# «Confidence W. Amadi cwamadi is an Associate Professor of Finance, Department of Accounting, Finance, and Economics, Walter R. Davis ...»

Confidence W. Amadi cwamadi@mail.ecsu.edu is an Associate Professor of Finance,

Department of Accounting, Finance, and Economics, Walter R. Davis School of

Business and Economics, Elizabeth City State University.

Abstract

The estimation of the cost of equity has been the most daunting of all the

variables essential for asset valuation. The most widely used methods for its

determination are the Gordon model, the capital asset pricing model (CAPM), the cost

of debt plus a risk premium, the arbitrage pricing theory (APT) and the residual income model (RIV). The Gordon model is limited to firms that pay dividends that are assumed to grow at a constant rate. The CAPM is based on a diversified portfolio, and the long- run concept in which the required, expected and realized returns are the same. The cost of debt plus risk premium assumes a levered firm, but does not specify how the risk premium should be determined. The APT is an extension of the CAPM with multiple, albeit, unspecified risk factors. The RIV model solves for the discount rate that equates the present value of the “excess income” to the current stock price. The objective of this paper is to introduce a model for the cost of equity that is based on the relationship between the earnings, the opportunity cost of funds and the concept of risk as the chance of a loss.

1 Introduction The Gordon model is limited to firms that pay dividends that are assumed to grow at a constant rate. It implicitly assumes that the current stock price is the intrinsic value of the firm. Explicitly, the Gordon model calculates the return that equates the present value of the expected dividend stream to the current stock price. The CAPM is based on a diversified portfolio, and the long-run concept in which the required return, expected return and realized return converge to the same value, a concept that only holds in a static environment. In the accounting profession, a variance of the Gordon model, the residual income model (RIV) provides a means for calculating the cost of equity based on a terminal value of the stock price, the earnings of the firm and the current stock price.

Aside from the obvious difficulty in estimating the terminal value of the firm‟s stock price, the derived cost of equity relies on the current stock price. By definition, the current stock price is what the last purchaser of the stock paid for the last share of stock bought. Investors sell or buy stock based on the relationship between the current price and the investors‟ perceived intrinsic value of the stock. In general, trades occur when either the participants believe at the minimum that the security is fairly priced or they have opposing views of over-valued (the seller) and under-valued (the buyer). For listed securities, there exists a wide range of prices based on various investors‟ perception of the value of the security. The current market price is merely the view of the last purchaser. Like all assets, the demand curve of securities must be downward sloping.

Conversely, the supply curve is upward sloping. Unlike other assets, where the intersection of the demand and supply curve represents the equilibrium quantity and price, for securities, the market price is the maximum price consumers are willing to pay to buy a unit of the security at the point in time. As a result, using the current stock price in the determination of the cost of equity raises the question of “whose cost of equity”?

The cost of debt plus a risk premium is another model that is gaining some attention (Lasher, 2002). Under this model, the cost of equity is the firm‟s cost of debt plus a risk premium. “The increment in risk between debt and equity is about the same for high-risk and low-risk firms. That increment tends to command an additional premium of between 3 percent and 5 percent” (p.398). The use of the model is limited to firms that have debt in their capital structure. Moreover, since firms issue debt with various maturities and covenants, the choice of the appropriate cost of debt becomes a concern.

2 There is a universal agreement that the higher the risk, the higher the required return. Unfortunately, there is no single accepted definition of risk. Alles (1995) presents an overview of the various investment risk concepts and how they are measured. Although standard deviation of returns is often viewed as a measure of risk, the introduction of portfolio diversification and its attendant systematic risk has brought into question its validity. Megginson (1997) defines risk as “the chance of a financial loss”. He goes on to say that “the term risk is used interchangeably with uncertainty to refer to variability of returns associated with a given asset” (p95). Thus, in both the capital asset pricing model and the arbitrage pricing theory, it is the later definition of risk as variability from expected return that has prevailed. This concept of risk has very serious limitation.

Consider an investment whose return on investment (ROI) has the following equally likely outcome: 12 percent, 15 percent, 20 percent, 25 percent, 35 percent and 40 percent. This investment return has variability, but is it risky? The risk-free rate is the required return for deferring consumption. It is purely the time value component of return. In the example above, if the risk-free rate were 7%, it can be seen that the investment has no risk. This is because the worst possible outcome is 5% higher than the risk-free rate. This investment has all possible returns that are higher than the required return based purely on time preference for consumption. As a result, its cash flow should be discounted using the risk-free rate of return. This is the basis for this paper.

The objective of this paper is to present a model of asset pricing that is based on

**the chance of a loss definition of risk. The rest of the paper is organized as follows:**

Section 2 gives a brief literature review of the role of earnings in security pricing;

Section 3 presents a brief overview of the most common asset pricing models, and Section 4 presents the chance of a loss (COL) model of risk measurement and asset pricing, Section 5 is on data and methodology, Section 6 presents the results and Section 7, analysis and conclusion.

Brief Role of Earnings Literature Review The purpose of this review is to focus or draw attention to the role of earnings in security valuation. Its role in valuation of assets, via dividend payout and retention ratio, is a well established in finance textbooks and academic literature.

3 Botosan and Plumlee (2005), explore the relationship between five measures of the cost of equity and firm risk with the objective of determining the most appropriate method for calculating the firm specific cost of equity. In their study, the arbitrage pricing theory (APT) is used to calculate the risk premium. The risk factors were identified as unlevered beta from the capital asset pricing model, leverage, “information risk”, market value of equity, book-to-price ratio, and earnings growth. The five methods used for deducing the cost of equity only differed by the method of estimating the terminal value of the firm‟s stock price. Thus, the estimated cost of equity was only as valid as the forecast of the terminal values. Even then, only two of the models were “consistently and predictably related” to their definition of risk.

