ECO 3104

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ECO 3104

• Lecture 21

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Valuing Costs and Benefits that Occur Over Time

• Many decisions involve tradeoffs between present
  and future consumption
• business investments in plant and equipment
• government public works projects
• investments by individuals in their human
  capital
• consumer purchases of durable goods

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Valuing Costs and Benefits that Occur Over Time

• Need method for comparing present and future
  costs and benefits
• must account for the fact that people prefer
  having things now rather than later
• a dollar received tomorrow is not the same as a
  dollar received today
• earlier availability gives individual more
  consumption choices
• if have good today, can consume now or wait until
  future

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Valuing Costs and Benefits that Occur Over Time

• Marginal rate of time preference (MRTP)
• extra future consumption you are willing to forgo
  to get additional consumption now, expressed as a
  percentage of the change in current consumption
• MRTP (-?C1-?C0)/?C0
• example ?C0 10, ?C1 -12
• MRTP (12-10)/10 .2 20

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Valuing Costs and Benefits that Occur Over Time

• Interest rate
• premium borrowers must pay lenders to obtain use
  of funds now
• price of earlier availability, expressed in
  percentage terms
• r (V1-V0)/V0

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Valuing Costs and Benefits that Occur Over Time

• Intertemporal exchange
• if there are no transaction costs or other
  distortions (eg. taxes), individuals will make
  exchanges between present and future consumption
  (borrow and lend) until their MRTP equals the
  market interest rate
• at the margin, market interest rate represents
  the tradeoffs each individual is willing to make
  between present and future consumption

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Net Present Value

• To compare alternative projects that include
  future benefits costs and benefits can find
  equivalent net value today or net present value

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Net Present Value

• PV of a single payment 1 year in the future

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Net Present Value

• PV of a single payment T years in the future
• t2
• in year 1, PV1V2/(1r)
• in year 0, PV0 PV1/(1r)
• V2/(1r)/(1r)
• V2/(1r)2

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Net Present Value

• PV of a single payment T years in the future

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Net Present Value

• PV of a stream of annual payments for T years in
  the future
• simply sum up PV of each payment

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Net Present Value

• If there are both future costs, C, and benefits,
  B, then net present value is

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Net Present Value

• Effect of time on NPV
• the further in the future the benefit, the lower
  the net present value
• Effect of interest rate on NPV
• the higher the interest rate, the lower the net
  present value

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PV of 1 Paid in the Future
Interest Rate 1 Year 2 Years 3 Years 4 Years 5
Years 6 Years

• 0.01 0.990 0.980 0.951 0.905 0.820 0.742
• 0.02 0.980 0.961 0.906 0.820 0.673
  0.552
• 0.03 0.971 0.943 0.863 0.744 0.554
  0.412
• 0.04 0.962 0.925 0.822 0.676 0.456
  0.308
• 0.05 0.952 0.907 0.784 0.614 0.377
  0.231
• 0.06 0.943 0.890 0.747 0.558 0.312
  0.174

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Net Present Value

• Valuing Payment Streams
• Choosing a payment stream depends upon the
  interest rate.

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Two Payment Streams
Today 1 Year 2 Years

• Payment Stream A 100 100 0
• Payment Stream B 20 100 100
• NPV of Stream A 100 100/(1r)
• NPV of Stream B 20 100/(1r) 100/(1r)2

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NPV of Payment Streams
r .05 r .10 r .15 r .20

• NPV of Stream A 195.24 190.90 186.96 183.33
• NPV of Stream B 205.94 193.54 182.57 172.77

Note that the NPV of A relative to B increases
as r increases since future payments are
discounted more heavily and stream Bs benefits
are weighted toward future periods
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Adjustments for Risk

• Determining the discount rate for an uncertain
  environment
• This can be done by increasing the discount rate
  by adding a risk-premium to the risk-free rate.
• Owners are risk averse, thus risky future cash
  flows are worth less than those that are certain.

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Adjustments for Risk

• Diversifiable versus non-diversifiable Risk
• Diversifiable risk can be eliminated by investing
  in many projects or by holding the stocks of many
  companies.
• Non-diversifiable risk cannot be eliminated and
  should be entered into the risk premium.

