Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Rational Functions: A New Asset Pricing Model
Nilanjana Chakraborty & Mohammed M Elgammala,b1
This paper develops a theoretical framework according to which the stock return is
found to be a rational function of two consecutive observations of nine independent
variables based on six parameters – stock volume, index price, index volume, past
performance, other market factors and time. The returns generated by this Rational
Function (RF) model increase but tend to flatten out for increasing risk, similar to the
extant empirical evidence which had remained unexplained until now. Thus, this paper
indicates that the stock returns don't add linearly in a portfolio since returns are
rational functions and we must hence model prices instead of returns when using
linear regression techniques.
Keywords: Asset Pricing, Returns; Volumes; Rational Functions; RF Model
1. Introduction
The widespread asset pricing models (e. g CAPM and FF3FM) are based on a basic
fundamental assumption that is stock returns add linearly in a portfolio. Therefore, the
models estimate stock returns by regressing them linearly against index returns and
sometimes against other additional market and accounting parameters. The origins of
this assumption can be traced to the modern portfolio theory of Markowitz (1952).
Based on this theory, Sharpe (1964) and Lintner (1965) introduced the Capital Asset
Pricing Model (CAPM) which suggests a linear relationship between the asset returns
and the returns of the market portfolio. However, numerous empirical studies in the last
sixty years introduce evidence against the CAPM’s ability to estimate the actual asset
returns (e.g. 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; Dempsey 2013).
The apparent demise of the CAPM led to the design of several factor models to
explain the behavior of asset returns. This work largely began with Fama and French
(1992) who proposed a three-factor model, augmenting the market beta with factors
relating to size and value. These premia were argued to capture systematic risk that the
market portfolio could not. However, according to Fama and French (2004) themselves,
these models are based on variables that have been included chiefly through empirical
motivation, the theoretical rationales for which have not been probed in depth. In fact
1
Corresponding Author: M. Elgammal, Email: m.elgammal@qu.edu.qa, Tel: +974 77190503, Fax: +974
44035081.
Affiliations:
A
Department of Finance and Economics, College of Business and Economics, Qatar University (Duhil,
Doha, 2713, Qatar).
b
Faculty of Commerce, Menoufia University, Egypt
1
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
there is a controversial debate on whether these factors capture rational risk or
behavioral phenomena.
This paper explains the discrepancies between the extant theoretical estimations and the
actual values of the asset returns by questioning the basic assumption behind the existing asset
pricing models. Do the stock returns really add linearly in a portfolio? Are the asset returns
linearly related to the index returns? Are there any other factors that should be considered in
modeling asset returns? This paper addresses these questions by studying the most basic asset
and market variables. First, it revisits the CAPM and discusses the findings of the extant
empirical literature based on this model. Next, it initiates the theoretical re-examination of the
stock market behavior by considering an individual security in isolation and then moving on to
incorporate the whole market portfolio and time trends into the analysis. This paper launches a
new Rational Function (RF) model which argues that assets returns don‟t add linearly in a
portfolio. This means that we should not model asset returns when using linear regression
techniques but rather model the prices. Our RF model also indicates that we must consider
variables other than the index price, which is the most widely used single variable, and include
variables like preceding asset price, asset volume, index volume and time trends to model asset
prices.
2. Literature Review
According to the Sharpe-Lintner CAPM equation, the relationship between the expected
return of an asset „i‟, denoted as E(Ri), and its coefficient for market premium βi,m is linear and
given by
E(Ri) = Rf + βi,m [E(Rm) – Rf ],
(1)
The asset „i‟ could represent either an individual stock or a portfolio, while Rf and E(Rm)
denote the risk-free rate of return and the expected market return respectively. However, over
the years various empirical studies based on the CAPM have consistently found that the
intercept is greater than the average risk free rate Rf and the coefficient on beta is less than
the average excess market return. This led to the articulation of the now academically
prominent Joint Hypothesis Problem by Fama in 1970, whereby he attributed the
discrepancies between the actual asset returns and the CAPM returns to either or both of the
following reasons: a) flawed asset pricing model and b) inefficient markets that do not adhere
to the idealistic assumptions used in the theory development. Thus, though there happens to
be a positive relationship between actual values of Ri and βi,m, it is found to be too flat as
compared to the relationship predicted by the CAPM equation. The findings of these studies
have been summarized in Fama and French (2004) and the Figure 1 shown below depicts a
rough sketch of the empirical evidence reported in these studies. In the Figure 1, the solid
line represents the average portfolio returns predicted by the CAPM equation while the dots
represent the actual average portfolio returns observed.
2
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Average Portfolio Returns
(%)
•
••
•
••
•
•
•
•
•
Average CAPM Returns (%)
Betas
Figure 1: Plot of average Actual and CAPM Returns against Betas (Fama and French 2004).
Hence, according to the joint hypothesis problem, the CAPM equation does not fully
capture the empirical reality though it very well points in the right direction, while the other
conclusion could be that the capital markets are inefficient and hence do not exactly fit into the
theoretically accurate models.
Nevertheless, a significant part of the literature believes that the CAPM is an ex ante
concept, whereas the so-called „tests‟ of the CAPM are conducted ex post. Roll (1977)
concludes that the so-called „tests‟ of the CAPM were invalid because they used inefficient
benchmark portfolios, whereas a valid test of the CAPM requires that the benchmark be efficient.
Many finance practitioners and researchers believe that ex ante risk matters and thus an ex ante
risk premium exists, even if ex post their belief defies empirical confirmation (see Roll and Ross
(1994), Kandel and Stambaugh (1995), Jagannathan and Wang (1996), Feldman (2007) and
Diacogiannis and Feldman (2011). At various times beta has been declared dead (Fama and
French, 1992), yet as the CAPM is proving extremely difficult to test and remains unverified, we
hold onto the concept that unavoidable investment risk is priced because a plausible alternative
has not been found.
Fama and French (1993) provided empirical evidence that a single factor encapsulating risk
does not adequately explain cross-sectional differences in stock returns and this finding
motivated a lot of research on asset pricing, reigniting the debate on the fundamental relationship
between risk and return, and challenging the widely-accepted CAPM. Fama and French (1993,
1996, 2004) introduce a three factor model including size premium (the difference between the
returns of the diversified portfolios of small and big stocks sorted on the basis of their sizes) and
the value premium (the difference between the returns of the diversified portfolios of high and
low B/M stocks), to explain variations in stock returns. However, according to Fama and French
themselves, the Fama French 3 Factor (FF3F) model suffers from the limitation that its two
additional factors were included mainly through empirical evidence while their underlying
rationales remain unrevealed. However, the FF3F model has motivated a lot of studies
3
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
investigating the explanation for these two variables (some recent studies include Dempsey,
2013; Bod‟a and Kanderova, 2014; Elgammal and McMillan 2014). The theoretical and
empirical studies examining the cross-sectional variation in stock returns found patterns
unexplained by the CAPM that are commonly known as anomalies. These studies on anomalies
led to various conclusions about the differences between the actual empirical evidence and the
theoretical estimates obtained by the CAPM and the FF3F.
