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Finance

 growingannuity
 present value of a growing annuity

 Calling Sequence growingannuity(cash, rate, growth, nperiods)

Parameters

 cash - amount of first payment rate - interest rate growth - rate of growth of the payments nperiods - number of payments

Description

 • The function growingannuity calculates the present value at period=0, of an annuity of nperiods payments, starting at period=1 with a payment of cash. The payments increase at a rate growth per period.
 • Since growingannuity used to be part of the (now deprecated) finance package, for compatibility with older worksheets, this command can also be called using finance[growingannuity]. However, it is recommended that you use the superseding package name, Finance, instead: Finance[growingannuity].

Examples

I hold an investment that will pay me every year for 5 years starting next year. The first payment is 100 units, and each payment is expected to grow by 3% each year. If the interest rate is 11%, what is the present value of the investment.

 > $\mathrm{with}\left(\mathrm{Finance}\right):$
 > $\mathrm{growingannuity}\left(100,0.11,0.03,5\right)$
 ${390.0340764}$ (1)

This can also be calculated as follows.

The cash flows are given by:

 > $\mathrm{cf}≔\left[100,100\cdot 1.03,100{1.03}^{2},100{1.03}^{3},100{1.03}^{4}\right]$
 ${\mathrm{cf}}{≔}\left[{100}{,}{103.00}{,}{106.0900}{,}{109.272700}{,}{112.5508810}\right]$ (2)

or equivalently as

 > $i≔'i':$
 > $\mathrm{cf}≔\left[\mathrm{seq}\left(\mathrm{futurevalue}\left(100,0.03,i\right),i=0..4\right)\right]$
 ${\mathrm{cf}}{≔}\left[{100.0}{,}{103.00}{,}{106.0900}{,}{109.272700}{,}{112.5508810}\right]$ (3)
 > $\mathrm{cashflows}\left(\mathrm{cf},0.11\right)$
 ${390.0340762}$ (4)

Here, we deal with a more complicated example illustrating differential growth. We have an investment that will pay dividends of 1.12 units starting one year from now, growing at 12 % per year for the next 5 years. From then on, it will be growing at 8%. What is the present value of these dividends if the required return is 12%? Solution: first part, the present value for the first 6 years is a growing annuity

 > $\mathrm{part1}≔\mathrm{growingannuity}\left(1.12,0.12,0.12,6\right)$
 ${\mathrm{part1}}{≔}{6.000000000}$ (5)

The fact that this is 6 times the present value of the first dividend is because the growth rate is equal to the required return. The second part, is a (deferred) growing perpetuity. Six years from now, the dividends will be

 > $\mathrm{div_6}≔\mathrm{futurevalue}\left(1.12,0.12,5\right)$
 ${\mathrm{div_6}}{≔}{1.973822685}$ (6)

 > $\mathrm{div_7}≔\mathrm{futurevalue}\left(\mathrm{div_6},0.08,1\right)$
 ${\mathrm{div_7}}{≔}{2.131728500}$ (7)

Its value 6 years from now is

 > $\mathrm{part2_6}≔\mathrm{growingperpetuity}\left(\mathrm{div_7},0.12,0.08\right)$
 ${\mathrm{part2_6}}{≔}{53.29321250}$ (8)

Which has a present value of

 > $\mathrm{part2}≔\mathrm{presentvalue}\left(\mathrm{part2_6},0.12,6\right)$
 ${\mathrm{part2}}{≔}{27.00000000}$ (9)

Therefore the investment has a present value of

 > $\mathrm{part1}+\mathrm{part2}$
 ${33.00000000}$ (10)

33 units.

Compatibility

 • The Finance[growingannuity] command was introduced in Maple 15.