KIERULFF’S NOTES ON THE COMPONENT COSTS OF CAPITAL
The cost of capital is the rate of return required to induce sufficient investment in a given
project or company by suppliers of capital. You use it to discount cash flows in present
value analyses and in comparison with the internal rate of return (IRR) or modified IRR
(MIRR) of an investment. The term “cost of capital” is generally used to refer to the
discount rate for the company as a whole. The term “hurdle rate” is used for project,
division, department, or other asset valuations.
This cost is a future oriented, incremental concept—what will be the cost of debt and the
required rate of return on equity for the next investment. Often, you estimate it by using
some average of past data assuming an efficient market (where companies are fairly
valued). If you use past data, remember that you are assuming that the required rate of
return = expected rate of return = average actual (past) rate of return unless you make
adjustments to reflect your view of the future. Also, you should recognize that the further
back you go into the past to obtain data, the more data points you have; the more data
points you have the greater is the statistical accuracy. On the other hand, the further back
you go, the less relevant to a forecast are the data points and the more likely they are to
mislead.
The cost of capital must be sufficient to support the following:
a. An inducement to forego liquidity. Investors tend to prefer liquidity since it gives
them the options to save, spend now on consumer goods, or invest in another
(presumably better) future opportunity. Historically, investors have required a
one to three percent real return on capital (non-inflationary) as encouragement to
invest. This factor is often referred to as liquidity preference or as the real rate of
interest—real because it does not include inflation or risk.
b. The rate of inflation expected over the investment’s life assuming the cash flow
estimate includes inflation.
c. Maturity or investment rate risk. Risk is most often understood as variation in
returns. The greater the expected variation, the greater is the risk. Since the
prices of government bonds vary slightly over time as interest rates change, even
these “safe” bonds are subject to risk. Longer-term government bond prices vary
more than short-term government bonds.
d. Market risk, expressed as a percentage and added to the above three factors.
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Factors a, b, and c are termed the “risk free” rate of return even though c denotes risk.
The proxy for the risk free rate is generally the current Treasury bond yield that matches
the maturity of the project (current rule of thumb, 20 years for a company). Factor d is
termed the risk premium (RP) or the market return less the risk free rate (Rm – Rf).
The liquidity and inflation factors in the riskless rate are multiplicative because they
are independent of one another; the maturity or investment rate risk is additive:
Risk free rate = {[(1+Liquidity %) * (1+Expected Inflation %)] – 1} + Maturity
(Investment Rate) Risk.
This is important because sometimes cash flows are forecast in constant rather than
current values, leaving out the effect of inflation. If that is the case, you must adjust the
risk free rate by dividing out expected inflation from the risk free rate rather than
subtracting it: (1 + Treasury bond yield) / (1+Expected Inflation) - 1
Capitalization and the cost of capital
The capitalization or Enterprise Value of the company (its total operating capital) is
made up of all interest bearing debt associated with operations, preferred stock, common
stock, and retained earnings. Retained earnings belong to the common stockholders so is
a part of common equity and included in the return to equity (Ke). Actually, the required
return on (cost of) retained earnings is less than the required return on new common stock
because it costs to float a stock issue—five to ten percent of the issue itself to pay for
lawyers, investment bankers, printers to print the offering materials and distribute them,
etc.
The currency value of the capitalization (interest bearing debt + equity) exactly equals the
value of the operating assets of the company minus the operating current liabilities.
Otherwise the balance sheet will not balance. A balance sheet T account can demonstrate
this. On the left hand side is net operating working capital (current operating assets
minus current operating liabilities), fixed operating tangible assets, and operating
intangible assets such as patents, copyrights, trademarks and the like. Current operating
working capital excludes such non-operating items as excess cash, loans to employees,
and current interest bearing debt. On the right hand side of the T account balance sheet
are all interest bearing debt + equity. Equity includes preferred stock, common stock,
retained earnings (cumulative profit and losses after taxes, cumulative after tax gains and
losses on the sale of assets, and cumulative gains and losses on currency transactions),
and the repurchase of stock by the company (treasury stock) with after tax cash flows.
