I was doing some catch up exercise on Khan academy and was given this seemingly simple looking problem
Find the compound interest and the total amount after 4 years and 6 months if the interest is compounded annually.
Principal = £100,000
Rate of interest 10% percent per annum
Calculate total amount and compound interest
I calculated it using compound interest formula:
$$ 100000(1.1)^{4.5} = 153556.10346 $$
But this turned out to be the wrong answer, the correct answer, as presented by Khan academy was this:
khan academy answer
153730.5
I can also arrive at this value by sort of using the compound interest formula for first 4 years, but then calculating interest for the last 6 months manually (0.1/2):
$$ 100000(1.1)^{4} = 146410 $$
$$ 146410 + ( 146410 \cdot 0.05 ) = 153730.5 $$
I still feel a bit unsatisfied, and feel I am not really understanding what's going on here and why would calculating the last step manually give a different answer.
Can you provide an explanation on why this formula should not apply on this case?
Best Answer
The Khan academy answer seems to be derived from assumptions about how financial institutions operate. The various assumptions may reflect real-life finance (but not the mathematical viewpoint) or the mathematical viewpoint (but not real-life finance), and at least one is arguably based on information missing from the problem statement.
We assume there are no changes to the principal (such as deposits or withdrawals) during the four and a half years other than the crediting of interest.
At the end of four years, immediately after the fourth year's interest is credited, the balance in the account should be, as you computed,
$$ 100000(1.1)^4 = 146410. $$
If you wait another six months and check the account balance again, I would expect still to see a balance of $146410.$ However, if you are allowed to withdraw the entire balance of the account at that time, you might be entitled to receive interest for the last six months. (Many real-life investments such as savings bank accounts allow this.)
If you are entitled to interest for the last six months, the usual practice (as far as I know) is to prorate the interest, that is, if exactly half a year has passed since the previous interest payment then you receive exactly half of one year's interest. That is $5\%$ of the balance after the last regular interest payment, in this case.
Since the question says nothing about whether the funds are withdrawn (or not) at the end of the four and a half years, however, the question is ill-formed. While the answer might plausibly be a result that could occur in real life (if you can still find an investment that pays interest only annually, allows the investment to be liquidated in the middle of the year, and pays prorated interest for the final partial year), there's no way really to guess which of at least two plausible interpretations is meant. It's a bad question.