Download Wealth and risk tolerance dynamics

Investment Performance
Measurement, Risk Tolerance and
Optimal Portfolio Choice
Marek Musiela, BNP Paribas, London
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Joint work with T. Zariphopoulou (UT Austin)
2

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Contents
3
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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Investment banking and martingale theory
4

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Investment banking and utility theory
5

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
The classical formulation
6

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Remarks
7

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
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
 
 

EP U X t , t Fs  U X s , s

0st
There exists a self-financing strategy for which the associated (discounted) wealth satisfies

*
  
*
EP U X t , t Fs  U X s , s
8


0st
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Example - value function

Value function


V  x, t   sup  EP u X T , T Ft , X t  x


*


*
 
0 s t T
Value function defines dynamic a performance process
U x, t   V x, t 
9
0t T
Dynamic programming principle
V X s , s  E P V X t , t Fs


x  R,
0t T
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Weaknesses of such specification
10

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Alternative approach – an example
11

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Differential inputs

Utility equation
ut u xx

Risk tolerance equation
rt 
12
1 2

ux
2
1 2
r rxx  0,
2
r ( x, t )  
u x  x, t 
u xx  x, t 
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
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


dSti  Sti ti dt   ti  dWt ,
dBt  rt Bt dt
13
i  1,..., k
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Market inputs

Using matrix and vector notation assume existence of the market price for risk process which
satisfies
 t  rt 1   tT t

Benchmark process
dYt  Yt t  t dt  dWt ,

Y0  1,
Views (constraints) process
dZ t  Z tt  dWt ,

Z0  1
Time rescaling process
dAt   t t t  t    t dt ,
2
14
 t t t   t
A0  0
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Alternative approach – an example

Under the above assumptions the process U(x,t), defined below is a dynamic performance
 x
U x, t   u
 Y , At
 t



Zt

It turns out that for a given self-financing strategy generating wealth X one can write
Z
XZ


dU t  dU  X t , t    u x   u x
  U   dWt
Y
Y

t
2
 X

1
1

 u xx Z    
 R   R      dt
2
Y

 Y
t
X

Rt  r  t , At 
 Yt

15
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Optimal portfolio

The optimal portfolio is given by
*



1 *
  Xt
 t   t  
 Rt*  t  Rt* t  t 
Yt

  Yt

 X t*


R  r
, At 
 Yt

1
rt  r 2 rxx  0,
2
*
t

16
r  x,0   r0  x 
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.
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Wealth and risk tolerance dynamics

The dynamics of the (benchmarked) optimal wealth and risk tolerance are given by
 X t* 
*


t  t    t  t   t dt  dWt 
d

R


t
t
t
 Y 
 t 


 X t*
  X t* 
dR  rx 
 Y , At 
d 
 Y 

 t
  t 
*
t
17

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
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,
 t t  t    t  0

t

18

 X t* 
  0,
d 
 Yt 
dRt*  0
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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
No benchmark and no views

The optimal allocations, given below, are expressed in the discounted with the riskless asset
amounts

 t*  Rt* t t ,
Rt*  r X t* , At

2
dAt   t t dt

t
rt 
19
1 2
r rxx  0,
2
r  x,0   r0  x 

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
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
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,
 t  0,

20
t  t

 t*  0
Other constraints can also be incorporated by the appropriate specification of the benchmark
and of the vector of views
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
No riskless allocation

Take a vector such that
1   t t  0

Define
1  1  t t
t 
t ,

1  t  t

 t   t t t  t 
The optimal allocation is given by
 t*  X t* t t  t 

21
It puts zero wealth into the riskless asset. Indeed,


1  1  t t
*

1   X 1  




X
t
t
t



1  t  t


*
t
*
t

t
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Space time harmonic functions

Assume that h(z,t) is positive and satisfies
1
ht 
hzz  0
2

Then there exists a positive random variable H such that
1


h z , t   E exp  zH  tH 2 
2



22
Non-positive solutions are differences of positive solutions
RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Risk tolerance function

Take an increasing space time harmonic function h(z,t)

Define the risk tolerance function r(z,t) by

r z, t   hz h 1 z, t , t

It turns out that r(z,t) satisfies the risk tolerance equation
1 2
rt 
r rzz  0
2
23

RISK TOLERANCE AND OPTIMAL PORTFOLIO CHOICE
Example

For positive constants a and b define
h z , t  

Observe that
b
 1

exp   a 2t  sinh az 
a
 2


r  z , t   a 2 z 2  b 2 exp  a 2t
24


The corresponding u(z,t) function can be calculated explicitly

The above class covers the classical exponential, logarithmic and power cases