last study topics understanding of statements qualification of statements numerical

Last Study Topics

• Understanding of Statements• Qualification of Statements• Numerical

Today’s Study Topics

• Numericals• Company and Project Costs of Capital• Measuring the Cost of Equity

Continue

• b. Could Percival do even better by investing equal amounts in the corporate bond portfolio and the index fund? – The correlation between the bond portfolio and

the index fund is +.1.– rp = x1r1 + x2r2– rp = (0.5 × 0.09) + (0.5 × 0.14) = 0.115 = 11.5%

Continue

• σP2 = x1

2σ12 + 2x1x2σ1σ2ρ12 + x2

2σ22

• σP 2 = (0.5)2(0.10)2 + 2(0.5)(0.5)(0.10)(0.16)(0.10) + (0.5)2(0.16)2

• σP2 = 0.0097

• σP = 0.985 = 9.85%

Understanding

• Therefore, he can do even better by investing equal amounts in the corporate bond portfolio and the index fund.

• His expected return increases to 11.5% and the standard deviation of his portfolio decreases to 9.85%.

Explain• “There may be some truth in these CAPM and

APT theories, but last year some stocks did much better than these theories predicted, and other stocks did much worse.” Is this a valid criticism?

– No. Every stock has unique risk in addition to market risk. The unique risk reflects uncertain events that are unrelated to the return on the market portfolio. The Capital Asset Pricing Model does not predict these events.

Explain

• a. The APT factors cannot reflect diversifiable risks.– True. By definition, the factors represent macro-

economic risks that cannot be eliminated by diversification.

• b. The market rate of return cannot be an APT factor.– False. The APT does not specify the factors.

Continue

• c. Each APT factor must have a positive risk premium associated with it; otherwise the model is inconsistent.

– True. Investors will not take on non-diversifiable risk unless it entails a positive risk premium.

Continue

• d. There is no theory that specifically identifies the APT factors.

– True. Different researchers have proposed and empirically investigated different factors, but there is no widely accepted theory as to what these factors should be.

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• e. The APT model could be true but not very useful, for example, if the relevant factors change unpredictably.

– True. To be useful, we must be able to estimate the relevant parameters.

– If this is impossible, for whatever reason, the model itself will be of theoretical interest only.

APT Model– Consider the following simplified APT model:

Factor Risk Exposures

FACTOR EXPECTED RISK PREMIUM

Market 6.4%

Interest rate -0.6%

Yield Spread 5.1%

MARKET INTEREST RATE YIELD SPREAD

STOCK (b1) (b2) (b3)

P 1.0 -2.0 -0.2

P2 1.2 0 .3

P3 .3 .5 1.0

Continuea. Calculate the expected return for the

following stocks. Assume rf = 5%.– For Stock P r = (1.0)×(6.4%) + (-2.0)×(-0.6%) + ⇒

(-0.2)×(5.1%) = 6.58%

– For Stock P2 r = (1.2)×(6.4%) + (0)×(-0.6%) + ⇒(0.3)×(5.1%) = 9.21%

– For Stock P3 r = (0.3)×(6.4%) + (0.5)×(-0.6%) + ⇒(1.0)×(5.1%) = 6.72%

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• b. What are the factor risk exposures for the portfolio?– Factor risk exposures:

• b1(Market) = (1/3)×(1.0) + (1/3)×(1.2) + (1/3)×(0.3) = 0.83

• b2(Interest rate) = (1/3)×(-2.0) +(1/3)×(0) + (1/3)×(0.5) = -0.50

• b3(Yield spread) = (1/3)×(-0.2) + (1/3)×(0.3) + (1/3)×(1.0) = 0.37

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• b. What is the portfolio’s expected return?

• rP = (0.83)×(6.4%) + (-0.50)×(-0.6%) + (0.37)×(5.1%) = 7.5%

Three Factors Model

• The following table shows the sensitivity of four stocks to the three Fama–French factors in the five years to 2001. Estimate the expected return on each stock assuming that the interest rate is 3.5%, the expected risk premium on the market is 8.8%, the expected risk premium on the size factor is 3.1%, and the expected risk premium on the book-to-market factor is 4.4%.

