## Abstract

This study presents a new method for calculating beta through a back-solving process, which assumes the capital asset pricing model (CAPM) to be absolute. This process has improved asset pricing abilities and allows for the discovery of the “one true” market returns. The market portfolio returns required for CAPM to be accurate are then calculated and compared with eight popular financial distributions and five market proxies. The overall best distribution to use for CAPM market returns is the Student *t*-distribution. This study also contributes to the literature that the market proxies used are inefficient and adversely affect the results used in other studies to discredit the CAPM.

**TOPICS:** Portfolio theory, portfolio construction, statistical methods

This study assumes the capital asset pricing model (CAPM) to be absolute in terms of the expected returns required being accurately priced with the average actual returns for asset *i*. Relying on this assumption, the back-solved market returns (*R _{m}
*) are calculated and assumed to be the “one true” market tangency portfolio from the CAPM theoretical assumptions. The tangency portfolio is then compared with popular proxies used in empirical studies, in lieu of the tangency portfolio, using

*t*-tests and correlations of returns. The tangency portfolio is also compared using two-sample Kolmogorov–Smirnov (KS) tests with 1,000 runs, likelihood tests, and plots of the probability density functions to simulated returns of popular financial distributions used to model tests relying on the CAPM framework.

Since William Sharpe [1964] published his pivotal paper on the CAPM, academia has subjected the model to a battery of tests. Early on, most of these tests supported the CAPM’s main predictions. One of the earliest tests was performed by Black, Jensen, and Scholes [1972], whose study found a positive relationship between returns and beta, in support of the CAPM. MBA and finance graduates are taught that the CAPM has many applications, such as estimating cost of capital, evaluating a portfolio manager’s performance, and solving market expected returns for an asset. The model is intuitive and simple to implement, without requiring advanced econometric modeling. Its ease of use and the empirical results defending the model have propelled the model to the realm of finance stardom. Research conducted by Graham and Harvey [2001] found that 75% of U.S. chief financial officers (CFOs) almost always or always use the CAPM. Globally, the lowest usage of the CAPM among CFOs was still a mighty 30%.

The Sharpe–Lintner (SL) CAPM relies on a key assumption that investors agree on the joint distribution of asset returns from time *t* − 1 to *t*
_{0} in order to obtain market clearing prices. Thus, the distribution from which the market returns are selected to empirically test the model is the same as the true distribution. In other words, modern portfolio theory, developed in part by Markowitz and Lintner, concludes that the tangency portfolio (the optimal portfolio in which all investors want their wealth invested) must inevitably be the market portfolio. Fama and French [2003] suggested that the shortcomings of the CAPM may be explained by the poor proxies for market returns on invested wealth, which play a pivotal role in the CAPM’s predictive power.

Countless CAPM studies have brought attention to its sensitivity to the choice of a market proxy. Many papers concluded that the CAPM model may be correct, but the proxy used (typically the S&P 500) is not representative of all publicly traded assets as theory dictates it ought to be. For example, Roll and Ross [1994] found that betas and mean returns having little relation may be due to mean–variance-inefficient market proxies. The SL CAPM implies mean–variance efficiency of the market index, and the results are sensitive to the index proxy’s location inside the ex ante mean–variance frontier (Roll and Ross [1994]).

Many studies support the CAPM (Black [1972]; Black, Jensen, and Scholes [1972]; Fama and MacBeth [1973]). Numerous other studies have challenged the CAPM, finding no relation between betas calculated using a market proxy and returns, including Montier [2007]; van Rensburg and Robertson [2003]; and Strugnell, Gilbert, and Kruger [2011]. Finally, some studies challenge the CAPM challengers, citing, among other reasons, poor empirical results being the fault of the market proxy (Amihud, Christensen, and Mendelson [1992]; Black [1993]; Breen and Korajczyk [1993]; Jagannathan and Wang [1993]; Kothari, Shanken, and Sloan [1995]). Studies have created global wealth proxies (Ibbotson and Siegel [1983]), forward-looking market returns (as opposed to the conventional historical mean), nonlinear market returns (French [2016]) and everything in between. Fama and French [2003] concluded that future work may show that the CAPM’s problems are resolved with the appropriate market proxy, although they are pessimistic about such a discovery.

