Estimating the Discount Rate

1
Estimating the Discount Rate

• P.V. Viswanath
• Based on Damodarans Corporate Finance

2
Inputs required to use the CAPM

• According to the CAPM, the required rate of
  return on an asset will be
• Required ROR Rf b (E(Rm) - Rf)
• The inputs required to estimate the required ROR
  are
• (a) the current risk-free rate
• (b) the expected market risk premium (the premium
  expected for investing in risky assets over the
  riskless asset)
• (c) the beta of the asset being analyzed.

3
The Riskfree Rate

• For an investment to be riskfree, i.e., to have
  an actual return be equal to the expected return,
  there must be
• No default risk this usually means a
  government-issued security but, not all
  governments are default free.
• No uncertainty about reinvestment rates.
• In practice, therefore, the riskfree rate is the
  rate on a zero coupon government bond matching
  the time horizon of the cash flow being analyzed.
• Theoretically, this means using different
  riskfree rates for each cash flow - the 1 year
  zero coupon rate for the cash flow in year 1, the
  2-year zero coupon rate for the cash flow in year
  2 ...
• Practically, if there is substantial uncertainty
  about expected cash flows, it is enough to use a
  single riskfree rate for all flows.

4
The Bottom Line on Riskfree Rates

• Using a long term government rate (even on a
  coupon bond) as the riskfree rate on all of the
  cash flows in a long term analysis will yield a
  close approximation of the true value.
• For short term analysis, it is appropriate to use
  a short term government security rate as the
  riskfree rate.
• If the analysis is being done in real terms
  (rather than nominal terms) use 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.

5
Measurement of the risk premium

• The risk premium is the premium that investors
  demand for investing in an average risk
  investment, relative to the riskfree rate.
• As a general proposition, this premium should be
• greater than zero
• increase with the risk aversion of the investors
  in that market
• increase with the riskiness of the average risk
  investment

6
Estimating Risk Premiums in Practice

• Survey investors on their desired risk premiums
  and use the average premium from these surveys.
• Surveying in practice is difficult because there
  is no way to ensure that the numbers that
  participants provide are the ones they use in
  their own decision making.
• Assume that the actual premium delivered over
  long time periods is equal to the expected
  premium - i.e., use historical data
• Estimate the implied premium in todays asset
  prices.

7
The Historical Premium Approach

• This is the default approach used by most to
  arrive at the premium to use in the model
• In most cases, this approach does the following
• it defines a time period for the estimation
  (1926-Present, 1962-Present....)
• it calculates average returns on a stock index
  during the period
• it calculates average returns on a riskless
  security over the period
• it calculates the difference between the two
• and uses it as a premium looking forward
• The limitations of this approach are
• it assumes that the risk aversion of investors
  has not changed in a systematic way across time.
  (The risk aversion may change from year to year,
  but it reverts back to historical averages)
• it assumes that the riskiness of the risky
  portfolio (stock index) has not changed in a
  systematic way across time.

8
Historical Average Premiums for the United States
9
What is the right historical premium?

• Go back as far as you can, so as to reduce the
  standard error of the estimate. The standard
  error is roughly equal to
• Standard Error in Risk premium Annual Standard
  deviation in Stock prices / Square root of the
  number of years of historical data
• With an annual standard deviation in stock prices
  of 24 and 25 years of data, for instance, the
  standard error would be
• Standard Error of Estimate 24/ v25 4.8
• Be consistent in your use of a riskfree rate --
  if you use the T.Bill(T.Bond) rate, use the
  spread over the T.Bill (T.Bond) rate.
• Use arithmetic averages for one-year estimates of
  costs of equity and geometric averages for
  estimates of long term costs of equity.

10
Implied Equity Premiums

• If we use a basic discounted cash flow model, we
  can estimate the implied risk premium from the
  current level of stock prices.
• For instance, if stock prices are determined by
  the simple Gordon Growth Model
• Value Expected Dividends next year/ (Required
  Returns on Stocks - Expected Growth Rate)
• Plugging in the current level of the index, the
  dividends on the index and expected growth rate
  will yield a implied expected return on stocks.
  Subtracting out the riskfree rate will yield the
  implied premium.

