Mutual Fund Expense Ratios: How High is Too High?

Transcription

1 Winner o the 2004 Journal o Financial Planning Call or Papers Competition Published in the Journal o Financial Planning in September 2004 This Drat: November 14, 2003 Mutual Fund Expense Ratios: How High is Too High? By Eric E. Haas * ABSTRACT This paper derives a quantitative means or determining the highest expense ratio a particular mutual und can have such that its prospective inclusion in a portolio is expected to increase the portolio s risk-adjusted returns. The approach provides a useul decision aid to assist in assembling portolios o mutual unds. While the approach is most applicable to index unds, it can also be applied to non-index unds, to the extent that an index exists which is suiciently representative o the und s characteristics. * Eric E. Haas is ounder and Chie Investment Oicer o Altruist Financial Advisors LLC in Holland, MI. He can be contacted by calling or by ehaas@altruista.com The current version o this paper can be ound at the website: Copyright 2003 Eric E. Haas 1 o 29

2 Investors are oten involuntarily restricted to relatively high-ee mutual unds with which to implement some o their asset allocation desires. One common example is the typical 401(k) retirement savings plan. Most plans primarily oer high ee actively managed mutual unds, oten in addition to a single (lower ee) S&P 500 index und. I an investor in that plan desires small cap stock exposure, or example, would the high ees associated with the only available small cap stock und completely negate the anticipated increase in their portolio s risk-adjusted return? I so, it may be more beneicial to either abandon the idea o diversiying into small cap stocks entirely, or perhaps it may be best to get exposure to small cap stocks elsewhere in one s extended portolio (e.g., in an IRA). Even i there were no such restrictions on available investing options, there are some asset classes where there do not exist any truly low cost mutual unds. Examples include commodity utures and international small-cap stocks. One o the principal attractions o passively managed mutual unds (e.g., index unds) is their low expense ratios. All else being equal, a und with lower expenses must have higher perormance than a similar und with higher expenses. 1 Investors need to know how low a mutual und s expense ratio must be in order or it to have a beneicial eect on their portolios; they need to know how high the expense ratio can be beore it completely eliminates the anticipated increase in the portolio s expected risk-adjusted return. This question is central to the selection o mutual unds or implementation o an asset-allocation plan. Copyright 2003 Eric E. Haas 2 o 29

3 This paper attempts to answer the question: For mutual und expense ratios, how high is too high? Speciically, the paper derives an equation which answers the question, What is the highest expense ratio I should be willing to tolerate when adding a und to my portolio? The problem is solved both or the simple case (where the existing portolio consists o one und) and the more general case (where there are arbitrarily many unds in the existing portolio and there are arbitrarily many changes to portolio composition being contemplated). The approach provides a methodology or validating or denying the prudence o choosing any particular investment vehicles to implement an asset allocation plan. Investors who are sensitive to investing expenses tend to naturally gravitate towards index unds. Thereore, unless noted otherwise, all reerences to mutual unds in this paper are to index unds, though the approach may apply equally well to actively managed unds. Copyright 2003 Eric E. Haas 3 o 29

4 The Problem Investors are increasingly embracing the tenets o Modern Portolio Theory ( MPT ). 2 Strategic asset allocators typically apply MPT principals at the asset class level when designing portolios. They desire to build their portolios with asset classes which have been shown to (or are expected to) have low correlations with each other. This is usually done by comparing representative indexes and modeling portolios made up o those indexes. Once a desired asset allocation is determined, an appropriate investment vehicle must be identiied or each asset class. Portolio optimization implies minimizing expenses when selecting each investment vehicle. For mutual unds, the principal net ongoing investing expenses are represented by the annual expense ratio, 3 which pays or certain management, distribution, and administrative expenses, as well as providing proit or the mutual und sponsor. There are several asset classes which might be considered desirable rom a diversiication standpoint where virtually all available mutual unds have relatively high expense ratios. 4 At some point, the expense ratio or a prospective mutual und will eliminate any expected beneit rom using it to diversiy one s portolio. How high can that expense ratio be in order to still expect to realize increased risk-adjusted portolio returns? Copyright 2003 Eric E. Haas 4 o 29

