Compounding: This shit runs your whole entire life.
We live in a world designed around a sick compounding system. It wasn’t always this way, but in the 1930s we came up with a code in the US national commercial banking scheme to standardize the way money is created via loan issuance. Prior to the 1930s, loans were typically repaid via a balloon payment structure. You would borrow money from the bank to purchase a house or some other type of loan, and every month you would pay a fraction of that principal amount as interest, until the payment date came along. On that payment date, you would pay the entire principal you owed. If you couldn’t come up with the money, tough luck. Basically, your only options were to make the balloon payment, default, or find another lender to give you money so you could rollover your debt and be perpetually broke. To this day, U.S. Treasury bonds issued to finance the government’s spending are paid via this same structure. Thus, the balloon payment is not an extinct species, but actually is thriving like never before, with $37 trillion in government debt and counting.
As you can probably see, this was deemed too risky of a standard practice. The plebes couldn’t be allowed to borrow in the same style as the benevolent and all-knowing government. Instead, it was replaced with a loan payment structure that we call amortization. In an amortized loan, the borrower pays a fixed amount once a month until they pay off the entire principal. Every monthly payment includes a portion that goes to pay down interest, and another portion that goes to pay down the principal, but the total sum of these two does not change.
One of the best examples of compounding in action is the 30-year fixed mortgage, where lenders design payments to minimize the variance in monthly payments. But here’s the secret: if you understand the math behind it, you can use compounding against the lender and in your favor.
The Math Behind a 30-Year Mortgage
A standard mortgage is structured using an amortized loan formula, ensuring that payments remain constant while the balance steadily decreases. The formula for the monthly payment $A$ is:
$$ A = P\frac{i(1+i)^n}{(1+i)^n-1} $$
Where:
- $P$ is the initial principal (loan amount).
- $i$ is the monthly interest rate (annual rate divided by 12).
- $n$ is the total number of payments (e.g., 360 for a 30-year loan).
- $A$ is the fixed monthly payment.
At first, a significant portion of each payment goes toward interest. Over time, as the principal decreases, the interest portion shrinks, and a larger share of the payment goes toward principal. This dynamic is an inverse of how compound interest works in investing—here, it’s compounding against you rather than for you.
But what if you could flip the game in your favor?
Derivation of the Mortgage Payment Formula
To see how the bank stacks the deck, let’s derive the amortized payment formula.
Let’s define:
- $A_k$ as the $k$-th monthly payment (fixed at $A$).
- $P_k$ as the remaining loan balance after $k$ payments.
- $Q_k$ as the interest portion of the $k$-th payment.
Step 1: First Month Calculation
- Initial principal: $ P_1 = P $
- Interest portion: $ Q_1 = iP = iP_1 $
- Principal portion of payment: $ A−Q_1 = A−iP_1 $
- Remaining balance after payment:
- $$ P_2 = P_1−(A−Q_1)=P−(A−iP) $$
- $$ P_2 = P(1+i)−A $$
Step 2: nth Month Calculation
Using the recursive formula,
$$ P_3=P_2(1+i)−AP_3 = P_2(1+i)−A $$
Substituting
$$ P_2 = P(1+i) − AP_2 = P(1+i) − A $$
We get
$$ P_3 = [P(1+i)−A](1+i) – A $$
$$ P_3 = P(1+i)^2−A(1+i)−A $$
$$ P_3 = P(1+i)^2 − A\sum\limits_{j=0}^1(1+i)^j $$
Generalizing this pattern for any month $n$,
$$ P_n = P(1+i)^{n−1} − A\sum\limits_{j=0}^{n-2}(1+i)^j $$
Since the summation is a geometric series,
$$ \sum\limits_{j=0}^m(1+i)^j = \frac{(1+i)^{m+1} – 1}{(1+i) – 1} = \frac{(1+i)^{m+1} – 1}{i} $$
Applying this to our equation,
$$ P_n = P(1+i)^{n−1} − A\frac{(1+i)^{n-1} – 1}{i} $$
At the final month $n+1$, the balance must be zero:
$$ P_{n+1} = 0 = P(1+i)^n−A\frac{(1+i)^{n} – 1}{i} $$
Solving for $A$:
$$ A = P\frac{i(1+i)^n}{(1+i)^n−1} $$
This equation ensures that the loan is fully paid off after $n$ payments.
