KIERULFF’S NOTES ON FINDING THE COST OF EQUITY WITH
DUFF & PHELPS
The Duff & Phelps Risk Premium Report (D&P) helps the user develop cost of equity
(Ke) calculations by analyzing the components of risk in the equity number. The Report
considers risk associated with size by examining eight company size measures and risk in
general by examining three risk measures. The combination of size and risk leads to a
more accurate cost of equity. The size and risk measures are found in Exhibits A, B, C,
and D at the end of the Risk Premium Report.
You can calculate the cost of equity in a number of ways using build-up models (BUM)
and BUM with CAPM (the capital asset pricing model). Build-up models add layers of
risk elements on top of the risk-free rate to estimate cost of equity capital. When it is
necessary to use your company’s data, three years of past data are required as a
minimum. Five years or more are preferred.
For each team to get the most learning out of this project, I would suggest assigning team
members as follows:
1. Each team member should do the calculations for at least one of the eight size
measures (levered and unlevered) described below. Actually, you will have
data for only five of the measures with our case.
2. Each team member should deal with one of the three risk measures (levered or
unlevered). All of you should understand how they are calculated.
3. The team should decide the weights (equal weights or otherwise) to put on the
measures.
4. The team should decide together about the equity risk premium (ERP)
Adjustment and Company Specific Risk Premium (CSRP) described below.
You can, of course, ignore the country risk premium (CRP).
You need a sum beta for the CAPM approach. A sum beta is the sum of the beta
produced by regressing the company ROI against the ROI of a market proxy such as the
S&P 500 in the same months plus the beta produced by regressing the market proxy ROI
lagged one month against the company ROI. For example, a November market proxy
ROI is paired with a December company ROI, an October market proxy ROI is paired
with a November company ROI and so on. The two betas are added together because
studies have shown that the sum beta better estimates the actual beta, especially for
smaller firms. The market proxy ROI is lagged a month because smaller firms typically
react more slowly to market changes than larger firms. As a result, for example, the
December ROI of a smaller company is likely to be influenced by factors that influenced
the November market proxy ROI.
You should come out with four measures of the cost of equity. The purpose of this
lecture note is to explain how to get these.
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1. BUM 1 unlevered with the risk premium relevered to your company’s D/E
ratio. Use the C Exhibits that measure market risk plus the size factor but
with the Ke’s unlevered. They will need to be relevered to your subject
company D/E ratio.
2. BUMCAPM. Use the B and C Exhibits.
3. BUM 2. Use the B exhibits.
4. BUM 3 unlevered; with the risk premium relevered to your subject company’s
D/E ratio. Use the D Exhibits.
The size measures in Exhibits A, B, and C are:
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market value of common equity
book value of common equity
five-year average of net income
market value of invested capital (MVIC)
total assets
five-year average earnings before interest, taxes, depreciation, and amortization
(EBITDA)
sales
number of employees
D&P breaks the selected NYSE stocks into 25 size categories for each of the eight
measures and then fills selected Amex and NASDAQ stocks into their appropriate size
categories. This is redone annually. The result is 25 X 8 or 200 unique portfolios of
stocks. Many companies fall below the average of the 25th size category, but you can still
use that size category if they fall within the 25th size category range. Projecting below
the 25th size category range is frowned upon by statisticians, but often done anyway. The
risk is that you are projecting outside the range of your data.
The risk measures are in Exhibit D (levered and unlevered) and include:

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
operating margin
coefficient of variation in operating margin
coefficient of variation in return on equity
D&P follows the same procedure with the risk measures as they do the eight size
measures, breaking the stocks into 25 categories. However, the risk measures companies
are ranked by the three risk measures, not size. Since size and risk are associated but not
always well correlated, some of the largest companies may not rank in the lowest risk
category while some of the smallest companies may demonstrate less risk than some
larger firms. Also, the D Exhibits include a factor for unsystematic risk
(industry/company risk) while the A, B, and C Exhibits do not.
The idea is to identify the row in each of the size and risk Exhibits that most closely
matches your company’s size or risk measure and work with the data in that row. You an
also interpolate between size and risk ranks in most cases to get a better estimate. It is
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important to understand the column headings and how the data relate to your
measurement objectives.
