Economic Questions (Calculation Based)

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TECON 480 – Class 8

August 14, 2019

Part 2 - Formulating the cost-benefit model

• Assignment #2 due Aug 16

• Instructions available on Canvas

Announcements / Reminders

Last Class

• Discussed how to evaluate benefits and costs in secondary

markets affected by a policy/project

• Briefly introduced the idea of considering the timing of costs

and benefits

Today’s Class

• Discuss how to properly account for the timing of costs and

benefits

• This material relates to Ch. 6 in Boardman et al.

Introduction

• Often projects and policies have impacts over a long time period

� Therefore, need to be able to compare the values today and in the future

• We focus on practical issues one must know to compute the net present value of a project

• Assume the social discount rate is given

• The rate is often specified by an oversight agency

• Congressional Budget Office (CBO)

• Environmental Protection Agency (EPA)

• etc.

Projects With Lives of One Year

Example:

• A city government has the opportunity to buy a parcel of land

for $10 million

• This parcel will be sold for $11 million one year from now

• If the city does not buy the land, it would invest this money in

bonds at an interest rate of 5%

• Should the city buy the land now?

• There are three ways to answer this question

• Each approach provides the same answer

Projects With Lives of One Year

Future Value Analysis

• Choose the project with the highest future value

• The future value FV in one year of an amount X invested at

interest rate i is:

• Investment in bonds:

� If the city compares the value of the land in one year and the

future value of the investment in bonds, it should decide to buy

the land

( )1FV X i= +

( )10 1 0.05 10.5FV = + =

Projects With Lives of One Year

Present Value Analysis

• Choose the project with the highest present value

• The present value PV of an amount Y received in one year with a

discount rate i is:

• Investment in land:

� We can compare this value to the $10 million that can be invested

in bonds. Investing in the land today is comparable to an investment of

$10.48 million in bonds today. Again, the land is a better investment.

( )1 Y

PV i

= +

( ) 11

10.48 1 0.05

PV = = +

Projects With Lives of One Year

Net Present Value Analysis

• Choose the project with the highest net present value

• The sum of the present values of all the benefits and costs of a

project (including the initial investment):

��=��(�)−��(�)

• Investment in land:

9 ( )

( ) ( )

11 10.48

1 0.05 0.48

PV B

NPV

PV C

 = = + =

= 

Projects With Lives of One Year

Note that this is the

opportunity cost of

buying the land

Projects With Lives of One Year

Net Present Value Analysis

• If there is only one new potential project and its impacts are

calculated relative to the status quo:

• It should be selected if NPV > 0

• It should not be selected if NPV < 0

• Here the status quo is the investment in bonds (NPV = 0)

Projects With Lives of One Year

Net Present Value Analysis

• If the impacts of multiple mutually exclusive alternative

projects are calculated relative to the status quo:

• Should choose the project with the highest NPV

• …as long as this project’s NPV > 0

• If the NPV < 0 for all alternative projects to the status quo, one

should maintain the status quo 12

Compounding & Discounting Over Multiple Years

• Assume now that the city can invest the $10 million in bonds

for five years at an annual rate of interest of 5%

• If there is simple interest, then the city will receive $500,000

per year

� Therefore, the future value would be $12.5 million

• If the interest is compound annually, the value is different

because each year you are “re-investing” the interest you have

earned to date 13

Compounding & Discounting Over Multiple Years

Compound interest: interest on reinvested interest

Note: calculations above are based on interest rate = 4%

Future Value Analysis

• The future value of an amount X invested for n years with

interest compounded annually at rate i is:

• Investment in bonds for 5 years @ 5%:

• If interest rate is 7%:

Compounding & Discounting Over Multiple Years

( )1 nFV X i= +

( )510 1 0.05 12.76FV = + ≈

( )510 1 0.07 14.02FV = + ≈

• How does compound interest work?

• 5 year investment, interest rate at 7% on a principal of $10 million:

•

• Original capital amount increases > 40% after 5 years

• Under simple interest: would have increased by 35%

• Over longer periods, the gap between these outcomes becomes

significant and increases with time

Compounding & Discounting Over Multiple Years

Present Value Analysis

• The present value of an amount Y received in n years, with

interest compounded annually at rate i is:

• The term is called the discount factor

• Discount factor can be thought of as the weight given to costs

and benefits occurring at different time periods 17

Compounding & Discounting Over Multiple Years

( )1 n Y

= +

( ) 1

1 n

Compounding & Discounting Over Multiple Years

Discount factors:

How much is a dollar received in year t worth today?

