Risk Measurement and
Hurdle Rates in Practice
04/09/08
Ch. 4
Investment decision
 Firms should invest in projects that creates
value for the firm’s shareholders
 These are projects that yield a return greater
than the minimum acceptable hurdle rate with
adjustments for project riskiness.
2
Investment decision
 Components of the investment decision
making process



Determine the appropriate hurdle rate for the
firm (Ch. 4)
Calculate the returns on investments (Ch. 5)
Evaluate project interactions (Ch. 6)
3
What is a hurdle rate?
 The hurdle rate for the firm represents the minimum
rate of return that the firm as a whole must generate
on its investments to satisfy its investors.
 This is sometimes referred to as the weighted
average cost of capital (WACC) or simply the cost
of capital.
 This hurdle rate is a function of (among other things):


Type and mix of investors (equity, debt, preferred
stock)
Riskiness of the firm
4
Cost of equity
 Required rate of return for equity investors (or
shareholders) is also referred to as the cost of
equity
 For publicly traded firms, we initially assume that the
these equity investors are diversified investors.
Consequently,


Only the firm’s risk relative to the market is relevant
(systematic risk)
Firm-specific risk is assumed to be diversified away
 We will later relax our diversified investor assumption
5
Cost of equity
 With our diversified investor assumption, the appropriate risk
measure for a firm is beta (β) which measures the firm’s
systematic risk
 Although it has its limitations, the appropriate model to estimate
the cost of equity is the Capital Asset Pricing Model (CAPM)
re  rf   (rm  rf )
where re is the cost of equity for a particular stock, rf is the risk
free rate, and rm is the return on the market (S&P 500 index for
our purposes)
6
Cost of equity
 To calculate the cost of equity for a firm, we need
estimates for each of its components:



Risk-free rate
Market return, or alternatively the market risk premium,
(rm-rf)
Firm’s beta
7
Risk-free rate
 With the risk-free asset, the actual return and
expected return do not vary.
 This asset assumes:


No default risk
No reinvestment risk
 Ideally, this means that you should use a risk-
free asset whose maturity matches the timing
of cash flows
8
Risk-free rate
 Realistically, using a long-term government
bond (even with coupon), such as a 30-year
Treasury bond return, is adequate for longterm analyses.
 For short-term analyses, short term
government securities, such as the 3-month
Treasury bill, are appropriate.

The current rates for these instruments are
available from the Yahoo (or other financial
information sources) and are represented by
the “yield”.
9
Market risk premium
 The market risk premium represents the extra return
beyond that of a risk-free asset that an investor
demands for moving their funds from the risk-free
asset to the risky (market portfolio) asset.
 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” risky
investment
10
What is your risk premium?
 Assume that stocks are the only risky assets and that you are
offered two investment options:


a riskless investment (say a Government Security), on
which you can make 6%
a mutual fund of all stocks, on which the returns are
uncertain
How much of an expected return would you demand to shift your
money from the riskless asset to the mutual fund?
 6% or less
 7%
 8%
 9%
 10%
 11% or more
11
Risk Aversion and Risk Premiums
 The risk premium is a weighted average of
the risk premiums demanded by each and
every investor.
 The weights will be determined by the
magnitude of wealth that each investor has.
 As investors become more risk averse, you
would expect the “equilibrium” premium to
increase.
12
Risk Premiums do change..
Go back to the previous example. Assume now
that you are making the same choice but that
you are making it in the aftermath of losing
your job. Would you change your answer?
 I would demand a larger premium
 I would demand a smaller premium
 I would demand the same premium
13
Estimating the market risk premium
 There are three methods to estimate the
market risk premium.



Survey premium
historical risk premium
Implied risk premium
*Note: The author has data available for much of these analyses
at http://pages.stern.nyu.edu/~adamodar/
14
Survey premiums
 Surveys conducted of fund managers and
economists can provide us with one estimate
for the current market risk premium.