Keep in mind that valuators are analyzing the value of an operating company, division,
joint venture, etc. not counting the assets and liabilities that do not contribute to
operations. Non-operating assets and liabilities are valued separately. Since the balance
sheet in this form must balance, you can value the assets by valuing the operating debt +
equity.
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It is important to note the distinction between market value and book value. The value of
the assets on the books of the company often does not reflect the market value of the
assets because of the difficulty in estimating intangible value. Most valuation authorities
recommend using the market value of the debt and equity. However, some academics
have suggested that market values fluctuate radically and may be poor measures of value
at times. Also, they note that executives setting policy using ratio measures such as debt
to equity do so in book value terms. When terms such as Enterprise Value, Market Value
of Invested Capital (MVIC), Total Enterprise Value (TEV), and Market Cap are used
market instead of book value is meant.
When estimating the cost of capital, the weighted average cost of capital (WACC)
approach has you weighing the debt and equity components by the percentage each
makes up of the total. WACC represents the discount factor (i, or sometimes r) in
discounted cash flow using the Income Method. Including the cost of preferred stock that
most writers exclude or treat as debt for simplicity:
WACC = [(Kd * D/V * (1 – T)] + (Kp * P/V) + (Ke * E/V) = i.
Where:
Kd = cost of debt (the interest rate)
D = amount of debt in the capital structure
V = value of the operating assets as measured by debt + equity
T = expected tax rate
Kp = required return on preferred stock
P = value of preferred stock
Ke = required return on equity
E = value of common stock
As you add debt, the cost of capital will begin to decline at the point investors realize the
risk of adding debt. It will continue to decline at least up to the point where debt
becomes so burdensome that risk begins to increase disproportionately. This
phenomenon is due to the tax deductibility of debt and was explained in a
groundbreaking article by Modigliani and Miller (now known as M&M).
The M & M insight
Does it matter whether you use debt, preferred stock, common stock or some
combination to finance a company or project? The returns to holders of these securities
have to cover their liquidity preference, inflation, maturity risk, and market risk. These
four factors stay the same regardless of the form of financing—so long as the risk
associated with the assets does not change. Another way of looking at it is this: the way
assets, including intangible assets, are used determines the value of a firm or division or
business unit and the risk associated with it. The market value of a set of assets is not
dependent upon how those assets are financed.
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On the other hand, debt is typically less expensive than preferred stock, which is less
expensive than common stock. In the event of financial difficulties, holders of these
securities are paid off in the above order, so creditors are more likely than equity holders
to get their money back if the balance sheet is restructured or the company is liquidated.
Furthermore, debt principal and interest must be paid when due or creditors can take
legal action. The rules on common stock and common stock dividends are not so strict.
It follows that when you add debt to a capital structure, you might expect that the overall
cost of capital would drop rather than stay the same because debt is less expensive than
equity.
So you are faced with a dilemma. Does adding debt not matter because the risks
associated with the way the assets are deployed in the market determines their required
rate of return? Or will adding debt reduce the overall cost of capital because debt is less
expensive than equity?
Modigliani and Miller (now nicknamed M&M) came up with an answer by
demonstrating that as you add debt, the risk to both debt holders and stockholders
increases because the requirement to pay principal and interest may cripple the company
and depress its stock in hard times when cash is limited. This risk is called financial risk
to distinguish it from market risk and operating risk. The company is said to be adding
financial leverage (called gearing in the UK), and as the risk associated with this
leverage increases with the addition of debt, so must the cost of equity. Similar
reasoning applies to the addition of preferred stock since it has many of the
characteristics of debt.
M&M showed that the reduction in capital costs coming about by using more debt is
exactly offset by the increase in the cost of debt and equity. This makes sense, since the
market risk remains constant at a point in time no matter how the company is financed.
Financing has nothing to do with market risk. If the cost of equity capital (its price in the
market) is below the rate needed to satisfy liquidity preference, inflation, investment risk,
and market risk, the demand for the equity will increase. Eventually, this will drive the
cost of equity up following the law of supply and demand. The same reasoning holds
true for suppliers of debt and investors in preferred stock. The reverse happens if the cost
of capital is too high. In other words, arbitrage will happen.
