CE 350 Introduction to Transportation Planning

1
CE 350Introduction to Transportation Planning

• TPH Chapter 9a
• Economic Evaluation Tools
• pp 312 - 316

2
Lecture Objective

• Understand the discount formulas and how to apply
  them to a transportation economic evaluation
  project.

3
Lecture Points

• Know the terminology
• Review interest equations
• Graphically show payments
• Develop B/C ratio for a project

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Terminology

• F future amount of money
• P present amount of money
• r discount rate (interest rate)
• N periods of repayment or project life
• (years)
• A uniform payments

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Money has time value...

• Even when you dont do anything with it.
• Why?
• Inflation is one simple reason.
• Also, human nature.

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Interest Rate (r)

• The interest rate transforms money today into
  money tomorrow.
• It is the rate at which money grows when
  invested.
• The interest rate is said to be simple if
  interest is paid only on the initial investment.
• The interest rate is said to be compound if
  interest is paid not only on the initial
  investment, but also on any interest re-invested
  in the previous periods.
• In the business world, compound interest is used
  exclusively.

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FUTURE VALUE
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Equation 1 Future Value

• F ( P) (1r) nThe future value of P in n years
  at a discount (interest) rate of r
• Invest 10,000 at 10 for 1 yearF (10,000)
  (10.1)1 11,000

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OR

• (1r)n is the compound amount factor (caf ?)
• ? F P (caf ?)

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Compound Interest

• Suppose the compound interest rate in the market
  is 10 per annum. What does it mean?
• It means that your deposit at the start of a
  period grows by 10.
• What will be the value of 100 after one year?
• 110.
• After two years?
• 121.

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Future Value of a Cash Flow

• If the compound interest rate is r , then the
  future value of P dollars one period from today
  is P(1r), two periods from today is P(1r)2, and
  t periods from today is P(1r)t.

• We refer to todays value as P (for present
  value)
• and to the future value P(1r)t Ft (for
  future value at
• time t).

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Example 1 Future Value

• Suppose the compound interest rate is 12 per
  annum. What is the future value after one year
  of 150?
• F1 150(1.12) 168
• After two years?
• F2 150(1.12)2 188.16
• After t years?
• Ft 150(1.12)t

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Example 2 Future Value

• Suppose r 8 and P 8000. What is the future
  value (F) of the amount after seven years? After
  twenty years?
• F7 8000(1.08)7 13,712
• F20 8000(1.08)20 37,287.66

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Manhattan Island

• Peter Minuet, the first director general of New
  Netherlands province, purchased Manhattan Island
  from the local Canarsee Indians for approximately
  24 in 1626.
• It is often claimed that the Indians got a raw
  deal.
• What do you think?

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Manhattan Island

• The question is What would 24 in 1626 be worth
  in 2001? Certainly more than 24 if it was
  invested prudently for 375 years.

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Manhattan Island

• A lot could happen to 24 if properly saved or
  invested for 375 years.
• Suppose we decide to make a rough assessment of
  the value of that 24 today if it were invested
  in a conservative project that earned 8 per year
  for 375 years.

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Manhattan Island

• What is the future value of 24 if it is invested
  at 8 for one year?
• That is, how much is it worth including interest?
• FV1 24(1.08) 25.92

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Manhattan Island

• After two years?
• FV2 24(1.08)2 28.00

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Manhattan Island

• After fifty years?
• FV50 24(1.08)50 1125.64

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Manhattan Island

• After 375 years?
• 82,057,739,561,528 82 trillion plus!

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82 trillion
1626
2001
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Equation 1 Future Value

• F ( P) (1r) nThe future value of P in n years
  at a discount (interest) rate of r
• F P (caf ?)

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PRESENT VALUE
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Present Value Concept

• Now let us consider the converse of a future
  value. Suppose the compound interest rate in the
  market is 10 per annum. You are told that one
  year from now you will have 110. What is the
  present value of this amount?
• Of course, 100. Why? Because 1 today is 1.10
  after one year if the interest rate is 10.
• What if you are told that one year from now you
  will have 100?
• The present value of 100 one year from now is
  90.91. Why? Because 90.91 grows to 100 one
  year from now.

