The Impact of Ross’s Exploration of APT on Our Research

By

Edwin J. Elton*

Martin J. Gruber**

December 3, 2017

* Professor Emeritus and Scholar in Residence, Stern School of Business, New York University, 44 West 4th Street, New York, NY 10012, USA; phone 212-998-0361; fax 212-995-4233; e-mail: eelton@stern.nyu.edu

** Professor Emeritus and Scholar in Residence, Stern School of Business, New York

University, 44 West 4th Street, New York, NY 10012, USA; phone 212-998-0333; fax 212-995-4233; e-mail: mgruber@stern .nyu.edu

The Impact of Ross’s Exploration of APT on Our Research

December 3, 2017

ABSTRACT

Steve Ross’s research has had a major impact on the theory and practice of Financial

Economics. In this paper, the authors concentrate on one of his contributions: Arbitrage Pricing

Theory. After reviewing the theory, they discuss Ross’s contribution to their research. In

particular, they review research on the number of factors present in the return generating process

and in expected returns, the use of macroeconomic variables in the APT setting, and implications

of APT for the performance and predictability of the performance for mutual funds.

Keywords: Arbitrage pricing theory, factor structure, mutual funds JEL Codes: B17, G11, G12

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We were pleased to be asked to contribute an article to this volume in honor of Steve

Ross. Ross has made significant contributions to the field of Financial Economics, many of

which we are sure will be reviewed in other essays in this volume. We decided to discuss one,

Arbitrage Pricing Theory, both because it has had a major impact on the academic and business

community and because it has been the basis for so much of the research we have published over

our careers. Without Ross’s contribution, the profession would be intellectually poorer.

In this essay, we will start with a brief review of APT models starting with the theoretical

structure and approaches to identifying elements of the structure. Ross’s work has had profound

influence on our research. In the rest of this essay, we review some of our research that has been

impacted by Ross’s contribution. This paper is divided into five sections. In the first section, we

discuss the basic APT model and approaches to implementing it. In the second section, we

discuss our contribution to estimating APT based in part on Roll and Ross’s use of factor

analysis to develop an APT model. In the third section, we discuss our application of the insight

from the work of Ross and others on employing economic variables as factors. In the fourth

section, we present some insight we have gained into mutual fund behavior based on Ross’s

classification of the return generating process and APT. The fifth section contains our

conclusions.

I. The APT Model

The APT model was fully described in Ross (1976). The APT model assumes a linear

return generating process for security returns of the following form:

Ri = ai + bi1 I1 + bi2 I2 + … + bij Ij + ei

where

1. ai is the expected return on security i if all indices have a value of zero;

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2. Ij is the value of the jth index;

3. bij is the sensitivity of security i to index j;

4. ei is a random error term with mean equal to zero and variance equal to σ2ei

In addition:

1. E (ei ej) = o for all i and j where i ! j

2. E (ei (Ij- Ij) ) = o where the bar indicates expected value.

Assuming the above results, the equilibrium model for expected return is

Ri = ⋋o + ⋋1 bi1 + ⋋2 bi2 + … + ⋋j bij (1)

where

1. ⋋o = RF

2. ⋋j = Rj - RF

The theory is both eloquent and correct but it still leaves us with a problem of identifying

the I’s, b’s and ⋋’s.

The general approaches to estimating these variables fall into three categories:

1. The use of factor analysis or principal components analysis to identify both the

betas and I’s simultaneously, see for example Roll and Ross (1980) and Elton and Gruber

(1988).

2. The specification of the I’s as macroeconomic influences, see for example Chen,

Roll and Ross (1983) and Elton and Gruber (1994).

-

-

-

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3. The specification of the I’s as a set of well-diversified portfolios that span the set

of securities of interest, see for example Sharpe (1982); Fama and French (1993); and Elton,

Gruber, Das and Hlavka (1993).

One of the results of the APT was an explosion in the development of multi-index

models. Ross showed that some of the indexes in a multi-index model could be priced and

therefore be part of an equilibrium model, but as Ross pointed out, indexes that are not in the

equilibrium model could still be a source of risk and affect realized returns over a period of time.

