Risk and uncertainty

Capital Budgeting

Risk and Uncertainty

Copyright ©2003 Stephen G. Buell

Risk and Uncertainty
Risk è the possibility that actual returns will
deviate from expected returns
Risk è situations in which a probability
distribution of possible outcomes can be
estimated
Uncertainty è worse, not enough information
available


Probability distribution of
expected outcomes


1

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 risk
Coefficien t of variation (cv) puts dispersionon a relativebasis
σ
cv =
CF
Consider σ x = 300 and CF x = 1000 versus σ y = 300 and CF y = 4000
Intuitively x is riskier. Need to show that.
300 300
cv x = = .300 while cv y = = . 075
1000 4000


Forecasted cash flows
State of Economy C Fj Pj

j = 1 Recession 100 30%

j = 2 Normal 300 50%

j = 3 Boom 800 20%


2

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


Required hurdle rate k'
σ
k ' = f ( risk) = f ( )
CF

Required rate of return k ' is a function of the forecasted risk of
the project

" Penalize" a riskier project by requiring a higher hurdle rate
for it to be acceptable


Alternate methods for incorporating
risk into capital budgeting
(1) Risk-adjusted discount rate

(2) Certainty equivalents


3

Risk-adjusted discount rate schedule
k'
.13

.10

.07

.04

0 σ
cv =
CF
.5 1.0 1.5

Risk-adjusted discount rate
4% is the risk-free rate
Curve is a risk-return trade-off function
Curve is an indifference curve
Firm 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

Certainty Equivalents
Convert the expected cash flows to their
certainty equivalents
Discount the certainty equivalents at the risk-
free rate of interest
Risk-free rate is the yield on a US Treasury
bond of the same maturity as the project


4

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


NPV w/ Certainty Equivalents

Let's 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


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


5

Two projects and the firm

Firm
A
Project B
Firm
Project A

B


Which is the more attractive project?

Project A is cyclical like the overall firm
Project B is counter cyclical
In isolation σA = σB but for this firm, B is the
more attractive project
Project B is highly negatively correlated with
the firm’s other assets so addition of Project
B reduces overall risk


Correlation and diversification
Difficult to find projects with high negative
correlation
However, if projects whose returns are
uncorrelated are combined, overall risk can
be reduced and even eliminated
Firms seek to diversify into other areas
Firms try to build a portfolio of assets


6

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


What's 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

III II I
k port
Which portfolio will be selected?

σ port

7

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


Risk of an individual asset in a portfolio context

Excess return (or risk premium) on security j is the difference
between the yieldon the securityand the yieldon risk- free
treasurysecurities (R f )
( P − Pj ,t ) + D j ,t +1
risk premium on security j = (k j - R f ) = j ,t +1 − Rf
Pj ,t
The market' s risk premium can be defined similarly :
(Pm,t +1 − Pm,t ) + Dm,t +1
risk premium on the market = ( k m - Rf ) = − Rf
Pm,t


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


8

Characteristic Line
k j − Rf

β
1

km − Rf


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


Beta and systematic risk
Beta is an indicator of systematic risk or
market risk: interest rate risk, inflation,
panics
β>1: stock is aggressive, more volatile than
the overall market, e.g., airlines, steel, tires
β<1: stock is defensive, less volatile than the
overall market, e.g., utilities


9

Unsystematic risk
Variations in security j’s return not due to
market forces
Unique to the firm, e.g., financial and
operating leverage, managed by crooks
Eliminated by diversification
Only systematic risk matters to a firm with a
diversified portfolio


What's a firm to do?
McDonald's and Sears are contemplat ing
going into the pizza business
McDonald's 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"
'


10

Risk and uncertainty

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