Valuation: IntroductionEstimating Inputs: Discount Rates
Critical ingredient in discounted cashflow valuation. Errors in estimating the discount rate or mismatching cashflows and discount rates can lead to serious errors in valuation.
At an intuitive level, the discount rate used should be consistent with both the riskiness and the type of cashflow being discounted.
Equity versus Firm: If the cash flows being discounted are cash flows to equity, the appropriate discount rate is a cost of equity. If the cash flows are cash flows to the firm, the appropriate discount rate is the cost of capital.
Currency: The currency in which the cash flows are estimated should also be the currency in which the discount rate is estimated.
Nominal versus Real: If the cash flows being discounted are nominal cash flows (i.e., reflect expected inflation), the discount rate should be nominal
While discount rates are a critical ingredient in discounted cashflow valuation, I think we spend far too much time on discount rates and far too little on cashflows. The most significant errors in valuation are often the result of failures to estimate cash flows correctly….
As companies increasingly become global, and multiple listings abound (Royal Dutch has six equity listings in different markets) the consistently principle becomes very important. The currency used in estimating cash flows should also be the currency in which you estimate discount rates - Euro discount rates for Euro cashflows and peso discount rates for peso cash flows.
Recently, I came across a valuation of a Mexican company, where the cashflows were in nominal pesos but the discount rate used was the dollar cost of capital of a U.S. acquirer…. As a result, the value was inflated by more than 300%….
Aswath Damodaran
Cost of Equity
The cost of equity should be higher for riskier investments and lower for safer investments
While risk is usually defined in terms of the variance of actual returns around an expected return, risk and return models in finance assume that the risk that should be rewarded (and thus built into the discount rate) in valuation should be the risk perceived by the marginal investor in the investment
Most risk and return models in finance also assume that the marginal investor is well diversified, and that the only risk that he or she perceives in an investment is risk that cannot be diversified away (I.e, market or non-diversifiable risk)
Re-emphasizes a key assumption that we make in risk and return models in finance. It is not risk that matters, but non-diversifiable risk and the cost of equity will increase as the non-diversifiable risk in an investment increases.
This view of the world may pose a problem for us when valuing private companies or closely held, small publicly traded firms, where the marginal investor (owner, venture capitalist….) may not be diversified.
Aswath Damodaran
Model Expected Return Inputs Needed
CAPM E(R) = Rf + (Rm- Rf) Riskfree Rate
Beta relative to market portfolio
Market Risk Premium
APM E(R) = Rf + j=1j (Rj- Rf) Riskfree Rate; # of Factors;
Betas relative to each factor
Factor risk premiums
Multi E(R) = Rf + j=1,,Nj (Rj- Rf) Riskfree Rate; Macro factors
factor Betas relative to macro factors
Macro economic risk premiums
Regression coefficients
Lays out the four basic models and how non-diversifiable risk is measured in each model:
The capital asset pricing model makes the most restrictive assumptions (no transactions costs, no private information) and arrives at the simplest model to estimate and use.
The arbitrage pricing model and multi-factor model make less restrictive assumptions but yield more complicated models (with more inputs to estimate)
The proxy model is dependent upon history and the view that firms that have earned higher returns over long periods must be riskier than firms that have lower returns. The characteristics of the firms that earn high returns - small market cap and low price to book value, for example in the Fama-French study - stand in as measures for risk.
Aswath Damodaran
Consider the standard approach to estimating cost of equity:
Cost of Equity = Rf + Equity Beta * (E(Rm) - Rf)
where,
In practice,
Short term government security rates are used as risk free rates
Historical risk premiums are used for the risk premium
Betas are estimated by regressing stock returns against market returns
While this equation is set up in terms of the capital asset pricing model, the issues raised with the CAPM apply to the more complex models as well -the APM and the multi-factor model
Aswath Damodaran
Short term Governments are not riskfree in valuation….
On a riskfree asset, the actual return is equal to the expected return. Therefore, there is no variance around the expected return.
For an investment to be riskfree, then, it has to have
No default risk
No reinvestment risk
Thus, the riskfree rates in valuation will depend upon when the cash flow is expected to occur and will vary across time.
In valuation, the time horizon is generally infinite, leading to the conclusion that a long-term riskfree rate will always be preferable to a short term rate, if you have to pick one.
Treasury bills may be default free but there is reinvestment risk when they are used as riskless rates for longer-term cashflows. A 6-month T.Bill is not riskless when looking at a 5-year cashflow.
Would a 5-year treasury be riskfree? Not really. The coupons would still expose you to reinvestment risk. Only a 5-year zero-coupon treasury would be riskfree for a 5-year cash flow.
If you were a purist, you would need different riskfree rates for different cashflows. A pragmatic solution would be to estimate the duration of the cashflows in a valuation and use a treasury of similar duration. (Since the duration is the weighted average of when the cashflows come in, this should be fairly long, especially when you count in the fact that the terminal value is the present value of cashflows forever).
Aswath Damodaran
Riskfree Rates in 2004
You cannot compre interest rates in different currencies but you can when they are in the same currency. Among the three 10-year Euro bonds, the German Euro bond is the best candidate for the riskless rate in Euros since it is the lowest. There is a default spread, albeit small, in both the Italian and Greek Euro bonds.
The Brazilian dollar denominated C-bond indicates how large the default risk is in Brazilian bonds….
The Mexican government 10-year rate is not a riskless peso rate since the ratings agency attaches a rating of A to these bonds. The true riskless rate is likely to be lower.
Aswath Damodaran
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Estimating a Riskfree Rate when there are no default free entities….
Estimate a range for the riskfree rate in local terms:
Approach 1: Subtract default spread from local government bond rate:
Government bond rate in local currency terms - Default spread for Government in local currency
Approach 2: Use forward rates and the riskless rate in an index currency (say Euros or dollars) to estimate the riskless rate in the local currency.
