(more practice with capital budgeting)

• SG Company currently uses a packaging machine that was purchased 3 years ago. This machine is being depreciated on a straight line basis toward a $400 salvage value, and it has 5 years of remaining life. Its current book value is $2500 and it can be sold for $3500 at this time.

• SG is offered a replacement machine which has a cost of $10,000, an estimated useful life of 5 years, and an estimated salvage value of $1000. This machine would also be depreciated on a straight line basis toward its salvage value. The replacement machine would permit an output expansion, so sales would rise by $1500 per year; even so, the new machine’s much greater efficiency would still cause before tax operating expenses to decline by $1800 per year. The machine would require that inventories be increased by $2000, but accounts payable would simultaneously increase by $750. No further change in working capital would be necessary over th4 e5 years. SG’s marginal tax rate is 40%, and its discount rate for this project is 12%. Should the company replace the old machine? (Assume that at the end of year 5 SG would recover all of its net working capital investment, and the new machines could be sold at book value at the end of its useful life).

Risk & Return

• Chapter 9: 3,12,13,17• Chapter 10: 3,5,13,17,22,27,34,38

• Note - In chapter 10, skip the following sections: – Efficient set (section 10.4)– Efficient set for many securities: skip the first part of

section 10.5, page 270 to middle of 271– The optimal portfolio, p. 278-280.

Measuring historical returns

Total return = dividend income + capital gains% total return = Rt+1 = (Divt+1+ Pt+1- Pt)/Pt

Geometric mean returns (1+ R)T = (1+R1)(1+R2)…(1+Rt)…(1+RT)

RA = [(1.15)(1.00)(1.05)(1.20)](1/4)-1 .0972 = 9.72%RB = [(1.30)(0.80)(1.20)(1.50)](1/4)-1 .1697 = 16.97%

Arithmetic mean returns: R = (R1 + R2 + …+ RT)/TRA = [.15 + .00 + .05 + .20]/4 = .10 = 10%RB = [.30 + -.20 + .20 + .50]/4 = .20 = 20%

Year RA RB

1998 15% 30%1999 0% -20%2000 5% 20%2001 20% 50%

Measuring total riskReturn volatility: the usual measure of

volatility is the standard deviation, which is the square root of the variance.

Var(R)=1

T -1( R - R ) + ... + ( R - R )1

2T

2

Calculating historical risk & return: example

• The variance, ² or Var(R) = .0954/(T-1) = .0954/3 = .0318

• The standard deviation, or SD(R) =.0318 = .1783 or 17.83%

Year Actual Return

Average Return

Return deviation

Squared deviation

1 0.1 0.05 0.05 0.00252 -0.07 0.05 -0.12 0.01443 0.28 0.05 0.23 0.05294 -0.11 0.05 -0.16 0.0256

Total 0 0.0954

0.0% 40.0%2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0% 22.0% 24.0% 26.0% 28.0% 30.0% 32.0% 34.0% 36.0% 38.0%

S&P 500 TRS&P 500 TR

U.S. Small Stk TRU.S. Small Stk TR

U.S. LT Corp TRU.S. LT Corp TR

U.S. LT Gvt TRU.S. LT Gvt TR

U.S. IT Gvt TRU.S. IT Gvt TR

U.S. 30 Day TBill TR

U.S. 30 Day TBill TR

U.S. InflationU.S. Inflation

Return (AM) Risk (S.D.)

January 1926 - December 2003Risk vs. Return

Historical Perspective

Capital Market History: Risk Return Tradeoff (Ibbotson, 1926-2003)

Investment Geometric Mean Arithmetic Mean Standard Deviation

Large company stocks 10.20% 12.20% 20.50%Small company stocks 12.1 16.9 33.2Long-term corporate bonds 5.9 6.2 8.7Long-term government 5.5 5.8 9.4Intermediate-term government 5.4 5.6 5.8U.S. treasury bills 3.8 3.8 3.2

Risk premium = difference between risky investment's return and riskless return.

