问一道题：NO.PZ2018123101000109 [ CFA II ]

NO.PZ2018123101000109 问题如下 Lebeva asks Kowalski to analyze a three-yebon issueVraiRive S.A., using arbitrage-free framework. The bons coupon rate is 5%, with interest paiannually ana pvalue of 100. In her analysis, she makes the following three assumptions:■ The annuinterest rate volatility is 10%.■ The recovery rate is one-thirof the exposure eaperio■ The hazarrate, or contionprobability of fault eayear, is 2.00%.Selecteinformation on benchmark government bon for the VraiRive bonis presentein Exhibit 2, anthe relevant binomiinterest rate tree is presentein Exhibit 3.Baseon Kowalski’s assumptions anExhibits 2 an3, the cret spreon the VraiRive bonis closest to: A.0.6949%. B.0.9388%. C.1.4082%. C is correct. The cret spreccalculatein three steps:Step 1 Estimate the value of the three-yeVraiRive bonassuming no fault. Baseon Kowalski’s assumptions anExhibits 2 an3, the value of the three-yeVraiRive bonassuming no fault is 100.0000.Supporting calculations:The bonvalue in eano is the value of next perios cash flows scountethe forwarrate. For the three nos on te 2, the bonvalues are follows:105/1.081823 = 97.0584.105/1.066991 = 98.4076.105/1.054848 = 99.5404.For the two nos on te 1, the two bonvalues are follows:[0.5 × (97.0584) + 0.5 × (98.4076) + 5.00]/1.060139 = 96.9052.[0.5 × (98.4076) + 0.5 × (99.5404) + 5.00]/1.049238 = 99.0948.Finally, for the no on te 0, the bonvalue is[0.5 × (96.9052) + 0.5 × (99.0948) + 5.00]/1.030000 = 100.0000.Therefore, the VNfor the VraiRive bonis 100.0000.Step 2 Calculate the cret valuation austment (CVA), anthen subtrathe CVA from the VNfrom Step 1 to establish the fair value of the bon The CVA equals the sum of the present values of eayear’s expecteloss anis calculatefollows:Supporting calculations:The expecteexposures eate are the bonvalues eano, weightetheir risk-neutrprobabilities, plus the coupon payment:te 1: 0.5 × (96.9052) + 0.5 × (99.0948) + 5.00 = 103.0000.te 2: 0.25 × (97.0584) + 0.5 × (98.4076) + 0.25 × (99.5404) + 5.00 = 103.3535.te 3: 105.0000The loss given fault (LG on eate is 2/3 of the expecteexposure.The probability of fault (PO on eate is follows:te 1: 2%te 2: 2% × (100% – 2%) = 1.96%.te 3: 2% × (100% – 2%)2 = 1.9208%.The scount factor on eate is 1/(1 + spot rate for the te) raiseto the correpower.Finally, the cret valuation austment eayeis the proof the LGtimes the POtimes the scount factor, shown in the last column of the table. The sum of the three annuCVis 3.7360.So, the fair value of the VraiRive bonis the VNless the CVor VN– CVA = 100 – 3.7360 = 96.2640.Step 3 Baseon the fair value from Step 2, calculate the yielto maturity of the bon ansolve for the cret spresubtracting the yielto maturity on the benchmark bonfrom the yielto maturity on the VraiRive bon The cret spreis equto the yielto maturity on the VraiRive bonminus the yielto maturity on the three-yebenchmark bon(whiis 5.0000%). Baseon its fair value of 96.2640, the VraiRive bons yielto maturity (YTM) is96.2640=5/(1+YTM)+5/(1+YTM)2+105/(1+YTM)3Solving for YTM, the yielto maturity is 6.4082%. Therefore, the cret spreon the VraiRive bonis 6.4082% – 5.0000% = 1.4082%. PV=-96.26PMT=5FV=105N=3求I/Y哪里错了？

NO.PZ2018123101000109问题如下 Lebeva asks Kowalski to analyze a three-yebon issueVraiRive S.A., using arbitrage-free framework. The bons coupon rate is 5%, with interest paiannually ana pvalue of 100. In her analysis, she makes the following three assumptions:■ The annuinterest rate volatility is 10%.