Fundamental Ratios and Fundamental Indexation; Constructing the Optimal Portfolio.
Wisse Koedam
University of Groningen June 2015
Abstract
This paper confirms the results of previous studies that portfolios of firms with low price-to-book (P/B) and low price-to-earnings (P/E) ratios have higher returns than portfolios of firms with high P/B and high P/E ratios. The portfolios are formed in a new way that combines fundamental and market variables: the portfolio weights are determined by three proportional financial ratios of the firms. Using this weighting strategy I find that a portfolio of firms with low P/E and P/B ratios has annual excess returns of 9.52% over the market return during a 20-year period (from 1995 to 2015).
1. Introduction
Most investors try to construct portfolios of assets that have a higher expected return than other portfolios for a similar level of risk. As a result academics and investors have devoted substantial effort to finding variables that can predict future stock prices and identify under- or overvalued stocks. Although there are arguments against finding variables that can forecast under- or overvalued stocks, several studies find that firms with high financial ratios tend to be overvalued, for example Basu (1977), De Bondt and Thaler (1985), Lakonishok, Shleifer and Vishny (1994) and Danielson and Dowdell (2001).
In this paper I use the price-to-earnings ratio (P/E ratio) and price-to-book ratio (P/B) to construct four different portfolios. The firms are classified, using classifications of
Danielson and Dowdell (2001), as either growth firms, value firms, turnaround firms or declining firms. I use a new way to determine the weight of each stock in the portfolio, by using a combination of fundamental variables and market variables. Arnott, Hsu and Moore (2005) show that weighting a portfolio by fundamental variables eliminates “noise” created by speculative investors and gives higher risk-adjusted returns compared to market
capitalization weighted portfolios. Nevertheless, unlike fundamental variables, market variables take into account expectations about future cash flows. Since both have attractive properties and pitfalls, I combine fundamental variables and market variables into
“Proportional Financial Ratio” portfolio weights. The portfolios that are formed using this procedure, are referred to as Proportional Financial Ratio (PFR) weighted portfolios in this paper.
The paper shows the performance of PFR weighted portfolios from 1995 to 2015 and tests whether growth, value, turnaround and declining firms earn excess returns relative to several benchmark indexes. I adjust the returns for risk using several measures such as volatility, downside risk and systematic risk. In addition I run a regression with the risk factors of Fama and French (1993) and Carhart (1997), to see whether the Carhart four-factor regression model shows that PFR weighted portfolios earn excess returns relative to the benchmark portfolios. The period from 1995 to 2015 is interesting because it contains the Internet Bubble in the early 2000s, in which stock prices of Internet firms rose sharply until the bubble burst and the financial crisis (2007 to 2009).
2. Related literature
Portfolios can be constructed in several ways. The simplest weighting strategy is equal weighting, where the investor invests the same amount of money in every stock of the
portfolio. The best-known and most widely used method is weighting by the market capitalization (market-cap weighting) of a firm. Market-cap weighting assigns portfolio weights proportional to the relative market capitalization of each firm. Arnott, Hsu and Moore (2005) propose a new way of determining the weights of stocks in a portfolio, which they call fundamental indexation. The authors state that market-cap weighting is not the optimal weighting strategy. They argue that market-cap weighting assigns the highest weight to firms with the largest market capitalization and this strategy is therefore likely to give a large weight to overvalued stocks. They overcome the problem of overweighting overvalued stocks by using measures of the so-called fundamentals of a firm to determine the weights in a portfolio. Fundamentals are all data about a firm beside the stock trading data; examples are accounting variables of a firm such as the sales, net profit, number of employees or book value of the firm. The disadvantage of stock prices as indicator of firm performance is that they contain noise. Noise is movement in stock prices caused by speculative investors. Arnott, Hsu and Moore (2005) argue that the use of fundamentals eliminates this noise, because the fundamentals of a firm can only change by real changes in the performance of the firm. The fundamentals that determine their portfolio weights are: dividends, book value, revenues, sales, employment and profit. In their fundamental indexation approach, the portfolio weight of a stock increases if the fundamentals of that stock increase; for example when the book value of a firm increases, the weight of this firm’s stock in the portfolio increases. They compare the performance of portfolios that use different individual fundamentals to determine the portfolio weights, and a portfolio that uses a composite weight determined by all
fundamentals. This composite weight is the average portfolio weight a firm receives based on all of the fundamentals they use. The performance differs for each individual fundamental portfolio weights but the composite portfolio weight has the best performance. In addition they compare the returns of their composite fundamental indexation strategy with multiple other weighting strategies. They find that over a 42-year period (from 1962 to 2004), their fundamental indexation approach produces a significant annual excess return of 1.94% over the annual return of the market-cap weighted S&P 500 index.
stocks are priced correctly and reflect all current information about a firm. A different theory is the arbitrage-pricing theory (APT) by Ross (1976), that relies on the absence of arbitrage opportunities (i.e. an opportunity to make a risk-free profit without committing any capital to the transaction) to explain why portfolios (or other assets) are on average correctly priced.
The Capital Asset Pricing Model (CAPM) of Treynor (1961), Sharpe (1964) and Lintner (1965), states that a stock’s expected return in equilibrium is equal to the risk-free rate plus compensation for the amount of systematic risk the firm faces. In this model systematic risk is measured by “beta”, which is the standardized covariance of the return of the stock and the return on the market portfolio.
