Three Factor Model and Real Options

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					Three Factor Model and Real Options: Stock Excess Returns vs. Investment and Operating Flexibility
© Spring 1997

Abstract This study seeks to identify the relationship between two of the three factors specified in the Fama and French's (1992,1995,1996) Three Factor Model and the firm's real options, i.e., investment and operating flexibility. It is argued that size and distress premia represent the observable market-based proxies for the implicit real options embedded in the firmbased strategic investment and operating decisions which are non-tradable in the market. The proposed two accounting-based proxies for real options are: 1) fixed-assets-to-totalassets ratio (FTA) to measure the level of investment flexibility, and 2) degree of operating leverage (DOL) to measure the level of operating flexibility. While the relationship between these two measures are generally positive, their impacts on net real options value are inverse. As a result, size and distress premia are suspected to be correlated with the firm's net real options. This conjecture forms two hypotheses to be tested independently: H1: Size is positively correlated with FTA and DOL resulting in positive net real options. H2: Distress is negatively correlated with FTA and DOL resulting in positive net real options. The ordinary least squares (OLS) model is employed to estimate the coefficients of determination (R2) of the two time-series regressions. The data source and methodology of data transformation are the same as that of Fama and French (1996) for consistency and comparison purposes. It is expected that both hypotheses are statistically significant and not rejected.

Conventional capital asset pricing model (CAPM) specified independently by Sharpe (1964), Lintner (1965), Mossin (1966), and extended by Black (1972) relates the firm's expected rate of return to the term premium of the riskless interest rate and the risk premium of the market. Since its inception, this ex ante model is criticized by many empirical researchers, most notably Fama and French, who attempted to test its predictive power and found that CAPM was not as robust as it is alleged to be. Ross (1975) derives a more generalized version of the CAPM called the arbitrage pricing theory (APT) which allows multiple factors to explain the variability of the firm's expected return, but does not specify any explanatory variable to model it. Relying on the logic behind the APT and the results of their widely quoted test of market efficiency, Fama and French (1992, 1995, 1996) build on the CAPM by specifying two other variables in addition to the market risk premium namely, size premium (market equity value) and distress premium (book-to-price ratio). Based upon this three factor model (TFM), they conclude that securities markets maintain their informational efficiency and that the TFM has a higher explanatory power than the CAPM. However, the inclusion of these two accounting-based variables into the CAPM does not provide the theoretical construct behind the vanished return anomalies and enough reasons as to why size and relative distress can cause the realized abnormal stock returns. Since size and distress are measured from the firm's reported book and market equity values, their linkage to the firm's assets value can also be established in order to see how the firm's investment and operating activities might have some impacts on them. The changes in book and market equity values could signal the investors about the firm's future earnings prospect as a result of its internal strategic investment and operating decisions. Therefore, the search for this type of private information

contained in size and relative distress will enable the investors to form a more accurate probability belief and rational expectations about the future stock price and returns, thereby increasing the predictive power of the CAPM. This paper is divided into four sections. Section one deals with the literary review on TFM and its rationale. The appropriate paradigm to analyze the value of the firm's investment and operating flexibility, or real options, in relation to its size and relative distress is discussed in Section two along with my research hypotheses. Section three characterizes the data set appropriate for the test and proposes the research methodology for testing the hypotheses. Finally, conclusion and recommendation for the operationalization of this study are given in the Section four.

Review of the Three Factor Model Back to Top
Fama and French (1992, 1995, 1996) developed the TFM to explain and capture the cross-sectional excess stock returns. Their studies suggest that size and book-to-price ratio (B/P) add more explanatory power to the standard CAPM in deriving the stock returns after finding out that their combined effects seems to overshadow the effects of leverage, earning-to-price ratio (E/P), and cash-flow-to-price ratio (C/P). Size contributes to the excess returns in the following manner. Small stocks tend to have lower earnings on book equity than do big stocks, controlling for B/P ratios. However, Banz (1981) finds a contradiction that average returns on small stocks are too high given their market b estimates, and average returns on big stocks are too low. Huberman and Kandel (1987) find evidence that there is covariation in the returns on small stocks that is not captured by the market risk premium and is compensated in average returns. Size premium which turns out to have a negative relationship with the stock returns should offer an additional explanation on the cross-sectional behavior of average returns. B/P ratio, on the other hand, is suggested by Chan and Chen (1991) to serve as a proxy in indicating the relative level of distress of the firm. Stattman (1980), Rosenberg, Reid, and Lanstein (1985), and Chan, Hamao, and Lakonishok (1991) conclude that distress premium and average stock returns are strongly positively correlated and are not captured by the market return. Thus, Fama and French model and test these two factors along with the market risk premium and find them able to capture most of the market anomalies in average stock returns which are not explained by the standard CAPM. The TFM's ex ante cross-sectional expected excess returns is given by: E[ri] - rf = bi(E[rm]-rf) + si(SMB) + hi(HML) where E[ri] - rf = expected excess return on portfolio i E[rm] - rf = expected excess return on a market portfolio m SMB = return difference between small-stock and big-stock portfolios HML = return difference between distressed and strong stock portfolios bi, si, hi = coefficients of market risk, size, and distress premium factors Expressing the above equation in terms of ex post time-serial excess returns gives: rit - rf = ait + bit(rmt-rf) + sit(SMB) + hit(HML) + eit where rit - rf = realized excess return on portfolio i at time t rmt - rf = realized excess return on market portfolio m at time t ait = intercept term of the regression line eit = random residual return on portfolio i at time t The rationale for TFM is provided by Fama (1994) based on a generalized mean-variance efficient

