Bootstrapping the Curve

Spot Rates by Construction

Spot-rate displays are constructed from exact discount factors.

highlighted = computed this step

Spot rates from discount factors

A spot rate is another way to express a discount factor. For year 1, the annualized spot rate is exact 1/24, matching discount factor 24/25.

dt=1(1+st)td_t=\frac{1}{(1+s_t)^t}

Multi-year annualization

For the second and third maturities, annualizing the exact discount factors takes roots. The rounded annualized spot displays are 5.41% and 5.98%, while the exact inputs remain 9/10 and 21/25.

st=(1dt)1/t−1s_t=\left(\frac{1}{d_t}\right)^{1/t}-1
Forward, spot, and par summaryForwards and par yield are recomputed from discount factors.Forward, spot, and par summaryYearDiscount factorForward rateSpot annualized3y par yield124/251/24 (4.17%)1/24 (4.17%)29/101/15 (6.67%)5.41% rounded321/251/14 (7.14%)5.98% rounded8/135 (5.93%)

Rounded display

The second and third annualized spot rates are irrational annualizations of exact discount factors. We show them rounded and do not pretend they are exact rational inputs.

exact spine=dt,rounded display=st\text{exact spine}=d_t,\quad \text{rounded display}=s_t