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# Risk and uncertainty

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### Risk and uncertainty

1. 1. Capital Budgeting Risk and Uncertainty Copyright ©2003 Stephen G. Buell Risk and UncertaintyRisk è the possibility that actual returns will deviate from expected returnsRisk è situations in which a probability distribution of possible outcomes can be estimatedUncertainty è worse, not enough information available Copyright ©2003 Stephen G. Buell Probability distribution of expected outcomes Copyright ©2003 Stephen G. Buell 1
2. 2. Initial measure of risk Standard deviation of expected cash flowsσ m σ = ∑ (CF j =1 j − CF ) 2 Pj m CF = ∑CF P j =1 j j m = number of possible outcomes CF j = j possibleoutcome th Pj = probability ©2003CFj G. Buell Copyright of Stephen occurring Improved measure of riskCoefficien t of variation (cv) puts dispersionon a relativebasis σcv = CFConsider σ x = 300 and CF x = 1000 versus σ y = 300 and CF y = 4000Intuitively x is riskier. Need to show that. 300 300cv x = = .300 while cv y = = . 075 1000 4000 Copyright ©2003 Stephen G. Buell Forecasted cash flows State of Economy C Fj Pj j = 1 Recession 100 30% j = 2 Normal 300 50% j = 3 Boom 800 20% Copyright ©2003 Stephen G. Buell 2
3. 3. Computing coefficient of variation CF = . 30(100) + . 50( 300) + . 20( 800) = 340 σ = (100 − 340) 2 . 30 + (300 − 340) 2 .50 + (800 − 340 ) 2. 20 σ = 245 .76 σ 245 . 76 CV = = = . 72 CF 340 Copyright ©2003 Stephen G. Buell Required hurdle rate k σk = f ( risk) = f ( ) CFRequired rate of return k is a function of the forecasted risk ofthe project" Penalize" a riskier project by requiring a higher hurdle ratefor it to be acceptable Copyright ©2003 Stephen G. Buell Alternate methods for incorporating risk into capital budgeting (1) Risk-adjusted discount rate (2) Certainty equivalents Copyright ©2003 Stephen G. Buell 3
4. 4. Risk-adjusted discount rate schedule k .13 .10 .07 .04 0 σ cv = CF .5 1.0 1.5 Copyright ©2003 Stephen G. Buell Risk-adjusted discount rate4% is the risk-free rateCurve is a risk-return trade-off functionCurve is an indifference curveFirm is indifferent to a cv=.5 and k=7% or a cv=1.0 and k=10%Select k based on risk from a predetermined schedule and compute NPV Copyright ©2003 Stephen G. Buell Certainty EquivalentsConvert the expected cash flows to their certainty equivalentsDiscount the certainty equivalents at the risk- free rate of interestRisk-free rate is the yield on a US Treasury bond of the same maturity as the project Copyright ©2003 Stephen G. Buell 4
5. 5. Certainty equivalents coefficients αt is the certainty equivalent coefficient for the cash flow at time t 0 < αt = 1 αt falls as risk increases αt is determined subjectively by the firm or obtained from a predetermined schedule ^ ^CF t = α t CF t CFt is the certaintyequivalent, periodt Copyright ©2003 Stephen G. Buell NPV w/ Certainty EquivalentsLets say α 1 = 1.00, α 2 = . 95, α 3 = .82, ..., α 10 = . 50 and i = 4% α 1 (CF 1 ) α 2 (CF 2 ) α 3 (CF 3 ) α (CF 10 )NPV = −CF0 + + + +L + 10 (1.04)1 (1.04) 2 (1 .04)3 (1. 04)10 1.00(5800) .95(5800) .82(5800) .50(19800 )NPV = −26000+ + + + L+ (1. 04)1 (1. 04) 2 (1.04)3 (1 .04)10 Copyright ©2003 Stephen G. Buell Risk in a portfolio context Consider the potential investment, not in isolation (as we have been doing), but in a portfolio context Look at the relationship between the investment and the firm’s existing assets and other potential investments Copyright ©2003 Stephen G. Buell 5
