Download FREC 424 -- Resource Economics Discounting and Benefit/Cost

FREC 424 -- Resource Economics
Discounting and Benefit/Cost Analysis
Compounding and Discounting
In financial markets, the rate of interest i charged on borrowed money reflects the supply of loanable funds
relative to demand: i is the equilibrium price of borrowed money. On the supply side, i must account for the
lender's perceived risk of default by the borrower (d), the lender's inflationary expectations e (the lender
wants to offset any decline in the loaned money's purchasing power), and an underlying rate of time
preference or "discount rate" r (not to be confused with the Discount Rate the Fed charges member banks for
overnight borrowing): i = d + e + r.
The discount rate reflects a universal human preference for having good things sooner rather than later and
postponing bad things. In theory, the social rate of discount equals market interest rates for risk-free
investments (e.g., Treasury notes) minus inflationary expectations. Theoretically, r should be stable through
time. Unfortunately, i is well-defined but variable, so r is difficult to gauge because market expectations
determining d and e are variable and ill-defined.
Individuals' personal discount rates depend on age, income, education level and other socioeconomic factors.
Empirical analyses of investment behaviors and expenditures on durable goods suggest that older, wealthier and
beter-educated people tend to have longer planning horizons and lower rates of discount, than younger, poorer
and less-educated people.
Discounting provides a mechanism for comparing benefits and costs incurred in different time periods. It is
basically the reverse of compounding, so let’s explain compounding first. When you deposit some money
(“principal”) in a bank savings account, the bank pays you interest on it which is added (“compounded”) into
the principal. Over time, your original principal plus the interest it keeps accumulating earn more interest in
each successive time period. If you put $100 in the bank at 5 percent annual interest, its value will be
…$100(1.05) = $105 after one year
…$100(1.05)2 = $105(1.05) = $110.25 after the second year (the first year's interest also earned interest)
…$100(1.05)3 = $110.25(1.05) = $115.76 after the third year, etc.
Compounding and discounting are exponential functions, so over long time horizons different interest rates or
discount rates can yield dramatically different results. If you compound $100 at 2% annually over 40 years you
will end up with about $221, more than doubling your money. If you compound at 5% annually, you will have
about $704. If you compound at 8% annually, you will have $2,172, increasing your money 21-fold.
The left graph below shows how today’s value (the “present value”) of a payment that you will receive some
time in the future is similarly sensitive to choice of discount rate. The right graph shows how $1,000 40 years
from now is discounted under discount rates ranging from zero to 15%.
Some common compounding and discounting formulas
We now review some basic financial calculation formulas, and pose sample questions to let you try each of
these. As the simple compounding example above indicates, the future value at time T of some current amount
$PV, growing at annual rate r compounded annually for T years, will be:
$FV = $PV(1+r)T
Question: If you put $100 in a savings account earning 4 percent annual interest compounded annually (you
leave the interest you earn in the account so that it earns interest too) how much money would you have in your
account after 5 years?
Discounting is simply the inverse of compounding. The present value of a single future payment $FV to be
received T years from now, discounted at annual rate r, is:
$PV = $FV/(l+r )T
Question: The Treasury is auctioning notes which will pay the holder $10,000 in 20 years. If a bank wants a 5
percent return on its money, what is the highest amount it would bid for a note?
The future value of current amount $PV, growing at annual rate r compounded monthly for T years, will be:
$FV = $PV(1+r/12) I2T
(If there are n periods per year, the rate per period is simply r/n and the term is nT periods.) So daily
compounding will yield
$FV = $PV(1+r/365 ) 365T
Question: If you have $100 earning 4 percent annual interest compounded monthly, how much money would
you have after 5 years? How much would you have if the interest is compounded daily? How much would you
have if the interest is compounded every minute?
Economists often use the simpler formula
$FV = $PVerT
where e is the root of the natural log (approximately 2.71828). This implies instantaneous compounding. You
can use the inverse of this formula to discount future payments to present values:
$PV = $FVe-rT
Question: If you have $100 earning 4 percent annual interest compounded instantaneously, how much would
you have after 10 years? What is the present value of a $10,000 Treasury note maturing in 10 years, discounted
at 5 percent?
You can use either discrete-period or continuous discounting to calculate internal rates of return on
investments by solving the appropriate discounting formula for r. If you invest $PV today in order to receive a
payment $FV at maturity, the implicit discount rate is
IRR = r = ($FV/$PV)(1/T) - 1
IRR = r = ln($FV/$PV)/t
Note that the implicit discount rate from the discrete-interval compounding formula (left) will be slightly higher
than the implicit discount rate from the continuous compounding formula (right).
