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# Strats and Quants Interview Questions

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Thought it would be good to have a bunch of brainteasers that bulge brackets like asking at interviews. Mainly applies to quants, strats and S&T but I know they ask these in some tech interviews as well.

Here are a couple:

You have two non-identical non-uniform ropes. Each rope takes exactly one hour to burn from one end to the other. Using just these two ropes and fire, how can you time exactly 45 minutes?

When the minute hand and the hour hand cross over for the first time after midday, what angle do they make with the vertical? (give an exact solution)

Pic not related
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>>991379
Idk. Off the top of my head the second one is 32 degree angle. Amirite?
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>>991385
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>>991379

Measure the distance of the rope burning for the first 15 min. Take that distance and put the fire out when the length matches on the other side
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>>991379
Why do they ask Quants these type of questions during interviews when they'll only be solving partial differential equations and modelling stuff in MATLAB?

Here's another one:

A certain stock is worth a 100 dollars on day one. On any given day, the stock can go up with 30%, or down with 25%, both with the same probability. What's the stock worth after a 1000 days?
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>>991398
Oh. Exact exact? To the second? Something like 32.083333 degrees. I think.
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>>991400
You only have the ropes and fire so there's no external way of knowing when fifteen minutes has passed.
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>>991404
I think they ask it to see general problem solving skills and the ability to think outside the box. Also, I specifically picked two that are different but they usually have some probability involved, so a lot more relevant.

As for your question. Increase by 30% is same as multiplying by 1.3. Decrease by 25% is same as multiplying by 0.75. These are equally likely so the expected value is a multiplication by 1.025. So the value after 1000 days is 100*(1.025^1000) which is a stupidly large number. About 5x10^12. Am I right?
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>>991409
So you can't just fold the rope in half? Then fold it again? To mark 15 minute points?
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>>991405
Here's an easy way to get the solution. The minute hand crosses the hour hand at about 1:05, 2:10, 3:15,..., 9:50, 10:55 and it would at 11:60 but that's 12:00. So in a 12 hour period they cross 11 times. And it goes through 360 degrees so the exact answer is 360/11.
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>>991415
From the question, the rope is non uniform. So what is half of the rope visually may not burn at the same rate. You're kind of along the right lines though.
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Fold a rope in half, then again, put another rope over that spot and burn the first so it burns through 3/4 of itself before setting the second on fire. Should be roughly 45 min...

They will meet at 13.05.05 i guess, so that would be like 30.5 degrees from the vertical...

Feeling like I'm wrong in both lol
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>>991414
You're wrong

Let R_i either be the 1.3 increase or 0.75 decrease on day i.

Then the price on day i is:

R_1 x R_2 x .... x R_i x 100

Because every day the price is the price from the last day, times the increase or decrease of today.

Because these appear about equally we get

0.75 x 1.3 x 0.75 x 1.3 .... x 100

0.75 x 1.3 = 0.975 (does it?)

So we get

0.975^500 x 100 equals approx. 0
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>>991420
I answered the clock question here (>>991416).

Here's the answer to the rope one:

You burn one rope from one end and the other rope from both ends. The one you burn from both ends will burn out in exactly half an hour and this means the first rope is burnt halfway through. Now we light the first rope from the other end which takes fifteen minutes.
Half an hour + fifteen minutes = 45 minutes
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>>991421
Oh of course! That gives a sensible answer as well. Thanks for showing the working as well
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>>991424
Huh, guess I've overthought the clock problem thinking of a second - exact answer..

Thanks a lot for both of these problems! The rope one is really nice and interesting :)
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>>991379
Take two ropes. Burn one from both ends and the other from one end. When the one that burns from both ends has burned completely, half an hour has elapsed and the other rope is half burned. At this point burn the other rope at the other end. It takes another 15 minutes for it to burn. 45 minutes.

Clock one happens after 1 o clock. The minute hand makes an angle 6x degrees where x is the number of minutes after 1 o clock. The hour hand makes an angle of 30(1 + x/60). We want these to be equal:
30(1+x/60) = 6x
30(60+x) = 360x

1800 + 30x = 360x
1800 = 330x
x = 5.454545 minutes

So the angle is 32.7272

>>991404
The expected value of the stock after 1 day is 130 + 75 divided by 2, which is 102.5

The answer is thus 100 x (1.025)^1000
T. somebody who has done a lot of these quant interviews and been accepted
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>>991421
According to this logic, the expected value after 2 days would be 97.5.

According to mine ( >>991431 ), it would be 100 x 1.025^2

Let's examine that.

