Interest Rates: Where the Price of Money Comes From

The first three modules priced the risk that a borrower doesn’t pay. But every one of those loans was priced over something — a funding cost, a benchmark, a “risk-free rate” that we treated as given. From here on, that shortcut stops working. The structural models of Module 6 discount at a rate $r$; the reduced-form models of Module 7 define credit risk as a spread over a curve. And in practice — as a lender, a borrower, or an analyst — you constantly meet finished rates with no explanation attached: a mortgage at 6.58%, a card at 23.8%, a CETES auction at 6.5%.

The goal of this module is that no rate ever looks like a single number to you again. Every interest rate is a sum of named parts, each part compensates a specific risk, and you can estimate each part from things you can observe. By the end you’ll rebuild real mid-2026 rates — a US mortgage, a Mexican mortgage, a car loan, a credit card — from their components, and see exactly where every percentage point lives.

An interest rate is a price — and prices itemize

An interest rate is the price of moving money through time: what you must pay to have money now, or what you earn for accepting later. Like any price, it compensates the seller — the lender — for everything they give up or take on. The CFA curriculum’s canonical decomposition, which we’ll use as the spine of this module, names five components:

• Real risk-free rate ($r_{\text{real}}$) — compensation for postponing consumption in a world with no inflation and no risk. Driven by slow forces: demographics, productivity, the global balance of saving and investment. Nobody sets it; it emerges. Think 1–2% in developed markets, historically higher in emerging ones.
• Inflation premium (IP) — compensation for the expected loss of purchasing power over the horizon, plus a slice for inflation uncertainty. This is the Fisher equation at work: $(1 + r_{\text{nominal}}) = (1 + r_{\text{real}})(1 + \pi)$. Crucially, IP is about the future, not the latest CPI print — and about credibility. A country that produced 100%+ inflation within living memory pays an inflation premium long after its actual inflation converges. Hold that thought; it is most of the Mexico story below.
• Default risk premium (DRP) — compensation for the possibility the borrower doesn’t pay. This is where the whole credit track plugs in, and we’ll open it up fully in a moment: DRP is built from PD and LGD.
• Liquidity premium (LP) — compensation for holding something you can’t exit quickly at fair value. A US Treasury has essentially none; a loan to a private company that no one else will buy has a lot.
• Maturity premium (MP) — compensation for locking in longer: more years of price risk if you must sell, more years for inflation and default to surprise you. In the market this is called the term premium, and it’s what makes yield curves slope up on average.

The first two components — real rate plus inflation premium — make up the nominal risk-free rate: what a default-free, liquid, short-dated instrument pays. The last three are the risk add-ons that a specific instrument stacks on top. Every rate you will ever meet is this sum with different terms switched on and different sizes plugged in.

The anchor: who sets the risk-free part

For the shortest horizon — overnight — the price of money is not discovered, it is administered. Central banks set a policy rate: the Fed funds target in the US (3.50–3.75% as of mid-2026), Banxico’s tasa objetivo in Mexico (6.50%, where it has sat since the cutting cycle ended). They enforce it operationally — paying interest on reserves, lending and borrowing at rates that bracket the target — so overnight money trades where the committee wants it.

How do they choose the number? Modern central banks are inflation targeters — Banxico’s target is 3% ±1 point, the Fed’s is 2% — and their reaction function is well captured by the Taylor-rule intuition: sit near the neutral rate, push above it when inflation runs hot, below it when the economy slumps. You don’t need the coefficients; you need the logic, because the market uses it to forecast the committee, and those forecasts are the yield curve.

Everything longer than overnight is built from two ingredients:

The first ingredient is the expectations hypothesis — a long rate is the average of the short rates the market expects, enforced by arbitrage. The second is the maturity premium from our decomposition, wearing its market name. Build a curve yourself and watch the shapes appear:

The vocabulary: normal (upward — stable expectations, positive premium), steep (hikes expected, or premium blown out), inverted (long below short — only possible if the market expects cuts hard enough to overwhelm the premium, which is why inversion is the classic recession signal).

One housekeeping note: the short-rate benchmarks contracts actually reference are now transaction-based overnight rates — SOFR in dollars (repo-based; it replaced LIBOR after the manipulation scandals), TIIE de Fondeo in pesos. When Module 7 discounts a CDS leg “at the risk-free curve,” it means a curve built on these.