5 The Solution In order to ensure we are actually improving the risk-adjusted perormance o your portolio, we need to have some means o measuring it. The most popular means o measuring risk-adjusted perormance is the Sharpe Ratio (see Sharpe [1966] and Sharpe [1994]). 5 In this section, the Sharpe Ratio is used to derive a solution. 6 A more general version o the solution at the portolio level is derived in Appendix B. Assumptions I assume that: The investor s goal is to improve their portolio s risk-adjusted return through prospective addition o an additional mutual und. A relevant index (or blend o indexes) adequately representative o each und s respective investment style is identiiable. Note that this criterion naturally suggests that index unds are best suited to the approach described, but actively managed unds are not necessarily excluded. The returns and volatilities o each relevant index, as well as the correlations between them, are known. The investor can either use known values o the past or predicted values o the uture, as is their preerence. Copyright 2003 Eric E. Haas 5 o 29

6 I the initial portolio contains more than one und, each und is proportionally divested in order to diversiy into the new und (i.e., the relative proportions o each und in the initial portolio remain the same in the proposed new portolio containing the proposed new und). Example D illustrates this assumption well. Note that this assumption is relaxed or the more general solution derived in Appendix B. Costless rebalancing occurs each period when modeling returns o theoretical portolios made up o various indexes. This only aects how the statistics used as inputs to the equation are generated. Other rebalancing requencies can be assumed without altering the validity o the overall approach. The Solution: I you deine the perormance dierential as the dierence between a portolio s total return and the return o the risk-ree asset, the Sharpe Ratio (as shown in Equation 1) is deined as the mean o the perormance dierential divided by its standard deviation. 7 r p r S = (1) σ p Copyright 2003 Eric E. Haas 6 o 29

7 Where r p is the realized return on the portolio, r is the risk-ree rate o return, 8 and σ p- is the standard deviation o the perormance dierential [r p r ]. Let the initial portolio, P, represent some existing mutual und you already own. Let subscript n represent some other new und you are considering combining with your existing und to orm a proposed new portolio, represented by a prime (e.g., P ). You propose to have the new und make up a percentage o your contemplated new portolio represented by the ractional weight w. Set a constraint that the Sharpe Ratio o the proposed new portolio must be at least as high as the original portolio s Sharpe Ratio: By substituting Equation 1 into Equation 2, you get: S S (2) rp r σ p r p r σ p (3) Mutual und returns generally do not exactly match their representative indexes. There is some dierential return ( DR ) representing the dierence between the index s perormance and the und s perormance: 9 r p = r DR (4) I Where r I is the return on the index (or combination o indexes) representing the portolio P and DR is the dierential return o the associated Copyright 2003 Eric E. Haas 7 o 29

8 mutual und (or the weighted average dierential return o a portolio o mutual unds). Thus Equation 3 becomes: ri DR r σ p r I DR r σ p (5) Note that the dierential return o the proposed portolio is just a weighted average o the new und s dierential return and the original und s (or the original portolio s weighted average) dierential return: ( ) + ( w)( DR) DR = w DRn 1 (6) Substituting Equation 6 into Equation 5 and multiplying by the denominators, we get: ( r DR r ) σ ( r w( DR ) ( w)( DR) r ) p I p I n 1 σ (7) Solving or DR n yields the ollowing: DR n r I σ σ p w p ( 1 w) DR r ( r DR r ) I (8) Equation 8 shows the maximum dierential return you should be willing to tolerate in a proposed new mutual und you are considering adding to your portolio. It is robust or all cases, supposing that the Sharpe Ratio remains a valid measure o risk-adjusted perormance. Copyright 2003 Eric E. Haas 8 o 29

9 In order to relate this solution to a und s expense ratio, a relationship between dierential return and expense ratio must be determined. Relatively little empirical research has been done in modeling index und returns. However, the studies to date make a strong case that an index und s dierential return can be well approximated by the und s expense ratio gives: DR ER (9) Where ER is a und s expense ratio. Substituting Equation 9 into Equation ER n r I σ σ p w p ( 1 w) ER r ( r ER r ) I (10) This equation shows the highest expense ratio that you should be willing to tolerate or the und you are considering adding to your portolio. I the und being considered has an expense ratio higher than that value, the resulting portolio would have a lower expected risk-adjusted return than the original portolio and the prospective und should be rejected in avor o the original portolio or some other better alternative. Copyright 2003 Eric E. Haas 9 o 29

