You have issued a $5-$year CDO which is based on ten equally$-$weighted companies which you assumeto have a probability of default of $15\%$ p$.$a$.$ The CDO was split into three tranches, the junior trancheconsisting of $70\%$ of the CDO, a mezzanine tranche of $20\%$ and the senior tranche of $10\%.$ Over thelast four years five companies have defaulted. You want to value the mezzanine tranche of this CDOat the beginning of its final year, assuming that all defaults occur in the middle of the year. The currentrisk free rate is $1\%$ p$.$a$.$

Q$4.$Briefly answer the following questions:

a$)$ Outline the basic idea behind the valuation of credit derivatives.

b$)$ Explain how the spread of collaterized debt obligations depends on the various parametersused in its theoretical valuation model.