Investment Performance Measurement, Risk Tolerance and Optimal Portfolio Choice Marek Musiela, BNP...

Investment Performance Measurement, Risk Tolerance and Optimal Portfolio Choice

Marek Musiela, BNP Paribas, London

RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE

2

Joint work with T. Zariphopoulou (UT Austin)

Investments and forward utilities, Preprint 2006

Backward and forward dynamic utilities and their associated pricing systems: Case study of

the binomial model, Indifference pricing, PUP (2005)

Investment and valuation under backward and forward dynamic utilities in a stochastic factor

model, to appear in Dilip Madan’s Festschrift (2006)

Investment performance measurement, risk tolerance and optimal portfolio choice, Preprint

2007


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Contents

Section 1 Investment banking and martingale theory

Section 2 Investment banking and utility theory

Section 3 The classical formulation

Section 4 Remarks

Section 5 Dynamic utility

Section 6 Example – value function

Section 7 Weaknesses of such specification

Section 8 Alternative approach

Section 9 Optimal portfolio

Section 10 Portfolio dynamics


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Investment banking and martingale theory

Mathematical logic of the derivative business perfectly in line with the theory

Pricing by replication comes down to calculation of an expectation with respect to a martingale

measure

Issues of the measure choice and model specification and implementation dealt with by the

appropriate reserves policy

However, the modern investment banking is not about hedging (the essence of pricing by

replication)

Indeed, it is much more about return on capital - the business of hedging offers the lowest

return


5

Investment banking and utility theory

No clear idea how to specify the utility function

The classical or recursive utility is defined in isolation to the investment opportunities given to

an agent

Explicit solutions to the optimal investment problems can only be derived under very restrictive

model and utility assumptions - dependence on the Markovian assumption and HJB equations

The general non Markovian models concentrate on the mathematical questions of existence of

optimal allocations and on the dual representation of utility

No easy way to develop practical intuition for the asset allocation

Creates potential intertemporal inconsistency


6

The classical formulation

Choose a utility function, say U(x), for a fixed investment horizon T

Specify the investment universe, i.e., the dynamics of assets which can be traded

Solve for a self financing strategy which maximizes the expected utility of terminal wealth

Shortcomings

The investor may want to use intertemporal diversification, i.e., implement short, medium

and long term strategies

Can the same utility be used for all time horizons?

No – in fact the investor gets more value (in terms of the value function) from a longer term

investment

At the optimum the investor should become indifferent to the investment horizon


7

Remarks

In the classical formulation the utility refers to the utility for the last rebalancing period

There is a need to define utility (or the investment performance criteria) for any intermediate

rebalancing period

This needs to be done in a way which maintains the intertemporal consistency

For this at the optimum the investor must be indifferent to the investment horizon

Only at the optimum the investor achieves on the average his performance objectives

Sub optimally he experiences decreasing future expected performance


8

Dynamic performance process

U(x,t) is an adapted process

As a function of x, U is increasing and concave

For each self-financing strategy the associated (discounted) wealth satisfies

There exists a self-financing strategy for which the associated (discounted) wealth satisfies

tssXUFtXUE sstP 0,,

tssXUFtXUE sstP 0,,**


9

Example - value function

Value function

Dynamic programming principle

Value function defines dynamic a performance process

TtxXFTXuEtxV ttTP 0,,sup,

TtsFtXVEsXV stPs 0,,**

TtRxtxVtxU 0,,,


10

Weaknesses of such specification

Dynamic performance process U(x,t) is defined by specifying the utility function u(x,T) and

then calculating the value function

At time 0, U(x,0) may be very complicated and quite unintuitive

It depends strongly on the specification of the market dynamics

The analysis of such processes requires Markovian assumption for the asset dynamics and

the use of HJB equations

Only very specific cases, like exponential, can be analysed in a model independent way


11

Alternative approach – an example

Start by defining the utility function at time 0, i.e., set U(x,0)=u(x,0)

Define an adapted process U(x,t) by combining the variational and the market related inputs to

satisfy the properties of a dynamic performance process

Benefits

The function u(x,0) represents the utility for today and not for, say, ten years ahead

The variational inputs are the same for the general classes of market dynamics – no