10 Examples A) Assume you currently own the Vanguard 500 Index Fund (VFINX). You are considering diversiying internationally by putting 40% o your portolio into the Vanguard Developed Markets Index Fund (VDMIX). VFINX tracks the S&P 500 index, while VDMIX tracks the MSCI EAFE Net index. By examining Table 1, we see that you ought to reject the proposed und because its expense ratio is so high that it would result in a reduced Sharpe Ratio or the resulting portolio (i.e., 0.34% is greater than the maximum allowed 0.18%). B) Assume you currently own the Vanguard 500 Index Fund (VFINX). You are considering diversiying into smaller stocks by putting 20% o your portolio into the Bridgeway Ultra-Small Company Market Fund (BRSIX), which roughly tracks the CRSP 10 index. The Bridgeway und has a relatively high 0.75% annual expense ratio. Is it worth it? Table 1 clearly suggests that you should accept BRSIX as 20% o the new portolio (i.e., 0.75% is less than the maximum 2.55% allowed). C) Assume you currently own the Vanguard 500 Index Fund (VFINX). You are considering diversiying into commodities by putting 10% o your portolio into the Oppenheimer Real Asset Fund (QRAAX), which Copyright 2003 Eric E. Haas 10 o 29

11 roughly tracks the Goldman Sachs Commodity Index (GSCI). QRAAX has a seemingly high expense ratio o 1.68%. Is it worth it? This is an example o applying the methodology to an actively managed und. QRAAX, while it has an explicit (though non-binding) goal o having at least a 90% correlation with the GSCI, is basically an actively managed und. However, rom its inception through the end o 2002, its quarterly returns have a correlation o 98.5% with the GSCI. This und tracks its benchmark almost as well as a true index und. We re conident that the GSCI is adequately representative o this und s investing style. Table 1 shows that you should accept QRAAX as 10% o the new portolio (i.e., 1.68% is less than the maximum 6.16% allowed). 11 The degree to which the und s expense ratio is below the maximum allowed suggests that commodities are an extremely eective diversiier. D) This procedure works equally well i you currently own a portolio o unds and you are considering adding an additional und. Suppose that you presently owned a portolio consisting o 80% Vanguard 500 Index Fund (VFINX) and 20% Vanguard Developed Markets Index Copyright 2003 Eric E. Haas 11 o 29

12 Fund (VDMIX). You are considering diversiying 30% o the portolio into the Vanguard Value Index Fund (VIVAX) (so that the resulting portolio is 56% Vanguard 500 Index Fund, 14% Vanguard Developed Markets Index Fund, and 30% Vanguard Value Index Fund). The Vanguard Value Index Fund tracks the S&P 500/Barra Value index. Table 1 shows that the proposed und s expense ratio is below the maximum allowed it is worth it to diversiy your portolio as proposed (i.e., 0.22% is less than the maximum 1.28% allowed). E) Some mutual unds are only available to individuals i obtained through certain inancial advisors. For example, Dimensional Fund Advisors unds are distributed to the retail market in that ashion. Individuals oten ask whether it is worth it to pay a inancial advisor an additional percentage annually in order to gain access to those unds. The techniques introduced in this paper can be used to help answer that question. Assume that your current portolio consists o the ollowing unds in equal percentages: Vanguard 500 Index Fund (VFINX); ER = 0.18% Vanguard Value Index Fund (VIVAX); ER = 0.22% Vanguard Small Cap Index Fund (NAESX); ER = 0.27% Copyright 2003 Eric E. Haas 12 o 29

13 ishares Russell 2000 Value Index Fund (IWN); 12 ER = 0.25% Further assume that you are considering transitioning your portolio to the ollowing unds, also in equal percentages: DFA US Large Company Portolio (DFLCX); ER = 0.15% DFA US Large Cap Value Portolio (DFLVX); ER = 0.31% DFA US Micro Cap Portolio (DFSCX); ER = 0.56% DFA US Small Cap Value Portolio (DFSVX); ER = 0.56% Note that the weighted average expense ratio o the initial portolio is %. For the proposed new portolio, it is %. Since the entire portolio is being replaced, w = 1.0. Are the generally higher expense ratios o the DFA unds worth it? I so, how much should you be willing to pay the inancial advisor? Table 1 holds the answer. It does indeed seem worth it to switch to the DFA unds (i.e., 0.395% per year is less than the maximum o 1.68% per year allowed). Additionally, even i the inancial advisor provides no value to the investor other than giving them access to the DFA unds (an assumption which is hopeully not valid in the majority o cases), it is still Copyright 2003 Eric E. Haas 13 o 29

