1 Testing Investment Portfolios A Permutation Test Testing - - PDF document

• 1

filename 1

Using R for Evaluating Trading Using R for Evaluating Trading Strategies Strategies

Patrick Burns Patrick Burns http://www.burns http://www.burns-

• stat.com

stat.com

June 2006 June 2006

filename 2

Summary Summary

• R is a good thing

R is a good thing

• Random portfolios are useful

Random portfolios are useful

• More information at

More information at http://www.burns http://www.burns-

• stat.com

stat.com

filename 3

Backtest Backtest Results Results

Wealth 1.00 1.10 1.20 1998 1999 2000 2001

filename 4

Emerging Market Fund Emerging Market Fund

1997 1998 1999 2000 2001 2002 2003 2004 2005 20 16 12 8

• 2

filename 5

Testing Investment Portfolios Testing Investment Portfolios

• Test if result is better than a guess

Test if result is better than a guess

• Computers exist

Computers exist

• Implies a random permutation test

Implies a random permutation test

filename 6

A Permutation Test A Permutation Test

• An amount of money in each asset

An amount of money in each asset (typically including a lot of zeros) (typically including a lot of zeros)

• Permute the amounts among the

Permute the amounts among the assets assets

• Takes at most 6 lines of R

Takes at most 6 lines of R

filename 7

There There’ ’s a Problem s a Problem

• Portfolios are not a haphazard

Portfolios are not a haphazard collection of assets collection of assets

• The permuted portfolios are not

The permuted portfolios are not realistic realistic

• In particular volatility is too large

In particular volatility is too large

filename 8

Typical Constraints Typical Constraints

• Non-negative weights (no short selling)
• weights less than some limit
• weights within some limit of benchmark

weights

• Country constraints (linear)
• Industry constraints (linear)
• Liquidity constraints

• 3

filename 9

Practical Constraints Practical Constraints

• Limit turnover

Limit turnover

• Limit number of assets traded

Limit number of assets traded

• Limit number of assets in portfolio

Limit number of assets in portfolio

• Threshold constraints

Threshold constraints

filename 10

Random Portfolios Random Portfolios

• Sample from the set of portfolios that

Sample from the set of portfolios that

• bey all constraints
• bey all constraints
• This is non

This is non-

• trivial

trivial

• Uses a genetic algorithm typically

Uses a genetic algorithm typically

filename 11

So Why is R Still Important? So Why is R Still Important?

• Now have a whole pile of portfolios

Now have a whole pile of portfolios

• Want to step through time in

Want to step through time in backtests backtests

• Want to graph results

Want to graph results

filename 12

A Valid A Valid “ “Permutation Permutation” ” Test Test

• Generate a random sample of

Generate a random sample of portfolios that satisfy given constraints portfolios that satisfy given constraints

• Compare actual result to distribution

Compare actual result to distribution from random portfolios from random portfolios

• 4

filename 13

Backtest Backtest Results Results

Wealth 1.00 1.10 1.20 1998 1999 2000 2001

filename 14

The Random Paths The Random Paths

Wealth 0.4 0.6 0.8 1.0 1.2 1998 1999 2000 2001

filename 15

Random Random Quantiles Quantiles

Wealth 0.4 0.6 0.8 1.0 1.2 1998 1999 2000 2001

filename 16

Whole Period from Random Starts Whole Period from Random Starts

0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Theoretical Quantiles Whole Period P-values

• 5

filename 17

10 10-

• day Non

day Non-

• overlapping p
• verlapping p-
• values

values

Combined p-value 0.0 0.2 0.4 0.6 0.8 1.0 1998 1999 2000 2001

filename 18