However, M&M’s proof only works in the absence of taxes. In practice, the government
participates in the market risk to the extent of the tax deductibility of interest. Dividends
on preferred and common stock are not so blessed. As a result, the more debt you use,
the cheaper the overall cost of capital, but the greater the financial risk. At some point,
the increasing cost of debt and equity offsets the advantage of tax deductibility and the
cost of capital rises. Sadly enough, there is no formula to tell you where this debt/equity
point is for any given company or industry at any given time. Expected economic
conditions and changes in the structure and strategic use of company assets influence this
point.
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The components of the cost of capital
1. Cost of Debt: Kd
a. Debt includes all interest bearing securities. Some theorists would exclude shortterm debt, i.e., debt that must be paid off in a year or less and is not rolled over
(extended) on a routine basis.
b. You may estimate the cost as a percentage by finding: (1) the required yield
(IRR) on the interest bearing debts of the subject company assuming the yield will
stay the same; or (2) the interest rate paid on similar debt instruments by
companies with similar risk characteristics; or (3) CAPM (see below).
c. The Kd is usually shown as the interest rate charged by the lender. It does not
take account of taxes; it is a pre-tax number. But the company does not pay the
pre-tax cost of debt—it pays the after tax rate because interest is tax deductible.
This lower cost should be recognized by the (1 – T) factor where T is the federal
and state income tax.
d. Debt is cheaper than preferred stock and common stock because of the tax factor,
but also because debt-holders get paid off before the others in the case of
bankruptcy. Also, interest must be paid and principal repaid on schedule or the
firm faces serious difficulties. There are many different kinds of debt and
repayment arrangements, and you should be familiar with the differences.
2. Cost of Preferred Stock: Kp
a. The cost of preferred stock may be estimated by dividing the expected (or past)
preferred dividend per share by the expected price of the preferred stock per share
(or the total preferred dividend by the total value of the preferred stock). Again,
the past doesn’t necessarily reflect the future.
b. Although there are significant differences between preferred stocks and corporate
bonds, you usually treat preferred stocks as bonds for the purpose of determining
cost of capital.
3. Cost of Equity (COE or Ke) measures the required return to holders of common
stock. The numerous measures of this factor all have strengths and drawbacks. The
most well known and practical measures are shown here:
a. (D1+P1-Po) / Po: Shareholder Return Model, i.e., dividends expected for the next
period plus expected change in stock price (capital gains) divided by beginning
stock price. This is referred to as return on shareholders’ equity or investment.
You use the next period if the dividend for this period has already been paid. You
may estimate this equity cost by using previous periods’ data, if you realize that
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you are violating the incremental, future value concept—i.e., the future may not
be like the past. If a stock split occurs before the end of a given period being
measured, the formula becomes, for an X for Y split (say 2 for 1):
Ke =
(x/y*D1) + [((x/y*P1) - Po)]
Po
b. Dividend growth model: Ke = (D1 / Po) + g.
Where D1 equals the next expected dividend (or free cash flow) to equity per
share, Po the current price of the stock, and g, the expected equity dividend (or
free cash flow to equity) growth forever. You estimate dividend or free cash
flow to equity growth (g) by (1) a time series analysis of past data using
regression, (2) a compound growth forecast based upon past data, (3) a subjective
estimate, and / or (4) the following sustainable growth rate (SGR) formula based
upon ROE:
SGR = Rate the company can grow without increasing the debt / equity ratio.
SGR = ROE * [1- (Dividends per share / Earnings per share)]
c. Equity build-up models (BUM), including the capital asset pricing model
(BUMCAPM). Measures of market risk, specific industry and/or company risk, and
country risk are added to the risk free rate to produce Ke. Market risk measures
have in common some form of beta, explained below.
Measuring Market Risk with Beta
Market risk is measured as the difference between the return on investments (ROI)
generated by a proxy market index such as the Standard and Poor’s (S&P) 500 Index or
various stock exchanges such as the New York Stock Exchange (NYSE) Index shown as
market risk (Rm) and the risk free rate (Rf). Market risk is often noted as (Rm - Rf) or
ERP (Equity Risk Premium). These indexes are assumed to reflect the entire market for
investments (a notion currently under challenge by practitioners and academics).