11
Implied Equity Premiums

• This is a market neutral approach to estimate the
  implied spread, i.e., it assumes that the current
  level of the market is correct)
• For instance, if the SP 500 is at 1100, expected
  dividends next year on the index is 33 and the
  expected growth in the earnings/dividends is 7
• 1100 33/(r - .07)
• Solving for r, we find r 10
• If the treasury bond rate is 7, the implied
  Equity Premium 10 - 7 3
• The problems with this approach are
• the discounted cash flow model used to value the
  index has to be correct.
• the inputs on dividends and expected growth have
  to be correct
• it implicitly assumes that the market is
  currently correctly valued

12
In Summary...

• The historical risk premium is 6.6, if we use a
  geometric risk premium, and much higher, if we
  use arithmetic averages.
• The current implied risk premium is much lower.
  Even if we use liberal estimates of cashflows
  (dividends stock buybacks) and high expected
  growth rates, the implied premium is about 4 and
  probably lower.
• Implied risk premium using the SP 500 Index
  value is 1!
• We will use a risk premium of 5.5, because
• The historical risk premium is much too high to
  use in a market, where equities are priced with
  with premiums of 4 or lower.
• The implied premium might be too low, especially
  if we believe that markets can become overvalued.

13
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.

14
Estimating Performance

• The intercept of the regression provides a simple
  measure of performance during the period of the
  regression, relative to the capital asset pricing
  model.
• Rj Rf bj (Rm - Rf)
• Rf (1-bj) b Rm ........... Capital Asset
  Pricing Model
• Rj aj bj Rm ........... Regression
  Equation
• If aj gt Rf (1-bj) ..Stock did better than
  expected during reg period
• aj Rf (1-bj) ..Stock did as well as
  expected during regr period
• aj lt Rf (1-bj) ..Stock did worse than
  expected during reg period
• Jensen's alpha, a measure of stock performance,
  is measure as aj - Rf (1-b)

15
Firm Specific and Market Risk

• The R squared (R2) of the regression provides an
  estimate of the proportion of the risk (variance)
  of a firm that can be attributed to market risk.
• The balance (1 - R2) can be attributed to firm
  specific risk.

16
Setting up for the Estimation

• Decide on an estimation period
• Services use periods ranging from 2 to 5 years
  for the regression
• Longer estimation period provides more data, but
  firms change.
• Shorter periods can be affected more easily by
  significant firm-specific event that occurred
  during the period
• Decide on a return interval - daily, weekly,
  monthly
• Shorter intervals yield more observations, but
  suffer from more noise.
• Noise is created by stocks not trading and biases
  all betas towards one.
• Estimate returns (including dividends) on stock
• Return (PriceEnd - PriceBeginning
  DividendsPeriod)/ PriceBeginning
• Included dividends only in ex-dividend month
• Choose a market index, and estimate returns
  (inclusive of dividends) on the index for each
  interval for the period.

17
Choosing the Parameters Boeing

• Period used 5 years
• Return Interval Monthly
• Market Index SP 500 Index.
• For instance, to calculate returns on Boeing in
  May 1995,
• Price for Boeing at end of April 27.50
• Price for Boeing at end of May 29.44
• Dividends during month 0.125 (It was an
  ex-dividend month)
• Return (29.44 - 27.50 0.125)/27.50
  7.50
• To estimate returns on the index in the same
  month
• Index level (including dividends) at end of April
  514.7
• Index level (including dividends) at end of May
  533.4
• Dividends on the Index in May 1.84
• Return (533.4-514.71.84)/ 514.7 3.99

18
Boeings Historical Beta
Boeing versus SP 500 10/93-9/98
10.00
Regression
line
5.00
Returns on Boeing
0.00
-25.00
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
20.00
-5.00
Beta is slope of this line
-10.00
-15.00
Each point represents a month
of data.
-20.00
Returns on SP 500
19
The Regression Output

• ReturnsBoeing -0.09 0.96 ReturnsS P 500
  
• R squared29.57
• Intercept -0.09
• Slope 0.96

20
Analyzing Boeings Performance

• Intercept -0.09
• This is an intercept based on monthly returns.
  Thus, it has to be compared to a monthly riskfree
  rate.
• Between 1993 and 1998,
• Monthly Riskfree Rate 0.4
• Riskfree Rate (1-Beta) 0.4 (1-0.96) .01
• The Comparison is then between
• Intercept versus Riskfree Rate (1 - Beta)
• -0.09 versus 0.4(1-0.96) 0.01
• Jensens Alpha -0.09 -(0.01) -0.10
• Boeing did 0.1 worse than expected, per month,
  between 1993 and 1998.
• Annualized, Boeings annual excess return
  (1-.0001)12-1 -1.22