Do the analysis in real terms (rather than nominal terms) using a real riskfree rate, which can be obtained in one of two ways –
from an inflation-indexed government bond, if one exists
set equal, approximately, to the long term real growth rate of the economy in which the valuation is being done.
Do the analysis in a currency where you can get a riskfree rate, say US dollars.
Approach 1: The Mexican peso bond rate is 11.5% and that the rating assigned to the Mexican government is A. If the default spread for A rated bonds is 1.5%, the riskless peso bond rate would be 10%.
Riskless Peso rate = Mexican Government Bond rate – Default Spread
= 11.5% - 1.5% = 10%
Approach 2: If the current spot rate is 38.10 Thai Baht per US dollar, the ten-year forward rate is 61.36 Baht per dollar and the current ten-year US treasury bond rate is 5%, the ten-year Thai risk free rate (in nominal Baht) can be estimated as follows
In general, while many practitioners use the last approach (working with a different, more stable currency) to avoid having to deal with inflation in the local currency, they shift the problem of making these estimates to the cashflows from the discount rates.
Aswath Damodaran
A Simple Test
You are valuing Embraer, a Brazilian company, in U.S. dollars and are attempting to estimate a riskfree rate to use in the analysis. The riskfree rate that you should use is
The interest rate on a Brazilian Real denominated long term bond issued by the Brazilian Government (15%)
The interest rate on a US $ denominated long term bond issued by the Brazilian Government (C-Bond) (10.30%)
The interest rate on a US $ denominated Brazilian Brady bond (which is partially backed by the US Government) (10.15%)
The interest rate on a dollar denominated bond issued by Embraer (9.25%)
The interest rate on a US treasury bond (4.29%)
The correct riskfree rate is the treasury bond rate. The C-Bond has a default spread component -the additional country risk can be built into the equity risk premium..
Aswath Damodaran
Everyone uses historical premiums, but..
The historical premium is the premium that stocks have historically earned over riskless securities.
Practitioners never seem to agree on the premium; it is sensitive to
How far back you go in history…
Whether you use T.bill rates or T.Bond rates
Whether you use geometric or arithmetic averages.
For instance, looking at the US:
Arithmetic average Geometric Average
Stocks - Stocks - Stocks - Stocks -
1928-2004 7.92% 6.53% 6.02% 4.84%
1964-2004 5.82% 4.34% 4.59% 3.47%
1994-2004 8.60% 5.82% 6.85% 4.51%
While everyone uses historical risk premiums, the actual premium in use can vary depending upon
How far back you go in time
Whether you T.Bills or T.Bonds
Whether you use arithmetic or geometric averages
This table was developed using data that is publicly accessible on the S&P 500, treasury bills and 10-year treasury bonds on the Federal Reserve of St. Louis web site.
Dataset: histretSP.xls
An interesting issue that has been raised by some researchers is that there may be a selection bias here. The U.S. stock market was undoubtedly the most successful equity market of the 20th century. Not surprisingly, you find that it earned a healthy premium over riskless rates. A more realistic estimate of the premium would require looking at the ten largest equity markets in the early part of the century and estimating the average premium you would have earned over all ten markets. A study at the London Business School that did this found an average equity risk premium of only 4% across these markets.
Aswath Damodaran
If you choose to use historical premiums….
Go back as far as you can. A risk premium comes with a standard error. Given the annual standard deviation in stock prices is about 25%, the standard error in a historical premium estimated over 25 years is roughly:
Standard Error in Premium = 25%/√25 = 25%/5 = 5%
Be consistent in your use of the riskfree rate. Since we argued for long term bond rates, the premium should be the one over T.Bonds
Use the geometric risk premium. It is closer to how investors think about risk premiums over long periods.
The noise in stock prices is such that you need 100-150 years of data to arrive at reasonably small standard errors. It is pointless estimating risk premiums with 10-20 years of data.
Consistent with our argument of using a treasury bond rate as a riskfree rate, we would estimate the premium over the treasury bond rate.
If you wanted a risk premium for the next year, you would use the arithmetic average - it is the single best estimate of next year’s premium. If you want a risk premium to use in a cost of equity which will be compounded over time, you should use a geometric average.
Aswath Damodaran
Risk Premium for a Mature Market? Broadening the sample
The problem with using the risk premium in the United States is that there is a survivor bias - the U.S, was the most successful equity market of the 20th century…
In “The Triumph of the Optimists”, Dimson and his co-authors examined risk premiums in the largest equity markets of the 20th century and concluded that the average equity risk premium across these markets is about 4%.
Aswath Damodaran
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Two Ways of Estimating Country Equity Risk Premiums for other markets..
Default spread on Country Bond: In this approach, the country equity risk premium is set equal to the default spread of the bond issued by the country (but only if it is denominated in a currency where a default free entity exists.
Brazil was rated B2 by Moody’s and the default spread on the Brazilian dollar denominated C.Bond at the end of August 2004 was 6.01%. (10.30%-4.29%)
Relative Equity Market approach: The country equity risk premium is based upon the volatility of the market in question relative to U.S market.
Total equity risk premium = Risk PremiumUS* Country Equity / US Equity
Using a 4.82% premium for the US, this approach would yield:
Total risk premium for Brazil = 4.82% (34.56%/19.01%) = 8.76%
Country equity risk premium for Brazil = 8.76% - 4.82% = 3.94%
(The standard deviation in weekly returns from 2002 to 2004 for the Bovespa was 34.56% whereas the standard deviation in the S&P 500 was 19.01%)
Many analysts use the default spread on the country bond as a measure of the extra risk premium to charge for that market. In fact, many leave this default spread in the riskless rate, thus adding it as a constant to every company in that market. Some double count it, by including it both in the riskless rate and by using a larger risk premium.
The equity standard deviation approach is promising but less liquid (and very risky markets) often have low standard deviations. Using this approach, you can end up with a lower standard deviation for these markets than you would for the US. (The standard deviations were estimated using 2 years of weekly prices.)
