by Aswath Damodaran

**Introduction to Valuation**

Valuing a financial asset requires discounting the cash flows that it is expected to generate over time. Although a quantitative exercise in nature, such modeling involves subjective judgments on firm-specific and market-wide inputs. Increased complexity does not necessarily improve accuracy. More detail, after all, increases the risk of error and obscures the information that matters most. The best models reveal a great deal about the key determinants of value.

Discounted cash flow (DCF) valuation is the foundation upon which all other valuation approaches are built (including the most popular method in the real world, relative valuations). DCF valuations are easiest to use when future cash flows can be reasonably estimated and a reliable proxy for risk (i.e. discount rate) is available. They are more difficult for distressed, cyclical, private, and recently restructured firms. Relative valuations derive the value of an asset from the price of comparable assets. It assumes other firms are correctly priced by the market on average. Comparable firms, however, are often a challenge to find and differences must be controlled for.

**Basics of Risk**

Risk in finance is defined as the probability that returns on an investment will be different than expected. It is measured as the distribution of actual returns around an expected return. Since the marginal investor is assumed to be diversified, only market risk – not firm-specific risk – is rewarded. (Diversification can be obtained with as few as 10-20 stocks). In practice, the spread of actual returns around the expected return is measured by the variance or standard deviation on past returns.

The default model for measuring non-diversifiable market risk is the capital asset pricing model (CAPM). An alternative approach is the arbitrage pricing model (APM). Whereas the CAPM captures market-wide risk in the market portfolio (e.g. S&P 500 index), the APM measures the sensitivity of investments to multiple macroeconomic factors. These factors, however, are left unspecified. Multi-factor models, on the other hand, attempt to identify specific factors in economic terms, such as industrial production, default premiums, term structures, inflation, and real rates of return. While the CAPM is a simpler model to estimate and use, the APM will yield a better estimate of risk when an investment is highly sensitive to specific economic factors, such as the price of oil for an oil company.

Default risk on corporate debt is a function of a firm’s ability to generate cash flows and satisfy its financial obligations (i.e. interest and principal payments). The conventional measure of default risk is a firm’s bond rating, which is assigned by a ratings agency and influenced by key ratios, such as interest coverage and debt-to-equity.

For an asset to be free of risk there must be no default risk or reinvestment risk. The only qualifying asset is therefore a default-free (i.e. government) zero-coupon bond. Since the governments of many economies are perceived as capable of defaulting, determining a risk-free rate is more involved. The first option is to take the most credible long-term corporate borrowing rates (in local currency) and subtract 50 basis points. The second option is to start with the government bond rate (in local currency) and add the default spread. The third option is to use the interest rate parity formula to back out the implied risk-free rate. (When long-term forward rates are unavailable, obtain the one-year borrowing rate, determine the spread over the one-year Treasury bill rate, and add that spread to the long-term Treasury bond rate).

Both the CAPM and multi-factor model assume that expected returns are measured as the risk-free rate plus an additional return to compensate for non-diversifiable market risk. They differ in how they capture market risk.

The CAPM uses a local stock market index as a proxy to estimate the risk premium earned on stocks over long-term risk-free securities. The standard approach to estimate this premium is by looking at geometric historical returns over a long time period. Although historical risk premium estimates range from 4% to 12% based on different inputs, a good rule of thumb is 5.5%.

An alternative method is to back out the implied required return on equity using a two-stage dividend discount model on the local market index. The first stage of growth can be based on consensus earnings estimates by analysts and the second stage of growth set equal to the Treasury bond rate. Since the implied premium changes over time it is worth looking at the average rate over 10-15 years. An even more rigorous approach is to relate the equity risk premium to fundamental macroeconomic data using a regression model.

For non-US markets that lack large, diversified stock markets, modified approaches to measuring historical risk premiums should be used. The first method is to add a default risk spread to the standard CAPM based on the country’s credit rating assigned by S&P, Moody’s or Fitch. These spreads can be found on Damodaran’s website (here) or at bondsonline.com (here).

Cost of Equity = Risk-free Rate + Beta * (US Risk Premium) + Default Spread

The second method is to scale the risk of one market against another to obtain a relative standard deviation. The equity risk premium of the country is then the product of the US risk premium and the calculated relative standard deviation. Ideally, the standard deviations should both be estimated in dollar terms.