EXPECTED (vs. Historical) RETURNS & VARIANCES

State of Economy

pi Probability

of state

ri Return in

state

+1% change in GNP .25 -5%

+2% change in GNP .50 15%

+3% change in GNP .25 35%

E(R) = p x ri=1

S

i i

i (pi x ri)

i = 1 -1.25%

i = 2 7.50%

i = 3 8.75%

Calculating the Expected Return:

Expected return = (-1.25 + 7.50 + 8.75) = 15%

EXPECTED (vs. Historical) RETURNS & VARIANCES

State of Economy

pi Probability

of state

ri Return in

state

+1% change in GNP .25 -5%

+2% change in GNP .50 15%

+3% change in GNP .25 35%

Calculating the variance:

Var(R) = p x r - r2

i=1

S

i

2

i

E(R) = r = 15% = .15

i (ri - r)² pi x (ri - r)²

I = 1 .04 .01

I = 2 0 0

I = 3 .04 .01

Var(R) = .02

PORTFOLIO EXPECTED RETURNS & VARIANCES

Portfolio weights: 50% in Asset A and 50% in Asset B

E(RP) = 0.40 x (.125) + 0.60 x (.075) = .095 = 9.5%Var(RP) = 0.40 x (.125-.095)² + 0.60 x (.075-.095)² = .0006SD(RP) =.0006 = .0245 = 2.45%

Note: E(RP) = .50 x E(RA) + .50 x E(RB) = 9.5%BUT: Var(RP) ≠ .50 x Var(RA) + .50 x Var(RB) !!!!

State of the economy

Probability of state

Return on asset A

Return on asset B

Return on portfolio

Good 0.4 30% -5% 12.50%Bad 0.6 -10% 25% 7.50%

PORTFOLIO EXPECTED RETURNS & VARIANCES

State of the economy

Probability of state

Return on asset A

Return on asset B

Return on portfolio

Good 0.4 30% -5% 10%Bad 0.6 -10% 25% 10%

E(RP) = 10%SD(RP) = 0 !!!!

New Portfolio weights: put 3/7 in A and 4/7 in B:

Covariance and correlation: measuring how two variables are related

Covariance is defined:

AB = Cov(RA,RB)

= Expected value of [(RA-RA) x (RB-RB)]

Correlation is defined (-1< AB<1):

AB = Corr(RA,RB) = Cov(RA,RB) / (A x B) = AB / (A x B)

Portfolio risk & return

If XA and XB the portfolio weights,

The expected return on a portfolio is a weighted average of the expected returns on the individual securities:

Portfolio variance is measured:

Expected return on portfolio = X R + X RA A B B

Var(portfolio) = X + X + 2 X X A2

A2

B2

B2

A B AB

Portfolio Risk & Return: Example

Depression -0.2 -0.375 0.140625 0.05 -0.005Recession 0.1 -0.075 0.005625 0.2 0.145

Normal 0.3 0.125 0.015625 -0.12 -0.175Boom 0.5 0.325 0.105625 0.09 0.035

---- ------ ----(sum) 0.7 0.2675 0.22

0.011375------

-0.0195

(RB-RB)²

(RB-RB)

0.001875-0.010875-0.021875

------0.0529

0.0000250.0210250.0306250.001225

RB (RB-RB) (RA-RA) xState RA (RA-RA) (RA-RA)²

RA = (-0.20 + 0.10 + 0.30 + 0.50)/4 = 0.175 Var(RA) = ²A = .2675/4 = .066875SD(RA) = A = .066875 = .2586RB = (0.05 + 0.20 - 0.12 + 0.09)/4 = 0.055 Var(RB) = ²B = .0529/4 = .013225SD(RB) = B = .013225 = .1150AB = Cov(RA,RB) = -0.0195/4 = -0.004875AB = Corr(RA,RB) = AB / AB = -0.004875/(.2586x.1150) = -.1369

Benefits of diversificationConsider two companies A & B, and portfolio

weights XA = .5, XB = .5

Stock A Stock B

E(RA)=10%E(RB)=15%

A=10% B=30%

Case 1: AB = 1 (AB = AB/AB)Portfolio Weights E(RP) SD(RP)

1 XA=1, XB=0 10% 10%2 XA=.5, XB=.5 12.50% 20%3 XA=0, XB=1 15% 30%

Benefits of diversification

Stock A Stock B

E(RA)=10%E(RB)=15%

A=10% B=30%

Case 2: AB = 0.2 (AB = AB/AB)

Portfolio Weights E(RP) SD(RP)1 same same same2 same same 16.73%3 same same same

Benefits of diversification

Stock A Stock B

E(RA)=10%E(RB)=15%

A=10% B=30%

Case 3: AB = 0 (AB = AB/AB)

Portfolio Weights E(RP) SD(RP)1 same same same2 same same 15.80%3 same same same

Intuition of CAPM

Components of returns: Total return = Expected return + Unexpected return

R = E(R) + UThe unanticipated part of the return is the true risk of any investment.

