Bootstrapping the Zero Curve

The previous lesson handed you a ready-made zero curve and asked you to price a bond with it. In real life the curve is not handed to anyone — the market trades almost no pure zero-coupon bonds beyond a few months' maturity. What you actually see are the prices of ordinary coupon bonds (and short-term deposits, and later, futures and swaps). The zero curve has to be reverse-engineered from those prices, one maturity at a time. That reverse-engineering is called bootstrapping, and it is the first thing a rates desk's systems do every morning before any bond, swap or derivative on the desk can be priced.

The idea: solve for one new rate at a time

Start at the short end, where pure discount instruments (T-bills, bank deposits) genuinely exist — their price directly reveals a zero rate, no algebra required. Then move out along the curve one bond at a time. Each new bond has one cash flow at a brand-new, unknown maturity, and possibly some earlier coupons landing on maturities you have already solved for. Since every earlier zero rate is already known, the bond-pricing equation

B = \sum_{i=1}^{n} C_i\, e^{-R(t_i)\,t_i}

has exactly one unknown left — the zero rate for the new bond's final maturity — and can be solved for it directly. Repeat, moving further out the curve each time, and the whole term structure emerges point by point. This only works if you proceed strictly shortest maturity first: solving a longer bond before its own coupons' zero rates are known leaves you with more unknowns than equations.

Worked example: four instruments, four points on the curve

Suppose these are today's quotes (all face value $100):

Instrument	Maturity	Coupon	Price
T-bill	3 months	none (pure discount)	98.75
T-bill	6 months	none (pure discount)	97.25
Bond	1 year	5% semiannual ($2.50 every 6 mo.)	99.20
Bond	2 years	6% annual ($6 once a year)	99.30

Step 1 — the 3-month T-bill. A pure discount instrument's price is a zero rate in disguise: 98.75 = 100\,e^{-R(0.25)\times 0.25}, so

R(0.25) = \frac{\ln(100/98.75)}{0.25} \approx 5.03\%.

Step 2 — the 6-month T-bill. Same trick, next maturity:

R(0.5) = \frac{\ln(100/97.25)}{0.5} \approx 5.58\%.

Step 3 — the 1-year bond. Now the first coupon-bearing instrument. Its $2.50 coupon at 6 months discounts at the R(0.5)=5.58\% we just found; only R(1) is unknown:

99.20 = 2.50\,e^{-0.0558 \times 0.5} + 102.50\,e^{-R(1) \times 1} = 2.431 + 102.50\,e^{-R(1)}.

Solving: e^{-R(1)} = (99.20 - 2.431)/102.50 = 0.94409, so R(1) \approx 5.75\%.

Step 4 — the 2-year bond. Its one coupon lands exactly at the 1-year point we just solved, so again only one unknown remains, R(2):

99.30 = 6\,e^{-0.0575 \times 1} + 106\,e^{-R(2) \times 2} = 5.665 + 106\,e^{-2R(2)}.

Solving: e^{-2R(2)} = (99.30 - 5.665)/106 = 0.88335, so R(2) \approx 6.20\%.

Four instruments, four bootstrapped points:

Maturity	Bootstrapped zero rate
3 months	5.03%
6 months	5.58%
1 year	5.75%
2 years	6.20%

This curve is now ready to price any other cash flow landing within these four maturities, exactly as in the previous lesson — bootstrapping is the machine that produces the curve that lesson simply handed you.

Our worked example bootstraps purely off bonds for clarity, but a real curve-construction desk stitches together several different instrument families, because each is most liquid (and most trustworthy) over a different stretch of maturity:

Cash deposits / repo for the very short end (out to a few months);
interest-rate futures — covered later in this module — for the next stretch, roughly 3 months to 2 years;
swap rates for everything beyond that, out to 30 years or more.

Because these instruments' maturities rarely line up on the exact dates you need, real systems also interpolate between bootstrapped points (linearly, or with a smoother spline) — and the choice of interpolation method is itself a modeling decision with real economic consequences: it can create small, arbitrageable wiggles in forward rates if done carelessly.

Two failure modes to recognize:

Wrong order. Try to solve the 2-year bond before the 1-year zero rate is known, and its pricing equation has two unknowns (R(1) and R(2)) but only one equation — it cannot be solved on its own. Bootstrapping must proceed strictly from the shortest maturity outward.
Misaligned coupon dates. The method above needs each new bond's earlier coupons to land exactly on maturities you've already solved. Real bonds rarely cooperate that neatly, which is exactly why practitioners interpolate the curve between bootstrapped points rather than insisting on a perfectly matched instrument at every date.