Fama and French (1993) agree with the EMH, but argue that the CAPM fails to empirically explain a large part of the variation in expected stock returns. They find that the CAPM is significantly improved when two additional explanatory variables are included in the regression: the Small (market capitalization) Minus Big (SMB) factor and the High (book-to-market) Minus Low (HML) factor. The SMB variable measures the historical excess returns of firms with a small market capitalization over firms with a large market
capitalization. The HML variable measures the historical excess returns of firms with a high book-to-market ratio over firms with a low book-to-market ratio. They argue that a firm with a high book-to-market ratio has higher returns than a firm with a low book-to-market ratio, because the former has a larger probability of becoming financially distressed and therefore has a higher risk. Furthermore they state that larger firms produce lower returns because they have a smaller probability of getting into financial distress than small firms, and consequently have lower risk.
Carhart (1997) extends this regression model by adding a fourth explanatory variable, which he names the “momentum” factor. He finds that if the price of a stock rises it tends to continue to rise in the near future and if a stock price declines it tends to continue to decline the near future.
Notwithstanding the arguments of the EMH and APT, against finding over- or
underpriced stocks, several studies find evidence suggesting that under- or overvalued stocks can be identified ex ante.
and measure expected performance by price-to-earnings and price-to-cash flow ratios. Declining stocks are then shares in firms that have a low price-to-book ratio (P/B ratio), whereas growth stocks have a high P/B ratio. Lakonishok, Shleifer and Vishny (1994) confirm the findings of Fama and French (1993) that firms with a high book-to-market ratio (or low P/B ratio) have higher returns. Moreover they find that declining stocks have on average a 10% to 11% higher return than growth stocks over a 22-year period from 1968 to 1990. However, Fama and French (1993) argued that the higher return for high book-to-market ratio (or low P/B ratio) firms stems from the higher risk these firms face and is therefore in line with the EMH. Contrarily, Lakonishok, Shleifer and Vishny (1994) find that declining stocks do not have higher risk than growth stocks. Although declining stocks do have a higher beta, this beta results mainly from upside risk. In recession periods, declining stocks have less negative returns than both the market and growth stocks, and in periods of economic growth, declining stocks have substantial higher returns than both growth stocks and the market. The high excess returns in times in which the market’s returns increase thus cause the higher beta for declining stocks. Investors like this upside risk because it is the risk of having a higher return than expected. These findings are similar to those of De Bondt and Thaler (1985), who find that a portfolio of past losing stocks earns an excess return of 25% over a portfolio of past winning stocks during a period of three years. The large returns of declining stocks can thus not be explained by the EMH and CAPM, because declining stocks have higher returns and lower risk than growth stocks.
There are several studies using a behavioral approach that try to explain this anomaly. Lakonishok, Shleifer and Vishny (1992) find that institutional investors engage in herding, which means institutional investors buy the same stocks that other institutional investors buy. Moreover they find that institutional investors do positive feedback trading, which is buying past winners and selling past losers. Barber, Odean and Zhu (2009) find that individual investors also engage in herding and find that individual investors collectively buy stocks that performed well in the past. They test several explanations and find strong evidence that this behavior is caused by psychological biases. Moreover the systematic behavior of individual investors influences asset prices and overvalues past winners, thereby reducing their
subsequent returns. Various studies show that growth firms have a higher demand for their stocks caused by the heuristics affect, as in Cooper, Dimitrov and Rau (2001) and Statman, Fisher and Anginer (2008), availability, as in Barber and Odean (2008) and
(2009), that influence investors’ behavior in a systematic way. Growth stocks tend to be firms that give investors positive affect (affect heuristic), grab the attention of investors due to abnormal returns, abnormal trading volume and news coverage (availability heuristic), and are expected by investors to perform well in the future because of their superior past
performance (representativeness heuristic). This systematic behavior of investors overvalues growth firms and thereby lowers the future return of these stocks, because the stock price has to increase by even more than originally expected to realize the high expected return. This is the other way around for declining firms that give negative affect to investors, do not grab the attention of investors and have poor past performance. Demand is low for these stocks, which causes the stock prices of these firms to be undervalued enabling these firms to realize a higher return easier than growth stocks.
3. Data and Methods
3.1 The return stages model
This paper uses the approach of Danielson and Dowdell (2001) to classify firms as growth, value, turnaround or declining firms. In their approach they combine price-to-book ratios (P/B ratios) with price-to-earnings ratios (P/E ratios) in what they call “the return stages model”. The authors classify firms into four different types of firms depending on their past year P/B ratio, past year P/E ratio and the expected return on equity (k). Fig. 1 below shows how I classify firms using this approach.
Figure 1: Firm classification by P/B and P/E ratios
P/B>1 P/B < 1
P/E > 1/k Growth firms Turnaround firms
P/E < 1/k Value firms Declining firms
The return stages model divides the future earnings of a firm into two periods; the first 𝜏 years are the growth phase and all years after 𝜏 are the equilibrium phase. The model
assumes that a firm will reinvest 100% of all future earnings during the growth period.
and that new investments have the same risk as the existing assets of the firm. The final assumption of the return stages model is that the expected return on equity (k) is fixed over time and 10% for all firms. Although this is a strong assumption it simplifies the construction of the portfolio and it has another convenient property. Because firms are separated into the four classifications by an expected return on equity (k) of 10%, this implies that a growth firm has to earn a return on existing and new equity of at least 10% to achieve a stock return of 10%. This is because investors expect that the return on both new and existing equity of growth firms is equal or above 10%. Declining firms do not have to earn a return on existing or new equity of at least 10% to achieve a stock return of 10%, because investors expect them to have a decreasing return on existing or new equity. If the actual return on existing or new equity is above the expectation of investors (which can be below 10%) then the stock return will be above 10%, since the stock price was calculated on a lower return on existing or new equity. Altering or lowering the expected return on equity can make large differences in the classifications of firms. It would therefore be appropriate to do a sensitivity analysis to see, whether changing the expected return on equity (k) increases or decreases the risk-adjusted returns of the portfolios of growth, value, turnaround or declining firms. However such a sensitivity analysis is unfortunately beyond the scope of this paper.