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Present Value of a Cash Flow

• In general, if the discount rate is r , then the
  present value of P dollars obtained one period
  from today is P/(1r)
• Obtained two periods from today is P/(1r)2
• Obtained t periods from today is P/(1r)t

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Example 3 Present Value

• Suppose the discount rate is 9 per annum. What
  is the present value (P) of 100 obtained one
  year from today?
• P 100/(1.09) 91.74
• Of 100 obtained two years from today?
• P 100/(1.09)2 84.17
• Of 100 obtained t years from today?
• P 100/(1.09)t

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Equation 2 Present Value

1/(1r)n is the present worth factor (pwf ?) P
F (pwf ?)
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Example 4 Net Present Value

• Suppose a project requires an initial investment
  of 60,000.
• At the end of the first year you expect to lose
  20,000.
• At the end of the second year (which is also the
  end of the project) you expect to gain 100,000.
• You assess that, given the risk of the project, a
  cost of capital of 12 is appropriate. Should
  you accept the project?

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Example 4

• On a time line, the cash flows would look like

-60,000
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Example 4

• To find the present value, we must first discount
  all cash flows back to the present

-60,000
?
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Example 4

• Using the NPV formula

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Example 4

• Or, on the time line

-60,000
-17,857.14
79,719.39
1,862.25
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Equation 2 Present Value

1/(1r)n is the present worth factor (pwf ?) P
F (pwf ?)
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Uniform Annual Series
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Uniform Annual Series Concept

• Now let us consider having a uniform annual
  payment or receipt
• This could be the case when
• We pay an annual amount each year into a
  retirement fund
• We draw out an annual amount from our winnings in
  the lottery

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Uniform Annual Series
Future Value
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Equation 3 Uniform Annual Series

• F A
• F A ( caf )

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Example
Given i 4, what is the future value of 25
annual payments of 5175 ? F (5175) (caf-4-25)
(5175) (41.646) 215,518
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Uniform Annual Series
Present Value
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Equation 4 Uniform Annual Series

• P A
• P A ( pwf )

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Example
Given i 4, what is the present value of 25
annual payments of 5175 ? P (5175) (pwf-4-25)
(5175) (15.622) 80,844
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Equation 5 Uniform Annual Series

• A F
• A F ( sff )

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Example
Given i 4, and 188,675 in year 25, what can
the annual distribution be? F (215,518)
(sff-4-25) (215,518) (0.02401) 5175
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Equation 6 Uniform Annual Series

A P (capital recovery factor) P (crf)
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Example
Given i 4 and an initial investment of
80,860 what amount can we draw out in 25 annual
payments? A ( 80,844) (crf-4-25) ( 80,844)
(0.06401) 5175
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Equation 7
Net Present Value (NPV)

If NPV ? 0 indicates net benefits for the project
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Equation 8
Benefit Cost Ratio (i.e. B/C Ratio)

B/C is the Sum of the Benefits ? Sum of the Costs
If B/C ? 1 the project should not be
considered If B/C ? 1 the project should be
carried forward
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Example
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BENEFIT COST RATIO

• Costs 153,185
• Benefits 47,520
• Ratio 47,520 0.31
• 153,185

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Two Payment Schedules

• Single Payments
• Uniform Annual Series
  OrEnd-Of-Year Payments

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Key Relationship
FIND KNOWING
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REMEMBER?
F future amount of money P present amount of
money r discount rate N periods of repayment
or project life (years) A uniform payments
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EXAMPLE!
If we want to find the future amount of an
investment (in n years), knowing the present
payment (amount we deposit today), what
relationship do we use?
Finding
Knowing We want to find the future knowing the
present F
P
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Payment Schedules

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Payment Schedules
Single Payments

Uniform Payments
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Where Do We Find The Factors?

• We use interest tables!!
• Lets s look!
• Note In the tables, S Our F
• R Our A
• But, the acronyms are the same!!

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Interest Table

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In the Table We Look For
P F
P A
Use pwf
Use pwf
F P
Use caf
F A
Use caf

A F
Use ssf
A P
Use crf
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The Analysis Steps

• What do you know?
• What do you want to calculate?
• Apply the ratio
• Set up an equation
• Insert values from interest tables
• Calculate the values

F
P
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Lecture Objective

• Understand the discount formulas and how to apply
  them to a transportation economic evaluation
  project.

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What do you think?
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Project
Benefits()

Costs (-)
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CE 350Introduction to Transportation Planning

• TPH Chapter 9a
• Discount and Capital Recovery
• pp 312 - 316