Daily compounding interest of an investment with monthly contributions that are a function.

Question

What is the future value of an investment that earns 3 percent interest a year compounded daily with monthly contributions of 850 dollars that increase by 2.5 percent a year after 50 years?

Answer 1

"Ordinary Annuities" (and "Annuity Dues") are sequences of identical payments made regularly over time (e.g. monthly, quarterly, etc) . But why not allow payments that grow or shrink geometrically (by constant percentage) over time? If inflation made prices go up at a fixed percentage, I could ask for my monthly payments to grow at the inflation rate. This problem asks for analysis of such a sequence of payments, as well as some other standard problems finding future values of annuities and lump sums.

Let’s assume that the payments are made for 50 + y years, where y is a whole number. y of years. And payments are at the end of the month. And assume that "3% annual interest compounded daily” (for the first 50 years) is computed as 3%/12 monthly interest compounded 30 times per month. So every month’s growth factor is

m = (1+.03/360)^30.

So the value of these payments 50 years (600 months) from now is the future value of the 50-year, 600-payment annuity

A₆₀₀=850 * (m⁶⁰⁰-1)/(m-1) [obtained using a well-known formula for future value of an annuity].

and the value of those 600 payments y years late, at time 50+y, is A₆₀₀*m^12y [using a well-known formula for the future value of a lump sum].

For t from 1 to y, the 12 monthly payments in the t^th year after the 50 year mark are all 850 * (1+.025)^t, so the value at the end of year 50+t of those 12 payments is the future value of the 12-month annuity for that year, which is

P_t = 850 * (1+.025)^t * (m¹²-1)/(m-1) .

Note that P_t+1 is a constant multiple of P_t. For every t from 1 to y-1, P_t+1 = P_t * [1 + .025] .

Treat that annuity's future value (the value after the t^th year) as a single annual deposit made at the end of year 50+t. So these y deposits are a geometrically increasing sequence of payments with money earning interest at a constant rate (in this case, not the same rate as the growth of the sequence of payments).

And the future value of that deposit at time 50+y (after y-t years of interest) is P_t * m^12(y-t).

And the future value of all y of those annual P_t is

∑{for t from 1 to y} (P_t * m^12(y-t)).

= ∑{for t from 1 to y} (850 * (1+.025)^t * (m¹²-1)/(m-1) * m^12(y-t)) *

=∑{for t from 1 to y} [850 * (m¹²-1)/(m-1) * m^12y] *[(1+.025) / m¹²]^t

(a geometric sum, since the values in both sets of square brackets do not depend on t )

= 850 * m^12y * (m¹²-1)/(m-1) * u * (1 – u^y-1) / (1 – u)

where u = (1+.025) / m¹²

and m = (1+.03/360)³⁰.

So the future value of all 50+y years of payments is:

A₆₀₀*m^12y + 850 * m^12y * (m¹²-1)/(m-1) * u * (1 – u^y-1) / (1 – u).