Many practitioners still use the CAPM as a primary forecast tool. The prediction power of the model depends on the input of expected market returns. Correctly specifying the distribution of market returns is therefore an important topic in verifying the validity of the CAPM. This study seeks to expand the literature in three main ways: (1) estimating the distribution of returns of the one true market; (2) exploring popular, all-star financial distributions’ comparability to the one true distribution; and (3) analyzing how well common market proxies used in empirical studies compare with the one true market.

**METHODOLOGY AND DATA**

The CAPM defines the relationship between risk and return.

Let *E*(*R _{i}
*) be the expected return investors require for asset

*i*;

*R*be the risk-free rate of return, using the historical period average; and

_{f}*R*

_{m}be the market portfolio return, using the historical period average. These investor-supplied parameters make up Equation (1), and all returns should have the subscript

*t*, which is omitted for abbreviation. The parameters in Equation (1) are the averages of the historical data for various types of assets.

where

2In this study several assumptions are made regarding the CAPM:

1. Many identical investors are price takers.

2. All investors plan to invest over the same time horizon.

3. No taxes or transaction costs are applied.

4. Borrowing and lending occur at a risk-free rate.

5. Investors only care about expected return and variance:

a. they prefer high mean but low variance;

b. the market consists of all publicly traded assets;

c. the market portfolio equals the value-weighted index of all publicly traded wealth.

Assuming CAPM is correct and the fault lies in the proxy used for the market returns, the average expected returns *E*(*R _{i}
*) would have zero difference and be equal to the average actual returns

*A*(

*R*

_{i}). Thus,

Given Equation (3) to be true, Equation (1) can be rewritten to solve for the required risk premium needed for the model to be accurate—Equation (4). The market and risk-free returns in Equation (4) are user supplied and based on required returns, historical data, or some other benchmark, so long as the *R _{m}
* is greater than the

*R*

_{f}. The study used conservative figures of 7% and 1%, respectively.

With the required beta needed for CAPM to accurately predict actual returns, the one true market returns can then be computed using Equation (5). These market returns from Equation (4) can be used to derive the back-solved beta. This back-solved beta, when used with the same user-supplied *R _{m}
* and

*R*

_{f}, will get the same average:

The calculated market returns that would be needed for 10 different portfolios are calculated from Equation (5). Assuming the CAPM is correct and beta is representative of the sole pricing factor, then the market returns in Equation (5) are the unmeasurable returns of the tangency portfolio. The tangency portfolio returns, calculated in Equation (5), include human capital; consumption; economic, behavioral, organizational, social, political, regulatory, and aggregated individual risk preferences; intertemporal choice; and future expectation factors. This equation allows the market returns to be form-fitted and exactly what they need to be so that the expected returns equal the actual returns (Equation [3]). The back-solved one true *R*
_{m} captures all unobservable factors. In countless CAPM studies, researchers have strained to find appropriate proxies and methods for measuring intertemporal choice (Merton [1973]), household consumption (Mehra and Prescott [1985]), risk aversion (Pratt [1964]; Arrow [1965]), and human capital (Mayers [1972]). The method in this research is incredibly simple and accurate, which are two characteristics any successful pricing tool requires. These form-fitted returns, or back-solved returns, are calculated using in-sample historical data just as the traditional CAPM uses. Furthermore, this market portfolio is efficient because no ratios involving stock prices (price-to-earnings ratio [P/E] and B/M) could improve the expected returns because nothing was missed by the back-solved market beta.

The calculated market returns in Equation (5) produce a beta and average market premium to solve Equation (1) and keep average expected returns and average actual returns equal within the sample (Equation [3]). This remains true if the derived market returns are multiplied by powers of 10 (10, 100, 1,000, etc.). With powers of 10, the beta and average market returns are transformed but still produce expected returns equal to that of the average portfolio returns. Yet if logarithms of −1, −2, −3 (0.1, 0.01, 0.001) are used, they do not work because the market returns then are smaller than the risk-free rate (1%) and cause the market premium (*R _{m}
*−

*R*

_{f}) to become negative. This negative market premium no longer allows for the expected returns from Equation (1) to equal the actual portfolio returns. In addition, for the market portfolio to be efficient, the risk premium (

*R*−

_{m}*R*

_{f}) must be positive (Fama and French [2003]). The market returns calculated and used in this study are therefore those with the smallest variance.

