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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 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%. 
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
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
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
$FV = $PV(1+r/365 ) 365T
Economists often use the simpler formula $FV = $PVerT
$PV = $FVe-rT
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
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
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
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]
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 food-borne 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. 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:
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. 
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