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Summary

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?

Calculations

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

$$G(r)=\sum\limits_{i=1}^{V-1}\frac{\frac{Nr}{K}}{(1+\frac{r+s}{K})^{i}}+\frac{N(1+\frac{r}{K})}{(1+\frac{r+s}{K})^{V}}.$$

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

$$G(r)=\sum\limits_{i=1}^{V-1}\frac{\frac{Nr}{K}}{(1+\frac{r+s}{K})^{i}}+\frac{N(1+\frac{r+s}{K})}{(1+\frac{r+s}{K})^{V}}-\frac{\frac{Ns}{K}}{(1+\frac{r+s}{K})^V}\\=\sum\limits_{i=1}^{V-1}\frac{\frac{Nr}{K}}{(1+\frac{r+s}{K})^{i}}+\frac{N}{(1+\frac{r+s}{K})^{V-1}}-\frac{\frac{Ns}{K}}{(1+\frac{r+s}{K})^V}.$$

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

$$G(r)=N-\sum\limits_{i=1}^V\frac{\frac{Ns}{K}}{(1+\frac{r+s}{K})^i}.$$

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

$$F(r)=\sum\limits_{i=1}^V\frac{\frac{Ns}{K}}{(1+\frac{r+s}{K})^i}$$.

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?

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