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Calculate the amount if 20,000 is compounded for 2 years and 4 months @12% p.a.

I try solving this problem but my answer was wrong.

We apply formula, $A = P(1 + R)^n $ $= 20,000(1 + .12)^{2 + 4/12}$ $= 20,000(1.12)^{2.3334}$

Calculating this on a calculator gives around 26055.8 but this is not correct answer. Answer is given to be 26091.52 which is around 35 units greater. I want to know what's the mistake in my work?

I know a different method to solve it, and can solve it on my own, just posting it to know mistake in this method.

  • Does this different method arrive at the 'correct' answer? And what is this method? – Magdiragdag Jun 01 '22 at 11:40
  • @Magdiragdag that's what you have added in your answer. –  Jun 01 '22 at 13:51
  • The simple answer is that the answer is $26091.52$ because in real life that is how a bank would value the account. If you were asking about a purely exponential growth function (deriving from some problem in science or engineering, perhaps) then you would likely get yet another answer. – David K Jun 01 '22 at 16:26
  • (Note on my previous comment: in actual real life I doubt you'll find a bank that compounds interest annually nowadays. But apparently this is how fiinance professors think.) – David K Jun 01 '22 at 23:21

2 Answers2

1

The only way I can arrive at $26091.52$ is giving 12% interest the first two years and then 12%/3 = 4% for the next four months. I don't see how this makes any sense at all, but it is true that $20000 \cdot 1.12^2 \cdot 1.04 = 26091.52$.

The original answer, obtained by $20000 \cdot 1.12^{28/12}$, seems correct to me.

Magdiragdag
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  • I still didn't understand what's wrong with my method –  Jun 01 '22 at 13:52
  • Your method is fine. The method that arrives at 26091.52 makes no sense. – Magdiragdag Jun 01 '22 at 15:21
  • This answer and the method in the question represent a form of continuous compounding such that the annual percentage rate comes out to $12%.$ It is a pure exponential growth curve. However, when a bank offers an account at $12%$ p.a. with annual compounding, they do not use this method. They use a method that in effect is a piecewise linear function. So $26091.52$ is the real-life answer, while $26055.8$ is a mathematical interpretation that has never been used (to my knowledge) in any actual case of an interest-bearing account. – David K Jun 01 '22 at 16:21
  • Yeah 26091.52 is correct answer. –  Jun 01 '22 at 16:48
  • @DavidK I'm quite curious to see how you'd arrive at 26091.52 without the trick of changing the effective interest rate after 2 years. I tried to come up with some way of interpreting the numbers such that continuously compounded interest comes up with that number, or such that monthly compounding comes up with that number, and some other uniform methods, but I didn't manage. I only managed to get to 26091.52 by treating the first two years differently from the last 4 months. – Magdiragdag Jun 01 '22 at 18:01
  • The problem statement says annual compounding. Therefore there is no compounding whatsoever between the start and end of the year. Not continuous, not monthly. The difference between the first two years and the last four months is that the first two years don't need to be pro-rated. They each get the full $12%,$ but the last four months only get $4/12$ of the annual interest, so you have your $1.12\cdot 1.12\cdot 1.04.$ It doesn't make much sense to a pure mathematician but it's how these things are customarily done (and I think there may be legal requirements for it too). – David K Jun 01 '22 at 18:43
  • Funny. So if you'd manage to convice a bank to do this three times in a row, there'd be a difference between 12% per year for 7 years and 3 times (2 years + 4 months at 12% per year). I guess over here banks always do compounding on a daily basis; then this issue doesn't show up. – Magdiragdag Jun 01 '22 at 19:36
  • Even better, go in every 4 months, withdraw the full principal and interest and deposit it in a new account. But I have no idea where one finds a bank that compounds annually nowadays. Maybe for a certificate of deposit, but then they likely charge an early withdrawal fee if you take it out before the full term. So maybe this is not a realistic question at all. I remember banks advertising continuous compounding. – David K Jun 01 '22 at 23:10
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12% of 20000 is 2400 so 2400 interest will be earned the first year. Since this is 'compounded annually' that is added to the principal making 22400.

12% of 22400 is 2688 so the interest for the second year wouold be 2688. But this is only invested for 4 months or 1/3 of the second year so we divide by 3: 2688/3= 896. Adding that to the 22400 we will have 22400+ 896= 23296 at the end of one year and four months.

George Ivey
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