The risk of any individual stock can be separated into two components.

1. Systematic or market risks (nondiversifiable).2. Unsystematic, unique, or asset-specific (diversifiable risks).

R = E(R) + U = E(R) + systematic portion + unsystematic portion

Measuring systematic risk: betaRm = proxy for the "market" return

Portfolio beta =weighted ave of individual asset’s betas

Stock $ Invested % of portfolio Beta

(1) (2) (3) (4) (3)x(4)IBM 6,000 50 0.75 0.375GM 4,000 33 1.01 0.336

Dow Chem 2,000 17 1.16 0.197Portfolio 12,000 100 0.908

R

)R,RCov( =

M2

MAA

Portfolio risk (beta) vs. return

Consider portfolios of:

Risky asset A, ßA = 1.2, E(RA) = 18%

Risk free asset, Rf = 7%

Proportion invested in Asset A (%)

Proportion invested at riskfree rate

(%)

Portfolio expected return (%)

Portfolio beta (ß)

0 100 7 025 75 9.75 0.350 50 12.5 0.675 25 15.25 0.9100 0 18 1.2125 -25 20.75 1.5

Proportion invested in Asset A (%)

Proportion invested at riskfree rate

(%)

Portfolio expected return (%)

Portfolio beta (ß)

0 100 7 025 75 9.75 0.350 50 12.5 0.675 25 15.25 0.9100 0 18 1.2125 -25 20.75 1.5

Market equilibriumReward/risk ratio = E(Ri) - Rf = constant!

ßi

The line that describes the relationship between systematic risk and expected return is called the security market line.

Market equilibrium

The market as a whole has a beta of 1. It also plots on the SML, so:

E( R ) - R = E( R ) - R

M fi f

i

E( R ) = R + [E( R ) - R ]xi f M f i

Using the CAPM: risk free rate and risk premium

Investment Geometric Mean Arithmetic Mean Standard Deviation

Large company stocks 10.20% 12.20% 20.50%Small company stocks 12.1 16.9 33.2Long-term corporate bonds 5.9 6.2 8.7Long-term government 5.5 5.8 9.4Intermediate-term government 5.4 5.6 5.8U.S. treasury bills 3.8 3.8 3.2

Historic Returns and Equity PremiaArithmetic GeometricAverage Average

Stocks1802-2002 9.50% 8.10%1900-2002 11.40% 9.50%1926-2002 11.80% 9.70%1982-2002 13.20% 12.00%

Treasury bonds1802-2002 5.20% 5.00%1900-2002 5.20% 4.90%1926-2002 5.90% 5.50%1982-2002 13.00% 12.30%

Equity premium1802-2002 4.38% 3.01%1900-2002 6.23% 4.37%1926-2002 5.90% 4.03%1982-2002 0.20% -0.20%

Using the CAPM: estimating beta

Regression output

Data providersBloomberg, Datastream, Value Line

R

)R,RCov( =

M2

MAA

Estimating beta: Continental Airlines

Estimating beta: Continental Airlines

Estimating beta: Continental Airlines

Estimating beta

• How much historical data should we use?

• What return interval should we use?

• What data source should we use?

DETERMINANTS OF BETA: Operating vs. financial leverage

Sales

- costs

- depr

EBIT

- interest

- taxes

Net income

Determinants of beta: financial leverage

With no taxes, beta of a portfolio of debt & equity = beta of assets, or

If Debt is not too risky, assume D = 0 , so

or

In most cases, it is more useful to include corporate taxes:

EDA ED

E

ED

D

EA ED

E

E

DAE 1

E

DTAE )1(1

Example: equity betas vs. leverage McDonnell Douglas (pre merger)equity (levered) beta 0.59 D/E .875%Tax rate = 34% risk premium = 8.5%T-Bill = 5.24%Unlevered beta = current beta/(1 + (1-tax rate)(D/E)

= .59/(1+(1-.34)(.875) = .374

D/(D+E) D/E Levered beta Cost of equity (%)0 0 0.37 8.42

0.1 0.11 0.4 8.650.2 0.25 0.44 8.940.3 0.43 0.48 9.320.4 0.67 0.54 9.82

Estimating betas using betas of

comparable companies Continental Airlines, 1992 restructuring

D/(D+E) D/E Equity beta

American Airlines 0.598 1.49 1.45Delta Air Lines 0.38 0.61 1.1United Airlines 0.43 0.75 1.25USAir Group 0.74 2.85 1.65 Average 0.537 1.16 1.36

= 0.77=1.36/(1+(1-0.34)*(1.16))

Unlevered beta of comparable companies = asset beta for Continental =

Example: estimating betaNovell, which had a market value of equity of $2 billion

and a beta of 1.50, announced that it was acquiring WordPerfect, which had a market value of equity of $1 billion, and a beta of 1.30. Neither firm had any debt in its financial structure at the time of the acquisition, and the corporate tax rate was 40%.