These insights led to a change from using single-index models to multi-index models for

purposes of evaluating investment performance.

The approaches to estimating APT are described very well by Ross and others in a

Practitioner’s Guide to Factor Models (1994). In that article, Burmeister, Roll and Ross (1994)

recognized “three alternative approaches to estimating an APT model.”1 They are: 1) the

statistical approach as used in Roll and Ross; 2) the specification of a set of well-diversified

portfolios; and 3) the use of Economic Theory to specify factors. While they continued in this

essay to use economic theory to develop the factors, they stated that the use of portfolios can lead

to insight “especially if the portfolios represent different strategies that are feasible for an

investor to pursue. For example, “if K (the number of portfolios) were equal to 2, one might use

small and large capitalization portfolios to substitute for the factors.”2 The debate as to which of

these techniques works best in any given set of circumstances, while an interesting topic and one

which has received a lot of attention, is beyond the scope of this essay. We have used all three

approaches in our research as described in this essay.

1 Burmeister, Roll, and Ross (1994), page 7. 2 Burmeister, Roll, and Ross (1994), page 7.

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II. Estimating the Number of Factors in the APT

Roll and Ross (1980) used maximum likelihood factor analysis to investigate the number

of factors present in the return generating process as well as an examination of how many of the

factors were priced. Roll and Ross (1980) concluded that five factors were present in the return

generating process and that it is likely that three factors are priced. Friend, Drymes & Gultekin,

(1984) questioned whether their conclusion was due to flaws in the methodology or real

influences. In particular, was there more than one factor or just one factor such as would be the

case with the capital asset pricing model? To test this, Cho, Elton, and Gruber (1994) reran the

Roll and Ross models using returns generated by the capital asset pricing model but with

parameters consistent with the individual stock data. When this was done, the Roll and Ross

method correctly identified the existence of only one priced factor. However, three or four were

priced when the Roll and Ross tests were repeated with the actual data used in the test above but

without forcing the data to fit the CAPM model. This is strong evidence of factors beyond the

market being priced and was consistent with the findings of Roll and Ross.

We continued our examination of the Roll and Ross (1980) methodology by examining a

new market; namely, the Japanese stock market and in particular the 400 stocks comprising the

NRI 400 stock index (Elton and Gruber, 1988). Our findings were four and possibly five factors

were present in the return generating process. Like Roll and Ross, we generated factor solutions

for several groups. We then used canonical correlations of the factors to show that the factors

found in the different groups were simply linear combinations of each other.

One criticism of Roll and Ross methodology is that the return generating process

generated in one period may be unique to that period and thus contain useful information about

future periods. One test of this is to examine whether an estimated factor structure in one period

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improves decision making in a subsequent period. In our paper examining the Japanese market

(1988), we used quadratic programming to form a portfolio of assets based on using an estimated

factor structure to find a portfolio of stocks from the NRI 400 index that most closely matched

the Nikkei 225 stock index. This was done for portfolio sizes of 25, 50, and 100 stocks. We

then examined how well these portfolios matched the index in the future. The forecast error

based on a four-factor model had less than one-half of the error based on a one-index model.

This demonstrated that the factor structure of the four-index model captured at last some of the

relevant aspects of return and was stable across periods.

There is another way to test the number of factors present in the return generating

process. Ross (1976) suggested that a specification of the return generating process should result

in the correlation between residuals from the model being very close to zero. A test of this type

is usually performed across portfolios formed on some basis such as size, beta, or industry.

Portfolios are used to improve the estimates of the betas.

In Elton, Gruber and Blake (1999), we explored alternative specifications of the return

generating process using a different set of portfolios (mutual funds) in the hope that these

portfolios had greater dispersion across the relevant factors and allowed us to better estimate a

return generating process.

