A common class of memes complains about banks’ “practice” to have homeowners and other borrowers pay interest first, and pay the principal only later.

These memes suggest that there must be some sort of conspiracy between bankers to subjugate borrowers. After all, why can’t I pay the principal first, thereby reducing the balance?

A common response to the meme is that homeowners entered this arrangement willingly. While that is generally true, it misses the point entirely. The payment schedule, with higher interest rate payments first, is mathematically determined and necessary. For a level-payment, fixed-rate, fixed-term, fully-amortizing loan, the payment schedule is dictated by facts and logic! So, not only is there no conspiracy, and not only did borrowers voluntarily agree to the terms of a loan, but banks simply have no choice about the payment schedule!

Simply put, for a standard fixed-rate, fixed-term, level-payment fully amortizing loan, the interest portion is larger at the start because interest is computed on the outstanding balance, which is largest at the start.

We can prove this mathematically. Let’s consider a simple loan of $B_0$ in your favorite currency. The loan has a fixed periodic interest rate $r$ and term of $N$ periods. (None of these assumptions are crucial, but they simplify the exposition tremendously.)

Let $B_n$ reflect the balance after $n$ payments, where $n \in \{0, 1, \ldots, N\}$. Initially, the balance is $B_0$; and after all $N$ payments we want the loan paid off, so $B_N = 0$. How does the balance change from one period to the next? Simple. Interest accrues on the current balance, and then the payment $P$ is subtracted. Mathematically,

\begin{equation} B_{n+1} = (1 + r)B_n - P. \label{loanrate} \end{equation}

Equation \eqref{loanrate} is what mathematicians call a first-order linear difference equation. Its unique solution can be shown to be $B_n = c(1+r)^n + \frac{P}{r}$. Since we know that $B_0$ is the initial principal and $B_N = 0$, it follows after some simple calculations that $P$ must, mathematically have the following value:

\begin{equation} P = \frac{rB_0(1+r)^N}{(1+r)^N - 1}. \label{payment} \end{equation}

Under this setup, this is the single possible payment! It is literally dictated by the laws of mathematics. Let’s inspect it further. In period $n$, interest owed is $I_n = rB_{n-1}$ and principal paid is $\Pi_n = P - I_n$. Substituting the solution for $B_{n-1}$:

\begin{align*} I_n &= (rB_0 - P)(1+r)^{n-1} + P\\ \Pi_n &= (P - rB_0)(1+r)^{n-1} \end{align*}

If we look at $I_{n+1} - I_n = r(rB_0 - P)(1+r)^{n-1}$, we find that this expression is actually negative. (This is because $P > rB_0$ to amortize the loan over time, so that the loan is actually paid off at $N$.) The implication is that the unique possible payment schedule indeed starts off with a "high" interest portion that subsequently trails off.

However, $\Pi_{n+1} - \Pi_n = r(P - rB_0)(1+r)^{n-1}$ is positive! Thus, it is correct that, over time, the portion paid to the principal increases.

So the next time someone complains about banks’ “practice” of having you “pay interest first:” There is no practice. There is no policy. There is no choice. The payment schedule is not a decision anyone made. It is a mathematical necessity, as unavoidable as $2 + 2 = 4$.

What if you tried to “pay principal first”?

Suppose you demanded that your bank let you pay principal first, meaning you wanted $\Pi_n$ to start high and decrease over time, rather than the other way around. What would happen?

For $\Pi_n = (P - rB_0)(1+r)^{n-1}$ to be decreasing, we would need $P - rB_0 \lt 0$, i.e., $P \lt rB_0$. But look at what this implies for the balance. After the first period, $B_1 - B_0 = rB_0 - P > 0.$

The balance is increasing. Your payment does not even cover the interest! This is called negative amortization, where the debt grows rather than shrinks. Far from paying the loan off faster, you’re falling further behind!

Worse still, with $P \le rB_0$, the loan cannot be paid off in finite time. The boundary condition $B_N = 0$ becomes impossible to satisfy for any finite $N$. You would have to either increase your payment (back above $rB_0$, restoring the "interest-first" structure), extend the term to infinity (possible only for $P = rB_0$), or default. No bueno. And once again, this is simply just mathematics. No politician can save you from plain mathematics.

In other words, “pay principal first” is not an alternative payment schedule. The assumptions that define a standard amortizing loan (a fixed rate, fixed term, fixed payment, balance paid off at maturity) mathematically require the interest-heavy-first structure. You cannot violate it without violating one of those assumptions.

It is true that borrowers are often allowed to pay extra principal early (prepayments/curtailments). That changes the balance path and reduces total interest, which may of course be sensible. But it is not mathematically possible to merely increase the percentage of $P$ being spent on $\Pi_n$ without violating important assumptions of the model. Similarly, borrowers can often opt for interest-only payments to temporarily reduce payments. Once again, there is no free lunch: such arrangements require either higher payments later, an extended term, or a balloon payment at maturity. Other loan structures can violate other assumptions of the model, which may be individually acceptable, but the fundamental equations dictated by mathematics do not and cannot change.