However, in using Duff & Phelps (D&P) data you need to distinguish between the ERP
and Rm or Rm –Rf (usually total market risk minus the risk free rate), and RP (risk
premium). The latter three usually refer to past data that may or may not have a factor
added for the expected future depending upon the author’s definition. In D&P, ERP
refers to the past actual Rm plus an ERPadjustment to reflect your estimate of what the future
may bring. RP refers to historical data in D&P.
Variations, both down and up, in this index over time constitute market risk—referred to
as systematic risk. Systematic risks are those that affect all companies more or less
equally and require a risk premium (RP or ERP) as part of the cost of capital. The
business cycle, interest rate changes, social upheavals, and similar events and trends
qualify as systematic risks. Since each company will react differently to these risks, a
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special qualifier called beta, signified by β, is multiplied by the ERP in the D&P data. A
beta larger than one indicates that the company is riskier than the average company in the
proxy market; a beta smaller than one indicates that the company is less risky. It is
important to remember that systematic risks do not cover all risks faced by a firm.
Unsystematic or unique risks associated with individual companies or industries will also
affect future company returns. Unsystematic risks are specific to a firm or industry,
examples being the sudden death of the CEO, the unexpected acquisition of a large order,
a lawsuit, positive and negative events that affect one industry more than others, and
similar events. Unsystematic risks must be captured separately to develop total company
risk.
Private data collection organizations such as Morningstar publish data estimating industry
risk (Morningstar’s Ibbotson SBBI Valuation Yearbook). However, industry risk is not
always the same thing as individual company risk and this fact must be considered. If
your company is riskier or less risky than the industry average, the industry average must
be adjusted.
Mathematically, the measure of systematic risk is the covariance of the independent and
dependent variables divided by the variance of the independent variable. The
independent variable is the ROI of the proxy variable while the dependent variable is the
ROI of the target company, division, or other asset. The ROI for the dependent variable
may be the return on shareholders equity, the return on debt (the yield measured by IRR),
or the return on both together. Using the return on shareholder equity will give you the
equity beta; using return on debt will give the debt beta; using return on debt + equity
will give the asset beta assuming preferred stock is treated as debt. Also, beta equity is
equivalent to the standard deviation of the asset’s return (e.g., the company ROI) divided
by the standard deviation of the proxy’s ROI times the correlation of the company’s ROI
with the proxy’s ROI.
In addition, beta may be found as the slope of the regression line when the Ke or Kd is
regressed as the dependent variable against a market proxy as the independent variable.
If the formula for simple linear regression (often referred to as ordinary least squares or
OLS) is shown as: Y = a + bX, then the b is the slope or beta. The vertical axis
(dependent variable or Y) is the company return on shareholder equity while the
horizontal axis (independent variable or X) is the proxy for the market. The measure here
is the extent to which the company ROI changes in relation to changes in the proxy.
Again, a change greater than one in the company return suggests that the company is
riskier than the average company in the proxy. In Excel you would calculate the beta as:
=SLOPE(range of company stockholders’ returns,range of proxy returns). Note that
there is no space between returns and range.
Beta is easy to calculate on Excel, but when you calculate it using the variance, you want
to use the variance of the population, i.e. VARP, not VAR. VAR in Excel is variance
based upon a sample; instead you are considering an entire population of data. When you
calculate the standard deviations of the asset and the proxy, use STDEVP, not STDEV;
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when correlation is calculated, use CORREL. In all cases, you are referring to
populations, not samples.
There are several ways to get the data. A good source is Yahoo Finance. Here are the
steps:
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Log on to http://finance.yahoo.com/.
Enter the stock symbol or name of the your company next to where it has
“Look up” in the upper left corner. Push return or enter.
In the column on the left of the new window there will be many selections
of topics about the company. These are good to explore for information.
Select historical prices and push return.
