Problem 1

A parabola is given by the following equation:

If represents the value of where the function has a minima and if , and are constants then the solution to the above equation can be written as:

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

(Note : This function is quite often used in quantitative finance, especially in the area of market risk management and modelling local volatility surface. The minima of the function will be at a point where the first derivative of a function, slope, is equal to zero )

Problem 2

A utility function of an investor is a twice differentiable function of his wealth w such that it satisfies non-satiation (first mathematical derivative of the utility function is greater than zero) and risk aversion (the second derivative of the utility function is less than zero). Which of the following is NOT a very robust (well defined) utility function?

1. 
2. 
3. 
4. , k is a constant.

( Note: Utility functions are widely used in portfolio optimization problems ).

Problem 3

A function is given by: . Then the value of the ratio is:

1. 0
2. 1
3. -1
4. None of the above.

Problem 4

The nature of utility functions of any investor has to satisfy these two conditions: it has to be necessarily increasing in levels of wealth ("more is better" which is that the first derivative has to be greater than zero) and risk aversion (the function has to be concave). With these definitions and with the level of wealth being represented by what restrictions will a quadratic utility function of the type face?

1. the function becomes discontinuous at a certain level of wealth ;
2. the function does not allow the level of wealth to exceed some fixed threshold;
3. the function does not satisfy the risk aversion condition;
4. None of the above.

Problem 5

An investor allocates funds between two assets using a simple rule. He chooses two assets, A and B and starts off by putting 50% of his wealth in A and 50% of his wealth in B.

His rule is simple: over a certain period (say, a week)

1.  if stock A goes up by 5% or more and stock B does not change (go up or down) by more than 5% then he takes out 5% or more from his allocation to B and allocates that amount to A;
2.  if stock B goes up by 5% or more and stock A does not move (go up or down) by more than 5% then he takes out 5% or more from his allocation to A and allocates that amount to B;
3.  if both stocks A and B don't move by more than 5% then he does not change is asset allocation mix;
4.  if one stock goes up by 5% or more and the other goes down by 5% or more then he does not change his asset allocation mix;

Suddenly, over weekly periods, the market displays large movements of 15% or more. Should the investor change his asset allocation rule? If yes, write the modified set of rules.

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