ContinueFactor Sensitivities

FACTOR COCA-COLA EXXON MOBILE PFIZER REEBOK

Market .82 .50 .66 1.17

Size -0.29 .04 -.56 .73

Book-to-Market

.24 .27 -.07 1.14

rCoca-Cola = 3.5% + (0.82 × 8.8%) + (-0.29 × 3.1%) + (0.24 × 4.4%) = 10.87%rEXXON= 3.5% + (0.50 × 8.8%) + (0.04 × 3.1%) + (0.27 × 4.4%) = 9.21%

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• rPFizer = 3.5% + (0.66 × 8.8%) + (-0.56 × 3.1%) + (-0.07 × 4.4%) = 7.26%

• rReebok = 3.5% + (1.17 × 8.8%) + (0.73 × 3.1%) + (1.14 × 4.4%) = 21.08%

Summary

• Qualification of Statements.• Examples.

– CAPM– APT– Three Factors Model

Principles of Corporate

Finance

Sixth Edition

Richard A. Brealey

Stewart C. Myers

Chapter 9

McGraw Hill/Irwin

Capital Budgeting and Risk

Today’s Study Topics

• Company and Project Costs of Capital• Measuring the Cost of Equity• Capital Structure and COC• Discount Rates for Intl. Projects• Estimating Discount Rates• Risk and DCF

Introduction

• LONG BEFORE THE development of modern theories linking risk and expected return, smart financial managers adjusted for risk in capital budgeting.

• How they should treat the element of risk with respect to each and every projects of different class?

Continue

• Various rules of thumb are often used to make these risk adjustments. – For example, many companies estimate the rate

of return required by investors in their securities and then use this company cost of capital to discount the cash flows on new projects.

COMPANY AND PROJECT COSTS OF CAPITAL

• The company cost of capital is defined as the expected return on a portfolio of all the company’s existing securities.

• It is used to discount the cash flows on projects that have similar risk to that of the firm as a whole.

Continue

• We estimated that investors require a return of 9.2% from Pfizer common stock.

• If Pfizer is contemplating an expansion of the firm’s existing business, it would make sense to discount the forecasted cash flows at 9.2 %.

• The company cost of capital is not the correct discount rate if the new projects are more or less risky than the firm’s existing business..

Company Cost of Capital

• A firm’s value can be stated as the sum of the value of its various assets.

• Each project should in principle be evaluated at its own opportunity cost of capital.

• For a firm composed of assets A and B, the firm value is;

PV(B)PV(A)PV(AB) valueFirm

Continue• Here PV(A) and PV(B) are valued just as if they

were mini-firms in which stockholders could invest directly.

• Investors would value A by discounting its forecasted cash flows at a rate reflecting the risk of A.

• They would value B by discounting at a rate reflecting the risk of B. – The two discount rates will, in general, be

different.

Continue

• This means that Pfizer should accept any project that more than compensates for the project’s beta.

• In other words, Pfizer should accept any project lying above the upward-sloping line that links expected return to risk in Figure 1. – If the project has a high risk, Pfizer needs a higher

prospective return than if the project has a low risk.

Company Cost of Capital• A company’s cost of capital can be compared

to the CAPM required return

Required

return

Project Beta1.26

Company Cost of Capital

13

5.5

0

SML

Understanding

• In terms of Figure1, the rule tells Pfizer to accept any project above the horizontal cost of capital line, that is, any project offering a return of more than 9.2%.– The company cost of capital rule, which is to

accept any project regardless of its risk as long as it offers a higher return than the company’s cost of capital.

Understanding

– It is clearly silly to suggest that Pfizer should demand the same rate of return from a very safe project as from a very risky one.

– If Pfizer used the company cost of capital rule, it would reject many good low-risk projects and accept many poor high-risk projects.

• Many firms require different returns from different categories of investment.

Understanding

10%nologyknown tech t,improvemenCost

COC)(Company 15%business existing ofExpansion

20%products New

30% ventureseSpeculativ

RateDiscount Category

For example, discount rates might be set as follows:

Perfect Pitch and the Cost of Capital• The true cost of capital depends on project

risk, not on the company undertaking the project. – So why is so much time spent estimating the

company cost of capital?

