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a single account, or split the money up. In the end, you still have every single one of your
$5,000, together with all the compound interest each dollar accumulates.
This observation is the key to the bucket approach. Since it makes no difference whether
or not the money is all kept in a single account, it will not change anything if we pretend
that an account is split into separate “buckets” whenever it helps us to do this.
The following example will illustrate:Example 4.7.1 Brandon has $1,457.06 in his savings account, which pays 3.6%
interest compounded monthly. Each month he deposits $75 into this account. How
much will he have in his savings account after 2 years?Brandon’s monthly payments are an annuity, but we also have to deal with the $1,457.06
which was in the account from the start. To handle this, we will pretend that the original
$1,457.06 is kept in a separate account from his new deposits.Bucket 1: The original $1,457.06, earning compound interest.FV PV(1 i)nFV $1,457.06 (^) 1 0.036__ 12
24
FV $1,565.67
Bucket 2: The monthly $75 payments.FV PMT s _n (^) | (^) i
FV $75 s __ 24 | (^). 036
⁄ 12
FV $1,863.49
Of course, Brandon’s actual account contains all the money we pretended was separated into
those two buckets. So his total future value will be $1,565.67 $1,863.49 $3,429.16.
“Annuities” with an Extra Payment
The same approach works nicely to deal with additional payments made to an annuity as well.
Example 4.7.2 Kevin deposited $1,000 each year into a savings account that earned
a 5% effective rate for 10 years. In the third year, though, he was able to deposit
$5,000 instead of his usual $1,000. How much did he have at the end of the 10 years?
We need to break Kevin’s payments up into buckets that can be handled either as an annuity
or as a single sum of money growing at compound interest. There are several ways to do this,
but the most effi cient one is to make the fi rst bucket an annuity of $1,000 each year (pretend-
ing that the third-year payment was $1,000, like all the others). The second bucket, then,
would be an extra $4,000 in the third year. (The second bucket is $4,000 and not $5,000
because $1,000 in the third year was included in the fi rst bucket annuity).
Bucket 1: $1,000 per year for 10 years.
FV PMT s _n (^) | (^) i
FV $1,000 s 10 __ (^) | (^) .05
FV $12,577.89
Bucket 2: The extra $4,000 paid in year 3. This money will earn compound interest for the
last 7 years. So:
FV PV( 1 i)n
FV $4,000(1.05)^7
FV $5,628.40
Putting the two buckets together, we see that Kevin’s future value was $12,577.89
$5,628.40 $18,206.29.
“Annuities” with a Missing Payment
What if instead of an extra payment, a payment is missed? This approach can handle that
sort of situation as well.
4.7 Future Values with Irregular Payments: The Bucket Approach (Optional) 197