A window comes up that offers a start date and end date for historical
prices. Set your range. If you want five years of monthly ROI data you
will need 61 data points because the first ROI requires the second data
point be divided by the first, and so on. If you plan to calculate sum beta
(discussed below) you will need 62 data points. Just to the right you can
select daily, weekly, monthly prices, or dividends only. Make your
selection, generally monthly, and push “Get Prices.” Scroll down to see
the matrix with date, opening price, high price for the day, low price for
the day, closing price for the day, average volume, and the closing price at
the end of the day adjusted for stock splits and dividends. You want the
adjusted closing price.
At the bottom of the page, you will see a command “Download in
Spreadsheet.” Do not use this because you will automatically download
an Excel spreadsheet that has comma-separated values (csv). These will
not line up in columns you can use in Excel. Instead, copy the data for the
periods of time desired and paste to a separate Excel spreadsheet.
If your stock pays dividends or has splits, these will show as separate
columns in your spreadsheet. To compare your data with a proxy variable
such as the S&P 500, you need to remove these columns. Run your
regressions on the ROI’s from the “Adj Close*” columns.
Follow the same procedure to get a proxy variable. For example, to get
the S&P 500 Index, enter ^GSPC in the “Get Quotes” area. There are
many versions of the S&P and you want ^GSPC.
There are at least seven different variations of beta for calculating the company’s overall
cost of capital. You should know about five at least—Harris-Pringle, Hamada, the sum
beta, the practitioners’ beta, and the Duff & Phelps (D&P) beta.
The Harris-Pringle beta formula is: βa = (D/V * βd) + (E/V * βe), where: D = Debt in the
capital structure; E = equity in the capital structure. V = D + E, the enterprise value. βa
measures the risk associated with both debt and equity, i.e., the market risk associated
with the assets of the company. It is a weighted average of debt and equity. The
formula: βa = [βe + (βd*D/E)] / (1+D/E) is often used in practice and is the algebraic
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equivalent.
Using algebra to obtain the equity beta when you have the asset beta, you get:
βe = [βa - (D/V * βd)] / (E/V); alternatively shown as: βe = βa + (D/E) * (βa – βd).
Many firms and investment services use the Hamada Method, a beta that includes taxes.
It assumes that the firm has debt beta of zero, D/V is constant, and that the discount rate
used to calculate the interest tax shield (ITS) is the same as the cost of debt (Kd).
Hamada uses the market value of equity also. Including the tax rate raises the levered
beta over its unlevered counterpart, indicating more risk.
βe = Ba * [1 + (1–T) * (D/E)]
Where:
βa = unlevered beta or asset beta
T = marginal tax rate
D / E = ratio of market value of interest‐ bearing debt to the market value of equity
Beta Unlevered or Beta Assets would be:
βa = βe / [1 + (1–T) * (D/E)].
The third beta you should know is the sum beta. The smaller the company, the larger is
the difference between calculated beta and what beta has turned out to be historically.
Company size affects risk and although smaller companies’ betas are higher than those of
larger companies, the calculated betas are still not high enough to explain the differences
in actual rates of return. To deal with the underestimation of small company rates of
return, many agencies such as D&P and Morningstar (Ibbotson) use the sum beta.
You create the sum beta by first finding beta by one of the three methods discussed
above. Then you lag the proxy data by one period compared with the company variable
and find a second beta. For example, if you were working with monthly data over a
period of one year starting January 2012, you would run the January-December, 2012
proxy variable ROI as the independent variable against the January-December, 2012 ROI
data for the company to get a beta. Then you would run the November-December 2012
company ROI variable as the dependent variable against the December, 2011-November,
2012 independent proxy ROI data to get a second beta. Finally, you add the two betas to
obtain the sum beta.
The idea is that smaller company stocks will react to changes in the economy slower than
the stocks of large companies. So what happens in November 2012 to the proxy ROI will
influence the December 2012 ROI of the smaller company. The company ROI is said to
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“lag” the proxy variable ROI. Unfortunately, even the sum beta still does not completely
explain the past differences between calculated cost of equity and actual equity costs, so
further size adjustments must be made.