• First, many (maybe, most) projects can be treated as average risk, that is, no more or less risky than the average of the company’s other assets.– For these projects the company cost of capital is

the right discount rate.

Continue

• Second, the company cost of capital is a useful starting point for setting discount rates for unusually risky or safe projects. – It is easier to add to, or subtract from, the

company cost of capital than to estimate each project’s cost of capital from scratch.

• Anyone who can carry a tune gets relative pitches right.

Business People

• are used to, but not about absolute risk or required rates of return.

• Therefore, they set a companywide cost of capital as a benchmark.

• This is not the right hurdle rate for everything the company does.– But adjustments can be made for more or less

risky ventures.

MEASURING THE COST OF EQUITY

• Suppose that you are considering an across-the-board expansion by your firm.

• Such an investment would have about the same degree of risk as the existing business.

• Therefore you should discount the projected flows at the company cost of capital.

• Companies generally start by estimating the return that investors require from the company’s common stock.

Continue

• Used the capital asset pricing model to do the working. This states;– Expected stock return =rf + Beta(rm – rf)

• An obvious way to measure the beta (B) of a stock is to look at how its price has responded in the past to market movements.

Measuring Betas

• The SML shows the relationship between return and risk.

• CAPM uses Beta as a proxy for risk.• Other methods can be employed to determine

the slope of the SML and thus Beta.• Regression analysis can be used to find Beta.

Dell Computer Stock

• Calculated monthly returns from Dell Computer stock in the period, after it went public in 1988, is given on the next slide.

• Also plotted returns against the market returns for the same month, is given too.

• We have fitted a line through the points. – The slope of this line is an estimate of beta.

Measuring BetasDell Computer

Slope determined from plotting the line of best fit.

Price data – Aug 88- Jan 95

Market return (%)

Dell return (%

)

R2 = .11

B = 1.62

Measuring BetasDell Computer

Slope determined from plotting the line of best fit.

Price data – Feb 95 – Jul 01

Market return (%)

Dell return (%

)

R2 = .27

B = 2.02

Other Stocks

• The next diagram shows a similar plot for the returns on General Motors stock, and the

• Third shows a plot for Exxon Mobil. • In each case we have fitted a line through the

points. • The slope of this line is an estimate of beta.

– It tells us how much on average the stock price changed for each additional 1% change in the market index.

Measuring BetasGeneral Motors

Slope determined from plotting the line of best fit.

Price data – Aug 88- Jan 95

Market return (%)

GM

return (%)

R2 = .13

B = 0.80

Measuring BetasGeneral Motors

Slope determined from plotting the line of best fit.

Price data – Feb 95 – Jul 01

Market return (%)

GM

return (%)

R2 = .25

B = 1.00

Measuring BetasExxon Mobil

Slope determined from plotting the line of best fit.

Price data – Aug 88- Jan 95

Market return (%)

Exxon Mobil return (%

)

R2 = .28

B = 0.52

Measuring BetasExxon Mobil

Slope determined from plotting the line of best fit.

Price data – Feb 95 – Jul 01

Market return (%)

Exxon Mobil return (%

)

R2 = .16

B = 0.42

Understanding• Diagrams show plots for the three stocks

during the subsequent period, February 1995 to July 2001.

• Although the slopes varied from the first period to the second, there is little doubt that Exxon Mobil’s beta is much less than Dell’s or that GM’s beta falls somewhere between the two. – If you had used the past beta of each stock to

predict its future beta, you wouldn’t have been too far off.

Understanding

• Only a small portion of each stock’s total risk comes from movements in the market.

• The rest is unique risk, which shows up in the scatter of points around the fitted

• lines in Diagrams. – R-squared (R2) measures the proportion of the

total variance in the stock’s returns that can be explained by market movements.

Table 1• Estimated betas and costs of (equity) capital for a sample of

large railroad companies and for a portfolio of these companies. • The precision of the portfolio beta is much better than that of

the betas of the individual companies—note the lower standard

error for the portfolio.