Mutual funds have some interesting properties compared to constructing portfolios on

some characteristic such as size. The important characteristic in constructing portfolios is that

the portfolios have a spread on beta for each factor effecting returns. There is good reason to

believe mutual funds have this characteristic. One important implication of modern portfolio

theory is that an investor should select an exposure (beta) to each factor, a level of risk adjusted

return, and a level of residual risk. The mutual fund industry has an incentive to offer an array of

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exposures to systematic factors in order to span investors’ differing objectives. If all mutual

funds offer similar sensitivities to a factor, then a fund deviating from this sensitivity would

attract substantial cash flow. Thus, there should be an array of different sensitivities to any

factor.

What are the characteristics of a good model? If the model is a good model, as Ross

pointed out, the correlation of residuals across funds should be close to zero (all co-variances or

correlations captured by the sensitivity to indexes) and for each index there should be many non-

zero betas. Since correlation of residuals can be negative or positive, we examined the absolute

value of the correlations. We also examined the number of large correlations.

We initially compared the market model to a four-index model consisting of a market

index, a value minus growth index, small-cap minus large-cap index, and a bond index. The first

three indexes have been well documented in the literature as affecting return. A bond index was

added because of the presence of bonds in mutual funds and the evidence that bond returns affect

stock returns, particularly for stocks in the financial sector.

Consistent with Ross, we found the four index stochastically dominated the market model

when correlation of residuals was examined. Likewise, for most funds, the betas on each of the

indexes in the four-index model were significantly different from zero.

We then investigated whether a fifth index could be identified that could improve results.

We tried four candidates, a sentiment factor, a momentum factor, a growth factor, and the first

principal component from the residuals of the four-factor model. When we used the first

principal component, we had a five-factor model that stochastically dominated the four-factor

model with most betas significant. This suggests a fifth factor is present. The question is: Are

any of the usual suspects the fifth factor?

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Using the Lee, Scheifer, and Thaler (1991) procedure to estimate a sentiment factor from

closed end funds did not result in an increase in significant betas or a decrease in correlation

across funds. Using Carhart’s (1997) momentum factor did result in a statistically significant

reduction in correlations between residuals but much less than the principal component factor.

We tried one more factor: a growth factor. When we substituted this factor for the principal

component factor, it captured all of the reduction in the correlation coefficients which had been

captured by the principal component factor. Why could this happen when we already have a

value minus growth factor? When we use the standard value minus growth factor, we are

implicitly assuming that growth and value are equally important. This is not true for by

separating their effects, the results improved significantly.

Although we initially examine mutual fund returns to decide on an appropriate model, we

also tested the model on industry portfolios and size portfolios and found that the model worked

for both, and our results were unchanged.

One final observation: the objective in constructing portfolios to determine return

generating processes or APT models is to get a large spread on the betas with each factor.

Examining the spread of betas on mutual funds compared to standard ways of grouping showed a

larger spread. This suggests mutual funds are a useful way to define portfolios for these types of

tests.

III. Applications to Other Areas

The logic behind Ross’s use of fundamental economic variables as APT factors was so

compelling that it was quickly applied in other areas. In this section, we discuss two examples

from our research: its use in estimating the cost of capital for regulated industries and its use in

bond mutual fund evaluation.

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Before the use of Ross’s APT to estimate the cost of capital, the principal approaches to

estimating the cost of capital and the fair rate of return for regulated utilities was either a

dividend discount model or the capital asset pricing model.

In Elton, Gruber, and Mei (1994), we discussed, estimated, and applied an APT model to

determine the cost of capital for nine New York utilities.3 The technique we used to derive an

APT was to specify factors that are likely to affect returns for common stocks. Following the

research of Chen, Roll, and Ross (1983), Burmeister and McElroy (1988), and Burmeister, Wall,

and Hamilton (1986), we used economic factors in the APT model. What are the variables that

are important in effecting return to investors? As pointed out by Chen, Roll and Ross, there are

two types of variables: those that affect expected cash flow and those that affect the discount

rate. Furthermore, expectations about these variables should be incorporated in price and it

should be changes in expectations of these variables that affect returns. We employed a set of

variables similar to those used by Chen, Roll and Ross, but we had several innovations. The

discount rate should be affected by both the level and term structure of interest rates. Changes in

the level can be captured by changes in the return on treasury bills and spread changes by the

difference between returns on long bonds and short bonds. These are our first two variables. We

used three variables to capture cash flow impacts, changes in the forecast of real GNP, changes

in forecasts of the GNP deflator and, since many firms have international operations, we also

used changes in forecasts of exchange rates. This was the first time expectational data about

variables was used rather than historical data on the variables of interest. Expectational data was

collected from several suppliers of professional forecasts as well as from the Federal Reserve