Aswath Damodaran
And a third approach
Country ratings measure default risk. While default risk premiums and equity risk premiums are highly correlated, one would expect equity spreads to be higher than debt spreads.
Another is to multiply the bond default spread by the relative volatility of stock and bond prices in that market. In this approach:
Country Equity risk premium = Default spread on country bond* Country Equity / Country Bond
Standard Deviation in Bovespa (Equity) = 34.56%
Standard Deviation in Brazil C-Bond = 26.34%
Default spread on C-Bond = 6.01%
Country Equity Risk Premium = 6.01% (34.56%/26.34%) = 7.89%
In this approach, we consider the country bond and the equity market in an emerging market to be competitors for investor funds. We argue that the equity market has to deliver a larger premium than the bond market to attract money since it is riskier.
The standard deviations in equity and the C-Bond are based upon two years of weekly returns. (One is in real and the other is in dollars. If this is a potential problem, you could convert one or another into another currency using exchange rates).
I am scaling up the country default spread by the relative volatility of equity. I would add this country risk premium (7.67%) to the risk premium I have for a mature equity market (4.82% for instance).
There is a chance that I am double counting some risk since you could argue that the 4.82% risk premium in U.S. already reflects some of the standard deviation in the treasury bond. This may be possibility and the country risk premium may look high, but I would change this risk premium over the forecast period - reducing it as I move towards the terminal year.
Aswath Damodaran
Can country risk premiums change? Updating Brazil in January 2004
Brazil’s financial standing and country rating improved dramatically towards the end of 2004. Its rating improved to B1. In January 2005, the interest rate on the Brazilian C-Bond dropped to 7.73%. The US treasury bond rate that day was 4.22%, yielding a default spread of 3.51% for Brazil.
Standard Deviation in Bovespa (Equity) = 25.09%
Standard Deviation in Brazil C-Bond = 15.12%
Default spread on C-Bond = 3.51%
Country Risk Premium for Brazil = 3.51% (25.09%/15.12%) = 5.82%
Aswath Damodaran
From Country Equity Risk Premiums to Corporate Equity Risk premiums
Approach 1: Assume that every company in the country is equally exposed to country risk. In this case,
E(Return) = Riskfree Rate + Country ERP + Beta (US premium)
Implicitly, this is what you are assuming when you use the local Government’s dollar borrowing rate as your riskfree rate.
Approach 2: Assume that a company’s exposure to country risk is similar to its exposure to other market risk.
E(Return) = Riskfree Rate + Beta (US premium + Country ERP)
Approach 3: Treat country risk as a separate risk factor and allow firms to have different exposures to country risk (perhaps based upon the proportion of their revenues come from non-domestic sales)
E(Return)=Riskfree Rate+ (US premium) + (Country ERP)
ERP: Equity Risk Premium
While many analysts use the first approach, it strikes us as unreasonable to tar all companies - large and small, domestic and export oriented - with the same brush.
The second approach tends to work well for all but the most extreme examples - firms that derive the bulk of their revenues outside an emerging market.
For companies that derive most of their revenues from markets other than their domestic market, the third approach offers the most promise, though we are left with the question of how to estimate lambda. Lambda will look a lot like beta, averaging one across all stocks. A stock with a lambda greater than one is more exposed to country risk than the average stock.
Aswath Damodaran
Estimating Company Exposure to Country Risk: Determinants
Source of revenues: Other things remaining equal, a company should be more exposed to risk in a country if it generates more of its revenues from that country. A Brazilian firm that generates the bulk of its revenues in Brazil should be more exposed to country risk than one that generates a smaller percent of its business within Brazil.
Manufacturing facilities: Other things remaining equal, a firm that has all of its production facilities in Brazil should be more exposed to country risk than one which has production facilities spread over multiple countries. The problem will be accented for companies that cannot move their production facilities (mining and petroleum companies, for instance).
Use of risk management products: Companies can use both options/futures markets and insurance to hedge some or a significant portion of country risk.
This is not meant to be an all inclusive list. While there are undoubtedly other factors to consider, you also have to be able to obtain this information not only on your firm but on other firms in the market. In fact, of the three items listed above, revenue sources may be the only publicly available information on companies.
Aswath Damodaran
Estimating Lambdas: The Revenue Approach
The easiest and most accessible data is on revenues. Most companies break their revenues down by region. One simplistic solution would be to do the following:
= % of revenues domesticallyfirm/ % of revenues domesticallyavg firm
Consider, for instance, Embraer and Embratel, both of which are incorporated and traded in Brazil. Embraer gets 3% of its revenues from Brazil whereas Embratel gets almost all of its revenues in Brazil. The average Brazilian company gets about 77% of its revenues in Brazil:
LambdaEmbraer = 3%/ 77% = .04
LambdaEmbratel = 100%/77% = 1.30
There are two implications
A company’s risk exposure is determined by where it does business and not by where it is located
Firms might be able to actively manage their country risk exposures
This is a simplistic approach, but it is the easiest one to use. You can usually get the percent of revenues that your firm gets in the country from its annual report.
Notice that Embraer has a lambda close to zero since it gets so little of its revenues in Brazil. Since its factories and work force are still in Brazil, this estimate seems to be too low.
Aswath Damodaran
Estimating Lambdas: Earnings Approach
More ammunition for the argument that Embraer is less exposed to country risk than Embratel. Note that Embratel’s earnings swing as the country bond default spread for Brazil widens (indicating investor unease with the country) whereas Embraer’s earnings are relatively unscathed.
Aswath Damodaran
ReturnEmbraer = 0.0195 + 0.2681 ReturnC Bond
ReturnEmbratel = -0.0308 + 2.0030 ReturnC Bond
Extends the regression approach used to estimate betas to estimating lambdas. All of the caveats of regression estimates apply - the estimate has a large standard error and reflects your company as it used to be. But it is perhaps our best shot at coming up with a comprehensive estimate of lambda for Embraer.
