Actuarial Life Contingencies & Insurance Mathematics
Actuarial Commutation Dx, Nx, Mx & Life Premium Formula
Calculate Life Annuity-Due a_ddot_x = N_x / D_x, Death-Benefit Commutation M_x = D_x - (i / (1 + i)) * N_x, Whole-Life Net Single Premium Face * (M_x / D_x), and Net Level Annual Premium Face * (M_x / N_x).
Connect life annuities and whole-life insurance instantly via M_x = D_x - d * N_x!
Legacy and statutory life-insurance valuation tables store discounted survivorship columns called Actuarial Commutation Functions: D_x = v^x * l_x (discounted survivors at age x) and N_x = SUM_{k>=0} D_{x+k}! Dividing N_x by D_x gives the Whole-Life Annuity-Due factor a_ddot_x = N_x / D_x. Even without a separate C_x mortality summation column, the fundamental actuarial identity A_x + d * a_ddot_x = 1—where effective discount rate d = i / (1 + i)—yields Death Commutation M_x = D_x - d * N_x identically! From D_x, N_x, and M_x, every whole-life policy reduces to simple division: Net Single Premium NSP = Face * (M_x / D_x) and Net Level Annual Premium P_x = Face * (M_x / N_x)!
Verify on a Statutory Mortality Table—Valuation_Interest_Rate i = 0.05 (5.0% -> discount rate d = 0.05 / 1.05 = 1/21, B2), Age_x_Commutation D_x = 1,000.00 (B3), Cumulative_Commutation N_x = 14,700.00 (B4), and Death_Benefit_Face = $100,000 (B5): Life Annuity-Due a_ddot_x is 14,700 / 1,000 = 14.7000 (E2); Death Commutation M_x is 1,000 - (1/21) * 14,700 = 1,000 - 700 = 300.00 (E3); Whole-Life Net Single Premium is $100,000 * (300 / 1,000) = $30,000.00 (E4); and Net Level Annual Premium is $100,000 * (300 / 14,700) = $2,040.82/yr (E5)!
| Actuarial Commutation Metric (i=5%, Dx=1,000, Nx=14,700, Face=$100,000) | Formula | Verified Output |
|---|---|---|
| Life Annuity-Due Factor a_ddot_x | N_x / D_x | 14.7000 (10.5000 @ Nx=10,500) |
| Death Commutation Function M_x | D_x - (i / (1 + i)) * N_x | 300.00 (500.00 @ Nx=10,500) |
| Whole-Life Net Single Premium | Face * (M_x / D_x) | $30,000.00 ($50,000 @ Nx=10,500) |
| Net Level Annual Premium ($/yr) | Face * (M_x / N_x) | $2,040.82/yr ($4,761.90 @ Nx=10,500) |
Actuarial Commutation Net Level Annual Premium Formula
Validate i > 0, D_x > 0, N_x >= D_x, Face > 0, and strict mortality feasibility d * N_x < D_x (since a mortal life annuity cannot exceed a perpetuity-due 1/d)!
=LET(rt_i, B2, d_x, B3, n_x, B4, fc_b, B5, IF(OR(rt_i <= 0, d_x <= 0, n_x < d_x, fc_b <= 0), "Invalid Commutation Table", LET(ds_d, rt_i / (1 + rt_i), m_x, d_x - ds_d * n_x, IF(m_x <= 0, "Invalid Commutation Table", ROUND(fc_b * (m_x / n_x), 2)))))Older Issue-Age Mortality Check (Set N_x = 10,500)
At an older issue age where N_x drops to B4 = 10,500 (a_ddot_x = 10.50): M_x rises to 500.00, Net Single Premium hits $50,000.00, and Annual Premium rises to $4,761.90/yr!
=IF(ISTEXT(E5), E5, ROUND(B5 * (E3 / B3), 2))Try the example
Use the original values or change them to test your understanding.