Transcript 125491457843

Including Risk in
Investment Evaluation
Lecture No. 26
Chapter 10
Fundamentals of Engineering Economics
Copyright © 2008
Probability Concepts for Investment
Decisions

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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
Illustration of Investment Risk
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
Aggregating Risk Over time
E[ PW ( r )] 
N

n 0
V [ PW ( r )] 
N

n 0
E ( An )
(1  r ) n
V ( An )
(1  r ) 2 n
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
Probability Ranges for a Normal
Distribution
Example 10.5 Computing the Mean and
Variance of an Investment Opportunity
$1,000
$2,000
0
1
$2,000
2
Solution:
$1, 000 $2, 000

 $723
2
1.06
1.06
2
2
$200
$500
V [PW(6%)]  $100 2 

 243, 623
2
4
1.06
1.06
 ($494) 2
E[PW(6%)]  $2, 000 
Pr( X  x)  Pr(PW(6%)  0)
 X  
 0  723 
 




   (1.4636)
  
 494 
 1   (1.4636)
 1  0.9283
 7.17%
NPW Distribution
Estimating Risky Cash Flows
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Three-Point Estimate
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Est0(An) = H (optimistic estimate)
Est0 (An) = L (pessimistic estimate)
Est0 (An) = M0 (most-likely estimate)
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Mean:

Variance:
H  4M 0  L
E[ An ] 
6
H L
Var[ An ]  

 6 
2
Example 10.6 Developing a Present Worth
Distribution for Capstone’s Investment Project
Step 1 – Calculating the Means and
Variance for Periodic Cash Flows
Step 2: Calculating the Mean and
Variance of the NPW
Step 3: Obtaining The NPW Probability
Distribution for Capstone’s MicroCHP
Project
Risk-Adjusted Discount Rate
Approach
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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
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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
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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.
Investment Strategies
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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
Example 10.8 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% (?)
Example 10.8 Broader Diversification
Increases Return
Option
1
2
Amount
Investment
$10,000 Bond
Expected
Return
7%
-100%
Value in
25 years
$54,274
$2,000
Lottery tickets
$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
Total
$0
$96,280
Key Points
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Project risk—the possibility that an investment
project will not meet our minimum requirements
for acceptability and success.
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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.
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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.
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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.
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The end goal of a probabilistic analysis of project
variables is to produce a NPW distribution.
All other things being equal, if the expected returns
are approximately the same, choose the portfolio with
the lowest expected risk (variance).
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.