Day Count Conventions • fixedincome

library(fixedincome)

Interest rates are quoted per year, but money is lent for some number of days. To compound a rate we need the period in years, and the day count convention (the daycount) is the rule that turns a period between two dates into a fraction of a year.

The same annual rate, for the same two dates, gives different amounts of interest with different daycounts. That is why every SpotRate in fixedincome carries its daycount.

How a daycount works

A daycount has two parts, written as "<days>/<days in base>":

how the days between two dates are counted: actual days, business days, or 30 days per month;
the days in base: how many of these days make one year.

The year fraction is the number of days divided by the days in base.

dc <- daycount("actual/360")
dc
#> <daycount: actual/360 >
dib(dc)
#> [1] 360

fixedincome supports these daycounts:

Daycount	Days counted	Days in base	Used for
`actual/360`	calendar days	360	money markets in USD and EUR
`actual/365`	calendar days	365	GBP money markets, many bonds
`business/252`	business days	252	Brazilian rates (DI, Selic, bonds)
`30E/360`	30 days per month	360	Eurobonds

Other bases also work with actual and business, for example actual/366 or business/250.

Counting the days between two dates

For actual/... and business/... the days are counted with the calendar of the rate, using the bizdays package. The "actual" calendar counts every day, and "Brazil/ANBIMA" counts only business days in Brazil.

d1 <- as.Date("2024-01-31")
d2 <- as.Date("2024-07-31")
term(d1, d2, "actual")
#> [1] "182 days"
term(d1, d2, "Brazil/ANBIMA")
#> [1] "125 days"

With 30E/360 every month has 30 days, and a date on the 31st is moved to the 30th, so from 2024-01-31 to 2024-07-31 there are exactly 6 × 30 = 180 days. The calendar is not used.

Same rate, different daycounts

Let’s compound a simple rate of 10% from d1 to d2 with each daycount.

compound(spotrate(0.10, "simple", "actual/360", "actual"), d1, d2)
#> [1] 1.050556
compound(spotrate(0.10, "simple", "actual/365", "actual"), d1, d2)
#> [1] 1.049863
compound(spotrate(0.10, "simple", "business/252", "Brazil/ANBIMA"), d1, d2)
#> [1] 1.049603
compound(spotrate(0.10, "simple", "30E/360", "actual"), d1, d2)
#> [1] 1.05

The year fractions are 182/360, 182/365, 125/252 and 180/360. So actual/360 pays the most interest: it counts every day, but its year has only 360 days. A rate of 10% is not the same rate under different daycounts.

Equivalent rates

To compare rates with different daycounts, find the rate that gives the same compounding factor for the same dates. implied_rate() returns the rate for a compounding factor and a year fraction.

Here a Brazilian rate of 10% (business/252) is converted to an actual/365 rate for the same period.

sr <- spotrate(0.10, "discrete", "business/252", "Brazil/ANBIMA")
factor <- compound(sr, d1, d2)
factor
#> [1] 1.048412

years <- as.numeric(term(d1, d2, "actual")) / 365
rate <- implied_rate("discrete", years, factor)
sr365 <- spotrate(rate, "discrete", "actual/365", "actual")
sr365
#> [1] "0.0994538 discrete actual/365 actual"
compound(sr365, d1, d2)
#> [1] 1.048412

Both rates give the same factor for these dates, but the equivalent rate depends on the dates: another period has another number of business days.

Rates for other periods

SpotRate objects are always annual, but many rates are quoted for other periods: inflation per month, the DI rate per day, coupons per semester. The period argument of spotrate() takes the rate for one period and stores the equivalent annual rate: the rate with the same compounding factor over that period. The period is a Term, or a string like "1 month", and the daycount of the rate turns it into a fraction of a year.

inflation <- spotrate(0.0045, "discrete", "business/252", "Brazil/ANBIMA",
  period = "1 month"
)
inflation
#> [1] "0.05535675 discrete business/252 Brazil/ANBIMA"
compound(inflation, term(1, "month"))
#> [1] 1.0045

With business/252 one month is 21 business days, but calendar months have 17 to 23 business days in Brazil. For a rate tied to a specific month, use a Term created from two dates: the rate is converted with the business days of that month, so compounding between those dates gives back the monthly rate.

jan <- term(as.Date("2024-01-01"), as.Date("2024-02-01"), "Brazil/ANBIMA")
jan
#> [1] "22 days"
inflation_jan <- spotrate(0.0045, "discrete", "business/252", "Brazil/ANBIMA",
  period = jan
)
compound(inflation_jan, as.Date("2024-01-01"), as.Date("2024-02-01"))
#> [1] 1.0045

Real interest rates

The real rate is the nominal rate discounted by inflation: $(1 + \text{real}) = (1 + \text{nominal}) / (1 + \text{inflation})$ . With discrete compounding this holds for any period, so it can be computed directly with annual SpotRate objects that have the same compounding, daycount and calendar.

nominal <- spotrate(0.1065, "discrete", "business/252", "Brazil/ANBIMA")
real <- (1 + nominal) / (1 + inflation) - 1
real
#> [1] "0.04846063 discrete business/252 Brazil/ANBIMA"

The real rate for one month is the same with both ways of computing it.

compound(real, term(1, "month")) - 1
#> [1] 0.003951371
compound(nominal, term(1, "month")) / compound(inflation, term(1, "month")) - 1
#> [1] 0.003951371

Converting Terms

A Term can be in days, months or years, and the daycount says how many days there are in one year (or in one month, which is one twelfth of a year). todays(), tomonths() and toyears() convert Terms with a daycount.

todays(daycount("business/252"), term(1, "year"))
#> [1] "252 days"
todays(daycount("actual/365"), term(1, "year"))
#> [1] "365 days"
toyears(daycount("30E/360"), term(90, "days"))
#> [1] "0.25 year"

with_daycount() evaluates operations between Terms with different units.

with_daycount("business/252", term(1, "month") + term(12, "days"))
#> [1] "33 days"
with_daycount("actual/360", term(6, "months") == term(180, "days"))
#> [1] TRUE

The article Working with Terms has more about Terms.

Things to watch

The daycount and the calendar must agree. For actual/... and business/... the days come from the calendar, so an actual/360 rate with the "Brazil/ANBIMA" calendar counts business days, and gives a smaller factor than the actual/360 rate above.

compound(spotrate(0.10, "simple", "actual/360", "Brazil/ANBIMA"), d1, d2)
#> [1] 1.034722

Use the "actual" calendar with actual/... daycounts, and a business calendar with business/....

Rates with different daycounts can’t be combined. Arithmetic and comparison between two SpotRates require the same compounding, daycount and calendar. Convert one of them first, as in “Equivalent rates”.

sr1 <- spotrate(0.10, "discrete", "business/252", "Brazil/ANBIMA")
sr2 <- spotrate(0.10, "discrete", "actual/365", "actual")
try(sr1 - sr2)
#> Error : SpotRate objects have different slots (compounding, daycount or calendar)

Only 30E/360 is supported among the 30/360 variants. The US (NASD) and ISDA versions treat the end of the month differently, and daycount() rejects them.

try(daycount("30/360"))
#> Error in daycount("30/360") : 
#>   Unsupported daycount (30/360): only actual/..., business/... and 30E/360 are implemented

Dates of a curve use the calendar. The terms of a SpotRateCurve are kept in days, and functions that turn them into dates, like maturities(), use the calendar of the curve, even with 30E/360.