Question: If you bid $5,000 for a zero-coupon bond paying $10,000 at its maturity in 10 years, what is the
implicit interest rate or internal rate of return?
The present value of a stream of future (variable) annual payments $X0... $XT to be received in years 0 through
T, discounted at annual rate r, is:
$PV = $X0/ (1+r) 0 + $X1/ (1+r)1 + $X2/ (1+r)2 +... + $XT/ (1+r )T
or (approximately)
$PV = $X0 + $X1e-r + $X2e-2r + ... + $XT e-Tr
Question: Which of the payment streams below has the higher present value at a discount rate of 5 percent?
Which has the higher present value at a discount rate of 10 percent?
Year
0
1
2
3
4
Stream A
$0
$50
$100
$150
$150
Stream B
$150
$100
$75
$50
$25
The present value of a finite annuity paying a constant $X per year for T years, discounted at annual rate r, can
be calculated directly from the integral of the continuous discounting formula:
PV = $X/r[1-e-rT]
Question: What is the present value of a lottery jackpot paying $1million per year for 20 years, discounted at 6
percent?
The present value of a perpetual annuity paying $X per year, discounted at annual rate r, is:
$PV = $X + $X/(1+r) + $X/(1+r)2 + .... = $X/r
Question: A parcel of land yields an annual rent of $100 per acre. If you capitalize this perpetual stream at a 5
percent annual rate of discount, how much is this parcel worth per acre?
The present value of a perpetual stream of payments starting at $x and increasing at compound rate g annually,
discounted at annual rate r, is:
$PV = $X + $X(1+g)/(1+r) + $X(1+g)2/(1+r)2 + ... = $X/(r-g)
Question: A parcel of land yields a perpetual stream of rents which are increasing (in real terms) by 2 percent
annually. If you capitalize this rent stream at a 5 percent annual rate of discount, how much is this parcel worth
per acre? (Note: if real rents increased at the rate of discount or faster, the parcel would theoretically have
infinite value.)
The annual payment required to amortize a loan of $X at r percent over T years is:
$pmt = $Xr/[1-(1+r)-T]
Monthly payments would be
$pmt = $X(r/12)/[1-(1+r/12)-12T]
Question: You have a $10,000 student loan which you will be paying back over 5 years at 6 percent annual
interest. What will your monthly payments be?
The basics of benefit-cost analysis
B-C analysis was formalized in the 19th century, and its has played a prominent role in the evaluation of US
public works projects (including water projects undertaken by the US Army Corps of Engineers) since the
1930’s. More recently it has been used to evaluate environmental protection policies.
The basic premise of benefit-cost analysis is that society should spend its resources where the benefits match or
exceed the costs. In theory, B-C analysis should be pretty straightforward. The analyst calculates the benefit
and cost schedules for each candidate policy over the appropriate time horizon, discounts these, and sums them
to obtain comparable present values of the total benefit stream versus the total cost stream for each option. In
choosing between a set of discrete policy options, the usual decision process involves calculating a B-C ratio for
each option and immediately ruling out options where B/C < 1. At this point, the optimal choice may be the
option with this highest B-C ratio (if that option is scalable), or the option with the largest B-C difference (if
the options are not scalable).
In practice, unfortunately, B-C analysis is rarely straightforward, particularly when used in environmental
policy decision-making. In fact, the practical difficulties of doing B-C analysis have created serious political
mistrust of the process. Many Federal environmental statues include wording that actually discourages the use
of B-C analysis. The following discussion lays out some important limitations of B-C analysis, and
summarizes the challenges facing economists who undertake to do such analyses.
One hindrance to public acceptance of B-C analysis of environmental policies the public dismay that
environmental risks cannot be eliminated entirely. Politicians are highly sensitive to the public’s irrational risk
perceptions. Indeed, most people believe they are entitled to a risk-free, pollution-free environment, and are
unwilling to countenance any political compromising of these rights. These unrealistic expectations are rarely
challenged. Consequently, we have had various economically inefficient policies such as the Delaney
Amendment (1958, only recently repealed, which banned food additives that could be shown to cause tumors in
lab animals at any dosage) that rule out any comparison of benefits versus costs. Although the costs of the
Delaney Amendment (preventable obesity, nutritional deficiencies, spoilage, illnesses and deaths from foodborne pathogens) almost certainly exceed its benefits, it took 40 years for Congress to muster the political
courage to repeal it.