Accordingly, based on these anomalies, various investment styles have been developed. The
four major investment styles are of size (Banz 1981, Fama and French 1993), value/growth
(Basu 1977; Fama and French 1992, 1993), momentum (Jegadeesh and Titman 1993, 2001) and
equity liquidity (Haugen and Baker, 1996; Datar, Naik, and Radcliffe, 1998). However, these
models cannot provide a robust economic explanation for their additional risk factors.
Another alternative for the CAPM is the consumption CAPM (CCAPM) which is a
theoretically motivated model which substitutes the CAPM market factor with consumption
growth. Therefore, the asset‟s risk can be defined as the covariance between the asset return and
consumption growth. According to the model, an asset with a positive covariance with
consumption (i.e., performing poorly in recessionary periods) is less desirable than an asset
which has a negative covariance with the consumption (i.e., performing poorly in expansionary
periods). Therefore, the first type of asset will required a higher return compared to the second
type of asset. Avramov (2008) argues that theoretically, the CCAPM is better than the traditional
CAPM as it considers the dynamic nature of portfolio decisions and incorporates other forms of
wealth in additional to the financial asset. Notably, consumption should convey a better measure
of economic states as investors consume less if they think future income will be low. However,
empirically the CCAPM cannot outperform the CAPM (Avramov, 2008).
In spite of the numerous factor models developed in the extant financial economics literature,
we have provided an empirical testing of the RF model in section 6, by comparing its
performance with that of only two models, which are CAPM and the FF3F, as these models are
the most popular and widely used today.
3. Asset Price in Equilibrium
We start the theoretical development of the RF model by revisiting the basics of price
determination of a publicly traded product in a free market environment. The supply and demand
functions control the pricing of an asset through the quantity, „volumes‟, being traded. In general,
the supply curve represents the quantities of an asset that the suppliers or sellers are willing to
sell at different prices while the demand curve represents the quantities of the asset that the
consumers or buyers are willing to buy at different prices ceteris paribus.
4
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Supply Curve
Price
Demand Curve
Equilibrium Price P*
Equilibrium Volume V*
Volume
Figure 2: The Supply and Demand Curves for a security
We mathematically express the exponential supply curve of a stock „i‟ on a day„t‟ as follows:
ps,t = as + bs [exp (vs,t )]
(2)
where, ps,t is the price corresponding to the volume vs,t, as is the minimum price at which the
supply curve starts and bs is the linear regression coefficient. The logarithmic demand curve for
stock „i‟ on day„t‟ may be defined as follows:
pd,t = ad - bd [ln (vd,t )]
(3)
where, pd,t is the price corresponding to the volume vd,t, ad is the minimum price at which the
demand curve starts and bd is the linear regression coefficient. At the point of market equilibrium,
equations (2) and (3) intersect each other, then:
ps,t = pd,t = pi,t ; and vs,t = vd,t = vi,t
(4)
Traditionally, the equations (2), (3) and (4) have been used to compute the equilibrium price
pi,t and equilibrium volume vi,t. However, it would be more interesting to have a mathematical
relationship between the equilibrium price and volume of the asset. So, taking an average of the
equations (2) and (3) at the point of equilibrium and substituting equation (4) lead us to equation
(5):
pi,t = (as+ ad) / 2 + (bs /2)[exp (vi,t )] – (bd /2)[ln (vi,t )]
(5)
Equation (5) could be further generalized by representing the constants and the slope
coefficients by single terms, as follows:
pi,t = ai + bi1 [exp (vi,t )] – bi2 [ln (vi,t )]
(6)
5
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Here, „ai‟ is a constant, while „bi1‟ and „bi2‟ are the coefficients of the exponential and
logarithmic values of the stock volume. This equation, thus, expresses the basic relationship
between the equilibrium price and the equilibrium volume of a stock „i‟, by taking both the
supply and demand forces into consideration.
4. Considering the Market Portfolio and Other Factors
In the above section, we have considered the specialized case of only a single asset being
traded in the market. However, in reality there are numerous stocks being traded simultaneously
in the market. Some of these stocks may influence the demand and/or supply of each other due to
various common economic, business or technological factors connecting them. In such a case,
both the prices and volumes of these stocks may be correlated amongst themselves. Therefore,
we expand our analysis by including the market portfolio pm,t the price of which on a day „t‟ is
given by:
pm,t = ∑ qj pj,t
w
(7)
j=1
where, j = 1 to w, including „i‟, thus representing all the stocks contained in combination
„m‟, while „qj‟ denotes the corresponding weight of the stock „j‟ in combination „m‟. Thus,
w
∑ qj = 1
(8)
j=1
From equation (7), it can be seen that pm,t is a linear aggregate of pi,t and other stock prices.
However, the prices of some of the stocks may be correlated with that of stock „i‟ and then pi,t
would vary in response to a change in the price of any one of these stocks. Thus, by substituting
the linear relationships between various stock prices with pi,t in equation (7), an exclusive linear
relationship between pi,t and pm,t, emerges as follows:
pi,t = ci + di pm,t
Stock
Price
(pi,t)
(9)
pi,t = ci + di pm,t
Stock
Volume
(vi,t)
vi,t = ei + fi vm,t
Angle = tan-1(d)
Intercept (c)
Angle = tan-1( f )
Intercept (e)
Market Price (pm,t)
6
Market Volume (vm,t)
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Figure 3 (a): Price Relationship
Figure 3 (b): Volume Relationship
Here again, „ci‟ is the intercept, while „di‟ is the slope as shown in Figure 3 (a). The equation
(9) indicates the joint movement of pi,t and pm,t when we consider only linear associations
between the various stock prices through regressions.
Equation (9) thus indicates that the price of a stock „i‟ is linearly related with the price of the
market combination „m‟ and varies in direct proportion with it. Similarly, the volume of the
market combination „m‟ is also an aggregate of the individual stock volumes, each being
considered according to its weight. Hence, the volume of the combination „m‟ on a day „t‟ is
given by:
w
vm,t = ∑ qj vj,t
(10)
j=1
Again, from equation (10) it can further be shown that the volume of stock „i‟ is also linearly
related with the volume of the market combination „m‟ and varies in direct proportion with it as:
vi,t = ei + fi vm,t
(11)
As can be seen, in equation (11), „ei‟ is the intercept and „fi‟ is the slope as shown in Figure 3
(b). Moving to the relationship between the price and volume for the market combination by
itself, one can see that the price pm,t and volume vm,t of the market combination „m‟ are respective
weighted averages of the prices and the volumes of all the stocks constituting the combination
„m‟. Taking the weighted average of the stock prices from equation (6), we get:
w
w
w
w
j=1
j=1
j=1
j=1
∑ qj pj,t = [∑qj aj + ∑ qj bj1 {exp (vj,t )} – ∑ qj bj2 {ln (vj,t )}]
(12)
Substituting equations (7) and (11) in equation (12) after replacing „i‟ with „j‟, we get,
w
w
w
pm,t = [∑qj aj + ∑qj bj1 exp{( ej+ fj vm,t)} – ∑qj bj2 ln{( ej+ fj vm,t)}]
j=1
j=1
(13)
j=1
On the basis of the various mathematical approximations, the equation (13) can be rewritten
as:
w
w
w
pm,t = ∑qj aj + ∑qj bj1{kej+mej exp(vm,t)} – ∑qj bj2{klj+mlj ln(vm,t)}
j=1
j=1
(14)
j=1
On clubbing the constants together and adding the slope coefficients of the above equation,
we get:
pm,t = om + um1 exp(vm,t ) – um2 ln(vm,t )
(15)
7
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
As apparent, om is a constant while um1 and um2 are slope coefficients in the above equation.