14 worth it to pay the inancial advisor as much as 1.285% o assets managed annually (1.68% % = 1.285%). Discussion The derived equations are robust and quite useul to investors to the extent that the assumptions are valid. It might be useul to examine some o those assumptions in order to assess the validity o the approach. One assumption was that the investor s goal is to improve their portolio s risk-adjusted return. Most would agree in principle with this assumption. However, the exact nature o risk which each investor is most sensitive to 13 is a matter o debate and it is urther not clear whether there exists a single best way to adjust perormance igures to relect that risk. The risk-adjusted return measure used here is the most widely used over the last three decades. In Equation 9, an index und s expense ratio was assumed to be a good estimate o the und s uture dierential return. While this assumption is supported by what little research exists in the literature, Elton, et al. [2003] notes that an index und s past dierential return may be a slightly better predictor o its uture dierential return than is its expense ratio. 14 While that may be the case, a signiicantly long track record is necessary in order to provide a useul estimate o that DR. I such a long track record is available, the investor can use it to estimate a und s expected dierential return and use Equations 8 or 11 to decide Copyright 2003 Eric E. Haas 14 o 29

15 whether the expected dierential return is excessive. However, this doesn t directly answer the question o whether a und s current expense ratio, which is more readily observable, is excessive. Further, it is an unortunate act that ew index unds have track records at this point in time which are long enough to be analyzed in this ashion. Thereore, it may be best or investors to use the current expense ratio as a pragmatic irst order estimate o uture dierential return. 15 While there has been relatively little research in modeling index unds, there has been signiicantly more activity in attempting to model returns o actively managed unds. Equation 9 may hold equally well or actively managed unds: there is strong evidence that there exists a signiicant negative correlation between expense ratio and perormance or actively managed unds. Carhart [1997] estimated that, or every 100 basis points o expense ratio, perormance decreased by 153 basis points. Malkiel [1995] ound that 100 basis points o expense ratio decreased perormance by 192 basis points, but that the coeicient was not signiicantly dierent rom For bond unds, the coeicient appears to be almost exactly -1.0 (i.e., 100 bp o expense ratio results in 100 bp o reduced perormance). 16 When applying this paper s techniques to actively managed unds, it is crucial to identiy an index (or a blend o indexes) that is adequately representative o the active und s investing strategy. I a und exhibits signiicant style drit over time (as actively managed unds tend to do), the results o this paper s analyses progressively lose signiicance as the und s style drits Copyright 2003 Eric E. Haas 15 o 29

16 urther rom that assumed during the analysis. The act that style drit can be completely avoided with an index und is one reason why the investor might preer to avoid actively managed unds altogether. Assuming Past is Prologue One o the assumptions in our approach was that perormance and volatility values or relevant indexes are known. This is obviously only true or past values. Perect knowledge o past perormance in general only guarantees a portolio optimized or past conditions (see Ang, Chua, and Desai [1980]). However, most investors are more interested in optimizing a portolio s uture perormance. In order or the approach described herein to be useul or optimizing a portolio s uture perormance, one or both o the ollowing must be true: Past index perormance, volatility, and correlations must be at least somewhat representative o the index s uture behavior; or Future index perormance, volatility, and correlations can be estimated with some accuracy. In general, the second condition is only true to the extent that the irst is true. The irst condition is only true over long periods o time and even then, the persistence o index returns, volatility, and correlations is noisy, at best. I a suiciently long data series or each o the indexes in question is not available, the techniques Copyright 2003 Eric E. Haas 16 o 29

17 suggested in this paper are o little use unless some means o accurately orecasting index perormance in the uture is available. 17 This criticism applies not only to our treatment in this paper, but to any analysis which uses the Sharpe Ratio in order to draw conclusions about what sort o investment activity might be prudent in the uture. Indeed, even the amous Black-Scholes ormula or option valuation (see Black and Scholes [1973]) requires an estimate o a security s uture volatility. I the Black-Scholes ormula has utility despite imperect knowledge o uture volatility (and most would agree that it does), then this paper s approach might similarly have utility, despite its dependence on estimated perormance parameters. To the extent that the measurable past (or the otherwise predictable uture) is a relevant predictor o the uture, the approach described herein provides a decision aid which is useul or portolio design and implementation. As imperect as it may be, the approach is an improvement on the alternative, which is simply guessing whether or not a particular und s expense ratio will detract rom the und s perormance so much as to outweigh the expected beneits o including that und in a portolio. Conclusions and Suggestions or Further Research Investors are increasingly turning to index mutual unds to implement their asset allocation plans. This is oten done in order to realize expected Copyright 2003 Eric E. Haas 17 o 29