Under these assumptions the return stages model can be expressed as follows:
𝑃!!!= 𝐼!!! !! ! !!!! !!! ! ! ! !"!,! (1)
Where 𝐼!!! is the book value of existing equity and 𝑅! is the return on existing equity, which remains the same until the end of the growth period (𝑡 = 𝜏). New investments earn a return equal to 𝑅!. Finally during the equilibrium period new investments earn a return of 𝑅! equal to the required return on equity 𝑘. The equilibrium phase therefore assumes that full
competition ensures that no excess returns are earned anymore. From this moment on a firm can choose to pay out dividends instead of reinvesting all earnings since no excess returns can be earned. The weighted average return on equity (𝑊𝑅!,! ) can be different from k and is the
The P/B ratio can be derived from the return stages model by dividing Eq. (1) by the book value of equity (𝐼!!!).
! != !!!! !!!!= !! ! !!!! !!! ! ! ! !"!,! (2)
The P/B ratio shows whether investors expect a firm to be profitable in the future. If the P/B
ratio is above one the firm is perceived by investors as profitable in the future and conversely
if the P/B ratio is below one, the firm is perceived to be not profitable in the future. Eq. (2) shows that profitability can result from either the value of expected excess returns from existing assets !!
! , or the value of expected excess returns from investments during the
growth period !!!!
!!! !
or the adjustment for possible excess returns in the equilibrium period !!
!"!,! resulting from 𝑅! and 𝑅!.
The P/E ratio is derived, by dividing both sides of Eq. (1) by the earnings at 𝑡 = 1 (𝐸!!!). ! ! = !!!! !!!!= ! ! !!!! !!! ! ! ! !"!,! (3)
The P/E ratio shows whether a firm’s earnings will increase in the future or decrease in the future; if the P/E ratio is smaller than 1/ 𝑘, profits will be in the near future and if the P/E
ratio is larger than 1/k, profits will be in the later future. Eq. (3) shows that the P/E ratio can
be greater than one either due to new investments in the growth period !!!!
!!! !
or due to an improvement in the return on the firm’s assets over time !!
!"!,! . This could for example result from a restructuring of the firm. It is important to emphasize that the P/E ratio does not show whether a firm is profitable, it only shows whether earnings will increase in the future or will decrease in the future, and not whether the firm is profitable or not.
In this paper the return stages model is the theoretical support for the use of P/E and
P/B ratios to classify firms as growth, value, declining or turnaround firm. The P/E and P/B ratios in this paper are however calculated in a different way, since there is no data on
share of the previous year. The firms are assumed to stay within their classification for at least five years to simplify the portfolio construction. This assumption is not realistic since there are firms that move to other classifications during these five years. However the transaction costs associated with removing and adding entire holdings of firms every year could dampen the returns of the portfolio. A sensitivity analysis to see whether a potential increase in risk-adjusted returns is larger than the increase in transaction costs would be appropriate, but is beyond the scope of this paper.
3.2 Data
I collect annual book values per share, annual revenues per share, annual EPS, and total return index per share for all firms of the NYSE and NASDAQ from January 1995 to January 2015. In addition I collect the stock prices in the years 1995, 2000, 2005 and 2010 of all firms. All data is obtained from Thomson Reuters Datastream and the data set includes shares that stopped trading during this period. I use the annual book values per share, annual revenues per share, and annual EPS announced in the 3rd quarter report of the previous year since these are the most recent values of earnings per share and book value per share publicly available. Using data of the year-end report is wrong because this information is not available at the 1st of January, which is the day that I construct the portfolios. Furthermore I use the fundamentals per share because there is more and higher quality data available on per share basis.
3.1 A new portfolio weighting approach
Despite the excess returns that Arnott, Hsu and Moore (2005) find for their
(2005) and is therefore excluded from my portfolio weighting approach. Finally their
approach only uses fundamental variables without taking advantage of the predictive power of market variables. Fundamental variables are determined by past performance of the firm and do not contain expectations about the future performance of the firm. Although market values contain noise, they are based on the expectations of future cash flow and could therefore contain information that is not captured in fundamental variables. In this paper I use a new way of determining the weights in the portfolio that combines fundamental variables and market variables. I call portfolios constructed using this procedure, Proportional Financial Ratio (PFR) weighted portfolios.
3.2 Derivation of Proportional Financial Ratio (PFR) portfolio weights
To calculate the PFR portfolio weights I use the annual EPS, annual book value per share (BVPS), and annual revenues per share (RPS) of the previous year. EPS are noisy due to accounting practices, therefore I smooth the earning per share (EPS) by using the average of the previous five-year earnings per share data; so for the 1995 EPS weights I use the average annual EPS of the period of 1990 until and including 1994 (Campbell, Shiller, 1998 & 2001).
First I compute financial ratios for all fundamentals of each stock by dividing the fundamental variables (EPS, BVPS or RPS) by the price (P) of the stock. Secondly, I divide the financial ratio (e.g. 𝐵𝑉𝑃𝑆!,!/𝑃!,!) by the sum of all stocks’ financial ratios (e.g.
𝐵𝑉𝑃𝑆!,!/ !
!!! 𝑃!,!) to determine the partial weight of the firm. The derivation of the partial PFR portfolio weights is shown below in Eq. (4), Eq. (5) and Eq. (6).
