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# Inter-portfolio diversification.

Discussion in 'Risk Management' started by Arty, Dec 29, 2010.

1. ### ArtyGuest

Hi everyone.

I tried to find some good informations about inter-portfolio diversification on this forum but I didn't really found anything useful.

What is a good coefficient number for a diversified portfolio?
Is good between -40 and +40 ?

2. ### ArtyGuest

Just to add to the previous post, I wanted to say correlation instead of diversification.

3. ### MGJ

I assume you trade currencies (FX). If so you can construct the 60 by 60 matrix of correlation coefficients for the 60 most liquid currency pairs, in less than a day. Then it's just a matter of discarding some of them (deleting both row "j" and column "j" from the correlation matrix) until you get a portfolio of the size you desire.

If your goal is low correlation, I think you won't have a very difficult time selecting a subset of 20 currency pairs whose average correlation (mean value of the 20 by 20 matrix neglecting the diagonal entries) is below +0.25.

You could start with a simplistic greedy algorithm, it may give you something you find acceptable: Sum each row of correlation coefficients, find the row with largest sum ("the most correlated currency pair"), and delete that currency pair from the portfolio. Repeat N times and you've deleted N pairs. What's left is the (60 - N) pairs having the lowest correlation. Simple, greedy, and possibly very effective.

If that doesn't produce suitably satisfying results, you can move on to more powerful weapons: quadratic programming, simulated annealing, etc.

4. ### ArtyGuest

MGJ, thanks for the replay.

I want to know the correlation inside the portfolio just because I want to see if/what is the relation between the daily NAV and the daily correlation between N days. It's just for some of my curiosities

Also you already responded to a question that I was about to ask. "Whether I have to use the diagonal numbers(1) for the mean calculation or not. My common sense was telling me that I should not use the diagonal when calculating the mean but I wasn't very sure.

Now another question please. Again my common sense is telling me that probably I'm right but I want confirmation from someone else who know better: When calculating the correlation between every 2 instruments, I have to pay attention whether I'm long or short on every instrument, right?

5. ### Steven.Davis

To do some real analysis, like Value at Risk, you need to analyze the whole matrix using Eigenvalues/Singular Values. For example, if the long-term treasury spread were to widen by 1%, these five positions would lose 5% each.

You are asking for a simple heuristic. My first pass, when looking at a portfolio is similar to a post above. Ignore the diagonal ones. What is the highest correlation (ignore sign)? How many correlations are similarly high? For example, if the top four values are 82%, 78%, 62%, 51%, then you are in trouble. As a first-pass heuristic, I would call that an 80% correlated portfolio. Better to drop a market or two or four.

As far as the sign, holding a short reverse the signs of the underlying market. Unless you are intentionally balancing your position sizes, this probably isn't helping much.

Once you have the off-diagonal correlation to less than 40%, ideally 20%. Take a look some shorter time periods, such as historically stressful events. Unfortunately, otherwise uncorrelated markets have a habit of going bad together.

There are some seriously good books on the subject if you want to get serious about it.

6. ### ArtyGuest

Steven,

What number of days do you consider to be appropriate when calculation correlation ? 1 month? 3 ?
Let's suppose that I trade medium-longer term.

Yes I agree that when we need the most diversification(economic crisis/recession) that is when the markets become more correlated.

thanks.

7. ### Steven.Davis

There is value to using very short correlation (30 day) and very long correlation (10 year.) The principle reason for using short correlations is to minimize the affect of "ghost features". Ghost features are artifacts of an outlier data point seriously affecting the data in disproportion to its likelihood. The principle reason for using long correlations is to be able to understand how modest correlation relates to macro-economics.

It is normal to use at least exponential weighting if not a full GARCH. A smoothing constant of 0.94 (half-life of 25 days) is popular. This has the benefit of a short correlation period while still technically complying with the Basal requirements.

In addition to weighing, people usually use the return = log difference and linear Pearson correlation.

Depending upon your needs it may not matter much. Here are examples of two highly correlated markets and two rather uncorrelated markets:
Code:
```DJIA vs S&P Futures Correlation
Year          Price                Log Price          Log Diff/Return
Linear     Rank       Linear     Rank       Linear     Rank
2009     99.30%     99.40%     99.30%     99.40%     98.10%     96.70%
2008     99.50%     98.90%     99.50%     98.90%     98.40%     97.60%
2007     92.90%     94.40%     92.80%     94.40%     98.00%     96.60%
2006     97.20%     92.30%     96.90%     92.30%     95.20%     94.30%
2005     79.50%     75.60%     79.50%     75.60%     94.80%     93.30%
2004     85.50%     88.60%     85.60%     88.60%     96.30%     95.10%
2003     99.30%     99.20%     99.30%     99.20%     97.80%     97.00%
2002     98.00%     96.40%     98.30%     96.40%     97.20%     96.80%

S&P 500 vs Silver Correlation
Year          Price                Log Price          Log Diff/Return
Linear     Rank       Linear     Rank       Linear     Rank
2009     83.30%     84.70%     79.90%     84.70%     16.80%     25.60%
2008     84.80%     61.40%     87.70%     61.40%     10.20%      0.90%
2007      6.80%      8.00%      7.70%      8.00%     23.20%     24.90%
2006     55.80%     56.90%     55.40%     56.90%      8.10%      5.70%
2005     64.40%     48.90%     63.10%     48.90%      2.40%      1.30%
2004     34.30%     29.50%     34.10%     29.50%      7.30%      5.50%
2003     75.00%     75.70%     73.70%     75.70%    -18.40%    -21.40%
2002     -8.40%     -8.00%     -7.20%     -8.00%    -17.80%    -18.80%
```
Clearly the return = difference of logs is the most useful for risk measurements (not so for pair-trading.)

When you get into the details, one needs to worry about separately generated volatility predictions and getting a positive definite matrix.

Carol Alexander provides a readable text on the subject http://www.amazon.com/Market-Models...9755/ref=sr_1_5?ie=UTF8&qid=1294201096&sr=8-5. Her newer books (since 2001) are probably even better.

The other point was stress matrices. How would your portfolio correlation look if the LTCM meltdown recurred or black Friday?

We have only been talking about correlation itself and not all of the details of factor analysis and Value at Risk. These are valuable things too depending upon your purposes. There are whole companies who do just this time of analysis.

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