Economists : 4 – 7%
Manager surveys (recent): 3.5%
These estimates can be volatile, may be short
term and may be based on other
computational methods
15
The historical premium approach
 In most cases, historical market risk premiums can
be estimated as follows:

define a time period for the estimation (1926-Present, 1962Present....)

calculate average returns on a stock index during the period;
with longer-term analyses use geometric returns

calculate average (geometric) returns on a risk-free security
over the same period

The market risk premium is then the difference between
the two, i.e., rm - rf.
16
Historical premium limitations
 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
17
Implied premium approach
 Dividend-based approach:
 The implied risk premium approach estimates a risk
premium based on current market values, dividends
and growth rates.

We can use a basic dividend discount model (DDM) to
estimate the implied risk premium:
Index value =

Expected index dividends
(Required return on the index – growth rate in dividends)
The implied risk premium would then be:
RR on index – current risk-free rate
18
Implied premium approach
 Earnings-based approach:
 Under the earnings-based approach, the earnings yield
(E/P) of the market is assumed to approximate the real
return on the market.

Note that the earnings yield is the inverse of the market
P/E multiple.

The implied risk premium would then be:
earnings yield – current risk-free rate
19
Implied premium limitations
 The limitations of this approach are:


It assumes that the DDM is correct to value
the market.
It assumes that the market is currently
correctly valued
20
What should we use as a risk
premium?
 The historical risk premium tends to be higher than
the implied risk premium.
 An average of the two measures may serve as a
good estimate for the risk premium because the
historical risk premium is much too high to use in a
market, where equities are currently priced with
premiums that are closer to 3%.
21
Estimating betas
 A firm’s beta represents the level of systematic risk
inherent in the firm.
 It can be measured in two ways:


Historical measure (or top-down approach) – measured
by regressing historical stock returns of the firm on index
(or market) returns
Bottom-up approach – estimated by measuring the
average beta for firms within the same industry after
adjusting for financial leverage
22
Historical betas
 The standard procedure for estimating historical betas is to regress
stock returns (r) against market returns (rm) -
r  a  brm
where a is the regression intercept and b is the slope of the regression.
 The slope of the regression (b) corresponds to the beta of the stock,
and measures the riskiness of the stock. Regression parameters are
always estimated with error. The error is captured in the standard error
of the beta estimate
 Note: Do not confuse this equation with the CAPM. This is simply a
linear representation of the relationship between the return for the firm
and the market return.
23
Setting up for the estimation
 Decide on an estimation period
 Services (such as Value-Line) 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,
monthly returns tend to work well.
 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.
 Run the regression
24
Applying the approach
 Data for periodic individual stock prices and market
index values can be found on Yahoo and other
financial websites

In general, closing stock prices should be used.

The S&P 500 serves as a good market index.

Ensure that the returns are calculated including dividends.


This would mean using the adjusted closing prices for the index
and stock in Yahoo
You should include the dividends paid by the company in the
month in which it was paid (if monthly returns are calculated).
25
Forward looking beta adjustment
 The regression-estimated betas represent betas that are calculated
using historical (or past) stock prices and index values.
 Since the purpose of calculating these betas is to provide us with a
hurdle rate for future decisions, sometimes these values are adjusted to
be forward-looking.
 When making the forward-looking adjustment, we assume that as firms
mature, their betas tend towards 1 (market beta).
Forward-looking beta = Regression beta * 0.67 + 1 * 0.33
 Many financial reporting agencies, such as Bloomberg use this kind of
adjustment.
26
Estimating performance
 The regression output (intercept and slope)provides a simple
measure of performance during the period of the regression,
relative to the capital asset pricing model.

