You need to accumulate $10,000. To do so, you plan to make deposits of $1,950 per year - with the first payment being made a year from today - into a bank account that pays 8.05% annual interest. Your last deposit will be less than $1,950 if less is needed to round out to $10,000. How many years will it take you to reach your $10,000 goal? How large will the last deposit be?
Formula for FV of annuity can be used to compute number of periods as:
FV = P x [(1+r) n – 1/r]
P = Periodic cash flow = $ 7,000
r = Rate per period = 8.05 % or 0.0805 p.a.
n = Numbers of periods
$ 10,000 = $ 1,950 x [(1+0.0805) n – 1/0.0805]
$ 10,000 x 0.0805 = $ 1,950 x [(1.0805) n – 1]
$ 805 / $ 1,950 = (1.0805) n – 1)
0.412820513 = (1.0805) n – 1
0.412820513 + 1 = (1.0805) n
(1.0805) n = 1.412820513
Taking logarithm of both sides we get,
n x log 1.0805 = log 1.412820513
n x 0.03362477121 = 0.15008699188
n = 0.15008699188/0.03362477121 = 4.463584033 or 5 periods
Five years need to deposit to accumulate $ 10,000
But the last deposit will be less than $ 1,950.
Let’s compute the last deposit.
Year |
Cash Flow (C) |
FV Factor computation |
FV Factor @ 8.05% (F) |
FV (=C x F) |
1 |
$1,950 |
(1+0.0805)^4 |
1.3630101341401 |
$2,657.869762 |
2 |
$1,950 |
(1+0.0805)^3 |
1.2614624101250 |
$2,459.851700 |
3 |
$1,950 |
(1+0.0805)^2 |
1.1674802500000 |
$2,276.586488 |
4 |
$1,950 |
(1+0.0805)^1 |
1.0805000000000 |
$2,106.975000 |
5 |
A |
(1+0.0805)^0 |
1 |
A |
Total PV |
$10,000 |
Let the last deposit be A.
Total FV = $2,657.869762 + $2,459.851700 + $2,276.586488 + $2,106.975000 + A
$ 10,000 = $ 9,501.2829488 + A
A = $ 10,000 - $ 9,501.2829488 = $ 498.7170512 or $ 498.72
The last deposit will be of $ 498.72
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