dynamic asset allocation introduction - stanford...

October 2009 © Gerd Infanger, all rights reserved. 1

Dynamic Asset Allocation for Hedging Downside Risk

Gerd InfangerStanford University

Department of Management Science and Engineeringand

Infanger Investment Technology, LLC


Outline

• Background and Concepts• Stochastic Dynamic Programming for Multi-period Portfolio

Optimization• (Active Asset Allocation Management)• Strategies for Hedging Downside Risk• Evaluation via Back-testing


Dynamic Asset Allocation

• The basic decision problem is how to allocate funds among various asset classes over time so as to achieve given investment objectives.

• An inherent trade-off exists between long-term growth and short-term value preservation.

• Optimal asset allocations change over time in response to:– Changes in conditional return distributions (active management)– Realized returns (attained wealth), depending on investor risk preferences

• The problem is best and most generally analyzed through multi-period portfolio optimization.


Theoretical Background

• Samuelson (1969) and Merton (1969, 1990): The optimal investment strategy is independent of wealth and constant over time if:

– Asset return distribution is iid – Utility function is CRRA (power) (If the utility function is logarithmic, non-iid

asset returns result in a constant strategy as well)– No transaction costs

• Mossin (1968), Hakanson (1971): Invest in each period as if all future investments were only in the risk-free asset if:

– Asset return distribution is iid – Utility function is HARA (power, exponential and generalized logarithmic)– No transaction costs– Absence of any borrowing and short sales constraints

• More recently, analytical solutions have been obtained also for HARA utility functions with borrowing and short sale constraints (Cox and Huang (1999), Campbell and Viceira (2002)).


Situations Requiring Numerical Solution

• Utility functions other than HARA (downside risk)• Non i.i.d. return processes (active management)• General side constraints• Transaction costs


Numerical Approaches to Dynamic Asset Allocation

• Stochastic programming– Can efficiently solve the most general model. Successfully used for asset

allocation and asset liability management.• Stochastic dynamic programming (stochastic control)

– Discrete state space (e.g., Musumeci and Musumeci (1999), Brennan, Schwartz and Lagnado (1998))

– When the state space is small, say, up to 3 or 4 state variables, “value function approximation” methods show promise .(e.g., De Farias and Van Roy (2003))


The WealthiORTM Approach

• Applies stochastic dynamic programming to a rich problem representation (Infanger (2006))

– Parameterized terminal utility functions representing various types of investor risk aversion (e.g., increasing and decreasing RRA)

– Downside risk control– Normal, lognormal, and empirical return distributions (using

bootstrapping from historical observations)– (Possible extensions to a restricted class of autoregressive processes)– Linear side constraints and bounds on holdings– Arbitrary cash flows


Increasing and Decreasing Relative Risk Aversion

Represented as piecewise CARA approximation, see Infanger (2006)

CARA

CRRA

W

ART

Increasing RRA

Decreasing RRA


Utility Functions for Hedging Downside Risk


Stochastic Dynamic Programming Recursion


In Practice, In-sample/Out-of-sample Approach


Example of a Downside Risk Protected Strategy

• Start conservatively and take on riskier positions as wealth increases.

Unfavorablereturns

Expected returns

Favorablereturns


Example of a Downside Risk Protected Strategy (cont.)

• Scale back risk when losses push wealth towards target wealth.

Strategy 2009 Strategy 2013

October 2009

Robustness Analysis

• In order to test robustness, we first estimated a model of asset class returns based on historical data and computed the optimal dynamic strategy.

• We then simulated the performance of the obtained strategy using different means of asset class return by varying each estimated mean as a fraction of the estimated standard deviation.

• We compared the (% change of) certainty equivalent wealth of the dynamic strategy compared to the best fixed-mix strategy as a function of the mean disturbance in fractions of standard deviations.

• (For the effect of errors in the parameter estimation on (single-period) optimal portfolio choice, see Chopra and Ziemba (1993).)

© Gerd Infanger, all rights reserved. 14


Robustness Analysis (cont.)

• The dynamic asset allocation model is robust with respect to wide ranges of changes in asset class means, with increasing improvement.

