A loan is an annuity wearing a different hat
You have already met the hardest idea in this guide — you just met it from the other side. In the last guide an annuity-immediate was a stream of equal payments you *receive*; a level loan is that very same stream, only now you are the one *making* the payments and the lender is the one collecting them. Flip the chair around and the mathematics is identical. This is the quiet secret that makes loans feel suddenly easy: there is nothing new to learn about the cash flows, only a new question to ask of them.
So what makes a repayment plan fair? The same principle that governs every deal in this rung: the equation of value. The money the lender hands you today must equal the present value of everything you promise to pay back. For a level loan of L repaid by n equal payments at rate i per period, that single equation says L equals the payment times the annuity factor a-angle-n. Rearranged, the payment is simply L divided by a-angle-n — the loan amount sliced into a stream that is worth exactly the loan today, no more and no less.
Where each payment really goes
Here is the question that trips up almost every first-time borrower. You pay the same amount every month for thirty years — so surely each payment chips away an equal slice of the debt? Not even close. Every payment is quietly split into two parts that shift their proportions as the years pass. The rule is beautifully simple: interest is always charged on whatever you currently owe, and whatever is left of the payment after that interest goes to reducing the principal.
Trace it through. At the start you owe the full loan, so the interest piece is large and only a sliver of your payment touches the principal. But that sliver does shrink the balance — so next period's interest is computed on a slightly smaller number, leaving slightly more of the fixed payment to attack the principal. The principal portion therefore *grows* every period, accelerating as the balance falls, while the interest portion shrinks to match. The total payment never changes; only its internal split does. On the very last payment, the leftover principal is exactly the balance, and the debt lands on zero.
Loan 10,000 at i = 6% per year, 5 equal annual payments. Payment = 10,000 / a-angle-5 (at 6%) = 10,000 / 4.21236 = 2,373.96 Yr | Balance start | Payment | Interest (6%) | Principal | Balance end 1 | 10,000.00 | 2,373.96 | 600.00 | 1,773.96 | 8,226.04 2 | 8,226.04 | 2,373.96 | 493.56 | 1,880.40 | 6,345.64 3 | 6,345.64 | 2,373.96 | 380.74 | 1,993.22 | 4,352.42 4 | 4,352.42 | 2,373.96 | 261.15 | 2,112.81 | 2,239.61 5 | 2,239.61 | 2,373.96 | 134.38 | 2,239.58 | 0.03* *rounding; the exact schedule lands on 0.
The outstanding balance: what you still owe, today
Stop the clock partway through the loan and ask the most practical question of all: if I wanted to clear this debt today, what cheque must I write? That figure is the outstanding balance, and there are two honest ways to compute it that must always agree. The *prospective* method looks forward: the balance is the present value of the payments you have *not yet made* — exactly the lender's-annuity idea again, just for a shorter remaining term. The *retrospective* method looks backward: take the original loan accumulated forward with interest, and subtract the payments you *have* made, each one also accumulated forward with interest.
Both routes land on the same number because they are two windows onto one equation of value. In the table above, after the third payment the prospective balance is two payments of 2,373.96 discounted back, which is 4,352.42 — exactly the balance the row shows. This equality is more than a tidy check; it is also the reason the amortization schedule never drifts. And it underlines a subtle warning worth carrying: the balance is *not* the loan minus the bare sum of payments you have made. That naive subtraction ignores the interest those past payments would have earned, and it always understates what you still owe.
The other road: the sinking-fund method
Amortization is not the only honest way to retire a debt. The sinking-fund method takes a completely different posture: leave the original loan untouched at full value the whole time, and instead build a separate savings pot — the sinking fund — that grows to exactly the loan amount by the maturity date, when you settle the whole debt in one lump. So the borrower does two things every period. First, pay the lender interest on the *full* original loan; since the principal never falls, this interest payment is the same every single period. Second, deposit a fixed amount into the sinking fund.
How big is that deposit? The fund is itself an annuity — equal deposits that must accumulate to the loan amount L by time n — so the deposit is L divided by the accumulation factor s-angle-n, computed at the *fund's* own rate of growth. And that little phrase is the whole point. The crucial twist is that the sinking fund may earn a *different* rate than the loan charges. If the fund happens to earn exactly the loan rate, the sinking-fund method costs precisely the same as amortization, dollar for dollar — they are secretly the same deal. But if the fund earns a lower rate than the loan (the common case, since safe savings usually pay less than a loan charges), the borrower must deposit more to reach the target, and the sinking-fund method turns out more expensive.
There is one conceptual subtlety worth pinning down. Under amortization, the balance you owe really does fall every period. Under the sinking-fund method, the *debt* sits stubbornly at full face value the entire time — what falls is the borrower's *net* position, which is the loan minus the steadily growing fund. The two are economically comparable but not identical in their bookkeeping, and an actuary is careful never to confuse the fund's balance with a reduction in the debt itself. Corporations historically used sinking funds to retire bond issues, and the method survives wherever a lump-sum debt must be met on a fixed future date.
Why this is the doorway to mortgages and bonds
Everything in this guide pays off the moment you look at the two biggest financial instruments in the world. A mortgage is simply a level loan amortized over many years, usually with a nominal rate convertible monthly — borrow 200,000 at a nominal 6 percent compounded monthly (0.5 percent a month) over 360 months, and the payment is 200,000 divided by a-angle-360, about 1,199 a month. Every amortization idea here applies unchanged: the early payments are nearly all interest, the outstanding balance is the present value of what remains, and paying down early saves disproportionate interest.
A bond turns the picture inside out: now *you* are the lender. You pay a price today and receive a stream of equal coupons plus a redemption of face value at maturity — which is structurally a sinking-fund-style loan from the issuer's side, with coupons playing the role of the interest payments and the face value the lump owed at the end. The bond's price is just the present value of those coupons and that redemption, and the constant rate that makes that present value equal the price you paid is its yield — the yield to maturity, an internal rate of return. The very same equation of value, read from the buyer's side instead of the borrower's.
One honest caveat before you move on. The clean payment formula assumes a single, fixed, truthfully quoted interest rate — and the real world is messier. Advertised loan rates can hide fees, and a nominal rate compounded monthly costs more than its headline suggests; the only trustworthy way to find a loan's true cost is to write the full equation of value including every fee, not to trust the number on the poster. This humility is the whole spirit of interest theory: the formulas are exact, but they are only as honest as the cash flows and the rate you feed them. Next you will turn to how that rate is itself measured and how it varies with the length of the loan.