Thus, equation (15) indicates that, under equilibrium conditions, the daily price and volume of
the market combination „m‟ are also linked similar to those of the individual stocks, even though
pm,t and vm,t are weighted averages. Next, in order to find the relationship between the price of the
individual stock and the volume of the market combination, we substitute equation (11) in
equation (6) to get,
pi,t = ai + bi1 exp( ei + fi vm,t) – bi2 ln( ei + fi vm,t)
(16)
Again, substituting mathematical approximations in equation (16) and then generalizing, we
get,
pi,t = gi + hi1 exp(vm,t ) – hi2 ln(vm,t )
(17)
Equation (17) outlines the relationship between the price of an individual stock „i‟ and the
volume of the market combination „m‟ under economic equilibrium between prices and volumes
in the market. Besides the market portfolio, the asset prices may be shaped by other market
factors, like firm size, book to market equity ratios etc. (Fama and French, 1993, 1996),
depending upon the socio-economic makeup of the investor mindsets of a particular market.
Furthermore, Jegadeesh and Titman (1993) document the momentum effect that is the asset
prices are influenced by their own past performance. Moreover, a great part of literature reports
time and seasonal trends in the movement of stock prices and trading volume (for example
Gallant et al. 1992, Chen et al. 2001, Pisedtasalasai and Gunasekarage 2007). There are many
explanations for such trend including the seasonality of the sales or the business or the regularity
of certain anticipated events, like earnings or dividend announcements, planned on the basis of
financial calendars. Accordingly, the linear and nonlinear (quadratic) time trend components in
the price of a stock „i‟ may be expressed as the log function of the chronological order of the
observation within the data sample and its square. Incorporating market factors, momentum, and
time trend in Equations (6), (10), (17), and then generalizing the resulting equation to get the
factors defining the stock price pi,t as follows:
pi,t = αi+βi1pm,t+βi2exp(vi,t)+βi3ln(vi,t)+βi4exp(vm,t)+βi5ln(vm,t)+βi6ln(tt)+βi7{ln(tt)}2+βi8MFi,t+βi9pi,t-1
(18)
Where MFi,t reflects the influence of other market factors, pi,t-1 is the preceding asset price.
ln(tt) and {ln(tt)}2 are linear and quadratic time-trend components respectively. In order to determine
the intercept and slope coefficients of equation (18), the values for „tt‟ are taken to be positive
integers starting at 1 and extending upto the total number of observations being considered for
the analysis. We have considered ln(tt) instead of plain tt values because the logarithmic values
have a more uniform distribution and their successive changes are more in proportion with the
successive changes in stock prices.
The above equation gives the best possible linear fit for the price of stock „i‟ using the basic
firm, market and time parameters. Nevertheless, for empirical analysis, the values for exp(vi,t)
and exp(vm,t) might be very large for large values of vi,t and vm,t and hence computationally
intractable. This can be resolved if we change the base of the exponential function from e (=
2.7183) to (1+10-N) where N is any number greater than 0, such that (1+10-N)x describes the
8
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
empirical data as best as possible while the values of (1+10-N)x are low enough to allow
meaningful computations. Thus, we may rewrite equation (18) as,
vi,t
vm,t
pi,t=αi+βi1pm,t+βi2(1+10-N) +βi3ln(vi,t)+βi4(1+10-N) +βi5ln(vm,t)+βi6ln(tt)+βi7{ln(tt)}2+βi8MFi,t+βi9pi,t-1
(19)
Similarly, it can be shown that the stock volume vi,t is given by,
pi,t
pm,t
vi,t=γi+δi1vm,t+δi2ln(pi,t)+δi3(1+10-N) +δi4ln(pm,t)+δi5(1+10-N) +δi6ln(tt)+δi7{ln(tt)}2+δi8MFi,t+δi9pi,t-1
(20)
From Equation (19) we may infer that the stock return Ri,t is not a linear polynomial function of
the return of the market combination Rm,t but rather a quotient of two polynomials. This follows
simply from the definition of return which requires it to be a ratio of the change in price on a day
„t‟ to the price on day „t-1‟. Thus, we have,
vi,t
vm,t
[{αi+βi1pm,t+βi2(1+10-N) +βi3ln(vi,t)+βi4(1+10-N) +βi5ln(vm,t)+ βi6ln(tt)+βi7{ln(tt)}2+ βi8MFi,t+ βi9pi,t-1} –
vi,t-1
vm,t-1
{αi+βi1pm,t-1+βi2(1+10-N)
+βi3ln(vi,t-1)+βi4(1+10-N)
+βi5ln(vm,t-1)+βi6ln(tt-1)+βi7{ln(tt-1)}2+βi8MFi,t-1+βi9pi,t-2}]
Ri,t = ___________________________________________________________________________________________________________
(21)
vi,t-1
vm,t-1
[αi+βi1pm,t-1+βi2(1+10 )
+βi3ln(vi,t-1)+βi4(1+10-N)
+βi5ln(vm,t-1)+βi6ln(tt-1)+βi7{ln(tt-1)}2+βi8MFi,t-1+βi9pi,t-2]
-N
Thus, from equation (21), we can see that the actual relationship between Ri,t and Rm,t is not
linear, as suggested by the CAPM equation, but nonlinear since Ri,t happens to be a rational
function of two consecutive readings of the stock volume, the index price and volume of the
market combination „m‟, the relevant market factors, preceding asset prices and the time trend.
To corroborate the theory developed above, a numerical example is being provided in the next
section along with graphs to clarify the difference between the plotting of a linear function as
compared to a rational function on the risk-return plane. However, before we proceed to plotting
graphs of the stock returns obtained by the Rational Function (RF) model as outlined in equation
(21), it would be necessary to develop a single measure of the inherent risk of the stock against
which the returns could be plotted. For this we would require a common coefficient of risk for all
the factors defining the stock price given in equation (19). Accordingly, we may rewrite the
equation (21) as follows:
vi,t
vm,t
pi,t = αi + βi,t [ pm,t+(1+10-N) +ln(vi,t)+(1+10-N) +ln(vm,t)+ln(tt)+{ln(tt)}2+MFi,t+pi,t-1]
(22)
Substituting simple notations in „x‟ for the independent variables in equation (27), we get,
9
pi,t = αi + βi,t (∑xi,n,t )
(23)
n=1
vi,t
vm,t
where, xi,1,t=pm,t; xi,2,t=(1+10-N) ; xi,3,t=ln(vi,t); xi,4,t=(1+10-N) ; xi,5,t=ln(vm,t); xi,6,t= ln(tt)
xi,7,t= {ln(tt)}2; xi,8,t=MFi,t; and xi,9,t=pm,t-1;
(24)
On comparing the equations (19) and (23), we get,
9
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
9
9
βi,t = (∑ βi,n x i,n,t ) / (∑ x i,n,t )
n=1
(25)
n=1
Again, substituting equation (23) in equation (21), we get the following for Ri,t:
9
9
9
n=1
n=1
Ri,t = { βi,t ( ∑ x i,n,t ) - βi,t-1 ( ∑ x i,n,t-1 )} / { αi + βi,t-1 ( ∑ x i,n,t-1 )}
n=1
(26)
On comparing equations (24), (25) and (26) and solving for βi, we get,
9
9
9
n=1
n=1
9
9
9
βi = αi∑βi,n(xi,n,t-xi,n,t-1)/[αi∑(xi,n,t-xi,n,t-1)-{(∑βi,nxi,n,t∑xi,n,t-1)-(∑βi,nxi,n,t-1∑xi,n,t)}]
n=1
n=1
n=1
(27)
n=1
The equations (26) and (27), in conjunction with the substitution-set (24), provide the return
and risk of stock „i‟ respectively.
