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梦梦 · 2024年11月12日

如果没用折现因子计算，哪里错了呢

  1. FRM I
  2. Valuation And Risk Models

NO.PZ2023091701000170

问题如下：

1.1 A fixed-income portfolio manager is using discount factors to price a sovereign bond. The bond is a coupon bond with the following cash flows:

  • 3 months from today: USD 67,500
  • 6 months from today: USD 67,500
  • 9 months from today: USD 4,567,500

The manager observes the following market price quotes for three zero-coupon bonds issued by the same country:

Based on the discount factors of the zero-coupon bonds, what is the present value of the cash flows from the sovereign coupon bond?

选项：

A.USD 3,728,209

B.USD 4,059,055

C.USD 4,349,780

D.USD 4,436,915

解释：

C is correct. First, the discount factors for the zero-coupon bonds must be calculated:

  • 3-month bond: 97.8012/100 = 0.978012
  • 6-month bond: 95.2375/100 = 0.952375
  • 9-month bond: 92.3805/100 = 0.923805

These are then applied to the corresponding cash flows from the coupon-paying bond:

  • 3-month CF: 67,500 * 0.978012 = 66,015.81
  • 6-month CF: 67,500 * 0.952375 = 64,285.31
  • 9-month CF: 4,567,500 * 0.923805 = 4,219,479.34

The present value of the bonds cash flows is therefore 66,015.81 + 64,285.31 + 4,219,479.34 = 4,349,780.46.

A is incorrect. This adjusts each of the discount factors from the zero-coupon bonds by raising them to the power of the respective number of quarterly periods to maturity, before applying them to the coupon paying bond’s cash flows:

  • 3-month adjusted factor: 0.978012^1 = 0.978012
  • 6-month adjusted factor: 0.952375^2 = 0.907018
  • 9-month adjusted factor: 0.923805^3 = 0.788390

B is incorrect. This adjusts each of the discount factors from the zero-coupon bonds by progressively multiplying all of the previous factors, before applying them to the coupon paying bond’s cash flows:

  • 3-month adjusted factor: 0.978012
  • 6-month adjusted factor: 0.978012 * 0.952375 = 0.931434
  • 9-month adjusted factor: 0.978012 * 0.952375 * 0.923805 = 0.860464

D is incorrect. This uses the correct discount factors from the zero-coupon bonds, but then raises them to the power of the respective portion of 1 year when computing the coupon paying bond’s cash flows:

  • 3-month CF: 67,500 * 0.978012^(3/12) = 67,125.85
  • 6-month CF: 67,500 * 0.952375^(6/12) = 65,873.05
  • 9-month CF: 4,567,500 * 0.923805^(9/12) = 4,303,916.01

老师，如果没想到折现因子，求零息债的spot rate，再算债券的pv，不等于答案呢，您看看我的解题过程哪里错了吗？

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1 个答案

李坏_品职助教 · 2024年11月12日

嗨，努力学习的PZer你好：


这道题已经明确要求要用折现因子计算了。


你的计算过程里，3-spot rate求出来的0.74，意思是一个月折现率是0.74%。折现的时候应该是67500 / (1+0.74%)^3 = 66023.407

