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王思祺 · 2020年08月17日

问一道题：NO.PZ2016031001000081

  1. CFA I
  2. Fixed Income

问题如下：

A two-year floating-rate note pays 6-month Libor plus 80 basis points. The floater is priced at 97 per 100 of par value. Current 6-month Libor is 1.00%. Assume a 30/360 day count convention and evenly spaced periods. The discount margin for the floater in basis points (bps) is closest to:

选项：

A.

180 bps.

B.

236 bps.

C.

420 bps.

解释：

B is correct.

The discount or required margin is 236 basis points. Given the floater has a maturity of two years and is linked to 6-month Libor, the formula for calculating discount margin is: 

PV=(Index+QM)×FVm(1+Index+DMm)1+(Index+QM)×FVm(1+Index+DMm)2+⋯+(Index+QM)×FVm+FV(1+Index+DMm)4PV=\frac{\frac{(Index+QM)\times FV}m}{{(1+\frac{Index+DM}m)}^1}+\frac{\frac{(Index+QM)\times FV}m}{{(1+\frac{Index+DM}m)}^2}+\cdots+\frac{\frac{(Index+QM)\times FV}m+FV}{{(1+\frac{Index+DM}m)}^4}

where:

PV = present value, or the price of the floating-rate note = 97

Index = reference rate, stated as an annual percentage rate = 0.01

QM = quoted margin, stated as an annual percentage rate = 0.0080

FV = future value paid at maturity, or the par value of the bond = 100

m = periodicity of the floating-rate note, the number of payment periods per year = 2

DM = discount margin, the required margin stated as an annual percentage rate

Substituting given values in:

97=(0.01+0.0080)×1002(1+0.01+DM2)1+(0.01+0.0080)×1002(1+0.01+DM2)2+⋯+(0.01+0.0080)×1002+100(1+0.01+DM2)497=\frac{\frac{(0.01+0.0080)\times100}2}{{(1+\frac{0.01+DM}2)}^1}+\frac{\frac{(0.01+0.0080)\times100}2}{{(1+\frac{0.01+DM}2)}^2}+\cdots+\frac{\frac{(0.01+0.0080)\times100}2+100}{{(1+\frac{0.01+DM}2)}^4}

97=0.9(1+0.01+DM)2)1+0.9(1+0.01+DM2)2+0.9(1+0.01+DM2)3+0.9+100(1+0.01+DM2)497=\frac{0.9}{{(1+\frac{0.01+DM)}2)}^1}+\frac{0.9}{{(1+\frac{0.01+DM}2)}^2}+\frac{0.9}{{(1+\frac{0.01+DM}2)}^3}+\frac{0.9+100}{{(1+\frac{0.01+DM}2)}^4}

To calculate DM, begin by solving for the discount rate per period:

97=0.9(1+r)1+0.9(1+r)2+0.9(1+r)3+0.9+100(1+r)497=\frac{0.9}{{(1+r)}^1}+\frac{0.9}{{(1+r)}^2}+\frac{0.9}{{(1+r)}^3}+\frac{0.9+100}{{(1+r)}^4}

r = 0.0168

Now, solve for DM:

0.01+DM2=0.0168\frac{0.01+DM}2=0.0168

DM = 0.0236

The discount margin for the floater is equal to 236 basis points.

这里面的30/360没用到吧~~

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1 个答案

吴昊_品职助教 · 2020年08月17日

同学你好：

是的哦，没有用到。

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NO.PZ2016031001000081问题如下 A two-yefloating-rate note pays 6-month MRR plus 80 basis points. The floater is price97 per 100 of pvalue. The current 6-month MRR is 1.00%. Assume a 30/360 y count convention anevenly spaceperio. The scount margin for the floater in basis points (bps) is closest to: A.180 bps.B.236 bps.C.420 bps. B is correct.The scount or requiremargin is 236 basis points. Given the floater ha maturity of two years anis linketo 6-month MRR, the formula for calculating scount margin is: where:PV = present value, or the priof the floating-rate note = 97Inx = referenrate, stateannupercentage rate = 0.01QM = quotemargin, stateannupercentage rate = 0.0080FV = future value paimaturity, or the pvalue of the bon= 100m = periocity of the floating-rate note, the number of payment perio per ye= 2 = scount margin, the requiremargin stateannupercentage rateSubstituting given values in:To calculate , begin solving for the scount rate per perior = 0.0168Now, solve for :(0.01+)/2=0.0168 = 0.0236The scount margin for the floater is equto 236 basis points.考点浮动利率债券解析浮动利率债券的Coupon Rate = Referenrate + QuoteMargin，比如一个每半年付息一次的浮动利率债券，其Coupon Rate是6-month MRR + 50 bps，50 bps（息差Sprea就是QuoteMargin。这道题中QuoteMargin是80bps。Referenrate和QuoteMargin共同决定Coupon Rate。给浮动利率债券未来现金流折现时，使用的折现率是Referenrate + scount margin。基准利率和scount margin共同构成对这个浮动利率债券的要求回报率。所以在基准利率的基础上，加上一个scount Margin后，折现未来现金流可以得到当前浮动利率债券。或者知道当前浮动利率债券价格和基准利率，可以反求scount Margin。本题知道浮动利率债券的Referenrate和Quotemargin，也就知道分子的Coupon rate；也知道浮动利率债券当前的债券价格，所以可以反求出来折现率，从而进一步求scount margin为236bp。 为看来题目解析和其他人的提问，但是对这句话不是很理解the current 6-month MRR is 1%， 这句话为什么可以直接理解一年的MRR呢？

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