I am currently studying interest rate risk management, and i can't seem to get the derivation right, and I would like to do all of the steps, to be sure that I understand what is going on.
Let Pz (t, T ) be the price of a zero coupon bond at time t with maturity T and continuously compounded interest rate r.
Duration = $-\frac{1}{P} \frac{d P}{d r}$
Let A and a be two constants and x be a variable. Let $F(x)=A \times e^{a x}$ be a function of x. Then, the first derivative of F with respect to x, denoted by $\frac{d F}{d x}$, is given by
Derivative of F(x) with respect to $x=\frac{d F}{d x}=A \times a \times e^{a x}=a \times F(x)$
The book shows (duration of zero coupon bond): $D_{z, T}=-\frac{1}{P_{z}(t, T)}\left[\frac{d P_{z}(t, T)}{d r}\right]$
$=-\frac{1}{P_{z}(t, T)} \times\left[-(T-t) \times P_{z}(t, T)\right]$
$=T-t$
Because I know the theory this makes total sense, but I cannot derive it. Does someone know how to do this?