Cost of Capital Estimation

Methods for Benchmarking the Cost of Equity Capital1. Capital Asset Pricing Model using beta from a

regression analysis (top-down method)2. Capital Asset Pricing Model weighting beta

estimates for individual divisions using industry betas (bottom-up method)

3. Implied Cost of Equity using stock valuation model, given stock price and expected growth rates

Only Systematic Risk is Priced in the Capital Asset Pricing Model The key result of the CAPM is that the relevant risk of

any asset is the risk that it adds to the market portfolio—the systematic risk Systematic risk is exposure to market factors that affect all

securities to a greater or lesser degree (e.g. inflation, GDP growth, interest rates, political events, etc.)

In well diversified portfolios exposure to firm-specific (unsystematic) events is diversified away (e.g. management changes, product announcements, litigation, etc.)

The systematic risk is measured by the security’s co-movement with a broadly diversified portfolio

Risk measures

Standard deviation (stdev)

Covariance (covar.s)

Correlation

Coefficient (correl)

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rrn

1i

2^

i

2n

rrrr

)AB(Cov

n

1i

B

^

BiA

^

Ai

BAAB

)AB(Cov

The standardization confines the ρ to values between –1 and +1

Portfolio standard deviation for a two-security portfolio:

ABBA2B

2B

2A

2Ap COVww2ww

Two-Security PortfolioVariance-Covariance Matrix The two-security portfolio contains two

covariance (market risk) terms and two variance (stand-alone risk) terms

2A CovA.B

CovB.A 2B

Three-Security PortfolioVariance-Covariance Matrix

2A CovA.B CovA.C

CovB.A 2B CovB,C

CovC,A CovC.B 2C

Standard deviation and risk

The standard deviation of a single security includes both systematic and unsystematic risk

For well diversified portfolios, the standard deviation includes systematic risk only

Efficient Portfolios

Combining assets with less than perfect correlation improves portfolio efficiency by reducing unsystematic risk

An efficient portfolio is one that offers: the most return for a given amount of risk, or the least risk for a give amount of return.

The collection of efficient portfolios is called the efficient frontier

ExpectedPortfolio Return, E(Rp)

Risk, p

Efficient Frontier

See two-security example

Capital Asset Pricing Model

The CAPM is an equilibrium model that specifies a linear relationship between risk and required rate of return for assets held in well-diversified portfolios.

When a risk-free security with return rRF is added, investors can create portfolios that combine this security with a portfolio of risky securities.

Since the risk-free asset has zero variance, its covariance is also zero Thus the standard deviation of a 2-security

portfolio of the risk-free asset and the market portfolio, M, is: wmσm

Adding a risk-free security

mmw mmw

What impact does rRF have onthe efficient frontier?

Both the standard deviation and expected return are linear functions of the weights wrf and wm

The straight line connecting rRF with M (market), the tangency point between the line and the old efficient frontier, becomes the new efficient frontier.

M

rRF

M Risk, p

Efficient Frontier with a Risk-Free Asset

The Capital MarketLine (CML):

New Efficient Frontier

.rM^

ExpectedReturn, rp

rp = rRF +^ p.rM - rRF^

M

p is determined by selecting weights on:the risk-free security (wrf)the market portfolio (wm)

The equation for the Security Market Line, the principal result of the Capital Asset Pricing Model, gives the risk/return relationship for individual securities.

Substitute the contribution of an individual security’s risk to the market portfolio standard deviation, ρi,mσi , for σp:

The Security Market Line (SML)

m

im,irfm

^

rfi

^

]rr[rr

irfm

^

rfi

^

beta]rr[rr

Beta

2,

2,

,m

mi

m

mimi

m

imii

Covbeta

Beta intuition

Beta is simply a measure of relative systematic risk, or relative exposure to the economic variables that drive market returns.

For example, a security with a beta of 1.20 exhibits 20 percent more variability in response to market returns as compared with a typical security.

