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 FE Problem Set #11 Team LatteJul 05, 2006 Problem #1 Consider a three name portfolio A, B and C. The default time correlation matrix between the three names (assets) is given as follows: Is the above a valid correlation matrix? Explain. (Hint : Look at the correlation between the second and the third asset and see if it fits. Also, try to decompose the correlation matrix into a cholesky matrix by using the equation , where C is the correlation matrix and A is the cholesky matrix. Or else, find out the determinant of the matrix, see what happens. ) Problem #2 Two names (assets) are 100% correlated with each other. Name A has 75 basis points CDS spread and name B has 140 basis points spread. Both the names have the same expected recovery rate of 40% but there is uncertainty (volatility) in the recovery rate of 30%. Assuming three different recovery rates (to accommodate for the volatility of recovery rate) of 0%, 40% and 90% estimate the first-to-default and second-to-default premiums for the basket. (Hint: Set up two equations in two unknowns for the recovery rate  one for the mean and one for the volatility (square of the volatility)  and estimate the respective probabilities for each of the states of recovery rate, i.e. p(0%), p(40%) and p(90%). After that the problem becomes trivial .) Problem #3 Assume a 5 year CDO structured on a portfolio of five reference assets (names), each with a principal value (debt) of \$1 million and a recovery rate of 30%. A credit structurer wants value this portfolio, not using a Copula (default time) approach but structural approach using Merton's model. He looks up the balance sheet of all the five names and finds that the asset value (on the balance sheet) of each name is \$1.5 million. Further, the asset volatility of each name is 25% and the risk free rate of interest in the economy is 5%. The CDO is split into three tranches, equity, mezzanine and senior with a notional amount of \$1 million, \$1 million and \$5 million respectively. The structurer is trying to estimate the fair spread on each tranche (premium). Should he use a closed form solution or rather take the Monte Carlos simulation approach? If he takes the MC simulation approach then how should he proceed with the problem? Outline the algorithm and implement it in Excel spreadsheet. What are the advantages and disadvantages of this approach over the copula (default time) approach? According to you, which method should give a higher spread on the tranches. (Hint: Structural models are generally not used in CDO valuation by practitioners, though if applied properly with relevant and robust input parameters such as balance sheet data and asset volatilities, it can be a very simple and yet powerful technique to value CDOs. The problem is with the data quality of the companies. Secondly, the Copula approach (default time modelling) has become the industry standard for CDO valuation since David Li's publication of the seminal paper in the year 2000. These two approaches are vastly different since one follows the stochastic asset path and then estimates default using balance data and the other models default time of the company .) Problem #4 A fund manager is directed by the plan sponsors to allocate \$1 million between stocks of a company A and the Convertible Bonds (CBs) on that company such that the expected return on the portfolio in one year's time is 10%. The stocks have a transaction cost of 125 basis points and an expected return of 12% whereas the CBs have a transaction cost of 85 basis points and an expected return of 6%. The one year portfolio allocation will be: 11% in stocks and 89% in CBs; 23% in stocks and 77% in CBs; 43% in stocks and 57% in CBs; 63% in stocks and 37% in CBs; Explain your answer. Problem #5 The retail division of a bank in a business day will make a large number of cash payments and receipts whereby customers will be accessing their savings and checking accounts. These will ultimately flow down to the cash desk of the Treasury (of the bank) who will have to fund the net payments or lend out the net receipts. Since the treasury will have to keep its books square within limits it needs to decide how much money will be borrowed or lent and for what period of time. The remaining balances are invariably sourced into the bank via the overnight market. The overnight rates fluctuates quite a bit from day to day. What kind of a swap is suitable for such situations? Explain with and example. What is the relationship between a log contract (the natural logarithm of the ration of the terminal value of an asset to the current spot value), a forward contract and put and call options? In which kind of a swap is this equivalence relation used for pricing? Any comments and queries can be sent through our web-based form. More on FE Problem >> back to top More from FE Problems More from Articles Searching for the Most Beautiful Equation in Finance Where does a Black Swan Come from? Napoleon on Wall Street: Advent of the Stochastic Volatility Models The Remarkable Power of the Monte Carlo Method Mean Reversion and the Half-Life of Interest Rates 