Finally I calculate the composite weight by taking the equal weighted average of the three partial weights (book value weight, revenues weight and earnings weight). The resulting formula is shown below in Eq. (7).
𝑃𝐹𝑅 𝑤𝑒𝑖𝑔ℎ𝑡 =!!∗ !"#!,! !!,!∗ !!!!!"#!,!!!,! +!!∗ !"#$!,! !!,!∗ !!!!!"#$!,!!!,! +!!∗ !"#!,! !!,!∗ !!!!!"#!,!!!,! (7)
The PFR weighting approach combines the predictive power of share prices with the noise eliminating power of fundamental values. The PFR weighting approach assigns a portfolio weight proportional to the relative size of the financial ratios of the firm.
Consequently this approach assigns the highest weight to the best value for money stocks and thereby implicitly identifies and overweighs undervalued stocks.
I rebalance the portfolio weights on an annual basis because rebalancing more than this results in higher transaction costs than benefits according to Arnott, Hsu and Moore (2005). The portfolio weights are thus determined on January the 1st of each year and are not rebalanced during the year until they are again determined on January the 1st of the
subsequent year.
3.3 Calculations of the portfolio returns
I obtain annual total return index data for each stock from Thomson Reuters
Datastream. The total return index of a stock assumes that all dividends are reinvested in the stock and therefore it captures both the capital gain and income return of the stock. The PFR portfolio weights, determined at January 1st, change during the year at the same rate as the total return index changes during the year. This is similar to market-cap weighting because the weights now increase when the price of the stock increases and when the stock pays a dividend, because the dividend is reinvested in the same stock. I then divide the product of the weights at the end of the year (t=1) and the total return index values at t=1, by the product of the January 1st weights and the total return index values at the beginning of the year (t=0). I take the logarithm of the returns to normalize them, and make them comparable and additive. 3.4 Testing the trading strategy
returns of the fundamental indexation weighted RAFI US 1000 index1. I adjust the returns for three types of risk in particular volatility, systematic risk and downside risk. Furthermore I run a regression using the four risk factors of Fama and French (1993) and Carhart (1997) as explanatory variables, to see whether their model can explain the returns of the PFR weighted portfolios and test whether the PFR weighted portfolios earn excess returns (positive four-factor alpha) over several benchmark indexes.
The Sharpe ratio measures the risk-adjusted return by dividing the excess return of a stock by its volatility (Sharpe, 1966).
𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =𝑅 − 𝑅! 𝜎 𝑅 = Average return of the portfolio
𝑅! = Risk-free rate (one-month T-bill rate)
𝑆 = Sharpe ratio
𝜎 = Volatility of the return
If the Sharpe ratio of, for example, the PFR weighted portfolio of declining stocks is higher than the Sharpe ratio of the portfolio of growth firms then this implies that the portfolio of declining firms has a higher return adjusted for its volatility.
Since most investors are mainly concerned with systematic risk, the returns of the portfolios should be adjusted for the betas of the portfolios. The Treynor ratio developed by Jack L. Treynor divides the excess return of a portfolio by the beta of the portfolio.
𝑇𝑟𝑒𝑦𝑛𝑜𝑟 𝑟𝑎𝑡𝑖𝑜=!!!! !
𝑅 = Average return of the portfolio 𝑅𝑓 = Risk-free rate (one-month T-bill rate) 𝛽 = Market beta
The higher the Treynor ratio, the higher the return of the portfolio adjusted for systematic risk.
Lakonishok, Shleifer and Vishny (1994) argue that investors are mainly concerned with downside risk because this can give investors liquidity concerns and substantial losses, whereas upside risk gives investors unexpected higher returns. Moreover Kahneman, Knetch and Thaler (1991) find that investors have loss aversion, meaning that they are more sensitive to losses than to gains. This implies that comparing downside risk is more relevant to evaluate portfolio performance than by comparing the Sharpe ratios of portfolios. In 1983 Brian M. Rom constructed the Sortino ratio, which calculates the downside risk-adjusted return of a portfolio.
𝑆𝑜𝑟𝑡𝑖𝑛𝑜 𝑟𝑎𝑡𝑖𝑜 =𝑅 − 𝑅! 𝐷𝑅 𝑅 = Average return of the portfolio
𝑅! = Risk-free rate (one-month T-bill rate). 𝐷𝑅 = Downside deviation 𝐷𝑅 =1 𝑛 (𝑅!− 𝑅!)!∙ 𝐹(𝑅) ! !!! n = Number of observations
𝑅! = Return of the portfolio at time t 𝐹(𝑅) = 1 if 𝑅!− 𝑅! < 0
𝐹(𝑅) = 0 if 𝑅!− 𝑅! ≥ 0
Finally I do an Ordinary Least Squares (OLS) regression of the returns of each of the portfolios against the four factors of Fama and French (1993) and Carhart (1997).
𝑅 = 𝛽! 𝑅!− 𝑅𝑓 + 𝛽!∙ 𝑆𝑀𝐵 + 𝛽!∙ 𝐻𝑀𝐿 + 𝛼 𝑅! = Return of the market (or return of other benchmark index)
𝑅𝑓 = Risk-free rate (one-month T-bill rate)
𝛽! = Coefficient for market risk factor (market beta)
𝛽! = Coefficient for small market capitalization (SMB) factor
𝛽! = Coefficient for low price-to-book ratio (HML) factor 𝛼 = Four-factor alpha
Besides the four factors, the Carhart equation has a constant named the four-factor alpha. This alpha is the unexplained excess return of the portfolio over the benchmark index after
adjusting for the three risk factors. The four-factor alpha indicates whether a portfolio earned an excess return relative to the benchmark index. When the four-factor alpha is negative the portfolio did not earn an excess return relative to the benchmark index and when the four-factor alpha is positive the portfolio did earn an excess relative to the benchmark index. I use the database of author Kenneth R. French’s website for the data on the four factors of Fama, French and Carhart.
