36Answers:1. B.The yield spread is the "singlepoint" difference between the yield to maturity (YTM)and the corresponding riskless spot rate; in this case, yield spread ~= 4.0% 1.61% ~=2.39%. As the endpoints at year 2.0 are near to each other, this yield is only slightly lessthan the zspread of 2.42%, due primarily to the fact that the upper blue line endpoint mustnecessarily be slightly greater than the 4.0% YTM. If the corresponding riskless swap rate isnecessarily interpolated, then this yield spread is an ispread.In regard to (A), (C), and (D), each are TRUE.2. C. $151.16We shock the zspread by one basis point, from 3.00% to 2.99%:$5.00*exp[(1.0%+2.99%)*0.5] + $5.00*exp[(2.0%+2.99%)*1.5] + $105.00*exp[(3.0%+2.99%)*1.5] = $105.6350,which is a price increase of about $0.01512 and per $1.0 million of par = $0.01512 *1,000,000/100 = $151.162.Note, per Malz, we can be more precise by shocking + 0.5 basis point and 0.5 basis point,but the result is nearby at $151.15 (due to convexity, including the upyield shock implies aslightly lower DVCS if we are more precise)3. C. 39.35%Lambda = 6 defaults/1 year = 0.5 defaults per month.The "waiting time" probability is characterized by the exponential distribution:P[next default within one month] = 1 EXP(lambda*T), where T =1 is given by 1 EXP(0.5*1) = 1 EXP(0.5) = 39.35%P[no default within one next month] = EXP(lambda*T) = EXP(0.5*1) = EXP(0.5) = 60.65%Note equivalently, we can view lambda as 6 defaults per year, with T = 1/12, such that:P[next default within one month] = 1 EXP(6*1/12).4. D.The conditional oneyear PD is equal to 8.6%, same as the unconditional oneyeardefault probability, and near to the hazard rate (9.0%) since the hazard rate is aninstantaneous conditional PD.To paraphrase Malz: the difference between the two and oneyear default probabilities—theprobability of the joint event of survival through the first year and default in the second—is7.87%.The conditional oneyear default probability, given survival through the first year, is thedifference between the two probabilities (7.87%), divided by the oneyear survival probabilityof exp(9.0%*1) = 91.39% = 1 8.61%. So the conditional oneyear PD = 7.87%/(18.61%) =8.61%, which in this example is equal to the unconditional oneyear default probability.In regard to (A), (B) and (C), each are TRUE.In regard to (A), the unconditional oneyear default probability = 1 exp(9.0%*1) =8.61%.In regard to (B), the unconditional twoyear default probability = 1 exp(9.0%*2) =16.47%.In regard to (C), the probability of joint event of survival through the first year anddefault in the second year = 16.47% 8.61% = 7.87%.
