Risk tolerance and optimal portfolio choice
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Transcript of Risk tolerance and optimal portfolio choice
Corporate
Bankingand Investment
Risk tolerance and optimal portfolio choice
Marek Musiela, BNP Paribas, London
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and Investment
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
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
ShortcomingsThe 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 investmentAt 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
There exists a self-financing strategy for which the associated (discounted) wealth satisfies
tssXUFtXUE sstP 0,,
tssXUFtXUE sstP 0,,**
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Example - value function
Value function
Dynamic programming principle
Value function defines dynamic a performance process
TtxXFTXuEtxV ttTP 0,,sup,
TtsFtXVEsXV stPs 0,,**
TtRxtxVtxU 0,,,
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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
BenefitsThe function u(x,0) represents the utility for today and not
for, say, ten years aheadThe variational inputs are the same for the general classes
of market dynamics – no Markovian assumption required The market inputs have direct intuitive interpretationThe 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
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
tTttt r 1
ttttttttt YdWdtYdY ,1, 0
1, 0 ZdWZdZ tttt
0, 0
2 AdtdA tttttt
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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
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Optimal portfolio
The optimal portfolio is given by
Observe thatThe optimal wealth, the associated risk tolerance and
the optimal allocations are benchmarkedThe optimal portfolio incorporates the investor views or
constraints on top of the market equilibriumThe 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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Portfolio dynamics
Assume that the following processes are continuous vector-valued semimartingales
Then, the optimal portfolio turns out to be a continuous vector-valued semimartingale as well. Indeed,
ttttttttt
tt
t
RRY
X
Y
**
**1
0,,, ttttttt
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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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Canonical variables
The wealth and risk tolerance dynamics can be written as follows
Observe that
Introduce the processes
ttttttttt
tttt
txttt
t
t
dWdtdM
dMdRAY
XrdRdMR
Y
Xd
**
***
,,
ttdAMd
11
1
1
,, *2
*
1
tt
t
t
AAA
A MtwRtxY
Xtx
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Canonical dynamics
The previous system of equations becomes
It turns out that it can be solved analytically
zrxtdwtxttxrtdx
y
xzzxtdwtxtdx
x 02212
121
0,,
0,
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Linear equation
Let h(z,t) be the inverse function of
It turns out that h(z,t) solves the following linear equation
1
00 ,
1,,
,
1
z
z
zdu
turthdu
turz
1
0
0
0
10,
0,2
1
2
1
z
zxzzt
duur
h
htzrhh
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Explicit representation
Solution to the system of equations is given by
One can easily revert to the original coordinates and obtain the explicit expressions for
t
x
z
twdsszrzhtz
ttzhtx
ttzhtx
0 01
0
2
1
,2
1
,
,
**
, tt
t RY
X
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Optimal wealth
The optimal (benchmarked) wealth can be written as follows
dtMddA
dWdtdM
MdAAzrzhAz
dutur
thAAzhAxY
X
ttttttt
ttttttttt
tss
t
xt
ztttt
t
2
00
10
1
1
*
,2
1
,
1,,
0
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Corresponding risk tolerance
The risk tolerance process can be written as follows
dtMddA
dWdtdM
MdAAzrzhAz
dutur
thAAzhAxR
ttttttt
ttttttttt
tss
t
xt
zttztt
2
00
10
1
2*
,2
1
,
1,,
0
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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,
01 tt
tttttttt
ttt
,1
11
ttttt X **
***
1
1111 tt
tt
tttttt XX
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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