Jan's initial investment is $6000. She is going to deposit $500/mo ($6K/yr) for the next 45 years and get 7% compound interest annually. How is the answer $1,778,831? Can you please provide the formula and how this answer comes to this amount?

1 Answer
Dec 5, 2017

This sum can be expressed in the formula:
#S = A*[(1+P)^(N+1)-1]/P#
where
#A# - initial and annual investment
#P# - annual interest
#N# - number of years

Explanation:

My first assumption was that the interest is paid annually at #7%# rate. The result for #45# years of accumulation was smaller than your amount.
Much closer result was with semi-annual payment of interest at 3.5% each half a year, which is equivalent to #7.1225%# annually.

Here is the theory.
At year #0# we put amount #A# in the bank for annual interest (paid annually) of #P#.
At year #1# we added the same amount #A# for the same interest #P#.
Do the same up to year #N#.
What is the final sum?

Since the initial amount #A# grows for #N# years, it accumulates into #A*(1+P)^N#.
Additional amount #A# contributed at the end of the first year grows for #N-1# years and accumulates into #A*(1+P)^(N-1)#.
Continue this process for #N# years. The total sum of all investments with interest will be
#A*(1+P)^N + A*(1+P)^(N-1) + A*(1+P)^(N-2) + ... + A*(1+P)^1 + A*(1+P)^0#

This sum can be expressed in the more compact formula using the formula for a sum of geometric progression with factor #(1+P)#:
#S = A*[(1+P)^(N+1)-1]/[(1+P)-P] = A*[(1+P)^(N+1)-1]/P#

For #A=6,000#, #P=0.07# and #N=44# this formula gives
#S=$1,714,496# (not your amount)
For #N=45# the result is
#S=$1,840,511# (also not your amount)

If we assume that the interest is compounded semi-annually (as most bonds do) at the rate #7/2%=3.5%# per half a year, this would result in an annual factor of greater than #1.007#:
#(1+(7/200))^2 = 1.071225#

For #A=6,000#, #P=0.071225# and #N=44# this formula gives
#S=$1,778,491# (almost your amount)
The difference between this and your amount might be attributed to rounding during the calculations.