From the curve to the rate you’re charged

The build-up above prices traded instruments. But most of the rates people actually meet — mortgages, car loans, cards — are set by a lender, not read off a screen. A bank prices a loan the way any business prices a product: cost, plus expected losses, plus a return on the capital the product consumes. Written out:

This is the banker’s version of the CFA build-up, and the components map one-to-one. The funding cost is the nominal risk-free rate at the matching maturity plus the bank’s own credit and liquidity premia (banks are not risk-free borrowers). The expected-loss term is the default risk premium. The capital charge is the price of unexpected loss — Module 1’s EL/UL distinction showing up in the price tag: you charge for EL in the rate, you hold capital against UL, and the capital has a cost that also lands in the rate.

Look closely at the expected-loss term, because this is where the credit track’s master equation becomes a price component:

Two borrowers with the same PD pay different rates if their LGDs differ. A 3% PD with 15% LGD (well-collateralized mortgage) needs ~45 bps of annual compensation; the same 3% PD with 85% LGD (unsecured card) needs ~255 bps. Collateral doesn’t change whether you default — it changes what the lender loses when you do, and the rate prices the product of the two. This is why the single fastest way to cut your own borrowing cost is not a better score but better security: the same person, moving the same debt from a card to a collateralized loan, can drop 20+ percentage points, nearly all of it LGD.

Now let’s rebuild real rates. All numbers as of mid-2026; they’ll drift, but the decomposition is the durable part.

Case study 1: the same house, two countries

A 30-year fixed mortgage in the US averaged 6.58% in late July 2026 (Freddie Mac). A Mexican bank mortgage averaged 11.5%, with the best banks quoting 9.9–11.5% (national comparison, 2026). Five percentage points apart, for the same product secured by the same kind of asset. Where does the gap live? Stack the build-ups side by side:

The punchline hides in the last row: the spreads are nearly the same. Mexican banks add roughly two points over their government curve; American lenders add roughly two points over theirs. Almost the entire five-point difference between countries lives in the sovereign benchmark — that is, in the first two components of the build-up, before any credit risk enters:

• Inflation premium, mostly. Not today’s inflation — mid-2026 headline inflation is actually lower in Mexico (3.55%) than in the US (4.2%). The premium prices expected inflation over decades plus inflation risk, and those are priced from history and credibility. Mexico lived 100%+ inflation in the 1980s and high single digits within the 2020s; the market charges for the memory. Central-bank credibility is a national asset measured in percentage points, and every Mexican borrower pays rent on its absence.
• A higher real rate and term premium. Emerging-market sovereigns pay more real interest (scarcer capital, currency risk for foreign holders) and a fatter premium for long lockups.

And inside the ~2-point spreads, the composition differs even though the totals match:

• The US spread is mostly not credit. A US 30-year fixed mortgage carries a prepayment option — the borrower can refinance whenever rates fall, which is exactly when the lender least wants the money back. Investors in mortgage-backed securities charge for writing that option; it’s the biggest slice of the 1.9 pp, alongside servicing and the guarantee fees of the agency securitization machine. Actual expected credit loss is a thin slice — recoveries on foreclosed US homes are substantial and the guarantors absorb the tail.
• The Mexican spread is more credit and cost. Foreclosure in Mexico has historically been slower and costlier — years of process eat the recovery, which raises LGD even when the collateral holds value. Banks defend against it with lower loan-to-value ratios and stricter aforos (bigger down payments = smaller LGD = the borrower self-insuring the lender). Add higher per-loan origination and servicing costs over a smaller lending base, and less refinancing pressure (Mexican mortgages are mostly fixed with limited prepayment culture, so less option cost but also less competition). Interesting detail: the sharpest Mexican banks quote as low as 0.7 pp over the Bono M — for their best-collateralized, lowest-LGD clients, mortgage lending is nearly a rates business, exactly as the equation predicts.

Case study 2: one borrower, three products

Same person, same month, three offers: a mortgage at ~6.6%, a new-car loan at ~6.7% (US average; 5.2% super-prime to 13%+ subprime), and a credit card at ~23.8% APR (average new-offer rate). Identical borrower, identical FICO — so the borrower’s creditworthiness can’t explain the ordering. The components can:

Three lessons hiding in the table:

1. PD is a property of the product, not just the person. The same borrower defaults on different debts at different rates because default is a choice made under stress, and people triage: mortgage last, card first. Lenders know the payment hierarchy and price it. Your card rate assumes the worse version of you; your mortgage rate assumes the better one.
2. LGD does the heavy lifting. Walk the LGD column and the rate column together. The mortgage and the car loan have nearly identical rates despite the car being worse collateral — because the car loan is shorter (less MP) and repossession is fast (LGD contained), roughly offsetting. The card has no collateral at all: when a card defaults, the lender recovers cents. An unsecured revolving product must charge the expected-loss premium of its worst realistic cohort, all through the DRP term.
3. Small loans carry big cost loadings. A chunk of the card’s 23.8% isn’t risk at all — it’s opex: fraud losses, network costs, rewards given back to the very borrowers who never revolve, and the servicing of millions of small balances. Expensive product, not just risky borrower.

