Bootstrapping the Curve

Bootstrapping the Curve

Coupon bond prices recover discount factors by forward substitution.

highlighted = computed this step

Coupon bonds as inputs

Use three annual coupon bonds with face $100.00 and coupon rate 5%. Their prices are $100.80, $99.30, and $97.50.

Pk=∑j=1kCjdjP_k=\sum_{j=1}^{k} C_jd_j

Forward substitution

Bootstrapping solves the earliest unknown first, then carries it forward. The prior coupon present values are $0.00, $4.80, and $9.30.

dk=Pk−coupon∑j<kdjface(1+c)d_k=\frac{P_k-\text{coupon}\sum_{j<k}d_j}{\text{face}(1+c)}
Bootstrap ladderCoupon bond prices recover discount factors by forward substitution.Bootstrapping the discount factorsMaturityBond pricePrior coupon PVFinal cash flowDiscount factor1$100.80$0.00$105.0024/252$99.30$4.80$105.009/103$97.50$9.30$105.0021/25

Recovered discount factors

The ladder recovers exact discount factors 24/25, 9/10, and 21/25. These are the exact rational spine of the curve.

(d1,d2,d3)=(24/25,9/10,21/25)(d_1,d_2,d_3)=(24/25,9/10,21/25)