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mitmit · 2020年10月01日

问一道题：NO.PZ2018062006000081

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.

老师，6个月的libor,不应该算PMT的时候是除以4吗，因为是2 years, m不应该是4吗？

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吴昊_品职助教 · 2020年10月01日

同学你好：

pays 6-month Libor plus 80 basis points，说明半年付息一次，题目中给出的利率都是年化利率，需要去年化，既然是半年付息一次，那么每一期拿到的的coupon就需要在coupon rate的基础上除以2。如果是一个季度付息一次，就除以4了。

m = periodicity of the floating-rate note, the number of payment periods per year，代表的是一年中的付息次数，这道题中m=2。

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