Brazil’s Bus252 Convention and Curve Calibration#

The day count convention in Brazil is unconventional. Annual interest rates are typically defined with compounded rates with day count fractions dependent upon business days. This differs to the simple period rate definition often used in G10 IRS. The following are equivalent (where f is usually set to 1 for annualized rates):

\[\underbrace{1 + d r}_{\text{simple rate}} = \underbrace{\left (1+\frac{r^{irr}}{f} \right)^{fd}}_{\text{compounded rate}}\]

Setup#

When defining a Curve to calibrate, we specifically add the convention ‘bus252’, and will add a calendar specifically designed for Brazil in 2025 and 2026.

In [1]: holidays = [
   ...:     "2025-01-01", "2025-03-03", "2025-03-04", "2025-04-18", "2025-04-21", "2025-05-01",
   ...:     "2025-06-19", "2025-09-07", "2025-10-12", "2025-11-02", "2025-11-15", "2025-11-20",
   ...:     "2025-12-25", "2026-01-01", "2026-02-16", "2026-02-17", "2026-04-03", "2026-04-21",
   ...:     "2026-05-01", "2026-06-04", "2026-09-07", "2026-10-12", "2026-11-02", "2026-11-15",
   ...:     "2026-11-20", "2026-12-25",
   ...: ]
   ...: 

In [2]: bra = Cal(holidays=[dt.strptime(_, "%Y-%m-%d") for _ in holidays], week_mask=[5, 6])

In [3]: curve = Curve(
   ...:     nodes={
   ...:         dt(2025, 5, 15): 1.0,
   ...:         dt(2025, 8, 1): 1.0,
   ...:         dt(2025, 11, 3): 1.0,
   ...:         dt(2026, 5, 1): 1.0,
   ...:     },
   ...:     convention="bus252",
   ...:     calendar=bra,
   ...:     interpolation="log_linear",
   ...:     id="curve",
   ...: )
   ...:

Instruments#

The Instruments used for calibration replicate the rates on DI1 futures. These are zero coupon swaps, ZCS, whose fixed rate definition is a compounded rate.

The implied rates from the futures data are assumed to be 14%, 13.7% and 13.5%.

In [4]: zcs_args = dict(frequency="A", calendar=bra, curves="curve", currency="brl", convention="bus252")

In [5]: solver = Solver(
   ...:     curves=[curve],
   ...:     instruments=[
   ...:         ZCS(dt(2025, 5, 15), dt(2025, 8, 1), **zcs_args),
   ...:         ZCS(dt(2025, 5, 15), dt(2025, 11, 3), **zcs_args),
   ...:         ZCS(dt(2025, 5, 15), dt(2026, 5, 1), **zcs_args),
   ...:     ],
   ...:     s=[14.0, 13.7, 13.5]
   ...: )
   ...: 
SUCCESS: `func_tol` reached after 4 iterations (levenberg_marquardt), `f_val`: 5.0189675302507837e-17, `time`: 0.0164s

Plotting#

In [6]: curve.plot("1b")
Out[6]: 
(<Figure size 640x480 with 1 Axes>,
 <Axes: >,
 [<matplotlib.lines.Line2D at 0x11a0b6490>])

(Source code, png, hires.png, pdf)

_images/z_bus252_convention-1_00_00.png

This Curve demonstrate the traditional stepped interest rate structure. This is because the ‘log_linear’ interpolation has been applied on a business day basis in accordance with the ‘bus252’ convention and provided calendar. But don’t forget these are simple rates. We adjust these in the final plot on this page.

As a demonstration of the difference in discount factors these can be plotted both for this curve and a conventional Curve with the same node values. Under a business day, and not calendar day, style the discount factors remain constant on a curve for a date which is not a business day.

In [7]: conventional = Curve(
   ...:     nodes={
   ...:         dt(2025, 5, 15): 1.0,
   ...:         dt(2025, 8, 1): curve[dt(2025, 8, 1)],
   ...:         dt(2025, 11, 3): curve[dt(2025, 11, 3)],
   ...:         dt(2026, 5, 1): curve[dt(2026, 5, 1)],
   ...:     },
   ...:     convention="act365f",
   ...:     calendar=bra,
   ...:     interpolation="log_linear"
   ...: )
   ...: 

In [8]: fig, ax = plt.subplots(1, 1)

In [9]: x, y1, y2 = [], [], []

In [10]: for date in bra.cal_date_range(dt(2025, 5, 15), dt(2026, 6, 15)):
   ....:     x.append(date)
   ....:     y1.append(curve[date])
   ....:     y2.append(conventional[date])
   ....: 

In [11]: ax.plot(x, y1)
Out[11]: [<matplotlib.lines.Line2D at 0x11c9887d0>]

In [12]: ax.plot(x, y2)
Out[12]: [<matplotlib.lines.Line2D at 0x11c988690>]

(Source code, png, hires.png, pdf)

_images/z_bus252_convention-2_00_00.png

The rates displayed in plots depend upon their definition and day count fractions. The DFs on both the curve and conventional Curves are the same but their O/N plots are quite different because of this. We can even convert the O/N rates to compounded rates manually as an additional comparison.

In [13]: fig, ax, lines = curve.plot("1b", comparators=[conventional])

In [14]: y2 = ((1 + lines[0]._y / 25200)**252 - 1) * 100

In [15]: ax.plot( lines[0]._x, y2)
Out[15]: [<matplotlib.lines.Line2D at 0x11ab179d0>]

In [16]: ax.legend(["Bus252", "Act365", "Bus252 Compounded"])
Out[16]: <matplotlib.legend.Legend at 0x11ab17b10>

(Source code, png, hires.png, pdf)

_images/z_bus252_convention-3_00_00.png
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