Estimate compounding

Question

Hi,

In one part of the interview, this is a solution:

The interviewee should be able to estimate the fact that $100*(1 + 20%)^5 is something like $250 off the cuff. Without compounding, it’s obviously $200 exactly, so the answer “something a bit more than $200” is good. Many people answer $200 exactly and, when prompted “is it more or less than $200” choose “less”, which is clearly madness.

Three questions:

1) How should I be able to estimate that this is 250$?

2) If I estimate in that the worth in 5 years will be 100$ + 5*20%(100$), how should I know if it the number should be higher or lower than 200$?

3) In this case, would you do the whole math or simplify it?

Answer 1

As quick answer, of course, is great "something a bit more than $200, we can assume $250" (you can say here $240, $230 or anything higher than 200 but lower than 300, use common sense).

No interviewer will ever ask you the exact number, but with a few seconds more you can really crack it. The most relevant addends in the whole compound interest formula (1+p)^n are the first three:

1
n*p
n*(n-1)/2 * p^2

Therefore in our problem we have: 1 + 5*0.2 + 5*4/2 * 0.2^2 = 1 + 1 + 0.4 = 2.4 -> $240

After a little practice, you will find this formula very easy and immediate, but you will impress every interviewer you are going to face with.

Try it in similar problems and let me know :)

Best,

Antonello

Answer 2

Hello,

Here's an easy way to look at this:

Assuming simple interest: Assuming simple interest of 20% for 5 years, that's $100 * 20% = $20 per year => $100 for 5 years

This will be $100+$100 = $200.

Adding the extra bit for compounding:

20% of 20 is 4. Doing that for 4 years is $16 - lets call this A

Doing that for 3 years is $12 - B

Doing that for 2 years is $8 - C

Doing that for 1 year is $4 - D

Now, A + B + C + D = $40

This gives you a ‘simple interest estimate for compounding’ at $40. Thus with true compounding, the answer would be a little over $240.

Answer 3

Adding some thoughts..

1. Try using Rule of 72 as a proxy for CAGR numbers - see how long it takes to double a number and then add / subtract 1-2 years from there.

2. You can just take linear growth and then adjust from there. You know it must be higher, if linear growth already gets you to 200. Compounding effect is stronger, so it can't be less than the 200.

3. For simple examples with “pleasant” growth rates like 10%, 20% or 50% and a relatively short timeframe, can just quickly do it manually. Here what it would look like is: 1) 100 x 1.2 = 120 x 1.2 = 144 x 1.20 = 173 x 1.2 = 207 x 1.2 = 248 → adding 20% increments should be fairly quick

Answer 4

what I don't get is the consumption. why is he drinking just 72 bottles a year after year 5? initially he said 8 bottles to drink and 2 as a gift = 10 bottles a month = 120 bottles consumption a year. but if he buys only 72 a year and consumes 120 the graph should fall….. where do I have a thinking gap?