Risk Measurement and Hurdle Rates in Practice

04/09/08Ch. 4

2

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.

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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)

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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

5

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

6

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)

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)

)( fmfe rrrr

7

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

8

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

9

Risk-free rate

Realistically, using a long-term government bond (even with coupon), such as a 30-year Treasury bond return, is adequate for long-term 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”.

10

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

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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

uncertainHow 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

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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.

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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

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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/

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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

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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, 1962-

Present....)

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.

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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

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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

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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

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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

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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%.

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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

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Historical betas The standard procedure for estimating historical betas is to regress

stock returns (r) against market returns (rm) -

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.

mbrar

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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

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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).

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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.

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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:

The regression model is a measure of the actual return for the stock:

mf rrr )1(

mbrar

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Estimating performance If

a > rf (1-β) .... Stock did better than expected during regression perioda = rf (1-β).... Stock did as well as expected during regression perioda < 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.

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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 firm-specific risks.

For undiversified investors, a total beta measure which accounts for both risks is more appropriate.

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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?

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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.

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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.

33

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.

34

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.

35

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-to-

equity 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.

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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 cash-adjusted 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.

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Equity betas and leverage

The following equation provides us with the mathematical relationship between the unlevered and levered beta:

where

L = Levered Beta

u = Unlevered Beta

t = Corporate marginal tax rate

D = Debt Value

E = Equity Value

EDtL

u /11

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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.

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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

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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.

For private firms where we do not have stock returns to run a regression, the R2 of comparable firms will suffice.

2RL

T

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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

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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

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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

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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.

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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)

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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

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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.

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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

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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)

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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)

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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)

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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 S&P Rating> 8.5 AAA6.50 - 8.50 AA5.50 – 6.50 A+4.25 – 5.50 A3.00 – 4.25 A-2.50 – 3.00 BBB2.25 – 2.50 BB+2.00 – 2.25 BB1.75 - 2.00 B+1.50 – 1.75 B1.25 – 1.50 B-0.80 – 1.25 CCC0.65 – 0.80 CC

0.20 – 0.65 C < 0.20 D

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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:

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.

ps

psps P

Dr

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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.

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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.

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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).

Market value of bonds (D) ndBTr

incipal

)1(

Pr

BTr

)BTr+(11

- 1

C = d

nd

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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.

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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

into 2 distinct components: Straight debt, whose value is

Conversion option = Current bond price – straight debt value

ndBTr

incipal

)1(

Pr

BTr

)BTr+(1

1 - 1

C = d

nd

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Accounting for convertible bonds

The straight debt component is included in debt.

The conversion option component is included in equity.

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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

*rd represents an after-tax cost of debt

Divisional costs of capital may be calculated if firms have distinct major divisions of operation

PSED

PSr

PSED

Dr

PSED

ErWACC psde

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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.

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Readings

Text, Ch. 4 We will not be covering risk-free rate for

emerging markets or historical premiums in other markets.