# Synthetic Replication of a Mutual Fund Using “Off-the-Shelf” Alpha, Beta, and Leverage

By: Alpha Male

A friend recently asked me to analyze his firm’s equity mutual fund on the basis of it’s alpha component.  What proportion of the fund could be represented by an ETF, by cash (or borrowed cash), and by a market neutral overlay (a.k.a. “the embedded hedge fund”)?  If we assume a smal fee for the embedded ETF, what is the implicit fee for the embedded hedge fund?  How does this compare to other equity funds in the same class? To answer these questions, I used a technique similar to that used by Professor Ross Miller at the State University of New York in August 2005.  I post my rough notes below to spark discussion on this technique, not to present any final conclusions or results.  The techniques used form the basis of a white paper in progress and a companion analytical tool currently under development.

Fund X Canadian Equity Fund Back of the Envelope Alpha/Beta Analysis

Fund X Canadian Equity Fund Back of the Envelope Alpha/Beta Analysis

Fund Return (since Jan. 1, 2000)

Fund
* Compounded Annual Return (01/00-09/05): 5.83%
* Annualized Monthly Standard Deviation (01/00-09/06): 11.95%
* Correlation: 0.93, Beta: 0.77

50/50 Blend of S&P/TSX and S&P500 (rebalanced monthly)
* Compounded Annual Return (01/00-09/05): 0.98%
* Annualized Monthly Standard Deviation (01/00-09/06): 14.48%

Using Fund X’s bea to infer the amount of the fund made up of an ETF and the remining (active) portion, we have:

By construction, the contribution from beta (above) has a benchmark correlation of 1 and the active contribution has a benchmark correlation of 0.

The active contribution is associated with only a portion of the fund.  If the active contribution was a stand-alone fund, we would have:

In other words, the monthly returns would be:

Fund X’s “Embedded Hedge Fund” (EHF) would have the following characteristics
* Proportion of Original Fund: 23% (1-Beta)
* Compounded Annual Return (01/00-09/05): 21.30%
* Annualized Monthly Standard Deviation (01/00-09/06): 18.80%
* Correlation: 0.00, Beta: 0.00

While Fund X’s “EHF” has a Sharpe ratio of aproximatley 1, it has a relatively high return and volatility.  This makes finding a real “off-the-shelf” hedge fund problematic.  However, this embedded hedge fund can also be represented as a less volatile (real) hedge fund fund that is simply levered-up (or down) at the risk-free rate (i.e. along the capital market line).

For example, if you want to represent this embedded hedge fund as a fund with a volatility that matches the market’s volatility, then you might pick a lower volatility fund (the red diamond in the chart below) and lever it up by approximately 30% to achieve the required volatility.

While the exact location of the red diamond (above) is completely arbitrary, it is constrained by the investor’s ability to safely borrow (ideally using only the ETF position as collateral).

Thus, Fund X can be represented as the combination of an ETF, an EHF and a negative cash position (i.e. borrowed money invested in the EHF).

Fund X’s new Embedded Hedge Fund (the Lower-Volatility Version that needed to be levered up to meet our volatility requirements) has the following characteristics:
* Proportion of Original Fund: 29.9% =(1-Beta)x(1+Leverage%)
* Compounded Annual Return (01/00-09/05): 17.45%
* Annualized Monthly Standard Deviation (01/00-09/06): 14.47% (by constriction, the same as Benchmark)
* Correlation: 0.00, Beta: 0.00

So while the original fund had the following cumulative returns…

…we now know it can also be represented as…

Note that the sum of the cumulative returns in the chart above exactly equal the cumulative returns in the original fund

Fee Implications

If 77% of Fund X can be replicated with an ETF and an ETF costs, say 17 bps, what must the management fee be on the remaining 23% – the embedded hedge fund?

Assuming the fund costs 2.07% per annum (which it does)

77% * 17 bps + 23% * x bps = 207 bps

x = 8.43% annual MER

When you run the top 35 largest Canadian equity funds through this model (using the TSX as the benchmark, not a 50/50 blend as above), you get the following list of MERs for the embedded hedge funds.  A variety of conservative assmuptions were required to derive the following data (e.g. risk-free rates, leverage-limits, target-volatility of the embedded hedge fund etc.) (also note that, to protect the innocent, Fund X may or may not be on the list).

But some managers (like Sprott) might argue that they charge high fees, but produce high returns, so the fees are worth it.  To get a sense of the fee for return for each fund, you can simply calculate the manager’s share of returns and re-rank.  For Fund X, the MER on the hedge fund is 7.8% (using the TSX as a benchmark) and the return on the hedge fund was 7.1% (again, using the TSX as a benchmark for the calculation).  So the pre-fee return was about 15% and the manager of Fund X essentially took about 50% of the profits (7.8%/15%).

This ranking is according to the manager’s share of profits from the embedded hedge fund.

As usual, the choice of benchmark is critical here.  A discussion of benchmark choice is beyond the scope of this note.  It is assumed that the appropriate benchmark is one that satisfies the mandate as defined by the investor (in this case, Canadian equities â€“ not Canadian equity small cap growth, or Canadian equity large cap technology etc. etc.).

The bottom line is that most bank mutual funds effectively take your money, invest it in an ETF, create a hedge fund (using the ETF as collateral) and take up to 93% of the profits from it.

Parting observation:  If you re-cast the portion of the fees taken by the manager in the form of 2% MER plus the remainder as a performance fee, initial “back-of-the envelope” calculations suggest that the average Canadian equity fund charges 2% management fee and an 18% performance fee.  This is remarkabley close to the “2 and 20” fee structure often cited by long-only managers as a problem with hedge fund managers.

– Alpha Male

Be Sociable, Share!