■ The recovery rate is one-thirof the exposure eaperio■ The hazarrate, or contionprobability of fault eayear, is 2.00%.Selecteinformation on benchmark government bon for the VraiRive bonis presentein Exhibit 2, anthe relevant binomiinterest rate tree is presentein Exhibit 3.Baseon Kowalski’s assumptions anExhibits 2 an3, the cret spreon the VraiRive bonis closest to:A.0.6949%. B.0.9388%. C.1.4082%. C is correct. The cret spreccalculatein three steps:Step 1 Estimate the value of the three-yeVraiRive bonassuming no fault. Baseon Kowalski’s assumptions anExhibits 2 an3, the value of the three-yeVraiRive bonassuming no fault is 100.0000.Supporting calculations:The bonvalue in eano is the value of next perios cash flows scountethe forwarrate. For the three nos on te 2, the bonvalues are follows:105/1.081823 = 97.0584.105/1.066991 = 98.4076.105/1.054848 = 99.5404.For the two nos on te 1, the two bonvalues are follows:[0.5 × (97.0584) + 0.5 × (98.4076) + 5.00]/1.060139 = 96.9052.[0.5 × (98.4076) + 0.5 × (99.5404) + 5.00]/1.049238 = 99.0948.Finally, for the no on te 0, the bonvalue is[0.5 × (96.9052) + 0.5 × (99.0948) + 5.00]/1.030000 = 100.0000.Therefore, the VNfor the VraiRive bonis 100.0000.Step 2 Calculate the cret valuation austment (CVA), anthen subtrathe CVA from the VNfrom Step 1 to establish the fair value of the bon The CVA equals the sum of the present values of eayear’s expecteloss anis calculatefollows:Supporting calculations:The expecteexposures eate are the bonvalues eano, weightetheir risk-neutrprobabilities, plus the coupon payment:te 1: 0.5 × (96.9052) + 0.5 × (99.0948) + 5.00 = 103.0000.te 2: 0.25 × (97.0584) + 0.5 × (98.4076) + 0.25 × (99.5404) + 5.00 = 103.3535.te 3: 105.0000The loss given fault (LG on eate is 2/3 of the expecteexposure.The probability of fault (PO on eate is follows:te 1: 2%te 2: 2% × (100% – 2%) = 1.96%.te 3: 2% × (100% – 2%)2 = 1.9208%.The scount factor on eate is 1/(1 + spot rate for the te) raiseto the correpower.Finally, the cret valuation austment eayeis the proof the LGtimes the POtimes the scount factor, shown in the last column of the table. The sum of the three annuCVis 3.7360.So, the fair value of the VraiRive bonis the VNless the CVor VN– CVA = 100 – 3.7360 = 96.2640.Step 3 Baseon the fair value from Step 2, calculate the yielto maturity of the bon ansolve for the cret spresubtracting the yielto maturity on the benchmark bonfrom the yielto maturity on the VraiRive bon The cret spreis equto the yielto maturity on the VraiRive bonminus the yielto maturity on the three-yebenchmark bon(whiis 5.0000%). Baseon its fair value of 96.2640, the VraiRive bons yielto maturity (YTM) is96.2640=5/(1+YTM)+5/(1+YTM)2+105/(1+YTM)3Solving for YTM, the yielto maturity is 6.4082%. Therefore, the cret spreon the VraiRive bonis 6.4082% – 5.0000% = 1.4082%.想问下求解spreaberate 用的是同期限的prate吗，为什么不用spot rate

NO.PZ2018123101000109问题如下Lebeva asks Kowalski to analyze a three-yebon issueVraiRive S.A., using arbitrage-free framework. The bons coupon rate is 5%, with interest paiannually ana pvalue of 100. In her analysis, she makes the following three assumptions:■ The annuinterest rate volatility is 10%.■ The recovery rate is one-thirof the exposure eaperio■ The hazarrate, or contionprobability of fault eayear, is 2.00%.Selecteinformation on benchmark government bon for the VraiRive bonis presentein Exhibit 2, anthe relevant binomiinterest rate tree is presentein Exhibit 3.Baseon Kowalski’s assumptions anExhibits 2 an3, the cret spreon the VraiRive bonis closest to:A.0.6949%. B.0.9388%. C.1.4082%. C is correct. The cret spreccalculatein three steps:Step 1 Estimate the value of the three-yeVraiRive bonassuming no fault. Baseon Kowalski’s assumptions anExhibits 2 an3, the value of the three-yeVraiRive bonassuming no fault is 100.0000.Supporting calculations:The bonvalue in eano is the value of next perios cash flows scountethe forwarrate. For the three nos on te 2, the bonvalues are follows:105/1.081823 = 97.0584.105/1.066991 = 98.4076.105/1.054848 = 99.5404.For the two nos on te 1, the two bonvalues are follows:[0.5 × (97.0584) + 0.5 × (98.4076) + 5.00]/1.060139 = 96.9052.