Ibbotson Associates/Morningstar also publishes data on size premiums for companies
since size is said by many to greatly influence the cost of capital (the smaller the
company, the greater the risk and the higher the cost of capital). However, the D&P Risk
Premium Report, based on Standard & Poor’s Compustat database, is now regarded as
the single best source for cost of capital data. Most experts confirm that D&P’s use of
historical returns since 1963 produces market risk premiums that are more consistent with
current estimates than the Ibbotson 1926-Present data. However, you are encouraged to
consult the Ibbotson Stocks, Bonds, Bills, and Inflation Valuation Yearbook (SBBI) in the
library because it still plays an important role in valuation. We will use it to estimate the
industry risk premium (IRP). Also, it has several excellent chapters on the cost of capital.
The Exhibits have information on average debt/market value of invested capital (MVIC).
MVIC is the total company capitalization (all interest bearing debt + equity). If you have
the debt/equity ratio you can find the debt/MVIC by dividing the percentage of the debt
by 1 + percentage of the debt. If you have the debt/MVIC ratio, you can find the
debt/equity ratio by dividing the debt/MVIC ratio by 1 – the debt/MVIC ratio. If the
debt/equity ratio were 30%, the debt/MVIC would be 30%/(1+30%) or 23.08%.
Working the other way, if the debt/MVIC is 23.08%, then the debt to equity would be
calculated as .2308/(1-.2308) = 30%.
It is important that you have a clear understanding of the definitions used. The
definitions will be given in the body of this paper but it may be helpful for you to see
them briefly defined together:
Risk Free Rate (Rf): Historical return on a risk free rate accounting for the historical real
rate of interest (return) on investment, the maturity risk associated with the “riskless”
return, and inflation.
Risk Premium over the risk free rate (RPm+s): the average historical difference (1963 to
the present) between the risk free rate and the average actual return on equity of
companies in each of 25 size categories. It adds a risk premium associated with the
market as a whole (RPm) to a risk premium associated with size in 25 categories. The
market risk premium (m) stays constant across the spectrum of size; the risk factor (s)
gets greater as size diminishes.
Risk Premium associated with the market (RPm): risk premium associated with the market
as a whole, such as the S&P 500 market proxy. It is an average from 1963 to the present
without adjustment for size—the m in RPm+s above. The RPm is a measure of past returns,
not expected future returns.
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Size Premium (RPs): the percentage that accounts for the historical difference in the
RPm+s due to company size. Again, smaller companies tend to have larger returns
indicating greater risk. The size premium is also used in BUMCAPM and BUM 2.
Equity Risk Premium (ERP): The risk premium chosen by the analyst. Up to December
31, 2012 the rate recommended by D&P was 5.5%. In February 2013 it was reduced to
5% where it continues up to this time (August, 2013). See
www.DuffandPhelps.com/CostofCapital for updates.
ERP Adjustment: a percentage that increases or decreases the ERP based upon the
analyst’s belief about whether the future will be better or worse than the average of the
1963-current period. RPm+s does not include the ERP Adjustment since the ERP
Adjustment is future oriented and RPm+s is based upon past data. The ERP Adjustment
may be found by subtracting RPm+s from the ERP.
Industry Risk Premium (IRP or RPu): RPu stands for risk premium unique, historical
unsystematic risk associated with an industry. D&P does not publish these data, but they
are available from Ibbotson Associates/Morningstar—the Ibbotson ‘SBBI’ book.
Company Specific Risk Premium (CSRP): company specific risk, unsystematic risk
associated with a specific company. The IRP and CSRP may be combined when you
think it appropriate, i.e., when your company risk is the same or practically the same as
the industry risk.
Country Risk Premium (CRP): some countries are riskier than others. Estimates of this
difference in risk are available from a number of sources including Aswath Damodaran,
Pablo Fernandez, BVR, and Ibbotson/Morningstar.
BUM AND CAPM USING THE SIZE EXHIBITS
We will not be using the levered build-up exhibit, Exhibit A. We can get the same results
with less effort by relevering the unlevered exhibits to the subject company’s D/E ratio in
Exhibit C.
BUM 1 Unlevered (D&P Size Study using C Exhibits, RPm+s). The C Exhibits
measure the difference between the historical return of equities (common stock)
in each of the 25 size portfolios and the risk free rate associated with the average
long term Treasury bonds as measured by Ibbotson & Associates. You build up
the risk premium by adding the following:
1. Risk free return or risk free rate (Rf). This rate accounts for liquidity
preference (the real rate of interest), the variation (risk) associated with Treasury
bonds, and the geometric average of inflation over the period being measured.
Many analysts and service providers today use a 20-year “normalized” Constant
Maturity U.S. Treasury bond, as does Duff & Phelps (D&P), as a proxy subject to
investment rate risk. They avoid using the current Treasury rate that is commonly
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employed because it is sometimes abnormally low when the Federal Reserve is
combatting recessions with monetary policy or abnormally high when the Fed is
combatting inflation.