5. Illustrating with hypothetical examples
According to Sharpe (1964) the daily rate of stock return Ri,t is given by:
Ri,t = [{(pi,t + Di ) – pi,t-1} / pi,t-1 ]
(28)
where, Di is the ex-dividend paid on the day „t‟. Even so, many economists and financial
analysts define Ri,t to be a plain ratio of the price of the stock „i‟ on day „t‟ to that on day„t-1‟,
when using it with the CAPM or other regression models. This works out fine with the linear
regression models because the difference gets adjusted in the constant term. This can be
explained by substituting equation (28) in equation (1) as follows:
[{(pi,t + Di ) – pi,t-1} / pi,t-1 ]
= Rf + βi,m [{(pm,t – pm,t-1 ) / pm,t-1 } – Rf]
(29)
The equation (29) can be simplified by clubbing the constants together as follows,
[(pi,t + Di ) / pi,t-1 ] = (1+Rf )(1- βi,m ) + βi,m (pm,t / pm,t-1 )
(30)
The equation (30) is the more popular version of the CAPM found in the literature because of
its simplicity and easier computational attributes. However, for our illustrations, we shall
consider a broader definition of the relationship between stock returns and market returns as
given by the Ordinary Least Square (OLS) model, which is a generalization of the CAPM and is
given as follows,
[{(pi,t + Di ) / pi,t-1} - 1] = κ i + βi,m {(pm,t / pm,t-1 ) - 1}
(31)
Further, for comparing the OLS model with the RF model using hypothetical data, βi,1 of the
latter is taken to be equal to βi,m of the former for the sake of the numerical illustrations presented
next. Having thus clarified the definition of return and the logic used for assigning values to the
coefficients βi,m and βi,1 when comparing the behavior of the RF model with that of the OLS
equation, we now provide numerical illustrations in the next section across two formats – returns
across increasing risk and returns across increasing time.
10
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
5.1 Illustrations for behavior of Returns across Increasing Risk: For illustrating the
behavior of returns across risk, we shall look at two cases: pm,t > pm,t-1; and pm,t < pm,t-1.
5.1.1: pm,t > pm,t-1: We now compute the values of the stock return Ri,t by using both OLS
model, as explained in equation (31), as well as the Rational Function (RF) model as given in
equations (17) and (24). Assuming pm,t = 2300, pm,t-1 = 2000 (thus considering a 15% increase in
index price in one day) and αi = 100. Next, for n = 2 to 9 let ∑βi,nxi,n,t-1 = 409.9, ∑βi,nxi,n,t = 450,
∑xi,n,t-1 = 51.95, ∑xi,n,t = 52.15 and finally let the constant κ i = 0.02. Further, we consider a data
series of fifteen βi,m and βi,1 values, starting at 0.2 and increasing by 0.1. Then, we get the
different values for Ri,t(OLS), βi and Ri,t(RF), as listed in Table (1) and plotted in Figure 4, through
the equations (31), (27) and (26) respectively. Graph (3) indicates that the RF model for stock
returns seems to provide a better description of the empirical evidence on the basis of the theory
discussed in the previous sections. We can see that Ri,t(RF) values are higher than OLS values for
lower betas and lower than OLS values for higher betas, thus producing the „flattening‟ effect. It
should further be noted that increasing values of market risk βi,m, as used in OLS, results in a
gradual increase in values of the overall price risk βi of the RF model as well. Thus, this
numerical illustration indicates the possibility that when Fama and French (2004) identified the
portfolios for increasing market risk βi,m, they obtained portfolios with increasing overall risk βi
as shown here, and the actual returns of those portfolios gradually flattened out just like the plot
shown here, since the portfolio returns are rational functions.
βi,m
βi
Ri,t (RFM)
Ri,t (OLS)
(OLS)
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
5.00%
6.50%
8.00%
9.50%
11.00%
12.50%
14.00%
15.50%
17.00%
18.50%
20.00%
21.50%
23.00%
24.50%
26.00%
0.15
0.20
0.25
0.30
0.36
0.42
0.48
0.54
0.61
0.68
0.76
0.84
0.93
1.02
1.12
Stock Returns
Returns
(RFM)
11.00%
11.72%
12.22%
12.59%
12.87%
13.09%
13.28%
13.42%
13.55%
13.66%
13.75%
13.83%
13.90%
13.96%
14.02%
30.00%
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
β i,m
Table (1): Values of returns Ri,t(OLS)
and Ri,t (RFM) for given βi,m values for
pm,t > pm,t-1.
Figure (4): Graph plots for Ri,t (OLS) and Ri,t (RFM)
11
values for pm,t > pm,t-1 , where triangles show
CAPM and the squares show the RF model values.
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
5.1.2 pm,t < pm,t-1: in the case of decreasing returns, we continue with the same values for
different variables in the above example, except that here pm,t = 1700 while pm,t-1 = 2000, thus
considering a 15% decrease in index price in one day. Then, again, for βi,m values starting at 0.2
and increasing by 0.1, we get the returns as listed in Table (2) and plotted in Figure (5). From
the graph it is obvious that for negative market returns, the stock returns decrease linearly
according to the OLS model. In comparison, the RF model shows that stock returns decrease
much slower thus giving a flattening effect and finally plateau off after the βi values reach a
certain level.
βi,m
Ri,t (OLS)
(OLS)
βi
Ri,t (RFM)
Stock Returns
Returns
(RFM)
0.20
0.30
0.40
0.50
0.60
-1.00%
-2.50%
-4.00%
-5.50%
-7.00%
0.01
0.02
0.03
0.05
0.06
-2.19%
-4.50%
-6.10%
-7.28%
-8.18%
0.70
-8.50%
0.08
-8.90%
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
-10.00%
-11.50%
-13.00%
-14.50%
-16.00%
-17.50%
-19.00%
-20.50%
-22.00%
0.09
0.10
0.12
0.13
0.15
0.16
0.18
0.19
0.21
-9.47%
-9.95%
-10.35%
-10.70%
-10.99%
-11.25%
-11.48%
-11.68%
-11.86%
0.00%
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
-5.00%
-10.00%
-15.00%
-20.00%
-25.00%
β i,m
Table (2): Values of returns Ri,t(OLS)
Figure (5): Graph plots for Ri,t
and Ri,t (RFM) for given βi,m values for
values for pm,t < pm,t-1., where triangles show OLS
pm,t < pm,t-1..
and the squares show the RF model values.