[0.5 × (98.4076) + 0.5 × (99.5404) + 5.00]/1.049238 = 99.0948.Finally, for the no on te 0, the bonvalue is[0.5 × (96.9052) + 0.5 × (99.0948) + 5.00]/1.030000 = 100.0000.Therefore, the VNfor the VraiRive bonis 100.0000.Step 2 Calculate the cret valuation austment (CVA), anthen subtrathe CVA from the VNfrom Step 1 to establish the fair value of the bon The CVA equals the sum of the present values of eayear’s expecteloss anis calculatefollows:Supporting calculations:The expecteexposures eate are the bonvalues eano, weightetheir risk-neutrprobabilities, plus the coupon payment:te 1: 0.5 × (96.9052) + 0.5 × (99.0948) + 5.00 = 103.0000.te 2: 0.25 × (97.0584) + 0.5 × (98.4076) + 0.25 × (99.5404) + 5.00 = 103.3535.te 3: 105.0000The loss given fault (LG on eate is 2/3 of the expecteexposure.The probability of fault (PO on eate is follows:te 1: 2%te 2: 2% × (100% – 2%) = 1.96%.te 3: 2% × (100% – 2%)2 = 1.9208%.The scount factor on eate is 1/(1 + spot rate for the te) raiseto the correpower.Finally, the cret valuation austment eayeis the proof the LGtimes the POtimes the scount factor, shown in the last column of the table. The sum of the three annuCVis 3.7360.So, the fair value of the VraiRive bonis the VNless the CVor VN– CVA = 100 – 3.7360 = 96.2640.Step 3 Baseon the fair value from Step 2, calculate the yielto maturity of the bon ansolve for the cret spresubtracting the yielto maturity on the benchmark bonfrom the yielto maturity on the VraiRive bon The cret spreis equto the yielto maturity on the VraiRive bonminus the yielto maturity on the three-yebenchmark bon(whiis 5.0000%). Baseon its fair value of 96.2640, the VraiRive bons yielto maturity (YTM) is96.2640=5/(1+YTM)+5/(1+YTM)2+105/(1+YTM)3Solving for YTM, the yielto maturity is 6.4082%. Therefore, the cret spreon the VraiRive bonis 6.4082% – 5.0000% = 1.4082%.请问VN以=5*0.970874+5*0.920560+105*0.862314得到吗？计算出来也是100；一定要用二叉树计算吗

NO.PZ2018123101000109 问题如下 Lebeva asks Kowalski to analyze a three-yebon issueVraiRive S.A., using arbitrage-free framework. The bons coupon rate is 5%, with interest paiannually ana pvalue of 100. In her analysis, she makes the following three assumptions:■ The annuinterest rate volatility is 10%.■ The recovery rate is one-thirof the exposure eaperio■ The hazarrate, or contionprobability of fault eayear, is 2.00%.Selecteinformation on benchmark government bon for the VraiRive bonis presentein Exhibit 2, anthe relevant binomiinterest rate tree is presentein Exhibit 3.Baseon Kowalski’s assumptions anExhibits 2 an3, the cret spreon the VraiRive bonis closest to: A.0.6949%. B.0.9388%. C.1.4082%. C is correct. The cret spreccalculatein three steps:Step 1 Estimate the value of the three-yeVraiRive bonassuming no fault. Baseon Kowalski’s assumptions anExhibits 2 an3, the value of the three-yeVraiRive bonassuming no fault is 100.0000.Supporting calculations:The bonvalue in eano is the value of next perios cash flows scountethe forwarrate. For the three nos on te 2, the bonvalues are follows:105/1.081823 = 97.0584.105/1.066991 = 98.4076.105/1.054848 = 99.5404.For the two nos on te 1, the two bonvalues are follows:[0.5 × (97.0584) + 0.5 × (98.4076) + 5.00]/1.060139 = 96.9052.[0.5 × (98.4076) + 0.5 × (99.5404) + 5.00]/1.049238 = 99.0948.Finally, for the no on te 0, the bonvalue is[0.5 × (96.9052) + 0.5 × (99.0948) + 5.00]/1.030000 = 100.0000.Therefore, the VNfor the VraiRive bonis 100.0000.Step 2 Calculate the cret valuation austment (CVA), anthen subtrathe CVA from the VNfrom Step 1 to establish the fair value of the bon The CVA equals the sum of the present values of eayear’s expecteloss anis calculatefollows:Supporting calculations:The expecteexposures eate are the bonvalues eano, weightetheir risk-neutrprobabilities, plus the coupon payment:te 1: 0.5 × (96.9052) + 0.5 × (99.0948) + 5.00 = 103.0000.te 2: 0.25 × (97.0584) + 0.5 × (98.4076) + 0.25 × (99.5404) + 5.00 = 103.3535.te 3: 105.0000The loss given fault (LG on eate is 2/3 of the expecteexposure.The probability of fault (PO on eate is follows:te 1: 2%te 2: 2% × (100% – 2%) = 1.96%.te 3: 2% × (100% – 2%)2 = 1.9208%.The scount factor on eate is 1/(1 + spot rate for the te) raiseto the correpower.Finally, the cret valuation austment eayeis the proof the LGtimes the POtimes the scount factor, shown in the last column