In the world of investing and corporate finance, the Efficient Market Hypothesis (EMH) casts a long shadow. EMH states that a sufficiently liquid market reflects the “correct” price at all times. Since efficient markets factor in all known and relevant information at all times, it is therefore practically futile to attempt to predict the future direction of market prices. In other words, a blindfolded monkey throwing darts at the Wall Street Journal has about the same chance as beating the market averages as any professional investor. At one extreme, the founder of Vanguard Investments Jack Bogle revolutionized the mutual fund industry around cheap indexing, which he posited as the solution to efficient markets. At the other, Warren Buffet’s seminal essay, *The Super-Investors of Graham and Doddes-ville*, defends the notion that right-headed investors can carve out a significant edge [1. The Super-Investors of Graham and Doddes-ville]. In the middle, you have the greater majority of investors who will likely cede that both extremes contain some amount of the truth. Even 2013 Nobel Laureate Eugene Fama, of the University of Chicago Booth School of Business, who is credited with developing EMH, has stated that “[asset prices] are typically right and wrong about half the time” [2. The Super-Brainy Quote]. Being able to determine when they are right and when they are wrong is *the* holy grail to traders and investors alike. In order to investigate how correctly assets prices reflect all known information, we must develop an intuition and methodology for estimating the fair value of an asset. As we will discuss, just because a methodology is *descriptive* does not mean it is *predictive* (i.e., correlation does not imply causation).

EMH can be tested by creating a fair value (FV) model for a given stock index. In this study, we will focus exclusively on the S&P 500. If the calculated fair value of the S&P 500 closely enough resembles the published index price, we may conclude that the model framework and variable selection are *descriptive*. For purposes of this study, we require that calculated fair value have at least a .9 correlation to the index with 99% confidence in order to be considered descriptive.

In addition, we will also be testing if the fair value calculation results in a predictive model. If found to be predictive, our model violates EMH. In theory, there should *no arbitrage*, meaning that a change in the fair value estimate should not be able to predict a change in the index. This sort of violation would indicate that our fair value model could be used to produce excess trading profits. In order to test for violations of no arbitrage, we will compare the lagged correlations in the fair value versus the index value. We lag variables by a week because information must available prior to actually being utilized in decisions. As a rule of thumb, and for our purposes, a lagged absolute correlation in excess of .1 is statistically significant and sufficient to produce excess returns. We will test lagged correlation at the 99% confidence level.

A simple fair model which utilizes the most relevant market information available demonstrates that the published S&P 500 index can be described with at least a .90 correlation with 99% confidence. The lagged correlation of the logarithmic changes between the calculated fair value and the index are not statistically significant. The results fail to discredit the EMH. The findings also corroborate a few important points on asset valuation:

- Discrete cash flows to capital investors, whether due to operations or asset sales, are the most important factor in determining the fair value of a capital project/investment;
- The discount rate represents the opportunity cost of pursuing an equally risky capital project/investment;
- Cash flows to equity should be discounted at the Cost of Capital; many investors mistakenly benchmark the discount rate to U.S. Treasury Yields; and,
- The broad market is
*mostly*efficient.

Exploring other facets of asset valuation, including the interpretation book (i.e., accounting) values, is beyond the scope of this post, yet our investigation will corroborate that discounted cash flow analysis is the most relevant and versatile valuation method available. More importantly, the third bullet above about the *right* discount rate is somewhat novel and will discussed at the conclusion of this study.

**Investment Valuation Framework**

In order to derive an appropriate fair value model of the S&P 500 Index, we must first understand what an investment is worth and how to value that worth. On the most basic level, the present value of any investment or capital project is equal to the sum of the discounted cash flows it throws off. For those who are unfamiliar with the concept of the “time value of money”, I suggest reviewing the eponymous Wikipedia post. In mathematical terms, the present value of streams of cash flows to be received in the future is equal to equation (1):

(1)
The value of an investment at time ‘‘;

The future value of a cash flow to be received at time ‘‘;

The discount rate which reflects a real (inflation adjusted) risk-free rate, opportunity costs, and risk premiums that compensate investors for bearing distinct types of risk [3. CFA Institute]; and,

The terminal time vector.