CAPM provides an expected return for the stock:
r  rf (1   )  rm

The regression model is a measure of the actual
return for the stock:
r  a  brm
27
Estimating performance
 If
a > rf (1-β) ....
Stock did better than expected during
regression period
a = rf (1-β)....
Stock did as well as expected during
regression period
a < rf (1-β)....
Stock did worse than expected during
regression period
 Alternatively, we can calculate Jensen’s alpha (α) :
α =a -
rf (1-β)
If this measure is greater (less) than 0, the company performed
better (worse) than investors expected during the period from
which the regression data was obtained.
28
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.
 Diversified investors are only concerned about market risk.
Undiversified investors care about both market and firmspecific risks.
 For undiversified investors, a total beta measure which
accounts for both risks is more appropriate.
29
The Relevance of R2
 You are a diversified investor trying to decide whether you
should invest in Home Depot or Bed, Bath and Beyond. Both
firms provide equal returns. They both have betas of 1.404, but
Home Depot has an R2 of 40% while Bed Bath and Beyond’s R2
is only 15%. Which one would you invest in?
 BBB, because it has the lower R2
 HD, because it has the higher R2

You would be indifferent
 Would your answer be different if you were an undiversified
investor?
30
Bottom-up betas
 The bottom-up approach relies on the fundamental
characteristics of the firm to determine the riskiness of the firm
and thus the firm beta.
 These fundamental characteristics include:
 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 with
lower fixed costs (as a proportion of total costs)

Financial Leverage: Firms that borrow more (higher debt, relative
to equity) will have higher betas than firms that borrow less.
31
Bottom-up betas
 The critical assumption we make in using the
bottom-up approach is that the riskiness associated
with the type of business and operating leverage
will be similar across firms that are in the same
industry and are of similar size.
 This assumption implies that the only difference in
riskiness between firms in a particular industry
comes from differences in financial leverage, or
debt/equity mix.
32
Bottom-up betas
 The first component of a firm’s risk is that
associated with its operations:

We can measure this risk by calculating the
firm’s unlevered beta (βU), i.e., a beta that
removes the effect of financial leverage.

This unlevered beta is also referred to as an
asset beta as it represents the riskiness of a
firm’s assets.
33
Bottom-up betas
 The second component of a firm’s risk is that
associated with its financial leverage

The greater the debt/equity ratio, the greater the financial
leverage, and therefore financial leverage risk
 The firm’s levered (or bottom-up) beta provides us
with an estimate of both operating and financial
leverage risks.
 This levered beta (βL), represents the firm’s risk and
is equivalent (but not necessarily the same in value)
to the historical beta calculated using the regression
approach.
34
Bottom-up betas
 The levered beta can be estimated by doing the
following:




Find out the industry that a firm operates in
Compute the beta (regression) for a set of
comparable firms
Calculate the average regression beta, debt-toequity ratio and tax rate for these firms using
simple averages.
Compute an average unlevered beta.

Note: If the firm has more than one distinct business
segment, compute a weighted average unlevered beta
with weights determined by segment revenues.
35
Bottom-up betas

Adjust the unlevered beta for comparable firm’s cash since cash
has a beta close to zero:
Cash-adjusted unlevered beta =
Unlevered beta
(1 – Cash/Firm Value)

Adjust for firm’s cash level by taking weighted average of cashadjusted unlevered beta from above and cash beta.

Lever up using the firm’s debt/equity ratio.
 This method is best for beta estimation for non-traded (private)
firms since stock prices are not available to measure a historical
beta.
36
Equity betas and leverage
 The following equation provides us with the mathematical
relationship between the unlevered and levered beta:
u 
L
1  1  t D / E 
where
L = Levered Beta
u = Unlevered Beta
t = Corporate marginal tax rate
D = Debt Value
E = Equity Value
37
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, for example,

the beta of a firm is the weighted average of the betas of the
firm’s distinct divisions

the beta of a firm after a merger is the market-value
weighted average of the betas of the companies involved in
the merger.
38
Is beta an adequate measure of risk for
a private firm?
 The owners of most private firms are not diversified. Beta
measures the risk added on to a diversified portfolio.
Therefore, using beta to arrive at a cost of equity for a
private firm will…



