Differential of Fisher-Weil Bond Price using discrete-time compunding I would like to ask you if my calculations are correct regarding the differential of the price of a bond using a sequence of spot rates as discount factors. 
\begin{equation}
P(i^S ) = \sum_{t=1}^{T} \frac{CF_t}{(1 + i_t^S)^t}
\end{equation}
is the Price of a Bond where the Cash Flows are discounted using spot rates. These spot rates vary for different maturities t. 
For example: for t=1 i=0.01, for t=2 i=0.015 etc.
Edit: Important assumptiom: Changes in i arise with the same magnitude for all maturities:
\begin{equation}
i_t^S + \Delta = (i_1^S + \Delta,..., i_T^S + \Delta)
\end{equation}
I'm not really a math wizard and not sure if my differential of this equation is correct or it is only possible using partial differentiation (?).
I need to submit my bachelor's thesis by monday and would be enormously thankful for any help! :)
\begin{align}
\frac{\partial P(i^S)}{\partial i^S} &= \sum_{t=1}^{T} \frac{(-t)CF_t}{(1 + i^S_t)^{t + 1}} \nonumber \\
&=\sum_{t=1}^{T} \frac{(-t)}{(1 + i^S_t)}\frac{CF_t}{(1 + i^S_t)^{t}}\nonumber \\
&=\sum_{t=1}^{T} \frac{(-t)}{(1 + i^S_t)}PV(CF_t).
\end{align}
\begin{align}
-\frac{1}{P(i^S)}\frac{\partial P(i^S)}{\partial i^S} &=   -\frac{1}{P(i^S)}\sum_{t=1}^{T} \frac{(-t)}{(1 + i^S_t)}PV(CF_t)\nonumber\\
&=   \sum_{t=1}^{T} \frac{t}{(1 + i^S_t)}\frac{PV(CF_t)}{P(i^S)}\nonumber\\
&= D^*_{\xi},
\end{align}
 A: Consider a nonflat term structure defined by the sequence of spot rates $i^S_t$, for $t = 1,\ldots, n$.
Denoting $\pmb i = (i^S_1, \ldots, i^S_n)$ as the vector of the spot rates and $P(\pmb i)$ as the price of the asset under the current term structure, we have
$$
P(\pmb i) =\sum_{t=1}^n\frac{C_t}{(1 + i^S_t)^t}
$$
Let $\pmb \Delta = (\Delta_1, \ldots, \Delta_n)$ denote the vector of shifts in the spot rates so that the new term structure is 
$$\pmb i + \pmb \Delta = (i^S_1+ \Delta_1, \ldots,i^S_n+ \Delta_n)$$
and the price of the asset under the new term structure is
$$
P(\pmb i+\pmb \Delta) =\sum_{t=1}^n\frac{C_t}{(1 + i^S_t+\Delta_t)^t}
$$
If, however, $\Delta_t=\Delta$ for $t = 1,\ldots , n$, then the term structure has a
parallel shift, then
$$
P(\pmb i+\pmb \Delta) =\sum_{t=1}^n\frac{C_t}{(1 + i^S_t+\Delta)^t}
$$
Using the first-order approximation in Taylor’s expansion
$$
\frac{1}{(1 + i^S_t+\Delta)^t}\approx \frac{1}{(1 + i^S_t)^t}-\frac{t\Delta}{(1 + i^S_t)^{t+1}}
$$
we can write
$$
P(\pmb i+\pmb \Delta) \approx \sum_{t=1}^n\frac{C_t}{(1 + i^S_t)^t}-\Delta \sum_{t=1}^n\frac{t C_t}{(1 + i^S_t)^{t+1}}
$$
which implies
$$
P(\pmb i+\pmb \Delta) -P(\pmb i)\approx -\Delta \sum_{t=1}^n\frac{t C_t}{(1 + i^S_t)^{t+1}}=-\Delta \sum_{t=1}^n\frac{t}{1 + i^S_t} \mathrm{PV}(C_t)
$$
where $\mathrm{PV}(C_t)=\frac{C_t}{(1 + i^S_t)^{t}}$.
Thus, we conclude
$$
-\frac{1}{P(\pmb i)}\left[\lim_{\pmb\Delta\to\pmb 0}\frac{P(\pmb i+\pmb \Delta) -P(\pmb i)}{\Delta}\right]=\sum_{t=1}^n\frac{t}{1 + i^S_t} \left(\frac{\mathrm{PV}(C_t)}{P(\pmb i)}\right)
$$
that is
$$
-\frac{1}{P(\pmb i)}\frac{\mathrm d P(\pmb i)}{\mathrm d \pmb i}=\sum_{t=1}^n\frac{t}{1 + i^S_t} \left(\frac{\mathrm{PV}(C_t)}{P(\pmb i)}\right)=D_{FW}
$$
The Fisher-Weil duration is a generalization of the Macaulay duration, with the present values of cash flows computed using a nonflat term structure.
