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kevinzhu · 2020年07月23日

问一道题：NO.PZ2016031001000081

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:

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

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:

$97=\frac{\frac{(0.01+0.0080)\times100}2}{{(1+\frac{0.01+DM}2)}^1}+\frac{\frac{(0.01+0.0080)\times100}2}{{(1+\frac{0.01+DM}2)}^2}+\cdots+\frac{\frac{(0.01+0.0080)\times100}2+100}{{(1+\frac{0.01+DM}2)}^4}$

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

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

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

r = 0.0168

Now, solve for DM:

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

DM = 0.0236

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

PMT=coupon payment=（Libor+0.8%)/2*100=(1%+0.8%)/2*100=0.9，FV=100，PV=-97，N=2*2=4，求出I/Y=1.681% 然后1.681%*2-1%=0.0236

半年付息一次，I/Y不是都要*2的吗？我这个算法是否正确？

（1%+ discount margin）/2=1.681% 可得DM=0.0236 这个式子不理解

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吴昊_品职助教 · 2020年07月24日

同学你好：

你的理解是对的。

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吴昊_品职助教 · 2020年07月24日

同学你好：

你用计算器按得I/Y=1.681%没有问题。现在题目要我们求的是Discount Margin，Discount rate = reference rate + required margin (or discount margin)，这个公式可以参考基础班讲义P166页。

现在discount rate=(0.01+DM)/2=0.01681，那我们求可以反求出discount margin=0.02362=236.2bp

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kevinzhu · 2020年07月24日

Discount rate = reference rate + required margin (or discount margin) 1.681%*2=1%+discount margin 这个理解对否?

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