Spot discount factor

Computes the discount factor implied by a spot rate for a given time, using compact actuarial notation.

Usage

discount_factor_spot(t, i, i_type = "effective", m = 1L, tidy = FALSE)

Arguments

t

Numeric vector of times in years. Each value must be greater than or equal to 0.

i

Numeric vector of spot-rate values.

i_type

Character vector indicating the spot-rate type. Allowed values are "effective", "nominal_interest", "nominal_discount", and "force".

m

Positive integer vector giving the conversion frequency for nominal spot-rate inputs. Observed values are validated as positive integers for every family, even when the conversion itself does not use m.

tidy

Logical scalar. If FALSE, returns a numeric discount factor. If TRUE, returns a tibble with intermediate calculations.

Value

If tidy = FALSE, a numeric vector of discount factors.

If tidy = TRUE, a tibble with input values, standardized rates, and discount factors.

Details

The spot rate may be supplied in FM-style notation:

• annual effective rate,

• nominal annual interest rate,

• nominal annual discount rate,

• force of interest.

Internally, the supplied spot rate is first converted to the equivalent annual effective rate using standardize_interest. The discount factor is then computed as $$v(t) = (1+i)^{-t}.$$

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

If \(t = 0\), the discount factor is 1. Valuation retains the original log accumulation rate. In tidy output, an extreme annual equivalent can round to -1 or overflow to Inf without invalidating a representable discount factor.

Input vectors must have length 1 or a common length. Missing values are propagated.

See also

standardize_interest, present_value, pv_flow

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

Examples

# Numeric discount factor
discount_factor_spot(
  t = 3,
  i = 0.05
)
#> [1] 0.8638376

# Vectorized example
discount_factor_spot(
  t = c(1, 2, 3),
  i = c(0.05, 0.055, 0.06)
)
#> [1] 0.9523810 0.8984524 0.8396193

# FM-style input with nominal annual interest
discount_factor_spot(
  t = 2,
  i = 0.08,
  i_type = "nominal_interest",
  m = 2
)
#> [1] 0.8548042

# Tibble output for teaching or auditing
discount_factor_spot(
  t = c(1, 2, 3),
  i = c(0.05, 0.055, 0.06),
  tidy = TRUE
)
#> # A tibble: 3 × 6
#>       t i_input i_type        m i_effective discount_factor
#>   <dbl>   <dbl> <chr>     <int>       <dbl>           <dbl>
#> 1     1   0.05  effective     1       0.05            0.952
#> 2     2   0.055 effective     1       0.055           0.898
#> 3     3   0.06  effective     1       0.06            0.840