18 beneits o diversiication with other elements o their portolios. Such investors are oten conlicted between the beneits o diversiication promised (by adding a particular asset class to their portolio) and a reluctance to pay the seemingly high expense ratios that sometimes accompany available index unds which invest in those asset classes. This paper provides a robust tool or assisting investors in resolving that dilemma. The tool s utilization depends on knowledge o the relationship o an index und s dierential return and its expense ratio. A irst order estimate o this relationship was presented and used or examples. Additional research in modeling index und perormance is clearly warranted. Improved models o the relationship between an index und s expense ratio and its dierential return will urther increase the utility o the approach suggested. Copyright 2003 Eric E. Haas 18 o 29

19 Notes The author thanks the anonymous reerees, Ed Tower, William Bernstein, William Reichenstein, and especially Allan Sleeman or reviewing the manuscript and making valuable suggestions. All errors are the author s. 1 See Sharpe [1991]. 2 See Markowitz [1952]. 3 There are other ees, such as ront or back end sales loads, internal transaction expenses, etc. which also generally ought to be minimized, but a detailed discussion thereo is beyond the scope o this paper. For more inormation on the relationship o mutual und transaction expenses and perormance, see Chalmers, Edelen, and Kadlec [2001]. 4 Examples include, but are not limited to: Commodity Futures, Small Cap Stocks, Emerging Market Stocks, Global Bonds, etc. 5 There exist several other viable measures o risk-adjusted perormance. The most oten used alternatives to the Sharpe Ratio are Jensen s Alpha and the Treynor Perormance Index. However, not all measures are amenable to use in the manner demonstrated in this paper. For example, the Upside-Potential Ratio is one such viable measure which is not well suited or this treatment. Copyright 2003 Eric E. Haas 19 o 29

20 6 Though unreported here, Jensen s Alpha and the Treynor Perormance Index were used to derive conceptually equivalent solutions. For the sake o brevity and clarity, those derivations aren t included here. In empirical tests also unreported here, both the Jensen and Treynor versions o the solution were less limiting (i.e., less conservative) than the Sharpe version. 7 Note that the ormula used here or the Sharpe Ratio diers somewhat rom that commonly used by others. Many others use σ p, the standard deviation o the portolio, as the denominator. We use the deinition advocated by Sharpe it gives a perormance measure adjusted or the volatility o the portolio perormance dierential (i.e., the volatility o the risky component o the return). See Sharpe [1994] or a more elaborate discussion o related issues. In practice, using either σ p or σ p- usually yields very similar results. 8 One month Treasuries are used as the risk-ree measure in the examples. 9 Note that this paper s deinition o dierential return is similar to what is oten reerred to as tracking error. However, some might deine dierential return (or tracking error) as the negative o this deinition. The deinition used here is useul because it somewhat simpliies the derivation. 10 See Frino and Gallagher [2001]: As expected, index unds earn signiicantly negative raw and risk-adjusted excess returns, and the margin o Copyright 2003 Eric E. Haas 20 o 29

21 underperormance is roughly equivalent to the average expense ratio. Also, see Elton, Gruber, and Busse [2003]: dierential return has a very high R 2 with past expenses (0.768). The relationship is signiicant at the one percent level. Furthermore, expenses on average lower dierential return by the amount o the expenses, since dierential return goes down by percent or every one percent increase in expenses. Note that Elton, et al. s [2003] deinition o dierential return is the negative o this paper s deinition. So, by our deinition, dierential return would increase with increases in expenses. 11 Note that this analysis ignores the initial sales commission o 5.75%. I the und s expense ratio were adjusted upward to relect the sales commission (perhaps by allocating it across a typical anticipated holding period o perhaps ive years), the und would still be worth it in the example. 12 This is actually an Exchange Traded Fund ( ETF ), which is somewhat dierent rom the open-ended mutual unds otherwise discussed here. This paper s approach should apply equally well to ETFs. This ETF was used in the example instead o the similar Vanguard Small Cap Value Index Fund (VISVX) due to the act that the Russell 2000 Value index (which IWN tracks) has a dramatically longer history (beginning in 1979) than the S&P/Barra Small Cap 600 Value index (which VISVX tracks, but starts as recently as 1994). This illustrates a limitation o the approach described in this paper: it requires indexes Copyright 2003 Eric E. Haas 21 o 29