(OLS)
and Ri,t
(RFM)
The previous graphs show that the RF model generates stock returns that tend to flatten out
towards the extremities of the beta axis However, the first case is the most practically applicable
case on the long run because over long periods of time, under equilibrium conditions, we find the
index values growing instead of shrinking, barring some exceptional incidents when the market
loses its equilibrium.
12
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
5.2 Illustrations for behavior of Returns across Increasing Time: We turn to compare the
behavior of the stock returns across increasing time as obtained from both linear and rational
functions for two cases of different proportions of linear and non linear components in the stock
price, i.e. low proportion of nonlinear component and high proportion of nonlinear component.
5.2.1 Low proportion of nonlinear component: Assume two data series, X and Y
representing the price sentiments and volume sentiments respectively of the market. Let there be
a third data series Z, comprising of the chronological rank of the data in the sample, representing
a time trend but which is an unknown factor for the estimation purpose.
Then, let the index price pm,t = 29+1.7 X, the index volume vm,t = ln(Y70), the time trend tt
= {ln(Z)}7 and the actual stock price is given by the pi,t = 0.003pm,t+0.002vm,t+0.0007tt+2 as
presented in Table (3). It is now obvious from the preceding description that pm,t is a linear
whereas vm,t and tt are nonlinear variables. However, since the time trend tt is an unknown
factor, the price is estimated by regressing pi,t on only two variables, pm,t and vm,t to get
(pi,t)RF-2 = a + b1 pm,t + b2 vm,t for computing the stock returns for the RF model. The returns
for OLS are computed by regressing actual stock returns on the actual market returns and then
using the intercept and the slope values with the actual market returns. In addition to the RF-2
values obtained from the two factor equation, a single factor version of the RF model has also
been used for drawing comparisons, called the RF-1 equation, according to which (pi,t)RF-1 =
ln(c + d1 pm,t), where c and d1 are obtained by regressing pi,t on only pm,t. After computing the
various returns using the OLS and the two RF model equations from the data as tabulated under
Table (3), they were further regressed on the actual returns, as obtained from pi,t, in order to
determine the extent of accuracy of the estimated returns. Table (4) summarizes the statistics of
results of fitting the estimated returns to actual returns:
13
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
t
X
Y
1
2000.00
5400.00
2
2090.00
5572.80
3
2236.30
4
2420.35
5
Z
tt
pi,t
(Case a)
(pi,t)RF-2 (pi,t)RF-1
Ri,t (Ri,t)RF-2 (Ri,t)OLS
pi,t (pi,t)RF-2 (pi,t)RF-1
Ri,t
(Ri,t)RF-2 (Ri,t)OLS
(Case a) (Case a) (Case a) (Case a) (Case a) (Case b) (Case b) (Case b) (Case b) (Case b) (Case b)
pm,t
vm,t
1
3429.00
601.59
0.00
13.49
12.93
2.57
794.36
793.79
6.68
2
3582.00
603.80
0.08
13.95
13.70
2.63
3.44%
5.97%
3.97%
797.68
797.43
6.68
5829.15
3
3830.71
606.94
1.93
14.71
14.93
2.71
5.40%
8.95%
5.81%
802.52
802.74
6.69
5695.08
4
4143.59
605.32
9.84
15.65
16.10
2.81
6.40%
7.88%
6.73%
801.35
801.80
6.70
2199.61
5626.74
5
3768.34
604.47
27.97
14.53
14.51
2.69
-7.12%
-9.90%
-6.10%
799.14
799.11
6.69
6
2129.22
5919.33
6
3648.68
608.02
59.29
14.20
14.25
2.65
-2.27%
-1.78%
-1.72%
803.41
803.46
7
2145.83
6043.63
7
3676.91
609.47
105.65
14.32
14.46
2.66
0.85%
1.48%
1.22%
805.42
805.56
8
2212.35
6079.90
8
3790.00
609.89
168.12
14.71
14.95
2.70
2.68%
3.40%
2.93%
806.35
0.42%
0.46%
0.22%
0.61%
0.67%
0.29%
-0.15%
-0.12%
0.32%
-0.28%
-0.34%
-0.15%
6.69
0.54%
0.54%
0.01%
6.69
0.25%
0.26%
0.12%
806.59
6.69
0.12%
0.13%
0.18%
9
2314.12
6347.41
9
3963.01
612.91
247.24
15.29
15.86
2.76
3.95%
6.07%
4.04%
810.84
811.41
6.70
0.56%
0.60%
0.23%
10
2251.64
6112.56
10
3856.79
610.27
343.17
15.03
15.25
2.72
-1.68%
-3.84%
-1.35%
807.16
807.38
6.69
-0.45%
-0.50%
0.03%
11
2398.00
6002.53
11
4105.59
609.00
455.84
15.85
16.19
2.80
5.47%
6.15%
5.45%
806.33
806.66
6.70
-0.10%
-0.09%
0.28%
12
2321.26
6170.60
12
3975.14
610.93
585.02
15.56
15.78
2.76
-1.87%
-2.52%
-1.72%
808.54
808.76
6.70
0.27%
0.26%
0.01%
13
2376.97
6380.40
13
4069.85
613.27
730.38
15.95
16.32
2.79
2.51%
3.43%
2.42%
811.97
812.34
6.70
0.42%
0.44%
0.17%
14
2334.19
6482.49
14
3997.12
614.38
891.54
15.84
16.09
2.77
-0.65%
-1.38%
-0.69%
813.31
813.56
6.70
0.16%
0.15%
0.05%
15
2490.58
6851.99
15
4262.98
618.26
1068.06
16.77
17.44
2.85
5.86%
8.35%
5.60%
819.28
819.94
6.70
0.73%
0.78%
0.28%
16
2555.33
7050.70
16
4373.06
620.26
1259.50
17.24
18.02
2.88
2.79%
3.34%
2.57%
822.34
823.12
6.71
0.37%
0.39%
0.17%
17
2468.45
6803.92
17
4225.36
617.77
1465.40
16.94
17.25
2.84
-1.76%
-4.27%
-1.87%
818.80
819.12
6.70
-0.43%
-0.49%
0.01%
18
2522.76
6382.08
18
4317.68
613.29
1685.30
17.36
17.34
2.86
2.49%
0.50%
2.27%
813.41
813.39
6.71
-0.66%
-0.70%
0.16%
19
2474.82
6548.01
19
4236.20
615.08
1918.75
17.28
17.12
2.84
-0.45%
-1.25%
-0.76%
815.66
815.50
6.70
0.28%
0.26%
0.05%
20
2468.64
6639.69
20
4225.68
616.06
2165.31
17.42
17.14
2.84
0.83%
0.12%
0.46%
817.07
816.78
6.70
0.17%
0.16%
0.09%
21
2508.13
6891.99
21
4292.83
618.67
2424.57
17.81
17.59
2.86
2.23%
2.60%
1.83%
820.84
820.62
6.71
0.46%
0.47%
0.14%
22
2572.34
6947.13
22
4401.98
619.23
2696.09
18.33
18.07
2.89
2.91%
2.75%
2.54%
822.09
821.83
6.71
0.15%
0.15%
0.17%
23
2697.62
6648.40
23
4614.95
616.15
2979.49
19.16
18.75
2.95
4.53%
3.72%
4.25%
818.92
818.51
6.71
-0.38%
-0.40%
0.23%
24
2472.90
6455.60
24
4232.94
614.09
3274.38
18.22
17.04
2.84
-4.92%
-9.08%
-5.52%
815.31
814.13
6.70
-0.44%
-0.53%
-0.13%
25
2368.79
6655.72
25
4055.95
616.23
3580.38
17.91
16.46
2.79
-1.72%
-3.44%
-2.47%
817.77
816.32
6.70
0.30%
0.27%
-0.02%
Table (3): Data for pm,t, vm,t, pi,t and values of estimated prices and returns across time for OLS, RF-2 and RF-1 models for 5.2.1 pi,t = 0.003pm,t + 0.002vm,t + 0.0007tt + 2 and
Case 5.2.2 pi,t = 0.003pm,t + 1.3vm,t + 0.0007tt + 2. The prices for (pi,t)RF-2 are estimated by regressing pi,t on pm,t and vm,t while (pi,t)RF-1 uses only pm,t values
14
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Table 4: Summary statistics of fitting the estimated returns to actual returns – Low proportion
of nonlinear component
Standard deviation of errors
OLS 0.01997
RF-2 0.08971
RF-1 0.07993
(R2) %
98.58
94.71
98.42
F-statistics
1525.44
394.21
1368.01
5.2.2 High proportion of nonlinear component: Next, we consider the same example as
above but with the proportion of the non-linear function vm,t increased in the actual price pi,t by
increasing its slope coefficient. For this, we have considered pi,t = 0.003pm,t+1.3vm,t+0.0007tt+2
and the resulting values have been given in Table (3). Again, on comparing the estimated
returns with the actual returns, results in Table (5) were obtained.