Francis et al (2004) examined the relationship between the cost of equity capital and seven attributes of earnings and concluded that accounting based attributes such as accrual earnings, has the largest impact on the cost of equity. Their results indicate that the cost of equity is related to earnings, even though the cost of equity under investigation is the Value-Line estimate of the cost of equity. Since Value-line does not disclose the methodology used in its estimate of the cost of equity, the explanatory power could possibly be due to multi-co linearity, if some of the same earnings attributes were used in the estimation.

Daske et al (2006), using the residual income valuation model, estimates a firm‟s expected cost of equity “derived from its stock price and analysts‟ consensus earnings forecasts”. The strength of the model is supposed to lie in the use of expected earnings. This is an ordinary conditional forecasting in which, according to Pindyck and Rubinfeld (1991) “the stochastic nature of the predicted value of the X‟s will lead to forecasts of Y which are less reliable than in the fixed-X case” (p195). In this case, the analyst‟s stock price estimate is the X-variable, while the cost of equity is the Y-variable.

Consequently, the estimated cost of equity is the implied cost of equity that is consistent with the analysts‟ forecast and its reliability is subject to the analyst‟s forecast. The authors in this case use the median of the analyst forecast.

Cheng (2005) investigated the determinants of residual income by analyzing the impact of value-creation and value-recording processes on abnormal return on equity.

They show that the abnormal return on equity (ROE) is positively related to accounting factors, industry monopoly power, and firm monopoly power.

Gode and Ohlson (2004) introduced stochastic interest rates into the “mark-tomarket” and “income-statement approach” to valuation. In the mark to market model, book value equals market value. The income-statement approach assumes that 4 earnings have sufficient information for valuation.They show that the residual income decreases with increase in the beginning period risk-free rate. They demonstrate that the information content of current accounting data decrease with increase in the level of interest rates.

Borgman and Strong (2006) by combining the cost of equity function of the capital asset pricing model and the constant growth model, and relaxing the zero growth inherent in the development of the weighted average cost of capital, show that increase in the earnings growth rate reduces the systematic risk of the firm and hence the required return on equity. Wei and Zhang (2006) using the data from 1976 to 2000 in the US stock markets show that “stock return volatility is negatively related to the return on equity and positively related to the volatility of the return on equity in cross-sections.

Nissim and Penman (2003) show that changes in interest rates have a dual effect on stock price. They show that changes in interest rates are negatively related to residual earnings by affecting the rate at which earnings are charged to the book value of equity used in generating the earnings. Changes in interest rate also affect the level of operating income.

In finance textbooks such as Ross et al (2006), students are taught that cash flow is supreme in the valuation and other operating decisions of the firm. While this may be true for capital budgeting purposes, Francis et al (2003) show that across the board, earnings dominate other performance measures (even in industries where other measures are preferred) in explaining “ex ante security returns”.

The preceding review highlights the significance of earnings in security valuation.

Most of the studies use earnings or its derivative as the numerator of the valuation equation. Although studies have documented the response of security prices to earnings announcement surprises (see Miller 2005), the role of determining the risk of firms has been confined to risk factors such as size, beta, book-to-market ratio, etc.

Earnings variability has not figured prominently in this endeavor.

**Factors Affecting the P/E (E/P) Ratio**

The price-earnings (P/E) ratio is commonly used as a measure of the growth potential and the riskiness of the firm‟s cash flow. The higher the P/E ratio, the higher the expected growth rate in earnings and or the lower the perceived riskiness of the firm‟s cash flow. Low P/E ratio stocks have been termed “value stocks”. Low P/E ratio could also be used to identify “fallen angels”. Low P/E ratio stocks characterize firms with poor growth potential and /or high risk cash flows. The chance of a loss model 5 (COL) relies on this attribute of the P/E ratio and is based on its reciprocal, the earnings yield (E/P).

Constand, Freitas and Sullivan (1991), examined the factors that affect the E/P ratios and market value of Japanese firms. They documented a positive relationship between changes in the standard deviation of earnings (as a proxy for risk), expected earnings growth rate and change in dividends per share. Amoako-Adu and Smith (2002) analyzed the causal relationship between the yield on the 3-month Treasury bill and the average P/E ratio of the Toronto Stock Exchange (TSE) and seven major Canadian industry indices. They found a very strong inverse relationship between the treasury yield and the average P/E ratio, with an adjusted R-squared of 95% for the TSE 300 Index. The utilities industry had the highest R-squared (98%), while the gold and silver industry had the lowest (84 percent).

By calculating the P/E ratio using earnings averaged over a period of eight years, Anderson and Brooks (2006) were able to significantly increase the return premium of value stocks over glamour stocks. Beneda (2002) show that for periods in excess of 14 years, high P/E portfolios out performed lower P/E ratio portfolios, but with an investment horizon of five years, the growth stocks (high P/E) lagged behind value (low P/E) stocks.

Bhargava and Malhotra (2006), studied the effect of price-earnings ratio on world and US indices. They conclude that (after adjusting for statistical estimation problems) price-earnings ratios did not have a significant impact on either future stock prices or earnings yield. Anderson and Brooks (2005) showed that the P/E ratio of a company was influenced by market-wide P/E ratio, the industry sector and the firm size. They showed that after adjusting for the effect of these variables, the ability of the P/E ratio to explain differences in future returns between “glamour stocks and value stocks” was doubled.