22 to have adequately long histories, unless uture values can somehow otherwise be estimated. 13 For example, the Sharpe Ratio uses standard deviation o the perormance dierential as a risk-proxy, while the Jensen and Treynor measures use beta. An excellent alternative measure o risk is downside risk See Van Harlow [1992]. 14 the coeicient o determination increases to when we substitute past dierential return or past expenses. Investors interested in dierential return can apparently choose an index und simply by looking at the past expense ratio, but can do even better by looking at past dierential return. 15 Note that Elton, et al. [2003] showed that expense ratios tend to persist: past expenses are almost perect predictors o uture expenses. We demonstrate the stability o expenses by noting that the coeicient o determination between past and uture expenses is with a slope o Using expense ratio instead o past dierential return as a predictor o uture dierential return has the desirable eect o making our analysis completely independent o any particular und s past perormance. This makes the calculation easier and it makes the approach easible or many relatively new unds, supposing that their corresponding indexes have adequately long operating histories. Copyright 2003 Eric E. Haas 22 o 29

23 16 See Blake, Elton, and Gruber [1993], Domian and Reichenstein [1997], Domian and Reichenstein [2002], and Reichenstein [1999]. 17 For example, i an index that a particular und is based on only has a history o two years, this paper s approach probably cannot be used with any degree o conidence. The longer the index data series, the better. Copyright 2003 Eric E. Haas 23 o 29

24 Reerences Ang, James S., Jess H. Chua, and Anand S. Desai. "Eicient Portolios versus Eicient Market." Journal o Financial Research, Fall 1980: Black, Fisher, and Myron Scholes. "The Pricing o Options and Corporate Liabilities." Journal o Political Economy, May/June 1973: Blake, Christopher R., Edwin J. Elton, and Martin J. Gruber. "The Perormance o Bond Mutual Funds." Journal o Business, Vol 66 No : Carhart, Mark M. "On Persistence in Mutual Fund Perormance." Journal o Finance, March 1997: Chalmers, John M.R., Roger M. Edelen, and Gregory B. Kadlec. "Fund Returns and Trading Expenses: Evidence on the Value o Active Management." Working Paper, University o Oregon, Domian, Dale L., and William Reichenstein. "Perormance and Persistence in Money Market Fund Returns." Financial Services Review, 1997 Vol 6: , and. "Predicting Municipal Bond Fund Returns." Journal o Investing, Fall 2002: Copyright 2003 Eric E. Haas 24 o 29

25 Elton, Edwin J., Martin J. Gruber, and Jerey A. Busse. Are Investors Rational? Choices Among Index Funds. Forthcoming in Journal o Finance, February Frino, Alex, and David R. Gallagher. Tracking S&P 500 Index Funds. Journal o Portolio Management, Fall 2001: Lintner, John. The valuation o risk assets and the selection o risky investments in stock portolios and capital budgets. Review o Economics and Statistics, February 1965: Malkiel, Burton G. "Returns rom Investing in Equity Mutual Funds 1971 to 1991." Journal o Finance, June 1995: Markowitz, Harry. Portolio Selection. Journal o Finance, March 1952: Reichenstein, William. "Bond Fund Returns and Expenses: A Study o Bond Market Eiciency." Journal o Investing, Winter 1999: Sharpe, William F. Capital asset prices: A theory o market equilibrium under conditions o risk. Journal o Finance, September 1964: Mutual Fund Perormance. Journal o Business, January 1966: Copyright 2003 Eric E. Haas 25 o 29

26 . The Arithmetic o Active Management. Financial Analysts Journal, January/February 1991: The Sharpe Ratio: properly used, it can improve investment management. Journal o Portolio Management, Fall 1994: Van Harlow, W. "Asset Allocation in a Downside-Risk Framework." Financial Analysts Journal, September/October 1992: Copyright 2003 Eric E. Haas 26 o 29

