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Equivalent interest rates in FM actuarial notation

Source: R/interest_equivalents.R
interest_equivalents.Rd

Converts a single interest-rate specification into equivalent actuarial rates for the same conversion frequency m.

Usage

interest_equivalents(
  i_type = c("effective", "nominal_interest", "nominal_discount", "force"),
  i,
  m = 1L
)

Arguments

i_type

Character string indicating the input interest-rate type. Must be one of "effective", "nominal_interest", "nominal_discount", or "force".

i

Numeric scalar giving the interest-rate value.

m

Positive integer scalar giving the conversion frequency for nominal rates.

Value

A tibble with columns:

family

Rate or factor family: "effective", "discount", "discount_factor", "force", "nominal_interest", or "nominal_discount".

notation

Actuarial notation for the equivalent rate.

m

Conversion frequency used for nominal rates.

description

Human-readable description.

value

Equivalent rate value.

Details

Internally, the supplied rate is first converted to the annual effective interest rate \(i\) using standardize_interest.

This function follows the compact actuarial notation used throughout tidyactuarial: i denotes the interest-rate value, i_type denotes the type of interest rate, and m denotes the conversion frequency for nominal rates.

Calculations retain the log accumulation rate of the original input. An extreme annual effective equivalent may round to -1 or overflow to Inf while other equivalent quantities remain representable.

Given the annual effective interest rate \(i\), the equivalents are:

Effective discount rate:

\(d = i/(1+i)\)

Discount factor:

\(v = 1/(1+i)\)

Force of interest:

\(\delta = \ln(1+i)\)

Nominal interest:

\(j^{(m)} = m[(1+i)^{1/m} - 1]\)

Nominal discount:

\(d^{(m)} = m[1 - (1+i)^{-1/m}]\)

See also

standardize_interest, discount_factor_spot

Other interest: accumulation_factor(), bootstrap_spot_curve(), discount_factor(), discount_factor_spot(), forward_curve(), forward_rate(), par_rate(), standardize_interest(), yield_curve()

Examples

interest_equivalents(i_type = "nominal_interest", i = 0.18, m = 4)
#> # A tibble: 6 × 5
#>   family           notation     m description                              value
#>   <chr>            <chr>    <int> <chr>                                    <dbl>
#> 1 effective        i           NA effective annual interest rate           0.193
#> 2 discount         d           NA effective annual discount rate           0.161
#> 3 discount_factor  v           NA annual discount factor                   0.839
#> 4 force            delta       NA force of interest                        0.176
#> 5 nominal_interest j^(4)        4 nominal annual interest rate convertibl… 0.18 
#> 6 nominal_discount d^(4)        4 nominal annual discount rate convertibl… 0.172
interest_equivalents(i_type = "nominal_discount", i = 0.10, m = 12)
#> # A tibble: 6 × 5
#>   family           notation     m description                              value
#>   <chr>            <chr>    <int> <chr>                                    <dbl>
#> 1 effective        i           NA effective annual interest rate          0.106 
#> 2 discount         d           NA effective annual discount rate          0.0955
#> 3 discount_factor  v           NA annual discount factor                  0.904 
#> 4 force            delta       NA force of interest                       0.100 
#> 5 nominal_interest j^(12)      12 nominal annual interest rate convertib… 0.101 
#> 6 nominal_discount d^(12)      12 nominal annual discount rate convertib… 0.1   
interest_equivalents(i_type = "force", i = 0.12)
#> # A tibble: 6 × 5
#>   family           notation     m description                              value
#>   <chr>            <chr>    <int> <chr>                                    <dbl>
#> 1 effective        i           NA effective annual interest rate           0.127
#> 2 discount         d           NA effective annual discount rate           0.113
#> 3 discount_factor  v           NA annual discount factor                   0.887
#> 4 force            delta       NA force of interest                        0.12 
#> 5 nominal_interest j^(1)        1 nominal annual interest rate convertibl… 0.127
#> 6 nominal_discount d^(1)        1 nominal annual discount rate convertibl… 0.113
interest_equivalents(i_type = "effective", i = 0.08)
#> # A tibble: 6 × 5
#>   family           notation     m description                              value
#>   <chr>            <chr>    <int> <chr>                                    <dbl>
#> 1 effective        i           NA effective annual interest rate          0.08  
#> 2 discount         d           NA effective annual discount rate          0.0741
#> 3 discount_factor  v           NA annual discount factor                  0.926 
#> 4 force            delta       NA force of interest                       0.0770
#> 5 nominal_interest j^(1)        1 nominal annual interest rate convertib… 0.08  
#> 6 nominal_discount d^(1)        1 nominal annual discount rate convertib… 0.0741

# Batch use with purrr
if (requireNamespace("purrr", quietly = TRUE) &&
    requireNamespace("tibble", quietly = TRUE)) {
  cases <- tibble::tibble(
    i_type = c("effective", "force"),
    i = c(0.08, 0.12),
    m = c(1, 1)
  )

  purrr::pmap(cases, interest_equivalents)
}
#> [[1]]
#> # A tibble: 6 × 5
#>   family           notation     m description                              value
#>   <chr>            <chr>    <int> <chr>                                    <dbl>
#> 1 effective        i           NA effective annual interest rate          0.08  
#> 2 discount         d           NA effective annual discount rate          0.0741
#> 3 discount_factor  v           NA annual discount factor                  0.926 
#> 4 force            delta       NA force of interest                       0.0770
#> 5 nominal_interest j^(1)        1 nominal annual interest rate convertib… 0.08  
#> 6 nominal_discount d^(1)        1 nominal annual discount rate convertib… 0.0741
#> 
#> [[2]]
#> # A tibble: 6 × 5
#>   family           notation     m description                              value
#>   <chr>            <chr>    <int> <chr>                                    <dbl>
#> 1 effective        i           NA effective annual interest rate           0.127
#> 2 discount         d           NA effective annual discount rate           0.113
#> 3 discount_factor  v           NA annual discount factor                   0.887
#> 4 force            delta       NA force of interest                        0.12 
#> 5 nominal_interest j^(1)        1 nominal annual interest rate convertibl… 0.127
#> 6 nominal_discount d^(1)        1 nominal annual discount rate convertibl… 0.113
#>