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# Math question...

I would like to create an investment portfolio with a low volatility output comprised of high volatility inputs. Read: a Fund of Funds product comprised of funds which individually have a high volatility return stream, however, I want the monthly performance of the FoF portfolio to be a low volatility return stream. Obviously, it goes without saying the the inputs (component Funds) will be highly diversified. Can anyone comment on the math involved here and the validity of creating such a portfolio idea?

2. ### MYDemaray

Correlation = (Covariance AB)/(StdDevA * StdDevB)

Variance of PortfolioAB = weightA^2 * StdDevA^2 + weightB^2 * StdDevB^2 + Correlation of AB * weightA * weightB *StdDevA*StdDevB

StdDeviation of Portfolio = SQRT(Portfolio Variance)

That's off the top of my head, but I think that's right. So basically your overall portfolio volatility will be a function of correlation and volatility -- which is very straightforward. You'd have to build a matrix to figure ot the total vol.

A great resource to calc this in Excel is Hoadley's option plugin -- it gives you a value at risk calc, which can also be viewd as the volatility of the portfolio -- often quoted as a 2 std dev format (95% confidence). If you can get your hands on the daily/monthly data of the underlying funds, I'd use Hoadley to figure out the value at risk for the portfolio...you could compare this to the overall std dev for each individual fund and see how it lowers to total vol. (www.hoadley.net)

3. ### vikana

FWIW, I've found the most of the classical "math" formulations don't works well in practice. I suspect it's because most of the classical statistical math assume normal distribution.

Here is what I do: I simply combine the equity curves, or better the trades if you have access to them, and do the math directly on the composite equity curve. The fewer assumptions you can make about distribution, the better you'll be off.

Edit: I should clarify my initial point. It's not that the math doesn't work, but that you have to be careful with the conclusions you draw from the calculations.

4. ### jrkob

This is absurd and makes no sense at all. A correlation coefficient, a variance or a standard deviation assume no distribution at all. They have nothing to do with each other. You just have to look at the formula to see that the distribution of the input data is completely irrelevant.

5. ### MYDemaray

Risk control is 80% common sense, 20% math, in my opinion. If you're trying to construct a Fund of Funds I would focus my attention more on understanding how each underlying fund makes their money -- make sure that your style bets are diversified.

Thank you for your helpful insight everyone. I know the strategies very well, however, my feeling is that too much emphasis is currently being placed on "low volatility strategies". The Fund managers have become whimy trying to compete against benchmarks and produce low volatility returns. I believe that truly attractive funds are those that will step out and take swings. With that said, as a Fund of Funds product, I need to produce low volatility returns to attract the institutional clients. Hence, my desire to create a portfolio of well diversified high volatility inputs seeking to create a low volatility product with enhanced returns. I am just trying to ascertain some reinforcement that you can in fact create a low volatility product utilizing high volatility components which is counter to standard logic currently deployed in the business. Beyond that, I want to ascertain how many diversified high volatility inputs I will require at a minimum to produce a low volatility earnings stream. I realize that it's at least as much of an art than a science. I've got the art and am looking for the science component. Thanks again!

7. ### Sparohok

This is an extremely well studied problem. You can go back to Markowitz' original 1952 paper, if you want, it's moderately readable. You can also look in any financial economics text. For that matter, here's a reasonable overview in Wikipedia:

http://en.wikipedia.org/wiki/Modern_portfolio_theory

There's an excellent treatment of portfolio theory in Capinski and Zastawniak's "Mathematics for Finance" p.107-117. They derive a closed form solution for the minimum variance portfolio subject to linear constraints which can be useful for straightforward optimizations.

If the closed form solution is too cumbersome or your constraints are more complex, you can use William Sharpe's gradient optimization method:

http://www.stanford.edu/~wfsharpe/mia/mia.htm
http://www.stanford.edu/~wfsharpe/mia/opt/mia_opt1.htm

This online text is written for practitioners. Sharpe provides sample code and walks you through the entire process.

More recent academic work focuses on statistical flaws of mean-variance optimization, the error sensitivity of the covariance matrix, and how to construct well-conditioned estimators. If nothing else, it is absolutely critical that you have a good grasp of the statistical sensitivity of the method and use robust data. For example, the number of historical data points should be much larger than the dimension of your covariance matrix, ideally 20x larger to ensure statistical significance. For a good summary of the statistical issues written for practitioners, I recommend Ledoit and Wolf's paper "Honey, I Shrunk the Sample Covariance Matrix." Yes, that's really what it's called. Between that paper and its references you can get a very good education on the pitfalls of mean-variance optimization.

http://www.ledoit.net/research.htm

Finally, anyone who thinks a fund of funds should be constructed with intuition or common sense, but without a deep understanding of the mathematics of portfolio optimization, has no business managing other people's money.

Martin

8. ### MoralHazard

As the previous poster points out, the theory is quite simple, but the devil is in the details, especially once you put in constraints and want to be fast about it.