Interview Questions at JPMorgan Sales and Trading

[euroazn]

@Ezra: If we use the X^(1/X) approach, that would suggest the answer is 2 though, always.
Can you please elaborate on your claim that at high t 3 is better? I am quite sure you are wrong...

 

[Stuart Truax]

@Euroazn

With the ($x^{\frac{1}{x}}$ ) approach, I think the logic would be the following:

1) Optimizing ($x^{\frac{1}{x}}$ ) would be the same as optimizing ($\frac{ln(x)}{x}$ ).

2) The first order condition (FOC) ($f'(x)=0$ ) yields ($e$ ) as the optimum. (We can verify it is a maximum through the second order condition).

3) Since the function ($x^{\frac{1}{x}}$ ) is: a) continuous, b)monotonically increasing for (x < e), c) monotonically
decreasing for (x > e), AND d) ($2^{\frac{1}{2}}=4^{\frac{1}{4}}$), it follows that ($3^{\frac{1}{3}}$) must be be greater than ($2^{\frac{1}{2}}$).

Hence the 3x every 3 years is the best option.

 

[Ezra]

@Ezra: If we use the X^(1/X) approach, that would suggest the answer is 2 though, always.
Can you please elaborate on your claim that at high t 3 is better? I am quite sure you are wrong...

I think it is possible we are both trying to answer different questions. I think there are a few different ways to interpret the question as posted. I was trying to answer the following question:

You are offered three different salaries:

1) a salary of x for year 1, where your salary doubles and is paid out every two years.

2) a salary of x for year 1, where your salary triples and is paid out every three years.

3) and a salary of x for year 1 where your salary quadruples and is paid out every 4 years.

Which one would you choose and why.

Now i don't think my interpretation of the question is the most accurate, or what was intended by the question. But that was the question I was trying to answer.

Did you interpret the question similarly?

 

[Ezra]

@Euroazn

With the ($x^{\frac{1}{x}}$ ) approach, I think the logic would be the following:

1) Optimizing ($x^{\frac{1}{x}}$ ) would be the same as optimizing ($\frac{ln(x)}{x}$ ).

2) The first order condition (FOC) ($f'(x)=0$ ) yields ($e$ ) as the optimum. (We can verify it is a maximum through the second order condition).

3) Since the function ($x^{\frac{1}{x}}$ ) is: a) continuous, b)monotonically increasing for (x < e), c) monotonically
decreasing for (x > e), AND d) ($2^{\frac{1}{2}}=4^{\frac{1}{4}}$), it follows that ($3^{\frac{1}{3}}$) must be be greater than ($2^{\frac{1}{2}}$).

Hence the 3x every 3 years is the best option.

This is an example where what would normally be though of as a "technical" question can also function very much like a "fit" question.

If you were interviewing at an S&T desk with a bunch of sports junkies and former college athletes, the above approach, although possibly correct, could signal that you are not an appropriate fit for that desk's culture.

However, if you are interviewing for a prop desks that creates quantitative black boxes to automatically trade exotic credit derivatives (hyperbole ), the above, if correct, would probably be more along the lines of the type of approach they would find more appealing.

So I think Andy put it best by saying its often not about the answer, but about your thought process and approach.

 

[Stuart Truax]

lol, you see. This is an example where what would normally be though of as a "technical" question can also function very much like a "fit" question.

If you were interviewing at an S&T desk with a bunch of sports junkies and former college athletes, the above approach, although possibly correct, could signal that you are not an appropriate fit for that culture.

However, if you are interviewing for a prop desks that creates quantitative black boxes to automatically trade exotic credit derivatives (hyperbole ), the above, if correct, would probably be more along the lines of the type of approach they would find more appealing.

So I think Andy put it best by saying its often not about the answer, but about your thought process and approach.

I totally agree with you. I just like using the LaTex equation functionality of the forum

 

[AlgoCan]

I agree with Ezra and Andy....each question might be interpreted in your way, they want your thought process not the result.

On Question 1: My take is the constant cash flow (mentioned by Ezra): your salary is fixed, say (\$x/year). Instead of receiving it each year you can opt to receive (\$2x/2year) etc. Using the present value calculation, assuming the risk free interest rate is ($r>0$) and continuous compounding:

case 1 (pay stream): ($2xe^{-2r}$), ($2xe^{-4r}$),(....),(${2xe^{-2nr}$), where ($n$) is the final year of contract (for permanent employee it can be considered the expected retirement).

