Costs of equity above 100% or below 7.2% are included in the percentile statistics because they provide valuable information to the reader. Costs of equity to such extremes are indicative of the cost of equity model failing due to the nature of the data for companies in the industry.

CAPM—Ordinary Least Squares (OLS)

where,
k_i = Cost of equity; 
R_f = Rate on risk-free asset; long-term government bond yield for March 31, 1997 (7.2%);
b _i = Levered beta of company i; and,
ERP =Expected equity risk premium. Long-horizon version from Ibbotson Associates' Stocks, Bonds, Bills, and Inflation 1997 Yearbook (7.5%).

CAPM—With Small Capitalization Premium

The CAPM does not fully account for the higher returns of small company stocks. Recent research has shown that the CAPM does not fully account for returns of medium sized companies either. Ibbotson Associates’ Stock, Bonds, Bills, and Inflation 1997 Yearbook gives a detailed analysis of company size and its impact on the CAPM. The column labeled "+ Size Premium" is the OLS form of the capital asset pricing model with the small capitalization premium added. Premiums are determined by the equity capitalization of a company and are simply added to the original CAPM equation. For this publication, the following premiums were added:

For mid-cap companies, whose equity capitalization is at or below $3,320,996,625 but greater than $774,452,250, a premium of 1.04% is added. For low-cap companies, whose equity capitalization is at or below $773,983,875, but greater than $201,911,250, a premium of 1.75% is added. For micro-cap companies, whose equity capitalization is at or below $201,169,500, a premium of 3.47% is added. No premium is added for large-cap companies.

For composites, the size premium is an equity capitalization weighted average of the size premiums of the companies included in the composite.

Three-Factor Fama-French Model

The Fama-French Three Factor Model is a multiple linear regression model developed by Eugene Fama and Kenneth French. The model is estimated by running a time series multiple regression for each company. The dependent variable is the company’s monthly excess stock returns over Treasury bill returns. The independent variables are as follows:

 

• The monthly excess return on the market over Treasury bills.
• SMB ("small minus big") – the difference between the monthly return on small-cap stocks and large-cap stocks.
• HML ("high minus low") – the difference between monthly returns on high book-to-market stocks and low book-to-market stocks.

The multiple regression formula can be written as follows.

The cost of equity is estimated as follows:

where,
k_i = Cost of equity; 
R_f = Rate on risk-free asset; long-term government bond yield for March 31, 1997 (7.2%); 
b_i = Market coefficient in the Fama-French regression; 
ERP = Expected equity risk premium. Long-horizon version from Ibbotson Associates' SBBI 1997
Yearbook (7.5%);
s_i = Small-minus-big coefficient in the Fama-French regression; 
SMBP =Expected small-minus-big risk premium, estimated as the difference between the historical average annual returns on the small-cap and large-cap portfolios, which is 3.70%;
h_i = high-minus-low coefficient in the Fama-French regression; and, 
HMLP =Expected high-minus-low risk premium, estimated as the difference between the historical average annual returns on the high market-to-book and low market-to-book portfolios, which is 5.04%. 

For further description reference: Fama, Eugene F. and Kenneth R. French, "Size and Book-to-Market Factors in Earnings and Returns," Journal of Finance, March 1995.

Analysts Single-Stage Discounted Cash Flow

where,
R_i = Cost of equity for company i;
D_i = Summation of prior twelve month’s dividend per share for company i;
g_i = Expected earnings growth rate for company i per the ACE database; and,
P_i = Most recent price per common share for company i. 

The single stage discounted cash flow model is the Gordon growth model which is stated as:

where,
P_i = Price per share for company i;
D_i = Dividend per share for company i at the end of year 1;
k_i = Discount rate for company i; and,
g_i = Dividend growth rate for company i.

In our model, we know all of the variables except k, so we have re-written the equation to solve for the discount rate or cost of equity. If a company pays no dividends, its cost of equity is its growth rate.

If individual company growth rates were not available in the ACE database, industry average growth rates were substituted.

Three-Stage Discounted Cash Flow

where,

CF_Ai0 = Average cash flow for company i;
CF_Ai5 = Expected cash flow for company i at the end of year five, which is equal to

CF_Ai0 (1 + g_i1)⁵;
CF_it = Cash flow for company i in year t;
NS_il = Net sales for company i in the most recent fiscal year;
NS_it = Net sales for company i in year t;
IBEI_it = Income before extraordinary items for company i in year t;
IBEI_Ai10 = Expected income before extraordinary items for company i at the end of year ten, which is equal to

[IBEI₅ (1 + g_i1)⁵] (1 + g₂)⁵;
g_i1 = Expected earnings growth rate for company i per the I/B/E/S or ACE database;
g₂ = Industry average of earnings growth rates;
g₃ = Economy wide long term growth rate per Ibbotson Associates. (This growth rate is comprised of the expected long-term inflation forecast and the historical GDP growth rate of 3.1%. The long-term inflation forecast is obtained from Ibbotson Associates' Optimizer Inputs for the first quarter of 1997. The current long-term inflation forecast is 4.81%.);
r_i = Cost of equity for company i;
DEP_it = Depreciation and amortization for company i in year t;
CE_it = Capital expenditures for company i in year t; and,
DT_it = Deferred taxes for company i in year t.

Solving for the cost of equity is an iterative process.

The Three-Stage model identifies three separate growth rates. A growth rate applicable to cash flows during the first five years of future performance, a growth rate applicable to cash flows over the sixth through tenth years, and a growth rate applicable to earnings for all future years following the first ten years. The first term of this model discounts to present those cash flows related to the first five years of growth. The second term discounts to present all cash flows in years six through ten. The third term discounts to present those earnings related to all future years beyond year ten.

Cash flows are used in place of dividends in the first and second terms because many companies do not pay dividends. Earnings are used in place of cash flows in the third term, because over extended periods of time it is assumed that capital expenditures and depreciation will be equal. Cash flow is defined as income before extraordinary items plus depreciation less capital expenditures plus deferred taxes. Normal cash flows and income before extraordinary items are used because of the potential for anomalous years. "Normal" cash flows and income before extraordinary items are estimated by multiplying the last five years average cash flow and earnings rates, both as a percent of sales, by the most recent year's sales. This reduces the effect that an anomalous recent year would have on a long-run expected cost of equity. A cost of equity number will be calculated under this model only if both cash flow and income before extraordinary items are positive values.

Companies included in the composite that do not have ACE growth rates are excluded from this calculation. In those instances where no companies in the composite have an ACE growth rate, the industry average is substituted.

In limited cases, no ACE growth rates will be available for any company in the industry. In this case, the industry average growth rate for the preceding SIC code is substituted. For example, if industry 7371 has no ACE growth rates for any of the companies included in the industry, the industry average growth rate for industry 737 is substituted.

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