.1 To fund an expansion in its operations, a company has just issued 5-year zero-coupon bonds with a total face value of f10 million, taking its total asset value up to f15 million. (i) Explain how the value of the bonds can be expressed in terms of a European put option. [3] (ii) Hence calculate the fair price of a holding of the company bonds with a face value of E100 using the Black-Scholes model, given that the price of a 5-year zero-coupon government bond is $77.88. Assume that the annualised volatility of the company's assets over the 5-year period is 25%. [4] (ili) Explain what is meant by a credit spread and calculate its value for the company bonds. [3] [Total 10] 2 Company X has just issued some 5-year zero-coupon bonds. A continuous-time two-state model tyle is to be used to model the status of the company and to calculate the fair price of the bonds. It is believed that the risk-neutral transition rate for failure of the company is 1 (t ) =0.002t , where t is the time in years since the issue of the bonds. The 5-year risk-free spot yield is 5.25% expressed as an annual effective rate. Calculate the risk-neutral probability that the company will have failed by the end of 5 years. [2] (ii) In the event of failure of the company, the bonds will make a reduced payment at the maturity date. The recovery rate for a payment due at time t is: 6(t) =1-0.05t Calculate the fair price to pay for E100 nominal of a Company X bond, taking into account the possibility of company failure. [3] (iii) An analyst is concerned that the estimate of 1 (t ) may be too simplistic. Explain the possible reasons for his concern and how the model could be developed to deal with this. [3] [Total 8] 3 (i) Explain what is meant by a default-free bond. (ii) State the possible outcomes of a default. (iii) List five types of credit event. (iv) Explain what is meant by the recovery rate for a bond.4 Claims occur on a portfolio of insurance policies according to a Poisson process with Poisson parameter 2. Claim amounts, X1, X2...., are assumed to be identically distributed with moment generating function My(t) . The insurer calculates premiums using a loading factor @ (>0). The insurer's adjustment coefficient, R , is defined to be the smallest positive root of the equation: A + er = AMx(r) where c is the insurer's premium income rate. (i) Using the above equation for R, or otherwise, show that, provided R is small, an approximation to R is R , where: R _ 2(c / 1-/) where #=E[X;] and o = var[X; ]. [4] (ii) Describe how the adjustment coefficient can be used to assess reinsurance arrangements on the basis of security. [3] (iii) The Poisson parameter, 1, for this portfolio is 20 and all individual claims are for a fixed amount of $5,000. The insurer's premium loading factor, #, is 0.15 and proportional reinsurance can be purchased from a reinsurer who calculates premiums using a loading factor of 0.25. Calculate the maximum proportion of each claim that could be reinsured so that the insurer's security, measured by R , is greater than the insurer's security without reinsurance. [9] [Total 16]
