Risk tolerance and
optimal portfolio choice
Marek Musiela, BNP Paribas, London
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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
Corporate
and optimal portfolio choice, Preprint 2007
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Contents

Investment banking and martingale theory

Investment banking and utility theory

The classical formulation

Remarks

Dynamic utility

Example – value function

Weaknesses of such specification

Alternative approach

Optimal portfolio

Portfolio dynamics

Explicit solution

Example
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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
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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
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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
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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
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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

*
t
  
*
s
EP U X , t Fs  U X , s

0st
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Example - value function
 Value function


V x, t   sup  EP u X T , T Ft , X t  x

0t T
 Dynamic programming principle

*


*
 
V X s , s  EP V X t , t Fs
0 s t T
 Value function defines dynamic a performance process
U x, t   V x, t 
x  R,
0t T
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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.
 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
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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 thinking about allocations in ways which are model
and preference choice independent
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Variational inputs
 Utility equation
ut u xx
1 2
 ux
2
 Risk tolerance equation
1 2
rt  r rxx  0,
2
u x  x, t 
r ( x, t )  
u xx x, t 
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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


dS  S  dt    dWt ,
i
t
i
t
i
t
i
t
i  1,..., k
dBt  rt Bt dt
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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,
 t t t   t
 Views (constraints) process
dZt  Ztt  dWt ,
Z0  1
 Time rescaling process
dAt   t t  t    t dt ,

t
2
A0  0
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Alternative approach – an example
 Under the above assumptions the process U(x,t),
defined below is a dynamic performance
x

U x, t   u , At Z t
 Yt

 It turns out that for a given self-financing strategy
generating wealth X one can write
XZ
 Z

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
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 Xt

Rt  r  , At 
 Yt

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Optimal portfolio
 The optimal portfolio is given by
*



X
1 *

*
*
t

 t   t  
 Rt  t  Rt t  t 
Yt

  Yt

*


X
*
t

Rt  r 
, At 
 Yt

1
rt  r 2 rxx  0,
2
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.
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Portfolio dynamics
 Assume that the following processes are continuous
vector-valued semimartingales
 ,

t t
 ,

t t
 ,

t t
t0
 Then, the optimal portfolio turns out to be a
continuous vector-valued semimartingale as well. Indeed,
1 *  X t*
* 
 t  
 Rt  t  t  Rt*  t t   tt
Yt
 Yt



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Wealth and risk tolerance dynamics
 The dynamics of the (benchmarked) optimal wealth
and risk tolerance are given by
 X t* 
  Rt*  t t t  t    t  t   t dt  dWt 
d 
 Yt 


*
*




X
X
*
t
t

dRt  rx 
, At d 
 Yt
  Yt 
 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
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Canonical variables
 The wealth and risk tolerance dynamics can be written
as follows
 X t* 
 X t*
 *
*
*
  Rt dM t ,
d 
dRt  rx 
, At dRt dM t
 Yt 
 Yt

dM t   t t t  t    t  t   t dt  dWt 


 Observe that
d M
t
 dAt
 Introduce the processes
x1 t  
X A*  1
t
YA 1
,
x2 t   RA*  1 ,
t
wt   M A 1
t
t
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Canonical dynamics
 The previous system of equations becomes
dx1 t   x2 t dwt ,
x1 0   z
dx2 t   rx  x1 t , t x2 t dwt ,
x
y
x2 0   r0  z 
z
 It turns out that it can be solved analytically
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Linear equation
 Let h(z,t) be the inverse function of


1
h, t    
du 
z0 r u , t 



1
z
du,
z0 r u , t 
z
 1
 It turns out that h(z,t) solves the following linear
equation
ht 
1
1
hzz  rx  z0 , t hz  0
2
2


1

h,0    
du 

z 0 r u 
0



 1
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Explicit representation
 Solution to the system of equations is given by
x1 t   h z t , t 
x2 t   hz  z t , t 
z t   h0
 1
1 t
 z   0 rx  z0 , s ds  wt 
2
 One can easily revert to the original coordinates and
obtain the explicit expressions for
X t*
,
Yt
Rt*
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Optimal wealth
 The optimal (benchmarked) wealth can be written as
follows
*
t
X
 x1  At   h z  At , At 
Yt
  1

h, t    
du 
z 0 r u , t 


 1
1 t
z  At   h0  z    rx  z0 , As dAs  M t
2 0
dM t   t t t  t    t  t   t dt  dWt 
 1

dAt  d M

t
  t t  t    t dt

t
2
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Corresponding risk tolerance
 The risk tolerance process can be written as follows
R  x2  At   hz  z  At , At 
*
t
  1

h, t    
du 
z 0 r u , t 


 1
1 t
z  At   h0  z    rx  z0 , As dAs  M t
2 0
dM t   t t t  t    t  t   t dt  dWt 
 1

dAt  d M

t
  t t  t    t dt

t
2
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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,
 t t t  t    t  0

 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
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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

t t
dAt   t  dt
1 2
rt  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,
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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,
 t  0,
t  t

 0
*
t
 Other constraints can also be incorporated by the
appropriate specification of the benchmark and of the
vector of views
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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 
 It puts zero wealth into the riskless asset. Indeed,
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


1

1


*
*

*
t t

1  t  X t 1  t  t 


X
t
t

1



t t


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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 2

hz, t   E exp  zH  tH 
2


Non-positive solutions are differences of positive
solutions
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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
 z, t , t 
r z, t   hz h
1
It turns out that r(z,t) satisfies the risk tolerance
equation
1 2
rt  r rzz  0
2
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Example
For positive constants a and b define
b
 1 2 
h z , t   exp   a t  sinh az 
a
 2 
Observe that

r z, t   a 2 z 2  b 2 exp  a 2t

The corresponding u(z,t) function can be calculated
explicitly
The above class covers the classical exponential,
logarithmic and power cases
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