Problem #1

The richest man on earth (not Mr. Bill Gates in this case), a multi-billionaire, is totally in love with his beautiful, young wife. He asks her what she wants as gift on her birthday - diamonds, a beach house, sports car, whatever she wants. He promises to give her all his wealth. However she asks him for a very modest gift. She tells him that she wants a dollar today (on her birthday), twice that amount (two dollars) the next month, twice that amount (four dollars) on the third month, twice that amount (eight dollars) on the fourth month and so on. She wants to start off with one dollar and keep doubling the amount every consecutive month. The billionaire agrees to that gift in writing and gives her a dollar on her birthday. By the end of five years:

1. the billionaire would be madly in love with his beautiful and humble wife;
2. the billionaire will realize that she is indeed very intelligent;
3. the billionaire will become bankrupt;
4. the billionaire will become bankrupt and his wife would become the richest person on earth;

( Hint : This example is that of a simple geometric progression and exponentials.)

Problem #2

A correlation matrix M between three random variables, x, y and z is related to the following lower triangular matrix:

1. Is it possible to deduce the correlation between the variables y and z? If so, then what is the correlation between the variables y and z and if not, then why not?
2. What is the relationship between the above lower triangular matrix C and the correlation matrix between the variables M?

( Hint : Yes, it is possible to get only an approximate value of the correlation between the two variables, if the above matrix is mathematically related to the correlation matrix. Correlation matrix is symmetric and the above matrix is lower triangular therefore the transpose of the above matrix times the matrix itself should yield an approximate correlation matrix, which may be mathematically inconsistent. )

Problem #3

A company is either solvent (can go on operating and pay its bills) or bankrupt (has no money left to pay its bills and operate any longer). Solvency is represented as State x where the company has value 1 and Bankruptcy is represented by State y where the company has value zero. If J is a Boolean set such that then which of the following mathematical sets represents J accurately?

( Hint : The set that represents J should yield numerically )

Problem #4

A stock is currently trading at $15. There is a financial contract on this stock which will pay off (generate a positive return on your investment if you buy this contract) if the stock price ever reaches $100 within a year's time. Which of the following statements is true regarding the price of this financial contract?

1. the financial contract will have some value more than zero today;
2. the financial contract will be worth zero today because the terminal value of $100 is too far away from the current price of $15;

Explain your answer.

( Hint : The above problem can be compared to that of heat conduction. If you place a body close to that of hot body there will be heat transfer. Even if the body is placed at a great distance there will always be some amount of heat that will be transferred to the new body. If there is a source of profit then the regions surrounding it should account for it; and even regions far from the source of profit should have non-zero value to account for the source of profit. One can look at it from a probabilistic angle as well. Now matter how far the target price of the stock is from the current price, there is always a finite probability, howsoever small, that the stock will touch the target price in the given period. Therefore, due to that very small finite probability the financial contract will have some value more than zero. )

Problem #5

You are driving a small car loaded with supplies in torrential rain. The car can just accommodate one other person besides you. At a junction you come across three people standing waving for a lift. One is your closest childhood friend who desperately needs to get to office for an urgent meeting; the second is your fianc?e/fianc? and would-be spouse all wet and drenched and very scared and the third person is a pregnant lady who is in extreme pain and in need of urgent medical attention. What would you do?

1. give a ride to your childhood friend so that he can save his job;
2. give a ride to your fianc?e/fianc? and save your marriage from a rocky start;
3. give a ride to the pregnant lady and take her to the hospital because that is the right thing to do;
4. there is a fourth solution;

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