.17.50PortfolioIndustry

.21.40PacificUnion

.26.52SouthernNorfolk

.24.46tionTransporta CSX

.20.64Northern Burlington

Error Standard.Beta

Summary

The Expected Return on Union Pacific Corporation’s Common Stock

• Suppose that in mid-2001 you had been asked to estimate the company cost of capital of Union Pacific Corporation.

• Table 1 provides two clues about the true beta of Union Pacific’s stock: – The direct estimate of .40 and the average

estimate for the industry of .50. • Use the industry average of .50

Continue• In mid-2001 the risk-free rate of interest rf was

about 3.5%. • 8% for the risk premium on the market, • You would have concluded that the expected

return on Union Pacific’s stock was about 7.5%.

– Expected stock return= rf +Beta(rm – rf)

= 3.5 + .5(8.0) = 7.5%

CAPITAL STRUCTURE AND THE COMPANY COST OF CAPITAL

• we need to look at the relationship between the cost of capital and the mix of debt and equity used to finance the company.

• Think again of what the company cost of capital is and what it is used for.

• We define it as the opportunity cost of capital for the firm’s existing assets; – we use it to value new assets that have the same

risk as the old ones.

Company Cost of Capitalsimple approach

• Company Cost of Capital (COC) is based on the average beta of the assets.

• The average Beta of the assets is based on the % of funds in each assets.– Example– 1/3 New Ventures B=2.0– 1/3 Expand existing business B=1.3– 1/3 Plant efficiency B=0.6

• AVG Beta of assets = 1.3

Capital Structure

• Capital Structure - The mix of debt & equity within a company

• Expand CAPM to include Capital Structure

R = rf + B ( rm - rf )Becomes;

Requity = rf + B ( rm - rf )

Union Pacific Corp.Example

100 valueFirmAssets Total

70ueEquity val

30Debt value100Assets

%75.12%157030

70%5.7

7030

30

assets

equitydebtassets

R

requitydebt

equityr

equitydebt

debtR

Understanding

• If the firm is contemplating investment in a project that has the same risk as the firm’s existing business, the opportunity cost of capital for this project is the same as the firm’s cost of capital; in other words, it is 12.75%.– What would happen if the firm issued an

additional 10 of debt and used the cash to repurchase 10 of its equity?

Union Pacific Corp.Example

100 valueFirmAssets Total

60ueEquity val

40Debt value100Assets

%75.12%156040

60%875.7

6040

40

assets

equitydebtassets

R

requitydebt

equityr

equitydebt

debtR

Understanding

• The change in financial structure does not affect the amount or risk of the cash flows on the total package of debt and equity.

• Therefore, if investors required a return of 12.75% on the total package before the refinancing, they must require a 12.75% return on the firm’s assets afterward.

Solve for Equity

• Since the company has more debt than before, the debt holders are likely to demand a higher interest rate.

• We will suppose that the expected return on the debt rises to 7.875%. – Now you can write down the basic equation for

the return on assets and solve for return on Equity. i.e.

Continue

• Return on equity = 16%– Increasing the amount of debt increased debt

holder risk and led to a rise in the return that debt holders required (rdebt rose from 7.5 to 7.875%).

– The higher leverage also made the equity riskier and increased the return that shareholders required (requity rose from 15 to 16 %).

equitydebtassets rV

Er

V

DR

Continue

• The weighted average return on debt and equity remained at 12.75%.

• What happen to cost of capital and return on equity, – If Co. has paid all of its debt and replace it with

equity?

equitydebtassets rV

Er

V

DR

How Changing Capital Structure Affects Beta

• The stockholders and debtholders both receive a share of the firm’s cash flows, and both bear part of the risk. – For example, if the firm’s assets turn out to be

worthless, there will be no cash to pay stockholders or debtholders.

• But debtholders usually bear much less risk than stockholders. Debt betas of large blue-chip firms are typically in the range of .1 to .3.

Continue

• The firm’s asset beta is equal to the beta of a portfolio of all the firm’s debt and its equity.

