# February 8

advertisement ```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()&lt;F\$10,1,0)
IF
Logical Test
Value if true
Value if false
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