This structure ensures that payments are predictable, minimizing risk for the lender. I find it amusing, and surely it is no coincidence, that this solution method to the amortization problem is uncannily similar to the solution method for the gambler’s ruin problem. We are all gamblers, the only difference between us is whether we want to admit it or not. In light of this fact, it is important to minimize our own financial risk whenever we can.
The Future Value of Payments: The Lender’s Perspective
Suppose you are a rich person, with a pile of cash, $P$, that you want to invest. If you are able to find an investment with an interest rate, $i$, the future value of your money after $n$ months is:
$$ F = P(1+i)^n $$
If instead of finding an investment, you found a borrower to whom to lend your pile of cash, $P$, at an interest rate, $i$, then set up an amortized payment structure, and finally reinvest the amortized cash flows at the same interest rate, $i$, the future value of your money after $n$ months is:
$$ F = A\sum\limits_{j=0}^{n-1}(1+i)^j $$
Applying the geometric series formula:
$$ F = A\frac{(1+i)^n−1}{i} $$
Substituting,
$$ A = P\frac{i(1+i)^n}{(1+i)^n−1} $$
we get,
$$ F = P\frac{i(1+i)^n}{(1+i)^n−1}\frac{(1+i)^n−1}{i} $$
Cancelling terms:
$$ F = P(1+i)^n $$
This result shows that the future value of the payments exactly equals the future value of the principal. In fact, for any loan payment structure, the future value of all payments discounted to the day that the entire principal is fully paid off, will exactly equal the future value of the principal. This is begging for a deeper explanation: the amortized loan structure is the result of an optimization problem.
Optimization Problem: Why Lenders Love Amortization
Lenders don’t just pick fixed monthly payments arbitrarily. They design the payment structure to minimize their risk—which in this case means minimizing the variance in payments.
Step 0: Define the Monthly Payments
Let $A_k$ represent the payment made at month $k$, for $k = 1, 2, …, n$. Our goal is to find a set of payments ${A_k}$ that minimizes the variance while ensuring the full repayment of the loan.
Step 1: Objective Function – Minimize Variance
The variance of payments is:
$$ Var(A) = \frac{1}{n}\sum\limits_{k=1}^n(A_k − \bar{A})^2 $$
where:
$$ \bar{A} = \frac{1}{n}\sum\limits_{k=1}^nA_k $$
This is the objective function. Minimizing variance ensures consistent, predictable payments.
Step 2: Constraint – Full Loan Repayment with Future Value Discounting
The total sum of all payments, discounted to the present, must equal the loan principal compounded forward:
$$ \sum\limits_{k=1}^nA_k(1+i)^{n-k} = P(1+i)^n $$
This formulation ensures that each payment is properly weighted according to its time value—early payments contribute more to future value, while later payments are more heavily discounted.
Step 3: Solve for the Optimal Payment Structure
To minimize variance, the trivial solution is when every $A_k$ is equal, meaning:
$$ A_1 = A_2 = ⋯ = A_n = A $$
Substituting into the constraint equation, $A$ drops out of the sum:
$$ A\sum\limits_{k=1}^n(1+i)^{n-k} = P(1+i)^n $$
Since the summation is a geometric series:
$$ \sum\limits_{k=1}^n(1+i)^{n-k} = \frac{(1+i)^n – 1}{i} $$
we obtain:
$$ A\frac{(1+i)^n – 1}{i} = P(1+i)^n $$
Solving for $A$:
$$ A = P\frac{i(1+i)^n}{(1+i)^n−1} $$
which is the standard amortized loan formula.
In other words, the amortized payment structure is mathematically optimal for minimizing risk—for the lender.
By formulating the constraint correctly with future value discounting, we see that the amortized structure emerges naturally as the optimal solution for minimizing variance in payments. This is why lenders structure loans in this way—it ensures stability while fully repaying the debt within the given timeframe.
