MAFS 5220 { Quantitative and Statistical Risk Analysis First Test, 2021 Time allowed: 2 hours Instructor: Prof. Y. K. Kwok [points] 1. Recall the time-continuous version of bond price B(i) with interest rate term structure i(0; t), where B(i) = Z T 0 c(t)ei(0;t)t dt; where T is the maturity date and c(t) is the continuous cash
ow rate, 0 t T . (a) Find the corresponding duration of B(i). What is the role of duration in analyzing sensitivity of bond price with respect to
uctuation in the interest rate [1] (b) The future value FH of the bond investment at the chosen investment time horizon H is given by FH(i) = B(i)e i(0;H)H : Suppose the whole term structure receives an uniform increase of , > 0, then B(i+ ) = Z T 0 c(t)e[i(0;t)+]t dt: Note that lnFH(i+ ) = lnB(i+ ) + i(0; H)H + H: The Immunization Theorem states that there exists a horizon H such that the hori- zon rate of return goes through a minimum at i. Explain why the rst order condition for immunization is given by d d lnFH(i+ ) =0 = 0. [1] (c) Find the horizon H that achieves immunization from the above rst order condition. [4] (d) Give the nancial interpretation of the result and its use in immunization. [2] 2. Is bond duration always an increasing function of maturity That is, longer lived bond always has higher duration. Prove or disprove the above statement. If it is not true, state precisely the condition under which longer lived bond has a higher duration. [5] 3. Suppose that each of two investments has 4% chance of a loss of $1 million, 2% chance of a loss of $2 million, and 94% chance of a prot of $10 million. These investment outcomes are independent of each other. (a) Find the VaR for one of the investments when the condence level is 95%. [2] 1 (b) Find the expected shortfall for the one-project investment when the condence level is 95%. [3] (c) What is the VaR for a portfolio consisting of the two investments when the condence level is 95% Does VaR satisfy the subadditivity condition in this case [4] 4. (a) In the extreme value theory formulation, the random loss distribution V is charac- terized by P [V > x] = nu n 1 + x u 1= ; where nu n is obtained by empirical data while the tail distribution is modeled by the Pareto distribution. What are the rationales of constructing P [V > x] using both empirical calibration and modeling by a tting distribution [1] (b) Recall that u is the threshold to be chosen by the modeler. What are the underlying considerations in choosing an appropriate value of u [1] (c) One may calculate VaR at a given condence level using P [V > x]. Alternatively, one may use counting observations, say VaR95% can be identied by the 5 th worst loss among 100 historical scenario events. What are the advantages of using the extreme value theory tting distribution [2] 5. Historical default probability values reported in Moody or Standard & Poor are typically lower than the bond price based default probability values inferred from traded bond prices. Suppose the reported historical 5-year default probability is 5 = 0:57%. On the other hand, the 5-year zero-coupon bond eB0;5 is traded with a credit spread of 50bps. (a) Assume recovery rate to be 40%. Find the 5-year bond price based default probability (denoted by Q5) from the credit spread. [2] (b) We dene Rd be the objective discount rate appropriate for this specic bond so that eB0;5 = (1 5)e5Rd : On the other hand, we recall eB0;5 = (1Q5)e5r: where r is the riskfree interest rate. Use 5 as given in the above data to nd R dr, which is considered as the extra return for holding this risky bond. [2] (c) How would you compare Rd r and credit spread of 50 bps as inferred from the observed bond price Explain the nancial rationales behind the discrepancy. [2] 6. A company has issued a 2-year bond with a coupon of 4% per annum payable annually. The yield on the bond (expressed with continuous compounding) is 6% and the risk-free rate is 3% with continuous compounding. The recovery rate is 20%. Defaults can take place halfway through each year. Find the default intensity, assuming to be constant for the whole 2 years. [6] 2 7. Consider the exponential model of joint defaults with 3 rms. Consider the matrix (aik)35 dened by (aik) = 0@ 1 0 0 1 10 1 0 1 0 0 0 1 0 1 1A ; where the rst 3 shocks are individual idiosyncratic shocks for the 3 rms, the 4th shock aects the rst and second rm, while the 5th shock aects the rst and third rm. The shock k is modeled by the Poisson process Nk with intensity k, k = 1; 2; : : : ; 5. Assume all shocks to be independent. (a) Explain how to nd i, i = 1; 2; 3, analytically, where i is the random default time of rm i. Find the joint survival function of the 3 rms: S(t1; t2; t3) = P [1 > t1; 2 > t2; 3 > t3]: [3] (b) The exponential survival copula associated with the survival function S(t1; t2; t3) is dened by C (u1; u2; u3) = S(S 1 1 (u1); S 1 2 (u2); S 1 3 (u3)); where ti = S 1 i (ui) or ui = Si(ti), i = 1; 2; 3. For xed i and j, explain why the two-dimensional marginal survival copula is given by C (ui; uj) = min uju 1i i ; uiu 1j j ; where i 2 (0; 1) and j 2 (0; 1). Find C (u1; u2) in terms of 1; 2; : : : ; 5. [6] Hint : For a pair of obligors i and j, the joint survival probability is S(ti; tj) = Si(ti)Sj(tj)min(e ti ; etj); where is the intensity of the common shock that knocks down rm i and rm j. (c) Find the default-event correlation coecient between the rst and third rms. [2] (d) How to perform simulation of i, i = 1; 2; 3, via simulation of the random arrival times of the shocks [2] | End | 3
