spot-rate-curve-interpolation.Rmd

---
title: "Spot Rate Curve Interpolation"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Spot Rate Curve Interpolation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  out.width = "100%"
)
```

The `fixedincome` package implements many interpolation methods for spot rate curve.
All interpolation methods inherits the S4 class `Interpolation`.
Once you instantiate an Interpolation object, it has to be set to a SpotRateCurve object and this
is done throught the curve's `interpolation<-` method.

There is a list with the interpolation methods implemented and their constructors:

- Flat Forward: `interp_flatforward`
- Linear: `interp_linear`
- Log-Linear: `interp_loglinear`
- Natural Spline: `interp_naturalspline`
- Hermite Spline: `interp_hermitespline`
- Nelson Siegel: `interp_nelsonsiegel`
- Nelson Siegel Svensson: `interp_nelsonsiegelsvensson`
- Flat Forward COPOM: `interp_flatforwardcopom`
  (thru [copom](https://github.com/wilsonfreitas/copom) package)

Here it follows an example on how to create and set an interpolation to a spot rate curve.

Let's start by creating a curve using data obtained with [rb3](https://github.com/wilsonfreitas/rb3)
package.

Firstly, the packages have to be loaded.

```{r message=FALSE, warning=FALSE}
library(rb3)
library(dplyr)
library(fixedincome)
```

In order to build a term structure formed only by futures maturities, the yield curve data and
futures data have to be mixed and this is done with the `rb3::yc_superset` function.
Once the superset is returned, the rows related to futures maturities can be filtered.
The first term, usually 1 business day term, is also used to anchor the curve's short part.

```{r}
refdate <- as.Date("2022-08-05")
yc_ <- yc_get(refdate)
fut_ <- futures_get(refdate)
yc_ss <- yc_superset(yc_, fut_)
yc <- bind_rows(
  yc_ss |> slice(1),
  yc_ss |> filter(!is.na(symbol))
) |>
  filter(!duplicated(biz_days))

yc
```

With the curve data prepared, the `spotratecurve` is created.

```{r}
sp_curve <- spotratecurve(
  yc$r_252, yc$biz_days,
  "discrete", "business/252", "Brazil/ANBIMA",
  refdate = refdate
)

sp_curve
```

From the output above it is possible to observe that this curve does not have an interpolation
method defined.

Let's, for example, define a flat forward interpolation for this curve.
The FlatForward object is created and set to the curve with the `interpolation<-` method.

```{r}
interpolation(sp_curve) <- interp_flatforward()

sp_curve
```

Now the output shows the curve with the interpolation defined.

## Interpolate with `[[`

The spot rate curve method `[[` is used to interpolte the curve.
The term is passed as a Term object or numeric and a spot rate curve is returned with all
interpolated values.

```{r}
sp_curve[[c(21, 42, 63)]]
```

The term 21 doesn't exist in the spot rate curve, so it is interpolated according to
the interpolation method defined.
Since the flat forward interpolation just connect the dots, the terms 42 and 63 have the same values
of the spot rate curve.

Other interpolation methods can be set with the `interpolation` method overriding any method set
previously.

```{r}
interpolation(sp_curve) <- interp_naturalspline()
sp_curve[[c(21, 42, 63)]]
```

## Unset interpolation

Set interpolation to `NULL` to unset the interpolation.

```{r}
interpolation(sp_curve) <- NULL
sp_curve[[c(21, 42, 63)]]
```

Note that for those terms in the `[[` method that don't have a related term in the spot rate curve,
`NA` is returned.

## Plot with interpolation

The `fixedincome::plot` method for the spot rate curve has an argument `use_interpolation` that
shows the interpolation together with the curve points.
This argument defaults to `FALSE`.

```{r fig.width=10, fig.height=5}
interpolation(sp_curve) <- interp_flatforward()
plot(sp_curve, use_interpolation = TRUE)
```

### Forward rates with and without interpolation set

Once the interpolation is set, the `plot` method uses it to calculate daily forward rates.
Otherwise, it uses the forward rates between the curve terms.
Set the `show_forward` argument to `TRUE` to show the forward rates.

```{r fig.width=10, fig.height=5}
interpolation(sp_curve) <- NULL
plot(sp_curve, show_forward = TRUE, legend_location = "bottomright")
```

The forward rates are drawn with step lines.

If the `use_interpolation` argument is `TRUE` then the daily forward rates are calculated with
the defined interpolation.

```{r fig.width=10, fig.height=5}
interpolation(sp_curve) <- interp_flatforward()
plot(sp_curve, use_interpolation = TRUE, show_forward = TRUE, legend_location = "bottomright")
```

It is possible to note that the flat forward daily rates are fairly close to curve terms forward
rates.

As the interpolation changes its effects can be viewed in the forward rates dynamic.

```{r fig.width=10, fig.height=5}
interpolation(sp_curve) <- interp_naturalspline()
plot(sp_curve, use_interpolation = TRUE, show_forward = TRUE, legend_location = "bottomright")
```