Aswath Damodaran
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Estimating a US Dollar Cost of Equity for Embraer - September 2004
Assume that the beta for Embraer is 1.07, and that the riskfree rate used is 4.29%. Also assume that the risk premium for the US is 4.82% and the country risk premium for Brazil is 7.89%.
Approach 1: Assume that every company in the country is equally exposed to country risk. In this case,
E(Return) = 4.29% + 1.07 (4.82%) + 7.89% = 17.34%
Approach 2: Assume that a company’s exposure to country risk is similar to its exposure to other market risk.
E(Return) = 4.29 % + 1.07 (4.82%+ 7.89%) = 17.89%
Approach 3: Treat country risk as a separate risk factor and allow firms to have different exposures to country risk (perhaps based upon the proportion of their revenues come from non-domestic sales)
E(Return)= 4.29% + 1.07(4.82%) + 0.27 (7.%) = 11.58%
Applies all three approaches to estimating the dollar cost of equity for Embraer. Note how much lower the cost of equity is with the lambda approach. Clearly, it will have a big impact on what value you arrive at for Embraer.
Aswath Damodaran
Valuing Emerging Market Companies with significant exposure in developed markets
The conventional practice in investment banking is to add the country equity risk premium on to the cost of equity for every emerging market company, notwithstanding its exposure to emerging market risk. Thus, Embraer would have been valued with a cost of equity of 17.34% even though it gets only 3% of its revenues in Brazil. As an investor, which of the following consequences do you see from this approach?
Emerging market companies with substantial exposure in developed markets will be significantly over valued by equity research analysts.
Emerging market companies with substantial exposure in developed markets will be significantly under valued by equity research analysts.
Can you construct an investment strategy to take advantage of the misvaluation?
Analysts will routinely undervalue these companies because they will be using too high a discount rate. A good strategy to follow would be to buy these companies, preferably after a crisis in the emerging market in question. Investors tend not to be discriminating during crises and mark down all stocks in an emerging market and correct themselves later.
Aswath Damodaran
Implied Equity Premiums
If we assume that stocks are correctly priced in the aggregate and we can estimate the expected cashflows from buying stocks, we can estimate the expected rate of return on stocks by computing an internal rate of return. Subtracting out the riskfree rate should yield an implied equity risk premium.
This implied equity premium is a forward looking number and can be updated as often as you want (every minute of every day, if you are so inclined).
The simplest analogy is to a bond. If you know the price of a bond, you can compute the yield to maturity as the discount rate that makes the present value of the expected cashflows on the bond equal to the price of the bond. Similarly, if you know the current market value of equities in the aggregate (or of an equity index), you can compute the discount rate that makes the present value of the expected cashflows on the index equal to the price of the index today.
There are two practical problems:
Unlike a bond, the cashflows on stocks are not promised but expected - you need an expected growth rate.
Unlike a bond, stocks have infinite lives. You have to consider cashflows forever.
Aswath Damodaran
Implied Equity Premiums
We can use the information in stock prices to back out how risk averse the market is and how much of a risk premium it is demanding.
If you pay the current level of the index, you can expect to make a return of 7.87% on stocks (which is obtained by solving for r in the following equation)
Implied Equity risk premium = Expected return on stocks - Treasury bond rate = 7.87% - 4.22% = 3.65%
Aswath Damodaran
Implied Risk Premium Dynamics
Assume that the index jumps 10% on January 2 and that nothing else changes. What will happen to the implied equity risk premium?
Implied equity risk premium will increase
Implied equity risk premium will decrease
Assume that the earnings jump 10% on January 2 and that nothing else changes. What will happen to the implied equity risk premium?
Implied equity risk premium will increase
Implied equity risk premium will decrease
Assume that the riskfree rate increases to 5% on January 2 and that nothing else changes. What will happen to the implied equity risk premium?
Implied equity risk premium will increase
Implied equity risk premium will decrease
The implied equity risk premium will decrease
As the index level increases
As earnings decrease
As the riskfree rate goes up
All three tend to move at the same time though in different directions. For instance, as the economy strengthens, earnings and earnings growth will increase (which should push up premiums) and interest rates will increase(which will push down premiums). The net effect will show up in the index.
Aswath Damodaran
Aswath Damodaran
Aswath Damodaran
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Implied Premiums: From Bubble to Bear Market… January 2000 to January 2003
A snap shot of premiums as the market retreated from its dot.com boom… When will the correction end? … When will the equity risk premium stop climbing?
Notice
The effect of 9/11 on implied premiums
The corrosive effect of accounting scandals at Enron, Global Crossing and Worldcom on equity risk premiums. Equity, as a class, becomes riskier if you trust firms less…
Aswath Damodaran
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Effect of Changing Tax Status of Dividends on Stock Prices - January 2003
Expected Return on Stocks (Implied) in Jan 2003 = 7.91%
Dividend Yield in January 2003 = 2.00%
Assuming that dividends were taxed at 30% (on average) on 1/1/03 and that capital gains were taxed at 15%.
After-tax expected return on stocks = 2%(1-.3)+5.91%(1-.15) = 6.42%
If the tax rate on dividends drops to 15% and the after-tax expected return remains the same:
2% (1-.15) + X% (1-.15) = 6.42%
New Pre-tax required rate of return = 7.56%
New equity risk premium = 3.75%
Value of the S&P 500 at new equity risk premium = 965.11
Expected Increase in index due to dividend tax change = 9.69%
In April 2003, the U.S. Congress passed legislation that made the tax rate on dividends equal to the tax rate on capital gains and set both to 15%. Since stocks are priced based upon the after-tax returns that investors hope to make, we can use the pricing of stocks on January 1, 2003 (when no one expected the tax law to change - President Bush’s proposal came 6 days later and was a complete surprise) to back out the implied equity rate of return (a pre-tax return).
