project_Onal_Wimberly.ppt

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



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



400

600

800

1000

1200

1400

1600

1800

2000

1 82 163 244 325 406 487 568 649 730 811 892 973 1054 1135



400

600

800

1000

1200

1400

1600

1800

2000

1 82 163 244 325 406 487 568 649 730 811 892 973 1054 1135



0

500

1000

1500

2000

2500

3000

3500

1 82 163 244 325 406 487 568 649 730 811 892 973 1054 1135



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 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

project_Onal_Wimberly.ppt

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