Problem 1

Which of the following is an example of "ergodicity"?

1. A fund manager making exceptional profits for a few years and then over a period of ten years he incurs huge losses and by the end of tenth year closes his shop;
2. You go to a casino and in only one bet of one hundred thousand dollars who make ten times that amount;
3. Price of a stock index, which has been flat for a month suddenly making jump of 500 points in reaction to some news event;
4. None of the above;

( Hint : Ergodicity essentially implies that over time, generally a long period of time, randomness gets eliminated from a time series* )

Problem 2

The model validation group in a bank has validated a derivatives pricing model and found that it has 5% failure rate (i.e. the model gives a wrong value of a derivative as compared to the market value). The trading team of the bank uses this model on 1,000 weekly trades (derivatives contracts) in its book and finds that the model affects 1/1,000 trades in the book (gives a wrong value). The risk manager on a given week picks up a trade at random to check the value. What is the probability that the model will give a wrong value for the chosen trade?

1. 0.1%
2. 2.0%
3. 5.0%
4. None of the above;

( Hint : You need to estimate the conditional probability that the model gives a wrong value for a trade chosen at random** .)

Problem 3

A matrix J is an n x m square matrix. X is another square matrix of eigenvectors of J and L is a diagonal matrix of the eigenvalues of J . Then the square root of the matrix J will be given by:

1. 
2. 
3. 
4. None of the above;

Problem 4

Which of the following is the only valid correlation matrix between three variables:

1. 
2. 
3. 
4. None of the above;

Problem 5

The newly appointed CEO of a loss making manufacturing company cuts costs by dramatically reducing labour costs (firing workers and employees) and salary costs. After a year the company makes a profit. What can be said about the CEO's skills and abilities and why?

1. It can be said that the CEO is a very bright strategist and the profitability of the company is directly and totally attributable to the CEO's strategy;
2. The CEO's strategy may or may not have had a direct relationship with the profitability of the company and therefore nothing can be said about the CEO's skills and abilities from this single strategy.

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*For a simple, yet brilliant, exposition of the concept of ergodicity see Nassim Taleb's Fooled by Randomness;
** This question is adapted from a famous quiz given to a set of medical doctors referenced in Deborah Bennett's Randomness which is quoted in Taleb's Fooled by Randomness.

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