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Unformatted text preview: Tools for Project Evaluation
Nathaniel Osgood 1.040/1.401J 2/11/2004 Basic Compounding
Suppose we invest $x in a bank offering interest rate i If interest is compounded annually, asset will be worth
$x(1+i) $x(1+i)2 $x(1+i)3 $x(1+i)n after 1 years after 2 years after 3 years .... after n years Opportunity Cost & The Time Value of Money
If we assume
money can always be invested in the bank (or some other reliable source) now to gain a return with interest later That as rational actors, we will never make an investment which we know to offer less money than we could get in the bank Then
money in the present can be thought as of "equal worth" to a larger amount of money in the future Money in the future can be thought of as having an equal worth to a lesser "present value" of money Notation
Cost
Time Revenue Simple investment Equivalence of Present Values
Given a source of reliable investments, we are indifferent between any cash flows with the same present value they have "equal worth" This indifferences arises b/c we can convert one to the other with no extra expense Future to Present Revenue
Given Future Revenue in year t... r Borrow a smaller amt now from reliable source
r/(1+i)t r Transforms future revenue to equivalent present revenue, at no additional cost burden
r/(1+i)t Future to Present Cost
Given Future Cost in year t...
c Invest a smaller amt now in reliable source
c c/(1+i)t Transforms future cost to equivalent present cost, with no additional cost burden
c/(1+i)t Present to Future Revenue
Given Present Revenue...
r Invest all now in reliable source; withdraw at t
r(1+i)t r Transforms present revenue to equivalent future revenue, at no additional cost burden
r(1+i)t Present to Future Cost
Given a present cost...
c Borrow = amt now from source; pay back at t
c c(1+i)t Transforms current cost to equivalent future, with no additional cost burden
c(1+i)t Summary
Given a reliable source offering annual return i we can shift costlessly between cash flow v at time 0 and v(1+i)t at time t Because we can flexibly switch from one such value to another without cost, we can view these values as equivalent The present value of a cash flow v at time t is just v/(1+i)t Notion of Net Present Value
Suppose we had
A collection (or stream) of costs and revenues in the future A certain source of borrowing/saving (at same rate) The net present value (NPV) is the sum of the present values for all of these costs and revenues
Treat revenues as positive and costs as negative Understanding Net Present Value
NPV (and PV) is relative to a discount rate
In our case, this is the rate for the "reliable source" NPV specifies the
Value of the cash stream beyond what could be gained if the revenues were returns from investing the costs (at the appropriate times) in the "reliable source"
The "reliable source" captures the opportunity cost against which gains are measured Key point: NPV of "reliable source" is 0
PV(revenue from investment)=PV(investment cost) Example: HighYield Investment
Assume reliable source with 10% annual interest Invest $100 in highrisk venture at year 0 Receive $121 back at year 1 What is the net present value of this investment? What is the net future value of this investment? What does this mean?
$121 $100 Example: Money in Mattress
Assume reliable source with 10% annual interest Place $100 in mattress at year 0 Retrieve $100 from mattress at year 1 What is the net present value of this investment? What does this mean?
$100 $100 Discounted Cash Flow
Computing Present Value (PV) of costs & benefits involves successively discounting members of a cash flow stream
This is because the value of borrowing or investment to/from the "reliable source" rises exponentially This notion is formalized through
Choice of a discount rate r
In the absence of risk or inflation, this is just the interest rate of the "reliable source" (gain through opportunity costs) Applying discount factor 1/(1+r)t to values at time t Outline
Session Objective Project Financing
Public Private Project Contractor Additional issues Financial Evaluation Missing factors Time value of money Present value NPV & Discounted cash flow Simple Examples Formulae IRR Develop or not Develop
Is any individual project worthwhile? Given a list of feasible projects, which one is the best? How does each project rank compared to the others on the list?
"Objective: Strive to secure the highest net dollar return on capital investments which is compatible with the risks incurred" We can EITHER Use NPV to
Evaluate a project against some opportunity cost
Use this opportunity to set the discount rate r
> NPV = < 0 Accept the project Indifferent to the project Reject the project Use NPV to choose the best among a set of (mutually exclusive) alternative projects Rates
Discount Rate:
Worth of Money + risk Minimum Attractive Rate of Return (MARR)
Minimum discount rate accepted by the market corresponding to the risks of a project Choice of Discount Rate Project Evaluation Example
Warehouse A Construction=10 months Cost = $100,000/month Sale Value=$1.5M Total Cost? Profit? Better than B? Warehouse B Construction=20 months Cost=$100,000/month Sale Value=$2.8M Total Cost? Profit? Better than A?
