Methods of Handling Project
Risk
Lecture No. 30
Professor C. S. Park
Fundamentals of Engineering Economics
Copyright © 2005
Probability Concepts for Investment
Decisions




Random variable: variable that
can have more than one
possible value
Discrete random variables: Any
random variables that take on
only isolated values
Continuous random variables:
any random variables can have
any value in a certain interval
Probability distribution: the
assessment of probability for
each random event
Expected Return/Risk Trade-off
Probability (%)
Investment A
Investment B
-30
-20
-10
0
10
Return (%)
20
30
40
50
Measure of Expectation
j
E[ X ]     ( p j ) x j
(discrete case)
j 1
 xf(x)dx
(continuous case)
Expected Return Calculation
Event
Return
(%)
Probability
Weighted
1
2
3
6%
9%
18%
0.40
0.30
0.30
2.4%
2.7%
5.4%
Expected
Return
10.5%
Measure of Variation
j
Var x   2x   ( x j   ) 2 ( p j )
j 1
x 
Var X
Var X   p x  ( p j x j )
2
j j
E X
2
 (E X )
2
2
Variance Calculation
Event
Deviations
Weighted Deviations
1
(6% - 10.5%)2
0.40(6% - 10.5%)2
2
(9% - 10.5%)2
0.30(9% - 10.5%)2
3
(18% - 10.5%)2
0.30(18% - 10.5%)2
( 2) = 25.65
Estimating the amount of
Risk involved in an
Investment Project
1.
2.
Probabilistic Cash Flow Approach
Risk-Adjusted Discount Rate Approach
1. Probabilistic Cash Flow Approach
E[ PW ( r )] 
N

E ( An )
(1  r ) n
N
V ( An )
(1  r ) 2 n
n 0
V [ PW ( r )] 

n 0
where r = a risk-free discount rate,
An = net cash flow in period n,
E[An ] = expected net cash flow in period n
V[An ] = variance of the net cash flow in period n
E[PW(r)] = expected net present worth of the project
V[(PW(r)] = variance of the net present worth of the project
Example
Period
Expected
cash flow
Estimated
standard
deviation
0
-$2,000
100
1
1,000
200
$1,000 $2,000
E[ PW (6%)]  $2,000 

 $723
2
1.06
1.06
2002 5002
2
V [ PW (6%)]  100 

 243,623
2
4
1.06 1.06
 [ PW (6%)]  243,623  $493.58
 = $494
2
2,000
500
-3
-$759
-2
-
0
= $723

2
3
$2,205
Present Worth Distribution

If we can assume that
PW(i) is a normal
distribution with mean
E[PW] and variance
V[PW], we can make a
more precise
probabilistic statement
of PW for the project.
 = $494
-3
-$759
-2
-
0
= $723

2
3
$2,205

0  E[ PW (i)] 
P[ PW (i )  0]  P  Z 


[
PW
(
i
)]


723 

 P Z 
  P Z  1.4648
493.58 

 0.0728
2. Risk-Adjusted Discount Rate Approach



An alternate approach to consider the risk elements
in project evaluation is to adjust the discount rate to
reflect the degree of perceived investment risk.
The most common way to do this is to add an
increment to the discount rate, that is, discount the
expected value of the risky cash flows at a discount
rate that includes a premium for risk.
The size of risk premium naturally increases with the
perceived risk of the investment
Example

You are considering a $1 million investment
promising risky cash flows with an expected value of
$250,000 annually for 10 years. What is the
investment’s NPW when the risk-free interest rate is
8% and management has decided to use a 6% risk
premium to compensate for the uncertainty of the
cash flows
Solution





Given: initial investment = $1 million, expected
annual cash flow = $250,000, N = 10 years, r = 8%,
risk premium = 6%
Find: net present value and is it worth investing?
First find the risk adjusted discount rate = 8% + 6%
= 14%. Then, calculate the NPW using this riskadjusted discount rate:
PW(14%) = -$1 million + $250,000 (P/A, 14%, 10) =
$304,028
Because the NPW is positive, the investment is
attractive even after adjusting for risk.
Comparing Risky Investment Projects Comparison Rule




If EA > EB and VA  VB,
select A.
If EA = EB and VA  VB,
select A.
If EA < EB and VA  VB,
select B.
If EA > EB and VA > VB,
Not clear.
Model
Type
E
(NPW)
Var
(NPW)
Model 1
$1,950
747,500
Model 2
2,100
915,000
Model 3
2,100 1,190,000
Model 4
2,000 1,000,000
Model 2 vs. Model 3  Model 2 >>> Model 3
Model 2 vs. Model 4  Model 2 >>> Model 4
Model 2 vs. Model 1  Can’t decide
Investment Strategies



Trade-Off between Risk and Reward
 Cash: the least risky with the lowest returns
 Debt: moderately risky with moderate returns
 Equities: the most risky but offering the
greatest payoff
Broader diversification reduces risk
Broader diversification increase expected return
Broader Diversification Reduces Risk
Broader Diversification Increases Return
Amount
Investment
Expected Return
$2,000
Buying lottery tickets
$2,000
Under the mattress
0%
$2,000
Term deposit (CD)
5%
$2,000
Corporate bond
10%
$2,000
Mutual fund (stocks)
15%
-100% (?)
Option
1
2
Amount
Investment
$10,000 Bond
Expected
Return
7%
-100%
Value in
25 years
$54,274
$2,000
Lottery tickets
$0
$2,000
Mattress
0%
$2,000
$2,000
Term deposit
(CD)
5%
$6,773
$2,000
Corporate bond
10%
$21.669
$2,000
Mutual fund
(stocks)
15%
$65,838
$96,280
Summary

Project risk—the possibility that an investment
project will not meet our minimum requirements
for acceptability and success.

Our real task is not to try to find “risk-free”
projects—they don’t exist in real life. The
challenge is to decide what level of risk we are
willing to assume and then, having decided on
your risk tolerance, to understand the
implications of that choice.
•Three of the most basic tools for assessing project
risk are as follows:
1. Sensitivity analysis– a means of identifying
the project variables which, when varied, have the
greatest effect on project acceptability.
2. Break-even analysis– a means of identifying
the value of a particular project variable that causes
the project to exactly break even.
3. Scenario analysis-- means of comparing a
“base –case” or expected project measurement (such
as NPW) to one or more additional scenarios, such as
best and worst case, to identify the extreme and most
likely project outcomes.

Sensitivity, break-even, and scenario analyses
are reasonably simple to apply, but also
somewhat simplistic and imprecise in cases
where we must deal with multifaceted project
uncertainty.

Probability concepts allow us to further refine the
analysis of project risk by assigning numerical
values to the likelihood that project variables will
have certain values.

The end goal of a probabilistic analysis of project
variables is to produce a NPW distribution.
•From the NPW distribution, we can extract
such useful information as the expected NPW
value, the extent to which other NPW values
vary from , or are clustered around, the
expected value, (variance), and the best- and
worst-case NPWs.
•All other things being equal, if the expected
returns are approximately the same, choose
the portfolio with the lowest expected risk
(variance).