Loan repayments

Suppose I take out a loan. It gains interest at rate $r$, compounded continuously. I repay the loan by making constant, continuous payments until time $T$. How does the repaid share of my loan vary over time? And how does it depend on $r$ and $T$?

Let $P_0$ be the initial value of my loan: the “principal.” Then my continuous payments $C$ must satisfy $$\begin{array}{cc}{P}_0& ={\int }_0^{T}C{e}^{-r\tau }d\tau \\ & =\frac{C}{r}(1-e^{-rT})\end{array}$$ and so the value of my remaining payments at time $t\in [0,T]$ equals $$\begin{array}{cc}{P}_t& \equiv {\int }_t^{T}C{e}^{-r(\tau -t)}d\tau \\ & =\frac{C}{r}(1-e^{-r(T-t)})\\ & =P_0\left(\frac{e^{-rt}-e^{-rT}}{1-e^{-rT}}\right)e^{rt}.\end{array}$$ If I don’t make any payments before time $t$ then the principal grows to $P_0{e}^{rt}$. Therefore, the value of my repayments up to time $t$ equals the difference $(P_0{e}^{rt}-P_t)$.

Now let $x\equiv t/T\in [0,1]$ be share of payments I’ve made up to time $t$. The chart below plots the corresponding share $$\frac{P_0{e}^{rt}-P_t}{P_0{e}^{rt}}{|}_{t=xT}=\frac{1-e^{-xrT}}{1-e^{-rT}}$$ of the loan that I’ve repaid. This share grows with $x$ at a decreasing rate. Intuitively, my repayment “slows down” because the interest on the principal and payments grows larger than the payments themselves. This slowing effect is stronger when the interest rate $r$ is larger and time horizon $T$ is longer.