Bank of New York. Even though these were the variables other researchers found were

3 As discussed before, there are many ways to estimate an APT model. Elsewhere we discussed and applied alternative ways to estimate it.

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important, there is a concern that all factors aren’t accounted for. Thus, we added a sixth factor:

the market with the effect of the five hypothesized factors removed.

Once we have specified the factors, the procedure is straightforward. We first estimated

the sensitivity to the factors (betas) for a large set of securities. Then each month we performed

a cross-sectional regression of each security’s return against the betas. The coefficients on the

betas are the lamdas of equation (1).

These are then averaged across the months to obtain a price of risk for each of our

factors. The cost of capital for any firm becomes the beta for the firm on each factor times the

price of risk for that factor summed up across all factors. In this article, we estimated the cost of

capital for each of nine New York utilities and compared these costs with the costs of capital for

a large set of utilities and a larger sample of stocks. The betas for utilities and industries were

consistent with what theory would suggest.

A second area that APT found application was to evaluate bond mutual funds. In Elton,

Gruber and Blake (1995), we estimated an APT model using both portfolio returns and

fundamental variables and then used this model to evaluate bond mutual funds. Both four- and

six-factor models were estimated. As discussed above, levels of variables affecting returns

should be incorporated in price. It is unexpected changes in these variables that cause excess

returns. Thus, for two of the variables, GNP and inflation, we used changes in professional

forecasts of these variables.

In addition, to these two expectational variables indexes, we used four return indexes.

These included an aggregate stock index and an aggregate bond index. These were the four

indexes in the four-index model. To obtain the six-index model, we added a default index (the

difference in return between high yield bonds and intermediate government bonds) and a

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mortgage index. The mortgage index was added to capture option elements. To estimate the

parameters of the model, we used a large set of bond indexes rather than a large set of individual

bonds. We made this choice because the duration on individual bonds changes over time while

for indexes it remains roughly constant over time.

Our four variable model explained 87% of the cross-section variation in bond funds

returns as opposed to 40.6% when only an aggregate bond index was used. When examining

expected return, expectational variables improved the explanation of cross-sectional variation by

24%. Once again, the insight of APT that variables other than market returns might explain

expected return lead to an improved understanding of what affects expected return. Our use of

changes in the expectations of professional forecasters served well as a measure of unexpected

change. On average, bond mutual funds had underperformance of about 11 basis points per

month.

IV. The Return Generating Process, APT, Mutual Fund Performance, and Separate

Account Performance

Ross’s emphasis on the return generating process and APT have revived the interest in

the development of the multi-index models. Single-index models are understood to not be

appropriate for explaining the behavior of financial institutions. In this section, we show how

Ross’s insights into the return generating process contributed to our research on the efficiency of

mutual funds and separate accounts.

We first became interested in examining mutual funds as a result of a widely read article

by Ippolito (1989). The results of his study are presented in Part A of Exhibit 1. Using a single

market index (S&P Index), Ippolito concludes that mutual funds on average outperform indexes

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(and therefore index funds) and that over a range of expenses the more mutual funds charged, the

higher the post expense alpha.

These findings are counterintuitive and are different from previous literature. What

accounted for this? Ippolito’s sample of mutual funds covered the period 1965-1995 and

included balanced funds and small stock funds as well as large stock funds. When indexes were

added to the model representing small stocks and bonds, the results changed (see Exhibit 1, Part

B). The average alphas on mutual funds became negative and the alphas were negatively related

to the expense ratio.