6. 6. Two projects and the firm Firm A Project B Firm Project A B Copyright ©2003 Stephen G. BuellWhich is the more attractive project?Project A is cyclical like the overall firmProject B is counter cyclicalIn isolation σA = σB but for this firm, B is the more attractive projectProject B is highly negatively correlated with the firm’s other assets so addition of Project B reduces overall risk Copyright ©2003 Stephen G. Buell Correlation and diversificationDifficult to find projects with high negative correlationHowever, if projects whose returns are uncorrelated are combined, overall risk can be reduced and even eliminatedFirms seek to diversify into other areasFirms try to build a portfolio of assets Copyright ©2003 Stephen G. Buell 6
7. 7. Portfolio definitions Portfolio è combination of assets Optimal portfolio è maximum return for a given degree of risk - or - minimum risk for a given rate of return Opportunity set è all possible portfolios Efficient frontier è locus of all optimal portfolios Efficient frontier è dominates all other portfolios of the opportunity set Copyright ©2003 Stephen G. Buell Whats out there?k port 4 2 500 sh Microsoft 100 sh XON-Mobil 8 T bills 5 10 gold ingots 12 soy bean contracts 1 .3 σ port Copyright ©2003 Stephen G. Buell III II Ik port Which portfolio will be selected? σ port Copyright ©2003 Stephen G. Buell 7
8. 8. Risk of an individual asset in a portfolio context capital gain + dividend yield on security j = original price (Pj ,t +1 − Pj ,t ) + D j ,t+1 kj = Pj,t Copyright ©2003 Stephen G. Buell Risk of an individual asset in a portfolio contextExcess return (or risk premium) on security j is the differencebetween the yieldon the securityand the yieldon risk- freetreasurysecurities (R f ) ( P − Pj ,t ) + D j ,t +1risk premium on security j = (k j - R f ) = j ,t +1 − Rf Pj ,tThe market s risk premium can be defined similarly : (Pm,t +1 − Pm,t ) + Dm,t +1risk premium on the market = ( k m - Rf ) = − Rf Pm,t Copyright ©2003 Stephen G. Buell Plot the last 60 months of observations Month kj km Rf k j − Rf k m − Rf 1 .05 .06 .03 .02 .03 2 .00 .02 .04 -.04 -.02 3 -.04 -.06 .04 -.08 -.10 4 .09 .08 .04 .05 .04 M M M M M M 60 -.01 -.02 .05 -.06 -.07 Copyright ©2003 Stephen G. Buell 8
9. 9. Characteristic Line k j − Rf β 1 km − Rf Copyright ©2003 Stephen G. Buell Equation of Characteristic Line k j − R f = α + β [k m − R f ] if α = 0, k j = R f + β [ km − R f ] Security j s expectedreturn is equal to the risk - free rateplus a risk premium This risk premiumis equal to the market risk premium times j s beta coefficien s t Copyright ©2003 Stephen G. Buell Beta and systematic riskBeta is an indicator of systematic risk or market risk: interest rate risk, inflation, panicsβ>1: stock is aggressive, more volatile than the overall market, e.g., airlines, steel, tiresβ<1: stock is defensive, less volatile than the overall market, e.g., utilities Copyright ©2003 Stephen G. Buell 9
10. 10. Unsystematic riskVariations in security j’s return not due to market forcesUnique to the firm, e.g., financial and operating leverage, managed by crooksEliminated by diversificationOnly systematic risk matters to a firm with a diversified portfolio Copyright ©2003 Stephen G. Buell Whats a firm to do? McDonalds and Sears are contemplat ing going into the pizza business McDonalds is only in fast foods - not diversified; ∴ they must be concerned with total risk σ k pizza = f ( ) use risk - adjusted discount ratemethod CF or certainty equivalent method to find NPV s Sears is diversified into department stores,autoparts, insurance, real estate, etc.; ∴ they are concerned only withsystematic risk k pizza = Rf + β pizza [ km − Rf ] use "Beta Model" Copyright ©2003 Stephen G. Buell 10