2. Risk Premium over the Risk Free Rate by Size: Risk Premium factor to
account for market risk (m) and company size (s) or (RPm+s). RPs indicates the
effect on the risk premium over the risk free rate caused by company size. RPm+s
measures the average historical difference (1963 to the present) between the risk
free rate and the average actual return on equity of companies in each of 25 size
categories. In other words, the risk premiums are adjusted by size using the
median of the eight measures described above. Because the total historical
difference is measured, the premium includes the beta automatically and produces
the entire historical risk value: ß(Rm-Rf). A separate beta is unnecessary and
results in double counting. In general, the smaller the company, the greater is the
ß(Rm-Rf) which suggests that smaller companies are riskier (have a higher beta)
and therefore require higher returns. RPm+s is unlevered in BUM 1 in the C
Exhibits, meaning that it demonstrates the risk in companies with no debt in their
capital structures. You will need to relever data in the C Exhibits to your
company’s D/E ratio using a formula from Harris-Pringle:
RPm+s, relevered = RPm+s, unlevered + [(Wd / We) * (βu – βd) * RPm]
The W refers to weight as a percentage, e.g., the percentage of debt. You get βu
from the C Exhibits portfolios that correspond to the size factor closest to yours.
For a more precise beta, you should interpolate between size factors. βu (beta
unlevered) is actually βa when there is no debt in the capital structure. βu is
simply a special name for βa when βa is used to calculate CAPM in an all-equity
financed firm. βa is always the same as βu since βa remains constant no matter
what the capital structure. However, βa will change when the market risk of the
assets change or management changes the assets themselves to reflect new market
realities. D&P assumes that the debt beta, βd, is 0.10. D&P also treats preferred
stock as debt since the preferred stock dividend, like debt, must be paid.
When using BUM to find a Risk Premium, you must decide whether to use the
Guideline Portfolio Method or the Regression Equation Method. To use the
Guideline Portfolio Method, you find the Smoothed Average Risk Premiums
in the eight size and three risk Exhibits that are closest to each of your eight
size measures or three risk measures. You then calculate the median of the
eight (or three) Premiums to obtain the RPm+s or RPm+u.
The Regression Equation Method is usually preferred because of added
accuracy. To obtain a smoothed average risk premium in each of the Exhibits
using the Regression Equation Method, you follow these steps:
a. Locate the Constant in the Regression Output table in the data exhibit
you are using. It is on the right hand side of the page.
b. Locate the X Coefficient in the Regression Output table.
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c. Find the log 10 of the factor you are researching. In Table C2, this is
the book value of your company. Note that D&P uses $millions.
Excel will find log10 for you if you enter =LOG10(). Put the value of
the factor between the parentheses and push return or enter on your
keyboard.
d. Multiply b by c and add a to get the Arithmetic Average Risk Premium
for the factor in the table.
e. Very small companies (below the 25th size average) can be analyzed
using the smoothed risk premium method, but great care should be
exercised because you are likely to be estimating outside the range of
the data (extrapolating beyond the range of the data), something that is
hard to justify but is common practice in some fields. There is a table
in each D&P report that shows the range of the data in Portfolio 25.
Table 5 on p. 25 of the 2012 D&P Report and Table 2, p. 22 in the
2013 Report show this range.
f. Do a through d for each of the eight available size measures and three
risk measures. Then take the median of the size measures (assuming
you believe that each should have equal weight) to get the Risk
Premium Over the Risk Free Rate for the size premiums. Do the same
for the risk premiums. Then make a judgment call on how to weight
the two medians—50%-50% or some other percentage—to get the
overall weight. Adjust if necessary considering the warning in e.
Alternatively, use the right hand columns of the C Exhibits to relate
the size premiums to the risk premiums. How to use the C Exhibits for
this purpose is explained below.
3. The ERP Adjustment: a fudge factor that may be applied to account for a
darker or rosier future than the average of the 1963-present period when using
BUM. Application is up to the judgment of the analyst. The ERP Adjustment
helps bring together past data with estimates of the future. The recommended
premium depends upon where you are in the cycle; at the peak of an
expansion, the number is lower. The ERP Adjustment is never used with
CAPM because CAPM uses beta and the beta is a future oriented factor that
should include the Adjustment. The ERP Adjustment is the difference
between the D&P or analyst recommended ERP and the total Risk Premium
over the Risk Free Rate recorded from 1963 to the present.
