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youtkr · 2022年08月04日

为什么lgd折回零时点用的是表2的spot rate,利率不是在二叉树里波动嘛?

NO.PZ2018123101000109

问题如下:

Lebedeva asks Kowalski to analyze a three-year bond, issued by VraiRive S.A., using an arbitrage-free framework. The bond’s coupon rate is 5%, with interest paid annually and a par value of 100. In her analysis, she makes the following three assumptions:
■ The annual interest rate volatility is 10%.
■ The recovery rate is one-third of the exposure each period.
■ The hazard rate, or conditional probability of default each year, is 2.00%.

Selected information on benchmark government bonds for the VraiRive bond is presented in Exhibit 2, and the relevant binomial interest rate tree is presented in Exhibit 3.

Based on Kowalski’s assumptions and Exhibits 2 and 3, the credit spread on the VraiRive bond is closest to:

选项:

A.

0.6949%.

B.

0.9388%.

C.

1.4082%.

解释:

C is correct. The credit spread can be calculated in three steps:
Step 1 Estimate the value of the three-year VraiRive bond assuming no default. Based on Kowalski’s assumptions and Exhibits 2 and 3, the value of the three-year VraiRive bond assuming no default is 100.0000.

Supporting calculations:
The bond value in each node is the value of next period’s cash flows discounted by the forward rate. For the three nodes on Date 2, the bond values are as follows:
105/1.081823 = 97.0584.
105/1.066991 = 98.4076.
105/1.054848 = 99.5404.
For the two nodes on Date 1, the two bond values are as 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 node on Date 0, the bond value is
[0.5 × (96.9052) + 0.5 × (99.0948) + 5.00]/1.030000 = 100.0000.
Therefore, the VND for the VraiRive bond is 100.0000.
Step 2 Calculate the credit valuation adjustment (CVA), and then subtract the CVA from the VND from Step 1 to establish the fair value of the bond. The CVA equals the sum of the present values of each year’s expected loss and is calculated as follows:

Supporting calculations:
The expected exposures at each date are the bond values at each node, weighted by their risk-neutral probabilities, plus the coupon payment:
Date 1: 0.5 × (96.9052) + 0.5 × (99.0948) + 5.00 = 103.0000.
Date 2: 0.25 × (97.0584) + 0.5 × (98.4076) + 0.25 × (99.5404) + 5.00 = 103.3535.

Date 3: 105.0000
The loss given default (LGD) on each date is 2/3 of the expected exposure.
The probability of default (POD) on each date is as follows:
Date 1: 2%
Date 2: 2% × (100% – 2%) = 1.96%.
Date 3: 2% × (100% – 2%)2 = 1.9208%.
The discount factor on each date is 1/(1 + spot rate for the date) raised to the correct power.
Finally, the credit valuation adjustment each year is the product of the LGD times the POD times the discount factor, as shown in the last column of the table. The sum of the three annual CVAs is 3.7360.
So, the fair value of the VraiRive bond is the VND less the CVA, or VND – CVA = 100 – 3.7360 = 96.2640.
Step 3 Based on the fair value from Step 2, calculate the yield to maturity of the bond, and solve for the credit spread by subtracting the yield to maturity on the benchmark bond from the yield to maturity on the VraiRive bond. The credit spread is equal to the yield to maturity on the VraiRive bond minus the yield to maturity on the three-year benchmark bond (which is 5.0000%). Based on its fair value of 96.2640, the VraiRive bond’s yield to maturity (YTM) is
96.2640=5/(1+YTM)+5/(1+YTM)2+105/(1+YTM)3
Solving for YTM, the yield to maturity is 6.4082%. Therefore, the credit spread on the VraiRive bond is 6.4082% – 5.0000% = 1.4082%.

如题

1 个答案

pzqa015 · 2022年08月04日

嗨,爱思考的PZer你好:


是EL折回零时点,而不是LGD折回零时点。

因为在计算EL的时候,已经考虑到利率的波动了(exposure是用二叉树计算的,考虑了利率波动),所以折现时就无需再考虑利率波动,也就不用二叉树,而用spot rate就可以了。

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就算太阳没有迎着我们而来,我们正在朝着它而去,加油!

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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%. 想确认一下,benchmark boncoupon跟债券的coupon(5%)是否一定要相等,还是只要期限匹配即可?题中3年期prate正好是等于5%,那如果不等于5%的情况,是否可以仍可以用这个3年期prate(例如4%)作为benchmark rate?

2024-03-13 13:26 2 · 回答

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2024-03-11 14:30 1 · 回答

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2024-03-04 17:08 1 · 回答

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2024-03-03 12:42 1 · 回答

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%. 老师为什么这里平价发行的政府债券的YTM就是其coupon rate呢?怎么计算出来的呢?

2023-10-20 20:45 1 · 回答