• The beta of this hypothetical portfolio is just a weighted average of the debt and equity betas:

equitydebtassets BV

EB

V

DBB Portfolio

Continue

• If the debt before the refinancing has a beta of .1 and the equity has a beta of 1.1, then; – Beta assets = .8

equitydebtassets BV

EB

V

DB

Continue

• What happens after the refinancing? The risk of the total package is unaffected, but both the debt and the equity are now more risky.– Suppose that the debt beta increases to 0.2.– Beta Equity = 1.2

equitydebtassets BV

EB

V

DB

Understanding• Financial leverage does not affect the risk or

the expected return on the firm’s assets, but it does push up the risk of the common stock.

• Shareholders demand a correspondingly higher return because of this financial risk.

• Figure on the next slide shows the expected return and beta of the firm’s assets. – It also shows how expected return and risk are

shared between the debtholders and equity holders before the refinancing.

Capital Structure & COC

Expected return (%)

Bdebt Bassets Bequity

Rrdebt=7.5

Rassets=12.75

Requity=15

Expected Returns and Betas prior to refinancing

Understanding

• After the refinancing.– Both debt and equity are now more risky, and

therefore investors demand a higher return.– But equity accounts for a smaller proportion of

firm value than before. – As a result, the weighted average of both the

expected return and beta on the two components is unchanged.

0

20

0 0.2 0.8 1.2

Capital Structure & COC

Expected return (%)

Bdebt Bassets Bequity

Rrdebt=7.875

Rassets=12.75

Requity=16

Expected Returns and Betas prior to refinancing

Capital Structure and Discount Rates

• The company cost of capital is the opportunity cost of capital for the firm’s assets.

• That’s why we write it as rassets.

• If a firm encounters a project that has the same beta as the firm’s overall assets, then rassets is the right discount rate for the project cash flows.

Continue• When the firm uses debt financing, the

company cost of capital is not the same as requity the expected rate of return on the firm’s stock; requity is higher because of financial risk.

– When the firm changes its mix of debt and equity securities, the risk and expected returns of these securities change; however, the asset beta and the company cost of capital do not change.

Continue• When companies discount an average-risk

project, they do not use the company cost of capital as we have computed it.

• They use the after-tax cost of debt to compute the after-tax weighted-average cost of capital or WACC.

equitydebt rV

ErTc

V

DWACC )1(

DISCOUNT RATES FOR INTERNATIONAL PROJECTS

• Foreign Investments Are Not Always Riskier;• Table 2 shows estimated betas for the

Egyptian market and for markets in Poland, Thailand, and Venezuela.

• The standard deviations of returns in these markets were two or three times more than the U.S. market, but only Thailand had a beta greater than 1.

International Risk

0.77.302.58Venezuela

1.39.482.91Thailand

0.81.421.93Poland

0.55.183.11Egypt

Betatcoefficien

nCorrelatioRatio

Source: The Brattle Group, Inc.

Ratio - Ratio of standard deviations, country index vs. S&P composite index

Continue

• The reason is low correlation. – For example, the standard deviation of the

Egyptian market was 3.1 times that of the Standard and Poor’s index, but the correlation coefficient was only .18.

– The beta was 3.1 x 0.18 = .55.

Understanding

• Table doesn’t prove that investment abroad is always safer than at home.

• But it should remind you always to distinguish between diversifiable and market risk.

• The opportunity cost of capital should depend on market risk.

DISCOUNT RATES FOR INTERNATIONAL PROJECTS

• Foreign Investment;• Suppose that the Swiss pharmaceutical

company, Roche, is considering an investment in a new plant near Basel in Switzerland.

• Since the project is risky, the company requires a higher return than the Swiss franc interest rate.– First measures the risk of the investment by

estimating Roche’s beta and the beta of other Swiss pharmaceutical companies in that country.

Continue• Beta of 1.1 and that the expected risk premium

on the Swiss market index is 6%.• Then Roche needs to discount the Swiss franc

cash flows from its project at 1.1 x 6 = 6.6%.– calculates these betas relative to the Swiss market

index.

• What if the project chooses to be in other country than Swiss?

• What would be the possible affect on the discount rate under this circumstances?

Understanding

• Risk cannot be considered in isolation; it depends on the other securities in the investor’s portfolio.

• Beta measures risk relative to the investor’s portfolio.