Table 5: Summary statistics of fitting
proportion of nonlinear component
Standard deviation of errors
OLS 0.01775
RF-2 0.00181
RF-1 0.01867
the estimated returns to actual returns – High
(R2) %
10.75
99.67
10.75
F-statistics
2.651
6565.98
2.649
On comparing the results obtained from the two cases, we can see that the accuracy of stock
returns estimated through the RF-2 model is higher than those through the OLS model for the
second case where the stock prices have a higher proportion of nonlinear constituents. The
average proportion of the nonlinear independent variable vm,t in stock price pi,t is only 7.65% for
Case 5.2.1 and a whopping 98.16% for Case 5.2.2. The standard deviations of the errors for the
RF-2 model are 0.08971and 0.00181 for the former and the latter cases while the standard
deviations of the errors for the OLS model are 0.01997 and 0.01775 respectively. Furthermore,
the F-statistic for the RF-2 model is quite significant even for the generally linear price structure
of Case 5.2.1 but is barely so for the OLS model for the more non-linear price structure of Case
5.2.2.
Since the true utility of an asset pricing model lies mainly in assisting investment strategies
by helping to choose the stocks with correct risk-return profiles in-keeping with the objectives of
a portfolio, the RF model cannot be dispensed with, since it offers the more accurate picture of
returns across risk as has already been discussed under Cases 5.1.1 and 5.1.2. It is possible that
not all the independent variables indicated in the full RF model as set out in equation (19) would
be empirically important in computing the returns. The empirical importance of the factors has to
15
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
be tested out using real life data from various markets. Hence a small empirical study has been
provided in the next section.
6. An Empirical Test
We expand our work by providing preliminary empirical test for the validity of RF model
using the Dow Jones Industrial Average (DJIA) index and its 30 constituent stocks as on April
30, 2013. The sample includes monthly closing prices and monthly trading volumes data for all
the 30 stocks as well as the DJIA index from May 30, 2003 to April 30, 2013. The financial data
were obtained from the database purchased from Norgate Investor Services, who are licensed
data providers of various stock markets all over the world. The empirics compare the
performance of the RF model, the Capital Asset Pricing Model (CAPM) (Sharpe 1964; Lintner
1965) and the Fama French 3 Factor model (FF3F) (Fama and French 1993, 1996, 2004). The
CAPM equation used for this example is the time adjusted model of equation (1).
Here, Rf,t is the risk free rate of return at time „t‟, E(Ri,t) and E(Rm,t) are the expected asset and
the expected market returns respectively while βi,m is the market risk of the asset „i‟. Similarly,
the FF3F model (Fama and French, 1996, 2004) is given by:
E(Ri,t) - Rf,t = βi,m [E(Rm,t) – Rf,t ) + βi,s E(SMBt) + βi,h E(HMLt)
(32)
Here, SMBt (small minus big) is the difference between the returns of the diversified
portfolios of small and big stocks sorted on the basis of their sizes. HMLt (high minus low) is the
difference between the returns of the diversified portfolios of high and low B/M stocks. The data
for the factor loadings of SMBt and HMLt were obtained from Kenneth French‟s website.
The literature introduces two different formats for studying the behavior of the stock prices
and volumes. The first format studies the average returns while the second format examines the
continuous returns. The average returns were found to be remarkably non-linear in their behavior
across increasing risk, whereas the time series of continuous returns were found to behave
„approximately‟ linearly and were hence modeled directly through linear regression. This
difference in behavior of average returns and continuous returns may be likened to the difference
in the shape of earth that is considered spherical when studied from far in space while it is taken
to be flat and linear for trigonometrical studies used for land surveys, standing on the surface.
Accordingly, for studying empirical average returns, the RFM equation (22) has been modified
and is as given below:
t
pi,t = αi + βi1[{(1 + ln(pm,t/ pm,t-1) pi,t-1}] + βi2[ln(vm,t/vm,t-1)] 2 + βi3(-0.1) t
(33)
The estimated average returns are then computed from two consecutive averages of estimated
prices. Thus, estimated prices are averaged from „t‟ to„t+n-1‟ intervals and from „t+1‟ to „t+n‟
intervals, to obtain the average returns from the ratio of these average prices. Where pi,t and pi,t-1
are the asset prices on days „t‟ and „t-1‟ respectively while pm,t and pm,t-1 are the index prices.
16
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Similarly, vm,t and vm,t-1 are the corresponding index volumes and tt is the time factor based on the
chronological rank of the day „t‟ in the whole sample. It may be noted that the role of the market
index in the above model is in shaping the price formation of the asset as a portion of the current
price pi,t varies in direct proportion with the market return as {1 + ln(pm,t/ pm,t-1) pi,t-1}.
Next, the RFM equation used for studying the empirical continuous returns is as follows:
Ri,t-Rf,t= βi,m(Rm,t–Rf,t)+βi,sSMBt+βi,hHMLt+βi,v(Vm,t)+βi,o(tt)2+βi,l(Ri,t-1)
(34)
The above equation is an enhanced version of the FF3F by including the additional
theoretically relevant variables, which are index volume, time and preceding value of the asset
return. It may be noted that the time factor used for modeling the continuous returns is different
from that used for modeling prices. This is because that while asset prices are influenced by the
exponential variation of the time-ranks, the continuous returns are shaped by the quadratic
variation of the time-ranks. It should be clarified here itself that, Ri,t=ln(pi,t/pi,t-1); Rm,t=ln(pm,t/pm,t1); and Vm,t=ln(vm,t/vm,t-1).
The 30 stocks of the DJIA were first sorted as per increasing risk that was measured by the
variances of their returns through a rolling time frame of the last 11 returns and then regrouped
into five smaller portfolios P1 to P5 consisting of 6 stocks each, with P1 containing the least
risky stocks while P5 contained the most risky stocks. Besides these 5 portfolios, the returns of
the full sample portfolio (P-full) consisting of all the sample stocks was also studied.
