In my calculations below I find that the effective duration(not spread duration, but interest duration) of a floating rate bond with yield spread can become negative. Do you see if they are correct? Does it happen in the real world that the effective interest duration of a floating rate bond can become negative?


I assume that we have a bond paying out at $K$ periods during the year, it has face value $N$ it has price $P$, we also assume that the current forward rates are all the same not denoted $r$. The first payment has been set to $c$, and it is $\Delta t_1$ time left ot the first payment, the are currently $V+1$ payments dates left.

We find the yield to maturity y, by solving the equation:

$$P = \frac{c}{(1+\frac{y}{K})^{\Delta t_1 K}}+\sum\limits_{i=1}^{V-1}\frac{\frac{Nr}{K}}{(1+\frac{y}{K})^{\Delta t_1 K+i}}+\frac{N(1+\frac{r}{K})}{(1+\frac{y}{K})^{\Delta t_1 K+V}}.$$

We can now calculate the yield-spread by $s=y-r$.

We now treat $r$ as a variable, but keep the yield-spread $s$ fixed. We denoted $G(r)$ as the price of the bond right after the first reset date. We then get


Our aim is to find a better expression for $G(r)$, by adding and subtracting $Ns/K/(1+(r+s)/K)^V$ we get


By doing then adding and subtracting $Ns/K/(1+(r+s)/K)^{V-1}$ we simplify it even further. By doing this trick again and again we end up with


We write $G(r)=N-F(r)$, where


Note that $F'(r)<0$.

Also note that the present value of the price is

$$\frac{(c+G(r))}{(1+\frac{r+s}{K})^{\Delta t_1 K}}.$$

Now we can calulate the duration

$$-\frac{1}{P}\frac{dP}{dr}=-\frac{\frac{d}{dr}\left(\frac{(c+G(r))}{(1+\frac{r+s}{K})^{\Delta t_1 K}}\right)}{\frac{(c+G(r))}{(1+\frac{r+s}{K})^{\Delta t_1 K}}}\\=-\frac{\frac{-\Delta t_1(c+G(r))}{(1+\frac{r+s}{K})^{\Delta t_1 K+1}}+\frac{G'(r)}{(1+\frac{r+s}{K})^{\Delta t_1 K}}}{\frac{(c+G(r))}{(1+\frac{r+s}{K})^{\Delta t_1 K}}}\\=\frac{\Delta t_1}{(1+\frac{r+s}{K})}-\frac{-F'(r)}{c+G(r)}\\=\frac{\Delta t_1}{(1+\frac{r+s}{K})}+\frac{F'(r)}{c+G(r)}.$$

Now since $F'(r)$ is negative, we see that when $\Delta t_1$ is small, this entire expression can be negative.

Do you see anything wrong with my argument, or is it correct?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.