4. Industry Risk Premium Unique (IRP) shown as RPu or Ns in D&P and other
publications. If the industry is composed of closely held small companies,
that fact increases risk and may not be accounted for in (1-3) above.
Company Specific Risk Premiums (CSRP) account for risk specific to your
company. Risk is not just downside risk. Risk as used here refers to variation
in expected returns. IRP and CSRP examples include the downside risk (bad
weather affecting agriculture (IRP), the president quitting suddenly or
someone making a costly mistake (CSRP) and the upside risk (a boom in the
commercial real estate industry (IRP), the market for a product being greater
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than expected or a competitor going out of business (CSRP). These
nonsystematic or unsystematic risks can be eliminated almost entirely if the
investor holds a well-diversified portfolio of investments. Being diversified
ensures that one industry’s or company’s disaster probably will be offset by a
fortuitous event(s) at another industry or company in the portfolio. Include
this measure only if the client portfolio is small enough to warrant including
unsystematic risk, i.e., the investor is not well diversified. CSRP and IRP are
alike only if the company has risk factors exactly like the average of the
industry. Ibbotson/Morningstar publishes IRP data by industry in the SBBI
Yearbook; D&P does not publish IRP data.
5. Country Risk Premium (CRP). The CRP measures the differences in
systematic risk among countries. Professor Pablo Fernandez publishes a CRP
survey of businesses, analysts, and academics; Professor Aswath Damodaran
also publishes an estimate. Companies such as BVR and Morningstar publish
country risk premia for a price. To get CRP, add the difference in risk premia
between the company’s country average and the country average for the U.S.
Include this measure if the company is located in or doing substantial business
in another country. You need to be careful, however. If the company is
multinational and is listed on a U.S. stock exchange, it is very likely that the
non-U.S. country risk premium will already be included in the multinational’s
stock price and therefore in its ERP. CRPs can be useful when considering
the purchase of a business or establishing a division or joint venture in a
foreign country.
Ke, levered = Rf + RPm+s, relevered + ERP Adjustment + CSRP/RPu + CRP
BUMCAPM (D&P Size Study using B Exhibits). Ke is the sum of the following:
1.
Risk free return (Rf). See above under BUM 1.
2. Beta (𝛽) times the Equity Risk Premium (ERP). It includes the historical RP
plus the ERP Adjustment. Other services may have different premiums. You
must provide the beta, which substitutes for the ERP Adjustment and
measures company systematic risk relative to the market proxy such as the
S&P 500. You estimate beta by developing the sum beta for your company or
by using an average or median of the sum betas of comparable companies
adjusted for size. Sum betas are included for each of the portfolios in the
Exhibits.
3. Size Premium (RPs) or beta adjusted size premium reflects risk associated
only with size and is added to the ERP. RPs is the s in BUM 1’s RPm+s. RPs =
RPm+s minus RPm. The CAPM, even with the sum beta, does not predict the
cost of equity very well for less-than-giant companies. Differences of 8-10
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percentage points and more are possible between large and small firms. The
smaller the company, the greater is the realized return and therefore the
greater is the expected risk since realized return is assumed to equal required
return. RPs is the difference between what the ERP portion of CAPM
predicted would happen between 1963 and the present and what historically
came to pass. That difference is presumed to be due to size or to causal
factors associated with size. Size premiums are found in the B Exhibits. You
smooth the Size Premium data using BUM 1 steps 2a through 2d above to
account for differences in your own data from the numbers in Exhibit B.
4. Industry Risk Premium (IRP) or Company Specific Risk Premium (CSRP) to
measure unsystematic or unique risk.
5. Country Risk Premium (CRP) where appropriate.
Ke, CAPM = Rf + (β * RPm) + RPs + RPu + CRP
BUM 2 (D&P Size Study using Industry Risk Premiums and the B Exhibits),
Ke is the sum of the following:
1. Risk free return (Rf). See above under BUM 1.
2. Equity Risk Premium (ERP) is given.
3. Size Premium (RPs) is given.
4. The Industry Risk Premium (IRP) and/or Company Specific Risk Premium
(CSRP) have been discussed.
5. Add Country Risk Premium (CRP) as appropriate.
Ke = Rf + ERP + RPs + RPu + CRP
BUM USING THE RISK EXHIBITS
BUM 3 levered and unlevered (Risk Study using D Exhibits).