Then as ($n\rightarrow\infty, pay=\sum\limits_{n=1}^{\infty}2xe^{-2nr}=\frac{2x}{1-e^{-2r}}$).

With the same token, case 2 yeilds: ($pay=\frac{3x}{1-e^{-3r}}$)

and case 3: ($pay=\frac{4x}{1-e^{-4r}}$).

We can actually find the interest rate level for which each scenario is optimal. Say for comparing case 1 and 2:

They will be equally good if (2(1-e^{-3r})=3(1-e^{-2r})). Use (v=e^{-r}), then solve (2v^3-3v^2-1=0).


What do you think?

 

[helpteye]

This is ecactly how I did it. The lowest common multiple. Simplest method.

1) I can pay you twice your money every two years, three times your money every three years or four times your money every four years. Which option do you choose and why?

answer..

can we take lcm of 2,3,4= 12

and then @ 2x multiples in 2 years implies 2^6= 64x in 12 years
@3x mltp in 3 years implies= 3^4= 81x in 12 years
@4x mltp in 4 years implies= 4^3=64x in 12 years

so it looks like you get highest returns by choosing second option @3x in 3 years. rest of the options give equal returns but not highest!!

let me know if the logic is correct.

Pramau

 

[Ezra]

I agree with Ezra and Andy....each question might be interpreted in your way, they want your thought process not the result.

On Question 1: My take is the constant cash flow (mentioned by Ezra): your salary is fixed, say (\$x/year). Instead of receiving it each year you can opt to receive (\$2x/2year) etc. Using the present value calculation, assuming the risk free interest rate is ($r>0$) and continuous compounding:

case 1 (pay stream): ($2xe^{-2r}$), ($2xe^{-4r}$),(....),(${2xe^{-2nr}$), where ($n$) is the final year of contract (for permanent employee it can be considered the expected retirement).

Then as ($n\rightarrow\infty, pay=\sum\limits_{n=1}^{\infty}2xe^{-2nr}=\frac{2x}{1-e^{-2r}}$).

With the same token, case 2 yeilds: ($pay=\frac{3x}{1-e^{-3r}}$)

and case 3: ($pay=\frac{4x}{1-e^{-4r}}$).

We can actually find the interest rate level for which each scenario is optimal. Say for comparing case 1 and 2:

They will be equally good if (2(1-e^{-3r})=3(1-e^{-2r})). Use (v=e^{-r}), then solve (2v^3-3v^2-1=0).


What do you think?

From a more intuitive standpoint, all pay the same total payout, except the 2x option is paid out more frequently, if interest rates are positive wouldn't this always be most desirable since we can reinvest earlier and more often? It is like comparing Bonds with different frequencies of coupon payments all else being equal, isn't it?

For example, all else being equal, shouldn't we always prefer the bond whose coupon is paid with highest frequency (i.e. quarterly coupon bond > semiannual coupond bond > annual coupon bond)?

(of course assuming interest rates positive)

And I believe your approach would bear this out. I think you just made an error with regard to the value of a perpetuity assuming continuous compounding, I believe it is (\frac{A}{e^{r}-1})

 

[ThinkDifferent]

first question is just about the time value of money and common sense....of course 2 years should be more preferable.

...after all, considering for example the third alternative, how on earth am I supposed to make a living during my first 4 years of employment???


all these solution with equations etc. shows how really people brains are distorted with all this financial engineering bollocks.....

 

[Ezra]

first question is just about the time value of money and common sense....of course 2 years should be more preferable.

...after all, considering for example the third alternative, how on earth am I supposed to make a living during my first 4 years of employment???


all these solution with equations etc. shows how really people brains are distorted with all this financial engineering bollocks.....

That was a point I was trying subtly (or rather not so subtly) to get at earlier.

 

[AlgoCan]

And I believe your approach would bear this out. I think you just made an error with regard to the value of a perpetuity assuming continuous compounding, I believe it is (\frac{A}{e^{r}-1})

No, I am using the fact:
($\sum\limits_{n=0}^{\infty}x^n=\frac{1}{1-x}$), for ($abs[x]$<1).

Also, contrary to the common sense approach, it turns out that 4x payment is a better answer (see attached graph- The x-axis is the interest rate and the y-axis the total payment. And without loss of generality I assumed annual salary is 1).

