[Math] Rate of Loan Principal being repaid function evaluation

Question

I am trying to obtain the function which gives the rate at which the principal of a compound interest loan is being repaid at. So over the life of the loan the total amount to be paid back is $Pe^{r(t)\tau}$ where $P$ is the original principal, $r(t)$ is rate of interest of the variable interest loan paid back in time $\tau$. Let's assume $\tau$ is a constant for this evaluation and the interest is compounded continuosly and the repayments are also being made continuously. So $$ \int_ 0^\tau x(r(t), t)dt = Pe^{r(t)\tau}$$ should define my repayment function for the whole loan, obviously only a part of this is going towards paying back the original principal. So my immediate thought was to define the interest-only repayment function like $$ \int_ 0^\tau y(r(t), t)dt = Pe^{r(t)\tau} – P$$ and then subtract then the difference of $x(r(t), t) – y(r(t), t)$ would give the rate of repayment of the principal, but i do n't think this is correct. How can i calculate this ? Any help with forming the correct equation would be much appreciated.

Answer 1

$Pe^{r(t)\tau}$ does not make a lot of sense. Ignoring the confusion caused by the so-called "continuous compounding interest rate" which is designed to mislead economics students, you have it as a function of $t$, something you are supposed to have integrated out, and you seem to be suggesting that the interest rate applies to the original principal throughout the period from $t=0$ to $\tau$ when for most of the period the amount outstanding is (one would hope) less than the original amount.

So let's write $x(t)$ as the rate at which the repayments (including interest) are made at time $t$, $Q(t)$ as the outstanding amount at time $t$, and $e^{r(t)}$ as the interest rate at time $t$. You start at $Q(0)=P$ and finish at $Q(\tau)=0$. So the amount outstanding at any point in time is determined by $$\frac{dQ(t)}{dt} = Q(t) e^{r(t)} - x(t)$$ and so $$\int_0^\tau \left( x(t) - Q(t) e^{r(t)} \right) \; dt = P \text{ or } \int_0^\tau x(t) \; dt = P + \int_0^\tau Q(t) e^{r(t)} \; dt $$ if the loan is to be finally repaid by time $\tau$.

If the variable interest rate is not foreseeable in advance then $x(t)$ may vary not just with $r(t)$ but also depend on previous interest rates through the amount outstanding and the remaining time $\tau-t$. It may also vary with the particular structure of the loan: one of the many issues of the sub-prime crisis was the use of initial "teaser" rates. One possibility for setting $x(t)$ is to make it such that if interest rates were to be constant from that point onwards, it too would be constant for the remainder of the period.

The rate at which interest is being paid at time $t$ is $Q(t) e^{r(t)}$, which needs to be less than $x(t)$ if the amount outstanding is to reduce; its integral is the total amount of interest paid over the period.