chapter 6 discounting. converting future value to present value making decisions having significant...
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CHAPTER 6
DISCOUNTING
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CONVERTING FUTURE VALUE TO PRESENT VALUECONVERTING FUTURE VALUE TO PRESENT VALUE
Making decisions having significant future benefits or costs means looking at consequences from where we are right now: converting future benefit/cost flows to
PRESENT VALUES
Making decisions having significant future benefits or costs means looking at consequences from where we are right now: converting future benefit/cost flows to
PRESENT VALUES
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DiscountingDiscounting
Future values are converted to present values by means of a discount rate.
That is, future nominal benefits are worth less than present benefits of equal magnitude -- the WIMPY principal- Inflation- Markets tell us that people demand
compensation for forgoing current consumption
Future values are converted to present values by means of a discount rate.
That is, future nominal benefits are worth less than present benefits of equal magnitude -- the WIMPY principal- Inflation- Markets tell us that people demand
compensation for forgoing current consumption
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Mechanics of Discounting IMechanics of Discounting I
PV = FV in year t / [1+r]^t
Where PV = Present ValueFV = Future Value (real or nominal)
t = Yearr = Discount Rate (real or nominal)
PV = FV in year t / [1+r]^t
Where PV = Present ValueFV = Future Value (real or nominal)
t = Yearr = Discount Rate (real or nominal)
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Mechanics of Discounting IIMechanics of Discounting II
For a Stream of Benefits from year 1 to year t, SUM [add up] all the present
values for all net future values
Where t = 3
PV = (FV in year 1 / [1+r]^1) + (FV in year 2 / [1+r]^2) + (FV in year 3 / [1+r]^3)
For a Stream of Benefits from year 1 to year t, SUM [add up] all the present
values for all net future values
Where t = 3
PV = (FV in year 1 / [1+r]^1) + (FV in year 2 / [1+r]^2) + (FV in year 3 / [1+r]^3)
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Three Ways to Find PVs
• Solve the equation with a regular calculator (or use FV tables from an accounting text).
• Use a financial calculator.
• Use a spreadsheet.
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10%
What’s the PV of $100 due in 3 years if i = 10%?
Finding PVs is discounting, and it’s the reverse of compounding.
100
0 1 2 3
PV = ?
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( )PV =
FV
1+ i = FV
11+ i
nn n
n⎛⎝⎜
⎞⎠⎟
PV = $1001
1.10
= $100 0.7513 = $75.13.
3
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Spreadsheet Solution
• Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, FV)
= PV(0.10, 3, 0, -100) = 75.13
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What is the PV of this uneven benefit stream?
0
100
1
300
2
300
310%
-50
4
90.91247.93225.39-34.15
530.08 = PV
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Spreadsheet Solution
Excel Formula in cell A3:
=NPV(10%,B2:E2)
A B C D E
1 0 1 2 3 4
2 100 300 300 -50
3 530.09
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PerpetuitiesPerpetuities
PV = NBF / rWhere NBF = a specified annual net-
benefit flow
For example:
$186k / .03 = $6.2m
PV = NBF / rWhere NBF = a specified annual net-
benefit flow
For example:
$186k / .03 = $6.2m
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Alternative Discount RatesAlternative Discount Rates
•Market rate = r + i + b + y
Where r = real, risk-free ratei = the expected rate of inflation
b = project specific (nondiversifiable) risk
y = income tax adjustment
•Nominal risk-free rate [n] = r + i
•Market rate = r + i + b + y
Where r = real, risk-free ratei = the expected rate of inflation
b = project specific (nondiversifiable) risk
y = income tax adjustment
•Nominal risk-free rate [n] = r + i
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Use of Alternative Discount RatesUse of Alternative Discount Rates
• Use real rate [r] with real FVs- For example, where you are using current costs to
estimate future costs
• Use nominal rate [n] with nominal FVs- For example, where you are making identical
nominal principal and interest payments each year
WHAT NOMINAL RATE SHOULD YOU USE?