6-spot rate求出来的0.8165也是一个月的折现率，所以折现的时候是67500 /(1+0.8165%)^6 = 64285.655

最后应该是 4,567,500 / (1+0.8844%)^9 = 4219513.855


三个加起来 = 4349822.917，最接近的是C选项。

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

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NO.PZ2023091701000170问题如下 1.1 A fixeincome portfolio manager is using scount factors to pricea sovereign bon The bonis a coupon bonwith the following cash flows:3 months from toy: US67,5006 months from toy: US67,5009 months from toy: US4,567,500The managerobserves the following market priquotes for three zero-coupon bon issuey the same country:Baseon thescount factors of the zero-coupon bon, whis the present value of thecash flows from the sovereign coupon bon A.US3,728,209B.US4,059,055C.US4,349,780US4,436,915 C is correct.First, the scount factors for the zero-coupon bon must calculate3-month bon 97.8012/100 = 0.9780126-month bon 95.2375/100 = 0.9523759-month bon 92.3805/100 = 0.923805These are thenapplieto the corresponng cash flows from the coupon-paying bon3-month CF: 67,500 * 0.978012 = 66,015.816-month CF: 67,500 * 0.952375 = 64,285.319-month CF: 4,567,500 * 0.923805 = 4,219,479.34The present valueof the bon cash flows is therefore 66,015.81 + 64,285.31 + 4,219,479.34 =4,349,780.46.A is incorrect.This austs eaof the scount factors from the zero-coupon bon raisingthem to the power of the respective number of quarterly perio to maturity,before applying them to the coupon paying bons cash flows:3-month austefactor: 0.978012^1 = 0.9780126-month austefactor: 0.952375^2 = 0.9070189-month austefactor: 0.923805^3 = 0.788390B is incorrect.This austs eaof the scount factors from the zero-coupon bon byprogressively multiplying all of the previous factors, before applying them tothe coupon paying bons cash flows:3-month austefactor: 0.9780126-month austefactor: 0.978012 * 0.952375 = 0.9314349-month austefactor: 0.978012 * 0.952375 * 0.923805 = 0.860464is incorrect.This uses the correscount factors from the zero-coupon bon, but thenraises them to the power of the respective portion of 1 yewhen computing thecoupon paying bons cash flows:3-month CF: 67,500 * 0.978012^(3/12) = 67,125.856-month CF: 67,500 * 0.952375^(6/12) = 65,873.059-month CF: 4,567,500 * 0.923805^(9/12) = 4,303,916.01 老师，我刚才发的解题图不用看了，您看看这个我是哪里算错了呀

2024-11-12 17:47 5 · 回答

NO.PZ2023091701000170问题如下 1.1 A fixeincome portfolio manager is using scount factors to pricea sovereign bon The bonis a coupon bonwith the following cash flows:3 months from toy: US67,5006 months from toy: US67,5009 months from toy: US4,567,500The managerobserves the following market priquotes for three zero-coupon bon issuey the same country:Baseon thescount factors of the zero-coupon bon, whis the present value of thecash flows from the sovereign coupon bon A.US3,728,209B.US4,059,055C.US4,349,780US4,436,915 C is correct.First, the scount factors for the zero-coupon bon must calculate3-month bon 97.8012/100 = 0.9780126-month bon 95.2375/100 = 0.9523759-month bon 92.3805/100 = 0.923805These are thenapplieto the corresponng cash flows from the coupon-paying bon3-month CF: 67,500 * 0.978012 = 66,015.816-month CF: 67,500 * 0.952375 = 64,285.319-month CF: 4,567,500 * 0.923805 = 4,219,479.34The present valueof the bon cash flows is therefore 66,015.81 + 64,285.31 + 4,219,479.34 =4,349,780.46.A is incorrect.This austs eaof the scount factors from the zero-coupon bon raisingthem to the power of the respective number of quarterly perio to maturity,before applying them to the coupon paying bons cash flows:3-month austefactor: 0.978012^1 = 0.9780126-month austefactor: 0.952375^2 = 0.9070189-month austefactor: 0.923805^3 = 0.788390B is incorrect.This austs eaof the scount factors from the zero-coupon bon byprogressively multiplying all of the previous factors, before applying them tothe coupon paying bons cash flows:3-month austefactor: 0.9780126-month austefactor: 0.978012 * 0.952375 = 0.9314349-month austefactor: 0.978012 * 0.952375 * 0.923805 = 0.860464is incorrect.This uses the correscount factors from the zero-coupon bon, but thenraises them to the power of the respective portion of 1 yewhen computing thecoupon paying bons cash flows:3-month CF: 67,500 * 0.978012^(3/12) = 67,125.856-month CF: 67,500 * 0.952375^(6/12) = 65,873.059-month CF: 4,567,500 * 0.923805^(9/12) = 4,303,916.01 老师，答案的折现因子具体是这样的吧？因为3个月一次coupon，所以100/｛（1+sp1/4）｝^1=97.8012，100/｛（1+sp2/4）｝^2=…对吧？sp1是一年期的spot rate，sp2是两年期的spot rate

2024-11-12 17:37 1 · 回答

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