of the table. The sum of the three annuCVis 3.7360.So, the fair value of the VraiRive bonis the VNless the CVor VN– CVA = 100 – 3.7360 = 96.2640.Step 3 Baseon the fair value from Step 2, calculate the yielto maturity of the bon ansolve for the cret spresubtracting the yielto maturity on the benchmark bonfrom the yielto maturity on the VraiRive bon The cret spreis equto the yielto maturity on the VraiRive bonminus the yielto maturity on the three-yebenchmark bon(whiis 5.0000%). Baseon its fair value of 96.2640, the VraiRive bons yielto maturity (YTM) is96.2640=5/(1+YTM)+5/(1+YTM)2+105/(1+YTM)3Solving for YTM, the yielto maturity is 6.4082%. Therefore, the cret spreon the VraiRive bonis 6.4082% – 5.0000% = 1.4082%. 老师，请问不考虑风险的债券YTM=5%是不是这么判断的表2中pcurve rates三年期为5%，等于债券coupon rate，说明债券是平价发行，因此YTM=5%

NO.PZ2018123101000109 问题如下 Lebeva asks Kowalski to analyze a three-yebon issueVraiRive S.A., using arbitrage-free framework. The bons coupon rate is 5%, with interest paiannually ana pvalue of 100. In her analysis, she makes the following three assumptions:■ The annuinterest rate volatility is 10%.■ The recovery rate is one-thirof the exposure eaperio■ The hazarrate, or contionprobability of fault eayear, is 2.00%.Selecteinformation on benchmark government bon for the VraiRive bonis presentein Exhibit 2, anthe relevant binomiinterest rate tree is presentein Exhibit 3.Baseon Kowalski’s assumptions anExhibits 2 an3, the cret spreon the VraiRive bonis closest to: A.0.6949%. B.0.9388%. C.1.4082%. C is correct. The cret spreccalculatein three steps:Step 1 Estimate the value of the three-yeVraiRive bonassuming no fault. Baseon Kowalski’s assumptions anExhibits 2 an3, the value of the three-yeVraiRive bonassuming no fault is 100.0000.Supporting calculations:The bonvalue in eano is the value of next perios cash flows scountethe forwarrate. For the three nos on te 2, the bonvalues are follows:105/1.081823 = 97.0584.105/1.066991 = 98.4076.105/1.054848 = 99.5404.For the two nos on te 1, the two bonvalues are follows:[0.5 × (97.0584) + 0.5 × (98.4076) + 5.00]/1.060139 = 96.9052.[0.5 × (98.4076) + 0.5 × (99.5404) + 5.00]/1.049238 = 99.0948.Finally, for the no on te 0, the bonvalue is[0.5 × (96.9052) + 0.5 × (99.0948) + 5.00]/1.030000 = 100.0000.Therefore, the VNfor the VraiRive bonis 100.0000.Step 2 Calculate the cret valuation austment (CVA), anthen subtrathe CVA from the VNfrom Step 1 to establish the fair value of the bon The CVA equals the sum of the present values of eayear’s expecteloss anis calculatefollows:Supporting calculations:The expecteexposures eate are the bonvalues eano, weightetheir risk-neutrprobabilities, plus the coupon payment:te 1: 0.5 × (96.9052) + 0.5 × (99.0948) + 5.00 = 103.0000.te 2: 0.25 × (97.0584) + 0.5 × (98.4076) + 0.25 × (99.5404) + 5.00 = 103.3535.te 3: 105.0000The loss given fault (LG on eate is 2/3 of the expecteexposure.The probability of fault (PO on eate is follows:te 1: 2%te 2: 2% × (100% – 2%) = 1.96%.te 3: 2% × (100% – 2%)2 = 1.9208%.The scount factor on eate is 1/(1 + spot rate for the te) raiseto the correpower.Finally, the cret valuation austment eayeis the proof the LGtimes the POtimes the scount factor, shown in the last column of the table. The sum of the three annuCVis 3.7360.So, the fair value of the VraiRive bonis the VNless the CVor VN– CVA = 100 – 3.7360 = 96.2640.Step 3 Baseon the fair value from Step 2, calculate the yielto maturity of the bon ansolve for the cret spresubtracting the yielto maturity on the benchmark bonfrom the yielto maturity on the VraiRive bon The cret spreis equto the yielto maturity on the VraiRive bonminus the yielto maturity on the three-yebenchmark bon(whiis 5.0000%). Baseon its fair value of 96.2640, the VraiRive bons yielto maturity (YTM) is96.2640=5/(1+YTM)+5/(1+YTM)2+105/(1+YTM)3Solving for YTM, the yielto maturity is 6.4082%. Therefore, the cret spreon the VraiRive bonis 6.4082% – 5.0000% = 1.4082%. 我这边exposure和答案算的一样，LGRR都是对的，但我算的CVA是4.068呀，不知道啥原因