3.3 Propositions to be examined
By combining the PFR weighting approach with the firm classification approach of Danielson and Dowdell’s (2001), I expect to arrive at an optimal investment strategy in which the PFR weighted portfolio consisting of declining firms has higher risk-adjusted returns than the PFR weighted portfolio consisting of growth firms.
Proposition 1: The PFR weighted portfolios consisting of declining firms has higher
risk-adjusted returns than the PFR weighted portfolio consisting of growth firms.
Moreover I expect all PFR weighted portfolios to have higher risk-adjusted returns than their market-cap weighted counterparts.
Proposition 2: All PFR weighted portfolios have higher risk-adjusted returns than their
market-cap weighted counterparts.
Proposition 3: The PFR weighted portfolios consisting of declining and turnaround
portfolios have higher risk-adjusted returns than the market portfolio and the RAFI US 1000 index.
Finally I expect that all the PFR weighted portfolios earn larger excess returns over the market portfolio during years where the market portfolio has negative returns, such as during the Internet Bubble (2000-2002) and the financial crisis (2007-2009).
Proposition 4: All PFR weighted portfolios have larger excess returns over the market
portfolio during years in which the market portfolio has negative returns, such as during the Internet Bubble (2000-2002) and the financial crisis (2007-2009)
4. Results
4.1 Declining and turnaround firms versus growth firms
Table 1 below shows the descriptive statistics of all PFR weighted portfolios for each five-year period and the full 20-year period. As expected, over the entire period of 1995 to 2015, the PFR weighted portfolio of declining firms has significantly higher annual returns (6.08%, p<0.01) than the PFR weighted portfolio of growth firms. Moreover when the returns are adjusted for risk, the PFR weighted portfolio of declining firms has the highest returns of all PFR weighted portfolios. Table 2 below shows that the PFR weighted portfolio of
declining firms has the highest Sharpe, Treynor and Sortino ratio of all PFR weighted
Table 1: Performance of all PFR weighted portfolios
This table summarizes the returns of each portfolio per five-year period in the total period of 1995 to 2015. The returns are the average annual logarithmic returns of the portfolios in that period.
1995-2000 2000-2005 2005-2010 2010-2015 1995-2015
Declining firms
Number of firms 77 246 27 299
Average annual log returns 12.5% 26.8% 11.7% 7.7% 14.7%
Standard deviation 7.2% 19.0% 27.6% 11.6% 18.2%
Growth firms
Number of firms 1236 1750 2530 1945
Average annual log returns 8.2% 13.3% 6.6% 6.3% 8.6%
Standard deviation 4.7% 12.35% 22.1% 6.6% 12.5%
Turnaround firms
Number of firms 220 285 218 595
Average annual log returns 11.6% 20.0% 8.9% 11.3% 13%
Standard deviation 6.8% 15.6% 33.7% 9.1% 18.3%
Value firms
Number of firms 129 190 52 274
Average annual log returns 5.5% 17.3% 4.3% 6.2% 8.3%
Standard deviation 6.6% 9.2% 16% 7.9% 11.1%
Total number of firms 1662 2471 2827 3113
Average annual log returns 9.7% 18.2% 7.7% 8.1% 10.9%
Standard deviation 5.9% 14.3% 25.8% 8.5% 15%
RAFI US 1000 index
Average annual log returns -* 1.07% -1.2% 5.7% 1.83%
Standard deviation -* 8% 14.4% 5% 9.7%
*The RAFI US 1000 index did not exist prior to 2000.
Table 2: Sharpe ratio, Sortino ratio and Treynor ratio of PFR weighted portfolios
Sharpe Sortino Treynor
4.1 Market-cap weighting
Table 5 below shows the descriptive statistics of all market-cap weighted portfolios for each five-year period and the full 20-year period. The average return of all market-cap weighted portfolios is higher (1.37%, p<0.05) than that of the market portfolio; this difference could be caused by a survivorship bias in the data. As this difference is significant this does indicate that the survivorship bias could have significant influence on the results. However the survivorship bias only has influence when comparing the returns of the portfolio with the market returns and not when returns between the portfolios are compared.
The differences in performance between portfolios of declining and growth firms are small when the portfolios are weighted by market capitalization. The market-cap weighted portfolio of declining firms does have an insignificant higher annual return (0.7%, p=0.80) than the market-cap weighted portfolio of growth firms over the full period, but the returns of the declining firm portfolio are also much more volatile. When the Sharpe ratio adjusts the returns for volatility shown in Table 4 below, the market-cap growth portfolio even has higher returns. However adjusted for systematic risk the market-cap weighted portfolio of declining firms does have a slightly higher return than the market-cap weighted portfolio of growth firms.
portfolio of all market-cap weighted portfolios, according to the four-factor model of Carhart (1997) shown in Table 9 below.