A related problem involves uncertainty. B-C analyses can only be as good as the scientific data they rely on,
and the chain of causality between the policy implementation and the desired outcome can be long and tenuous.
For example, the link between a policy (reduce emissions from a company's smokestack by fifty percent, say)
and its intended results (reduce the incidence of lung cancers by 2 in 10,000) depends on understanding (1) the
climatologic and engineering processes explaining how the smokestack controls will reduce ambient
concentrations of the pollutant, (2) the geographic and socioeconomic links explaining how reducing ambient
concentrations will reduce human exposures, and (3) the epidemiologic links between reduced exposures and
reduced incidences of cancers. Experts may be required to elucidate the causalities at each step, and the
uncertainties get compounded at each step.
(In the absence of data, everyone is an expert. True experts will generally qualify their findings with
appropriate caveats, disclose areas of uncertainty, and concede the technical limitations of their analyses
forthrightly. Unfortunately, a nuanced presentation may not carry the day against dramatic oversimplification
in a highly politicized debate. Public trust in experts appears to have declined over the last two decades. US
courts have been slow to reject the junk science, bad statistics and other nonsense peddled by so-called "expert
witnesses" who are paid to support one side or the other in litigation.)
Public risk perceptions are often inconsistent with experts' risk assessments. People naturally discount mundane
risks, including risks they undertake voluntarily, while they exaggerating the significance of statistically trivial
but exotic risks, particularly if they are incurred involuntarily or provoke dread. They ignore experts who say
they should focus less on carjackings and terrorist attacks, and more on quitting smoking and wearing seat belts.
The B-C framework is supposed to compare all benefits and costs associated with policy choices in monetary
units, but some elements such as the values of saving habitats for endangered species, preventing cancers or
saving human lives are difficult to translate into money terms. In fact, attempts to do so often elicit expressions
of moral outrage, and some opponents of B-C analysis misrepresent it as morally corrupt. The economic
valuation of lives does not involve any moral judgment by the analyst. Rather, it is based on observations of
statistical risks people incur voluntarily. Voluntary acceptance of risk implies a self-valuation of one's health or
life in statistical terms.
Another problem with B-C analyses is that they usually introduce normative issues associated with benefit
transfers. Some people win from the proposed project; others lose. The standard Hicks-Kaldor efficiency
criterion suggests that if the winners could compensate the losers and still be better off, the project is
worthwhile. Whether the winners actually should compensate the losers is another issue. These are politically
sensitive redistribution questions that politicians instinctively prefer to avoid. B-C analyses typically compare
costs and benefits without explicit regard to these distributional issues, but wrangling over such issues often
frustrates political implementation of policies that would clearly be good for society. Public mistrust of the
redistributive effects of government policies creates substantial political inertia that impedes adoption of
obviously beneficial policies.
In practice, B-C analyses of public works will list specific categories of beneficiaries and estimate the dollar
benefits each category will receive. (These are typically the people lobbying hard for the project.) The costs
are borne by the taxpayers, who are likely to remain “rationally ignorant” of the project and offer little
opposition to it. B-C analyses of environmental policies tend to be more contentious, because the parties who
will “pay” are industry groups that often have well-organized lobbies to counter the lobbying efforts of
environmental advocacy groups.
A final controversial element in B-C analyses is the choice of discount rate. In some cases the choice of
discount rate largely determines the outcome of the analysis. Here's an illustration:. Suppose we are trying to
determine the optimal use for a vacant parcel of public land. The local power company wants to use it for a
nuclear waste dump, a local developer wants to build condos on it, and a local environmental group wants to
preserve it for recreation and wildlife habitat. These uses would generate the following net social benefit
streams:
year
1
2
3
4
5
6
7
8
9
NPV's
r = 0.04
r = 0.06
r = 0.08
nuke dump condos preserve
$30
$110
$15
$35
$0
$15
$40
$0
$15
$15
$0
$15
$0
$0
$15
$0
$0
$15
$0
$0
$15
$0
$0
$15
$0
$0
$15
$109.59 $105.77
$104.92 $103.77
$100.56 $101.85
$111.53
$102.03
$93.70
The low discount rate (r = 0.04) favors the preserve option, since this yields the largest net benefits in the distant
future. The high discount rate (r = 0.08) favors the condo option, since its net benefits are realized soonest.
And the intermediate discount rate (r = 0.06) favors the nuke dump. The graph below shows NPV for each
option as a function of the discount rate applied. In this example the preferred options change at discount rate
thresholds of about 0.047 and 0.069.