Bootstrapping the Yield Curve

So far we’ve spoken of “the yield” as if a bond had one. But a 6% coupon paid in year 1 and the face value paid in year 10 should not be discounted at the same rate — money for one year and money for ten years cost differently. The market prices each maturity with its own spot rate (the yield on a pure zero-coupon claim at that horizon), and the collection of those rates is the spot curve. The trouble: almost no zeros trade at every maturity, so we can’t read the spot curve off the screen. We have to build it — by bootstrapping, stripping coupon bonds apart one maturity at a time. This is the single most important construction in fixed income; every discount factor, every valuation, every forward rate descends from it.

Spot rates: the price of money at each horizon

Analogy. Imagine renting money like you rent a car: a one-day rental and a one-year rental have completely different daily rates. The spot rate $z_t$ is the rental rate for money borrowed today and repaid in a single lump at time $t$ — it’s the yield on a pure zero-coupon bond maturing at $t$. There’s a different $z_t$ for every horizon, and together they form the term structure of interest rates.

Why spot rates are the “true” rates. A zero-coupon bond has exactly one cash flow, so its yield discounts that cash flow with no blending — it’s a clean, unambiguous price of money for that horizon. The discount factor for time $t$ is

and any cash flow at time $t$, from any bond, must be discounted by this same $DF_t$ — otherwise an arbitrageur could buy the cheap version and sell the dear one. Spot rates are the atoms; a coupon bond’s price is just a sum of its cash flows times the appropriate discount factors.

The bootstrap: solve one maturity at a time

Analogy. Bootstrapping is climbing a ladder where each rung must be built from the rungs below it. You can’t reach the 3-year spot rate until you’ve nailed the 1- and 2-year rates, because a 3-year coupon bond’s early coupons need to be discounted at those rates first. So you start at the bottom and climb.

The recursion. Given the prices of coupon bonds at maturities $1, 2, 3, \dots$:

1. Year 1. A 1-year bond has a single cash flow, so its yield is the 1-year spot rate $z_1$. Read it straight off.
2. Year 2. A 2-year bond pays a coupon at year 1 and coupon + face at year 2. Discount the year-1 coupon using $z_1$ (now known), subtract that present value from the bond’s price, and the remainder is the present value of the year-2 cash flow. Solve for $z_2$.
3. Year $t$. Every spot rate $z_1, \dots, z_{t-1}$ is known, so a $t$-maturity bond’s first $t-1$ coupons can all be discounted. Subtract them from the price, leaving a single equation in $z_t$. Solve.

Each step has exactly one unknown because every shorter rate is already locked in. That’s the magic — and the constraint: you must have a bond (or a price) at every maturity step, with no gaps.

A full worked bootstrap

Let’s bootstrap three rates from three par bonds (all priced at $100 par, annual coupons).

Step 1 — the 1-year spot. The 1-year par bond pays $104 in one year for $100 today:

Step 2 — the 2-year spot. The 2-year bond pays $5 (year 1) and $105 (year 2):

The first term is $4.808$, so $\frac{105}{(1+z_2)^2} = 95.192$, giving $(1+z_2)^2 = 1.1030$ and $z_2 = 5.02\%$.

Step 3 — the 3-year spot. The 3-year bond pays $6, $6, $106:

The first two terms are $5.769 + 5.439 = 11.208$, so $\frac{106}{(1+z_3)^3} = 88.792$, giving $(1+z_3)^3 = 1.1938$ and $z_3 = 6.08\%$.

Notice the spot curve (4.00, 5.02, 6.08) sits above the par/coupon curve (4, 5, 6) when the curve slopes upward — because each later par coupon was being “subsidised” by the cheap early discounting, and stripping reveals the true, higher long-horizon rate.

Discount factors and no-arbitrage

The deeper truth. Bootstrapping really recovers a set of discount factors $DF_1, DF_2, DF_3, \dots$ — the present value today of $1 delivered at each future date. Spot rates are just discount factors in disguise ($DF_t = (1+z_t)^{-t}$). Once you have the discount factors, you can price any cash flow stream — any bond, swap, or structured note — by multiplying each cash flow by its date’s discount factor and summing. The curve is a universal pricing kernel.

Why no-arbitrage forces it. If two instruments deliver the same cash flow at the same date but imply different discount factors, you could buy the cheap one and sell the dear one for a riskless profit. Markets stamp that out, so a single consistent set of discount factors must price every default-free cash flow. Bootstrapping is how you extract that unique set from observed prices.

The practical wrinkles

Real bootstrapping is messier than three tidy par bonds:

• Gaps and interpolation. Bonds don’t exist at every maturity, so you must interpolate the curve between traded points — linearly on the spot rates, on the log-discount factors, or with splines. Different interpolation choices give slightly different forward rates, which matters for derivatives.
• Coupon timing. Real coupons are semiannual and bonds have odd settlement dates, so you bootstrap on the actual cash-flow dates, not neat integer years.
• Instrument choice. Practitioners build the short end from money-market rates and futures, and the long end from swaps or government bonds — splicing several instrument types into one continuous curve.
• Errors compound. Because the recursion reuses earlier rates, a mispriced short bond contaminates every longer rate above it. Clean inputs at the short end matter enormously.

When to use it

You bootstrap whenever you need to discount cash flows correctly across maturities: pricing a bond off the curve rather than a single YTM, valuing a swap, marking a portfolio, or — crucially — extracting the forward rates the market has priced (the next lesson). Anytime a single yield would blur together cash flows of different horizons, you reach for the bootstrapped spot curve. It’s the foundation layer; nearly everything quantitative in fixed income sits on top of it.

Putting it together

A bond doesn’t have one yield — each of its cash flows lives at a different horizon and earns a different spot rate. Because zeros rarely trade at every maturity, we bootstrap the spot curve from coupon-bond prices: start at the short end, where the 1-year bond hands you $z_1$ directly, then climb, reusing every solved rate to strip the next bond down to a single unknown. The output is a set of discount factors that — by no-arbitrage — uniquely price every default-free cash flow. Mind the practical wrinkles (interpolation, coupon timing, error propagation), and you’ve built the curve the entire market trades off.

Next — spot, forward, and term-structure models — we use the spot curve to extract the forward rates the market has priced for future periods, then take a first quantitative look at how rates themselves evolve: the mean-reverting Vasicek and CIR models that bridge into stochastic processes.