The third method is to combine default premiums and relative standard deviations. The latter is calculated as the relative volatility of the equity market and the bond market of a country. This approach will yield the highest country risk premium of the three methods.

The weakness of the above approaches is that they assumes all firms in a country sell only to the local market and are therefore exposed entirely to country risk. However, many firms sell to the global market and are less exposed to country risk. The preferred method for correcting this is to differentiate a firm’s exposure to country and market risk. A firm’s exposure to each country in which it operates can be estimated as λ (= firm’s % revenues from country / average local firm’s % revenues from country).

**Estimating Risk Parameters and Costs of Financing**

The standard approach to measuring the beta of an asset is a regression of monthly returns on the asset against monthly returns on a market index over five years (inclusive of dividends). The R squared of the regression measures the goodness of fit of the estimate and shows the proportion of firm risk attributable to the market. The balance (1 – R squared) is attributable to firm-specific, non-diversifiable risk. The standard error reveals the degree of possible error in the estimate.

The intercept of the regression shows the performance of the investment when compared to Jensen’s alpha [=Rf * (1 – β), where Rf is the monthly risk-free rate]. The excess of the intercept over Jensen’s alpha is attributable to the entire sector and firm-specific actions. To capture only the latter, subtract the average excess returns of industry peers. Annualize the monthly rate if desired.

Since many emerging markets are dominated by a few large companies, their indices do not serve as good market proxies. It is better to use indices covering multiple economies in the same region (e.g. the MSCI Euro Index instead of a specific European country). The MSCI Global Index is also acceptable if the typical investor is assumed to be globally diversified.

Standard errors of most regression betas are high due to noise in the data. Equity risk in a company should therefore be determined by the average beta of 15-20 publicly-traded firms in the same type of business. Comparable firm betas must be adjusted for differences in financial and operating leverage. An unlevered beta is provided by the following equation, where the corporate tax rate is 35% and the debt-to-equity ratio is the average over 5 years.

If there are significant differences in operating leverage across firms, this can be controlled for as well. An approximate measure of operating leverage is the % change in operating profit divided by % change in sales. It can alternatively be estimated by dividing SG&A by other operating expenses. This assumes the former is fixed and the latter is variable.

For a company that has multiple businesses, beta should be estimated as the weighted average of unlevered betas for each business the firm operates in. The resultant beta must be re-levered based on current market values of debt and equity.

The cost of debt for a firm can be determined using the yield to maturity on outstanding long-term bonds. It can alternatively be estimated based on the firm’s credit rating and the average spread paid by firms with a similar rating. If a rating does not exist, a synthetic rating can be estimated by examining the financial characteristics of the firm. When calculating the after-tax cost of debt, the marginal corporate tax rate should be used (35% in the US).

The cost of preferred stock, if dividends are assumed to be constant forever, is estimated as the preferred dividend per share divided by the market price per preferred share. The cost of convertible bonds is determined by disaggregating the security into its debt and equity components. The straight bond component is the present value of a security with given coupon rate, maturity, and yield to maturity of similarly rated bonds (assume a par value of $1,000). The conversion option is the par value (i.e. $1,000) multiplied by the percent of par the convertible bond was issued at, less the straight bond component. Scale both preceding components to the full amount of the issued bond and add the respective amounts to the value of debt and equity.

When calculating WACC, weights should be based on market value and not book value. The value of debt includes short-term and long-term borrowings as well as the value of capitalized operating leases. For firms that have non-traded debt (e.g. bank debt), market value can be approximated by treating a firm’s entire debt as a single bond with the coupon set equal to annual interest expenses, par value equal to book value, maturity equal to the weighted average duration, and YTM equal to the current cost of debt for the company.

**From Earnings to Cash Flows**

When computing after-tax cash flows, the marginal tax rate (rather than the effective tax rate) should be applied since it will prevail in the long term. Alternatively, use the effective rate in the current period and converge to the marginal rate over time. For firms that have global operations, the tax rate can be the weighted average of the marginal tax rates in different locales or the marginal tax rate of the country in which the company is incorporated. It is also possible to apply the respective marginal tax rates to the income streams of each country. The chosen tax rate in any given year should be the same rate used to calculate the after-tax cost of debt in the WACC computation.