What went wrong with the use of the single index model? A number of mutual funds in

Ippolito’s sample were small stock funds and held stocks with an average capitalization well

below the average capitalization in the index used by Ippolito to measure performance. This was

a period when small capitalization stocks outperformed large capitalization stocks even after

accounting for differences in their beta. In addition, small stock mutual funds had high expense

ratios compared to mutual funds which held large stocks. Since small stock funds outperformed

large stock mutual funds measured against the S&P Index, this led to the erroneous conclusion

that high expense funds outperformed low expense funds.

Note also that the regression coefficient on the small stock and bond indexes are both

positive. This is another way to determine the holdings of mutual funds in addition to examining

their holdings directly. This use of multi-index models to diagnose the holdings of mutual finds

is an additional advantage of multi-index models. They lead to a better idea of the holdings of a

mutual fund. This has become known as return based style analysis and this idea is attributed to

Sharpe (1996).

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Having accepted the APT and the use of a set of portfolios that span the types of

investments held by any financial institution allows us to examine a set of interesting problems.4

1. Do active mutual funds outperform a set of indexes (and by analogy index funds) that

describe the return pattern on the fund or would investors be better off just buying index funds?

2. Do managers in general have information?

3. Can we identify sets of managers that have outperformed indexes?

4. Can we predict which managers will outperform indexes in the future?

5. Can the predictions lead to selecting managers who will have positive future alphas?

In Exhibit 2, we gain some insight into the problems. Note that in both of these non-

overlapping periods (1986-1995 and 1999-2009), the alpha for mutual fund managers was

negative but smaller than the expenses charged. This is a good news/bad news story. Bad news

because managers underperform a set of indexes with the same risk. Good news in that the

performance before fees is positive. This is consistent with what we showed earlier in Exhibit 1.

In Exhibit 1, using a multi-index model, we found a negative alpha of 1%. Exhibit 2 shows

negative alphas of -.65% and -.68%. The results of these studies have been consistently found in

other studies beyond the ones just discussed. Management seems to have an ability to

outperform indexes before expenses. They display real skill but after fees investors have

negative performance.5

4 An APT equilibrium model may have fewer indexes influencing expected return than a model that spans the types of investments a financial institution holds. However, if some of the factors are not in the equilibrium model, they will not be priced over different periods of time and will not affect expected return. However, they will affect risk and realized return. 5 When mutual funds alphas are computed using their prospectus benchmark, which is a single index model, they actually show a positive alpha. See Elton and Gruber (2014). The funds often do not have the same risk pattern as their benchmark and show a tilt relative to their prospectus benchmark factors which have a positive expected return.

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The next logical question to examine is: Is there predictability in mutual fund

performance? We have just established that management’s ability to select securities exists.

However, on average, expenses are larger than pre-expense alpha resulting in negative return to

the investor. Are their managers with sufficient skill to have a positive post-expense alpha and

does this performance allow us to predict future performance? A closely related question is can I

identify managers who will outperform indexes in the future?

Predictability

If management ability exists, it may lead to prediction since in the mutual fund industry

the price of the fund at any instant in time does not reflect management ability. A mutual fund

sells at net asset value whether it is managed by a good manager or a bad manager. Unlike a

security where the price is determined in part by the value of management, mutual fund prices

are independent of the value of management. As long as there is some consistency in

management ability over time, there should be predictability in mutual fund performance.

Berk and Green (2004) present the most compelling argument against predictability.

They have two principal arguments of why predictability will not exist. Management will

capture all excess return in fees and assets under management will grow and there are

diseconomies of scale.

The argument that management will capture all excess return in fees is not supported by

the data. Management fees are set by a schedule in the prospectus. These schedules generally

show declining fees as a percentage of assets as assets grow. It is very difficult to change these

schedules. In addition, other expenses such as audit, accounting, and administration don’t

increase one to one with size. Consistent with this, empirical evidence shows a decline in

expense ratios as funds grow in size. See Elton and Gruber (1990).