4.2 PFR weighting versus market-cap weighting
All PFR weighted portfolios have significantly higher annual returns than their market-cap weighted counterparts. The difference is largest for the modified portfolio of declining (6.59%, p<0.05) and the portfolio of turnaround (7.13%, p<0.05) firms. Moreover the Sharpe, Treynor and Sortino ratios of all the PFR weighted portfolios in Table 2 above are higher than those of the market-cap weighted portfolios in Table 4 below. Indeed the worst performing PFR weighted portfolio, that of growth stocks, has higher returns for every risk adjustment than the best performing market-cap weighted portfolio, the modified market-cap weighted portfolio of declining firms. Consequently Proposition 2, stating that all PFR weighted portfolios have higher risk-adjusted returns than their market-cap weighted counterparts, is confirmed.
4.3 PFR weighted portfolios versus the market portfolio
The PFR weighted portfolios all have significant larger annual returns than the market portfolio in the period of 1995 to 2015. The differences are again the largest for the PFR weighted portfolios of declining (11.63%, p<0.01) and turnaround (9.91%, p<0.01) firms, shown below in Table 5. The possible survivorship bias in the data could account for part of the difference, however this cannot explain the full 11.63% difference in annual returns.
Table 3: Average market capitalization of firms in the portfolios consisting of declining firms in millions of dollars
1995 2000 2005 2010 Average PFR weighted 307 427 410 827 493 Market-cap weighted 4836 26063 1612 25783 14573 Market-cap weighted without outliers 1143 217 333 117 453
Table 4: Sharpe ratio, Sortino ratio and Treynor ratio of market-cap weighted portfolios
Sharpe Sortino Treynor
Table 5: Performance of all market-cap weighted portfolios
This table summarizes the returns of each market-cap weighted portfolio per five-year period in the years 1995 to 2015. The returns are the average annual logarithmic returns of the portfolios in that period.
1995-2000 2000-2005 2005-2010 2010-2015 1995-2015
Declining firms
Number of firms 77 246 27 299
Average annual log returns 1.3% 10.5% -1.6% 7.2% 4.3%
Standard deviation 10.3% 24.6% 13.1% 7.4% 14.9%
Declining firms modified
Number of firms 75 239 25 270
Average annual log returns 3.87% 18.8% 2.24% 7.5% 8.09
Standard deviation 10.0% 21.3% 16.7% 8.98% 15.37
Growth firms
Number of firms 1236 1750 2530 1945
Average annual log returns 10.0% -1.5% -0.3% 6.4% 3.6%
Standard deviation 1.56% 8.4% 9.8% 3.9% 8.2%
Turnaround firms
Number of firms 220 285 218 595
Average annual log returns 6.29% 3.9% 7.3% 5.8% 5.8%
Standard deviation 8.26% 13.8% 32% 9.5% 17%
Value firms
Number of firms 129 190 52 274
Average annual log returns 2.0% 3.9% 1.8% 4.22% 3%
Standard deviation 6.1% 5.9% 13.8% 4.9% 7.9%
Total number of firms 1442 2185 2608 2518
Average annual log returns 8.8% 0.9% 1% 6.9% 4.4%
Standard deviation 1.2% 10.3% 15.9% 5.4% 9.7%
Market portfolio
Average annual log returns 8.8% -2.1% -0.8% 6.3% 3%
Standard deviation 2.1% 9.7% 12% 4.6% 8.9%
In Table 6 below, the differences in annual returns between the PFR weighted portfolios and the market portfolio are even larger in times of market downturn. The PFR weighted portfolio of declining firms has an annual return that is 29.31% (p=0.13) larger during the Internet Bubble (2000-2002) and 10.52% larger (p=0.53) larger during the
financial crisis (2007-2009) than the annual return of the market portfolio. All PFR weighted portfolios together have an average annual return that is significantly 19.98% (p=0.08) larger during the Internet Bubble (2000-2002) and 9.84% (p=0.51) larger during the financial crisis
Table 6: One-sided t-test for the differences in annual logarithmic returns between the PFR weighted portfolios and the market portfolio (1995-2015)
*** significant at 0.01 level, ** significant at 0.05 level, * significant at 0.1 level
The difference in annual returns between the PFR weighted portfolios and the market portfolio is even larger in times of market downturn, shown in Table 7 below. The PFR weighted portfolio of declining firms now has annual returns that are respectively 29.31% (p=0.13) larger than the returns of the market portfolio during the Internet Bubble and 10.52% (p=0.53) larger during the financial crisis. All PFR weighted portfolios together have an average annual return that is 19.98% (p=0.08) larger than the annual return of the market portfolio of during the Internet Bubble (2000-2002) and 9.84% (p=0.51) larger during the financial crisis (2007-2009).
Table 7: One sided t-test for the difference in annual log returns between PFR weighted portfolios and the market-cap weighted portfolios during the Internet Bubble (2000-2002) and the financial crisis (2007-2009)
Internet Bubble Financial crisis
Declining firms +29.31 +10.52
Growth firms +16.02* +9.38
Turnaround firms +20.84** +11.98
Value firms +21.69** +5.23
Average of all PFR weighted portfolios +19.98* +9.84
*** significant at 0.01 level, ** significant at 0.05 level, * significant at 0.1 level
The results of the regression using the four-factor model of Carhart (1997) are
presented below in Table 8 for the PFR weighted portfolios and in Table 9 for the market-cap weighted portfolios. The four-factor model explains a reasonable part of the variation in returns, resulting in values of the adjusted R-squared between 0.65 and 0.95. The small minus big (SMB) coefficient for the PFR weighted portfolios of declining and turnaround firms is high, indicating that part of the high returns of both portfolios is due to the inclusion of on average small firms.