The method of back-solving for the market returns, instead of using a market proxy, has increased prediction accuracy and would be even more accurate in shorter forecast horizons. This is crucial for practitioners who view one month as a long investing horizon and are not interested in a 10-year estimate of returns. Jagannathan and McGrattan [1995] concluded in their study that the CAPM has something to offer to investors who are interested in the long run, but the straight-line relationship—between beta and returns—breaks down in shorter periods. The back-solved beta corrects for this shortcoming of the model and requires no historical data other than that of the traditional SL CAPM.

Roll [1977] critiqued the CAPM’s market portfolio as having to include every possible asset—real estate, metals, jewelry, baseball cards, and beyond. Because data for the true market portfolio of invested wealth is unobservable, the CAPM has truly never been tested. Time-series and cross-section regressions used in an attempt to empirically disapprove have instead simply tested not the CAPM itself, but the efficiency of the specific market proxy used (Fama and French [2003]).

The hunt for the best market proxy spread into research by Gibbons, Ross, and Shanken [1986], in which they provided an F-test (GRS test) that tests whether the market proxy is the tangency portfolio.

Stambaugh [1982] contradicted the market proxy dilemma, stating that tests are not sensitive to broadening the market proxy because the stock returns consist of the majority of the volatility already. Lakonishok, Shleifer, and Vishny [1994] furthered the argument that adding additional factors into the market proxy would better explain why price ratios like P/E and B/M are not explained by the SL beta. If one stands on the pillar that CAPM is absolute, however, then back-solving for the unseen, untouchable, unmeasurable one true market returns needed to obtain the beta that can accurately predict the average returns of asset *i* is evidence of the market proxy problem and counters the aforementioned arguments.

**Data**

Two problems in CAPM tests have been well documented. First, estimates using individual securities are inaccurate and contain measurement error. Second, downward bias in estimates of the standard errors and slopes comes from spill-over effects—for example, industry effects in average returns. Therefore, following best practices, portfolios based on U.S. industry and Asian nations are formed.

This study uses 20 portfolios to back-solve the market returns in 10 different instances (five U.S. industry and five Southeast Asian markets) for both weekly and quarterly returns. The observations span from January 2005 to December 2014. The U.S. industry–replicating portfolios are utilities (UT), real estate (RE), industrial goods (IG), healthcare (HC), and basic materials (BM). The Asian portfolios represent countries that formed a free-trade bloc (the Association of Southeast Asian Nations [ASEAN]) and in 2016 began free trade. The ASEAN portfolios used are Indonesia, Malaysia, Philippines, Thailand, and Singapore. Each portfolio consists of 50 randomly selected companies, formatted in both weekly and quarterly returns. Bias may occur when portfolios are formed based on a sorted characteristic to explain returns, but when the grouping is based on an industry or country there is no such violation with regard to the characteristic sorting dilemma. Grouping according to industry or region to reduce error proceeds from the approaches used by Friend and Blume [1970] and Black, Jensen, and Scholes [1972]. The market portfolios are calculated individually for each specific portfolio because political, behavioral, human capital, consumption, and other components are country and market specific. Although future research could use a world portfolio of assets and the back-solved market returns would thus be priced with all unobservable factors on a global scale, to get the exact properties of the market portfolio one would need to construct the tangency portfolio for the world’s wealth of tradable assets. No such portfolio currently exists, but it would have a back-solved beta of one.

**Proxies Used**

Previous literature has relied on a proxy for *R*
_{m} when estimating the CAPM. The derived one true *R*
_{m}s are compared to those common proxies. Three U.S. and two global indexes, for a total of five proxies, are compared to the 10 sets of market returns derived by the back-solved method that used the 10 empirical portfolios discussed in the data section. The S&P 500 proxy, with a market capitalization of $18.5 trillion, has been used in the majority of CAPM studies and textbooks, which suggest using it for the market proxy. The New York Stock Exchange (NYSE), with a market cap of $16.6 trillion, has been used in CAPM studies (Fama and MacBeth [1973]; Jagannathan and McGrattan [1995]), which concluded that this proxy is efficient for the market portfolio. The Wilshire 5000 is a market-capitalization-weighted index that includes over 3,500 companies that are all headquartered in the United States (over-the-counter, penny, American depositary receipts, limited partnerships, and stocks of extremely small companies are excluded). The index currently includes over 22 trillion dollars of U.S. capital. The number of companies and capital size make it a more expansive U.S. index than the S&P 500 and NYSE.