Result of the CAPM

Expected return for stocks includes a risk-free component a risky component as determined by a risk premium

for the average stock, known as the ‘market risk premium’, (rm - rrf)

and the beta of the individual stock, which measures the degree of market risk exposure for the individual security

The expected (required) return on the stock is the issuing company’s cost of Equity

irfm

^

rfi

^

beta]rr[rr SML:

Security Markey Line

0%

5%

10%

15%

20%

25%

0 0.5 1 1.5 2 2.5 3

Beta

Exp

ecte

d R

etur

n

Risk-free rate

Expected return on market

= Rm - Rrf

Std. Deviation BetaSecurity A 20% 1.25Security B 30% 0.95

Which security has more unsystematic risk?

Which security has more systematic risk? Which security should have the higher

required return?

Estimating the CAPM Inputs

The beta of the security The expected market risk premium The current risk-free rate

irfm

^

rfi

^

beta]rr[rr

Estimating Beta (top-down approach) Standard approach is to regress stock returns

against those of a broad market index, where the slope of the regression line is the beta coefficient: most services use either 5 years (monthly returns) or

2-3 years (weekly) regressed on the S&P 500 A 5-year interval insures against possible aberrant

shocks to the beta due to unusual short-term events A shorter interval may better reflect the company’s

current risk profile if its business or operating environment have changed

Many services adjust beta toward 1.0 Example: Adj. Beta = 0.67*beta + 0.33*1.0

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Bottom-up Betas The beta of a portfolio is a market-value weighted

average of the betas of the assets comprising the portfolio the beta of a firm is a weighted average of the

individual divisions or projects in which the firm invests

A bottom-up beta estimates beta for each of the divisions using industry betas, and uses a weighted average of these division betas to estimate the corporate beta

Advantages of Bottom-up Betas

Since the procedure involves averaging across several regression betas, the estimation error is lower, and the estimates are more stable

The bottom-up beta may provide a better estimate of the true beta when the firm has reorganized or restructured itself substantially during the period of the regression Weight the division betas based on the current mix

Division betas are required to make investment decisions

Division Cost of Capital

Rate of Return (%)

WACC

Project H

Division H’s WACC

Risk

Project L Composite WACC for Firm A

13.0

7.0

10.0

11.0

9.0

Division L’s WACC

0 RiskL RiskAverage RiskH

The firm’s overall cost of capital cannot be applied to individual divisions or projects when there are differences in risk: 1) operating (business) risk; 2) financial risk

Operating Risk

Variability in Earnings Before Interest and Taxes Two sources:

1) Industry Effects on sales Cyclical companies have higher business risk than non-cyclical

firms Firms which sell more high-cost and discretionary products will

have higher business risk

2) Operating leverage effects: Operating leverage refers to the proportion of the total costs of the firm that are fixed.

Other things equal, higher operating leverage results in greater earnings variability

Operating leverage measure = % Change in EBIT / % Change in Revenues

Operating Leverage

Revenue 100 200 300

Variable costs (20%) (20) (40) (60)

Fixed costs (100) (100) (100)

EBIT (20) 60 140

Revenue 100 200 300

Variable costs (50%) (50) (100) (150)

Fixed costs (40) (40) (40)

EBIT 10 60 110

Financial Risk

As firms borrow, they create fixed costs (interest payments) that make their earnings to equity investors more volatile

This increased earnings volatility increases the equity Beta

As more variance is added, and some fraction of this variance is correlated with markets, beta increases

The Pure-Play approach to Beta estimation The typical approach is to find publicly traded

‘pure play’ companies operating primarily in division’s business Can expand to customers and suppliers if difficult to

find companies They should have levels of operating risk (EBIT

variance) that are comparable to that of the division since they are in the same industry

Their levels of financial risk, however, will vary due to differences in financing choice

The Pure-Play approach to Beta estimation Process for dealing with financial leverage

differences: Unlever the betas of the pure-play firms

removes the effects of their debt-equity mix on beta Take an average of these unlevered betas Relever the betas at the division’s target debt-to-stock

ratio

The Cost of Equity at Different Levels of Debt: Hamada’s Equation bL = bU [1 + (1 - t)(D/S)] bU is the beta of a firm when it has no debt (the unlevered beta) Use this equation to unlever the observed pure play betas