If we assume that the average tax rate paid on dividends by investors (individual as well as institutional) is 30% on 1/1/03 and the tax rate paid on capital gains is 15%, we can then re-estimate the after-tax return from the implied pre-tax return. Changing the tax rate and keeping the after-tax return fixed allows us to back out a new pre-tax return. Repricing stocks with this lower pre-tax return gives us the effect on stock prices.
In reality, the effect on stock prices is likely to be much larger since companies can be expected to increase the dividends, thus pushing up the dividend yield on the index.
Aswath Damodaran
Which equity risk premium should you use for the US?
Historical Risk Premium: When you use the historical risk premium, you are assuming that premiums will revert back to a historical norm and that the time period that you are using is the right norm. You are also more likely to find stocks to be overvalued than undervalued (Why?)
Current Implied Equity Risk premium: You are assuming that the market is correct in the aggregate but makes mistakes on individual stocks. If you are required to be market neutral, this is the premium you should use. (What types of valuations require market neutrality?)
Average Implied Equity Risk premium: The average implied equity risk premium between 1960-2003 in the United States is about 4%. You are assuming that the market is correct on average but not necessarily at a point in time.
Whenever the historical premium is higher than the current implied equity risk premium, you will be using a much higher discount rate than the market and consequently ending up with a much lower present value for stocks.
If you are required to be market neutral (as is the case with equity research analysits, equity mutual fund managers, acquisitions), where you are asked to value an individual stock and not pass judgment on the market, the best number to use is the current implied equity risk premium.
Aswath Damodaran
Implied Premium for the Indian Market: June 15, 2004
Level of the Index (S&P CNX Index) = 1219
This is a market cap weighted index of the 500 largest companies in India and represents 90% of the market value of Indian companies
Dividends on the Index = 3.51% of 1219 (Simple average is 2.75%)
Other parameters
Next 5 years = 18% (Used expected growth rate in Earnings)
After year 5 = 5.5%
Solving for the expected return:
Expected return on Equity = 11.76%
Implied Equity premium = 11.76-5.5% = 6.16%
This implied equity risk premium was estimated right after an election had delivered an unexpected (and unwelcome) surprise to markets. The expected growth rate was not available for all companies and the average used reflects only those companies which are larger and followed by multiple analysts. This may cause a bias but the direction of the bias is unclear.
Aswath Damodaran
Implied Equity Risk Premium for Germany: September 23, 2004
We can use the information in stock prices to back out how risk averse the market is and how much of a risk premium it is demanding.
If you pay the current level of the index, you can expect to make a return of 7.78% on stocks (which is obtained by solving for r in the following equation)
Implied Equity risk premium = Expected return on stocks - Treasury bond rate = 7.78% - 3.95% = 3.83%
Aswath Damodaran
Estimating Beta
The standard procedure for estimating betas is to regress stock returns (Rj) against market returns (Rm) -
Rj = a + b Rm
where a is the intercept and b is the slope of the regression.
The slope of the regression corresponds to the beta of the stock, and measures the riskiness of the stock.
This beta has three problems:
It has high standard error
It reflects the firm’s business mix over the period of the regression, not the current mix
It reflects the firm’s average financial leverage over the period rather than the current leverage.
The standard approach for estimating betas- regressions - leads to flawed and backward looking estimates of risk.
Aswath Damodaran
Beta Estimation: The Noise Problem
Note the standard error problem. Amazon has a beta estimate of 2.23, but the standard error of 0.50 results in a considerable range around this estimate.
Aswath Damodaran
Beta Estimation: The Index Effect
This beta looks much better (in terms of standard error) but it is misleading. Nokia dominates the Helisinki index (it was 70% of the index at the time of this regression).
The reason it is misleading is because Nokia’s largest single investor was Barclays, which manages one of the worlds’ largest global index funds. Barclays would not view the beta of this regression as a good measure of risk. (They would probably prefer a beta estimate against a global equity index like the Morgan Stanley Capital Index).
Aswath Damodaran
Modify the regression beta by
changing the index used to estimate the beta
adjusting the regression beta estimate, by bringing in information about the fundamentals of the company
Estimate the beta for the firm using
the standard deviation in stock prices instead of a regression against an index
accounting earnings or revenues, which are less noisy than market prices.
Estimate the beta for the firm from the bottom up without employing the regression technique. This will require
understanding the business mix of the firm
estimating the financial leverage of the firm
Use an alternative measure of market risk not based upon a regression.
Changing the regression parameters, which is what we do in the first approach, will yield such a large range of betas (with large standard errors) that it will leave you more confused about the true beta of a firm rather than less confused.
While you can use standard deviations to compute a relative standard deviation, you are assuming that market risk and total risk are perfectly correlated. With accounting earnings, the biggest limitation is that you will have relatively few observations in your regression.
We prefer bottom-up betas….
Aracruz ADR = 2.80% + 1.00 S&P Aracruz = 2.62% + 0.22 Bovespa
Aswath Damodaran
These are the three fundamentals that drive betas.
Firms that produce luxury goods (such as Gucci and Tiffanys) should have higher betas than firms that cater to more basic needs (Walmart). These firms tend to have revenues that are much more sensitive to changes in economic conditions.
Firms in businesses with high fixed cost structures (like airlines) should have higher betas than firms with more flexible cost structures.
Firms that borrow more money will have higher equity betas than otherwise similar firms that do not borrow money.
Aswath Damodaran
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In a perfect world… we would estimate the beta of a firm by doing the following
To do this, we have to be able to measure the fixed costs and variable costs of each of the firms that operates in the business. This is difficult to do, though we may be able to estimate fixed and variable costs by looking at operating expenses over time.