This research is focused on the developed United States; the study uses U.S. industry portfolios but also examines the developing markets in Southeast Asia. The emerging markets add robustness to the results and are interesting because they have few securities, are fairly illiquid, have a brief historical age, and are emerging market economies relative to the United States.

The two global indexes used are BlackRock Global Opportunities Portfolio (BROAX) and MSCI All Country World Index (ACWI) All Cap. The former fund has over $39 billion in average market capitalization with half invested in giant conglomerates, which themselves are multiasset and globally diversified. The fund averages no more than 15% in any one sector and has 12% in the Asian region, 34% in Europe and Africa/Middle East, and the remaining 52% in the Americas. The ACWI has more than 14,000 securities from 23 developed and 23 emerging countries. This index covers 99% of all worldwide equity investment opportunities with $41 trillion of market capitalization. Of all 10 sectors, the highest weight is in financials at 20%, and the lowest is utilities at 4%. For country weights, the United States makes up 52% of the weighting.

**Two-Sample Kolmogorov–Smirnov Test**

In addition to being compared to the market proxies, the solved one true market returns are also compared to popular financial distributions. The five popular (all-star) distributions used and tested in finance are scaled *t*, Student’s *t*, lognormal, Pareto, and beta. Simulated (random) returns were calculated from the means of 1,000 iterations, following each of the five distributions. Of the numerous CAPM studies, just a handful have used simulation techniques. Lackman [1996], Cho, Elton, and Gruber [1984], and Solnik [1977] also constructed portfolios using simulated data. Amsler and Schmidt [1985] used a Monte Carlo experiment to test how simulated returns work in the CAPM. The summary statistics of the simulated data are available in Exhibits A1 (weekly) and A2 (quarterly). Also in the exhibits are the back-solved market returns for the 10 portfolios that make up the study’s sample. Because of the small sample size in testing the quarterly data (<10,000), the exact P-values are obtained, except in the case of ties. The test follows the same methodology as proposed by Conover [1971].

There is no law that stock returns must fit any specific distribution class, but some are more popular than others. Previous evidence on the distribution of stock returns supports the five all-star distributions selected. Mandelbrot [1963] pioneered the examination of stock returns belonging to a non-normal stable class (referring to the class as *stable Paretian*). Fama [1965] and Blume [1968] found that returns also belong to a non-normal stable class. Lastly, in support of testing the Pareto distribution, Teichmoeller [1971] and Officer [1972] found daily returns to also have some, but not all, of the properties to fit a non-normal stable class.

Calculated returns often have traits of fat tails, high peaks, and skewed distributions, so the *scaled t*, which allows for leptokurtosis, is also selected. Aparicio and Estrada [1997] found strong support for the scaled *t* in six Scandinavian markets that could not be rejected at any significance level. In their study, the scaled *t* easily fit the data better than the logistic, exponential power, a two-model mixture, and normal distributions. Other researchers have also found the scaled *t* to fit stock returns better than competing models (Praetz [1972]; Blattberg and Gonedes [1974]; Gray and French [1990]; Peiró [1994]).

The *lognormal* is used often to describe stock prices, which cannot be negative in value. For this study, it is applied to returns on the one true market portfolio, which in the long run may also be non-negative. Stock indexes are historically upward trending (helped by inflation), and it may be plausible that the market portfolio, which prices the world of assets, may also have non-negative returns as money moves from equities (decreasing returns) to other assets (increasing returns), offsetting any net decreases.

The *beta* distribution, not the same as CAPM’s risk beta, is very popular in the finance literature that estimates recovery rates on fixed income portfolios. With the market portfolio including unobservable factors like human capital, which pays monthly dividends in the form of wages, a beta distribution may fit the returns well.