PenaMora 2003 Drawing out the examples
Project A
10 Months $1,500,000 $100,000 $100,000 $100,000 $100,000 ... $100,000 $2,800,000 Project B 20 Months $100,000 $100,000 $100,000 $100,000 $100,000 Outline
Session Objective Project Financing
Public Private Project Contractor Additional issues Financial Evaluation Missing factors Time value of money Present value NPV & Discounted cash flow Simple Examples Formulae IRR Interest Formulas
i = Effective interest rate per interest period (discount rate of MARR) t = Number of compounding periods PV = Present Value NPV = Net Present Value FV = Future Value A = Annuity Interest Formulas: Payments
Single Payment Compound Amount Factor
(F/P, i%, n) = (1 + i )n Single Payment Present Worth Factor
(P/F, i%, n) = 1/ (1 + i )n = 1/ (F/P, i%, n) Uniform Series Compound amount Factor
(F/A, i%, n) = (1 + i )n  1 / i Uniform Series Sinking Fund Factor
(A/F, i%, n) = i / (1 + i )n  1 = 1 / (F/A, i%, n)
PenaMora 2003 Interest Formulas: Series
Uniform Series Present Worth Factor
(P/A, i%, n) = 1/ i  1/ i (1 + i )n Uniform Series Capital Recovery Factor
(A/P, i%, n) = [i (1 + i )n] / [(1 + i )n 1] = 1 / (P/A, i%, n)
PenaMora 2003 Note on Continuous Compounding
To this point, we have considered annual compounding of interest Consider more frequent compounding
Interest is in %/year Fraction of interest gained over time t (measured in years)=it For n compounding periods/year, effective rate for entire year is As n we approach continuous compounding and quantity approaches ei Over t years, we have eit i 1 + n n Equipment Example
$ 20,000 equipment expected to last 5 years $ 4,000 salvage value Minimum attractive rate of return 15% What are the?
A  Annual Equivalent B  Present Equivalent
PenaMora 2003 Equipment Example Outline
Session Objective Project Financing
Public Private Project Contractor Additional issues Financial Evaluation Missing factors Time value of money Present value NPV & Discounted cash flow Simple Examples Formulae IRR Internal Rate of Return (IRR)
Identifies the rate of return on an investment
Example: Geometrically rising series of values Typical means of computing: Identify the discount rate that sets NPV to 0 IRR Investment Rule
r>
= r* < Accept Indifferent Reject "Accept a project with IRR larger than the discount rate."
Alternatively, "Maximize IRR across mutually exclusive projects."
PenaMora 2003 Internal RateofReturn Method (IRR) Example 0(r%) = 20,000 + 5,600 (P/A, r%, 5) + 4,000 (P/F, r%, 5) i = +/ 16.9% > 15% then the project is justified Internal RateofReturn Method (IRR) Graph IRR vs. NPV
Most times, IRR and NPV give the same decision / ranking among projects. IRR does not require to assume (or compute) discount rate. IRR only looks at rate of gain not size of gain IRR ignores capacity to reinvest IRR may not be unique (payments in lifecycle): Trust NPV: It is the only criterion that ensures wealth maximization
PenaMora 2003 Other Methods I
BenefitCost ratio (benefits/costs) or reciprocal
Discounting still generally applied Accept if <1 Common for public projects Does not consider the absolute size of the benefits Can be difficult to determine whether something counts as a "benefit" or a "negative cost" Costeffectiveness
Looking at noneconomic factors Discounting still often applied for noneconomic
$/Life saved $/QALY Other Methods II
Payback period ("Time to return")
Minimal length of time over which benefits repay costs Typically only used as secondary assessment Drawbacks
Ignores what happens after payback period Does not take discounting into account Discounted version called "capital recovery period" Adjusted internal rate of return (AIRR) Outline
Session Objective Project Financing
Public Private Project Contractor Additional issues Financial Evaluation Missing factors Time value of money Present value NPV & Discounted cash flow Simple Examples Formulae IRR What are we Assuming Here?
That only quantifiable monetary benefits matter Certain knowledge of future cash flows Present value (discounting) using equal rates of borrowing/lending Money is not Everything
Social Benefits
Hospital School Employment opportunities Intangible Benefits
New cafeteria Strategic benefits
Partnering with firm for longterm relationship We are missing critical uncertainties
Revenue Level of occupancy Elasticity and Level of cost Duration of project Postconstruction revenue Costs Construction costs
Sale of building Environmental conditions Labor costs Size of lowest bid Variable interest rates Energy costs How quickly items wear out Labor costs Maintenance costs ...
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 Spring '04
 DR.NATHANIELOSGOOD

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