Last Payment Adjustment#

When the product \(n \cdot m\) of a term and payment frequency is not an integer, the final sub-annual period is incomplete. Lactuca applies a proportional weight to that last period so that the present value reflects the true contracted term \(n\) — not the nearest aligned grid point.

Note

This adjustment applies only to discrete_precision mode for the standard payments_frac scaling on the full payment grid. In discrete_simplified with fractional n_eff and m > 1, the terminal fraction s = n_eff - k is valued on an exact m-thly tail (discrete_precision conventions) rather than via Woolhouse on the annual grid. Continuous modes do not implement fractional final-period scaling on the simplified path.

Annuity-due versus annuity-immediate#

Lactuca exposes separate methods for each payment convention:

Method family

Convention

Symbol

Fractional adjustment

äx, äxy, äxyz, äjoint

Annuity-due — payments at the start of each sub-period

\(\ddot{a}^{(m)}\)

None — last payment is at \(\lfloor nm \rfloor / m \le n\) with full weight

ax, axy, axyz, ajoint

Annuity-immediate — payments at the end of each sub-period

\(a^{(m)}\)

Last payment scaled by \(w\) when \(w > 0\)

For annuity-due methods (äx, äxy, äxyz, äjoint), the last payment is at \(\lfloor nm \rfloor / m \le n\) and is included with full weight — no scaling. The next payment would fall at \((\lfloor nm \rfloor + 1)/m > n\) and is simply excluded.

For annuity-immediate methods (ax, axy, axyz, ajoint), the final period ends at \((\lfloor nm \rfloor + 1)/m > n\), so the last payment is pulled back to \(n\) and proportionally reduced by \(w\).

The fractional weight#

The core quantity is the fractional part of \(n \cdot m\):

\[w = n \cdot m - \lfloor n \cdot m \rfloor \in [0, 1)\]

When \(w = 0\) the term is exactly aligned with the payment grid and no adjustment is needed. When \(w > 0\) the final period is incomplete and \(w\) is used as the scaling weight.

Note

When a time shift ts > 0 and deferment d are used together, \(w\) is computed from the effective term \(n_\text{eff} = \max\!\bigl(n - \max(ts - d,\, 0),\, 0\bigr)\) rather than from raw \(n\). When \(ts \le d\) the shift falls entirely within the deferment period and \(n_\text{eff} = n\).

Example 1 — aligned (\(n = 10.75\), \(m = 4\), quarterly):

\[nm = 10.75 \times 4 = 43, \qquad w = 43 - 43 = 0\]

No fractional adjustment is applied.

Example 2 — incomplete (\(n = 10.6\), \(m = 4\), quarterly):

\[nm = 10.6 \times 4 = 42.4, \qquad w = 42.4 - \lfloor 42.4 \rfloor = 0.4\]

The last quarter covers only \(w/m = 0.1\) year out of a full sub-period of \(1/m = 0.25\) years.

Annuity adjustment#

For ax, axy, axyz, ajoint (annuity-immediate) with \(w > 0\):

  • The last payment time is the exact term endpoint \(n_\text{eff} + d_\text{eff}\), where \(d_\text{eff}=\max(d-ts,\,0)\) is the effective deferment after the time shift (for \(ts=0\) or when the shift falls inside the deferment period, this reduces to \(n_\text{eff}+d\)).

  • The last payment amount is scaled by \(w\):

\[\text{PV contribution}_{\text{last}} = \frac{w}{m} \cdot {}_{n_\text{eff}}p_x \cdot v^{n_\text{eff}+d_\text{eff}} \cdot g(n_\text{eff}+d_\text{eff})\]

where \(g(\cdot)\) is the growth factor, \({}_{n_\text{eff}}p_x\) is the survival probability from the shifted age to \(n_\text{eff}\) (for multi-life methods axy, axyz, ajoint, this is the joint survival probability of all lives), and \(d_\text{eff}=\max(d-ts,\,0)\); see Prospective Reserves and the ts Parameter.

This is exposed in the payment_adjustment array returned by return_flows=True (see Inspecting Cash Flows): all elements are 1.0 except the last, which equals \(w\).

For äx, äxy, äxyz, äjoint (annuity-due), \(w\) does not produce a scaled final payment: the last payment was already made at \(\lfloor nm \rfloor / m\) with full weight.

Insurance adjustment#

For all insurance methods — Ax, Axy, Axyz, Afirst, and their term and deferred variants — the fractional-period scaling applies whenever \(w > 0\). Insurances have no due/immediate distinction (that concept is replaced by mortality_placement), so the adjustment fires regardless of any annuity-related convention:

  • The last period’s death probability is scaled by \(w\):

\[{}_{1/m}q_{x + t_{\text{last}}}^{\text{adjusted}} = w \cdot {}_{1/m}q_{x + t_{\text{last}}}\]

  • The last period’s discount time offset is also scaled:

\[t_{\text{discount,\,last}} = t_{\text{last}} + \delta_m \cdot w\]

where \(\delta_m\) is the mortality_placement offset (\(0\), \(1/(2m)\), or \(1/m\)).

Both effects are visible in the return_flows=True dict via the death_probability_adjustment and discount_time arrays — see Inspecting Cash Flows.

Irregular cashflows#

When using cashflow_times and cashflow_amounts instead of the n/m grid, the fractional adjustment is not applied — the programmer controls each payment’s timing and amount directly. See Irregular Cashflows for details.

Note

Annuity-due methods (äx, äxy, äxyz, äjoint) do not accept cashflow_times — they always use the standard n/m grid. For irregular schedules with multiple lives use the immediate counterparts (ax, axy, axyz, ajoint) with explicit cashflow_times / cashflow_amounts.

See also#

  • Inspecting Cash Flows — payment_adjustment and death_probability_adjustment arrays

  • Calculation Modes — discrete_precision applies payments_frac scaling on the uniform grid; discrete_simplified values fractional n_eff via an exact m-thly tail (not Woolhouse on the last period)

  • Irregular Cashflows — arbitrary cashflow timing

  • Configuration — force_integer_ts and other calculation settings

  • Notation and Glossary — \(\ddot{a}\), \(a\), \(m\), \(n\) symbol definitions