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Their second argument is much more compelling. Diseconomies of scale can set in

because as a fund increases in size, the fund either has to select inferior investments or place

more money in current investments.6 The latter may result in higher transaction costs.7 In

addition, as a fund grows, it needs more staff and this leads to problems with the span of control.

While we agree with Berk and Green that there may be diseconomies of scale, their

magnitude and the time span for the influences to become important and how fast expenses

decrease affects whether predictability exists.

To demonstrate predictability, we examined the relationship between future alpha and

past alphas, adjusting for a set of control variables which have been shown in the literature to

have an impact on mutual fund performance. This is shown in Exhibit 3. Note that the

coefficient of determination of the prediction of alpha one period in the future is .21. The

influencers in our model account for 21% of the variance in future alpha. The most striking

result is that past alphas are major predictors of future alphas, even when other influences are

controlled for. The coefficient on past alpha of .18 is statistically significant at the .01 level.

This supports that a 1% increase in past alpha results in an 18 basis points increase in future

alpha.

As expected, both expense ratios and turnover (a proxy for trading costs) are negatively

related to future alphas given the past level of alpha. The most important result which is

consistent with Berk and Green is the negative and statistically significant relationship between

cash flow and future alphas. Berk and Green were correct that cash flows do tend to erode future

6 As funds increase in size, they hire more analysts. This can mitigate or eliminate the problem since the fund may be able to find more mispriced securities. 7 The issue is do diseconomies of scale increase faster than expenses decline and, if they do, at what size does this occur? In addition, how much does a fund need to grow to incur diseconomies? If it takes time, predictability can exist.

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alphas, but the influence does not destroy the positive relationship between 1 year future alpha

and 1 year past alpha. However, size, unlike the hypothesis of Berk and Green, has virtually no

influence on the relationship between past and future alphas.

We repeated this regression using future alphas measured over two- and three-year

periods after the forecast period. The results are interesting. The relationship between past

alphas and future alphas grows weaker as we look at two- and three-year forecasting horizons.

However, it is still highly significant with a t value of 6.32 over a three-year horizon.

As we move to a three-year horizon, the statistical significance of expense ratios,

turnover, and cash flows become larger and more significant. While they erode the performance

over a three-year period, they do not destroy predictability over a period that an investor could

take advantage of.

Predictability can also be demonstrated by directly examining relationships between past

alpha and future alpha. This is done in Exhibit 4 and Exhibit 5. In each exhibit, funds are

divided into 10 groups in each year by alpha.8 Then the alpha for each group is recorded in the

following year. The realized alpha from the prediction group is then averaged over all years.

Exhibit 4 does this for the entire sample of mutual funds. Note that the future alphas are almost

perfectly correlated with the ranking in the past year. The funds that did better in the past have a

strong tendency to do better in the future.9

While there is strong predictability, the question is whether predictability is destroyed by

size. Exhibit 5 divides our sample into three groups by total assets under management. The

correlation between future performance and past performance remains very high. For each

8 In computing Exhibits 4 and 5, alpha was determined using a five-index model consisting of the market index, a small minus large size index, a growth minus value index, a momentum index, and a bond index. The bond index was included because of the presence of long-term bonds in the mutual fund holdings of many funds. 9 This is also shown in Gruber (1996) and Elton, Gruber and Blake (1996).

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group, this is true despite size differences across groups being examined. The average size is

168 million for the smallest group and 5.6 billion for the largest group. The worst funds tend to

do worse in the future and the best funds tend to do well.10 Size does not destroy predictability.

As shown in Exhibit 4, the alpha earned from buying the top 10% of the funds with the

best past performance is 156 basis points per year and, as shown in Exhibit 5, it is a large

positive number for each size group, with the largest funds showing no deterioration in achieved

alpha compared to the smaller funds.

Mutual funds are not the only portfolio of securities available to wealthy investors.

Wealthy investors can purchase a custom portfolio that is managed to meet their tax and social

concerns. These portfolios are called Separately Managed Accounts (SMAs) and are offered by

many firms that specialize in this service and also some mutual fund families. Another option

available to investors is comingled accounts offered by banks and trust companies. These

accounts are available for pension plans and are called Collective Investment Trusts (CITs).