• For example, you can now buy funds that specialize in investment in emerging capital markets such as Vietnam, Peru, or Hungary.

Continue

• As investors increase their holdings of overseas stocks, it becomes less appropriate to measure risk relative to the domestic market and more important to measure the risk of any investment relative to the portfolios that they actually hold.

SETTING DISCOUNT RATES WHEN YOUCAN’T CALCULATE BETA

• What should a manager do if the asset has no such convenient price record?

• What if the proposed investment is not close enough to business as usual to justify using a company cost of capital?– These cases clearly call for judgment.

Continue

• For managers making that kind of judgment, we offer two pieces of advice.

– 1. Avoid fudge factors. Don’t give in to the temptation to add fudge factors to the discount rate to offset things that could go wrong with the proposed investment. Adjust cash-flow forecasts first.

Example

• Project Z will produce just one cash flow, forecasted at $1 million at year 1.

• It is regarded as average risk, suitable for discounting at a 10% company cost of capital.

– P.V = = $909,100

Continue

• The most likely outcome is $1 million, • But you also see some chance that project Z

will generate zero cash flow next year.• Also, new worry is the ‘Technology’.

– There is some discount rate (10% plus a fudge factor) that will give the right value, but we don’t know what that adjusted discount rate is?

Continue• For many projects, the most likely cash flow is

also the unbiased forecast. • If there are three possible outcomes with the

probabilities shown below, the unbiased forecast is $1 million.

Continue

• This might describe the initial prospects of project Z.

• But if technological uncertainty introduces a 10 percent chance of a zero cash flow, the unbiased forecast could drop to $900,000:

Continue

• Now, recalculate the PV,

– PV = = $818,000

Continue

– 2. Think about the determinants of asset betas. Often the characteristics of high and low-beta assets can be observed when the beta itself cannot be.

• Cyclicality , Many people intuitively associate risk with the variability of book, or accounting, earnings.

• But much of this variability reflects unique or diversifiable risk.

Continue

• What really counts is the strength of the relationship between the firm’s earnings and the aggregate earnings on all real assets.

• We can measure this either by the accounting beta or by the cash-flow beta.– Firms with high accounting or cash-flow betas

should also have high stock betas—and the prediction is correct.

Continue

• This means that cyclical firms—firms whose revenues and earnings are strongly dependent on the state of the business cycle—tend to be high-beta firms.

– Thus you should demand a higher rate of return from investments whose performance is strongly tied to the performance of the economy.

Continue• Operating Leverage,• We have already seen that financial leverage

increases the beta of an investor’s portfolio.• Those who receive the fixed costs are like debt

holders in the project; they simply get a fixed payment. – Those who receive the net cash flows from the

asset are like holders of common stock; they get whatever is left after payment of the fixed costs.

Asset Betas

• How the asset’s beta is related to the betas of the values of revenue and costs?

)PV(revenue

PV(asset)B

)PV(revenue

cost) ePV(variablB

)PV(revenue

cost) PV(fixedBB

assetcost variable

cost fixedrevenue

)PV(revenue

PV(asset)B

)PV(revenue

cost) ePV(variablB

)PV(revenue

cost) PV(fixedBB

assetcost variable

cost fixedrevenue

Asset Betas• The betas of the revenues and variable costs

should be approximately the same, because they respond to the same underlying variable, the rate of output.

• Therefore, we can substitute Beta variable cost and solve for the asset beta.

• Remember that Beta fixed cost = 0.

Continue

• Thus, given the cyclicality of revenues (reflected in Beta revenue), the asset beta is proportional to the ratio of the present value of fixed costs to the present value of the project.

PV(asset)

cost) PV(fixed1B

PV(asset)

cost) ePV(variabl-)PV(revenueBB

revenue

revenueasset

PV(asset)

cost) PV(fixed1B

PV(asset)

cost) ePV(variabl-)PV(revenueBB

revenue

revenueasset

Continue

• Other things being equal, the alternative with the higher ratio of fixed costs to project value will have the higher project beta.

– Empirical tests confirm that companies with high operating leverage actually do have high betas.