Estimate the beta for Novell after the acquisition, assuming that the entire acquisition was financed with equity.

Assume that Novell had to borrow the $1 billion to acquire WordPerfect. Estimate the beta after the acquisition.

Example: estimating betaSouthwestern Bell, a phone company, is considering

expanding its operations into the media business. The beta for the company at the end of 1995 was 0.90, and the debt/equity ratio was 1. The media business is expected to be 30% of the overall firm value in 1999, and the average beta of comparable media firms is 1.20; the average debt/equity ratio for these firms is 50%. The marginal corporate tax rate is 36%.

a. Estimate the beta for Southwestern Bell in 1999, assuming that it maintains its current debt/equity ratio.

b. Estimate the beta for Southwestern Bell in 1999, assuming that it decides to finance its media operations with a debt/equity ratio of 50%.

Boeing – commercial aircraft divisionBoeing Grumman Northrop Lockheed

% revenue from defense 26% 87% 89% 85%

Estimated betas:

1. Statistical services Value line 1.00 0.95 1.00 1.10 Datastream 1.06 0.53 0.94 0.97

2. Using S&P 500 58 months 0.81 0.80 0.74 0.87 12 months 1.37 0.73 0.72 0.69 60 days 1.65 0.68 0.50 0.52

3. Using NYSE Composite 58 months 0.87 0.86 0.79 0.95 12 months 1.51 0.80 0.77 0.75 60 days 1.79 0.73 0.53 0.57

MV D/E 0.018 1.756 1.288 1.182

Boeing – commercial aircraft division

Using 58 months, NYSED/E (1+(1-t)D/E) Equity beta Asset beta

Grumman 1.7560 2.1590 0.8600 0.3983Northrop 1.2880 1.8501 0.7900 0.4270Lockheed 1.1820 1.7801 0.9500 0.5337

Average 1.4087 1.9297 0.8667 0.4491

Boeing total unlevered: Equity beta Asset beta0.0180 1.0119 0.8700 0.8598

Implied unlevered commercial beta:total boeing asset beta=(.26*betadef)+(.74*betacom)beta comm = 1.0041

Relever commercial beta:D/E (1+(1-t)D/E) Equity beta Asset beta

0.0180 1.0119 1.0160 1.0041risk free rate 0.0882Risk premium 0.0540cost of equity: 0.1431 =0.0882+1.016*0.054

WACC

• The key is that the rate will depend on the risk of the cash flows

• The cost of capital is an opportunity cost - it depends on where the money goes, not where it comes from.

WACC = (E/V) x Re + (D/V) x RD x (1 - T)

Cost of Equity: Dividend Growth Model

Price Dividend Growth Cost of12/27/2002 Yield Forecast Equity

Chevron-Texaco $65.90 4.2% 6.9% 11.1%Exxon-Mobil $34.64 2.7% 8.1% 10.8%Royal Dutch $43.41 2.9% 8.5% 11.4%British Petroleum $39.70 3.6% 7.4% 11.0%

Northwestern Corporation 8/04 - WACC

WACC = (E/V) x Re + (D/V) x RD x (1 - T)

Historical beta?

Sources for beta?

Northwestern Corporation - peers

Selection of Comparable Companies used factor including:

Sources?

Northwestern Corporation - peers

Northwestern Corporation - Beta

Northwestern Corporation – Cost of equity

re = rf + βe(rm – rf)

Levered beta = .41*(1+(1-.385)*1.381) = 0.75Ibbotson ’03*, (rm – rf) = 7%20 year bond 4/02 = 5.9%Re = 5.9% + 0.75*(7%) = 9.85%

Adding a 1.48% size risk premia (Ibbottson), and 2% company specific risk premia, cost of equity = 13.33%*Arithmetic mean, large stocks – long term treasury bonds, time period not specified

Northwestern Corporation - WACC

WACC = (E/V) x re + (D/V) x rD x (1 - T)

Cost of Debt 6.80%Cost of Equity 9.85%Cost of Equity (with additional premia) 13.33%

WACC 6.56%WACC (with additional premia) 8.02%