Aswath Damodaran
Adjusting for operating leverage…
Within any business, firms with lower fixed costs (as a percentage of total costs) should have lower unlevered betas. If you can compute fixed and variable costs for each firm in a sector, you can break down the unlevered beta into business and operating leverage components.
Unlevered beta = Pure business beta * (1 + (Fixed costs/ Variable costs))
The biggest problem with doing this is informational. It is difficult to get information on fixed and variable costs for individual firms.
In practice, we tend to assume that the operating leverage of firms within a business are similar and use the same unlevered beta for every firm.
The standard practice of using the same unlevered beta for all firms will overstate betas for firms that have low fixed costs because of strategic decisions that they have made over time (such as Southwest in the Airline business) and understate betas for smaller firms in large infrastructure businesses where fixed costs tend to be much higher early in a firm’s life cycle.
Aswath Damodaran
Equity Betas and Leverage
Conventional approach: If we assume that debt carries no market risk (has a beta of zero), the beta of equity alone can be written as a function of the unlevered beta and the debt-equity ratio
L = u (1+ ((1-t)D/E))
In some versions, the tax effect is ignored and there is no (1-t) in the equation.
Debt Adjusted Approach: If beta carries market risk and you can estimate the beta of debt, you can estimate the levered beta as follows:
L = u (1+ ((1-t)D/E)) - debt (1-t) (D/E)
While the latter is more realistic, estimating betas for debt can be difficult to do.
In some books, the unlevered beta is referred to as the asset beta. To derive this equation, set up a balance sheet with the tax benefit from debt shown as an additional asset:
Assets Value Liabilities Value
Operating Assets A ( =u) Debt D (=0)
Tax Asset tD (=0) Equity E ( =lev)
You can set the weighted averages of the two sides equal:
u (A/(A+tD) = lev (E/(D+E))
Substituting in A = D+E -tD, you get the first equation. If you set the beta of debt be d instead of zero, you will get the second equation.
The first equation assumes that debt carries no market risk and works reasonably well for investment grade firms. For firms with junk bonds (which tend to behave like equity and carry market risk), the second approach works better. You will need to estimate a beta for debt - I use betas that are a function of the rating of the firm, with debt betas increasing as the rating falls.
Aswath Damodaran
How broadly or narrowly do we define comparable firms?
We would argue for a broader rather than a narrow definition, because the savings in standard error increase with the number of firms.
How do we deal with differences in operating leverage and business mix that may persist across these firms?
Assume that there are no differences in operating leverage and business mix or that the differences average out.
Adjust for the differences quantitatively. For instance, you could decompose the unlevered beta further into a business beta and the operating leverage effect:
Unlevered beta = Business beta (1 + (Fixed Cost/Variable costs))
How do we compute an average - simple or weighted?
I prefer simple averages. Otherwise, your betas reflect those of the largest and most stable firms in the business.
Aswath Damodaran
Why bottom-up betas?
The standard error in a bottom-up beta will be significantly lower than the standard error in a single regression beta. Roughly speaking, the standard error of a bottom-up beta estimate can be written as follows:
Std error of bottom-up beta =
The bottom-up beta can be adjusted to reflect changes in the firm’s business mix and financial leverage. Regression betas reflect the past.
You can estimate bottom-up betas even when you do not have historical stock prices. This is the case with initial public offerings, private businesses or divisions of companies.
While we still use regression betas to compute bottom-up betas, the law of large numbers works in your favor. The average of a large number of imprecise betas is more precise than any one regression beta. The reason we unlever and relever is because different firms may have different debt ratios.
Even if a company is in a single business and has not changed its business over time, the bottom up beta should dominate a regression beta. If a company has changed its business mix, the argument for bottom up betas becomes irrefutable.
Aswath Damodaran
Disney in 2003
Estimate a levered beta for Disney
Market debt to equity ratio = 37.46%
Marginal tax rate = 37.60%
Levered beta = 1.1258 ( 1 + (1- .376) (.3746)) = 1.39
The unlevered betas (corrected for cash) and the enterprise value to sales ratios were obtained by looking at comparable firms in each of Disney’s businesses. Disney’s revenues from each business are from the 2002 financial year.
Should you adjust betas for a company’s cash holdings?
It depends on how you will approach valuation. If you are valuing the non-cash or operating assets of the firm first and adding the cash back at the end (which is the conventional practice in firm valuation), you should not adjust the equity beta for the cash holdings of the firm. If you are valuing cash as part of the firm, which is what you implicitly do when you use net income as your base for estimating dividends or free cashflows to equity, you should adjust the beta for the cash holdings of the firm. Very simply, the beta for the firm inclusive of cash will be
Beta of non-cash assets (Non-cash Assets/ Firm Value) + 0 (Cash/ Firm Value)
Aswath Damodaran
Aerospace 0.95 18.95% 1.07
Levered Beta = Unlevered Beta ( 1 + (1- tax rate) (D/E Ratio)
= 0.95 ( 1 + (1-.34) (.1895)) = 1.07
The unlevered beta is the beta for aerospace companies globally. The debt to equity ratio is Embraer’s current market debt to equity ratio and the tax rate used is the marginal tax rate in Brazil.
Aswath Damodaran
Comparable Firms?
Can an unlevered beta estimated using U.S. and European aerospace companies be used to estimate the beta for a Brazilian aerospace company?
Yes
No
I would say yes, as long as
there are no significant regulatory differences between aerospace companies in Brazil and aerospace companies in the United States
The product is not a discretionary product in one market and a non-discretionary product in another
Note, though, that using the same unlevered beta does not translate into using the same cost of equity for U.S. and Brazilian aerospace company because the country risk premium (estimated earlier) would increase the cost of equity for the latter.
