Use bootstrapping to derive zero-coupon rates.

FRM® Part I · Financial Markets and Products · Properties of Interest Rates · FRM1.FM.16.j

The idea

Bootstrapping builds the zero curve from bond prices, one maturity at a time, working from short to long. Short-term instruments such as bills have a single cash flow, so the zero rate follows directly. Take a six-month bill with price P and face 100, with the rate quoted semiannually. Semiannual zero rate = 2 × (100 ÷ P − 1). For the next coupon bond, the coupons that fall on earlier dates are discounted with the zero rates already found. What remains of the price, after subtracting those present values, equals the final cash flow discounted at the one unknown zero rate, so solve for it and then move to the next maturity. With annual compounding, n is the number of years to the final cash flow. Annual zero rate = (final cash flow ÷ remaining value)^(1/n) − 1. Common errors are using yields to maturity in place of zero rates, forgetting to subtract the coupons' present value, and omitting the n-th root.

Formulas

Six-month zero rate from a bill price

zero rate = 2 × (100 ÷ P − 1)

P: the bill price per 100 of face

Gives the rate with semiannual compounding. A bill has one cash flow, so no earlier rates are needed.

Bootstrapped zero rate, annual compounding

zero rate = (CFn ÷ V)^(1 ÷ n) − 1

CFn: the final cash flow, coupon plus principal
V: the price minus the present value of the earlier coupons at the zero rates already found
n: the years to the final cash flow

Work from short to long. Using yields to maturity in place of zero rates, or omitting the n-th root, are the usual errors.

A practice question

From the question bank, with the reason each wrong option is wrong.

A 6-month Treasury bill with face value 100 is priced at 95.00. The 6-month zero rate with semiannual compounding is closest to:

10.53% - correct
5.26%Why not: Reports the 6-month return without annualizing it.
10.26%Why not: Computes the continuously compounded rate instead of the semiannually compounded rate.
10.80%Why not: Compounds the 6-month return twice for annual compounding when the question asks for a semiannual quote.

Six-month return = 100/95 - 1 = 5.263%. Semiannually compounded rate = 2 x 5.263% = 10.53%.

A harder one

The 1-year zero rate is 3.0% (annually compounded). A 2-year bond with face value 100 and a 5% annual coupon is priced at 102.00. The 2-year zero rate implied by bootstrapping is closest to:

5.00%Why not: Reports the coupon rate, which is not a zero rate.
3.96% - correct
8.09%Why not: Forgets to take the square root of the 2-year growth factor to annualize it.
3.00%Why not: Repeats the 1-year zero rate, assuming a flat zero curve.

Present value of the first coupon = 5/1.03 = 4.8544. Remaining value = 97.1456 = 105/(1 + z2)², so z2 = 3.96%.

The hardest tier

Par bonds with annual coupons have yields of 3.0% for 1 year, 4.0% for 2 years and 5.0% for 3 years, each priced at 100. Bootstrapping gives a 3-year zero rate that is closest to:

5.07% - correct
5.00%Why not: Equates the 3-year zero rate with the 3-year par yield; with an upward-sloping curve the zero rate is above the par yield.
4.00%Why not: Averages the three par yields (3%, 4%, 5%) instead of bootstrapping.
15.99%Why not: Forgets to take the cube root of the 3-year growth factor.

1-year zero = 3.0%. 2-year: 100 = 4/1.03 + 104/(1 + z2)² gives z2 = 4.020%. 3-year: 100 = 5/1.03 + 5/(1 + z2)² + 105/(1 + z3)³ gives z3 = 5.069%.

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