27 Appendix A Data Sources Russell 2000 index provided courtesy o Dimensional Fund Advisors. Russell 2000 Value index provided courtesy o Dimensional Fund Advisors. S&P 500 index provided courtesy o Barra, Inc. S&P 500/Barra Value index provided courtesy o Barra, Inc. CRSP 9-10 index provided courtesy o Dimensional Fund Advisors. CRSP 10 index provided courtesy o Dimensional Fund Advisors. MSCI EAFE Net index provided courtesy o Morgan Stanley Capital International. Fama/French Large Cap Value benchmark index provided courtesy o Kenneth French. Fama/French Small Cap Value benchmark index provided courtesy o Kenneth French. Goldman Sachs Commodities Index Total Return provided courtesy o Campbell Harvey. One month Treasuries returns provided courtesy o Dimensional Fund Advisors. Copyright 2003 Eric E. Haas 27 o 29

28 Appendix B More Generalized Solution or Multi-Fund Portolios I you relax the assumption that the unds in the initial portolio remain in the same relative proportions in the proposed new portolio (i.e., you allow the proposed portolio s composition to be completely independent o the initial portolio s), it is possible to develop a more general (albeit somewhat more complex) solution at the portolio level (i.e., to better address situations like Example E). Suppose that the initial portolio can contain any number o unds, each with its weight varying rom 0% to 100% o the portolio. Their relative proportions need not remain ixed in the proposed new portolio. You wish to judge whether the weighted average expense ratio o a proposed new portolio is worth it, given the present composition o your portolio and the proposed composition o the new portolio. It can be shown that the Sharpe Ratio version o the general solution is: m σ m i i I I p i= 1 p ( w DR ) r r r r ( w DR ) i= 1 σ i i (11) Where m is the number o mutual unds in the mutual und universe being considered, w i is the raction o the initial portolio invested in the i th mutual und, w i is the raction o the proposed new portolio to be invested in the i th mutual und, and DR i is the dierential return o the i th mutual und Copyright 2003 Eric E. Haas 28 o 29

29 Copyright 2003 Eric E. Haas 29 o 29 Substituting in Equation 9 gives: ( ) ( ) = = m i i i I p p I m i i i ER w r r r r w ER 1 1 σ σ (12) Where ER i is the expense ratio o the i th mutual und. Equation 11 is robust or all cases. Equation 12 depends on the validity o Equation 9.

30 Table 1, Examples Existing Mutual Fund Proposed New Mutual Fund ER o, Expense Ratio o Existing Fund σ p-, Std Dev o initial perormance dierential σ p-, Std Dev o proposed new perormance dierential r I, Mean return o initial und s index r I, Mean return o proposed new portolio using representative indexes or each represented und (assuming costless rebalancing each period) r, Mean return o risk-ree asset (one month Treasuries) w, raction o proposed new portolio dedicated to proposed new index und Maximum ER n Actual ER n or proposed new mutual und Accept/Reject proposed new mutual und? A B C D E Vanguard 500 Index Fund (VFINX) Vanguard Developed Markets Index Fund (VDMIX) 0.015% per mo. (0.18% per year) Vanguard 500 Index Fund (VFINX) Bridgeway Ultra- Small Company Market Fund (BRSIX) 0.015% per mo. (0.18% per year) Vanguard 500 Index Fund (VFINX) Oppenheimer Real Asset Fund (QRAAX) 80% VFINX and 20% VDMIX Vanguard Value Index Fund (VIVAX) VFINX, VIVAX, NAESX, and IWN in equal proportions DFLCX, DFLVX, DFSCX, DFSVX in equal proportions 0.015% per mo % per mo % per mo. (0.18% per year) (0.212% per year) (0.2225% per yr) 4.58% per month 5.66% per month 4.58% per month 4.22% per month 4.57% per month 4.16% per month 6.30% per month 4.14% per month 4.20% per month 4.50% per month 0.964% per mo % per mo % per mo % per mo % per mo % per month 1.084% per month 0.973% per mo % per month 1.282% per month 0.521% per month 0.310% per month 0.521% per month 0.528% per month 0.537% per month % per mo. (0.18% per year) 0.028% per mo. (0.34% per year) Reject; Actual ER n is too high 0.212% per mo. (2.55% per year) 0.063% per mo. (0.75% per year) Accept; Actual ER n is below maximum 0.513% per mo. (6.16% per year) 0.14% per mo. (1.68% per year) Accept; Actual ER n is below maximum 0.106% per mo. (1.28% per year) 0.018% per mo. (0.22% per year) Accept; Actual ER n is below maximum 0.140% per mo. (1.68% per year) 0.033% per mo. (0.395% per year) Accept; Actual ER n is below maximum Data Period Used Copyright 2003 Eric E. Haas