BTW, TD, your common sense logic needs to be consistent...YOUR correct answer should be not to accept any of these options all together, how are you suppose to wait 2 years for your first payment? The interviewer will be impressed by your integrity ;-)

 

[pruse]

all these solution with equations etc. shows how really people brains are distorted with all this financial engineering bollocks.....

"bollocks"?? Please!

 

[Ezra]

No, I am using the fact:
($\sum\limits_{n=0}^{\infty}x^n=\frac{1}{1-x}$), for ($abs[x]$<1).

Also, contrary to the common sense approach, it turns out that 4x payment is a better answer (see attached graph- The x-axis is the interest rate and the y-axis the total payment. And without loss of generality I assumed annual salary is 1).

BTW, TD, your common sense logic needs to be consistent...YOUR correct answer should be not to accept any of these options all together, how are you suppose to wait 2 years for your first payment? The interviewer will be impressed by your integrity ;-)

I'm lazy about the math so I'm going to be honest and tell you I didn't think through your infinite sum, but according to wikipedia the PV of a perpetuity with continuous compounding is (\frac{A}{e^r-1})


http://en.wikipedia.org/wiki/Time_value_of_money#Continuous_compounding

If you want we can enter a swap, I'll pay you 4,000,000 every four years, and you can pay me 2,000,000 every two years. Then we can see who makes money

 

[alain]

If you want we can enter a swap, I'll pay you 4,000,000 every four years, and you can pay me 2,000,000 every two years. Then we can see who makes money

the counterparty risk is too high

 

[Ezra]

the counterparty risk is too high

:D This made me laugh.

 

[bob]

Wow, there is a serious lack of common sense prevailing in this thread.

Anyway, I find this question interesting...

4) What happens to pension liabilities when interest rates go up?

...since in some sense all three possible answers--that they decrease, remain the same, and increase--can be argued to be correct.

 

[Joy Pathak]

...since in some sense all three possible answers--that they decrease, remain the same, and increase--can be argued to be correct.

Hey bob can you explain why all three work?
Pension liabilities are very sensitive to interest rates. If interest rates fall the present value of future pension payments rise.Atleast I would assume. I don't know much about pension accounting, but that would sort of make sense right?

 

[tylor]

1.) i would choose twice every two years. i read this as my salary is x, and my options are 2x, 3x, and 4x. therefore i would choose 2x due to time-value-of-money, risk of inflation, etc.

2.) $50? 50/50 so should be 2 to 1?

3.) small hand to big hand.

4.) no idea

5.) short JPMorgan stock due to inability to realize extreme talent.

6.) none

 

[bob]

Hey bob can you explain why all three work?
Pension liabilities are very sensitive to interest rates. If interest rates fall the present value of future pension payments rise.Atleast I would assume. I don't know much about pension accounting, but that would sort of make sense right?

Defined-benefit pension schemes are essentially short a fixed-income instrument of very long duration. So, from a mark-to-market perspective, an increase in interest rates causes the NPV of the liabilities to decrease.

Moreover, since usually the duration of the liabilities far exceeds the duration of the assets, an interest-rate increase can look like a favorable move to the pension fund, since it makes the fund less underfunded. (Sad, but true: Virtually all pensions in the West have liabilities whose PV far exceeds the PV of their assets.)

But...mark-to-market isn't the only measure--or even the most important measure--of the pension's viability. The thing that really matters to them is cash flow. An increase in interest rates can push funding gaps further down the road, since it increases the rate of return on reinvested asset cash flows, but it doesn't actually change the liability cash flows themselves. So, in real sense, interest rates are irrelevant to the liability side of the balance sheet; no matter what the rates are, the promised payment is the same.

And then again...interest rates don't usually go up in a vacuum. Most often, an increase in the nominal interest rate is accompanied by an increase in inflation expectations. Many pension funds' liabilities are inflation-linked in the form of cost-of-living adjustments. An increase in inflation expectation means that the liability cash flows are likely to actually grow when interest rates rise.

 

[euroazn]

@tylor

2.) $50? 50/50 so should be 2 to 1?

This isn't adequate enough.
Suppose instead of 100 dollars, the interviewer offered to double your net worth if you win or take you all your money if you lose. According to your logic, that is fair. However, no person would take that - a) because they're risk averse and b) the marginal utility of money decreases.