WO2014108763A1

WO2014108763A1 - Rfm asset pricing model for investments

Info

Publication number: WO2014108763A1
Application number: PCT/IB2013/051519
Authority: WO
Inventors: Nilanjana NILANJANA CHAKRABORTY
Original assignee: Nilanjana Chakraborty Nilanjana
Priority date: 2013-01-10
Filing date: 2013-02-25
Publication date: 2014-07-17

Abstract

The assets considered here are publicly traded instruments of investment like stocks and their portfolios. It can be shown that asset prices are linear polynomials of four basic factors - asset volumes, index price, index volume and time and hence their returns are rational functions and do not add linearly in a portfolio, especially when the returns are averaged out over multiple time intervals. However, for time series data observed on single time interval basis, returns may be treated as 'approximately' linear and modeled directly through multiple regression using two or more of the four basic variables mentioned above, depending upon the market being studied. As a result, the returns obtained from this Rational Function Model (RFM) are more accurate than those obtained from the existing models.

Description

DESCRIPTION

TITLE : RFM Asset Pricing Model for Investments

A) FIELD OF THE INVENTION

This invention relates to investments in publicly traded assets in general and more specifically to the stock market. For reaping the utmost benefits from their investments the investors should be able to form accurate expectations of the future returns and their risks so as to choose the most efficient asset portfolios that give highest returns for minimum risks. The models invented allow the investors to estimate the asset returns more accurately than the existing linear asset pricing models like Capital Asset Pricing Model (Sharpe, 1964; Lintner, 1965) or its more generalized version called the Ordinary Least Squares Model. The models presented here are based on the Rational Function Model (RFM) theory developed by the inventor which has reduced the discrepancies between the returns estimated by the existing linear asset pricing models and the actually observed returns, across increasing risk for a given time period as well as across increasing risk on a continuous real time basis, to the minimum possible levels.

B) BACKGROUND OF THE INVENTION

For investments in publicly traded assets like stocks, the rational guiding precepts were given by Markowitz (1952) in his modern portfolio theory, mathematically defining risk for the first time as the variance in returns. He further gave the twin objectives of maximizing returns while minimizing risks as the goal of all rational management of such investments. However, the Markowitzian framework of portfolio management assumes that all stock returns add linearly in a portfolio and based on this the Capital Asset Pricing Model (CAPM), given by Sharpe (1964) and Lintner (1965), advocates a linear relationship between the asset returns and the returns of the market as represented by a proxy index. Though the CAPM led to the very important revelation of the connection between the asset returns and the market returns that helped investors in forming more accurate expectations about the asset returns, yet the empirical evidence did not quite match this theory. Subsequently a few other linear asset pricing models like the CAPM were put forth but the CAPM which had arrived the first remained the most popular. In the meanwhile, many studies reported that the actual empirical returns were consistently and systematically found to plot out much flatter than the theoretical CAPM returns across increasing market risk. Academicians and investors alike wondered about this anomaly but its underlying reasons remained unearthed. Hence, the actual asset returns were found to be positively related to the market returns, as stated by the CAPM, but much slower in increasing with increasing risk. As today's finance world becomes more and more sophisticated and demanding due to the huge progress in Telecommunications and Information Technology, it is of utmost importance that we should understand the workings of the asset markets and the theory should match the empirical reality. Accordingly, Chakraborty and Iyer (forthcoming) put forth a new theory that explained the above mentioned discrepancy between the theoretical and the empirical asset returns as being due to the non-linearity of the asset returns when averaged over multiple time intervals and also because of the interplay of other deciding factors like volume and time. This application provides preliminary empirical evidence from the US market similar to the Indian evidence provided by Chakraborty (forthcoming) that the stock returns are rational functions of index price, index volume and time and hence non-linear in their behavior across risk. Further, another model has been developed that could help 'improve' the forecasting of the asset returns, given that the markets are efficient (Fama 1970) and hence the returns cannot be forecasted with perfect certainty.

