Prospective Reserves and the ts Parameter#

The prospective reserve \({}_{t}V\) at elapsed time \(t\) for a contract issued to a life aged \(x\) is the difference between the actuarial present value of all future benefits and the actuarial present value of all future net premiums, both evaluated from the current attained age \(x + t\):

\[ {}_{t}V_x = \text{APV}(\text{future benefits at age }x+t) - \text{APV}(\text{future premiums at age }x+t) \]

In Lactuca, both present values are computed by calling the same äx, ax, Ax (or joint-life equivalents) with the effective age \(x + t\). The ts parameter is the mechanism that shifts the calculation to that age without requiring the caller to recompute \(x + t\) and adjust the remaining term manually.

The ts parameter (ts is short for time shift) represents the number of years elapsed since the reference age x. Two equivalent parameterisations are equally valid:

  • x = inception age, ts = total elapsed time (years since the contract start).

  • x = age at the last elapsed whole-year anniversary, ts = fractional remainder of the current policy year.

Both produce identical results because the engine only uses \(x + ts\) internally. It is available on all main life-contingency methods: äx, ax, Ax and their joint-life variants (äxy, äxyz, Axy, Axyz), as well as the pure endowment nEx.

ts enables prospective reserve calculations at any point in the life of a contract:

  • Integer ts (e.g. ts=3): the valuation falls exactly on a policy anniversary. This is the standard case for annual statutory reserves.

  • Fractional ts (e.g. ts=3.5): the valuation falls between two anniversaries. The survival and discounting calculations are mathematically exact. When a GrowthRate is also provided, Lactuca applies the actuarial convention that benefit revaluations occur at whole-year anniversaries; a UserWarning is emitted so the user can review that convention before accepting the result (see Growth rates and ts).

Note

ts vs t — two orthogonal concepts

In actuarial notation the symbol \(t\) typically denotes a future duration — the time span measured forward from a given age, as in the survival probability \({}_{t}p_x = l_{x+t}/l_x\). The parameter name ts in Lactuca is deliberately distinct: it represents elapsed time measured backward, from the reference point to the current valuation date. The engine converts ts into an age shift and a duration reduction before any survival calculation is performed. Once the effective age and duration are established, the internal survival functions use t (duration) in the classical sense.

How ts transforms the calculation#

The caller always passes the reference age x and the full remaining term n measured from that reference. The engine applies four internal transformations:

\[x_{\text{eff}} = x + ts\]
\[d_{\text{eff}} = \max(d - ts,\; 0) \qquad ts_{\text{eff}} = \max(ts - d,\; 0)\]
\[n_{\text{eff}} = n - ts_{\text{eff}}\]

The shift first absorbs any deferment: if \(ts \le d\), the deferment decreases and the duration is unchanged (\(ts_{\text{eff}} = 0\), \(n_{\text{eff}} = n\)). Only the portion of \(ts\) that exceeds \(d\) shortens the remaining duration. In the common case \(d = 0\) this simplifies to \(n_{\text{eff}} = n - ts\).

Note

Pure endowments (nEx, nExy, nExyz, nEjoint) do not accept a deferment parameter, so the formula simplifies to \(x_{\text{eff}} = x + ts\) and \(n_{\text{eff}} = \max(n - ts, 0)\).

The call signature is therefore:

lt.äx(x, n=n, ts=ts, ir=ir)          # d=0 → n_eff = n - ts
lt.äx(x, n=n, d=d, ts=ts, ir=ir)     # general case

When ts=0 (the default) no shift is applied; the result is the standard actuarial present value at age x — the initial cost of the contract.

Note

If \(ts \ge d + n\) the contract has fully elapsed and the present value is zero. Lactuca returns 0.0 immediately in that case without raising an error. Passing a negative ts raises ValueError.

For whole-life calculations (n=None) the effective duration is computed as \(n_{\text{eff}} = \omega - x_{\text{eff}} - d_{\text{eff}}\), where \(\omega\) is the terminal age of the table. The shift shortens the residual lifetime automatically; no finite n is required.