Aswath Damodaran
Gross Debt Ratio for Embraer = 1953/11,042 = 18.95%
Levered Beta using Gross Debt ratio = 1.07
Net Debt Ratio for Embraer = (Debt - Cash)/ Market value of Equity
= (1953-2320)/ 11,042 = -3.32%
Levered Beta using Net Debt Ratio = 0.95 (1 + (1-.34) (-.0332)) = 0.93
The cost of Equity using net debt levered beta for Embraer will be much lower than with the gross debt approach. The cost of capital for Embraer, though, will even out since the debt ratio used in the cost of capital equation will now be a net debt ratio rather than a gross debt ratio.
When you use the net debt ratio approach, the cost of capital will attach a much higher weight to the cost of equity. This will partially offset the lower cost of equity you will arrive at using the net debt ratio.
The two approaches will diverge because of differences in assumptions about tax rates (the net debt approach reduces the tax advantage of debt by offsetting it against the taxes paid on interest earned on cash) and default risk (the net debt approach assumes that debt is close to riskless).
Aswath Damodaran
Small Firm and Other Premiums
It is common practice to add premiums on to the cost of equity for firm-specific characteristics. For instance, many analysts add a small stock premium of 3-3.5% (historical premium for small stocks over the market) to the cost of equity for smaller companies.
Adding arbitrary premiums to the cost of equity is always a dangerous exercise. If small stocks are riskier than larger stocks, we need to specify the reasons and try to quantify them rather than trust historical averages. (You could argue that smaller companies are more likely to serve niche (discretionary) markets or have higher operating leverage and adjust the beta to reflect this tendency).
Aswath Damodaran
Aswath Damodaran
Estimating the Cost of Debt
The cost of debt is the rate at which you can borrow at currently, It will reflect not only your default risk but also the level of interest rates in the market.
The two most widely used approaches to estimating cost of debt are:
Looking up the yield to maturity on a straight bond outstanding from the firm. The limitation of this approach is that very few firms have long term straight bonds that are liquid and widely traded
Looking up the rating for the firm and estimating a default spread based upon the rating. While this approach is more robust, different bonds from the same firm can have different ratings. You have to use a median rating for the firm
When in trouble (either because you have no ratings or multiple ratings for a firm), estimate a synthetic rating for your firm and the cost of debt based upon that rating.
The cost of debt is not the rate at which you borrowed money historically. That is why you cannot use the book cost of debt in the cost of capital calculation.
While many companies have bonds outstanding, corporate bonds often have special features attached to them and are not liquid, making it difficult to use the yield to maturity as the cost of debt.
While ratings are often useful tools for coming up with the cost of debt, there can be problems:
A firm can have multiple ratings. You need a rating across all of a firm’s debt, not just its safest…
A firm’s bonds can be structured in such a way that they can be safer than the rest of the firm’s debt - they can be more senior or secured than the other debt of the firm.
Aswath Damodaran
Estimating Synthetic Ratings
The rating for a firm can be estimated using the financial characteristics of the firm. In its simplest form, the rating can be estimated from the interest coverage ratio
Interest Coverage Ratio = EBIT / Interest Expenses
For Embraer’s interest coverage ratio, we used the interest expenses from 2003 and the average EBIT from 2001 to 2003. (The aircraft business was badly affected by 9/11 and its aftermath. In 2002 and 2003, Embraer reported significant drops in operating income)
Interest Coverage Ratio = 462.1 /129.70 = 3.56
This is a simplistic approach but it uses the ratio that explains the largest proportion of the differences between ratings at non-financial service U.S. companies - I tried the 8 ratios that S&P said it depends upon the most to rate companies (which are available on its web site) and correlated them with bond ratings in 1999.
You could expand this approach to incorporate other ratios and create a score - similar to the Altman Z score - but you have to decide on whether the trade off is worth it - more complexity for less transparency.
Aswath Damodaran
Interest Coverage Ratios, Ratings and Default Spreads
If Interest Coverage Ratio is Estimated Bond Rating Default Spread(2003) Default Spread(2004)
> 8.50 (>12.50) AAA 0.75% 0.35%
6.50 - 8.50 (9.5-12.5) AA 1.00% 0.50%
5.50 - 6.50 (7.5-9.5) A+ 1.50% 0.70%
4.25 - 5.50 (6-7.5) A 1.80% 0.85%
3.00 - 4.25 (4.5-6) A– 2.00% 1.00%
2.50 - 3.00 (4-4.5) BBB 2.25% 1.50%
2.25- 2.50 (3.5-4) BB+ 2.75% 2.00%
2.00 - 2.25 ((3-3.5) BB 3.50% 2.50%
1.75 - 2.00 (2.5-3) B+ 4.75% 3.25%
1.50 - 1.75 (2-2.5) B 6.50% 4.00%
1.25 - 1.50 (1.5-2) B – 8.00% 6.00%
0.80 - 1.25 (1.25-1.5) CCC 10.00% 8.00%
0.65 - 0.80 (0.8-1.25) CC 11.50% 10.00%
0.20 - 0.65 (0.5-0.8) C 12.70% 12.00%
< 0.20 (<0.5) D 15.00% 20.00%
The first number under interest coverage ratios is for larger market cap companies and the second in brackets is for smaller market cap companies. For Embraer , I used the interest coverage ratio table for smaller/riskier firms (the numbers in brackets) which yields a lower rating for the same interest coverage ratio.
This table was last updated in 2004 (for interest coverage ratios and ratings). The numbers in the first column are the ones I would use for larger market cap companies (> $ 10 billion), whereas the numbers in the brackets are the ones for smaller or riskier firms. The default spreads get updated more frequently and the most recent ones can be obtained from http://www.bondsonline.com.
Aswath Damodaran
Cost of Debt computations
Companies in countries with low bond ratings and high default risk might bear the burden of country default risk, especially if they are smaller or have all of their revenues within the country.
Larger companies that derive a significant portion of their revenues in global markets may be less exposed to country default risk. In other words, they may be able to borrow at a rate lower than the government.
