We described an interest rate (r) is the rate of return that equates the value of different cash flows on different dates.
If you could pay 975 today to receive 1,000 in 1 year, you can consider the value of waiting 1 year to be 25 (1,000 – 975.) It can be expressed as an interest rate, r = 25/975 = 0.02564 or 2.56%.
Present and Future Value of a Single Sum of Money
If you could continue to invest at 2.56% for an additional year, the future value would increase by more than 25 because the interest earned in the first year would earn interest in the second year. The final value after two years would thus be 1,000(1.0256) = 1,025.60. So the future value equation for N periods can be simplified to:
FVN = PV(1+r)N
So, the 1,000 investment after five years would be 1,000(1.0256)5 = 1,174.32.
Similarly, you could calculate the present value of a future cash flow as PV = FVN/(1+r)N. So a future lump sum of 1,200 in 5 years would have a present value of 1,200/(1.0256)5 = 1,057.53.
Present and Future Value of a Series of Cash Flows
We can also calculate the present or future values of a series of cash flows. The common types of cash flow series are:
An ordinary annuity that pays a finite set of level cash flows beginning 1 year in the future.
An annuity due that pays a finite set of level cash flows beginning today.
A perpetuity that pays an infinite set of level cash flows.
An uneven series of cash flows that pays different amounts at different times.
Future and Present Value of an Ordinary Annuity
For an ordinary annuity, consider a plan where you invest 1,000 annually for 5 years beginning 1 year from today, with an interest rate of 3%. How much will you have after 5 years? We can do this by calculating the future value of each payment. We receive the first payment one year from now, and its value will increase each of the remaining 4 years to 1,000(1.03)4 = 1,125.51. The payment received in the second year will have 3 remaining years to grow, and so on, giving:
1,000(1.03)4 + 1,000(1.03)3 + 1,000(1.03)2 + 1,000(1.03)1 + 1,000 =
1,125.51 + 1,092.73 + 1,060.90 + 1,030 + 1,000 = 5,309.14.
Because the 1,000 payment is constant, it can be factored out, and we can derive a formula for the annuity factor:
((1+r)N – 1)/r
In this case, (1.035 - 1)/0.03 = 0.15927/0.03 = 5.30914, which when multiplied by the 1,000 payment gives the same answer of 5,309.14.
Now consider a situation where we are the recipient of the same series of cash flows: How much is it worth to us today? In this case, rather than compounding each payment, we discount them – the first payment by 1 year, the second by 2, and so on.
1,000/(1.03)1 + $1,000/(1.03)2 + $1,000/(1.03)3 + 1,000/(1.03)4 + 1,000/(1.03)5 =
970.87 + 942.60 + 915.14 + 888.49 + 862.61 = 4,579.71.
This, too, can be simplified into a present value factor:
$PV\ =\ A\left[\frac{1-\frac{1}{{(1+r)}^N}}{r}\right]$
In this case,
$PV\ =\ 1,000\left[\frac{1-\frac{1}{{(1+.03)}^5}}{.03}\right]=\ 1,000\left[\frac{0.13739}{0.03}\right]\ =\ 4,579.71$
To find the present value of a perpetuity, the equation simplifies further to A/r. In this case, a perpetual cash flow of 1,000 per year at 3% would be worth 1,000/0.03 = 33,333.33.