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Discounting & Finance

How should the future benefits of a project be weighed against present costs?

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Generic Group Project

You are making a recommendation about using catchment basins for groundwater recharge in LA. Costs now provide water in future, offsetting future water acquisition costs. [2001 Group Project on urban stormwater runoff]

Good idea? Big issue: comparing costs today with

benefits tomorrow

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Example 1: “Contractor wins

$314.9 million Powerball”

Winner opts for $170 million lump-sum payoff instead of 30 annual payments of about $10.5 million per year.

Question: Why would someone choose $170 million over $315 million?

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Example 1: “Contractor wins

$314.9 million Powerball”

Winner opts for $170 million lump-sum payoff instead of 30 annual payments of about $10.5 million per year.

Question: Why would someone choose $170 million over $315 million?

Answer: The time value of money. Future earnings must be discounted.

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Outline

What is discounting? Why do we discount? The mechanics of discounting. The importance & controversy of

discounting. Discounting in practice.

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What is discounting?

Public and private decisions have consequences for future:Private: Farmer invests in water-saving

irrigation. High up-front cost, benefits accrue over time.

Public: Dam construction/decommissioning, Regulating emissions of greenhouse gases, wetlands restoration, etc.

Need method for comparing costs & benefits over time.

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Why do we discount?

Simple answer: Put $100 in bank today, get about $103 next year.

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Why a positive discount rate?

Impatience – I would rather have that ice cream cone

today than tomorrow Decreasing marginal value of wealth

An extra dollar is less important if I am richer Productivity of capital

Letting Carol’s forest grow another year generates more lumber

Risk Will I live to see the money I put in the bank?

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Example 2: Carol’s Forest

Assume forest grows at a declining annual rate Annually: 4%, 3.9%, 3.8%,….

When should she cut her forest? If she’s patient: wait and get more wood If she’s impatient: cut now Tension: impatience to consume vs. waiting and

producing more Interest rate is an “equilibrium” between

impatience of consumers and productivity of the forest

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Combining impatience with capital productivity

Wo

od

to

day

Wood tomorrow

Indifference curvefor consumers

Slope=“rate of exchange” between wood today and wood tomorrow [eg, 1 cord today = 1.1 cords tomorrow]

ForestProductivity

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Mechanics of discounting

Money grows at rate r. Invest V0 at time 0:

V1=V0(1+r)

V2=V1(1+r),…

Future Value Formula: Vt=V0(1+r)t.

Present Value Formula: V0 = Vt/(1+r)t. Other formulae available.

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Example 3: The drip irrigation problem Farmer has to decide whether to invest in

drip irrigation system: should she? Basic Parameters of Problem:

Cost = $120,000. Water savings = 1,000 Acre-feet per year,

forever Water cost = $20 per acre foot.

Calculate everything in present value (alternatively, could pick some future date and use future value formula)

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Investing in drip irrigation (r=.05)

Year Costs Benefits CumulativeNet Gain

0 120,000 20,000 -100,000

1 0 19,048 -80,952

2 0 18,141 -62,811

3 0 17,277 -45,534

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When does she break even?

Drip Irrigation Project

-150000

-100000

-50000

0

50000

100000

150000

200000

0 5 10 15 20 25

Year

Net

Pay

off

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Concept of Present Value

(annual discount rate r)

What is the present value of a stream of costs and benefits, xt: x0, x1,

…,xT-1

PV= x1 + (1+r)-1x2+(1+r)-2x2+…+(1+r)-(T-1)xT-1

If PV > 0, stream is valuable

Annuity: Opposite of present value – convert a lump-sum into a steam

of annual payments

Eg: spend $1,000,000 on a dam which is equivalent to $96,000 per

year for 30 years (check it!)

Eg: Reverse mortgages for seniors

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Where does inflation come in?

Inflation is the increase in the cost of a “basket of goods” over time.

Your grandpa always says “An ice cream cone only cost a nickel in my day”….the fact that it’s now $2 is inflation.

Want to compare similar values across time by controlling for inflationCorrect for inflation: “Real”Don’t correct for inflation: “Nominal”

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What is the risk free rate?US Tresury Yield Curve January 16, 2009

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The “Consumer Price Index”

CPI is the way we account for inflation. CPIt = 100*(Ct/C0)

Ct = cost of basket of goods in year t.

C0 = cost of basket of goods in year 0.

E.g. Year CPI

1990 100

1991 104.2

1992 107.4

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Some other discounting concepts

Net Present Value (NPV): The present value of a stream of values over the life of the project (eg, NPV of B-C)

Internal Rate of Return (IRR): The interest rate at which project would break even (NPV=0).

Scrap Value: The value of capital at the end of the planning horizon.

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Importance of discounting

Discounting the future biases analysis toward present generation.If benefits accrue later, project less likelyIf costs accrue later, project more likelySpeeds up resource extractionEg, lower discount rate increases desirability

of reducing GHG now (WHY?) “Risk-adjusted discount rate”

Risky projects may justify increasing discount rate.

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Social vs. private discount rate Private discount rate—for private decisions

A positive concept Result of market – supply and demand

Social rate – for societal decisions A normative concept Usually lower than private rate How should we make intertemporal decisions? Moving resources between generations

different than between years for same person

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Intergenerational decisions

Choice of discounting an ethical decision We do have a preference for consumption earlier

rather than later We also tend to think a dollar is more important to a

poorer person than a richer person Result: r = η g + δ

η: elasticity of marginal utility of consumption wrt income

δ: pure rate of time preference g: growth rate of income

Example (η = 1; g = 2%; δ = 0%) r = 2%

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Social discount rate in practice Small increase in r can make or break

a project. Typical discount rates for public

projects range from 2% - 10%. Usually do “sensitivity analysis” to

determine importance of discount rate assumptions.

Be clear about your assumptions on r.

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What is the discount rate?Weitzman’s survey (2160 Economists)

“Taking all relevant considerations into account, what real interest rate do you think should be used to discount over time the benefits and costs of projects being proposed to mitigate the possible effects of global climate change?”

Mean = 4%, Median = 3%, Mode = 2%

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Discount Rate Choice

-100

0

100

200

300

400

500

-5 0 5 10 15 20 25 30

Discount Rate

Res

po

nse

s

Distribution of responses

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Far-distant costs or benefits

Many important environmental problems have costs and/or benefits that accrue far in the distant future.

Constant-rate discounting has 3 disadvantages in this case: Very sensitive to discount rate Far distant consequences have little or no

impact on current policy Does not seem to fit empirical or

experimental evidence very well

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Constant-rate discounting

NPV

0

200

400

600

800

1,000

1,200

0 0.02 0.04 0.06 0.08 0.1 0.12

Discount Rate, r

PV

of

1000

, T=

100

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Stern Review of Economics of Climate Change

Conclusion: Benefits of action outweigh costs of inaction

ControversiesLow discount rates for benefits

• What does this do?

Different discount rate for costs and benefits

• What does this do?

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What’s the big deal?So what if we do a little more climate protection than necessary?

Money diverted from educationDeveloping countries have less well-

educated population in 2050 Savings diverted from other projects

Research into cancer gets shortchanged