Borrowing rate on tax-exempt, general-purpose bonds of similar maturities
• Use real rate [r] with real FVs- For example, where you are using current costs to
estimate future costs
• Use nominal rate [n] with nominal FVs- For example, where you are making identical
nominal principal and interest payments each year
WHAT NOMINAL RATE SHOULD YOU USE?
Borrowing rate on tax-exempt, general-purpose bonds of similar maturities
In Project analysis
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Annualizing Capital CostsAnnualizing Capital Costs
• Since real government budgets are formulated one year at a time, the budget tends to be biased against delivery methods requiring up-front investments
• The proper solution is converting everything to PV
• However, there is a reasonable alternative, which is the annualizing capital costs
• Since real government budgets are formulated one year at a time, the budget tends to be biased against delivery methods requiring up-front investments
• The proper solution is converting everything to PV
• However, there is a reasonable alternative, which is the annualizing capital costs
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Mechanics of AnnualizingMechanics of Annualizing
Annual Cost of a Capital Asset
= P [r + d - a]
Where P = Purchase Price [replacement cost]
d = Depreciation rate
[wear and tear + obsolescence]
a = Appreciation rate
Annual Cost of a Capital Asset
= P [r + d - a]
Where P = Purchase Price [replacement cost]
d = Depreciation rate
[wear and tear + obsolescence]
a = Appreciation rate
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DOES THE CHOICE OF DISCOUNT RATE MATTER?DOES THE CHOICE OF DISCOUNT RATE MATTER?
• Yes – choice of rate can affect policy choices.
• Generally, low discount rates favor projects with the highest total benefits.
• High SDRs rates favor projects where the benefits are front-end loaded.
• Yes – choice of rate can affect policy choices.
• Generally, low discount rates favor projects with the highest total benefits.
• High SDRs rates favor projects where the benefits are front-end loaded.
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Appendix: Monte Carlo Simulation with ExcelAppendix: Monte Carlo Simulation with Excel
• Most spread sheets provide a function for generating random variables that are distributed uniformly from 0 to 1 [in Excel the function is RAND()]
• To generate uniform random variables with other ranges, one simply multiplies the draw from the uniformly distributed from 0 to 1 by the desired range and adds the minimum value [for SDRs with = 2% and a range from 0 to 4%, use the following formula: RAND()*.04]
• Alternatively you can combine functions for the inverse of the cumulative normal distribution and the uniform distribution: NORMSINV(RAND())
• The standardized normal distribution can be given any and through simple transformations: add a constant = and multiply by the square root of the desired variance.
• Most spread sheets provide a function for generating random variables that are distributed uniformly from 0 to 1 [in Excel the function is RAND()]
• To generate uniform random variables with other ranges, one simply multiplies the draw from the uniformly distributed from 0 to 1 by the desired range and adds the minimum value [for SDRs with = 2% and a range from 0 to 4%, use the following formula: RAND()*.04]
• Alternatively you can combine functions for the inverse of the cumulative normal distribution and the uniform distribution: NORMSINV(RAND())
• The standardized normal distribution can be given any and through simple transformations: add a constant = and multiply by the square root of the desired variance.
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Steps in Monte Carlo Simulation with ExcelSteps in Monte Carlo Simulation with Excel
1. Construct a row of appropriate random variables and the formulas that use them to compute net benefits (the last cell in the row should contain net benefits)
2. Copy the entire row N times (spreadsheets up to 10K -- use logic functions or macros to replicate)
3. Chart array of outcomes (the results in last cells), plot as histogram, calculate and
1. Construct a row of appropriate random variables and the formulas that use them to compute net benefits (the last cell in the row should contain net benefits)
2. Copy the entire row N times (spreadsheets up to 10K -- use logic functions or macros to replicate)
3. Chart array of outcomes (the results in last cells), plot as histogram, calculate and
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Monte Carlo SetupMonte Carlo Setup
LNG Navigation Safety Factor
1.000.200.04
=NORMINV(RAND(),C$10,(C$9-C$11)/3.29)
NORMINVProbabilityMeanStandard Deviation
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Monte Carlo SetupMonte Carlo Setup
=IF(RAND()<F$10,1,0)
IFLogical TestValue if trueValue if false
Probability of a Disaster Given a Massive Spill
10%