When the stream of discrete cash flows is constant and equally dispersed throughout time into perpetuity, equation (1) can be simplified into the perpetuity equation, equation equation (2):

(2)

While equations (1) and (2) are fairly straightforward as presented, in the real world cash flows are neither constant, perpetual, nor evenly dispersed throughout time. For this reason, we often introduce a variable to represent the growth rate of cash flows, G. The present value of growing cash flows can be represented by equation (3):

(3)
The growth rate of cash flows at time ‘‘; our approach will utilize a capitalization-analyst weighted average of “analysts’ long-term (3 to 5 year) EPS growth rate” for all companies in the S&P 500 index as it has been calculated using the Compustat North America Database.

Since the “real” world imposes several constraints on our ability forecast future cash flows and the growth rates thereof, we combine equations (1) and (3) into usable framework for valuation. Equation (4) represents the notion that analysts are able to only reliably predict forward earnings two years into the future and the growth rate thereof for an additional one to three years thereafter:

(4)

Unlike bonds, equity represents an indefinite claim on excess future cash flows. Although in reality, no corporate entity lasts forever, it is due to this indefinite claim that we are able to simplify our model by assuming perpetuity. If indeed we can assume perpetuity of cash flows, we can use equation (2) to rewrite equation (4) as follows:

(5)

Equation (5) gives us a workable valuation framework. We may with discretion extend the importance of the growth variable in the model by adding terms to equation (5) in which cash flows grow. We could also eliminate our assumptions about perpetuity by reverting to a discrete cash flow model, vis-a-vis equations (1) and (4). In spite all possible variations, we will continue to build on equation (5); three terms of discrete growth are appropriate since this represents analysts’ inability to forecast growth in excess of a few years. Even though equation (5) provides a good valuation framework, we still need to define how we derive our variables that represent cash flow and the discount rate.

**Cash Flows**

The variables representing cash flows and the discount rate to the right of the equalities in equations (1) through (5) can be broken into various components. For our purposes, we will assume that only discrete cash flows to equity count. The only types of discrete cash flows that are received by equity holders are dividends and cash changes to equity (e.g., stock offerings and buybacks). We represent cash flows (the ones that we care about) through equations (6) and (7):

(6)
where:

Analysts’ Forecast for Earnings Per Share (EPS) at time ‘‘; the weighted average of the forward EPS for years 1 and 2 is used past year 2.

(7)

For the purposes of this research, we assume that over the long-run, the difference between the stock repurchases and new offerings nets out to zero. In an idealized world, corporations raise capital by selling stock which at the time of issuance (excluding underwriting fees and taxes) is essentially money for free. As the company begins to generate cash flows, management may decide to return capital to investors through dividend and buybacks. When the stock price appreciates and the company buys back stock back at a price greater than it was issued, this increases the firm’s cost of capital. This scenario, however, is ideal. For the broad market, share issues and buybacks are just likely to create value as they are to dilute/destroy it.

Over the last few years, larger companies have been net buyers of their own shares. Although this sort of financial engineering has the potential to unlock shareholder value by shrinking the denominator of per share metrics like EPS, the recent surge of buybacks even has value-conscious Warren Buffett worried that corporations are buying their stock at the expense of investing in future growth. Moreover, the broader perspective on net payout to investors hinges heavily on how it is measured. Traditionally among academic circles, buybacks have been measured as the combination of two cash flows items ‘Equity Purchased’ less ‘Equity Issued’ [4. Gray, Vogel: Dissecting Shareholder Yield] [5. Boudoukh, Michaely, Richardson, Roberts: On the Importance of Measuring Payout Yield: Implications for Empirical Asset Pricing]. Academics typically measure buyback yields as follow: (8)

Data points for corporate actions which affect shares outstanding are typically found on the Statement of Cash Flows. These items, however, are hopelessly inadequate at painting an overall picture of the true flow of value. Although equity must typically be purchased at the market price, it can be issued for free (e.g., as a executive compensation) which can act as a hidden tax on capital investors. Although SEC rulings backs the conventional wisdom that stock options and restricted stock can be used as a cheap form of compensation which also incents management to make decisions which align with shareholder interests, recent controversy over the widening wealth gap has caused many to wonder whether the pendulum has swung too far. Moreover, some research suggest that while executive compensation leads to short-term earnings increases, it results in no operating cash flow benefit [6. Aboody, Hughes, Ke, Ross: Top Executives Insider Trading Profits and Firm Performance]. Cash is, after all, king. An alternative and more accurate method to demonstrate the long-term zero-sum effects on stock issuance and buybacks is demonstrated by examining the changes in shares outstanding from quarter to quarter. In order to compare on an apples-to-apples basis, historical shares must account for splits.