Figure 6: Plot of Average Portfolio Returns across increasing risk from P1 to P5 formed on
prior variances. The plot consists of Actual average returns and Estimated average returns
using different models.
17
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
After sorting the stocks, the ranked series of stock prices were reconstructed from the actual
stock returns for each rank, using some common base number (like 100), so as to avoid the
sudden abrupt changes in the prices of these rank-stocks after each sorting.
Table 6: Slope values and t-statistics for the CAPM, FF3F and RF models
P1
P2
P3
P4
P5
P-Full
Panel A: CAPM
βi,m
0.74(15.60)
0.77(17.59)
1.07(24.44)
1.27(20.42)
1.67(19.26)
1.10(56.62)
Panel B: FF3
βi,m
βi,s
βi,h
0.80(15.68)
-0.26 (-2.93)
-0.07(-0.89)
0.77(15.75)
-0.01(0.01)
-0.03(-0.41)
1.03(21.18)
0.12(1.47)
0.12(1.47)
1.25(18.07)
0.20 (1.66)
-0.07(-0.66)
1.46(16.85)
0.48(3.21)
0.56(3.96)
1.06(52.42
0.12(3.28)
0.08(2.41)
1.00(49.74)
-0.84(-0.92)
-35.92(-1.56)
1.01(107.96)
1.59(1.61)
-7.69(-0.31)
1.00(102.71)
1.73(1.80)
21.09(0.89)
0.97(61.12)
-1.54(-0.97)
7.14(0.18)
0.98(61.15)
-1.43(-0.80)
-109.00(-2.42)
0.75(0.99)
0.99(174.03)
-0.06(-0.14)
-24.62(-2.39)
0.77(14.28)
-0.24(-2.69)
-0.08(-0.93)
-0.011(-2.25)
1.7E-07(0.47)
0.02(0.35)
0.74(14.52)
0.02(0.18)
-0.02(-0.25)
-0.007(-1.47)
9.8E-07(2.87)
-0.03(-0.67)
1.04(20.16)
0.12(1.38)
0.09(1.11)
-0.006(1.18)
1.7E-07(0.50)
0.06(1.41)
1.23(16.47)
0.22(1.78)
-0.08(-0.70)
-0.007(-1.06)
-2.2E-07(-0.45)
0.01(0.23)
1.49(16.52)
0.49(3.27)
0.55(3.83)
-0.001(-0.05)
-1.6E-06(-2.70)
0.001(0.00)
1.05(48.85)
0.12(3.5)
0.08(2.31)
-0.004(-1.89)
-4.4E-08(-0.31)
0.002(0.12)
Panel C: RF1
αi
βi,1
βi,2
βi,3
Panel C: RF2
βi,m
βi,s
βi,h
βi,v
βi,o
βi,l
The table shows the estimates and t- statistics for five different portfolios sorted based on the risk (covariance) in additional
to all stocks portfolio. The estimates is driven using the CAPM (Assets Pricing Model, FF3F (Fama-French three factor
model), RF1 (Rational Function model: the average returns format), RF2 (Rational Function model: continuous returns
format). Figures in brackets stand for t-statistics.
Table 7: Correlations and Sum of Squared Errors (SSE) between actual and estimated
average returns
Model
CAPM
FF3F
RFM
Correlation
-81.74%
-81.75%
99.90%
Standard Errors
1.00E-04
1.25E-04
4.01E-07
Table 7 presents the correlation and the standard errors of the estimated average returns with actual average
returns from P1 to P5:
18
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Table (8): Correlations and Sum of Squared Errors (SSE) of the estimated continuous returns
with actual continuous returns.
Model
P1
P2
P3
P4
P5
Correlations with Actual Continuous Returns%
P-full
CAPM
83.7(0.043)
86.5(0.037)
92.1(0.037)
89.2(0.076)
89.0(0.141)
98.4(0.007)
FF3F
85.3(0.039)
86.6(0.0.037
92.4(0.036)
89.5(0.073)
91.9(0.110)
98.7(0.006)
RFM
86.0(0.037)
87.2(0.034)
92.8(0.034)
89.6(0.072)
91.8(0.102)
98.8(0.006)
Figures in brackets stand for the Sum of Squared Errors (SSE).
The above rank series data were then analyzed using the CAPM, the FF3F model and the
RFM equations as given in equations (1) and (32) to (34) for the two different formats „average‟
and „continuous‟ returns respectively. The estimated returns have been compared for their
empirical validity through their correlations with the actual returns as well as through their Sum
of Squared Errors (SSE) with respect to the actual returns for each of the six sub-portfolios P1 to
P5 and P-full. It should be mentioned here that for the sake of comparison of the estimates as per
RFM equation (33) across P1 to P5, the intercept αi is taken as common for all the sub-portfolios
P1 to P5. For this, the intercept αi is computed for the full sample P-full and then used in
common for estimating the prices for P1 to P5 as well. This way we have a common intercept
value for all the sub-portfolios P1 to P5 and P-full which enables us to compare their
performance across risk.
Findings reported in Tables 6-8 and the Figure 6 indicate that the RF model outperforms both
the CAPM and FF3F models for both average returns as well as for continuous returns. The
market return has the highest t-statistics in all the models. This implies that the market return is
the most important factor in estimating the asset returns. The t-statistics for other factors are
considerably lower in both FF3F as well as RFM equations even though these factors are
important in improving the accuracy of the estimates.
For the average returns, the RFM estimates are found to be the most accurate as compared to
both CAPM and FF3F estimates. The correlations of both CAPM and FF3F estimates are
negative, being -81.74% and -81.75% respectively while that of the RFM is 99.90%. The SSE
values also indicate the analytical superiority of the RFM equation (33) that has been used for
estimating average returns. The SSE for CAPM estimates is 1.00 E-04 and for FF3F estimates it
is 1.25 E-04 while for RFM it is only 4.01 E-07, which is more than 99% improvement over both
CAPM and FF3F estimates. These results are further substantiated very clearly by the Figure 6
which is a plot of the actual average returns across P1 to P5 and the estimates of the same. It can
be distinctly seen that only the RFM estimates closely follow the actual average returns, while
the estimates of the CAPM and FF3F provide misleading estimates. We can explain the out
performance of RFM by its ability to follow the non-linear curve of the actual average returns,
being a non-linear model itself, whereas the CAPM and FF3F estimates go on increasing even
19
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
when the returns for P3 to P5 are decreasing. This is a very important finding of this example
that clearly brings out the need to use RFM for estimating average asset returns across increasing
risk and thus make the right investment choices.
Furthermore, studying the continuous returns also shows that the RFM equation (34)
provides the best fitting for stock returns. Table 8 indicates that the continuous returns estimated
by RFM have higher correlation, though marginal, than the estimates of the CAPM and FF3F.
The higher analytical accuracy of the RFM equation (34) is further supported by results for SSE
given in brackets in Table 8 where one can see that the SSE is consistently lower for the RFM
estimates. The improvement of the RFM estimates over the FF3F estimates has the lowest value
of 1.27% for P4 while the highest value is that of 9.31% for P2. Thus the preliminary empirical
evidence provided here substantiates the RFM theory that we have been discussing so far.