Markovian assumption required

The market inputs have direct intuitive interpretation

The family of such processes is sufficiently rich to allow for analysis optimal allocations in

ways which are model and preference choice independent


12

Differential inputs

Utility equation

Risk tolerance equation

2

2

1xxxt uuu

txu

txutxrrrr

xx

xxxt ,

,),(,0

2

1 2


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Market inputs

Investment universe of 1 riskless and k risky securities

General Ito type dynamics for the risky securities

Standard d-dimensional Brownian motion driving the dynamics of the traded assets

Traded assets dynamics

dtBrdB

kidWdtSdS

ttt

tit

it

it

it

,...,1,


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Market inputs

Using matrix and vector notation assume existence of the market price for risk process which

satisfies

Benchmark process

Views (constraints) process

Time rescaling process

0, 0

2 AdtdA tttttt

tTttt r 1

ttttttttt YdWdtYdY ,1, 0

1, 0 ZdWZdZ tttt


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Alternative approach – an example

Under the above assumptions the process U(x,t), defined below is a dynamic performance

It turns out that for a given self-financing strategy generating wealth X one can write

ttt

ZAY

xutxU

,,

tt

tt

t

xx

tt

xxtt

AY

XrR

dtRRY

X

YZu

dWUY

XZu

Y

ZutXdUdU

,

1

2

1

,

2


16

Optimal portfolio

The optimal portfolio is given by

Observe that

The optimal wealth, the associated risk tolerance and the optimal allocations are benchmarked

The optimal portfolio incorporates the investor views or constraints on top of the market

equilibrium

The optimal portfolio depends on the investor risk tolerance at time 0.

xrxrrrr

AY

XrR

RRY

X

Y

xxt

tt

tt

tttttt

ttt

t

02

**

***

*

0,,02

1

,

1


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Wealth and risk tolerance dynamics

The dynamics of the (benchmarked) optimal wealth and risk tolerance are given by

Observe that zero risk tolerance translates to following the benchmark and generating pure

beta exposure.

In what follows we assume that the function r(x,t) is strictly positive for all x and t

t

tt

t

txt

tttttttttt

t

Y

XdA

Y

XrdR

dWdtRY

Xd

***

**

,


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Beta and alpha

For an arbitrary risk tolerance the investor will generate pure beta by formulating the

appropriate views on top of market equilibrium, indeed,

To generate some alpha on top of the beta the investor needs to tolerate some risk but may

also formulate views on top of market equilibrium

0,00 **

tt

tttttt dR

Y

Xd


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No benchmark and no views

The optimal allocations, given below, are expressed in the discounted with the riskless asset

amounts

They depend on the market price of risk, asset volatilities and the investor’s risk tolerance at

time 0.

Observe no direct dependence on the utility function, and the link between the distribution of

the optimal (discounted) wealth in the future and the implicit to it current risk tolerance of the

investor

xrxrrrr

dtdA

AXrRR

xxt

tttt

ttttttt

02

2

****

0,,02

1

,,


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No benchmark and hedging constraint

The derivatives business can be seen from the investment perspective as an activity for which

it is optimal to hold a portfolio which earns riskless rate

By formulating views against market equilibrium, one takes a risk neutral position and

allocates zero wealth to the risky investment. Indeed,

Other constraints can also be incorporated by the appropriate specification of the benchmark

and of the vector of views

0,0 * tttt


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No riskless allocation

Take a vector such that

Define

The optimal allocation is given by

It puts zero wealth into the riskless asset. Indeed,

***

1

1111 tt

tt

tttttt XX

01 tt

tttttttt

ttt

,1

11

ttttt X **


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Space time harmonic functions

Assume that h(z,t) is positive and satisfies

Then there exists a positive random variable H such that

Non-positive solutions are differences of positive solutions

02

1 zzt hh

2

2

1exp, tHzHEtzh


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Risk tolerance function

Take an increasing space time harmonic function h(z,t)

Define the risk tolerance function r(z,t) by

It turns out that r(z,t) satisfies the risk tolerance equation

ttzhhtzr z ,,, 1

02

1 2 zzt rrr


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Example

For positive constants a and b define

Observe that

The corresponding u(z,t) function can be calculated explicitly

The above class covers the classical exponential, logarithmic and power cases

aztaa

btzh sinh

2

1exp, 2

tabzatzr 2222 exp,

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