How well do these other options to purchase a portfolio of assets perform? In Elton,

Gruber and Blake (2013), we examined this question. To compute alpha, we used two models.

The first was the Carhart model which consists of the market model, a growth minus value index,

a small minus large index, and a momentum index. The second model used the same four

indexes plus a micro-cap index which consisted of the residuals from regressing the excess return

on micro-cap securities against the four-index model.

This was added because a number of SMAs specialized in portfolios in this micro-cap

sector. The results from these two models showed a negative alpha. However, when we

compared their performance with a matched set of mutual funds, they did much better with

10 In Elton, Gruber and Blake (2012), we also divide funds into categories based on different levels of TNA and the results are consistent with those just discussed.

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alphas of negative 17 basis points to negative 24 basis points for the two models. In contrast,

mutual funds had negative alphas of negative 77 basis points to 94 basis points. Part of the

reason for this difference is a difference in expense ratios. SMAs and CITs had expense ratios

12 basis points less than the matched set of mutual funds. This does not explain all of the

differences and the managers of these portfolios seem to do a better job of selecting individual

securities compared to mutual funds.

These results are very different from the results obtained by examining their performance

on the basis of a single-index model. When we examined the performance of these funds using

the benchmark they chose to use in reporting performance, we found these funds were highly

skilled in selecting a benchmark. They had positive performance against their benchmark. This

disappeared when the single benchmark that best fit their past performance was used as a single

index and, as discussed above, the sign on alpha was reversed when a more appropriate multi-

index model was used.

Conclusion

Ross developed the Arbitrage Pricing Theory. The idea that a single index model would

not serve all purposes and the fact shown in Roll and Ross (1980) that more than one index was

priced changed the way we think about financial markets. Ross’s research has had a major

impact on the profession and on our own work. In this essay, we have discussed areas where

Ross’s insight has affected our research as a way to illustrate some of the areas that his research

has impacted the profession. Much of the earlier research based on his ideas showed that

security returns were better described by a multi-index model rather than a single-market index.

The model that competed with the APT was the CAPM and, for the APT to better describe

expected returns, it was necessary that the market model wasn’t sufficient to explain return

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variation or expected return. In the second section, we discussed our exploration of this issue

using data in the U.S. and Japanese markets and data on stock mutual funds. These studies

supported the APT model and presented evidence on the number of factors in both the return

generating process and expected return.

The acceptance of the APT meant its application quickly spread to other areas. In the

second section, we discuss two of these areas. First, the APT as a way to describe expected

returns became one of the accepted ways to estimate the cost of capital. We illustrate this use by

estimating the cost of capital for nine New York utilities. Second, the same idea can be applied

to other financial markets. In the second essay in this section, we discuss how expected return

can be estimated in the bond market and how this can be used to evaluate bond mutual funds.

Finally, one of the most useful aspects of Ross’s insight and the research on return

generating processes it spurred is the general acceptance that one needs a multi-index model to

properly explain investment returns and in particular to evaluate mutual fund managers. In the

third section, we discuss the wide range of insight multi-index models have provided in

understanding mutual fund performance and private account performance.

Ross’s insight was the basis of much of our research and it has had a major impact on the

field of financial economics.

P a g e | 20

EXHIBIT 1*

Alphas for Individual Funds

Part A Single Index

Number Pos Mean Alpha T Value* Mean S&P

Beta Mean Small Stock Beta

Mean Bond Beta R2

87 0.61 3.14 0.94 0.74

Part B Multi-Index

34 -1.00 -7.50 0.94 0.21 0.13 0.83

* This exhibit is taken from Elton, Gruber, Das and Hlavka (1993). It showed the average alpha

for 143 funds examined by Ippolito (1989). Part A is the results using Ippolito’s procedure but

corrected for data errors. The single index used is the S&P Index. Part B reports results for a

three-index model including the S&P Index, a small stock index and a bond index. The small

stock index and the bond index have been orthoganalized from the S&P index so that the betas

on these indexes represent their effect after the effect of the S&P index is taken into account.