Lastly, the superstar among the all-stars is the Student’s* t*-distribution. The Student’s *t* is the most commonly referenced distribution in finance literature; its claim to fame comes from the slightly fatter tails that help estimate the probability of fat left-tail losses (consistent with financial data).

**Likelihood Test**

After viewing the results from the KS tests, likelihood tests were performed for robustness of the distribution-ranked results. The Student’s *t* outperformed the other four all-star distributions on all 10 markets based on the likelihood test. The Student’s *t* is then put against three new contenders for the throne. The Student’s *t*-distribution is a normal distribution with fatter tails. What part of this distribution made it so successful (tails or normalness)? To shed light on the matter, the probability of a normal distribution and the probability of the logistic model were tested using the likelihood method. Because of its fat tails, the logistic model was originally suggested by Smith [1981] to model returns. The Cauchy distribution is the third new contender, for a total of eight distributions, vying for top rank. The Cauchy distribution was selected to give the non-normal stable class a chance at redemption after Pareto was found to be the worst-fitting distribution.

**RESULTS**

To prevent data-snooping, visualizations of the distributions were performed after the distributions used in the statistical tests were selected. In Exhibits A9 and A10, the histograms, probability density, and empirical cumulative distribution functions paint the picture of the one true market portfolios. The weekly return distributions, with 521 observations, have tall, thin-fingered peaks. Both the weekly and quarterly density plots for Indonesia and Thailand are right-skewed bimodal. The plots for Philippines, Malaysia, UT, HC, BM, IG, and Real Estate market portfolios are all left skewed. Lastly, Singapore is the closest appearing to a normal (Gaussian) distribution. Modern portfolio theory, in addition to the CAPM and APT, depends on the assumption of normally distributed asset returns, whether it be explicit or implicit.

Exhibit 1 gives support to studies that indicate that the fault of poor empirical results lies with the market proxy and not the CAPM. When using the one true market return, the CAPM is able to correctly predict the average in-sample returns for each of the 10 markets, as seen in the Back *E*
_{r} column in Exhibit 1. The back-solved betas assess the individual portfolios as more risky (larger) than the SL betas. It is logical that the individual portfolios should be riskier than the tangency (market) portfolio of all traded assets in the world. Using the more volatile market indexes for the various countries has produced SL betas that are all <1. Being <1 does not indicate that the portfolios are less risky than the market indexes. On the contrary, they could have higher volatility, but the average of the co-movements is less than the variance of the indexes. However, by incorrectly assigning the individual portfolios with SL betas <1, the expected returns calculated would be less than that of the market returns.

The back-solved betas are much larger than the SL betas (Exhibit 1), and it would make sense that the portfolios are more risky than the real tangency portfolio. Research since the dawn of the CAPM has found that the actual line between average returns and beta should be much flatter than that predicted by the SL model. The incorrect steepness of the line causes predictions for low-beta stocks to be too low and the expected returns of high-beta stock to be too high (Friend and Blume [1970]). This is corrected in the back-solved betas, as seen in Exhibit 2. Exhibit 2 contains graphs of the 10-year actual average returns along with the 10-portfolio traditional SL and back-solved betas. The absolute value of the back-solved beta’s slope of 0.0011 is half that of the SL beta’s slope of 0.0025. Using the back-solved betas also produces an R^{2}, goodness of fit, of 1. The SL betas are using the market proxies for their respective countries, and some of these inefficient proxies had market premiums <0 during the 2005–2014 period. This is the cause for the downward slope, because beta (risk) causes a stronger decrease in portfolio returns. For example, when the markets go down 1%, a beta of 1.5 is expected to return −1.5%. Actual returns are independent of the method used to calculate beta and remain the same in both comparisons. With regard to the betas, the back-solved betas are larger than the SL betas, which stretches the line out flatter along the *x*-axis.