(9)

The trends of these two methods for measuring the effects of stock issuances and buybacks are shown below in Figure (1).

Figure (1)

The point of drudging through all of this is to justify how, over the long-run, the effects of share issues and buybacks net out to zero. Obviously, in the microcosms of single companies, well-timed issues and buybacks can create long-term shareholder value. However, these corporate actions are probably just as likely to destroy value, especially during times of market euphoria. In our building our fair value equation, we will assume that these types of corporate actions elicit no net effect on market valuations.

**Discount Rate**

The discount rate is an underlying theme in corporate finance. Determining the correct discount rate is truly the alchemy of finance. It is a hurdle which represents the opportunity cost and risk assumed for taking on a given capital project. For many investors, there is a lot of confusion over what the discount rate represents and how to derive the right one. Even Warren Buffett and Charlie Munger, two of the greatest investors ever, are unclear about the meaning and derivation of the discount rate. Although Buffett’s position that “you can’t compensate for risk by using a high discount rate” is spot on, he conflates the issue by using Treasury Yields in the denominator and “Margin of Safety” factors in the numerator [7. WACC Is Flawed, Use Warren Buffett’s Approach Instead] [8. Discounted Cash Flow: What Discount Rate To Use?] [9. Warren Buffett’s Discount Rate used in DCF]. This method is wrong. There is no difference between using a 50% margin of safety factor and doubling the discount rate. Moreover, while Treasury Yields *compete* with stock market yields for investor dollars (vis-a-vis, Alan Greenspan’s intuition which gave rise to the popular Fed Model), they are not the most appropriate benchmark for discounting capital projects and investments [10. Fed Model, Wikipedia]. Even though I harbor the deepest respect for the great value investors of old, the point is to stand on the shoulders of giants who came before us, not rest on their laurels.

U.S. Treasury yields do not provide an appropriate discount rate used in time value estimates for the vast majority of corporations and individuals. Aside from the largest banks and lending institutions, no one can borrow at those rates. Instead, the overwhelming majority of individuals and corporations must borrow at prevailing market rates. The total amount of cash used to recompense investors, in the form of dividends, buybacks, debt repayments, and interest forms the basis for the true corporate hurdle rate. The ramifications of this insight favors libertarianism since it would mean that the Federal Reserve’s monetary policy (i.e., setting interest rates for large lending institutions) has little direct effect on borrowing costs for the vast majority of the population nor on or employment. By setting the discount window rate to ultra-low levels, the Fed allows banks to borrow for essentially nothing, but has no recourse on them to set the rates at which banks can lend that money. The original idea, I’m sure, was to allow banks to compete for our business by offering lower interest rates on loans and mortgages, thereby lowering the cost of capital for everyone. Market forces seem to have elicited unintended consequences for top-down planners, as often is the case. Ultra low interest rate policies have likely exacerbated the widening wealth gap in the U.S [11. Citation Needed]. I digress.

Calculating the true cost of capital is actually somewhat simple. The total amount of cash used to recompense investors, in the form of dividends and interest forms the basis for the true corporate hurdle rate. This rate, depending on the context, is often referred to as the weighted average cost of capital (WACC). In our implementation of the market’s WACC, we ignore stock and debt issuance and retirement (i.e., buybacks) for two reasons: (1) the long-term net effect of capital issuance and buybacks are nil; and, (2) negative discount rates lead to illogical results; depending on how it is measured, the effect of new issues could result in extended periods in which the calculated discount rate is negative. Equation (10) converts these intuitions into a rate formula:

(10a)

Or, in terms more familiar to proponents of the Capital Asset Pricing Model (CAPM):

(10b)

Which, discretized according to earlier conventions is:

(10c)

where:

Suffice it to say that (10a) is the simplest expression for the intended metric. Proponents of CAPM need only determine unlevered cost of equity through past market volatility in order for aforesaid framework to coincide with the canonical asset pricing framework. It is this author’s opinion however that CAPM is a naive extrapolation mechanism for asset pricing which regards only expected mean and variance. Moreover, it is also this author’s opinion that observed fluctuations in market price contain almost no information pertaining to the fundamental cost of equity.