Alpha is expressed in percentage per year.

P a g e | 21

EXHIBIT 2*

Average Alphas for Individual Funds

b.p./year 1986 - 1995

270 funds

1999 – 2009

1,374 funds

Alpha -.65 -.68

Fees 1.13 1.18

Added value before fees .48 .50

* Entries in the exhibit are expressed as percent per year. The data for the period 1986-1995 is

from Gruber (1996), page 123. The data for the period 1999-2009 is from Elton, Gruber and

Blake (2012). The number of funds reported is the average number of funds per year while the

alpha data for the later sample is from Table 2 and the expense data from Table 5 in Elton,

Gruber and Blake (2012).

P a g e | 22

EXHIBIT 3*

Regression Output from Regressing Future Alphas on a Set of Prior Variables

(objective and year dummy variables included in regressions)

Dependent Variable Is Future Alpha

Adj. R. Square

Intercept

Ranking

Alpha

Cash

Flow %

Log TNA

Expected

Ratio

Turnover

Ratio

Family Size

Differential

0.2145 0.02257 (3.00)

0.1802 (23.53)

-0.00011 (3.95)

-0.00081

-1.15

-1.5654 (-6.18)

-0.0092 -7.58

0.00039 (0.67)

Dependent Variable is Average of First and Second Year Evaluation Alphas

Adj. R. Square

Intercept

Ranking

Alpha

Cash

Flow %

Log TNA

Expected

Ratio

Turnover

Ratio

Family Size

Differential

0.14566 0.00183 (0.34)

0.0753 (13.67)

-0.00009 (-4.46)

-0.00092 (-1.82)

-1.8159 (-9.96)

-0.0113 -12.96

0.00085 (2.05)

Dependent Variable is Average of First, Second and Third Year Evaluation Alphas

Adj. R. Square

Intercept Ranking Alpha

Cash Flow %

Log TNA

Expected Ratio

Turnover Ratio

Family Size

Differential

0.12108 0.01228 (2.83)

0.0279 (6.32)

-0.00008 (-4.82)

-0.0003 (-0.75)

-1.7974 (-12.31)

-0.0099 (-14.19)

0.00125 (3.76)

* Taken from Elton, Gruber, Blake (2012). This table shows regression results from regressing

next periods alpha on current alpha and a set of control variables. Alpha was computed using a

P a g e | 23

five-index model, the Carhart (1997) four-index model plus a bond index; t values are in

parentheses.

P a g e | 24

EXHIBIT 4*

Is there predictability?

Future alphas when Funds are Ranked by Past Alphas

Decile Future Alpha

Decile 1 (lowest) -2.49

Decile 2 -1.40

Decile 3 -1.04

Decile 4 -1.09

Decile 5 -.83

Decile 6 -.78

Decile 7 -.57

Decile 8 -.16

Decile 9 +.25

Decile 10 (highest) +1.56

Spearman Corr. 0.988

p-value <.0001

* Taken from Elton, Gruber, and Blake (2012). This table shows the average current alpha from

dividing mutual funds into 10 groups by past alphas. Alpha is expressed in percentage per year.

P a g e | 25

EXHIBIT 5*

Future Alphas when Funds are Ranked on Past Alphas

Rank Bottom half of TNA Top half of TNA Top ¼ of TNA

Decile 1 -2.90 -2.23 -1.51

Decile 10 +1.35 +1.77 +1.40

Average -.78 -.52 -.47

Correlation .96 .94 .93

P value 0.00001 <.0002 <.0001

Average size 108 2,867 5,563

* Taken from Elton, Gruber and Blake (2012). This table shows the average realized alpha when

mutual funds are divided into 10 groups by past alpha. Group 1 is the lowest past alpha group

and Group 10 the highest. Funds are divided in half by total net assets for columns 2 and 3.

Column 4 is the results for the upper quartile of total net assets. Alpha is expressed in

percentage per year.

P a g e | 26

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