Any deferred taxes (tax loss carryforwards) should be valued separately by discounting the assumed payments (savings) each year by the cost of capital. For firms that have significant R&D arms, the tax benefit of expensing rather than capitalizing R&D spending should be deducted from firm value. This is computed as the difference between R&D expense and the amortization amount, multiplied by the tax rate. In valuing firms with government tax subsidies, the benefit from the subsidy should be valued separately as the present value of expected tax savings.

Since cash flows to the firm are computed after reinvestments, it is necessary to forecast net capital expenditures and investments in working capital. To normalize capital spending, use the average over a number of years to determine gross capital expenditures. Subtract depreciation in the current period to determine the net amount. Spending on R&D should be added to capital expenditures and the pro forma amortization of R&D added to depreciation. Capital expenditures should include acquisitions, which should also be normalized over time.

Working capital is the difference between current assets and current liabilities. Cash and marketable securities should be excluded from current asset and all interest-bearing debt should be backed out from current liabilities. Working capital projections can be linked to expected changes in revenue (=change in working capital / change in revenues) based on the firm’s own history or the industry average. Forecasting working capital is easier and (generally) more accurate when done on a consolidated basis rather than separately by individual line-item. Changes in working capital should not be negative in the long run.

**Estimating Growth**

The growth rate is the most critical input in a valuation. It can be forecasted by using the historical growth rate, relying on analyst estimates, or examining a firm’s fundamentals. When reviewing a company’s past growth, the geometric average provides a more accurate measure than the arithmetic average since the former takes compounding into account. The slope coefficient of an OLS regression can also be used to measure growth. Since the growth measure will be specified in dollar terms, it can be converted to a percentage by dividing the slope coefficient by the average sales (or earnings) over the same time period. To capture the effect of compounding, a log-linear model must be implemented. More sophisticated time-series models, such as the AutoRgressive Integrated Moving Average (ARIMA), are better at predicting earnings in the short-term. However, their predictive power diminishes over longer time horizons.

Past growth in revenue or earnings is not a reliable predictor of future growth at most firms. High growth rates are especially hard to maintain over time as firms expand in size and attract competition. Following analyst predictions is an alternative method to forecasting growth. Since they closely monitor all available information on the firm, industry, and the overall economy, analysts should make better forecasts than models that rely solely on historical data. Empirical evidence suggests that analysts provide superior forecasts in the short term, but not in the long run. In valuation, it is the latter that is most important getting right. Analyst estimates should nonetheless be incorporated in a valuation when a firm has undergone significant changes. Note that most services report analyst estimates for earnings, not revenue.

The soundest way to forecast growth successfully is to base the estimation on the firm’s fundamentals. The simplest function of growth in earnings is the retention ratio (=retained earnings / net income) multiplied by the return on equity (ROE). Since the retention ratio assumes the firm reinvests everything that it retains and doesn’t issue additional equity to fund new projects, a revised measure can be estimated (=capital expenditures – depreciation + change in working capital – net debt issued). Since these items are volatile year-to-year, the average reinvestment rate over 3-5 years should be used. If a firm’s financial leverage has changed significantly over time, the historical returns on equity should be unlevered and then re-levered to reflect the firm’s current leverage (where debt-to-equity is based on book values, interest rate is interest expense / book value of debt, and ROC is NOPLAT / book value of debt and equity. NOPLAT is EBIT(1-t), excluding non-operating items).

To measure only the returns on newer investments, the marginal return on equity can be calculated as the change in net income in the current period divided by the change in book value of equity in the prior period. If a firm is expected to improve its ROE on all investments (new and old) in a given year, the next period growth rate can be calculated using the first (1) equation below (where ROEt is the improved return and ROEt-1 is the existing return). The growth rate in subsequent periods is calculated using the conventional formula (=b*ROEt). If the improvement in ROE is only on new investments, there is no one-time boost to growth. If the improvement is only on existing assets, the expected growth rate next period is determined using the second (2) equation below.

Determining growth in operating income is similar to that for earnings. The expected growth in EBIT is the product of the reinvestment rate and return on capital.

To adequately forecast the ROC of future investments, marginal returns should be used (=change in after-tax EBIT / change in capital invested). The tax rate should be the effective rate. A firm’s return on capital should be compared to its cost of capital. If the firm is generating excess returns, consider what competitive advantage the firm has and whether it can last. When calculating historical reinvestment rates, capital expenditures should include R&D expenses and capitalized operating leases (related expenses will additionally need to be backed out of EBIT and the implied amortization/depreciation amounts factored into depreciation). Industry averages likely provide the best estimates for future reinvestment rates and returns on capital since both are mean reverting.