And the car lease? A lease usually beats the loan rate for the same car and borrower, and the build-up explains why: the lessor owns the asset. There is no lien to enforce — a defaulted lease means taking back your own car, which is a structurally lower LGD than repossessing collateral. The lessor instead takes residual-value risk (what the car is worth at lease-end), which is a market risk, not a credit risk, and manufacturers routinely subsidize the implicit rate to move inventory. (Quoting quirk: US leases hide the rate in a “money factor” — multiply it by 2400 to get the approximate APR. A money factor of 0.0025 is ~6%.)

The Mexican version of this table is the same shape shifted up by the sovereign curve — with one violent exception. Bank credit cards in Mexico commonly run 30–80% CAT, and the fintech cards extending credit to thin-file customers run well past 100% (Nu ~146% CAT, Stori ~205%). Run the build-up on a 150% CAT and it still parses: an unbanked first-card customer with no credit history is a 15–30% annual PD proposition at ~90% LGD — that’s a 15–25 pp expected-loss premium alone — plus the opex of tiny balances, plus the funding curve, plus the losses of everyone who was approved and shouldn’t have been, all recovered from those who pay. Whether that price is fair is a real question (Module 12 takes up fairness formally); that it is decomposable is this module’s point. No rate is a mystery; some are just grim arithmetic.

From rates to discount factors: the object models actually use

Everything above was economics. What the models of Modules 5–12 consume is arithmetic: a machine that converts “$1 at time $T$” into “$x today.” That machine is the discount factor:

Three things to internalize about $D(T)$:

1. It’s the price of a promise. $D(5) = 0.78$ means the market pays 78 cents today for a riskless dollar in five years. A discount curve is a price list for future dollars.
2. Compounding conventions are packaging, not substance. The same $D(T)$ can be quoted as slightly different $y_T$ depending on convention. Continuous compounding wins in models because it makes rates additive: discounting at $r$ and adding a hazard $\lambda$ gives $e^{-(r+\lambda)T}$ — precisely the Duffie–Singleton trick waiting in Module 7.
3. The “risk-free” curve is a choice. Treasury, SOFR/OIS, CETES — they differ by basis points (collateral, taxes, scarcity). For this track, “risk-free” means “the best available curve with no default risk in it,” and the interesting object is always the spread over it — which, as of this module, you can now itemize.

The zero-coupon rate $y_T$ and the discount factor are the same information in two outfits; the workbook’s first sheet converts between them, builds a discount curve from a handful of zero rates, and computes the implied forwards — the exact curve the next module prices bonds off.

Where this connects

• Module 5 prices bonds as bundles of promises weighted by this module’s discount factors, and formalizes the credit spread you’ve now met as DRP + LP.
• The expected-loss premium PD × LGD is Module 1’s master equation divided by exposure and per year — the same three letters, now setting prices instead of measuring losses.
• Merton (Module 6) uses $r$ inside Black-Scholes and derives the DRP as an option premium; reduced-form models (Module 7) write it as $\lambda(1-R)$ — intensity times LGD, the credit triangle, which is this module’s expected-loss premium in continuous time.
• Rate risk itself — what happens to a bank when the curve moves — is the “S” in CAMELS (Module 9). The 2023 US regional-bank failures were curve risk, not credit risk.
• In the personal-finance track, the debt module’s payoff logic and the retirement module’s real-vs-nominal distinction are this module’s components doing household chores.

Try it yourself

Two exercises. First, the workbook below (shared with Module 5): enter a policy rate, an expected path, and a term premium to generate zero rates, discount factors, and forwards with visible formulas. Rebuild yesterday’s CETES or Treasury curve once from public data — afterward, every “the curve steepened” headline reads as a sentence about expectations and premia.

Second, no spreadsheet required: take one rate you’re personally paying — mortgage, car, card — and itemize it. Look up your country’s policy rate and 10-year yield, subtract, and write down what’s left as PD × LGD + opex + margin. Estimate the LGD from the collateral. If you can defend each line to yourself, this module did its job — and you’ll never see an interest rate as a single number again.

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