Floating Rate Note Duration from Discounted Cash Flows

# Answer by xing gao (score 1)


# Is this simple model used to calculate the interest rate duration and credit duration of a floating rate note? Other models?












I found this model for floating rate bonds in a book I am reading and I am wondering if it is used anywhere in practice?

$$MV=\frac{\frac{(Index+QM)\cdot FV}{PER}}{\left(1+\frac{Index+DM}{PER}\right)^1}+\frac{\frac{(Index+QM)\cdot FV}{PER}}{\left(1+\frac{Index+DM}{PER}\right)^2}+\cdots+\frac{\frac{(Index+QM)\cdot FV}{PER}+FV}{\left(1+\frac{Index+DM}{PER}\right)^N},$$

where MV= market value, Index = reference rate, QM = Quoted Margin, FV= face value, PER= periodicity, N=number of periods to maturity. The important point here is that we put in todays value of Index everywhere, even though it will change.

In this model I assume you have MV, Index is the current rate, and you have QM and FV, then you solve for DM. The calculated interest rate duration is:

$$\frac{MV(Index-\Delta Index)-MV(Index+\Delta Index)}{2\Delta Index \cdot MV},$$ and to calculate the credit duration you have

$$\frac{MV(DM-\Delta DM)-MV(DM+\Delta DM)}{2\Delta DM \cdot MV}.$$

Is this model and these equations used in the real world to calculate interest rate duraiton and credit duration? If not, could you please tell me which model is used and how these durations are calculated?

## Answer by xing gao (score 1)

https://quant.stackexchange.com/a/61481

I don't have enough repuation to commnet, but I think it is a general cashflow discount model for bond pricing, and the formula looks wrong. The last item should be discounting of principal and last period coupon, which should be like this:

PER here refers to coupon payment frequency. FV definitely cannot be divided by frequency.

## Answer by sqrt2sqrt2 (score 0)

https://quant.stackexchange.com/a/63185

Interest rate duration and credit duration are defined as the derivative of MV of a bond with respect to IR and credit. The first order derivative can be easily approximated using centered difference (which I would call a method rather than a model). The centered difference formula is the one you mentioned: it has the good property that the error goes to second order (you can easily see it by using Taylor expansion to the third order for MV), i.e. it is somehow the best numeric approximation of derivative to minimize the error. I confirm that these are the formulas used to determine duration in practice. Apart from mathematical formulas, it is useful to have clear intuition for IR and credit derivatives for a floating rate note (I assume reset in advance). If IR moves, all the cashflow apart from the first one (already fixed) will change accordingly (both denominator and numerator of each addedum from the second one change in the same way). So the IR sensitivity of your note is the same as the one of a fixed coupon note expiring at the next coupon date - e.g. 0.25 if the next coupon is in 3 months etc. If credit changes instead, only the discount factors changes (the denominator of each addendum in your formula). So the credit sensitivity of your note is the same as the sensitivity of a fixed coupon bond with the same implicit cashflows of the floating rate note - e.g. typically a duration in area 10 for a 10 year expiring note with the rule of thumb that the more QM>DM, the smaller the duration.