(MVE) portfolio concept of Merton's (1973) intertemporal capital asset pricing model (ICAPM). ICAPM investors not only are concerned with the risk-returns tradeoff but also interested in hedging more specific state-variable risks as a result of consumption-investment tradeoff. The optimal portfolios are, therefore, multifactor mean-variance (MMV) efficient. The MMV portfolios are spanned by the risk-free security and any three linearly independent MVE portfolios including two risky statecontingent market portfolios and a zero-covariance market portfolio. In terms of expected excess returns, the intercepts in the regressions are zeros. In terms of realized excess returns, the R 2 of the regressions are equal to one. The candidate MMV portfolios' risk premia identified for TFM are 1) the market premium, 2) the size premium, and 3) the distress premium. The size and distress factors are able to describe the returns based on E/P, C/P, and sales-rank index identified by Lakonishok, Shleifer, and Vishny (1994) as well. One advantage of TFM over the other models is that market, size, and distress premia are less correlated with one another. This makes the TFM regression coefficients easier to interpret. Other proxies for the MMV portfolios are not as robust as those included in the TFM. Sales rank, for example, suffers errorsin-variable problem because this proxy is not adequately diversified thereby confounding the model.

Theoretical Basis on Real Options and Research Hypotheses
Black (1986) links the market price of security to the fundamental value of the firm. Changes in the fundamentals have a direct impact on the expected return and the risk characteristic of the security price. Since the value of the firm is a function of the expected future cash flows discounted by the appropriate cost of capital, the market price of stock is therefore the function of the underlying cash flows of the firm. In the literature on investment under uncertainty, the net present value of the project is not only determined by the expected future cash flows stream and the discount rate, but also the value of the options to commit or delay committing the firm's resource in such investment. Conditioned on uncertain investment costs and future incomes, the net present value of the project has to be adjusted to account for the flexibility in waiting for more information to reduce the uncertainties before making any investment outlay. Therefore, the firm's project value which determines the future prospect of its earnings depends also on the options to invest, delay investment, or abandon an investment altogether. This investment flexibility is referred to as real options or options on real assets as opposed to financial options. Two main features of investment flexibility are irreversibility and deferrability. According to Pindyck (1988), irreversibility means that once the investment decision is made, the capital expenditures incurred in the project are partially or totally sunk costs. In other words, if the firm decides to abandon the project, it has to forego such investment partially or completely. This has an important implication to and impact on the firm's asset allocation decision since irreversibility imposes high exit penalty for ineffectual investment. The degree of irreversibility varies depending on how marketable or liquid the abandoned assets are from which the firm can retrieve some residual value. If it were relatively high, the project is perceived to have low flexibility and thus low real options. If the exit costs were low for the firm to abandon the project, its real options value increases. McDonald and Siegel (1986) treat deferrability as analogous to the time value of financial options in which the firm can opt to delay the investment until favorable market conditions materialize or new information improves the probability of project's success. This source of real options depends on how sustainable the investment opportunity is over a certain period. If the opportunity is either now or never, then deferrability has zero real options value. In contrast, the project can be delayed indefinitely, the real options value approaches infinity and the project may never be implemented. To be consistent with the option pricing paradigm, it is assumed that there exists a certain date that the project must be implemented, or exercise date. However, the investment costs on that exercise date cannot be determined at present the same way as the strike prices on financial options are. Therefore, the value of deferrability depends on exercise date and the information prevailing on that date.