As already mentioned, the CAPM, in the form of its more generalized version called the Ordinary Least Squares (OLS) model, is the most popular asset pricing model in use today in both halls of learning as well as in the industry and is given by the following equation (1):

R_u = R_f + fii,_m (R_m,t - R_f), i =1, 2, ,N. ...(1)

Here, Rj _t and R_{m t} are the asset and the market returns respectively while ?_;>m is the market risk. However, various empirical studies (Douglas 1968; Friend and Blume 1970; Miller and Scholes 1972; Blume and Friend 1973; Fama and MacBeth 1973; Stambaugh 1982; Fama and French 1992; Fama and French 2004 etc.) reported differences in the theoretical asset returns predicted by the CAPM and the empirical returns that were actually observed. They found that across increasing market risk β_{ί ιη}, the actual asset returns _i plotted out much flatter than the ones estimated through CAPM equation (1). Hence, the actual asset returns are higher than the CAPM returns for lower risk and lower than the CAPM returns for higher risk. Hence, though the actually observed asset returns were found to have a positive relationship with the market premium (R_m,t - Rf ), their chart is much flatter than the CAPM estimates of the returns. The findings of these studies were summarized by Fama and French (2004) who themselves provided very comprehensive empirical evidence based on the monthly returns of US stocks from 1928 to 2003 as shown in Fig I. The Fig I is a rough copy of the actual plot given by Fama and French (2004) whereby the average return of the portfolios are plotted against their post- ranking betas obtained by regressing the portfolio returns on the market returns. The solid line represents the average CAPM returns while the dots represent the average actual returns. Based on the above differences between the CAPM returns and the actual returns, Fama (Fama 1970, Fama and French 2004) put forward the Joint Hypothesis problem that these differences are due to either or both of two possibilities - a) a flawed asset pricing model that does not capture the empirical reality and/or b) an inefficient market that does not abide by the idealistic assumptions of the theoretical model. Further, Fama and French developed a three factor model (Fama and French, 1993, 1996) that utilizes two additional measures besides the index return, viz. the size of the firm and the book to market equity ratio which were found to reduce the discrepancies between the CAPM returns and the actual returns. However, according to Fama and French themselves, the three factor model suffers from the limitation that the two additional factors were included through empirical motivations while the underlying theoretical rationale remains unclear. C) BRIEF DESCRIPTION OF THE DRAWINGS

Fig I is a rough sketch of the plot of Actual returns and CAPM returns across ¾ as reported by Fama & French (2004).

Fig. II is a set of charts showing the results of analyzing the returns of the stock portfolios formed from the thirty stock components of the Dow Jones Industrial Average as on January 1 , 2013, using the new RFM model presented as Equation (3) in this application.

D) DETAILED DESCRIPTION

1) New Theory and Models

A new asset pricing theory and model was put forth by Chakraborty and Iyer (forthcoming), wherein they explained that the above discussed discrepancies between the theoretical and the actual returns were due to the fact that asset returns did not add linearly in a portfolio since they were rational functions (i.e. quotient of two polynomials of the asset prices) based on multiple factors like index price, index volume, asset volume and time. They accordingly gave the following Rational Function Model (RFM) for asset pricing:

Here, p_{m t} and p_{m 1} are the index prices on days 't' and -Γ respectively, while v_m , and v_{m i} are the corresponding index volumes. Similarly, v and v_; are the asset volumes while t_t and t_t.i are the time values based on the chronological ranks of the day 't' and 't- Γ respectively in the whole sample. The variable N is any number greater than 0, such that (1 +10^~N)^X describes the empirical data as best as possible while (1+10^"N)^X gives values low enough to allow meaningful computations.

The above RFM equation basically forms the conceptual foundation on which we may build the final models for practical or empirical applications after studying the actual data specific to a market. For this, Chakraborty (forthcoming) has bifurcated the application of the RFM theory into two broad areas depending upon the time intervals being considered for the analyses and given the following two empirically tailored models: a) Returns across Risk format - This format considers average asset returns (i.e. returns of average asset prices) that have been averaged across the whole of time period of study. These average asset returns are then sorted according to the increasing risk of the assets and studied. This format is useful for plotting the risk-return profile of the assets being studied and thus helps in choosing the most efficient asset portfolios. For this format the asset prices are computed by the following equation:

Pit = a'i + β'η[{(1 + R_m,_t) p_it-i}^aJ + β' α^η '(V_mJJ^b + β' ) + β ₄ [{(off] ■ · -(3)

Here, a'i is the intercept, β'η to β ₄ are the slope coefficients while a, b, c and d are appropriate exponential and base numbers as shown in the above equation. It may be noted that the variable for index price p_m,_t has been replaced by the term [{(1+R_{m t}) pu- ]}^a] as this term is a variant of the OLS model expressed in terms of prices and hence the index price information is contained in the variable R_{m t}. Besides, the term containing the information on the variable for index volume v_m is basically the inverse tangent function of the percentage change in the index volume between two consecutive observations, i.e. Vm.t = [(v_m,t - v_m,t-i) / Vm,t-i]■ This term was used for representing the index volume in the model since inverse tangent function plots similar to logarithmic function but was found to give slightly better results empirically. Finally, the variable for asset volume v_;>i was dropped from the empirical model since its contribution was found to be negligible. b) Returns across Risk-Time format - This format studies continuous asset returns across both increasing risk and time. This format is useful in describing the asset returns across risk and time and study the existing asset data. Since this format uses data on a single time interval basis, the empirical asset returns were found to behave 'approximately' linearly. This may be compared to the scenario whereby the Earth though spherical in shape as visible from space is treated as flat for trigonometrical calculations done for land surveys. Hence, for this format the asset returns were modeled directly through regression of the multiple factors identified by the RFM theory and the empirically suitable model comes out to be as follows:

°_i3 V_m,_t+fi °_;4[tan ' '(V_mJJ^b+fi °_i5(V_it)^c+fi °_ί6(ί +β (¾ · · · (4)

Here, again, a °- is the intercept, β ° to β °y are the slope coefficients while a, b, c, d and e are the exponential and base numbers as shown in the above equation. As explained in the previous point, the terms V_m,_t and are the percentage changes between two consecutive observations of the index volume and the asset volume respectively. It should be pointed out that even though some of the independent variables in equation (4) are themselves functions of other variables, still the ultimate independent variables are those that have been identified as the market drivers in the RFM theory, viz. asset volumes, index price, index volume and time.

Further, in addition to the empirical models (3) and (4) discussed above, this paper has developed another model for the purpose of forecasting of asset returns in order to help investors arrive at more accurate estimates of future asset returns in order to form more realistic expectations. As the markets are efficient and many of the events occurring are unpredictable, hence no perfect forecasting of asset returns is possible. However, on the basis of the historical data, some approximate values may be forecasted and the following model offers an improvement on the existing OLS model, especially when the asset returns are non-linear in their behavior across risk: Ri.t = α , + β"α,,[{1 + ln(p_m,_t/p_m ) + t,} (_Pi,_t-i)^a] ^b + β"α,, [ln(p_m,t/p_m,t-i)] + fi'Wtt) ^C- - -(5)

Here also, α" _;ί is the intercept, fi"n_rt to fi"n__t are the slope coefficients while a, b and c are the exponential numbers as shown in the above equation.

2) Methodology for the Empirical Tests

The main objective of this paper is to provide preliminary empirical results after testing the RFM concepts in the US stock market. A similar study (Chakraborty, forthcoming) has already been carried out for the Indian stock market. Accordingly, a sample of only 30 stocks that constituted the Dow Jones Industrial Average (DJIA) as on January 1 , 2013 was selected for this study. The DJIA is the oldest and one of the most popular stock indices in the US. The daily price and volume data of these stocks were collected from the website of NASDAQ for the last 10 years, i.e. from January 2003 to February 2013. The data on two indices - DJIA and S&P500 were collected from Yahoo! Finance website. About 120 monthly observations were considered from January 31 , 2003 to December 31 , 2012 while 95 daily observations were collected since the last change of components of the DJIA on September 14, 2012 to February 1 , 2013. The collected data were then analyzed for 120 monthly observations of the stocks and the two indices and again for 95 monthly observations starting from February 28, 2005 to December 31 , 2012 for the same set of entities. The analysis of 95 monthly observations was separately carried out so as to compare with the results obtained from the analysis of the 95 daily observations. Accordingly, six sets of stock and index data samples were constructed as listed in Table I and named S I to S6.

The stocks in each of these main samples were first sorted as per increasing risk as measured by the variance of their returns during the last 12 observations and then regrouped into five sub-portfolios PI to P5 of six stocks each, wherein PI consisted of the six stocks of lowest variance while P5 contained six stocks of the highest variance. Besides these 5 sub portfolios, the returns of the full sample portfolio (P-full) consisting of all the 30 stocks were also analyzed and studied. After sorting the stocks, the ranked series of stock prices were reconstructed from the stock returns series for each rank, using some common base number (like 100), so as to avoid sudden sharp changes in prices of these rank-stocks due to changes in stock rankings after each sorting. The above data for the samples S I to S6 were then analyzed using the OLS model (where R_{i t} = a _;- + fi_im R_{m t}) and the equations (3) and (4) for the two different formats based on risk and time as discussed above and their results have been reported in Tables II to VIII. The values for a, b, c, d or e were determined through the method of iterations such that it minimized the errors of the estimates as compared to the actuals. The portfolio returns estimated by the OLS and RFM models were compared with the actual returns and the correlations and sum of squared errors (SSE) between the actual returns and the estimated returns were computed for each sub-portfolio in each sample. The results of the analysis using equation (3) for the Risk Format has been plotted as six charts for the samples S 1 to S6 and given under Figure II. Finally, the forecasted asset returns as per the OLS model and the RFM equation (5) were compared with the actual returns and their Sum of Squared Errors (SSE) were computed and compared to determine the improvement offered by the RFM equation (5). These results have been given in Table IX.