21
Breaking down Boeings Risk

• R Squared 29.57
• This implies that
• 29.57 of the risk at Boeing comes from market
  sources
• 70.43, therefore, comes from firm-specific
  sources
• The firm-specific risk is diversifiable and will
  not be rewarded

22
Estimating Expected Returns December 31, 1998

• Boeings Beta 0.96
• Riskfree Rate 5.00 (Long term Government Bond
  rate)
• Risk Premium 5.50 (Approximate historical
  premium)
• Expected Return 5.00 0.96 (5.50) 10.31

23
How managers use this expected return

• Managers at Boeing
• need to make at least 10.31 as a return for
  their equity investors to break even.
• this is the hurdle rate for projects, when the
  investment is analyzed from an equity standpoint
• In other words, Boeings cost of equity is
  10.31.
• What is the cost of not delivering this cost of
  equity?

24
Fundamental Determinants of Betas

• Type of Business Firms in more cyclical
  businesses or that sell products that are more
  discretionary to their customers will have higher
  betas than firms that are in non-cyclical
  businesses or sell products that are necessities
  or staples.
• Operating Leverage Firms with greater fixed
  costs (as a proportion of total costs) will have
  higher betas than firms will lower fixed costs
  (as a proportion of total costs)
• Financial Leverage Firms that borrow more
  (higher debt, relative to equity) will have
  higher equity betas than firms that borrow less.

25
Determinant 1 Product Type

• Industry Effects The beta value for a firm
  depends upon the sensitivity of the demand for
  its products and services and of its costs to
  macroeconomic factors that affect the overall
  market.
• Cyclical companies have higher betas than
  non-cyclical firms
• Firms which sell more discretionary products will
  have higher betas than firms that sell less
  discretionary products

26
Determinant 2 Operating Leverage Effects

• Operating leverage refers to the proportion of
  the total costs of the firm that are fixed.
• Other things remaining equal, higher operating
  leverage results in greater earnings variability
  which in turn results in higher betas.

27
Measures of Operating Leverage

• Fixed Costs Measure Fixed Costs / Variable
  Costs
• This measures the relationship between fixed and
  variable costs. The higher the proportion, the
  higher the operating leverage.
• EBIT Variability Measure Change in EBIT /
  Change in Revenues
• This measures how quickly the earnings before
  interest and taxes changes as revenue changes.
  The higher this number, the greater the operating
  leverage.

28
A Look at The Home Depots Operating Leverage
29
Reading The Home Depots Operating Leverage

• Operating Leverage Change in EBIT/ Change
  in Sales
• 34.94/ 32.58 1.07
• This is similar to the operating leverage for
  other retail firms, which we computed to be 1.05.
  This would suggest that The Home Depot has a
  similar cost structure to its competitors.

30
A Test

• Assume that you are comparing a European
  automobile manufacturing firm with a U.S.
  automobile firm. European firms are generally
  much more constrained in terms of laying off
  employees, if they get into financial trouble.
  What implications does this have for betas, if
  they are estimated relative to a common index?
• European firms will have much higher betas than
  U.S. firms
• European firms will have similar betas to U.S.
  firms
• European firms will have much lower betas than
  U.S. firms

31
Determinant 3 Financial Leverage

• As firms borrow, they create fixed costs
  (interest payments) that make their earnings to
  equity investors more volatile.
• This increased earnings volatility which
  increases the equity beta

32
Equity Betas and Leverage

• 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)
• where
• ?L Levered or Equity Beta
• ?u Unlevered Beta
• t Corporate marginal tax rate
• D Market Value of Debt
• E Market Value of Equity
• The unlevered beta measures the riskiness of the
  business that a firm is in and is often called an
  asset beta.