Average annual return differences
Declining firms +11.63%***
Growth firms +5.55%**
Turnaround firms +9.91%***
Table 8: Four factor alpha, beta, HML, SMB, MOM and R-squared of PFR weighted portfolios with the market portfolio as benchmark index (1995-2015 and 2000-2015 for the RAFI US 1000 index)
Alpha Beta HML SMB MOM Adjusted R-squared
Declining firms 9.52*** 1.05*** 1.01** 1.81*** -0.30** 0.71
Growth firms 5.01*** 0.84*** 0.42** 0.99*** -0.29*** 0.86
Turnaround firms 7.62*** 1.27*** 0.71** 1.26*** -0.44*** 0.85
Value firms 4.65** 0.79*** 0.83*** 1.00*** -0.07 0.65
RAFI US 1000 -0.64 1.00*** 0.58*** -0.06 -0.07* 0.95
*** significant at 0.01 level, ** significant at 0.05 level, * significant at 0.1 level
The HML coefficient is large for the PFR weighted portfolio of declining firms. This means that the high returns of the PFR weighted portfolio of declining firms can be partly explained by the inclusion of many high book-to-market firms that have higher risk. Nevertheless after controlling for the four risk factors, all four-factor alphas are large, positive and significant. This indicates that according to the Carhart four-factor model, all PFR weighted portfolios earned excess returns over the market portfolio. The PFR weighted portfolios of declining and turnaround firms have the highest excess returns over the market portfolio with four-factor alphas of 9.52% (p<0.01) for the PFR weighted portfolio of declining firms and 7.62% (p<0.01) for the PFR weighted portfolio of turnaround firms.
Table 9: Four-factor alpha, beta, HML, SMB, MOM and R-squared of market-cap weighted portfolios with the market portfolio as benchmark index. (1995-2015)
Alpha Beta HML SMB MOM Adjusted R-squared
Declining firms 0.04 1.05*** 0.13 1.77*** -0.01 0.65
Declining firms modified 4.00 1.03*** 0.22 1.86*** -0.32 0.61
Growth firms 0.88 0.92*** 0.01 -0.10** -0.01 0.99
Turnaround firms 0.20 1.59*** 1.14** -0.35 -0.09 0.61
Value firms 0.72 0.61*** 0.12 0.51* 0 0.41
*** significant at 0.01 level, ** significant at 0.05 level, * significant at 0.1 level
All four-factor alphas of the market capitalization weighted portfolios are lower than those of the PFR weighted portfolios, although all four-factor alphas for the market-cap weighted portfolios are not statistically significant. This means that the market-cap weighted portfolios did not earn significant excess returns over the market portfolio according to the Carhart four-factor model.
Table 10: Four factor alpha, beta, HML, SMB, MOM and R-squared of PFR weighted portfolios with the S&P 500 Value index as benchmark index. (1995-2015)
Alpha Beta HML SMB MOM Adjusted R-squared
Declining firms 11.35*** 0.95*** 0.61 1.79*** -0.32** 0.68
Turnaround firms 11.04*** 1.31*** 0.18 1.29** -0.44*** 0.84
Value firms 5.96*** 0.76*** 0.52** 0.99*** -0.09 0.64
*** significant at 0.01 level, ** significant at 0.05 level, * significant at 0.1 level
4.4 PFR weighted portfolios versus the RAFI US 1000 index
All PFR weighted portfolios have significantly larger returns than the RAFI US 1000 index, shown below in Table 11. The difference in returns is largest for the PFR weighted portfolios of declining (13.56%, p<0.01) and turnaround (11.58%, p<0.01) firms.
Table 11: One sided t-test for the difference in annual log returns of PFR weighted portfolios over the fundamental value weighted RAFI US 1000 index.
Mean
Declining firms +13.56***
Growth firms +6.89***
Turnaround firms +11.58***
Value firms +7.44***
*** significant at 0.01 level, ** significant at 0.05 level, *significant at 0.1 level
In Table 2 above it appears that the RAFI US 1000 has lower returns for every risk
adjustment compared to the PFR weighted portfolios. Moreover in Table 8 the four-factor alpha is negative, although statistically insignificant, which implies that the RAFI US 1000 did not outperform the market in the period of 2000 to 2015. Finally I use the RAFI US 1000 index as benchmark index in the four-factor analysis. Using the RAFI US 1000 index as benchmark index improves the regression model, as all adjusted R-squared values increase. This means that with the RAFI US 1000 index as benchmark index, the model explains more of the variation in returns of the PFR weighted portfolios. The resulting coefficients are summarized below in Table 12.
Table 12: Four-factor alpha, beta, HML, SMB, MOM and R-squared of PFR weighted portfolios with the fundamental value weighted RAFI US 1000 index as benchmark index. (2000-2015)
Alpha Beta HML SMB MOM Adjusted R-squared
Declining firms 6.68** 0.96** 1.11* 2.26*** -0.31* 0.79
Growth firms 3.91** 0.78*** 0.23 1.28*** -0.27*** 0.91
Turnaround firms 6.92*** 1.25*** 0.23 1.60*** -0.38*** 0.89
Value firms 5.07** 0.84*** 0.27 1.32** -0.02 0.68
Although the four-factor alphas of all portfolios are smaller compared to the four-factor analysis with the market portfolio as benchmark portfolio, all four-factor alphas are still large and significant. This means that according to the Carhart four-factor model, all PFR weighted portfolios earn significant excess returns over the RAFI US 1000 index. Therefore proposition 3, stating that the PFR weighted portfolios of declining and turnaround portfolios have higher risk-adjusted returns than the market portfolio and the RAFI US 1000 index, is confirmed. 4.4 Liquidity effect
The four-factor analysis already shows that part of the high annual returns of the portfolio of declining firms is due to the inclusion of smaller firms. I want to test whether this small firm effect is partly due to liquidity risk. First, Table 13 below shows that the average size in terms of market capitalization as a matter of fact is lower for the PFR weighted declining portfolio compared to the other PFR weighted portfolios.