After 2 days there are two probabilities:

2 increases (P = 0.25, price = 169)
2 decreases (P = 0.25, price = 56.25)
1 increase, 1 decrease (P = 0.5, price = 97.5)

0.25(169) + 0.25(56.25) + 0.5(97.5) = 105.0625 = 100 x 1.025^2

This should help demonstrate why the expected value is 100 x 1.025^1000. What you are looking at is the modal average, mine is the mean, which is what is being asked.
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>>991440
>After 2 days there are two probabilities:

3 probabilities
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>>991431
You got the clock one right but there is a much more elegant solution.

The first time the hands crossover is about 1:05, then about 2:10 and so on until 10:55. At 11 they never cross because it becomes 12:00. So in one circle it crosses 11 times so the answer is simply 360/11 which is the same as the answer you got.
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>>991451
Nice.
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>>991440
Nope. Here is a more rigorous proof written by someone on /sci/, also in the thread some dude made a simulation in MATLAB and it went to zero really fast.

Here's the thread if you're interested:

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>>991457
No, you're wrong. You don't understand the mathematics in that.

What that image shows is that as n goes to infinity, the expected value (mean) goes to infinity, while it converges "almost everywhere" to 0.

This is an interesting mathematical fact, however it has nothing to do with the question at hand.

The expected value is 100 x 1.025^1000.

Expected value is what is generally assumed to be what is being asked in questions like this. And the expected value IS 100 x 1.025^1000

Your answer 100*(0.975^500) is the single most likely price after 100 days... also known as the modal average (the "mode")... but unless otherwise stated there is no reason to assume the question is asking for the mode.
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>>991464
>>991457
>>991404

Essentially the question is asked poorly. The question is "What's the stock worth after a 1000 days?". In the vast majority of cases this can be assumed to mean "What's the expected value". If you want to interpret the question "What's the stock worth after a 1000 days?", the only correct answer is "I don't know" because the stock price is dependent on several random variables.

It's silly to assume the question is asking for the mode though.
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>>991464
>Your answer 100*(0.975^500) is the single most likely price after 100 days...

Well that was the question
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>>991465
I agree, but if you read the thread on /sci/ the right answer ended up being zero.
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>>991465
If you want to interpret the question literally*

>>991466
No it wasn't, the question was "What's the stock worth after 1000 days", see: >>991404

There are a number of ways to interpret this:

-Literally, what is the stock worth after 1000 days? Answer: I don't know and neither do you. There are several possible answers to the question.

-Most likely, non trivial interpretation: What is the expected value? Answer: 100 x 1.025^1000
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>>991424
If the rope was non uniform than half the rope burning doesnt mean shit as you stated before
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>>991467
The right answer to a question that wasn't asked (the mode), sure.
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>>991471
If someone put a gun to your head right now and asked what's most likely the price of the stock after a thousand days, what would you answer?
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>>991473
The most likely price of the stock (aka the modal average) is approximately 0, but that's not what was asked:

>A certain stock is worth a 100 dollars on day one. On any given day, the stock can go up with 30%, or down with 25%, both with the same probability. What's the stock worth after a 1000 days?

>What's the stock worth after a 1000 days?

That doesn't say "What is the most likely price"
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>>991474
I don't see how these are two different things. I'm not a native speaker so they are literally the same thing if you ask me.
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>>991474
>>991473
And another thing, Almost everywhere convergence isn't even close to being true at n = 1000. With 515 positive changes and 485 negative changes the overall price of the stock increases.
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>>991475
They aren't the same thing.

>I'm not a native speaker

Maybe that's the problem.
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>>991470
It does. Half the rope by dimension doesn't mean anything since dimensions are arbitary (non-uniform). But since we are given that it takes an hour to burn through, we know that burning from both ends will take half an hour.
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>>991379
Heard on The Street: Quantitative Questions from Wall Street Job Interviews - Timothy Crack

Frequently Asked Questions in Quantitative Finance - Paul Wilmott

Quant Job Interview Questions And Answers - Mark Joshi, Nick Denson, Andrew Downes

A Practical Guide To Quantitative Finance Interviews - Xinfeng Zhou

Starting Your Career as a Wall Street Quant: A Practical, No-BS Guide to Getting
a Job in Quantitative Finance - Brett Jiu

Cracking the Coding Interview: 150 Programming Questions and Solutions - Gayle McDowell
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>>991379
That's pretty easy. Bend one of them in half, then attach to the other one. This way it'll take 1h 30m for it to burn completely. Once it burns half way, you have your 45 min.