Problem 1

You have come to meet an old friend of yours at his apartment building and need to park your car. You have two choices. You can park your car in his apartment building and the rate there is $25 fixed for one hour and $75 fixed beyond one hour. There is a 25% probability that you will leave his apartment within an hour and there is a 75% probability that you will stay on beyond an hour. The second option is for you to park your car on the street. If you do so you estimate that there is 25% probability that you will get a $20 ticket from the cops for illegal parking, a 50% probability that you will get a $100 ticket for obstruction of traffic and a 25% probability that you will get away scot-free for the entire duration of the parking. Ignoring the ethics of the problem, which option will be choose purely based on a financial analysis:

 a) park in the apartment building;
 b) park illegally on the street;

Give reasons for your answer.

( Hint: Calculate the probability weighted expected value and also the financial risk (variance) of each strategy and then make a comparison.)

Problem 2

You broker tells you that Stock A, which is currently trading at $25, is a great buy. You estimate that there is a 50% chance of the stock hitting $30 in a year's time and a 50% chance of the stock going down it $20 in a year's time. On the other hand, you friend advises you to buy Stock B which is a dot com stock. It is trading at $20 and once again you estimate that it has a 50% chance of hitting $30 in a year's time but a 25% chance of it going down to $10 and even a 25% chance of the stock tanking at $5. Which stock would you buy:

a) stock A
b)  stock B

Give reasons for your answer.

( Hint: Like Problem #1 calculate the probability weighted expected value and the financial risk and then make a comparison.)

Problem 3

Which of the following is a pure random number sequence:

a) a random number sequence that is generated by a computer program written by you;
b) a random number sequence that is generated by the Microsoft Excel^TM function library;
c) recording the chirruping of birds in a forest for a period of time;
d) the daily price of Dow Jones Industrial Average divided by 1000.

( Hint:Any random number generated by man either using a pen and a paper or a computer program is quasi-random; only God can generate a pure random number sequence. Why?)

Problem 4

The following mathematical identity is given:

Where, A and X are both square matrices (equal number of rows and columns) and is the transpose of A . Then we can say that:

a) A is a square root matrix;
b) A is a power matrix;
c) A is a diagonal matrix;
d) None of the above;

(Hint: A is a square root matrix. Try the operation by an example of simple 2 x 2 matrix.)

Problem 5

A dependent variable Y is perfectly determined by the independent variable X according to the following formula:

Which of the following statements about the correlation between X and Y is true:

a) X and Y are perfectly positively correlated (correlation is close to one);
b) X and Y are positively correlated;
c) X and Y have zero correlation;
d) The correlation between X and Y cannot be determined:

( Hint: Correlation is a linear concept. .)

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