7. Practical Implications
From the Figures 1 and 6 it can be seen that the plots of empirical average asset returns
across increasing risk are non-linear. The reason behind this is evident from the figures 4 and 5,
where the returns computed using the RF model are found to plot as curves that increase initially
with increasing risk but gradually flatten out for higher risk values. Apart from the mathematical
behavior of rational functions, the multiple factors used in estimating asset prices may increase
or decrease individually thereby influencing the asset returns further and thus adding to the nonlinearity of the returns across increasing risk, where risk is the simple variance of the returns
over a rolling time-frame. This shows very clearly that for making proper investment choices
that are efficient in terms of returns and risks, we must use the RF model equation (33) to
estimate the average asset returns that behave non-linearly across increasing risk. It can be seen
from the figure 6, that the actual average returns first increase from P1 to P2 but then they
decrease from P3 to P5. In such a case, extant linear models like the CAPM and the FF3F
generate misleading estimates whereby their estimated average returns keep increasing linearly
from P1 to P5. Only the RFM estimates follow the actual average returns accurately, having
99.9% correlation with the actuals. Thus mean-variance efficient investments can be chosen
accurately only with the RF model equation (33) and not by the extant linear asset pricing
models.
Further, the monitoring of the time series of continuous asset returns also would be more
accurate using the RF model equation (34) as compared to the CAPM and the FF3F models. This
would help investors in making better short-term buy-sell decisions for their investments.
20
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
8. Conclusions and future scope of work
The RF model developed in this paper is built on the most basic variables and hence hopes to
circumvent the shortcomings of its predecessors, like the CAPM and the FF3F model that were
built from derivative variables and so in turn depended on the behavior of their constituent
variables. In the process, the RF model indicates that asset returns, especially average returns,
don't add linearly in a portfolio since returns are rational functions. We must hence model prices
instead of returns when estimating average returns and further, for modeling asset prices, we
must also consider other factors in addition to the index price like asset volumes, index volume,
time trends, preceding asset prices and other relevant market factors if any.
The importance of the Rational Function (RF) model is further supported by the
numerical illustrations and the empirical example provided in the paper, which explain as to why
the actual average asset returns plot flatter across increasing risk as compared to the average
asset returns estimated by existing asset pricing models. This solves to some extent the 'Joint
Hypothesis Problem' of dual-testing required for the asset pricing model as well as the market
efficiency, by indicating that we need to refine our asset pricing model first. Further, the
numerical illustrations of the RF model also indicate that given the shape of the curve obtained
by plotting the asset returns against the common risk, it seems that the best investment policy for
the investors would be to focus on medium-risk assets since beyond a point, the changes in
returns for changing risks are negligible.
Like any other mathematical model describing empirical realities, the RF model also needs to
be tested out using real life data to truly assess its practicality even though we have offered an
empirical example in this paper substantiating the theory. It would be interesting to study the
mathematical behavior of the asset returns using more comprehensive empirical data from
different markets and the results would indicate the practical value of the various factors
identified in this paper. It is also possible that additional factors that influence the asset price
might be unearthed. If so, such factors need to be identified alongwith their underlying
theoretical rationales. Further, the RF model itself could be further refined through careful
modeling that improves upon the approximations that have been used in this paper. Hopefully,
there would be studies in the near future that would attempt to address these objectives.
References
Avramov, D.: “Empirical Assets Pricing”. PhD course, University of Maryland [Online].
Available at: http://www.rhsmith.umd.edu/finance/faculty/avramov.aspx (2008)
Banz, R. W.: The Relationship Be-tween Return and Market Value of Common Stocks. Journal
of Financial Economics. 9:1, 3-18 (1981)
Basu, S.: Investment Performance of Common Stocks in Relation to Their Price- Earnings
Ratios: A Test of the Efficient Market Hypothesis. Journal of Finance. 12:3, 129-56 (1977)
21
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Blume, M.E., Friend, I.: A New Look at the Capital Asset Pricing Model. Journal of Finance 28,
19-33 (1973)
Bod‟a, M., Kanderova, M..: Linearity of the Sharpe-Lintner version of the Capital Asset Pricing
Model. Procedia – Social and Behavioral Sciences 110, 1136-1147 (2014)
Chen, G., Firth, M., Rui, O.M.: The Dynamic Relation between Stock Returns, Trading Volumes
and Volatility. Financial Review 36, 153-173 (2001)
Datar, V., Naik, N., Radcliffe, R.: Liquidity and Stock Returns: An Alternative Test. Journal of
Financial Markets 1, 203–219 (1998)
Dempsey, M.: The Capital Asset Pricing Model (CAPM): The History of a Failed Revolutionary
Idea in Finance? Abacus 49, 44-50 (2013)
Diacogiannis, G., Feldman, D.: Linear Beta Pricing with Inefficient Benchmarks, Working
Paper, Australian School of Business, University of New South Wales. Available at SSRN:
http://ssrn.com/abstract=893702 (2011)
Douglas, G.W.: Risk in the Equity Markets: An Empirical appraisal of Market Efficiency.
University Microfilms, Inc., Ann Arbor, Michigan (1968)
Elgammal, M. M., McMillan, D.: Value Premium and Default Risk. Journal of Asset
Management, 15, 48-61 (2014)
Fama, E.F.: The Behavior of Stock Market Prices. Journal of Business, 38, 34-105 (1965)
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)
Feldman, D.: Incomplete Information Equilibria: Separation Theorems and Other Myths. Annals
of Operations Research, 151, 119-149 (2007)
Friend, I., Blume, M.E.: Measurement of Portfolio Performance under Uncertainty. American
Economic Review 60, 607-36 (1970)
Gallant, R.A., Rossi, P.E., Tauchen, G.: Stock Prices and Volume. Review of Financial Studies
5, 199-242 (1992)
Haugen, R.A., Baker, N.L.: Commonality in the Determinants of Expected Stock Returns.
Journal of Financial Economics 41, 401–439 (1996)
Jagannathan, R., Wang, Z.: The Conditional CAPM and the Cross-Section of Expected Returns‟,
Journal of Finance 51, 3-53 (1996)
22
Proceedings of 4th European Business Research Conference
9 - 10 April 2015, Imperial College, London, UK, ISBN: 978-1-922069-72-6
Jegadeesh, N., Titman, S.: Returns to Buying Winners and Selling Losers: Implications for Stock
Market Efficiency. Journal of Finance. 48:1, pp. 65-91 (1993)
Kandel, S., Stambaugh, R.F.: Portfolio Inefficiency and the Cross-Section of Expected Returns.
Journal of Finance 50, 157-184 (1995)
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)
Pisedtasalasai, A., Gunasekarage, A.: Causal and Dynamic Relationship among Stock Returns,
Return Volatility and Trading Volume: Evidence from Emerging markets in South-East Asia.
Asia Pacific Financial Markets 14, 277-297 (2007)
Roll, R.: A Critique of the Asset Pricing Theory‟s Tests. Journal of Financial Economics 4, 129176 (1977)
Roll, R., Ross, S.A.: On the Cross-Sectional Relation Between Expected Returns and Betas.
Journal of Finance 49, 101-121 (1994)
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)
23