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Including Risk in Investment Evaluation Lecture No. 26 Chapter 10 Fundamentals of Engineering Economics Copyright © 2008 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 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%)] $1002 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 Three-Point Estimate Est0(An) = H (optimistic estimate) Est0 (An) = L (pessimistic estimate) Est0 (An) = M0 (most-likely estimate) 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 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. 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 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 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. 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.