β ∼ N(b0, ν0)I I (t) = (P(t),S(t))I E

[β|F I

]∼ N(bt , νt )

2. Optimization conditional on (bt , νt ) (two additional state variables)

MODEL (3)

I More specifically, assume s is univariate and follows an O-U process

ds = κ(s − s)dt + σsdzs

I This gives the price processdPP = (µ + β(s − s))dt + σPdzP

I Let the true process for β follow an O-U process

dβ = λ(β− β)dt + σβdzβ

I Conditional on the investor’s filtrationdPP = (µ + β(s − s))dt + σPdzP

dW = (rW + xW (µ + b (s − s)− r)− c)dt + xW σPdzP (BC)

I We will have a KF state space representation for (b, ν), with long-runvariance of the posterior equal to ν

MODEL (3)

MODEL (4)SOME INTUITION

I The investor’s valuefunction

J(W , b, ν, s, t) = max{(c(τ):t≤τ≤T}

{∫ Tt e−ρt c1−γ

1−γ dτ + e−ρT W 1−γT

1−γ |F It

}s.t. (BC) and KF state space repr.

I The investor is trying to learn about a regression coefficient between stockreturn and s − s

I When s ≈ s the investor does not learn much about β

I If β is stochastic then the steady state value for ν, ν, has the followingproperties

I ν > 0

I ν depends on the path that s takes

I If β is a constant (λ = 0, σβ = 0) then

I The learning path still depends on s

I The investor eventually learns about the true value of β, that is ν = 0

J(W , b, ν, s, t) = max{(c(τ):t≤τ≤T}

1−γ |F It

I ν > 0

J(W , b, ν, s, t) = max{(c(τ):t≤τ≤T}

1−γ |F It

I ν > 0

J(W , b, ν, s, t) = max{(c(τ):t≤τ≤T}

1−γ |F It

I ν > 0

J(W , b, ν, s, t) = max{(c(τ):t≤τ≤T}

1−γ |F It

I ν > 0

OPTIMAL PORTFOLIO ALLOCATION

I Under isoelastic utility

J(W , b, ν, s, t) = e−ρt W 1−γ

1−γ φ(b, ν, s, t)

I The myopic term plus three hedging terms

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

σPσsρsP︸︷︷︸predictability

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

ν(s − s)︸︷︷︸parameter uncertainty

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

σPσbρβP︸︷︷︸β stochastic

ρsPdt = E (dzPdzs )

ρβPdt = E(dzPdzβ

J(W , b, ν, s, t) = e−ρt W 1−γ

1−γ φ(b, ν, s, t)

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

ρβPdt = E(dzPdzβ

J(W , b, ν, s, t) = e−ρt W 1−γ

1−γ φ(b, ν, s, t)

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

ρβPdt = E(dzPdzβ

J(W , b, ν, s, t) = e−ρt W 1−γ

1−γ φ(b, ν, s, t)

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

ρβPdt = E(dzPdzβ

J(W , b, ν, s, t) = e−ρt W 1−γ

1−γ φ(b, ν, s, t)

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

ρβPdt = E(dzPdzβ

OPTIMAL PORTFOLIO ALLOCATIONFOR MOST OF THE PAPER

J(W , b, ν, s, t) = e−ρt W 1−γ

1−γ φ(b, ν, s, t)

I The myopic term plus two hedging terms

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

I Direct effect of learning: parameter uncertainty

I Indirect effect of learning: φ depends upon a different set of state variablesat the presence of parameter uncertainty

I ∂ ln φ(b,s,ν,t)∂s

OPTIMAL PORTFOLIO ALLOCATIONFOR MOST OF THE PAPER

J(W , b, ν, s, t) = e−ρt W 1−γ

1−γ φ(b, ν, s, t)

I The myopic term plus two hedging terms

x∗ =µ + b(s − s)− r

γσ2P︸︷︷︸

myopic term

γσ2Pφ

I Direct effect of learning: parameter uncertainty

I Indirect effect of learning: φ depends upon a different set of state variablesat the presence of parameter uncertainty

I ∂ ln φ(b,s,ν,t)∂s

CALIBRATION

HORIZON EFFECT AS A FUNCTION OF STATE

HEDGING RELATED TO LEARNINGINTUITION

I Hedging component for parameter uncertainty:φb

γσ2P φ

ν(s − s)

I Positive if s < s

I Negative if s > s

I Strong horizon effect: zero at one month, 25 percent at 20 years(s = d

p = 2%)

I Suppose b0 > 0 and the investor observes s > s, then an expectedly highstock return means good news. The estimate b is revised upwards which isgood news. Positive correlation between good news and current stockreturn. As a hedge, the investor wants to sell something positivelycorrelated with news. Negative hedge demand.

I Converse for s < s

γσ2P φ

ν(s − s)

I Positive if s < s

I Negative if s > s

p = 2%)

γσ2P φ

ν(s − s)

I Positive if s < s

I Negative if s > s

p = 2%)

COMPARISON OF THREE INVESTMENT STRATEGIES

I KS: Myopic [Kandel and Stambaugh (1996)]

I BSL: Dynamic, but ignoring parameter uncertainty (ν0 = 0) [Brennan,Schwartz, Lagnado (1997)]

I O: Optimal strategy [this paper]

I Prior b0 = 0 ⇒ iid, unless the strategy is O

WITH A REALISTIC PRIOR THE ALLOCATION DEPENDS LESS ON S

LESS AGGRESSIVE MARKET TIMING (1)ALLOCATION CONDITIONAL ON D/P

LESS AGGRESSIVE MARKET TIMING (2)ALLOCATION OVER TIME

WELFARE COMPARISON

CONCLUDING REMARKS

I The intertemporal hedging component associated with parameteruncertainty is of the same magnitude as the one associated withpredictability

I The intertemporal hedging component associated with parameteruncertainty is particularly relevant at long horizons

I The optimal market timing is far less aggressive in the presence ofparameter uncertainty

l p : t effects of p u d apeople.stern.nyu.edu/svnieuwe/pdfs/phdpres2007/pres1_3.pdf · portfolio...

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