Estimating Discount Rate

Estimating Discount RateEstimating Inputs: Discount RatesCritical 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 nominalCost of EquityThe 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.Risk and Return ModelRegression or Proxy Model Implied Rate of ReturnEstimating the Cost of EquitySteps of risk- return model:Measuring RiskDiversifiable and Non-diversifiable RiskAssume the Marginal Investor is Well Diversified.

Type of Risk-return model:CAPMArbitrage Pricing ModelMultifactor Model

Risk and Return ModelThe Cost of Equity: Competing ModelsModelExpected ReturnInputs Needed

CAPME(R) = Rf + (Rm- Rf)Riskfree RateBeta relative to market portfolioMarket Risk PremiumAPME(R) = Rf + j=1j (Rj- Rf)Riskfree Rate; # of Factors;Betas relative to each factorFactor risk premiumsMulti E(R) = Rf + j=1,,Nj (Rj- Rf)Riskfree Rate; Macro factorsfactorBetas relative to macro factorsMacro economic risk premiumsLays 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.Risk Free RateNo default riskNo uncertainty about reinvestment rates

Risk PremiumWhat risk premium supposed to measureRisk aversion of investorsRiskiness of the average-risk investmentEstimating Risk PremiumSurvey PremiumHistorical PremiumTime period usedChoice of risk free securityArithmetic and Geometric averageImplied Equity PremiumBetaHistorical BetaFundamental BetaAccounting Beta

Estimating Parameters for Risk and Return ModelThe CAPM: Cost of EquityConsider the standard approach to estimating cost of equity:Cost of Equity = Rf + Equity Beta * (E(Rm) - Rf)where, Rf = Riskfree rateE(Rm) = Expected Return on the Market Index (Diversified Portfolio)

In practice,Short term government security rates are used as risk free ratesHistorical risk premiums are used for the risk premiumBetas 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 modelShort 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.

An investment to be riskfree, then, it has to haveNo default riskNo 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). 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:Approach 2: Use forward rates and the riskless rate in an index currencyForward Rate of Domestic Currency = Spot rate x (1+ Interest Rate of domestic currency)/(1+Interste Rate in Foreign Currency)Risk free rate = [(1+risk free rate in foreign currency) x (1+inflation rate in domesic economy)/(1+Inflation rate in foreign economy)] - 1

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.

Risk PremiumThe 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 historyWhether you use T.bill rates or T.Bond ratesWhether you use geometric or arithmetic averages.

For instance, looking at the US:Arithmetic averageGeometric AverageStocks -Stocks -Stocks -Stocks -Historical PeriodT.BillsT.BondsT.BillsT.Bonds1928-20147.92%6.53%6.02%4.84%1964-20145.82%4.34%4.59%3.47%1994-20148.60%5.82%6.85%4.51%While everyone uses historical risk premiums, the actual premium in use can vary depending uponHow far back you go in timeWhether you T.Bills or T.BondsWhether you use arithmetic or geometric averagesThis 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.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 years 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.Implied Equity PremiumsIf 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.Implied Equity PremiumsWe 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%

14Historical Market BetaLength of time periodReturn intervalMarket IndexFundamental BetaType of BusinessDegree of operating leverageDegree of financial leverageAccounting BetaCalculation of BetaEstimating BetaThe standard procedure for estimating betas is to regress stock returns (Rj) against market returns (Rm) -Rj = a + b Rmwhere 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 errorIt reflects the firms business mix over the period of the regression, not the current mixIt reflects the firms 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.Solutions to the Regression Beta ProblemModify the regression beta bychanging the index used to estimate the beta adjusting the regression beta estimate, by bringing in information about the fundamentals of the companyBloomberg beta = raw beta (.67) + 1 x (.33)

Estimate the beta for the firm using the standard deviation in stock prices instead of a regression against an indexaccounting 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 requireunderstanding the business mix of the firmestimating the financial leverage of the firm.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.Determinants of Betas

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.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. Adjusting for operating leverageWithin 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 firms life cycle.Equity Betas and LeverageConventional 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 ratioL = 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:AssetsValueLiabilitiesValueOperating AssetsA ( =u)DebtD (=0)Tax AssettD (=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.

Bottom-up Betas

Three measurement issues come up: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.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 firms 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.Regression or Proxy ModelImplied Rate of Return ModelEstimating the Cost of DebtThe 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 tradedLooking 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 firms debt, not just its safestA firms bonds can be structured in such a way that they can be safer than the rest of the firms debt - they can be more senior or secured than the other debt of the firm.

Estimating Synthetic RatingsThe 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 ratioInterest Coverage Ratio = EBIT / Interest Expenses

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.Interest Coverage Ratios, Ratings and Default SpreadsIf Interest Coverage Ratio isEstimated Bond RatingDefault Spread(2003)Default Spread(2004)> 8.50(>12.50)AAA0.75%0.35%6.50 - 8.50(9.5-12.5)AA1.00%0.50%5.50 - 6.50(7.5-9.5)A+1.50%0.70%4.25 - 5.50(6-7.5)A1.80%0.85%3.00 - 4.25(4.5-6)A2.00%1.00%2.50 - 3.00(4-4.5)BBB2.25%1.50%2.25- 2.50(3.5-4)BB+2.75%2.00%2.00 - 2.25((3-3.5)BB3.50%2.50%1.75 - 2.00(2.5-3)B+4.75%3.25%1.50 - 1.75(2-2.5)B6.50%4.00%1.25 - 1.50(1.5-2)B 8.00%6.00%0.80 - 1.25(1.25-1.5)CCC10.00%8.00%0.65 - 0.80(0.8-1.25)CC11.50%10.00%0.20 - 0.65(0.5-0.8)C12.70%12.00%< 0.20( $ 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.Cost of Debt computationsCompanies 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.Weights for the Cost of Capital ComputationThe 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 todays 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.