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Discount factor under conventional or time-varying interest

Source: R/discount_factor.R
discount_factor.Rd

Computes the discount factor from time t back to time s using exactly one of three interest specifications:

  • a conventional interest rate i;

  • an accumulation function a(t);

  • a time-varying force of interest delta(t).

Usage

discount_factor(
  s = 0,
  t,
  i = NULL,
  i_type = "effective",
  m = 1,
  a = NULL,
  delta = NULL,
  subdivisions = 100L,
  rel.tol = 1e-08,
  tidy = FALSE
)

Arguments

s

Numeric vector of valuation times. Defaults to 0.

t

Numeric vector of payment or accumulation times.

i

Optional numeric vector of conventional interest-rate values.

i_type

Character vector indicating the conventional interest-rate type. Allowed values are "effective", "nominal_interest", "nominal_discount", and "force". Used only when i is supplied.

m

Positive integer vector giving the conversion frequency for nominal rates. Used only when i is supplied.

a

Optional accumulation function of one numeric time argument.

delta

Optional force-of-interest function of one numeric time argument.

subdivisions

Positive integer giving the maximum number of subintervals used by stats::integrate() when delta is supplied.

rel.tol

Positive relative tolerance passed to stats::integrate() when delta is supplied.

tidy

Logical scalar. If FALSE, returns a numeric vector. If TRUE, returns a tibble with the inputs, accumulation factor, and discount factor.

Value

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

If tidy = TRUE, a tibble containing the interval, interest model, relevant intermediate quantities, accumulation factor, and discount factor.

Details

The discount factor is the reciprocal of the corresponding accumulation factor: $$v(s,t) = \frac{1}{A(s,t)}.$$

Therefore, $$v(s,t) = (1+i)^{-(t-s)}$$ for a constant annual effective rate, $$v(s,t) = \frac{a(s)}{a(t)}$$ for an accumulation function, and $$v(s,t) = \exp\left(-\int_s^t \delta(u)\,du\right)$$ for a time-varying force of interest.

Exactly one of i, a, or delta must be supplied. The requested discount factor is evaluated from the log accumulation, without requiring its reciprocal to be representable. It must be positive and finite. In tidy output, an unrepresentable intermediate accumulation factor may be Inf or zero; the discount factor is computed independently.

Numeric arguments may have length 1 or a common length. Scalars are recycled over the remaining scenarios, making the function suitable for use inside dplyr::mutate() pipelines.

This function handles general discounting over an interval \([s,t]\). For discount factors obtained specifically from a spot-rate curve, see discount_factor_spot.

See also

accumulation_factor, discount_factor_spot, present_value, pv_flow

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

Other time-value: accumulation_factor(), future_value(), fv_flow(), irr_flow(), irr_flow_multi(), plot_cash_flow(), present_value(), pv_flow(), solve_value()

Examples

# Constant annual effective interest
discount_factor(
  t = 5,
  i = 0.07
)
#> [1] 0.7129862

# One rate recycled over several times
discount_factor(
  t = c(1, 2, 5),
  i = 0.07
)
#> [1] 0.9345794 0.8734387 0.7129862

# Accumulation function
a_fun <- function(t) {
  exp(0.03 * t + 0.002 * t^2)
}

discount_factor(
  s = 2,
  t = 5,
  a = a_fun
)
#> [1] 0.876341

# Time-varying force of interest
delta_fun <- function(t) {
  0.03 + 0.004 * t
}

discount_factor(
  s = 2,
  t = 5,
  delta = delta_fun
)
#> [1] 0.876341

# Pipe-friendly use
if (requireNamespace("dplyr", quietly = TRUE) &&
    requireNamespace("tibble", quietly = TRUE)) {
  scenarios <- tibble::tibble(
    i = 0.07,
    t = c(1, 2, 3, 5)
  )

  scenarios |>
    dplyr::mutate(
      discount = discount_factor(
        t = t,
        i = i
      )
    )
}
#> # A tibble: 4 × 3
#>       i     t discount
#>   <dbl> <dbl>    <dbl>
#> 1  0.07     1    0.935
#> 2  0.07     2    0.873
#> 3  0.07     3    0.816
#> 4  0.07     5    0.713