The “Checklist” > 9c. Dynamic allocation: time series strategies > Expected utility maximizationThe objective

The objective

We assume that the utility function satisfies• “rich is better than poor”, i.e.

dutility(y)

dy> 0 (9c.33)

• “richer is better when poor”, i.e.

d2utility(y)

dy< 0 (9c.34)

Exponential utility

utility(y) = −e−λy (9c.35)

Power utility

utility (y) = y1−λ (9c.36)

where 0 < λ < 1.

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Expected utility maximizationOptimization

Optimization

The optimal allocation policy reads

hrisky(·) ≡ argmaxh(·)∈C

(E{utility(V stratthor )}) (9c.37)

constraint

Approaches to solve this problem• dynamic programming• martingale methods

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

The “Checklist” > 9c. Dynamic allocation: time series strategies > Expected utility maximizationOptimization

Optimization

The optimal allocation policy reads

hrisky(·) ≡ argmaxh(·)∈C

(E{utility(V stratthor )}) (9c.37)

Arithmetic Brownian motionSuppose that

• V riskyt ← arithmetic Brownian motion (9c.5)

• rrf ≡ 0

• utility(y) = −e−λy

Then the optimal solution is the buy-and-hold policy

hrisky(·) ≡ µ

λσ2(9c.38)

• small λ ⇒ risk-seeking• large λ ⇒ risk-averse

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update

The “Checklist” > 9c. Dynamic allocation: time series strategies > Expected utility maximizationOptimization

Optimization

The optimal allocation policy reads

hrisky(·) ≡ argmaxh(·)∈C

(E{utility(V stratthor )}) (9c.37)

Geometric Brownian motionSuppose that

• V riskyt ← geometric Brownian motion (9c.8)

• rrf > 0

• utility (y) = y1−λ

Then the optimal solution is the constant weight policy

wrisky ≡ 1

λ

µ− rrf

σ2(9c.39)

• λ ≈ 1 ⇒ risk-seeking• λ� 1 ⇒ risk-averse

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The “Checklist” > 9c. Dynamic allocation: time series strategies > Expected utility maximizationOptimization

Geometric Brownian motion

V stratt ∼ LogN (ln vstrattnow + ( µ

strat − 12(σstrat)2)(t− tnow ), (σstrat)2 (t− tnow )) (9c.40)

≡ rrf + wrisky(µ− rrf ) ≡ wriskyσ

ARPM - Advanced Risk and Portfolio Management - arpm.co This update: Mar-28-2017 - Last update