Years (t)

Discount Rate (i) 0 1 2 3 4 5 6 7 8 9 10

0.2 1.00 0.83 0.69 0.58 0.48 0.40 0.33 0.28 0.23 0.19 0.16

0.15 1.00 0.87 0.76 0.66 0.57 0.50 0.43 0.38 0.33 0.28 0.25

0.1 1.00 0.91 0.83 0.75 0.68 0.62 0.56 0.51 0.47 0.42 0.39

0.08 1.00 0.93 0.86 0.79 0.74 0.68 0.63 0.58 0.54 0.50 0.46

0.05 1.00 0.95 0.91 0.86 0.82 0.78 0.75 0.71 0.68 0.64 0.61

0.02 1.00 0.98 0.96 0.94 0.92 0.91 0.89 0.87 0.85 0.84 0.82

0.01 1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.91

0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

See ‘Discount factor example’ spreadsheet in the ‘Miscellaneous Files’

folder on Canvas to see how these values were calculated

Present Value Analysis

• When we compute the present value of costs and benefits for

a CBA, the Y’s and X’s differ from year to year

� How can we deal with this?

• We have to refine the formula to take into account these

differences

Compounding & Discounting Over Multiple Years

• Consider the stream of benefits over time

• Must take into account the benefit of each unique year,

multiply it by the associated discount factor, and then sum

across all years

• � �

represents the benefits at year t

• n is the lifespan of the project

Compounding & Discounting Over Multiple Years

( ) ( ) ( ) ( ) ( )

0 11

0 1 1 ...

1 1 1 1

n n

B B BB PV B

i i i i

− −

→ = + + + + + + + +

• This expression can be simplified as the following:

• Analogous expression applies to the stream of costs Ct of the

project:

Compounding & Discounting Over Multiple Years

( ) ( )0 1

B PV B

= +

∑

( ) ( )0 1

C PV C

= +

∑

Exercise • Which of the following alternatives would you recommend for

the government? Assume an interest rate of 8%.

Project I

Project II

Net Present Value Analysis

• We can use the present value of costs and benefits that we’ve

just determined to compute the NPV of a project:

• Alternatively, can compute the net benefits for each year:

� Then compute the NPV using the net benefits: 23

Compounding & Discounting Over Multiple Years

( ) ( ) ( ) ( )0 01 1

n n

t t

B C NPV PV B PV C

i i= =

= − = − + +

∑ ∑

t t t NB B C= −

( )0 1

NB NPV

= +

∑

Exercise

• Assume a library is considering purchasing a new information

system that would give users access to a number of online

databases for five years

• The benefits of the system are estimated to be $100,000 annually

• The system initially costs $325,000 to install

• It costs $20,000 each year to operate and maintain the system

• After 5 years the system will be dismantled and sold, resulting in

a net cash inflow of $20,000

• The appropriate discount rate is 7%

24Should the city provide the library with the

funds to purchase this system?

Comparing Projects with Different Time Frames

• Simplest scenario is when projects have the same time span

• However, must often compare projects with different time spans

• Analysts should refine the NPV to take into account the different time spans

• Analysts can use two methods to evaluate projects with different life spans:

• Roll-Over Method

• Equivalent Annual Net Benefit Method (not discussed here)

• These methods always lead to the same conclusion

Suppose that Tacoma Power is considering two new alternative sources of power:

1. A hydroelectric dam

• It would last 75 years

• NPV of the project is $30 million

2. A solar plant

• It would last 15 years

• NPV of the project is $24 million

• Discount rate = 8% per year

� NPV criterion requires selecting the hydroelectric dam (30 > 24)

Comparing Projects with Different Time Frames

Roll-Over Method

• However, suppose that the solar plant is chosen

• Also suppose that at the end of the life span of 15 years,

another solar plant can be built (and this process can continue

until the end of the life span of the hydroelectric dam)

� We would have 5 back-to-back solar plants over the 75 year

period

Roll-Over Method

• The net present value of five back-to-back solar plants is:

� This NPV is higher than that of the hydroelectric dam

� Therefore, should select the solar plant option

( ) ( ) ( ) ( )15 30 45 60 24 24 24 24

24 34.94

1.08 1.08 1.08 1.08

NPV = + + + + =

Hint: to help you understand the calculations here,

draw a “75-year timeline” for these two options

Other Considerations

• Shorter projects have an additional benefit (not included in

NPV calculation using the Roll-Over Method) because one

does not necessarily have to roll-over the shorter project

when it is finished

• A better option might be available at that time

• This additional benefit is called quasi-option value (i.e. the

value of flexibility) and will be discussed later in the course

Inflation & Real vs. Nominal Dollars

• Conventional private sector financial analysis measures monetary amounts in nominal dollars (sometimes called current dollars)

• But, due to inflation, one cannot buy as many goods and services with one dollar today as one could in the past

� Must control for inflation (i.e. general price increases)

• Control for inflation by converting nominal dollars to real dollars (sometimes called constant dollars)

• Typically use the consumer price index (CPI) deflator, but other measures are available

• U.S. inflation history:

Inflation & Real vs. Nominal Dollars

Inflation

• The CPI market basket:

Inflation

• The CPI market basket evolves over time:

Inflation

• Magnitude of CPI allows us to compare price levels over time

� Example: how much higher are average prices in 2016 than

they were in 2006?

• Rate of change of CPI indicates the rate of inflation

� Example: how much higher were average prices in the 4th

quarter of 2015 than in the 3rd quarter of 2015?

Inflation

• Can you think of any examples where inflation may be an

important part of your decision-making?

• When is inflation “good” or “bad” for you?

Inflation

My personal experiences:

1. My previous Canadian student loan:

• Borrowed $8,014 in 2000

• Began repayment in 2007

• Made further repayments in 2014

2. My existing mortgage payments are fixed at $1,840/month

for the next 27 years

Problems with the CPI

CPI is the most commonly used measure of inflation, though most economists think that the CPI overstates inflation due to:

• Substitution bias

• Market basket is fixed, but consumers will respond to changes in relative prices by purchasing different quantities of goods

• Difficulty in incorporating new goods and services

• Were smartphones included in the market basket?

• Changes in quality over time

• Compare $1000 spent on a TV in 1992 with $1000 spent on a TV in 2016 (even after adjusting for inflation)

Analyzing Future Benefits & Costs

• So how do we address inflation in practice?

• Analysts should either:

• Measure the benefits and costs in real dollars and discount using

a real discount rate

• Measure the benefits and costs in nominal dollars and discount

using a nominal discount rate

� Both methods would result in the same numerical answer

• Usually, we use real dollars for public-sector project evaluation

• Suppose the expected annual rate of inflation during the life

of the project is denoted by m

• Benefits or costs that are given in nominal dollars may be

converted to real dollars by discounting them at rate m using

the present value method:

Analyzing Future Benefits & Costs

( ) Nominal cost or benefit

Real cost or benefit

t t

= +

• If the discount rate is given in nominal dollars and is denoted by i, then it can be converted to a real discount rate, denoted by r using the expression:

• Note that r is approximately equal to i - m, especially if m is small

• There are many estimates/forecasts of expected future inflation

• Conclusion: try to be explicit (and consistent) about whether you are measuring values in real or nominal terms

Analyzing Future Benefits & Costs

i m r

− =

Long-Lived Projects & Horizon Values

• Some projects can continue to provide costs and/or benefits

well after the project is “finished” from an engineering or an

administrative perspective

• Extreme examples:

• The Great Wall of China continues to generate tourism benefits

and maintenance costs even though it was built to discourage

invasions many centuries ago.

• Preschool training programs may benefit participants throughout

their entire lives and these benefits may pass-through to their

children, their children’s children, etc.

�All of these impacts should be included in a CBA 42

Long-Lived Projects & Horizon Values

• However, in practice, often unclear how to handle costs and

benefits that arise far in the future

• Question: What is the appropriate value of n?

• In some cases, n → ∞:

( )0 1 t

NB NPV

∞

= +

∑

Time-Declining Discounting

• It is suggested that time-declining discount rates can be used

for long-lived and intergenerational projects

• Example: climate change impacts

• As an example, could use the following:

Year Discount Rate

0 – 50 3.5%

50 – 100 2.5%

100 – 200 1.5%

200 – 300 0.5%

> 300 0%

Sensitivity Analysis in Discounting

• Determining the appropriate discounting method and the

value of the social discount rate is often difficult, and this

creates a risk of bias (why?)

• Consequently, sensitivity analysis should usually be

conducted on the discount rate

• We will discuss this concept more generally later in the course

• Useful to plot the NPV of a project for several possible values

of the discount rate 45

Assignment #2

• Any questions?