Until now we have been assuming that future values arose from only one initial investment. What happens when there is a stream of multiple investments? An annuity describes a stream of equal annual cash flows. (Annuities come in two flavors: a stream of outflows invested to produce future returns or a stream of inflows of investment returns.)

Consider an example. What would be the future value of $1,000 invested at the end of each year in a bank account that paid 8% interest, compounded annually? (Notice the importance of whether the amounts are invested at the beginning or end of each year.)

Now consider what equal payment could be made at the end of each year if you started with $5,866.60, assuming the principal earned interest at an annual rate of 8%. You guessed it! An annuity of $1,000.

Simplifying calculation of annuity's future value

Annuity calculations are simplified by using tables, as well as calculators and spreadsheets. Look at the attached tables, created using a spreadsheet, which show:

• future value of $1 deposited at the end of each of n periods compounded at i percent
  [FV = SUM n=1 to n (1 + i)ⁿ]

• present value of $1 deposited at the end of n periods discounted at i percent
  [PV = SUM i=1 to n 1/(1 + i)ⁿ]

From the attached future value table, you will notice that if you invest $1 each year at 12% interest for 25 years (assuming annual compounding) you would have $133.33. Now imagine that you started a savings plan in which you put $2,000 each year into a mutual fund that on average has 12% annual returns, after fees and taxes. In 25 years, you would have $266,667.74. (Notice the power of steady savings and compounding!)

Or, if you are adventuresome, you can calculate the present value of an annuity using a formula (imbedded in the attached present value table):

Example

King Tobacco Company is negotiating to compensate the state of New Columbia for the state's health-cost payments related to tobacco use. The company says it will pay the State of New Columbia $200,000,000 for the payments that the state incurred over the last twenty year -- which have averaged $10 million per year.

New Columbia's attorney general understands that this offer does not take into account the time value of money and seeks a settlement that will reflect pre-judgment interest. What should the attorney general seek, assuming the following interest rates (averaged over the last 20 years):

• the risk-free rate of US Treasury notes/bills/bonds - 5.1%
• the state's cost of borrowing - 6.5%
• the tobacco companies' cost of debt - 8.9%
• the tobacco companies' cost of equity - 13.4%

Answer:

There are at least two steps in calculating pre-judgment interest. First, what interest rate is appropriate? Second, using this interest rate, what is the current value of the state's past losses (a future value computation)? See "Should Tobacco Companies Pay the Present Value of Damages?"

An interest rate that focuses on the state's loss may under-deter the culpable party, here King Tobacco. On the other had, an interest rate that looks at the loss from the tobacco company's perspective (particularly that of equity shareholders) may overstate the company's responsibility.

Furthermore, these computations assume a flat interest rate during the entire pre-judgment period. That is, payments made 20 years ago are treated the same as those last year, even though their future value when made may have been different given different interest expectations.