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Duration-based immunization with multiple assets

Source: R/inmunize_duration.R
immunize_duration.Rd

Computes asset weights that duration-immunize a stream of liabilities using two or more assets, using compact actuarial notation.

Usage

immunize_duration(L, t, P, D, i, i_type = "effective", m = 1L)

Arguments

L

Numeric vector with liability payments.

t

Numeric vector of the same length as L, giving the times at which each liability payment occurs.

P

Numeric vector with present values or prices of the immunizing assets, evaluated on the same yield basis.

D

Numeric vector with the Macaulay duration of each asset, expressed in the same time units as t.

i

Numeric scalar. Interest-rate input used to discount the liabilities.

i_type

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

m

Positive integer. Conversion frequency for nominal rates. Ignored for i_type = "effective" and i_type = "force".

Value

A tibble with:

w

Numeric vector of asset weights or units.

PV_L

Present value of the liabilities.

D_L

Macaulay duration of the liabilities.

PV_A

Present value of the immunized asset portfolio.

D_A

Macaulay duration of the asset portfolio.

n_assets

Number of assets used.

Details

The method enforces:

  1. present value of assets = present value of liabilities;

  2. Macaulay duration of assets = Macaulay duration of liabilities.

For exactly two assets, a closed-form solution is used. For three or more assets, a minimum-norm solution is computed by linear algebra.

This function follows the compact actuarial notation used throughout tidyactuarial: L denotes liabilities, t denotes payment times, P denotes asset prices or present values, D denotes asset durations, i denotes the interest rate, i_type denotes the interest-rate type, and m denotes the conversion frequency for nominal rates.

Let \(PV_L\) and \(D_L\) be the present value and Macaulay duration of the liability stream at yield \(i\). Let \(P_j\) and \(D_j\) be the price and duration of asset \(j\). The weights \(w_j\) are chosen so that:

$$\sum_j w_j P_j = PV_L$$

and

$$ \frac{\sum_j w_j P_j D_j}{\sum_j w_j P_j} = D_L. $$

For two assets, the closed-form solution is:

$$ w_1 = \frac{PV_L(D_L - D_2)}{P_1(D_1 - D_2)}, \qquad w_2 = \frac{PV_L - w_1 P_1}{P_2}. $$

For three or more assets, the minimum-norm solution of the linear system \(Aw = b\) is computed, where \(A\) is a \(2 \times r\) matrix with rows

$$(P_1,\ldots,P_r)$$

and

$$(P_1D_1,\ldots,P_rD_r),$$

and

$$b = (PV_L, PV_LD_L)^T.$$

The minimum Euclidean norm is taken over asset units w, not portfolio market-value fractions. It is evaluated by singular value decomposition of the row-scaled constraint matrix, without forming normal equations. Short positions are allowed. Matching present value and duration gives a zero value and first derivative of the surplus at the base yield, but does not ensure nonnegative surplus after rate shifts; the Redington convexity condition must be checked separately. Liability values retain the original log accumulation rate, and relative moments avoid monetary numerator overflow. Payment times include zero: a liability at time zero contributes value but no duration. Constraints are scaled by a common monetary magnitude and by moment magnitudes before products are formed. These are row transformations, preserving the norm over units; asset columns are not rescaled independently. Final values, weights and sensitivities must be finite, and nearly cancelled liability value is rejected. Rank and cancellation tests remain subject to double precision; reported portfolio moments should be inspected for ill-conditioned systems.

See also

immunize_duration_convexity, bond_duration, bond_convexity

Other immunization: immunize_duration_convexity(), plot_immunization_gap()

Examples

# Two-asset immunization
immunize_duration(
  L = c(5000, 8000),
  t = c(3, 7),
  P = c(100, 200),
  D = c(3, 7),
  i = 0.05
)
#> # A tibble: 2 × 6
#>       w   PV_L   D_L   PV_A   D_A n_assets
#>   <dbl>  <dbl> <dbl>  <dbl> <dbl>    <int>
#> 1  43.2 10005.  5.27 10005.  5.27        2
#> 2  28.4 10005.  5.27 10005.  5.27        2

# Three-asset immunization: minimum-norm solution
immunize_duration(
  L = c(5000, 8000),
  t = c(3, 7),
  P = c(100, 150, 200),
  D = c(2, 5, 8),
  i = 0.05
)
#> # A tibble: 3 × 6
#>       w   PV_L   D_L   PV_A   D_A n_assets
#>   <dbl>  <dbl> <dbl>  <dbl> <dbl>    <int>
#> 1  25.9 10005.  5.27 10005.  5.27        3
#> 2  26.0 10005.  5.27 10005.  5.27        3
#> 3  17.5 10005.  5.27 10005.  5.27        3

# Nominal annual interest rate convertible monthly
immunize_duration(
  L = c(5000, 8000),
  t = c(3, 7),
  P = c(100, 200),
  D = c(3, 7),
  i = 0.06,
  i_type = "nominal_interest",
  m = 12
)
#> # A tibble: 2 × 6
#>       w  PV_L   D_L  PV_A   D_A n_assets
#>   <dbl> <dbl> <dbl> <dbl> <dbl>    <int>
#> 1  41.8 9440.  5.23 9440.  5.23        2
#> 2  26.3 9440.  5.23 9440.  5.23        2