The same one-time increases to operating income as a result of improving ROC can be calculated similarly to those for ROE (see equations 1 & 2 above, substituting ROC for ROE). If the current ROC is expected to converge toward the industry average over several years, the expected short-term growth rate can be estimated as follows.

When forecasting future revenues using growth rates, keep track of the dollar revenues to ensure they are reasonable given the size of the sector that the firm operates in. Consider also the competitive environment and the capacity of the firm to scale. Operating margins can be estimated by looking at the firm’s own trends and those of comparable companies. To ensure reinvestment assumptions are internally consistent with revenue growth, look at the sales-to-capital ratio over time.

**Estimating Terminal Value**

Terminal value can be calculated assuming liquidation, a multiple, or a constant growth rate. Liquidation value (to equity investors) is estimated based on inflation-adjusted book value, less the value of debt outstanding in the terminal year. Multiples in the terminal year are estimated by looking at how peer companies are priced by the market. They can be based on sales, earnings, or book value. This approach is not recommended since it comingles relative and DCF valuation methods.

Choosing a constant growth rate to estimate terminal value requires subjective input. The growth rate of a firm, however, cannot be greater than the overall growth rate of the domestic economy (or global economy, if it operates multi-nationally). The stable growth rate should also not exceed the riskless rate used in the valuation. Whether the current growth rate will converge suddenly or gradually over time to the stable rate depends on the firm’s size, current growth rate, and competitive advantages. Firms with strong (moderate) advantages might maintain their high growth for 10 (5) years. Small high-growth firms are the most likely to maintain rapid growth in the near future since they have the most room to grow. The market share of a firm and the growth of its target market determine the potential total market for a firm’s products or services. Note that firms may already be in a steady state of growth.

As firms transition from high growth to stable growth, other firm characteristics should follow. Stable growth firms should be less risky (have a beta between 0.8 and 1.2), use more debt (have debt ratios similar to larger, more mature firms in the same industry), have low or no excess returns (have returns on capital that converge to industry averages or to the cost of capital), and reinvest less than high growth firms (but still enough to maintain stable growth rate). To maintain internal consistency when changing the debt ratio, re-estimate the cost of capital and monitor the synthetic credit rating over time. The stable growth rate, retention ratio (or reinvestment rate), and ROE (or ROC) should be linked together to ensure they are also consistent.

High growth rates can change abruptly to stable growth (two-stage model), stay high for a period then gradually converge to stable growth (three-stage model), or change each year from the initial period to stable growth (n-period model). Two-stage models work best when the current growth rate is very close to the long-term growth rate. They are also best suited for firms that will maintain a high growth rate for a specific period, after which point the current barriers will disappear (e.g. patents, legal regulations, or construction lead times). Three-stage models are ideal for high-growth firms that will have a transition phase. N-stage models should be used when changes are required in each year, such as with very young firms.

**Dividend Discount Model**

Although viewed by many analysts as outdated, the dividend discount model (DDM) remains a useful tool for valuating specific companies. The two inputs to the model are the expected dividends (based on future growth rates in earnings and payout ratios) and the cost of equity (determined by CAPM or multi-factor models). The Gordon growth model (GGM) is the simplest form of the DDM but assumes the company is already in a steady state. Good candidates for the GGM include those growing at a rate less than or equal to the overall economy and which have well established dividend payout policies. Qualifying firms should also be paying out as much as they can afford (look at dividends and stock buybacks as a % of FCFE).

Since stock buybacks are another form of returning cash to stockholders, they should be incorporated into the payout ratio (though averaged over a 4-5 year period to smooth out volatility). New debt should also be netted out since firms can use it fund buybacks. (Note that since stock buybacks reduce the book equity of firms, they can increase ROE dramatically. Re-estimating ROE by adding back stock buybacks in recent years can yield more reasonable estimates).

The H-model is a special case of the DDM, which starts with a high initial growth rate (gS) and declines linearly to a stable growth rate (gL). The variable H is the number of years of the high-growth period divided by two. The limitation is that it assumes the payout ratio remains constant. The three- and n-stage models remove this constraint at the cost of more inputs.

