Computing the Styles of the LS100 Funds




Style and Selection

Ultimately, a mutual fund provides returns for its investors. The return for a mutual fund is, of course, determined by its investments and the return on each of those investments. However, a relatively limited number of factors will greatly influence the return on a typical fund. Analysts find it useful to identify a key number of important factors, then attribute some of the return on a fund to the joint effect of its exposures to those factors and the performance of the factors. The remaining portion of the fund's return is attributed to the choice of specific securities as vehicles for obtaining the particular set of factor exposures.

We follow this general approach, using the returns on broad indices of particular types of securities as factors. Each such index represents a specific asset class. A fund's exposures to such asset classes constitutes its style. The proportion of a fund's return not due to style is termed its selection return.

Asset Classes

We use 15 asset classes for our analyses. The following list provides a description of each asset class and the index used to represent its returns.

These returns are expressed in U.S. Dollars. The returns on Japanese Yen and European/Asian deposits are estimated from the returns obtained from futures markets in currencies. Investing in a one-month U.S. treasury bill and agreeing to trade the proceeds for Yen at the end of the month (by "selling yen futures") is similar economically to converting U.S. dollars to Yen today, then investing the proceeds in a one-month Yen deposit account. In each case, the ultimate return in U.S. Dollars will depend on the rate at which Yen can be exchanged for Dollars at the end of the month. Since the FTA currency-hedged indices incorporate the results obtained from positions in currency futures, we utilize the difference between their returns and those of the unhedged indices to "back out" the effects of the futures positions, then incorporate the return on U.S. deposits to estimate returns on the two non-U.S. deposit asset classes.

Style Analysis

To estimate the exposures of a fund to a set of selected asset classes, we utilize the procedure known as style analysis (sometimes termed "returns-based style analysis").

For details, see:

The monthly returns on a fund over a period of time are lined up with those on the selected asset classes, then a computer is asked to find a mixture of the asset classes that "moved most" with the fund. More specifically, let x represent a specific portfolio made up solely of investments in the asset classes. Let Rx[t] be the return on this portfolio in month t and Rf[t] be the return on the fund in month t. Define e[t] = Rf[t]-Rx[t] as the tracking error of the fund relative to asset mix (portfolio) Rx[t] in month t. Finally, let V(e[t]) be the variance of e[t] over a chosen historic period. The goal of style analysis is to select from all allowed asset mixes (x's) the one that would have provided the minimum tracking variance . V(e[t]).

In practice we modify this general approach somewhat to place more emphasis on recent months than on more distant months. In particular, we seek to minimize the weighted tracking variance, with each month assigned a weight equal to 2^(1/60) times the weight assigned the prior month. In effect, each month receives slightly over 1% more weight than its predecessor. This procedure can be characterized as using a "60-month half-life", since a half the weight is assigned to month t-60 as is assigned to month t.

If no constraints were imposed on asset holdings, problems of this type could be solved using standard techniques of regression analysis. However, in this context, such an approach typically gives ineffective (and sometimes bizarre) results. By adding constraints reflecting minimal information about the investments the fund actually makes, one can usually obtain greatly improved results. Since such constraints typically include inequalities, quadratic programming methods must be utilized. Fortunately, software that can solve problems of this type is relatively widely available.

Using Excess Returns

Were it not for the use of non-US asset classes, we might follow the approach described in the papers referenced above, which employed the following constraints:

However, this would not make it possible to represent partial or complete hedging of the currency risk associated with non-US security positions. To do so in a parsimonious manner, we transform the problem to one involving (1) exposure to currencies (deposits) and (2) non-deposit, risky assets.

Each of our first three asset classes represents investment in a form that is relatively risk-free in its own currency (although the latter two are risky in terms of dollar returns). We retain these three classes in their original form and require that the quadratic program select exposures to them that sum to 1 and that each exposure lie between zero and one, inclusive. In a sense, the decision to invest in the currency of a foreign country involves acceptance of the risk of that country.

To reflect the decision to take risk within a country, we utilize excess returns for each of the remaining asset classes. In the case of U.S. securities, this involves subtracting each month's Treasury Bill return from the return on the asset class in question. In the case of Japanese Stocks, it involves subtracting the return on Yen deposits from the return on the index of stocks. For European/Asian stocks, we subtract the return on European/Asian deposits. Finally, the excess return on Non-US Government bonds is estimated by subtracting from the return on the original index a 50/50 blend of Japanese and European/Asian deposits.

An excess return can be considered the return obtained by borrowing money in the relevant country and investing the proceeds in the risky securities under consideration. In principle it takes no investment, per se. We sometimes term such an approach a zero-investment strategy. A very similar result could be obtained by taking positions in index futures contracts. In the latter case another investment may have to be "posted as margin" to guarantee performance if the investor must make a net payment to the counterparty in question.

While in principle a fund could take exposures in risky securities that sum to more than 1.0, this is relatively rare. We thus require that the sum of such exposures be less than or equal to 1.0. We also rule out short positions in any risky asset class.

Summarizing the Results

Although the formal style analysis is performed using deposits (including US dollars) and zero-investment strategies, the results can easily be transformed to exposures stated in more conventional terms. The exposure to each risky asset class is the same as the exposure to its excess return. The net exposure to a deposit equals that found in the initial analyses less the sum of the amounts of the deposit in question associated with the zero-investment strategies.

We use the following measures to summarize the estimated style of a fund:

The information on the style of each of the individual funds is presented in this form. The style of a portfolio of funds is computed by weighting the styles of the component funds by their relative values.

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