On an extended note, we use the market prices (when available) to determine the cost of capital since they represent the reality of the present in whatever time period they are examined. Even though this approach may cause the WACC to fluctuate wildly in times of asset price turmoil, “mark-to-market” is far more reliable than the alternative of “mark-to-myth”. Furthermore, in our actual implementation of the WACC, we ignore banks and diversified financial firms. Due to the way in which interest payment costs are reported on Compustat’s North American database, unraveling the conflation between interest payments to investors and the discount window is not currently feasible. Using the assumptions encapsulated in equation (8), Figure (2) demonstrates the wide disparity between yields on the Ten Year U.S. Treasury Note (the most popular Treasury benchmark) and the aggregated cost of capital for all companies in the S&P 500:

Figure (2)

**Fair Value**

Having defined the parameters of our fair value equation, we now fill in the parameters for equation (5). The result is then normalized using an estimate for the number of shares in the index. The index shares are necessary for direct comparison of our fair value calculation to the published index price. The actual calculation for shares in S&P Indices is somewhat complicated in order to account for index rebalancing and corporate actions. [12. S&P 500 Equal Weight Index Methodology] [13. S&P U.S. Indices Methodology] [14. Index Mathematics Methodology]. Rather than replicate S&P’s methodology, we infer the number of shares based on the sum of individuals companies’ market capitalizations and the published index price. The final fair value and index shares are described in our final two equations, equations (11) and (12):

(11) ;

and (12) ;

where:

= the sum of all individual market capitalizations in the index, and;

the published price of the index.

The result of the final fair equation equation results in a calculated index fair value which is directly comparable to the published index price. In the case of Figure (3), we are comparing the fair value of the S&P 500 versus its published value on a weekly frequency. Figure (4) compares the weekly natural logarithmic change of the calculated Index Fair Value versus the change of the published index.

Figure(3)

Figure(4)

**Statistical Results**

Table (1): Correlation Confidence Intervals of the Calculated Index Fair Value and the S&P 500 Price

Sample Size: 772

Sample Correlation (Pearson): 0.9545

Degrees of Freedom: 771

Alpha: 0.01

Correl Upper Limit: 0.9621

Correl Lower Limit: 0.9454

Table (2): Correlation Confidence Intervals for the Logarithmic Weekly Change of the Calculated Index Fair Value and the S&P 500 Price (No Lag)

Sample Size: 770

Sample Correlation (Pearson): 0.8344

Degrees of Freedom: 769

Alpha: 0.01

Correl Upper Limit: 0.8606

Correl Lower Limit: 0.8038

Table (3): Correlation Confidence Intervals for the Lagged Logarithmic Weekly Change of the Calculated Index Fair Value and the S&P 500 Price (One Week Lag)

Sample Size: 770

Sample Correlation (Pearson): -0.0124

Degrees of Freedom: 769

Alpha: 0.01

Correl Upper Limit: 0.0807

Correl Lower Limit: -0.1052

**Analysis of Results**

Our fair value model established in equations (5) through (12) results in graph (1) where we compare the calculated fair value for the S&P 500 Index versus the published price. Just through visual inspection, we can see that they strongly correlate. In statistical terms, they correlate at least .9454 with 99% confidence. These findings substantiate that our assumptions regarding variable selection and articulation are appropriate (i.e., our model provides a “good enough” approximation of the fundamentals of asset valuation).

Graph (2) and Tables (2) and (3) summarize the weekly logarithmic changes in the fair value estimate versus the published index. When there is no lag between the variables, a change in one highly correlates to a change in the other, as would be expected. However, this high level of correlation does not establish a causal link. In other words, we want to know if a change in fair value causes the index to change, or whether fair value is plainly just descriptive without any predictive power to it. In order to investigate causality, we lag the logarithmic changes in the published index by one week. We lag the index itself as opposed to the estimate of its fair value in order to ensure that we have time to act on changes to our estimate of fair value. This test produces no correlation that is statistically significant above the .1 level. This finding fails to discredit either EMH or the no arbitrage principle. In practical terms, it does not appear as though the fair value calculation would allow us to partake in excess trading profits.

**Discussion**

To recap everything, our fair value model corroborates the following assumptions on asset valuation:

- Discrete cash flows to capital investors, whether due to operations or asset sales, are the most important factor in determining the fair value of a capital project/investment;
- The discount rate represents the opportunity cost of pursuing an equally risky capital project/investment;
- Cash flows to equity should be discounted at the Cost of Capital; many investors mistakenly benchmark the discount rate to U.S. Treasury Yields; and,
- The broad market is
*mostly*efficient.