**Free Cash Flow to Equity**

Whereas the DDM measures how much cash is paid out to stockholders as dividends or buybacks, free cash flow to equity (FCFE) measures how much cash is potentially available to stockholders.

FCFE = Net income – (Capital expenditures – Depreciation) – (Change in non-cash working capital) + (New debt issued – Debt repayments)

Or, FCFE = NI – (CapEx – Depr)(1 – δ) – (ΔWC)(1 – δ), where δ is the portion of reinvestment funded from new debt. Historical FCFE can be recalculated this way using the average debt ratio [=avg net debt / (avg CapEx – avg Depr + avg ΔWC)]. Under this revised method, FCFE will be less volatile on an annual basis but average FCFE over the entire period will remain the same. Note that capital expenditures should include acquisitions.

The target or optimal debt ratio should be used for forecasting future periods. The debt-to-equity ratio used in FCFE should be consistent with the debt-to-equity ratio used in calculating levered beta. To forecast FCFE, first calculate a normalized FCFE in the current year to smooth out annual jumps (use average capital expenditures, but depreciation from the current year). The future growth rate of FCFE can be estimate as the product of the equity reinvestment rate and the non-cash ROE (where ROE is estimated using net income from the current year and book value of equity from the previous year).

The stable growth rate of FCFE should not exceed the economy in which the firm operates by more than 1-2%. Net capital expenditures should not be disproportionately large in steady state (i.e. the difference between capital spending and depreciation should narrow). Beta should converge to one and the stable reinvestment rate should be comparable to mature firms in the same industry. FCFE can be forecasted using constant growth or multi-stage models similar to the DDM. The value of the firm is obtained by discounting FCFE at the cost of equity.

**Free Cash Flow to Firm**

Free cash flow to firm (FCFF) differs from FCFE in that it includes cash flows associated with non-equity claims (e.g. debt). The FCFF approach is best for valuing firms with past or future leverage changes since cash flows relating to debt do not have to be explicitly estimated as with FCFE.

FCFF = EBIT * (1 – Tax rate) + Depreciation – Capital expenditures – Δ Working capital

The expected growth in FCFF (i.e. operating income) is the reinvestment rate multiplied by the return on capital (see the Estimating Growth section above for further details). Since financial leverage can augment FCFE, growth in FCFF is expected to be lower than FCFE. In steady state, however, the growth rates of FCFE and FCFF will converge. In the long run, the ratio of capital expenditures-to-depreciation should reflect industry averages and beta should be close to one. The discount rate used for FCFF is the WACC.

After using FCFF to value the operating assets of a firm, non-equity claims should be subtracted out and non-operating assets added back to determine the value of equity. Non-equity claims include debt, preferred stock (using market value), capitalized operating leases, expected liabilities on lawsuits (=probability x damages), unfunded pension obligations, and deferred tax liabilities (=present value of when tax obligations are expected to be paid). Non-operating assets include cash and marketable securities, holdings in other firms, unutilized assets, and overfunded pension plans.

Cash and marketable securities should be excluded from operating assets and valued separately since they represent financial assets. Since they are recorded at market value on the balance sheet, book value can be used. Holdings in other firms should be valued separately and their proportional ownership added to the equity value of the parent company. These investments can be valued using market prices, multiples of book value, or a separate DCF valuation. When a company owns 50% or more of another firm, the financials of the two firms are consolidated. The subsidiary’s financials must be stripped from the parent company’s financials in such cases to avoid double counting. The market value of unutilized assets (such as real estate holdings) can often differ greater from their book values. Making the appropriate revision to non-operating assets, however, requires detailed access to what a firm owns and uses that is generally not available. If a firm has an overfunded pension plan, the excess funds should be included in the valuation on an after-tax basis (the tax rate on early withdrawals from pension funds is 50% in the US).

**Equity Value per Share**

In determining equity value per share, the value of equity must be divided by the number of outstanding shares. Existing employee stock options should also be incorporated since they have a potentially dilutive effect on equity. One method is to divide the value of equity by the fully diluted number of shares. If the information is available, a better estimate can be obtained by only counting in-the-money and fully vested options. The alternative is to estimate the value of options using the Black Scholes option pricing model, subtract this amount from the value of equity, and divide by the undiluted number of shares outstanding. Future expected option grants can be estimated as a percent of revenues or operating income using the experience of previous years or more mature firms in the sector as benchmarks.