CHAPTER 6
DISCOUNTING
CONVERTING 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
Discounting
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
Mechanics of Discounting I
PV = FV in year t / [1+r]^t
Where PV = Present Value
FV = Future Value (real or nominal)
t = Year
r = Discount Rate (real or nominal)
Mechanics 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)
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.
What’s the PV of $100 due in
3 years if i = 10%?
Finding PVs is discounting, and it’s
the reverse of compounding.
0
PV = ?
10%
1
2
3
100
PV =
FVn
1
n = FVn
1+ i
1+ i
1
PV = $100
1.10
n
3
= $100 0.7513 = $75.13.
Spreadsheet Solution
• Use the PV function: see spreadsheet.
= PV(Rate, Nper, Pmt, FV)
= PV(0.10, 3, 0, -100) = 75.13
What is the PV of this uneven
benefit stream?
0
10%
1
2
3
4
100
300
300
-50
90.91
247.93
225.39
-34.15
530.08 = PV
Spreadsheet Solution
1
A
B
C
D
E
0
1
2
3
4
100
300
300
-50
2
3
530.09
Excel Formula in cell A3:
=NPV(10%,B2:E2)
Perpetuities
PV = NBF / r
Where NBF = a specified annual netbenefit flow
For example:
$186k / .03 = $6.2m
Alternative Discount Rates
• Market rate = r + i + b + y
Where r = real, risk-free rate
i = the expected rate of inflation
b = project specific (nondiversifiable) risk
y = income tax adjustment
• Nominal risk-free rate [n] = r + i
Use 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, generalpurpose bonds of similar maturities
In Project analysis
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
Mechanics 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
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.
Appendix: 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.
Steps 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
Monte Carlo Setup
LNG Navigation
Safety Factor
1.00
0.20
0.04
NORMINV
Probability
Mean
Standard Deviation
=NORMINV(RAND(),C$10,(C$9-C$11)/3.29)
Monte Carlo Setup
Probability of a
Disaster Given a
Massive Spill
10%
=IF(RAND()<F$10,1,0)
IF
Logical Test
Value if true
Value if false
0