Table 13: Average market capitalization of firms in PFR weighted portfolios in millions of dollars
1995 2000 2005 2010 Average
Declining 307 427 410 827 493
Growth 1636 2416 3849 4021 2981
Value 749 1291 1730 4324 2023
Turnaround 1392 185 925 512 754
To test whether this size difference causes a liquidity effect I use trading volume as a proxy for liquidity. Table 14 below shows that the trading volume of the PFR weighted portfolio of declining firms is indeed the lowest of all portfolios followed by the PFR weighted portfolio of turnaround firms.
Table 14: Average trading volume for PFR weighted portfolios over 1995-2013
Declining Growth Turnaround Value
Trading volume 1,456,925 5,851,944 2,788,727 3,875,468
on returns since investors want to be compensated for having less liquid stocks, such as in Amihud (2002). I check whether past trading volume has a negative effect on portfolio returns by using a lagged log trading volume change in the regression. The log trading volume
change now does have a negative effect on log returns, but this effect is statistically insignificant.
5. Discussion and limitations
Declining firms have a larger return(4%, p=0.16) than growth firms when the
portfolios are market-cap weighted. However the downside risk-adjusted return of the market cap weighted portfolio of growth firms is higher than that of declining firms, which is not in line with the findings of Lakonishok, Shleifer and Vishny (1994). A closer look at the returns of both portfolios shows that prior to the Internet Bubble (1995-2000), declining stocks have much lower returns than growth stocks, whereas this is reversed in the period in which the Internet Bubble bursts (2000-2005). The reason for this reversion is probably that declining firms did not profit from the large increases of stock prices of Internet (growth) firms between 1995 and 2000 and did not experience the large collapse of stock prices of Internet (growth) firms between 2000 and 2005. After 2005, the difference between the returns of the two portfolios becomes smaller, which could imply that investors discovered the undervaluation of declining firms and made it disappear.
However the results for the PFR weighted portfolios are different. The PFR weighted portfolio of declining firms has the highest risk-adjusted return of all portfolios for whatever risk investors want to be compensated for. According to the four-factor model of Carhart (1997) the PFR weighted portfolio of declining firms even earns a significant excess return of 9.52% over the market portfolio. This means that once the portfolios are weighted using this new approach the undervaluation of declining firms seems to persist and the results are in line with the findings of Lakonishok, Shleifer and Vishny (1994).
results from a higher demand for the stock which increases the return of the stock. A better proxy for liquidity would be the bid-ask spread as percentage of the stock price. However a complete bid-ask spread liquidity test is beyond the scope of this paper.
5.1 Limitations
All data in this paper is obtained from Thomson Reuters Datastream, which uses Worldscope data to report accounting variables. Most empirical papers that analyze US firms use data from Compustat. Ulbricht and Weiner (2005) find that there is no reason to assume that Compustat data are superior to Worldscope data. They do find that Compustat has a higher coverage of firms in North America in the period before 1997, but that Worldscope has broader coverage of firms in North America from 1997 to 2003. However they conclude that using Compustat or Worldscope has a negligible impact on research results.
The average returns of all market-cap weighted portfolios are significantly (1.37%, p<0.05) larger than the return of the market portfolio, whereas these should not be
significantly different from each other. This could explain part of the excess returns that this paper finds. Moreover the survivorship bias could possibly have a relatively larger effect on the portfolio of declining stocks since these stocks are most likely to go bankrupt and are possibly less covered in the Thomson Reuters Datastream due to their smaller size.
Nevertheless it seems unlikely that this survivorship bias explains the entire 11.63% annual mean excess log return of the PFR weighted declining portfolio.
The period of 20 years studied in this paper seems a long time but it is only a short period in history. If a portfolio proves to have significant higher returns during these 20 years it does not necessarily mean that this portfolio will continue to do so in the future.
Furthermore the results are based on only firms of the NYSE and the NASDAQ, which are probably not representative of all firms. However the NYSE and the NASDAQ together trade a total of more than 4700 firms accounting for one third of all firms publicly traded worldwide. Consequently it is likely that the findings of this paper do (or in a weaker form) apply to other stock indexes as well.
6. Conclusion and directions for future research
do confirm previous findings when the portfolios are weighted using a new approach. This new approach, called proportional financial ratio (PFR) portfolio weighting, combines fundamental and market variables to determine the portfolio weights. The portfolio of
declining firms that is weighted using the PFR approach has higher risk-adjusted returns than the portfolio of growth firms over the entire 20-year period (from 1995 to 2015) studied. This implies that part of the declining firms are still undervalued relative to growth firms and that the PFR weighting approach successfully identifies these undervalued stocks.
All PFR weighted portfolios in this paper have significant larger risk-adjusted returns than their market-cap weighted counterparts. Moreover, according to the four-factor model of Carhart (1997), the PFR weighted portfolio consisting of declining firms has significant excess returns over the market portfolio of 9.52% during a 20-year period. These excess returns are even larger during periods in which the market portfolio has negative returns. In conclusion, the new PFR portfolio weighting approach produces excess returns and works especially well for a portfolio consisting of declining firms.
Future research could apply the PFR weighting approach to stocks that are traded on other exchanges, in different countries. There can be significant differences in performance of the strategy due to institutional, legal and cultural differences. Besides, it would be interesting to examine whether the results are time-dependent by testing whether the PFR weighted portfolios consisting of declining firms earn similar excess returns in the years prior to 1995.
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