3) Results and Discussion

As already mentioned, the results have been reported in the Tables II to IX. On studying the values of the slopes in the Tables II a) and II b), it can be seen that the values of fi_im for the OLS model increase more steeply than the values of β'η for the RFM equation (3) across the sub portfolios PI to P5 which contain stocks of increasing riskiness. The β'α values appear to have very small ranges between their minimum and maximum values. However the values of βι_Μ are nearly the same as those of β^° _α of equation (4) as listed in Table VI. Further the t-statistics of the slopes indicate that the index price is the most important factor in the RFM equations (3) and (4). However, on comparing the results of OLS and RFM both in terms of correlations of the estimates with the actuals as well as in terms of Sum of Squared Errors between estimates and actuals, it is clear that the RFM results are better and hence it follows that index price by itself cannot provide the important improvements brought in by the other RFM factors.

The correlations between the actual returns and the RFM returns are consistently positive and above 99% for all the samples S 1 to S6 for the equation (3) used for the Risk Format as listed in Table III. On the other hand, the correlations of the OLS returns with the actual returns are even negative showing that the OLS model does not depict the true picture of average asset returns across increasing risk. The slopes of the estimates regressed on the actuals, assuming these to be equal, as reported in Table IV are nearly all 1 for RFM indicating that the RFM estimates are nearly same as the actual average returns. The Sum of Squared Errors of Averages (SSEA) values as given in Table V are also consistently smaller for the RFM equation. The graphs of Fig II also show that the RFM average returns estimated for the Risk format are nearly same as those of the actual returns. This proves the empirical validity of the RFM theory.

For the Risk-Time Format, as can be seen from the Table VII, the correlations between the actual returns and the RFM estimates are again consistently higher than those of the OLS estimates, though marginally so. Similarly, the Sum of Squared Errors (SSE) between the actual returns and the estimated returns for the RFM are all lesser than those of the OLS model as shown in Table VIII. Further, a paired t-test between the SSE values for the OLS model and the SSE values for the RFM equation (4) revealed that the null hypothesis that the OLS estimates are better or equal to the RFM estimates can be very safely rejected for the Risk-Time Format. This indicates that the RFM equation (4) is a definite improvement over the OLS model in estimating and describing the existing asset returns. Hence, even though the asset returns behave approximately linearly across increasing risk for single time interval analyses used in the Risk-Time format, yet the additional independent variables, apart from the index return, that have been considered in equation (4) are important in improving the accuracy of the estimated asset returns.

Finally, besides estimating returns from historical and existing data as outlined in RFM equations (3) and (4), this paper has developed a model for forecasting future returns as given in equation (5). It should be mentioned that given that the markets are efficient, no asset return can be predicted with perfect certainty. However, the portion of the variances in the asset returns that can be logically explained through various independent variables can be safely forecasted. Accordingly, the results of the analysis of the equation (5) as given in Table IX show that the forecasts obtained through the RFM equation are more accurate than those obtained from the OLS model especially if the asset returns are nonlinear over increasing risk and for shorter time-windows. This is so because the percentage improvements of the SSE_RFM values over the SSE_OLS values are higher for the monthly returns, ranging from 1.11% for P-full of S2 to 7.43% for P-full of S4. From the samples of monthly returns, it can be seen that the SSE values of the RFM forecasts are better for the shorter time-windows of 95 trading days for S3 and S4 as compared to those for a longer window of 120 trading days for SI and S2. Similarly, the percentage improvements for the monthly returns are higher than those of daily returns as can be seen from the results for S3 & S4 and S5 & S6. It should also be mentioned that forecasting for daily returns is rather risky and the investors would be well-advised to supplement these forecasts with actual valuations of the daily events and analysis of the fundamentals. Further, it can be seen from the Table IX, that the RFM forecasts are better for more comprehensive indices like the S&P 500 as compared to the DJIA. Lastly, it should be mentioned that the values of a, b and c would have to be ascertained on a case to case basis for different markets and also when considering different time -window sizes.