Thus, fixed mortgage payments aren’t just a convention; they’re a mathematical necessity when aiming for predictable and manageable loan repayments.
How the Borrower Wins by Varying Payments
The field of finance spans the methods, techniques and principles whereby humans move, allocate, and price money and risk. While varying payments does not move money between borrower and lender, it does move risk. When you pay off a loan early, you move risk from your account to the lender’s account. You may not gain an edge on the first moment of the distribution, but you certainly gain on the second moment. In fact, this edge in the second moment is quite significant. Neglecting or mishandling risk has been the downfall of many. Banks hate when borrowers pay off loans early, because it increases their risk profile. If a bank does not like you to pay off your loan early, you should probably consider that as a good sign that you actually should pay off your loan early.
- Early Lump Sum Payments Reduce Interest Costs
- If a borrower makes an early lump sum payment, they effectively reduce the outstanding principal sooner. Since interest is calculated on the remaining balance, less principal means less total interest paid.
- The amortization formula assumes the borrower will make the fixed payment A every month, maximizing the lender’s total interest earnings over time.
- By paying early, the borrower breaks this assumption, shifting the game in their favor. As we know, the future value of the payments stays the same, so you are not actually winning money back from the lender by paying earlier. But still, a penny saved is a penny earned.
- Lump Sum Payments Disrupt the Lender’s Expected Cash Flow
- Lenders model loan portfolios assuming regular payments with minimal variance.
- A large, early payment disrupts their expected return structure. Lenders prefer slow, steady interest payments rather than an early repayment that shrinks the principal too quickly.
- The Lender’s Risk Model Favors Fixed Payments
- Lenders prefer low-variance cash flows because they can predictably reinvest interest payments.
- If borrowers suddenly repay in a lump sum, the lender now has to find a new borrower or investment to replace that lost interest income—which may not be as profitable. Finding a borrower also incurs a non-zero cost: mortgage brokers don’t work for free. That cost is the cost of risk. If you do not reinvest your cash, you run the risk of not being able to pay your outgoing cash flows generated by your liabilities with your incoming cash flows generated by your assets.
- If the Game is Zero-Sum, Borrower Gains When the Lender Loses
- If we assume a zero-sum game, where every dollar the lender takes off in risk is a dollar the borrower puts on in risk, then reducing the lender’s asset account balance is an automatic win for the borrower.
- Varying payments (especially through early lump sums) hurts the lender’s risk management, meaning the borrower benefits.
Should You Vary Payments?
There are two interest rates you should keep your eye on. The first is the interest rate you are paying to the bank, $i_L$. The second is the interest rate available for you to invest in, $i_I$. There are three cases:
- $i_L > i_I$
- Yes, no question. In this case, a penny saved is a penny earned. The best interest rate available to you is not from your investments, but from paying down your loan balance. In addition, hurting the bank by making early lump sum payments is a financial advantage because it reduces borrower risk.
- $i_I > i_L$
- No, because your investment returns exceed the mortgage rate. If you have an alternative investment that grows faster than your mortgage interest rate, then keeping your mortgage and investing instead may be the better play. You should play the bank’s game, but be aware that you are gambling, and manage risk accordingly. Keep in mind the second moment of the distribution.
- $i_L = i_I$
- This case is confusing, if you haven’t done the preceding analysis. You should pay the loan balance, because while you don’t gain money on a spread, you do gain on the risk balance, at the expense of the bank. In fact, this is the case for a threshold interest rate differential, $i_T$, where you should pay the loan balance for $i_L + i_T \geq i_I$. This differential takes into account the cost of risk, which has the same units as expected earnings. This is a hard problem in mathematical finance that depends on multiple variables that vary from person to person and situation to situation, and falls outside the scope of this introductory essay.
In short: Varying your payments, especially by paying early, is a strong strategy for borrowers and a disadvantage for lenders.
Final Thoughts: Use Compounding to Your Advantage
Compounding is either your worst enemy or your best ally—depending on which side of the equation you’re on.
- If you’re borrowing, disrupt compounding by paying early.
- If you’re investing, harness compounding by reinvesting earnings.
The question isn’t if compounding is at play—it’s whose side it’s on.
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