ANOTHER LOOK AT RISK AND DISCOUNTEDCASH FLOW

• In practical capital budgeting, a single discount rate is usually applied to all future cash flows.– Among other things, the use of a constant

discount rate assumes that project risk does not change.

• This can’t be strictly true, for the risks to which companies are exposed are constantly shifting.

• It involves converting the expected cash flows to certainty equivalents.

Continue

• We will first explain what certainty equivalents are.

• Then we will use this knowledge to examine when it is reasonable to assume constant risk.– You are considering construction of an office

building that you plan to sell after one year for $400,000.

– Since that cash flow is uncertain, you discount at a risk-adjusted discount rate of 12% rather than the 7% risk-free rate of interest.

Continue

• This gives a present value of

– PV = 400,000/1.12 = $357,143

• Suppose a real estate company offers to fix the price at which it will buy the building from you at the end of the year, – PV = Certain Cash Flow / 1.07 = $357,143

Certain Cash Flow = $382,143

Understanding• A certain cash flow of $382,143 has exactly

the same present value as an expected but uncertain cash flow of $400,000. – The cash flow of $382,143 is therefore known as

the certainty-equivalent cash flow.• Compensation for uncertainty in-terms of returns is

equal to;

= $400,000-$357,143 = $42,857• To get rid of the risk, take a cut in the return of;

= $400,000 - $382,143 = $17,857

Continue

• Method 1: Discount the risky cash flow at a risk-adjusted discount rate r that is greater than rf. – The risk-adjusted discount rate adjusts for both

time and risk.

• Method 2: Find the certainty-equivalent cash flow and discount at the risk-free interest rate rf.

Risk,DCF and CEQ

• This is called the certainty equivalent of C1 denoted by CEQ1.

• Since CEQ1 is the value equivalent of a safe cash flow, it is discounted at the risk-free rate.

tf

tt

t

r

CEQ

r

CPV

)1()1(

Risk,DCF and CEQ

ExampleProject A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project?

Risk,DCF and CEQExample

Project A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project?

%12

)8(75.6

)(

fmf rrBrr

Risk,DCF and CEQExample

Project A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project?

%12

)8(75.6

)(

fmf rrBrr

240.2 PVTotal

71.21003

79.71002

89.31001

12% @ PV FlowCashYear

AProject

Risk,DCF and CEQExample

Project A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project?

%12

)8(75.6

)(

fmf rrBrr

240.2 PVTotal

71.21003

79.71002

89.31001

12% @ PV FlowCashYear

AProject

Now assume that the cash flows change, but are RISK FREE. What is the new PV?

Risk,DCF and CEQExample

Project A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project?.. Now assume that the cash flows change, but are RISK FREE. What is the new PV?

240.2 PVTotal

71.284.83

79.789.62

89.394.61

6% @ PV FlowCashYear

Project B

240.2 PVTotal

71.21003

79.71002

89.31001

12% @ PV FlowCashYear

AProject

Risk,DCF and CEQExample

Project A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project?.. Now assume that the cash flows change, but are RISK FREE. What is the new PV?

240.2 PVTotal

71.284.83

79.789.62

89.394.61

6% @ PV FlowCashYear

Project B

240.2 PVTotal

71.21003

79.71002

89.31001

12% @ PV FlowCashYear

AProject

Since the 94.6 is risk free, we call it a Certainty Equivalent of the 100.

Risk,DCF and CEQExample

Project A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project? DEDUCTION FOR RISK

15.284.81003

10.489.61002

5.494.61001riskfor

DeductionCEQFlowCash Year

Risk,DCF and CEQExample

Project A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project?.. Now assume that the cash flows change, but are RISK FREE. What is the new PV?

The difference between the 100 and the certainty equivalent (94.6) is 5.4%…this % can be considered the annual premium on a risky cash flow

flow cash equivalentcertainty 054.1

flow cashRisky

Risk,DCF and CEQExample

Project A is expected to produce CF = $100 mil for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of .75, what is the PV of the project?.. Now assume that the cash flows change, but are RISK FREE. What is the new PV?

8.84054.1

100 3Year

6.89054.1

100 2Year

6.94054.1

100 1Year

3

2

Summary