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The Net Present Value Criterionfor Capital
Investment Decisions

• Real versus nominal discount rates
• Assume price, cost, and profits are in real terms
• need to use real interest rate in calculations
• real rate nominal rate expected inflation

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The Net Present Value Criterionfor Capital
Investment Decisions

• Real versus Nominal Discount Rates
• If convert everything to nominal terms, get same
  result
• Gains, costs and discount factor all increased by
  rate of inflation
• NPV doesnt change
• usually best to work in real terms, thereby
  avoiding necessity to estimate expected inflation

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Investment Decisions by Consumers

• Consumers face similar investment decisions when
  they purchase a durable good.
• Compare future benefits with the current purchase
  cost
• Examples
• Cars
• Appliances
• TV sets and stereo equipment

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Investment Decisions by Consumers

• Benefits and Cost of Buying a Car
• S value of transportation services in dollars
• E total operating cost/yr
• Price of car is 20,000
• Resale value of car is 4,000 in 6 years

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Investment Decisions by Consumers

• Benefits and Cost

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The Value of Lost Earnings

• NPV can be used to determine the value of lost
  income from a disability or death.
• Example
• John Smith dies in an auto accident on January 1,
  2004 at 56 years of age.
• computer programmer, salary 85,000

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The Value of Lost Earnings

• What is the PDV of Smiths lost income to his
  family?
• Must adjust salary for predicted increase over
  time (g)
• Assume a 1.5 real rate of growth in future
  earnings

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The Value of Lost Earnings

• What is the PDV of Smiths lost income to his
  family?
• Must determine how much longer he would have
  worked
• account for probability of death from other
  causes and retirement
• can be determined from work life tables
• 8 years

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The Value of Lost Earnings

• What is the PDV of Jennings lost income to his
  family?
• need to determine real discount rate, r
• r (nominal rate) - (expected inflation rate)
• want nominal rate on government bonds of similar
  length
• need to determine expected rate of inflation
  based on recent inflation rates calculated from
  the consumer price index

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The Value of Lost Earnings

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Calculating Lost Wages
Income Discount Factor Present Value Year W0(1
g)t 1/(1 r)t W0(1 g)t/(1 r)t

• 2004 85,000 1.000 85,000
• 2005 86,275 .976 84,171
• 2006 87,569 .952 83,350
• 2007 88,883 .929 82,536
• 2008 90,216 .906 81,731
• 2009 91,569 .884 80,934
• 2010 92,943 .862 80,144
• 2011 94,337 .841 79,362
• Total 653,102

g0.015 r.025
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The Value of Lost Earnings

• Finding PDV
• The summation of column 4 will give the PDV of
  lost wages (653,102)
• The amount Jennings family could actually
  recover as compensation would also depend on
• Taxes the Jennings would have paid on his salary
• Fringe benefits they would have received (eg.
  health insurance)
• Home production (eg. yard work, childcare) Mr.
  Jennings would have provided
• Consumption by Mr. Jennings (food and clothing
  expenses if he would have lived)

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Alternatives to NPV

• Benefit-cost ratio
• problem
• will select projects with high relative net
  benefit, not necessarily greatest total net
  benefit

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Alternatives to NPV

• Internal rate of return (IRR)
• IRR is the rate of return (r) such that PV of
  benefits equals the present value of costs
• compute IRR and compare to market interest rate

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Net Present Value of a Factory
10
8
6
4
Net Present Value ( millions)
2
0
-2
-4
-6
0
0.05
0.10
0.15
0.20
Interest Rate, r
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Alternatives to NPV

• Internal rate of return (IRR)
• problems
• if negative cash flows at beginning and later in
  project can get multiple solutions

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Alternatives to NPV

• Internal rate of return (IRR)
• problems
• IRR gives discount rate at which net present
  value is zero, but at other discount rates the
  preferred project may be different

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Net Present Value of a Factory
10
8

6
4
Net Present Value ( millions)
2
0
-2
-4
Project 1
Project 2
-6
0
0.05
0.10
0.15
0.20
Interest Rate, R
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End of Lecture 21