4) Conclusions

The RFM estimates of the asset returns are more accurate than the OLS estimates across both across Risk as well as Risk-Time formats as indicated by the results from the equations (3) and (4). This shows that asset returns are non-linear in nature and hence do not add linearly in a portfolio. However, for practical purposes these asset returns could be treated as 'approximately' linear over time series data involving single time intervals. All the same, the additional factors apart from the index price, like the stock volume, index volume and time trends that have been identified by the RFM theory should be used for estimating the asset returns for both cross-sectional as well as time-series data. This proves that asset price and asset volume are complementary market forces, since the third basic factor - time is an uncontrollable and passive factor. It is quite possible that the accuracy of the RFM estimates would increase with increase in portfolio size and for assets which exhibit greater nonlinearity in their returns as compared to the market index. Further, since the asset returns are remarkably non linear when analyzed over long time horizons so for such cases the medium risk assets are the most mean-variance efficient investments which have reasonably proportionate increase in returns for the given increase in risks. Finally, the RFM forecasts of asset returns are more accurate than the OLS forecasts for both monthly as well as for daily returns, though the latter should be approached with caution. The RFM forecasts are better for assets with more pronounced non-linearity in their price structures, for shorter time -windows and for more comprehensive market indices. Thus, it seems that the RFM theory, if used judiciously, could help the investors to make more money from the stocks as compared to the existing asset pricing models. REFERENCES

Blume, M.E., Friend, I., "A New Look at the Capital Asset Pricing Model", Journal of Finance 28, 19-33 (1973)

Chakraborty, N., "Stock and Portfolio Returns are Rational Functions: Preliminary Empirical Evidence from India", (Forthcoming), Presented at International Finance Conference 2012 held during December 19-21 , 2012 at Indian Institute of Management - Calcutta, Kolkata, India.

Chakraborty, N., Iyer, P.P., "Price and Volume: Complementary Forces of a Market", (Forthcoming), Presented at the 4^th International Finance and Banking Society (IFABS) Conference held during June 18-20, 2012 at University of Valencia, Valencia, Spain. Douglas, G.W., "Risk in the Equity Markets: An Empirical appraisal of Market Efficiency", University Microfilms, Inc., Ann Arbor, Michigan (1968)

Fama, E.F., "Efficient Capital Markets: A Review of Theory and Empirical Work", Journal of Finance 25, 383-417 (1970)

Fama, E.F., French, K.R., "The Cross-Section of Expected Stock Returns", Journal of Finance 47, 427-65 (1992)

Fama, E.F., French, K.R., "Common Risk Factors in the Returns on Stocks and Bonds", Journal of Financial Economics. 33, 3-56 (1993)

Fama, E.F., French, K.R., "Multifactor Explanations of Asset Pricing Anomalies", Journal of Finance. 51 , 55-84 (1996)

Fama, E.F., French, K.R., "The Capital Asset Pricing Model: Theory and Evidence", The Journal of Economic Perspectives 18, 25-46 (2004)

Fama, E.F., MacBeth, J.D., "Risk, Return, and Equilibrium: Empirical Tests", Journal of Political Economy 81 , 607-36 (1973)

Friend, I., Blume, M.E., "Measurement of Portfolio Performance under Uncertainty", American Economic Review 60, 607-36 (1970)

Lintner, J.V., "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets", Review of Economics and Statistics 47, 13-37 (1965)

Markowitz, H.M., "Portfolio Selection", Journal of Finance 7, 77-99 (1952)

Miller, M.H., Scholes, M.S., "Rates of Return in Relation to Risk: A Reexamination of Some Recent Findings", In Jensen, M.C. (Ed.), Studies in the Theory of Capital Markets, Praeger, New York (1972)

Sharpe, W.F., "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk", Journal of Finance 19, 425-42 (1964)

Stambaugh, R.F., "On the Exclusion of Assets from Tests of the Two-Parameter Model: A Sensitivity Analysis", Journal of Financial Economics 10, 237-68 (1982)

Claims

CLAIMS What is claimed is:

1. A method, as embodied in Equation (3), for more accurately estimating (as compared to existing linear asset pricing models) asset or stock returns, when averaged over a given time period, across increasing risk, based on existing and historical data so that the results will enable the investors to choose the assets that give the highest returns for a given level of risk and to compare the assets across the different risk levels to decide if the increase in returns are proportionate to the increase in risk.

2. The method of Claim 1, modified as shown in Equation (4), for accurately estimating asset or stock returns, on a continuous time basis for a given time period, across increasing risk, based on existing and historical data so that the results will enable the investors to understand the behavior of the asset returns across risk and time and accordingly identify the relevant variables for monitoring to make more informed buy-sell decisions for their chosen assets of investments.

3. The method of Claim 1, modified as shown in Equation (5), to provide a thumb-rule guide for more accurately forecasting future asset or stock returns for the next observation, across increasing risk, based on existing and historical data so that the results will enable the investors to form more accurate expectations of the risk and returns of an asset and thus make more profitable choices of assets and more profitable buy-sell decisions for their chosen assets.