FREC 444 -- Economics of Environmental Management
Discounting


The rates of interest i charged on borrowed money in financial markets 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).

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' discount rates depend to some extent 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.  If I put $100 in the bank at 5 percent annual interest, its' value is $100(1.05) = $105 after one year, $100(1.05)2
= $105(1.05) = $110.25 the second year (the first year's interest also earned interest), $100(1.05)3  = $115.76 the third year, etc.

As the chart below indicates, compounding and discounting are exponential functions, so they can yield dramatically different results over long time horizons depending on the interest rate paid or discount rate applied.




Some common discounting formulas

The future value at time T of 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?

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
Stream A
Stream B
0
$0
$150
1
$50
$100
2
$100
$75
3
$150
$50
4
$150
$25

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 8 percent annual interest. What will your monthly payments be?