Over estimate the cost of equity for the private firm
Under estimate the cost of equity for the private firm
Could under or over estimate the cost of equity for the private firm
39
Total risk versus market (systematic)
risk
 For private firms (or firms where owners are not diversified), it
would be more appropriate to use the firm’s total risk rather than
just market risk.
 The adjustment to account for total risk is a relatively simple one,
since the R2 of the regression measures the proportion of the risk
that is market risk.
T 
L
R2
 For private firms where we do not have stock returns to run a
regression, the R2 of comparable firms will suffice.
40
Bottom-up or regression (top-down)
beta: Which one should we use?
 The bottom-up beta will give you a better
estimate of the true beta when


the firm is not substantially different
fundamentally (size, operational
characteristics, etc.) from the other firms in the
industry
the firm has reorganized or restructured itself
substantially during the period of the
regression
41
Summary of cost of equity estimation
 Determine an appropriate risk-free rate (the
30-yr T-bond rate will usually suffice).
 Estimate an appropriate market risk premium
 Calculate firm beta either by:
Using a top-down or regression approach
 Bottom-up approach
 Make adjustments if necessary to account
for undiversified investors or to make the
value forward looking

 Calculate the cost of equity using the CAPM
42
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 or hybrid securities
43
What is debt?
 General Rule: Debt generally has the following
characteristics:



Commitment to make fixed payments in the future
The fixed payments are tax deductible
Failure to make the payments can lead to either default or
loss of control of the firm to the party to whom payments are
due.
 As a consequence, debt should include
 Any interest-bearing liability, whether short term or long term.
 Any lease obligation, whether operating or capital.
44
Cost of debt vs. required rate of return
for debtholders
 The required rate of return for bondholders of a particular firm is
a function of:


Current interest rate for the risk-free asset (30-yr. T-bond yield)
Default risk associated with the firm, i.e., how likely is the firm to go
bankrupt (risk premium).
 Bondholders are compensated in interest payments (or coupon
payments) for this required rate of return. This represents the
before-tax cost of debt (BT rd)
 Because from the firm’s perspective interest expense is tax-
deductible, the after-tax cost of debt (rd) is:
BT rd * (1 – tax rate)
45
Determining the default risk
 Firms will have a higher default risk if:

They generate low cash flows relative to their
financial obligations

They have more volatile cash flows

They have less liquid assets
46
Determining the default risk
 Firms will have a higher default risk if:
 They generate low cash flows relative to their financial
obligations

They have more volatile cash flows

They have less liquid assets
 Rating agencies (S&P and Moody’s) evaluate these
factors when determining a rating for the firm and its
issues.
47
Estimating the cost of debt
 Depending on whether or not the firm in question has
bonds that are publicly traded and on available
information, there are three ways (in order of
preference) to estimate the before-tax cost of debt:



Look for prices and yields of bonds outstanding
Estimate the cost of debt from the firm’s credit rating
Estimate the cost of debt by calculating a synthetic credit
rating
48
Estimating the cost of debt
 If the firm has bonds outstanding, and the bonds are
traded, the yield to maturity (YTM) on a long-term,
straight (no special features) bond can be used as
the before tax cost of debt.
 The YTM incorporates the risk-free rate and firm-
specific default risk.
 Sources:


Look at the Corporate Bond excerpt in the WSJ or other
publications
Yahoo may also have this information
rd = YTM * (1-t)
49
Estimating the cost of debt
 If the firm is rated, use the credit rating and a typical
default spread on bonds with that rating to estimate
the cost of debt.