Beside investment flexibility, there are separate treatments of real options which involve operating flexibility. Operating flexibility refers to the latitude in the firm's decision to utilize its assets strategically to enhance earnings potential for the upside opportunities while hedging them for the downside risks. Majd and Pindyck (1987), Carr (1988), and Trigeorgis (1993) identify the staged investment option for the firms engaging in the long-term capital-intensive projects in which each stage in the development can be viewed as an option on the value of subsequent stages. Brennan and Schwartz (1985), McDonald and Siegel (1985), Trigeorgis and Mason (1987), and Pindyck (1988) study the scale alteration option for natural resource and cyclical industries in which the firms can expand or contract their operating scales based upon the expected market conditions. The abandonment option is extended from the scale alteration option by Myers and Majd (1990) that in case of prolonged decline in market conditions, the firms can abandon their current operations permanently and realize the resale value of their assets in secondary market. Margarbe (1978), Kensinger (1987), Kulatilaka (1988), and Kulatilaka and Trigeorgis (1993) classify the switch use option into two types: product flexibility and process flexibility. If prices of or demand for products change, the firms can alter the output mix of the facility. Alternatively, the same products can be produced using different mixes of inputs. The early or preemptive investments in R&D, undeveloped land, strategic acquisition, and infrastructure represent the growth option. Myers (1977), Trigeorgis (1988), Pindyck (1988), Brealey and Myers (1991), and Chung and Charoenwong (1991) find that these types of investment are prerequisite to a chain of interrelated projects which open up future growth opportunities. Finally, different kinds of operating flexibility can be combined as being the collective real options, both upside-potential enhancing calls and downside-protection puts. Their combined options interact and yield different value from the sum of separate option values. However, these real options cannot be easily detected or merely observed from the market-based data which impose even more difficulty on their proxies identification. Nonetheless, we may resort to certain accounting-based measures which are publicly available and objectively enough to proxy for investment and operating flexibility such as the ratio of fixed assets to total assets (FTA) and the degree of operating leverage (DOL). By definition, DOL means the potential use of fixed operating costs including depreciation of fixed assets to magnify the effect of changes in sales on the firm's earnings before interest and taxes (EBIT). It can be derived as follows: DOL = CM/(CM-FC) = %EBIT/%S where CM = contribution margin = sales - total variable costs FC = fixed operating costs S = dollar sales level Changes in fixed costs can affect operating leverage significantly. The firm can incur fixed costs rather than variable costs, or substituting one types of cost for the other. Increase in fixed costs would increase operating leverage and vice versa. FTA measures the level of the firm's commitment in fixed assets which can be proxied for relative size and degree of irreversibility. High FTA means larger fixed-assets base which could incur high exit costs to the firm. Firms with low FTA have a relatively smaller fixed-assets base which provides them with less exit burden and higher investment flexibility. DOL, on the other hand, measures how well the firm utilizes its fixed assets to generate higher earnings before interest and taxes which, in effect, reduce its level of distress and thereby offering high real options. Small firms tend to have lower intensity of fixed capital investment than large firms which provide them with higher investment flexibility and high average stock returns. Distressed firms which have employed more fixed capital investment experience higher operating flexibility and possess high real options which result in their high average stock returns. Both are not captured by market risk premium because they are firm-specific characteristics. This rationale helps shape my conjecture about the relationship between size premium and investment

flexibility and between distress premium and operating flexibility. The following hypotheses are proposed: H1 : Size is positively correlated with FTA and DOL. Small firms tend to have high investment flexibility due to low FTA but low operating flexibility because of low DOL. The net effect on real options is positive since investment flexibility outweighs operating flexibility. H2 : Distress is negatively correlated with FTA and DOL. Distressed firms tend to have low investment flexibility due to high FTA but high operating flexibility as a result of high DOL. The net real options is positive because operating flexibility outweighs investment flexibility. These two hypotheses shall be modified for the regression models in terms of size and distress premia to coincide the transformation of average returns to excess returns.