The D Exhibits use accounting measures of risk specific to your company—
operating margin variation, net income, and book equity—thus capturing
company specific risk regardless of size. In that way, they are significantly
different from the A, B, and C Exhibits. RPm+u replaces RPm+s in calculations; the
u stands for “unique” or CSR. Use the D Exhibits just like the BUM 1 and 2 B
and C Exhibits. Note that the levered Arithmetic Average Risk Premium and the
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unlevered Arithmetic Average Unlevered Risk Premium are in the same tables—4
and 5 columns from the right margin.
At this time, it is possible to use the Regression Method for RPm+u, levered only.
Data are not available to calculate the Regression Method for RPm+u, unlevered.
Operating margin is sales minus the sum of cost of goods sold, selling, general
and administration expenses, and depreciation. The coefficient of variation is the
standard deviation of the operating margin divided by the mean of the operating
margin. Return on equity is net income before extraordinary items divided by the
book equity value.
The formulas for levering and unlevering the RPm+u are:
RPm+u, unlevered = RPm+u, levered – (Wd / We) * (ßu – ßd) * RPm
RPm+u, levered, relevered = RPm+u, unlevered + (Wd / We) * (ßu – ßd) * RPm
The formulas for the cost of equity:
Ke, unlevered = Rf + RPm+u, unlevered + ERP Adjustment + CRP
Ke, levered, relevered = Rf + RPm+u, levered + ERP Adjustment + CRP
MORE ON COMPANY SPECIFIC RISK
Size affects risk; the larger the company, the smaller the risk tends to be. The largest
companies are in Portfolio 1 of the A, B, and C Exhibits; the second set of large
companies are in Portfolio 2 of A, B, and C; and so on. If your company appears in
Portfolio 1 in the A Exhibit for book value of equity, it must also appear in Portfolio 1 in
the B and C Exhibits for book value of equity. The same is true for the other seven
measures of size. However, there is no guarantee that a company appearing in Portfolio 1
in the market value of equity will also appear in Portfolio 1 in, say, the companies ranked
by the book value of equity or the number of employees. A given company may appear
in several different size portfolios.
Likewise, there is no guarantee that a company ranked in Category 1 in any or all of the
eight measures of size will wind up in Category 1 in any of the three risk measures in
Exhibit D. Size and risk do seem to correlate, but the correlation is not perfect. A small
company may have a niche market in an industry relatively impervious to the business
cycle and therefore have relatively small variations in its operating margin and ROI—the
risk characteristics measured by the D Exhibits. That company would tend to have a
large number portfolio rank (say, 23 or 25 indicating high risk) in the A, B, and C
Exhibits and at the same time find itself in a large number portfolio rank (say 21 or 22
indicating low risk) in the D Exhibits.
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You deal with such companies by using data in the A, C, and D Exhibits to modify the
median RP’s obtained from BUM 1, BUMCAPM, and BUM 2. Recall that we are dealing
with levered risk premiums. Suppose you were analyzing a company with total assets of
$2.148 billion. That would put it in Portfolio Rank 16 in the A-5, B-5, and C-5 Exhibits
for Companies Ranked by Total Assets. It would indicate an average operating margin of
9.40% when you looked at the third from the right-hand column of Portfolio Rank 16 in
the C-5 Exhibit (D&P, 2013). In the A-5 Exhibit (the levered Exhibit), Rank 16 would
indicate a levered average risk premium of 9.23%.
But what if the company itself had an average operating margin of 6.71%? The
Operating Margin data in the D-1 Exhibit, at 6.71%, would put the company in portfolio
rank 20. Rank 20 of the Exhibit D-1 Operating Margin has a smoothed average risk
premium of 11.24%. The portfolio in Exhibit D-1 that has an Operating Margin closest to
the C-5 Exhibit’s 9.40% is portfolio 16 with a smoothed average risk premium of 9.71%.
The difference between 11.24% and 9.71% is 1.53%. This suggests that, in the case of
the Operating Margin, your company is riskier than the A Exhibit’s 9.23% indicates by a
factor of 0 to 1.53%.
You perform the same tasks for the average coefficient of variation (CV) of Operating
Margin and average CV of ROE. Do this for all eight of the size factors. Use judgment
to determine the appropriate change in size risk based upon the risk study. Keep in mind
that if you use the C Exhibits to calculate the smoothed average risk premium you are
combining unlevered risk premiums with levered risk differences from the D Exhibits
and judge accordingly.
You may also use the same method to adjust the BUMCAPM. The only difference is that
the factors you use should be applied to the B Exhibits rather than the A Exhibits. In the
example above, the B Exhibits would suggest that the range of 0 to 1.53% be applied to
the 4.11% Smoothed Premium over CAPM in B-5 portfolio rank 16.