Improvement in Certainty Equivalent Wealth, Dynamic vs Best Fixed Mix

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Deviation of means (standard deviations)

Impr

ovem

ent

(% C

EW

)

Horizon 10 Years, dynamic downside risk, quadratic and linear penalty

36 years 16 years95% probability range


Example: International Asset Allocation, Hedging Downside Risk

• Asset classes: – Equities: US, EU, Japan, Asia– Long-term bonds, short-term bonds, cash

• Benchmark: MSCI World 40%, Long-term bonds 60%• Model calibration via implied means• Horizon of 5 years with monthly rebalancing• Active monthly return predictions, based on multiple factors

– Based on proprietary technique used in Infanger’s active management of about $150M of asset allocation funds

– In actual trading active management adds about 100 basis points per year over benchmark at comparable risk

• Strategy evaluation via back-testing


Performance Measures

• Results from 11 overlapping back-tests over 5 years from 1993-2007• Wealth Target = Max (0%, Money Market minus 1%)

• We will show results for 2008 later

Months 660Exp. Return Underperf Underperf Turnover

(% per annum) (# of months) (% of months) (% per month)

Target 2.49%Benchmark 7.83% 177 26.82%w/o predictions DR 5.04% 9 1.36% 3.56%w/o predictions QL 4.55% 3 0.45% 4.21%Active 10 DR 8.23% 16 2.42% 7.81%Active 10 QL 8.37% 7 1.06% 7.90%Active 20 DR 8.62% 14 2.12% 13.25%Active 20 QL 8.47% 3 0.45% 12.78%


Case Active 10 DR, Number of Underperformances

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Num

ber o

f Und

erpe

rform

ance

s

Month of Strategy

Number of Underperformances

Portfolio Benchmark


Case Active 10 DR, Frequency and Length of Underperformances

0

1

2

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Freq

uenc

y

Length of underperformance (months)

Downside risk (frequency of years below target)

Portfolio Benchmark


Case Active 10 DR, Cumulative Performance 2000-2004

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 12 24 36 48 60

Valu

e of

Por

tfolio

Years starting 12_31_99

Cumulative Performance

Portfolio Benchmark Target


Case Active 10 DR, Asset Allocation 2000-2004

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Asse

t wei

ghts

(%)

Time

Asset Allocation A_05_00

StxUS StxEU StxJA StxAS BdEU BdSh Cash

October 2009

Case Active 10 DR, Results for 2008


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 12 24 36 48 60

Valu

e of

Por

tfolio



Portfolio Benchmark Lower Bound

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 12 24 36 48 60

Valu

e of

Por

tfolio




0

0.2

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0.6

0.8

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1.2

0 12 24 36 48 60

Valu

e of

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tfolio




0

0.2

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0.8

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1.2

0 12 24 36 48 60

Valu

e of

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tfolio




2008

20082008

2008

October 2009

Case Active 10 DR, Results for 2008 (cont.)


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Asse

t w

eigh

ts (

%)

Time



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Asse

t w

eigh

ts (

%)

Time



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Asse

t wei

ghts

(%

)

Time



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Asse

t w

eigh

ts (

%)

Time



2008

2008 2008

2008

October 2009

Case Active 10 DR, Results for 2008 (cont.)


2008 # of Underperformances 60 MeanMonth Portfolio Benchmark Sample Portfolio Benchmark

1 0 3 5 1.009 0.9742 0 3 5 1.013 0.9653 0 3 5 1.014 0.9474 0 3 5 1.014 0.9685 0 3 5 1.012 0.9706 0 3 5 1.014 0.9327 0 3 5 1.020 0.9338 0 3 5 1.028 0.9589 0 4 5 1.027 0.924

10 0 5 5 1.027 0.88411 0 5 5 1.042 0.87712 0 5 5 1.053 0.866

Sum 0 43 Average 5.34% -13.37%


Summary

• A numerical approach to dynamic asset allocation based on stochastic dynamic programming handles a very rich problem representation.

• The approach may be used successfully for controlling downside risk in the short-run, while obtaining benchmark level expected returns in the long-run.

• As demonstrated through back-testing, downside risk was reduced considerably, while the cost of the downside risk protection was more than offset by employing active management based on predictions.

• The approach had performed well during the crisis of 2008 with no underperformances versus the target observed. This was due to the downside risk protection and the active predictions as well.

dynamic asset allocation introduction - stanford...

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