Standard & Poors, Moody’s and Fitch provide credit ratings
for firms. The first of these ratings can be found at :
www.standardandpoors.com

Default spreads can be found at www.bondsonline.com
(premium service) OR inferred from bond spreads of other
bonds with the same rating
rd = (30-yr. T-bond yield + spread) * (1-t)
50
Estimating the cost of debt
 If the firm is not rated,

estimate a synthetic rating for the company, and use the
synthetic rating to arrive at a default spread and a cost of
debt
rd = (30-yr. T-bond yield + spread) * (1-t)
51
Estimating synthetic credit 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
Interest Coverage Ratio
> 8.5
6.50 - 8.50
5.50 – 6.50
4.25 – 5.50
3.00 – 4.25
2.50 – 3.00
2.25 – 2.50
2.00 – 2.25
1.75 - 2.00
1.50 – 1.75
1.25 – 1.50
0.80 – 1.25
0.65 – 0.80
0.20 – 0.65
< 0.20
S&P Rating
AAA
AA
A+
A
ABBB
BB+
BB
B+
B
BCCC
CC
C
D
52
Cost of preferred stock
 The cost of preferred stock, which has some characteristics of debt
and equity (specified dividend, not tax deductible, infinite life) is
calculated as follows:
rps 
D ps
Pps
where rps, Dps, and Pps are the cost of preferred stock, dividend on
preferred stock and current price per share of preferred stock,
respectively.
 www.quantumonline.com has useful information about preferred
stocks.
53
Estimating capital weights
 The firm’s cost of capital is a function of the
cost of each type of financing the firm adopts
and of the weights of each type of financing
 Book values (obtained from financial
statements):




Equity: includes common stock and retained earnings
Debt: includes long term debt and the present value of
leases
Preferred Stock: includes preferred stock
Convertible bond value should be apportioned to equity
and debt.
54
Estimating capital weights
 Market Values:
 Market Value of Equity (E) and Preferred Stock (PS)
should include the following:

Market Value of Shares outstanding:
Shares outstanding * current stock price

Market Value of Warrants outstanding
 10-K filings (with the SEC) and annual reports should provide
information about the shares outstanding, market value of
warrants should be in the 10-Ks.

Conversion option value of outstanding convertible bonds
should be included in Equity.
55
Estimating capital weights
 Market value of debt (D) is more difficult to estimate
because few firms have only publicly traded debt.

Estimate the market value of debt from the book value by
treating the entire debt as one coupon bond, with a coupon
(C) set equal to interest expenses and maturity (n) equal to
the average maturity of all debt outstanding and using the
current before-tax cost of debt (BT rd).
1

1
 (1 + BTr ) n
d
Market value of bonds (D) = C 
BTr d



 Pr incipal

n
 (1  BTrd )

56
Converting leases to debt
 The “debt value” of leases is the present
value of the lease payments, at a rate that
reflects their risk.
 In general, this rate will be close to or equal
to the rate at which the company can borrow,
i.e., pre-tax cost of debt.
 Capital leases are included in the balance
sheet, operating leases are not.
57
Accounting for convertible bonds
 A convertible bond is a bond that can be converted into equity at
the option of the bondholder.
 To incorporate a firm’s outstanding convertible bond issues into
our cost of capital calculations, we separate the bond
1


into 2 distinct components:
1
 (1 + BTr ) n  Pr incipal
d
 Straight debt, whose value is = C 

n



BTr d
 (1  BTrd )

Conversion option = Current bond price – straight debt value
58
Accounting for convertible bonds
 The straight debt component is included in debt.
 The conversion option component is included in
equity.
59
Cost of Capital
 The weighted average cost of capital (or firm hurdle rate) is then
just the weighted average of the individual sources of capital
E
D
PS






WACC  re 
  rd 
  rps 

 D  E  PS 
 D  E  PS 
 D  E  PS 
*rd represents an after-tax cost of debt
 Divisional costs of capital may be calculated if firms have distinct
major divisions of operation
60
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.
61
Readings
 Text, Ch. 4

We will not be covering risk-free rate for
emerging markets or historical premiums in
other markets.
62