For the sake of concision, other aspects of asset valuation were not touched on. Perhaps the most important point is that a generalized model for estimating the fair value of a stock index may not be the most appropriate for valuing individual capital investments. Namely, the WACC for individual projects and companies is too idiosyncratic for use as a discount rate. Rather, the discount rate should represent the *required* rate of return on a comparable investment; using an adjusted market WACC to represent capital structure is appropriate:

(13)
where:

The required cash return on debt; can be based on credit ratings and/or an aggregate of peers’ cost of debt; and

The required cash return on equity; can be based on the required return for competing investments and/or an aggregate of peers’ cost of equity.

In addition, while measuring discrete cash flows tends to work for estimating the fair value of a broad stock index, this method will fail to estimate the value of equity which does not pay discrete cash flows. As of 10/20/2014, only 393 (79%) of S&P 500 companies have declared an upcoming dividend. Just because a firm does not plan to pay out discrete cash flows to equity investors does not mean that its equity is worth nothing. To circumvent this problem, investors try to measure the *ability* of an equity to pay out future cash flows rather than the discrete cash flows themselves. Cash flows which are not paid out to investors but are instead reinvested in growth projects have the ability to grow future potential *free cash flows*, meaning that earnings which are reinvested in growth are not worth nothing. This idea of measuring *Free Cash Flow* (FCF) has resulted in several valuation metrics by the same name [15. David Harper, Advanced Financial Statement Analysis: Cash Flow, Investopedia]. Economists’ and academics’ analogue for FCF is *Net Operating Profit After Taxes* (NOPAT). [16. David Harper, EVA: Calculating NOPAT, Investopedia]. Warren Buffett’s construct for *Owner Earning’s* additionally encapsulates the notion for measuring the *ability* of a firm to generate excess cash flows [17. Jae Jun, What is Owner Earnings?, Old School Value].

Estimating fair value by interpreting book (i.e., accounting) values is the other very important side of valuation, yet the nuanced nature of industry and corporate accounting practices makes this approach nearly impracticable from a systematic perspective. The model used herein to describe the fair value of the S&P 500 Index does not touch on the interpretation of the book values of assets; it is far more difficult to systematically estimate a stock index’s fair value based on the book values of individual companies. In theory, accounting entries are supposed to provide a fair depiction of asset values. However, most assets are recorded at cost and depreciated over time; accounting values often have nothing to do with an asset’s ability to generate profit. Moreover, the abilities of firms to generate cash flows are often the result of intellectual (i.e., intangible) properties; firms do have strong incentives to accurately value their intellectual properties in accounting statements. Technology driven firms typically undervalue intangible assets, whereas lending institutions often overvalue their intangible assets. If book value is supposed to represent liquidation value, the bottom line truly is that today’s liquidation value of equity is today’s market price. Since Generally Accepted Accounting Principles (GAAP) are highly specified depending on the particular business and type of business; the ability to read balance sheets to infer whether market prices correctly reflect ‘reality’ is an art best suited to highly trained and highly specialized industry analysts. In order to be reconcile how book values articulate with cash flow analyses in the greater scheme of asset valuation, interested parties could investigate the hypothesis that equity (debt) is a call (put) option on a firm’s value. Although the broader topic of asset valuation is undoubtedly a rich topic, this study lends weight to the position that discounted cash flow analysis may be the most relevant and versatile method available.

Perhaps a novel idea to some, the model derived and described above demonstrates that investors often mistakenly lend too much weight to Federal Reserve and U.S. Department Treasury interest rates when determining an appropriate discount rate. Talking heads on Bloomberg, CNBC, and elsewhere often mention the importance of interest rate policies from the Fed when in fact only banks and other larger lending institutions are able to borrow at these rates. For these large financial institutions, the cost of capital is essentially *nothing*; they borrow at Fed Funds/Treasury rates and lend at an altogether different rate. For the vast majority of enterprises, the rate at which firms are able to finance capital projects and investments provides a far more grounded estimate of the *true* discount rate. This novel point entails broader ramifications for watchers of the Federal Reserve. Particularly, the Fed Funds rate matters mostly to the largest lending institutions; I.e., top-down monetary policy exerts a negligible impact on the vast majority of businesses and investors. Moreover, it has been posited (correctly, I think) that the current regime of ultra-low interest rates simply allows a few of the already rich to get richer.