Proposed Data Source and Research Methodology
To be consistent with Fama and French's (1996) study, the monthly excess returns on four portfolios from NYSE, AMEX, and NASDAQ between 1963 and 1993 based on size and distress proxies are used. The size proxy is derived as follows. At the end of June of each year t, stocks from the three exchanges are allocated to two groups (Small or Big) based on whether their June market equity is below or above the median value. The distress proxy is derived in the same manner as in the size proxy by allocating stocks to two groups (Low or High) based on the breakpoints for the bottom 50% and top 50% of the values of B/P for NYSE stocks. Four size-distress portfolios are defined as the intersections of the two market equity values and the two B/P groups. SMB is the difference between the average returns on the two small-stock portfolios (S/L and S/H) and the average returns on the two big-stock portfolios (B/L and B/H). HML is the difference between the average returns on the two high-distressstock portfolios (S/H and B/H) and the average returns on the two low-distress-stock portfolios (S/L and B/L). For the two real options independent variables, the method of deriving the proxies are similar to the one employed on size and distress dependent variables. The investment flexibility proxy using the firms' FTA ratios are grouped into two: high investment flexibility (HIF) and low investment flexibility (LIF), based on the breakpoints for the bottom 50% and top 50% of the value of FTA. Notice that low FTA signifies HIF and high FTA means LIF. The operating flexibility proxy based on DOL is calculated from the percentage change in earnings before interest and taxes (EBIT) divided by the percentage change in sales. High DOL means high operating flexibility (HOF) while low DOL refers to low operating flexibility (LOF). DIF is the difference between HIF and LIF. DOF is the difference between HOF and LOF. The ordinary least squares (OLS) estimation model is proposed to be a statistical tool to find the relationships between TFM factors and the real options variables. The assumptions for the model are: 1. 2. 3. The time-series patterns of variance are constant. The time-series patterns of random errors are uncorrelated. The explanatory variables, i.e., FTA and DOL, are deterministic.

The regression model for testing H1 is given by: SMBt = t + (DIFt) + (DOFt) + t

The regression model for testing H2 is given by: HMLt = t + (DIFt) + (DOFt) + t where SMBt = return difference between small- and big-stock portfolios at time t HMLt = return difference between distress and glamour portfolios at time t DIFt = difference between high and low investment flexibility at time t DOFt = difference between high and low operating flexibility at time t

Conclusion and Recommendation
The contribution of this paper is for investment theorists, empiricists, and practitioners to gain more understanding about Fama and French's TFM as to why the two additional risk factors, i.e., size and distress, are more parsimonious than other factors identified by previously such as E/P and C/P. Undeniably, there should be an explanation behind these two factors' ability to capture abnormal returns in excess of market risk premium. One of the more distinct explanations is based on the analysis of real options possessed by small and/or distressed firms. Small firms benefit from not committing capital intensive investment which is translated into high investment flexibility, but lose from not having enough fixed expenditures to lever their earnings. The net effect on real options is positive since their investment flexibility has more value than operating flexibility. Distressed firms tend to make high capital investment in fixed assets thereby losing investment flexibility, but gain operating flexibility from potential earnings growth when their fixed assets contribute to the higher DOL. Their net real options value is also positive because operating flexibility in the future outweighs the current investment flexibility. If the hypotheses were true, they would imply that value stocks (i.e., small and/or distressed stocks) should not be undervalued based on their poor earnings and prospects. Their real options values will reveal their firm-specific risk premia to be captured by the market return. This should resolve the market anomaly or behavioral puzzle as to why value stocks earn higher average returns than growth stocks. The same reasons can be used to explain why growth stocks earn lower average returns. It is simply that large and/or glamour stocks do not have the embedded real options or that their real options have been used up or depleted already. These implications should lead to more extensive research about the intra-actions among different kinds of real options and inter-actions between real options and stock returns. The empirical test of my hypotheses will prove that the above implications are true, based on the ordinary least squares (OLS) statistical technique. However, the assumptions on constant variance and uncorrelated errors can be relaxed by using the generalized least squares (GLS) model. The results from both techniques may improve our understanding about the relationships between real options and size and distress premia. After this preliminary study is done, it is recommended that further investigation on the patterns of variance and errors be conducted using the generalized autoregressive conditional heteroskedastic (GARCH) model which will support the use of GLS model suggested previously. I believe the links between market-based and firm-based risk premia can be established through the real options analytical framework. The only problem is how we can identify the proper proxies for real options. My suggestion here is to use the accounting-based measures of fixed assets employment (FTA) and utilization (DOL). Other non-accounting measures may serve as the better proxies but may lack desirable properties of observability and objectivity which are very important for market-based asset valuation and risk-premium evaluation.

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* Worapot Ongkrutaraksa is a lecturer in Finance and Strategic Management at Maejo University's Faculty of Agricultural Business,
Chiang Mai, Thailand. He used to conduct his post-graduate research in financial economics at Kent State University and international political economy at Harvard University through the Fulbright sponsorship between 1995 and 1998.