Exhibit A8 shows the back-solved quarterly market portfolio returns (*R*
_{m}) for the 10 tested markets in decimal format. This table gives the reader a look into the behavior of the unmeasurable market portfolio. In 2007, the calculated unseen market returns for real estate and industrial goods were a net loss. By the first quarter of 2008, half of market returns began their downward spiral. This is very interesting to note because the inefficient proxies commonly used did not begin to decline until the fourth quarter of 2008 to early 2009. This timing indicates that the inefficient market proxies (stock indexes) lag up to one year behind their market portfolio that consists of human capital, consumption, production, behavior, social, and all other unseen market price factors that compose the tangency portfolio. This coincides with financial economics in that the often difficult to measure early warning signs begin to reprice assets before stock indexes collapse.

**Proxy Results**

The analysis of the results with the Asian market portfolios considered only the two global indexes. For the U.S. sectors, however, all five indexes were part of the analysis. For brevity, an excerpt of the market proxy’s correlations and *t*-test results with the true market portfolios is available in Exhibits A6 and A7. The five-index proxy’s summary statistics are available in Exhibit A3. The winning proxy based on the averages within each of the two regions was outlined for easier interpretation.

Of the five market proxies tested, the proxy that best matches the CAPM’s market portfolio for U.S. sectors is the Wilshire 5000. For both weekly and quarterly results, the Wilshire 5000 market proxy had substantially better correlations with the one true market returns than did the other two U.S. proxies. The Wilshire 5000 also had *t*-statistics fairly close to those of the two global proxies. The Wilshire 5000 had a correlation with the U.S. sectors of 0.9884 at the weekly data frequency. The best global index, based on correlation, is the ACWI, with a weekly data correlation of 0.9817 with the U.S. sectors and 0.4244 correlation with the Asian market portfolios.

Many asset pricing studies with economic variables used quarterly observations. In these studies both the S&P 500 and the NYSE are not adequate proxies, with correlations with the true market below 0.1. A simulation study by Kandel and Stambaugh [1987] found that the market proxy must be at least 0.7 correlated with the tangency portfolio. Only when this target is met is a rejection of the CAPM not dependent on the proxy used. The Wilshire 5000 was the only U.S. index that met or exceeded the correlation criteria for both the weekly and quarterly observations.

The NYSE was actually found to be detrimental to any study implementing the CAPM. The NYSE was also the smallest proxy used, with a one-fourth smaller market cap than the Wilshire 5000. This proxy had negative correlations with the U.S. market portfolios for both weekly and quarterly returns, and the weekly *t*-test strongly rejected the null hypothesis of equal sample means at the 99% confidence level. Any studies rejecting the CAPM using the NYSE proxy should be re-evaluated. This is opposite to the findings of Fama and MacBeth [1973]. Fama later became an opponent of the CAPM, using the NYSE as his market proxy. A home-team advantage did not exist within the market proxies, meaning the U.S. indexes were not much better than the global indexes for the U.S. market portfolios. As for the Asian markets, the U.S. indexes had results similar to those of the global indexes, sometimes even better. Ultimately, in the end, the two world indexes did well in both regions for both tests at the two data frequencies. This finding indicates that the two world proxies are robust and perform well for a variety of market portfolios.

**All-Star Distribution Results**

The two-sample KS tests used the calculated market portfolio returns (Equation [5]
*R _{m}
*) on the 10 portfolios and analyzed with the means of 1,000 iterations of simulated data for the five all-star financial distributions (Pareto, scaled, Student’s

*t*, lognormal, and beta). The graphs of the distributions of the simulated data are in Exhibits A9 and A10, and summary statistics are given in Exhibits A1 and A2. The weekly returns of the 10 years of data reveal that the scaled and Student’s

*t*-distributions are the all-star distributions most similar to the CAPM’s true market portfolio returns. The Student’s

*t*was about twice as significant as the scaled for the Thailand and Singapore markets. Of the 10 weekly portfolio returns, Singapore and the Student’s

*t*-distribution were from the same distribution at the 5% significance level (results given in Exhibit A4). The calculated

*R*s and simulated distributions were then solved for the quarterly returns (40 observations). With fewer data points, the distribution curves are smoothed out, which allows for better fits in the comparisons of two distributions. Of the five U.S. industry

_{m}*R*s, the Student’s

_{m}*t*was the best-fitting financial distribution with three wins and two ties (with scaled). When using the higher data frequency, the Student’s

*t*was the best fitting for the Asian

*R*

_{m}s; however, using the lower frequency (quarterly) data, the Student’s

*t*has one win and four ties (with scaled). For the complete results, see Exhibit A4.

