Bonds: Pricing the Promise

A bond is a loan that grew up and got a market price. That second part changes everything. A bank loan sits on a balance sheet at face value until something goes wrong; a bond is repriced continuously by strangers, and the price is an opinion about the borrower. That’s why this module sits where it does: the structural models of Module 6 and the reduced-form models of Module 7 both work by reading credit risk out of bond and derivative prices. To follow them, you need to know what a bond price contains.

Anatomy of the promise

Strip away the jargon and a standard (bullet) bond is a fixed schedule of cash flows:

• Face value (par, principal) — the amount repaid at the end. Quoting convention: prices are expressed per 100 of face.
• Coupon — the periodic interest payment, fixed at issue as a percentage of face. A 6% coupon on 100 face pays 6 per year (in most markets, 3 twice a year).
• Maturity — when the face value comes back.

Some useful variations: zero-coupon bonds pay nothing until maturity and sell at a discount (CETES and US T-bills are exactly this — buy at 97.5, receive 100, the 2.5 is your interest); floating-rate notes reset the coupon off a reference rate like TIIE or SOFR (Module 4); amortizing bonds repay principal along the way, like a mortgage. Issuers range from sovereigns in their own currency (Treasuries, Mexico’s Bonos M — udibonos if inflation-linked) through sovereigns in foreign currency, states, banks, and corporates — an ordering that will become a credit-risk gradient two modules from now.

Price is present value — nothing more

Take the discount curve from Module 4 and apply it to the schedule. Each cash flow is a promise; each promise has a price today; the bond’s price is the sum:

The middle expression is the honest one — each cash flow discounted off the curve at its own maturity. The right-hand one is the common shortcut: one single rate $y$ that discounts every cash flow. Run the equation backward — given the market price, solve for the $y$ that reproduces it — and you get the bond’s yield to maturity (YTM), the single most quoted number in fixed income.

From the formula, three facts follow immediately:

1. Price and yield move inversely. The cash flows are fixed; if the discount rate demanded by the market rises, the present value of those fixed flows falls. This is the seesaw at the center of all fixed income.
2. Par, premium, discount. Coupon above the market yield → price above 100 (premium); below → discount. And as maturity approaches, price converges to face — the pull to par — because a promise about tomorrow has nowhere else to go.
3. Long bonds swing harder. A yield change compounds over every remaining period, so the more distant the cash flows, the bigger the price reaction. That observation deserves its own number.

Duration: the seesaw, measured

Modified duration is the price sensitivity of the bond, in years:

A duration of 7 means a 1-percentage-point rise in yield costs about 7% of the bond’s value. Its sibling, Macaulay duration, is the value-weighted average waiting time of the cash flows — and the two are nearly the same number ($D_{\text{mod}} = D_{\text{Mac}}/(1+y)$), which is a genuinely elegant coincidence: how long you wait and how hard you fall are one quantity. Traders use the cash version, DV01 — dollars lost per basis point — because desks hedge in dollars, not percentages.

What drives duration is exactly what the present-value formula suggests: longer maturity → higher duration; higher coupon → lower duration (more of the value arrives early); zeros are the extreme case — duration equals maturity. A 30-year zero is pure, undiluted rate exposure.

The linear approximation bends, though. Price as a function of yield is a convex curve: each extra basis point of yield hurts slightly less, each basis point of relief helps slightly more. Convexity is the curvature correction — second-order, usually small, but systematically in the holder’s favor and decisive for long maturities and big moves.

Play with all of it — coupon, maturity, yield, and watch price, duration, and the curve respond:

Two experiments worth running. Set the coupon to zero and slide maturity out: duration marches up one-for-one, and the price-yield curve visibly bows. Then set a high coupon on the same maturity and watch duration drop — the early cash flows anchor the value. This is why the 2023 US regional-bank story (Module 9) was a duration story: portfolios of long, low-coupon bonds bought at the yield lows were maximally convex to exactly the hiking cycle that arrived.

The credit spread: where this track re-enters

Everything so far works for a default-free bond. Now price a corporate bond with the same cash-flow schedule and it always trades at a higher yield. The difference

is the credit spread — the bond market’s price for the possibility that the promise is not kept. (Measured properly against the whole curve rather than one point, it’s called a Z-spread: the constant shift of the risk-free curve that reprices the bond. Same idea, better plumbing.)

The temptation is to read the spread as a default probability with the serial numbers filed off, and to first order the decomposition does start there:

The first term is Module 1’s master equation per unit of exposure, per year. But the other two are not small: measured spreads run persistently wider than historical default losses justify — several times wider for investment grade. That gap (the “credit spread puzzle”) is compensation for the fact that defaults cluster in the worst states of the world, plus payment for illiquidity. Hold that thought; it becomes the risk-neutral vs. real-world distinction that Module 6 and Module 7 both turn into machinery.

One more piece of anatomy matters for credit: where the bond sits in the capital structure. Secured debt is backed by specific collateral; senior unsecured stands in the general queue; subordinated stands behind it; equity absorbs the first loss. Same issuer, one default event — very different recoveries. This is Module 1’s LGD wearing legal clothing, and it’s why one company’s bonds can carry different ratings at different seniorities (Module 8).

Where this connects

• The identity at the heart of Module 6 — risky debt = risk-free bond − put option — needs this module’s risk-free bond price and returns the credit spread as an option premium.
• Module 7 reprices this module’s risky bond as survival-weighted present value, and the credit triangle $s \approx \lambda(1-R)$ is the spread decomposition above with the risk premium folded into $\lambda$.
• Ratings (Module 8) are the agencies’ ordinal opinion of the same risk the spread prices continuously — the disagreements between the two are tradeable information.
• In the personal-finance track, retirement portfolios shift toward bonds precisely because of duration: shorter-duration assets make sequence-of-returns risk survivable, at the price of lower expected return.

Try it yourself

The workbook’s second sheet is a full bond calculator on top of the first sheet’s curve: enter coupon, maturity, and either a price or a yield, and it produces the other one, plus Macaulay and modified duration, DV01, convexity, and a price-yield table you can chart. The last block prices the same bond off the risk-free curve and off a spread-shifted curve, so you can see exactly how many pesos (or dollars) 50 bps of credit spread costs at each maturity. Do it once by hand before letting the models of the next two modules do it for you.

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