Why using compound interest formula gives (potentially) wrong answer in this instance

Question

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?

Answer 1

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.

Answer 2

You are correct and the Khan answer is wrong. The interest multiplier for six months should be $\sqrt {1.1} \approx 1.0488$. If you apply that twice you get $1.1$ for an annual multiplier as you should.

Answer 3

It is in the formulation:

the interest is compounded annually.

That means you get your interest only after a full year has passed, so the interest received after 4 1/2 years is the same as that after 4 years.

Answer 4

The period is 4.5 years.

We divide the calculation into 2 parts.

Part (A): The first 4 years - We can use the compound interest formula:

Compound Interest =$$ 100000 * (1.1)^{4} = 146,410.00 \tag A$$

Part (B): Treat the remaining 6 months using simple interest formula:

Six Months Interest = $$146410*(0.1)/2= 7,320.50 \tag B$$

The total amount at the end of 4.5 years is $(A)+(B)$:

$$ 146,410.00 + 7,320.50 = 153,730.5$$

Your issue was with:

then calculating interest for the last 6 months manually (0.1/5)

You divided by $5$ not by $2$. We divide by $2$ because its a half year.

Answer 5

If you compute and credit interest annually, you get an annual increment of interest. This is done for some financial instruments.

You can also compute and credit interest monthly. This used to be standard for savings accounts (on the "minimum monthly balance").

You can also compute interest daily (and typically, credit and compound monthly).

You could, now, with computer automation, compute and credit and compound daily, using fractional cents, since the small amount of interest credited daily would be lost if you discarded small amounts.

The difference between compounding daily and hourly is small. For numbers very close to 1, compounding interest is almost exactly the same as average interest. If the interest for a day is "1", then for a half day it is "1/2". In the limit, it becomes meaningless: you have to compound infinitely an infinitely small amount of money. But fortunately, the limit is bounded, and for a normal amount of interest over a year, the bound is very close to "calculated and compounded daily".

For this reason, we can use "compounding continuously" as an approximation for "compounding daily". In the limit, calculating "compounding continuously, to get 10% per annum", over a half year, the equation is $$ 100000(1.1)^{0.5} = 104880.884817 $$, and at the end of the second half of the year you would have 104881*104881= 110000.

Now let's try that compounding daily, at an interest rate of 0.026116 per day: $$ 100000(1.00026116)^{365} = 110000 $$ $$ 100000(1.00026116)^{182} = 104867 $$ $$ 100000(1.00026116)^{182.5} = 104880 $$ ... once you start dealing with days, half days and leap years start to matter, but the result for the half year is close to the result from the "compounding continuously" formula. To be exact, you just have to get the "0.5" years to match an exact number of days: you've got to be more exact than "0.5 years" (Note, the result for the full year is exact, because that's where we started: 10% per annum.)

If you don't match the fractional year to an exact number of days, you've got a fractional day, with the exact same problem you started with, except that difference between the "wrong" result and the "right" result is small. So lets try to see how big the error is:

$$ Interest for one day: 0.026116% $$ $$ Interest for half day (continuous): (0.00026116)^{0.5} = 0.00013057 $$ $$ Interest for half day (daily): (0.00026116)/2 = 0.00013058 $$ On £100000, that's 0.1 pence per half day: if you used the "wrong" formula to compare returns that compounded daily, that's the kind of error you'd get.

Savings banks still do "interest credited monthly" or "interest credited quarterly", and consequently still do "compounding monthly" or "compounding quarterly". "Compounding continuously" is easy to calculate only when the balance is constant. Once the balance starts moving, you have to do a separate interest calculation for every balance, so it's no easier than any other periodically credited account.

"Interest credited daily" is mostly only of value as a comparison rate for other short term investments. Very few financial instruments are actually constructed that way.

For real financial calculations, you have to know the interest rate, interest rate interval, the compounding interval, the date interval calculation (how long is a month?), the payment dates (not necessarily the same as the compounding or date-interval dates), the rounding rules (cents? dollars? lakh?) and balance calculation rules