CHAPTER 6

DISCOUNTING

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

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

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)

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)

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.

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 = ?

( )PV =

FV

1+ i = FV

11+ i

nn n

n⎛⎝⎜

⎞⎠⎟

PV = $1001

1.10

= $100 0.7513 = $75.13.

3

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

100

1

300

2

300

310%

-50

4

90.91247.93225.39-34.15

530.08 = PV

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

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

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

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

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

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

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.

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.

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

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

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%