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Quantitative Finance · Glosarium

Apa itu Compounding in arrears?

Definition 1.14 Markets II: Rates, FX and Credit · Bab 1 — Central Banks and the Short Rate

Over an interest period of NN calendar days with business days d1,…,dMd_1, \dots, d_M, fixings r1,…,rMr_1, \dots, r_M and weights nin_i, the calendar days from did_i to the next business day, the rate compounded in arrears is

R  =  [ ∏i=1M(1+ri niB)−1]BN,R \;=\; \Bigl[\,\prod_{i=1}^{M}\Bigl(1 + \frac{r_i\,n_i}{B}\Bigr) - 1\Bigr]\frac{B}{N},

with B=360B = 360 for the dollar and euro and 365 for sterling. It is known only after the last fixing. Three conventions bring the payment date forward: a lookback of pp business days uses rr from pp days before each did_i with the period’s own weights; an observation shift takes rates and weights from a period moved back by pp business days; a lockout freezes the last kk fixings at the one before them.

The illustrative fixings of  and the rate compounded from the start of the period to each day. The compounded rate absorbs a change only gradually: each day at the new level is one more weight in an average over all the days so far. Friday fixings weigh three days, the Friday before Labor Day four. The fixings are illustrative, not published SOFR. Data: the chapter’s tutorial.
Figure 1.3. The illustrative fixings of Example 1.16 and the rate compounded from the start of the period to each day. The compounded rate absorbs a change only gradually: each day at the new level is one more weight in an average over all the days so far. Friday fixings weigh three days, the Friday before Labor Day four. The fixings are illustrative, not published SOFR. Data: the chapter’s tutorial.

Contoh

Example 1.16 (September 2026, illustrated)

Take illustrative fixings of 3.62% up to 16 September, 3.87% from the hike, and 3.95% on the quarter-end day, 30 September (Figure 1.3). Over 1 September to 1 October (30 days, 21 fixings, Labor Day on 7 September) the day-weighted average is 3.7393% and the compounded rate 3.7448%: compounding adds 0.54 basis points, against the 0.56 of Proposition 1.15 for a flat 3.74%. With a five-day lookback the rate is 3.6836%, 6.12 basis points lower, USD 5 098 on USD 100 million: the lookback moves five days of the new rate into the next period, where the borrower will pay them.

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