The practitioners’ beta is simply the Harris-Pringle beta with the debt beta assumed to be
zero. The Duff & Phelps beta is the Harris-Pringle beta assuming the debt beta to be 0.1.
These beta assumptions are handy, but not entirely valid because the debt beta should
increase as the debt/value ratio increases. In practice, however, the change in the cost of
capital is small no matter what the debt beta, and the debt beta usually stays at zero when
the D/V is relatively low.
Professor Marshall E. Blume noted the tendency of beta to regress over time to the mean,
i.e., to 1.0, and developed an analytical approach called the Blume beta to deal with it. A
simplified model of this approach is used by many financial services. The one used by
Ibbotson-SBBI 2013 is: 0.371 + 0.635 X β; where β is the historical beta estimated by
ordinary least squares (OLS) regression. A second adjustment, the Vasicek adjustment,
is more complex, but possibly more accurate since it is equipped to deal with the standard
errors of individual securities. The Vasicek adjustment is discussed in the 2013 Ibbotson
SBBI Valuation Yearbook, p. 78.
Note the following: When βd = 0 (debt is considered riskless), then βa = E/V * βe
When D/V = 0 (the firm is 100% equity financed), then βa = βe
When D/V = 0, βa is defined as beta unlevered (βu)
One way to find the cost of capital for the firm with no debt is to use the asset beta—the
unlevered beta. The cost of capital for a firm that is entirely equity financed will equal a
risk free rate plus the asset beta times the risk in the marketplace.
The risk in the marketplace is usually based upon the historical average return of the
proxy market over a long period of time adjusted for expectations about the future minus
the selected Treasury bond yield. Thus, Cost of Equity Capital = Rf + βa * (Rm-Rf) or Rf +
(βa * RP); where Rf = Risk free rate, Rm = stock market return, RP = risk premium (RmRf). The asset beta (βa) is best understood as a measure of the risk associated with the
assets themselves. The asset beta will change as assets are redeployed within the
company or as markets change. βa will not change as the company moves from one level
of debt to another unless the company debt and consequent interest payment obligations
become so high that the solvency of the company is threatened.
Combining the costs of debt, preferred stock, and common stock produces the Weighted
Average Cost of Capital (WACC), the “i” or “r” that is used as the discount rate. This
rate includes a factor that accounts for the tax effect of debt and is shown as:
WACC = (Kd * (1 – T) * D/V) + (Kp * P/V) + (Ke * E/V)
Where:
Kd = interest rate on debt
T = tax rate
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D/V = percentage of debt in the capital structure
Kp = dividend per preferred stock share divided by price of preferred share
P/V = percentage of preferred stock in the capital structure
Ke = required return to common stock shareholders
E/V = percentage of common stock in the capital structure
Adjusted Present Value (APV): An Alternative to WACC
In theory at least, M&M demonstrated that the only reason the cost of capital declines as
risk is added is because debt is tax deductible. The more the debt, the greater is the
amount of deduction and the lower is the cost of capital. APV captures this by using the
unlevered Ke to discount the free cash flows and terminal value and adding the interest
tax shield discounted by the cost of debt. The interest tax shield is the periodic (annual)
interest amount in currency times the interest rate times the tax rate. You should
calculate a terminal value of the interest tax shield using the Gordon model if the
company will use a constant growth factor beyond the terminal year.
This approach makes sense only if the interest cost is estimated (assumed to be known)
up to the horizon period and (for the terminal value) beyond. Otherwise it is better to
assume a constant D/V ratio and use WACC. If you discount the interest tax shield at the
same rate as the unlevered Ke, you will in fact get the same present value as you would
with WACC.
Other Factors
Ke is influenced by three other factors not accounted for directly in either WACC or
APV. First, the cost of debt tends not to increase as the debt level rises from zero to
some relatively low D/V. Financial analysts speculate that investors pay little attention to
debt at low levels of leverage. Second, as suggested earlier, investors become concerned
about potential bankruptcy if debt levels get too high and they raise the bar on the Ke
more rapidly than the theory suggests. Finally, beta is very sensitive to company size.