NO.PZ2019011002000007
问题如下：
Bond B is a 4-year annual coupon bond with a par value of $1000, and coupon rate of 6%. The risk-neutral probability of default (the hazard rate) for each date for the bond is 1.50% and the recovery rate is 25%.
Li is a credit analyst in a wealth management firm. He is considering a future interest rate volatility of 20%.
The current spot rates and forward rates are shown in the table below:
He built a binomial interest rate tree by using his volatility estimation and the current yield curve. The Binomial interest rate tree is shown below:
According to the information above, what is the fair value of Bond B?
选项：
A.1098.14
B.1144.63
C.1251.35
解释：
A is correct
考点：使用二叉树对有风险的固定利率债券进行估值
解析：
首先利用二叉树模型，计算VND,(Value of the bond assuming No Default）;
得到债券的VND为：1144.63
下面就要计算债券的CVA。
第一步计算二叉树上每期的exposure，
如Date 4的exposure为1060;
Date 3的exposure为：
0.1250×980.75+0.3750×1005.54+0.3750×1022.86
+0.1250×1034.81+60=1072.60
Date 2的exposure为：
0.25×1008.76+0.50×1043.43+0.25×1067.73+60
=1100.84
Date 1的exposure为：
0.50×1063.57+0.50×1099.96+60=1141.76
有了每一期的Exposure，可以计算LGD（Loss given default），有公式：
LGD = exposure × (1-recovery rate)
已知Hazard rate为1.500%，则每一期的POS（Probability of survival）为：
(100%-1.5%)1=98.5%
(100%-1.5%)2=97.0225%
(100%-1.5%)3=95.5672%
(100%-1.5%)4=94.1337%
(100%-1.5%)5=92.7217%
已知每一期的POS，则可以算出每一期的POD（Probability of default）
折现因子（DF）可以题干信息中获得；最终PV of expected loss = Expected loss ×DF。
我们可以得到如下表格：
所以该债券的Fair value为：1144.63 – 46.4915 = 1098.1385
用spot rate将无违约的fair value计算出来，是1144.63，因为减去违约补偿，一定比1144.63小，所以选A，考试的时候是不是也可以类似这样选一个
