Example: Using Vectors to Replicate a Mutual Fund with an ETF and a Hedge Fund

By: Alpha Male

Another posting on this site introduces the idea of using vectors for analyzing active and passive management.  Specifically, that posting suggested that a mutual fund can be viewed as a “host” for a market ETF and an “embedded market neutral hedge fund” (or any other active fund with a different market correlation than the host).  Below, I’ve tried to work through a simple example of how vectors can be used to isolate these components.

Let’s go back to a basic example.  Let’s assume the market volatility is 20%, but let’s not concern ourselves with adding a fund to an index as we did adding B to A above.  Let’s just look at one fund.  Let’s call it the Active Opportunities Fund.  The Active Opportunities Fund has a market correlation of 0.5 and a volatility of 15% per annum.  Thus, it can be represented by the following chart:

                                       
If we simply follow the logic used in the previous example, we can quickly see that there are two ways of getting to point y, either via The Active Opportunities Fund or via a combination of a market ETF and a market neutral hedge fund.

                                        

So the question is: what are z and x.  Since the market correlation of the Active Opportunities Fund is 0.5, then x must by 7.5% (0.5×15%).  So the market contributes half of the volatility – or 7.5%.  As we have seen from the previous example, z can be defined as:
                                                                                             
 

That is to say: the volatility not explained by the market is 12.99%.  Put another way: the volatility of the market neutral hedge fund embedded within the Active Opportunities Fund is 12.99%.  We can only assume that the non-market volatility is the result of decisions that the manager has taken on her own, in other words skill for lack of a better word. 

So we now know how much of the total volatility is contributed by the market and how much of it is contributed by the manager’s skill.  Now let’s imagine you want to buy each portion separately.  Naturally, we’ll have to assume you can buy a skill portion in the form of a market neutral hedge fund with a zero market correlation and a 12.99% standard deviation.  But the perfectly correlated portion can be purchased by ETFs, futures, or swaps.  For simplicity, let’s assume you decide to buy your correlated portion with ETFs. 

You know that the ETF portion contributed 7.5% to overall volatility.  But how much ETF should you buy.  We know that the market has a volatility of 20%.  So if we had all of our money is ETFs, the volatility would be 20%.  That’s too high.  We only need to have 7.5% volatility drawn from the market.  So we only need 7.5% divided by 20% or 37.5% of our money to be in ETFs to get our required amount of volatility from the market. 

If this doesn’t make intuitive sense, consider it this way:  We know that Beta is a measure of the average percentage a fund would be expected to fluctuate given a 1% change in the market.  Assume for a moment that the manager’s skill is negligible and the volatility of the non-market-correlated portion of the fund is 0%.  A Beta of 0.5 for the host fund (ETF portion plus uncorrelated portion) would mean that the fund would fluctuate, on average, 0.5% for every 1% change in the market.  Of course this would only be possible if exactly 0.5 or 50% of the fund were made up of the market itself (via, say, an ETF).   Likewise, if you needed 37.5% of your fund to be invested in a market ETF, that’s the same as saying the beta is 0.375

We now know how much of our fund must be made up of a market ETF.  By definition, the remaining portion must have been allocated to the manager skill or zero-correlation component of the fund.  In other words.  62.5% of our fund has contributed 12.99% to the fund’s overall volatility. If there existed a fund into which we could invest 62.5% of our money to get 12.99% volatility, what would be that fund’s volatility. The answer is 12.99%/.625 or 20.784%. 

Thus, if we found a fund with a zero market correlation and a 20.784% standard deviation and invested 62.5% of our money in it, we would get the 12.99% standard deviation we are looking for in order to complete our synthetic mutual fund. 

– Alpha Male

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