33
Effects of leverage on betas Boeing

• The regression beta for Boeing is 0.96. This beta
  is a levered beta (because it is based on stock
  prices, which reflect leverage) and the leverage
  implicit in the beta estimate is the average
  market debt equity ratio during the period of the
  regression (1993 to 1998)
• The average debt equity ratio during this period
  was 17.88.
• The unlevered beta for Boeing can then be
  estimated(using a marginal tax rate of 35)
• Current Beta / (1 (1 - tax rate) (Average
  Debt/Equity))
• 0.96 / ( 1 (1 - 0.35) (0.1788)) 0.86

34
Boeing Beta and Leverage

• Debt to Debt/Equity Beta EffectCapital
  Ratio
  of Leverage
• 0.00 0.00 0.86 0.00
• 10.00 11.11 0.92 0.06
• 20.00 25.00 1.00 0.14
• 30.00 42.86 1.10 0.24
• 40.00 66.67 1.23 0.37
• 50.00 100.00 1.42 0.56
• 60.00 150.00 1.70 0.84
• 70.00 233.33 2.16 1.30
• 80.00 400.00 3.10 2.24
• 90.00 900.00 5.89 5.03

35
Betas are weighted Averages

• The beta of a portfolio is always the
  market-value weighted average of the betas of the
  individual investments in that portfolio.
• Thus,
• the beta of a mutual fund is the weighted average
  of the betas of the stocks and other investment
  in that portfolio
• the beta of a firm after a merger is the
  market-value weighted average of the betas of the
  companies involved in the merger.

36
The Boeing/McDonnell Douglas Merger

• Company Beta Debt Equity Firm Value
• Boeing 0.95 3,980 32,438 36,418
  
• McDonnell Douglas 0.90 2,143 12,555
  14,698

37
Beta Estimation Step 1

• Calculate the unlevered betas for both firms
• Boeing 0.95/(10.65(3980/32438)) 0.88
• McDonnell Douglas 0.90/(10.65(2143/12555))
  0.81
• Calculate the unlevered beta for the combined
  firm
• Unlevered Beta for combined firm
• 0.88 (36,418/51,116) 0.81 (14,698/51,116)
• 0.86

38
Beta Estimation Step 2

• Boeings acquisition of McDonnell Douglas was
  accomplished by issuing new stock in Boeing to
  cover the value of McDonnell Douglass equity of
  12,555 million.
• Debt McDonnell Douglas Old Debt Boeings
  Old Debt
• 3,980 2,143 6,123 million
• Equity Boeings Old Equity New Equity used
  for Acquisition
• 32,438 12,555 44,993 million
• D/E Ratio 6,123/44,993 13.61
• New Beta 0.86 (1 0.65 (.1361)) 0.94

39
Firm Betas versus divisional Betas

• Firm Betas as weighted averages The beta of a
  firm is the weighted average of the betas of its
  individual projects.
• At a broader level of aggregation, the beta of a
  firm is the weighted average of the betas of its
  individual division.

40
Bottom-up versus Top-down Beta

• The top-down beta for a firm comes from a
  regression
• The bottom up beta can be estimated by doing the
  following
• Find out the businesses that a firm operates in
• Find the unlevered betas of other firms in these
  businesses
• Take a weighted (by sales or operating income)
  average of these unlevered betas
• Lever up using the firms debt/equity ratio
• The bottom up beta will give you a better
  estimate of the true beta when
• the standard error of the beta from the
  regression is high (and) the beta for a firm is
  very different from the average for the business
• the firm has reorganized or restructured itself
  substantially during the period of the regression
• when a firm is not traded

41
The Home Depots Comparable Firms
42
Estimating The Home Depots Bottom-up Beta

• Average Beta of comparable firms 0.93
• D/E ratio of comparable firms
  (2002076)/16,232 14.01
• Unlevered Beta for comparable firms
  0.93/(1(1-.35)(.1401))
• 0.86
• If the Home Depots D/E ratio is 20, our
  bottom-up estimate of Home Depots beta is
  0.861(1-.35)(.2) 0.9718

43
Decomposing Boeings Beta

• The values were estimated based upon the revenues
  in each business and the typical multiple of
  revenues that other firms in that business trade
  for.
• The unlevered betas for each business were
  estimated by looking at other publicly traded
  firms in each business, averaging across the
  betas estimated for these firms, and then
  unlevering the beta using the average debt to
  equity ratio for firms in that business.
• Unlevered Beta Average Beta / (1 (1-tax rate)
  (Average D/E))
• Using Boeings current market debt to equity
  ratio of 25
• Boeings Beta 0.88 (1(1-.35)(.25)) 1.014

