FE Problem Set #14
Team Latte September 13, 2006
Problem #1
A slice of a VBA code is given below which uses a Do....Loop without any condition in the Do or Loop line.
Write the same slice of code with a While condition attached to the Do. How will the condition change if we have to write the loop with Until instead of While .
Problem #2
An investor wants to hold a portfolio of three stocks Troubled Motors, TopCom and BioSoft. Troubled Motors has an expected return of 7% and an annualized historical volatility of 18%. TopCom's expected return and volatility are 9% and 7% and BioSoft's expected return and volatility are 11% and 8%. The correlation matrix (of historical returns) of the three stocks (in the order of Troubled Motors, TopCom and BioSoft) is:
The risk free rate is 4%. If short selling is allowed the what is the optimum allocation (in percentage) that the investor should make to the stocks such that the portfolio lies on an efficient frontier? What would be the slope of the straight line representing the efficient set?
( Hint : One of the ways to construct an efficient frontier is to maximize the Sharpe ratio. The Sharpe ratio for the above problem is given by: . and you need to follow a simple maximization of the above objective function using a lagrangian multiplier and the usual constraints of portfolio weights, i.e. the sum of the weights is equal to one. The above problem can be reduced to a matrix equation involving the variancecovariance matrix of the form shown below:
Where all the above parameters are given in the problem and one needs to solve of the portfolio weights and the relationship between and is given by . Therefore, once we solve for the values of we can estimate the optimal portfolio weights
from them. This problem can be very easily solved using an Excel spreadsheet .)
Problem #3
The newly appointed head of market risk of a bank wants to set VaR limits of the trading desk. He wants to start from a blank sheet and disregard the limits set by his predecessor. The trading desk can be broadly split into two business units, A and B with an average correlation of 0.65. The total capital available to the trading desk is $1,000 and the bank follows the 1996 Basel recommendation to hold capital equal to the three times the square root of ten times the 99% one day VaR. The risk manager further estimates that business unit A is expected to return $2.5 per dollar of VaR and the business unit B is expected to return $1.8 per dollar of VaR.
Given the above estimates by the risk manager wants to set the Dollar VaR limits for business units A and B such that the total revenue is maximized. You are required to assist the risk manager in doing so.
( Hint : First calculate the one day 99% VaR limit from the available capital and the Basel formula. Then using the quadratic portfolio VaR equation express VaR of business unit A in terms of the business unit B, and this can be easily done because the correlation is given. Then find out the corresponding revenue for both the business units for different values of VaRs of each unit such that the total VaR is equal to that estimated above. Using interpolation find out the maximum revenue and the VaRs corresponding to that revenue are the required limits for individual VaRs for A and B .)
Problem #4
You are trying to model the worst case performance of a portfolio of vanilla (European style) options all written on one single underlying asset. The portfolio delta is given by and the portfolio gamma is given by . What will be the worst case scenario for the portfolio change in terms of the delta and the gamma?
( Hint : Expand in Taylor Series and take the minimum.)
Problem #5
A five year US government bond (zero coupon) has a principal of $1,000 and a current value of $825.25. A similar 5 year AAA corporate bond (also zero coupon) with the same face value has a current market value of $785. What do these numbers tell us about the risk of default of the AAA corporate bond?
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