(bL’s), then average the resulting bU’s Use the debt-stock mix (D/S) and marginal tax rates (t) of these companies

to unlever

Divide bL by term in brackets

Relever average unlevered beta at the division’s target capital structure (D/S) and marginal tax rate (t) Multiply resulting average bU by term in brackets

Plug relevered beta into CAPM to yield rs

See AOL example

Levered Beta Calculation

Division's target capital structure (D/S) = 0.6Tax Rate = 40%

Pure Play Actual

Market

ValueMarket

Value Unlevered

Company Beta (bL) Debt Equity D/S Beta (bu)

A 0.8 1000 1000 1.00 0.50

B 1.2 800 500 1.60 0.61

C 0.6 1500 2000 0.75 0.41

Average 0.51

Levered beta =

.51[1+(1-.4)*.6]

Result 0.69

Second CAPM Input:The Market Risk Premium The equity market risk premium is the

premium that investors demand for investing in an average risk investment, above the risk-free rate, (rm – rrf)

The expected rate of return on the average stock minus the risk-free rate at any point in time.

irfmrfi betarrrr ][^^

Approaches to Estimating the Market Risk Premium Assume that the actual premium delivered over

long time periods is equal to the expected premium - i.e., use historical data

Estimate the implied premium using today’s security prices and expected growth in earnings

Forecasts future stock returns based on fundamentals: payouts, multiples and growth

Survey data

The Historical Risk Premium Approach Defines a time period for the estimation Calculates the difference between average

returns on a stock index and average returns on a riskless security during the period

Uses the difference as a premium looking forward

The Historical Risk Premium Approach The limitations of this approach are:

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)

Assumes that the riskiness of the “risky” portfolio (stock index) has not changed in a systematic way across time.

Strange results, since after periods of high returns, you conclude that investors have become risk averse, when the opposite is probably true.

Risk Premium Based on Forecasted Fundamentals R = PAYOUT * E/P + G≈ 7.5% ≈ 50% 7% ≈ 4% (a P/E of 14) (2 real +

2 inflation)

Use P/E ratio to determine earnings yield, multiply by payout which includes dividends and repurchases

Add an expected LT growth rate for earnings to arrive at 7.5% forecast yield for large stocks (S&P 500)

Subtracting long-term Treasury yield of 1.8% yields an estimated risk premium of 5.7%

Implied Market Risk Premiums

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2011 survey data on the Market Risk Premium

Third CAPM Input:The Risk-free Rate U.S. Treasuries are used as the risk-free rate While open to debate, most favor using a long-term

Treasury rate for the following reasons: The LT rate reflects an average of future expected short-term

rates over the life of the investment The LT rate is much more stable than ST rates The cash flows underlying stocks are long-lived

The 10-year Treasury is commonly used The Treasury rates can be found on:

http://www.bloomberg.com/markets/rates/index.html

irfmrfi betarrrr ][^^

Implied Cost of Equity as another benchmark As an alternative to the CAPM approach, bottom-up

or top-down, for corporate finance purposes the cost of Equity can be estimated using the stock price and expected growth in Free Cash Flow to Equity

)/(10 FCFEgrFCFEP

Since stock price and consensus analyst growth forecasts are known, the company can back into an implied cost of Equity by solving for ‘r’ using a stock valuation model.

Constant growth version:

Advantages of Implied Cost

Market-based measure Do not have to estimate beta Do not have to estimate market risk

premium These assumptions are implicit in the

market’s valuation

Weighted Average Cost of Capital

The overall required rate of return % onInvested Capital (Debt + Stock)

WACC = (D/V)rd(1-T) + (S/V)rs

rd = % weighted average yield-to-maturity on debtrs = % required return on stock (cost of Equity)

D = $Debt market valueS = $Stock market valueT = % tax rateV = $Enterprise value = D + S

Key Terms

Capital Asset Pricing Model Systematic risk Beta (unlevered and levered) Market risk premium Operating risk Financial risk

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