The synthetic rating for Embraer is A-. Using the 2004 default spread of 1.00%, we estimate a cost of debt of 9.29% (using a riskfree rate of 4.29% and adding in two thirds of the country default spread of 6.01%):
Cost of debt
= Riskfree rate + 2/3(Brazil country default spread) + Company default spread =4.29% + 4.00%+ 1.00% = 9.29%
When estimating the cost of debt for an emerging market company, you have to decide whether to add the country default spread to the company default spread when estimating the cost of debt.
For smaller, less well known firms, it is safer to assume that firms cannot borrow at a rate lower than the countries in which they are incorporated. For larger firms, you could make the argument that firms can borrow at lower rates. In practical terms, you could ignore the country default spread or add only a fraction of that spread.
Aswath Damodaran
Synthetic Ratings: Some Caveats
The relationship between interest coverage ratios and ratings, developed using US companies, tends to travel well, as long as we are analyzing large manufacturing firms in markets with interest rates close to the US interest rate
They are more problematic when looking at smaller companies in markets with higher interest rates than the US.
To develop an interest coverage ratio/ratings table, you need lots of rated firms and objective ratings agencies. This is most feasible in the United States. As long as interest rates in another country are similar to those in the United States, the ratings yielded by the table are fairly reasonable. When interest rates are high, interest coverage ratios will come under downward pressure and the table may need to be adjusted to reflect this.
Aswath Damodaran
Weights for the Cost of Capital Computation
The weights used to compute the cost of capital should be the market value weights for debt and equity.
There is an element of circularity that is introduced into every valuation by doing this, since the values that we attach to the firm and equity at the end of the analysis are different from the values we gave them at the beginning.
As a general rule, the debt that you should subtract from firm value to arrive at the value of equity should be the same debt that you used to compute the cost of capital.
The rationale for using market values is simple. You are considering how much someone who would buy the company today should be willing to pay for the company. Since he or she can buy equity and debt at today’s market value, you use those as weights. However, you could very well push the market values towards your estimated values in the process of buying the company. If this is a concern, you can iterate the weights in the cost of capital calculation to make the values used in the weights converge on the values estimated in the analysis.
Aswath Damodaran
Equity
Debt
Market Value of Debt = 2,083 million BR ($713 million)
Cost of Capital
Cost of Capital = 10.70 % (.84) + 9.29% (1- .34) (0.16)) = 9.97%
The book value of equity at Embraer is 3,350 million BR.
The book value of debt at Embraer is 1,953 million BR; Interest expense is 222 mil BR; Average maturity of debt = 4 years
Estimated market value of debt = 222 million (PV of annuity, 4 years, 9.29%) + $1,953 million/1.09294 = 2,083 million BR
Most of Embraer’s debt is not traded. Many analysts assume that book value of debt is equal to market value. You can estimate the market value of debt fairly easily using the interest rate on the debt and the book value of debt.
Aswath Damodaran
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If you had to do it….Converting a Dollar Cost of Capital to a Nominal Real Cost of Capital
Approach 1: Use a BR riskfree rate in all of the calculations above. For instance, if the BR riskfree rate was 12%, the cost of capital would be computed as follows:
Cost of Equity = 12% + 1.07(4%) + 27 (7.%) = 18.41%
Cost of Debt = 12% + 1% = 13%
(This assumes the riskfree rate has no country risk premium embedded in it.)
Approach 2: Use the differential inflation rate to estimate the cost of capital. For instance, if the inflation rate in BR is 8% and the inflation rate in the U.S. is 2%
Cost of capital=
= 1.0997 (1.08/1.02)-1 = 0.1644 or 16.44%
The two approaches will give you different answers because they make different assumptions about the equity risk premium. In the first approach, the risk premium remains a constant as you switch from one currency to another. In the second, you scale up the risk premium for higher interest rate currencies. The second approach will give you more consistent valuations as you switch from currency to currency.
Aswath Damodaran
Dealing with Hybrids and Preferred Stock
When dealing with hybrids (convertible bonds, for instance), break the security down into debt and equity and allocate the amounts accordingly. Thus, if a firm has $ 125 million in convertible debt outstanding, break the $125 million into straight debt and conversion option components. The conversion option is equity.
When dealing with preferred stock, it is better to keep it as a separate component. The cost of preferred stock is the preferred dividend yield. (As a rule of thumb, if the preferred stock is less than 5% of the outstanding market value of the firm, lumping it in with debt will make no significant impact on your valuation).
Keep your cost of capital simple. If possible, consolidate all of your capital into either debt or equity. With convertible bonds, this is fairly simple to do. With preferred, you do have a problem since its non-tax deductible status makes it unlike debt and it certainly is not equity. Here, you would make the exception and allow for a third source of capital. (In practice, I would do this only if preferred stock were more than 5% of capital in market value terms. Otherwise, I would ignore it for purposes of analysis).
The cost of preferred stock is the preferred dividend yield.
Aswath Damodaran
Decomposing a convertible bond…
Assume that the firm that you are analyzing has $125 million in face value of convertible debt with a stated interest rate of 4%, a 10 year maturity and a market value of $140 million. If the firm has a bond rating of A and the interest rate on A-rated straight bond is 8%, you can break down the value of the convertible bond into straight debt and equity portions.
Straight debt = (4% of $125 million) (PV of annuity, 10 years, 8%) + 125 million/1.0810 = $91.45 million
Equity portion = $140 million - $91.45 million = $48.55 million
The alternative approach is to use an option pricing model to value the conversion option and use that value as the value of the equity portion.
In practice, here are a few problems that you may run into:
The bond may not be traded. The best solution here is to value the conversion option. If this is not feasible, use the face value as market value.
The value of the straight bond exceeds the market price. If this happens, treat the entire bond as debt since equity options cannot have a negative value.
The firm is not rated and has no straight debt. You have to use a synthetic rating to get a cost of debt.
Aswath Damodaran
Big picture of the cost of capital.