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Risk Management for Mutual Fund Portfolios
An Analysis of Linear Rebalancing Strategies
Mehmet ÖNAL
David WIMBERLY
Introduction
We seek to apply formal risk management methodology to the optimization of mutual fund portfolios
Our solution methodology is based on the CVaR approach outlined in Krokhmal et. al (2002)
Introduction
The problem is to find a strategy to allocate available fund to some number of accounts
These accounts have daily returns
Introduction
The solution approach outlined in Krokhmal et al (2002) is to maximize expected daily rate of return subject to constraints on risk measure
),.....,,( 21 nRisk xxx
Introduction
Krokhmal et al (2002) use several risk measures Mean Absolute Deviation (MAD) Conditional Drawdown at Risk (CDaR) Maximum Loss Conditional Value at Risk (CVaR)
In our work we choose CVaR to be our risk measure
Problem Formulation
Maximize
Subject to
][1
i
n
ii xrE
nixi ,...,1 10
11
n
iix
wxxxCVaR n ),.....,,( 21
Problem Formulation
where
accounts ofindex theisn 1,2,...,
iaccount toallocatedbudget ofamount
have tolike wouldrisk we ofamount maximum the
iaccount ofreturn daily
i
x
w
r
i
i
Problem Formulation
We can reduce this program to a linear program with scenarios of equal probability of occurrence
Each scenario is a vector of daily returns of accounts i=1,2,…,n
Problem Formulation
Maximize
Subject to
S
snn xsxsxs
S 12211 )()()(
1
nixi ,...,1 10
11
n
iix
Ssxsxsxsz
Ssz
wzS
nns
s
S
ss
,...,2,1 ))()()((
,...,2,1 0
1
1
1
2211
1
Problem Formulation
where
))(),.......,(),(),(())(,(
))(,( if 0
))(,( if ))(,( z
level confidence
s scenarioin iaccount ofreturn daily )(
riskat Value
scenarios available ofnumber
321
s
i
sssssxf
sxf
sxfsxf
s
S
S
Problem Formulation
As the days pass we obtain more information on the performances of the accounts
We suggest resolving this optimization as the daily data become available, i.e., as the scenarios to consider increase
Solution Approach
Starting with a sufficient number of scenarios (in this work it is one year), we suggest re-optimizing (rebalancing the portfolio) in every 20 business days with the updated scenarios
Solution Approach
optimization after the first year with
260 scenarios reoptimization after the second month with 300
scenarios
……………
reoptimization after one moth
with 280 scenarios
time
Scenarios start on this day
Solution Approach Begin with a sufficiently large number of
scenarios While you are controlling the funds
Run optimization on the in sample set Observe the performance of the portfolio for
the following 20 days Update the in-sample set by adding 20
business days’ data
Solution Approach
We have approximately 5 years’ data to test the performance of this algorithm
The data was obtained from Theta Research, Inc., a mutual fund research firm which monitors mutual fund managers and their portfolio results
Solution Approach
We first optimize with the scenarios obtained in the first year (in-sample set: data of 260 days)
We then regularly rebalance the portfolio every 20 business days, increasing the size of the in-sample set in each iteration
Results
We did all our calculations in MATLAB Optimal accounts found in the last in-
sample optimization and their historical performances in the last 5 years are presented in the next slides
The results were obtained with CVaRα<-0.995, α=0.90 (recall that we worked on the loss function)
Value of account 366
400
600
800
1000
1200
1400
1600
1800
2000
1 79 157 235 313 391 469 547 625 703 781 859 937 10151093 1171
Value of account 366
Value of account 333
0
500
1000
1500
2000
2500
1 64 127 190 253 316 379 442 505 568 631 694 757 820 883 946 1009 1072 1135
Value of account 333
Value of account 395
400
600
800
1000
1200
1400
1600
1800
2000
1 82 163 244 325 406 487 568 649 730 811 892 973 1054 1135
Value of account 395
Value of account 226
400
600
800
1000
1200
1400
1600
1800
2000
1 82 163 244 325 406 487 568 649 730 811 892 973 1054 1135
Value of account 226
Value of account 968
0
500
1000
1500
2000
2500
3000
3500
1 82 163 244 325 406 487 568 649 730 811 892 973 1054 1135
Value of account 968
Value of account 286
400
600
800
1000
1200
1400
1600
1800
2000
1 82 163 244 325 406 487 568 649 730 811 892 973 1054 1135
Value of account 286
value of account 342
0
500
1000
1500
2000
2500
1 78 155 232 309 386 463 540 617 694 771 848 925 1002 1079 1156
time
valu
e
account 342
value of accout 306
0
500
1000
1500
2000
2500
3000
3500
4000
1 82 163 244 325 406 487 568 649 730 811 892 973 1054 1135 1216
time
valu
e
account 306
Results Historical performance of our portfolio is
shown below
Value of the protfolio
850
900
950
1000
1050
1100
1150
1 44 87 130 173 216 259 302 345 388 431 474 517 560 603 646 689 732 775 818 861
days
valu
e
Series1
Results We were able to increase the performance
of our methodology if we make the CVaR constraint tighter and constrain that no more than 15 % of the funds can be allocated to any account
Historical performance of our portfolio with the additional constraints
value of the portfolio
940
960
980
1000
1020
1040
1060
1080
1100
days
days
valu
e of th
e pro
tfolio
value of the portfolio
Conclusion Despite some issues with the data
set, we were able to construct an efficient frontier and an optimal portfolio with the in-sample data
We were able to run out-of-sample calculations and reached an overall result of a 5% loss on the first run
On the second run, we were able to improve this to a breakeven position
We observed that we were dealing with funds which were all fund of funds
It appeared the managers were all benchmarking the S&P 500
But our results were better than the S&P 500 by a considerable margin