44
From Cost of Equity to Cost of Capital

• The cost of capital is a composite cost to the
  firm of raising financing to fund its projects.
• In addition to equity, firms can raise capital
  from debt

45
Estimating the Cost of Debt

• If the firm has bonds outstanding, and the bonds
  are traded, the yield to maturity on a long-term,
  straight (no special features) bond can be used
  as the interest rate.
• If the firm is rated, use the rating and a
  typical default spread on bonds with that rating
  to estimate the cost of debt.
• If the firm is not rated,
• and it has recently borrowed long term from a
  bank, use the interest rate on the borrowing or
• estimate a synthetic rating for the company, and
  use the synthetic rating to arrive at a default
  spread and a cost of debt
• The cost of debt has to be estimated in the same
  currency as the cost of equity and the cash flows
  in the valuation.

46
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
• Consider InfoSoft, a firm with EBIT of 2000
  million and interest expenses of 315 million
• Interest Coverage Ratio 2,000/315 6.15
• Based upon the relationship between interest
  coverage ratios and ratings, we would estimate a
  rating of A for the firm.

47
Interest Coverage Ratios, Ratings and Default
Spreads

• Interest Coverage Ratio Rating Default Spread
• gt 12.5 AAA 0.20
• 9.50 - 12.50 AA 0.50
• 7.50 9.50 A 0.80
• 6.00 7.50 A 1.00
• 4.50 6.00 A- 1.25
• 3.50 4.50 BBB 1.50
• 3.00 3.50 BB 2.00
• 2.50 3.00 B 2.50
• 2.00 - 2.50 B 3.25
• 1.50 2.00 B- 4.25
• 1.25 1.50 CCC 5.00
• 0.80 1.25 CC 6.00
• 0.50 0.80 C 7.50
• lt 0.65 D 10.00

48
Costs of Debt for Boeing, the Home Depot and
InfoSoft

• Boeing Home Depot InfoSoft
• Bond Rating AA A A
• Rating is Actual Actual Synthetic
• Default Spread over treas. 0.50 0.80 1.00
• Market Interest Rate 5.50 5.80 6.00
• Marginal tax rate 35 35 42
• Cost of Debt 3.58 3.77 3.48
• The treasury bond rate is 5.

49
Estimating Market Value Weights

• Market Value of Equity should include the
  following
• Market Value of Shares outstanding
• Market Value of Warrants outstanding
• Market Value of Conversion Option in Convertible
  Bonds
• Market Value of Debt is more difficult to
  estimate because few firms have only publicly
  traded debt. There are two solutions
• Assume book value of debt is equal to market
  value
• Estimate the market value of debt from the book
  value for Boeing, the book value of debt is
  6,972 million, the interest expense on the debt
  is 453 million, the average maturity of the
  debt is 13.76 years and the pre-tax cost of debt
  is 5.50.
• Estimated MV of Boeing Debt

50
Estimating Cost of Capital Boeing

• Equity
• Cost of Equity 5 1.01 (5.5) 10.58
• Market Value of Equity 32.60 Billion
• Equity/(DebtEquity ) 82
• Debt
• After-tax Cost of debt 5.50 (1-.35) 3.58
• Market Value of Debt 8.2 Billion
• Debt/(Debt Equity) 18
• Cost of Capital 10.58(.80)3.58(.20) 9.17

51
Choosing a Hurdle Rate

• Either the cost of equity or the cost of capital
  can be used as a hurdle rate, depending upon
  whether the returns measured are to equity
  investors or to all claimholders on the firm
  (capital)
• If returns are measured to equity investors, the
  appropriate hurdle rate is the cost of equity.
• If returns are measured to capital (or the firm),
  the appropriate hurdle rate is the cost of
  capital.

52
Back to First Principles

• Invest in projects that yield a return greater
  than the minimum acceptable hurdle rate.
• The hurdle rate should be higher for riskier
  projects and reflect the financing mix used -
  owners funds (equity) or borrowed money (debt)
• Returns on projects should be measured based on
  cash flows generated and the timing of these cash
  flows they should also consider both positive
  and negative side effects of these projects.
• Choose a financing mix that minimizes the hurdle
  rate and matches the assets being financed.
• If there are not enough investments that earn the
  hurdle rate, return the cash to stockholders.
• The form of returns - dividends and stock
  buybacks - will depend upon the stockholders
  characteristics.