Using the 521 weekly data observations for the calculated market portfolio returns (*R _{m}
*), the Student’s

*t*-distribution has the highest likelihood of being the distribution, over the seven other distributions. When the data are reduced to 40 quarterly returns, the Student’s

*t*-distribution is still the most likely in five out of ten market portfolios. The excellence of the Student’s

*t*is then divided into fat tails and normalness and tested again using two other distributions.

Comparing the fat-tailed logistic model and normal distribution, the logistic model has the higher likelihood on eight of the ten market portfolios. HC and UT market portfolios were the only two that ranked the normal distribution higher than the logistic, which indicates that the fat tails of the Student’s *t* over normality is what benefited this distribution the most. Unfortunately, yet again, the non-normal stable class, this time represented by the Cauchy distribution, had the lowest likelihood eight out of ten times and never had the highest likelihood. Exhibit A5 presents the results, and the winning distribution is bordered by a box for easier interpretation.

In addition to the 10 back-solved *R _{m}
*s are the two best-performing simulated scaled and Student’s

*t*-distributions, per the KS test, to allow for a cross comparison between the KS and likelihood tests. Both the KS and likelihood tests, using low- and high-frequency data, indicate that when performing tests of the CAPM, the Student’s

*t*is best over the other seven distributions. That is to say, the market (tangency) portfolio follows the Student’s

*t*-distribution the most closely of the eight popular financial distributions tested. The worst performing are the non-normal stable class (Pareto and Cauchy). The QQ-plots, omitted for succinctness, illustrated support for the KS and likelihood test results.

**CONCLUSION AND IMPLICATIONS**

The goal of this study is not to prove that the back-solved beta is superior to SL beta methodology; rather, it seeks to show that, given the CAPM to be absolutely true, the back-solved beta can provide details regarding the market return distributions. Back-solving for beta is sensitive to the user-supplied *R _{f}
* and

*R*benchmarks. The beta produced is dependent on these inputs, although regardless of the beta, the one true market returns (Equation [5]) will still have the exact same number of positive and negative returns in the same proportions, producing the one true market distributions. In this study, the user-supplied inputs were 7% for the annual

_{m}*R*and 1% for the

_{m}*R*

_{f}, and the simulated data were given equal parameters. For forecasting capabilities, the back-solved beta expected return is essentially the historical average method of forecasting. Back-solving beta has allowed this study to back-solve the market distributions needed for CAPM to be 100% accurate in-sample. These distributions were then used in comparison with eight widely used financial distributions and five common market proxies.

The summary statistics, histograms, density plots, and empirical CDFs were also provided to give a peek into the properties of the one true market. Per the QQ-plots, KS tests, and likelihood tests of high- and low-frequency data, the Student’s *t*-distribution best fits the CAPM’s market (tangency) portfolio. The Student’s *t*-distribution has rightly earned its place as the most commonly used distribution in empirical finance research. Future CAPM and tangency portfolio studies should implement tests that are valid given this distribution. The largest U.S. market proxy (Wilshire 5000) was found to be the best match overall for the U.S. market portfolios, and the smallest proxy (NYSE) was found to be an inefficient proxy. Because the more microtailored indexes performed worse for matching the true market portfolios, future research should investigate more world wealth indexes. This research found a bias in using the NYSE as a market proxy for CAPM studies at the weekly and quarterly frequency and the S&P 500 at the quarterly frequency. Finally, the two world indexes were found to be robust and to perform well in both the U.S. and Asian markets, with a slight edge to ACWI.

Practitioners should accordingly model tests based around the Student’s *t*-distribution assumption or some other similar distribution (e.g., using a generalized autoregressive conditional heteroskedasticity model that assumes returns to be fat-tailed, versus the autoregressive conditional heteroskedasticity model, that only assumes the returns to be heteroskedastic). Those investors or risk managers using the CAPM, or even a multifactor model that requires a market premium factor, should consider the larger Wilshire 5000 index (for the United States) or ACWI (for ASEAN) for their market proxy, and certainly should avoid using the NYSE.

**Appendix**

- © 2017 Pageant Media Ltd