# Estimate compounding

Oliver Wyman case: Setting up a Wine Cellar
New answer on Feb 18, 2022
7.2 k Views

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?

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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

Hi Antonello, does this formula work for any number of years and growth percentage ? Thanks a lot.

Hi Amandine, yes it is the best quick approximation for any number. The more little are the numbers the more accurate is this formula.

Hi Antonello, can you please clarify again how you got the three terms that you use to approximate (1 + 20%)^5? Just tell me how you calculate the first one for example

(edited)

Hi David, it's a classical Math formula. You can find here the theorem: https://en.wikipedia.org/wiki/Binomial_theorem

Good use of Binomial Theorem

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?

(edited)

Hi - here is my understanding of the case. In the "drinking section" - he buys 120 a year and consumes 120 a year, so the net impact is 0. In the "investment section" - he buys 72 a year and sell whatever has appreciated over 5 years. Except for Year 6 when he sells 192 (120 during kickstart + 72 during Year 1), in subsequent years he sells 72 per year. Hope that helps!

Antonello gave the best answer
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