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梦梦 · 2024年11月12日

哪里算错了

  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，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

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5 个答案

pzqa39 · 2024年11月14日

嗨，努力学习的PZer你好：


说反了！

N＝0.25 0.5和0.75是每年的年利率，因为它们的单位是年（比如3个月是0.25年）

N＝1 2和3是每季度的季度利率

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加油吧，让我们一起遇见更好的自己！

pzqa39 · 2024年11月14日

嗨，努力学习的PZer你好：


（1+10.25%）^0.5-1=5% 这里0.5次方不对啊


在你的设定里N=1是一个季度，N=2是两个季度，但是都是以“每季度”作为单位的，所以你N=2求出来的I/Y其实也是每个季度的I/Y。

（1+10.25%）^0.25-1这样就对了，不管N是多少，都是0.25次方把年化的折成每季度。

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努力的时光都是限量版，加油！

梦梦 · 2024年11月14日

我总结一下哈：3，6，9我用n=1，2，3求的是年利率？n=0.25，0.5，0.75，求的是季度利率？

pzqa39 · 2024年11月13日

嗨，努力学习的PZer你好：


在我们设置金融计算器的时候I/Y和N是需要保持一致的。如果单位是年那就都是年，如果是按照月的数量鹅那就都是月。

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

pzqa39 · 2024年11月12日

嗨，努力学习的PZer你好：


每半年付息一次的债券，n=2，pv，lv，pmt面值*coupon rate*0.5，计算I/Y，求出的是年化的YTM还是除以2以后的YTM？

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你描述的这种情况是除以2后的YTM,不是年化的


但是这道题里面你用的N=0.25按计算器呀

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加油吧，让我们一起遇见更好的自己！

梦梦 · 2024年11月12日

因为债券是3个月，哦，是不是 如果我n输入1，6个月n输入2，9个月，n输入3，求出的利率就可以对付息债券直接按1，2，3，次方折现了？ 如果输入的是0.25，因为时间已经年华了，是0.25年，所以求出的i/Y就是年化的。 原来i/Y取决于N

pzqa39 · 2024年11月12日

嗨，爱思考的PZer你好：


题目已经要求用discount factors了，这样是最便捷最不容易出错的方法，一个一个算spot rate会浪费很多时间，在考场上是来不及的，所以我们最好直接就按照题目要求的方法来计算。


另外，你这样设置N，算出来的I/Y是年化的，但是实际上计算的是三个月/六个月/九个月的，所以倒数第二行现金流折现分母不应该是一次方二次方三次方，应该是3个月对应0.25次方，6个月对应0.5次方，9个月对应0.75次方。这样算出来的答案和C是接近的。

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

梦梦 · 2024年11月12日

I/Y 计算的不是每期现金流吗？比如一年期限，每半年付息一次的债券，n=2，pv，lv，pmt面值*coupon rate*0.5，计算I/Y，求出的是年化的YTM还是除以2以后的YTM？我咋记得求的是除以2后的YTM

梦梦 · 2024年11月13日

我大致明白了，但是我n输入0.25，0.5，0.75，算出的利率分别是9.3%，10.25%和11.15%， 其中（1+9.3%）^0.25-1=2.4%，和输入N=1算出的利率是一样的，但是（1+10.25%）^0.5-1=5%，和输入N=2算出的利率2.47%就不一样了，11.15%也是如此，这又是哪里不对了

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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，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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2024-11-12 17:24 1 · 回答

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