Basic usage#

The most common case is an integer ts: the valuation falls on a future policy anniversary. The calculation reduces to a standard annuity starting from the attained age \(x + ts\):

from lactuca import LifeTable, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

# Standard anniversary reserve (ts = 0, no shift)
reserve_ann = lt.äx(50, n=15)
print(reserve_ann)

# Reserve at the 3rd anniversary: ts = 3 advances age to 53, remaining term to 12
reserve_3 = lt.äx(50, n=15, ts=3)
print(reserve_3)

config.reset()

Fractional ts is used when the valuation date falls between two anniversaries. The survival and discounting are computed exactly for the fractional age:

from lactuca import LifeTable, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

# Mid-year valuation: 0.5 years since last anniversary
reserve_mid = lt.äx(50, n=15, ts=0.5)
print(reserve_mid)

# Quarter-year valuation: 0.25 years since last anniversary
reserve_off = lt.äx(50, n=10, ts=0.25)
print(reserve_off)

# 2.75 years since inception: 2 full anniversaries elapsed, 0.75 yr into the third year
reserve_multi = lt.äx(50, n=10, ts=2.75)
print(reserve_multi)

config.reset()

ts vs d — key distinction#

See also

For a standalone introduction to the d parameter — concept, formulas, code examples, and joint-life products — see Deferred Life Contingencies.

ts and d look similar but serve different roles:

Parameter

Meaning

Affects

ts

Time already elapsed since last anniversary

Starting age, payment-grid origin, growth schedule

d

Future deferral before payments begin

First payment is at \(t = d_{\text{eff}}\) after current valuation age

The crucial algebraic interaction: \(ts\) consumes deferment first. The deferment that remains after the shift is \(d_{\text{eff}} = \max(d - ts, 0)\), and only the portion of \(ts\) beyond \(d\) reduces the term.

\(ts\) vs \(d\)

\(d_{\text{eff}}\)

\(n_{\text{eff}}\)

\(ts < d\) (shift is within deferment)

\(d - ts\)

\(n\) (unchanged)

\(ts = d\) (shift exactly exhausts deferment)

\(0\)

\(n\) (unchanged)

\(ts > d\) (shift extends past deferment)

\(0\)

\(n - (ts - d)\)

Example — a 5-year-deferred, 20-year annuity evaluated at ts=0.5 (mid-year):

from lactuca import LifeTable, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

# ts=0.5 < d=5 → deferment is only partially consumed: d_eff = 4.5, n_eff = 20
reserve_combined = lt.äx(50, n=20, d=5, ts=0.5)
print(reserve_combined)

# ts=6 > d=5 → deferment fully consumed: d_eff = 0, n_eff = 20 - (6-5) = 19
reserve_past_d = lt.äx(50, n=20, d=5, ts=6)
print(reserve_past_d)

config.reset()

Note

ts with explicit cashflow_times

For methods that accept an explicit payment schedule (cashflow_times), ts still applies but with two differences:

  1. Each payment time is shifted relative to the valuation date: payments at times \(< ts_{\text{eff}}\) (in the past) are silently dropped.

  2. Integer enforcement (config.force_integer_ts) is bypassed — when the caller controls the exact payment grid, fractional shifts never raise or warn.

Integer-only mode (force_integer_ts)#

By default, Lactuca accepts any non-negative float as ts. Without a GrowthRate, fractional ts is mathematically exact: survival probabilities and discount factors are computed at the precise fractional attained age. When a GrowthRate is active, the growth schedule is advanced by int(ts) complete anniversaries (see Growth rates and ts below). For portfolios where only anniversary-date reserves are required (e.g. end-of-year statutory convention), set config.force_integer_ts = True to make Lactuca raise a ValueError on any fractional ts:

from lactuca import LifeTable, config

config.force_integer_ts = True          # fractional ts now raises ValueError
config.decimals.annuities = 4

lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)
lt.äx(50, n=15, ts=0.5)
# ValueError: [äx] Parameter 'ts' (shift) must be an integer value (got ts=0.5).
# To allow fractional shifts, set 'Config.force_integer_ts = False'.

config.reset()                          # restore defaults after the example above

The following table summarises all input combinations:

ts value

force_integer_ts

GrowthRate present

Behaviour

Integer

False or True

Any

Accepted silently

Fractional

False

No

Accepted silently

Fractional

False

Yes

UserWarning emitted (see Growth rates and ts)

Fractional

True

Any

ValueError raised

Note

force_integer_ts is enforced by annuity and insurance methods (ax, äx, Ax, and all joint-life variants). Pure endowment methods (nEx, nExy, nExyz, nEjoint) currently do not enforce this setting — they accept fractional ts regardless of its value, because endowments never carry a GrowthRate.

Piecewise interest rate with ts#

For non-constant InterestRate term structures, the engine automatically calls ir.shifted(ts) to align the yield curve with the valuation date. No user action is needed:

from lactuca import LifeTable, InterestRate, config

config.decimals.annuities = 4

# Piecewise curve: 2.5% for years 0–5, 3.0% for years 5–10, 3.5% thereafter
# 2 terms → 3 rates required (last rate applies indefinitely)
ir = InterestRate(terms=[5, 5], rates=[0.025, 0.030, 0.035])
lt = LifeTable("PASEM2020_Rel_1o", "m")

# The yield curve is automatically shifted by ts=2.5 yr internally:
# effective curve from valuation date = 2.5% for 2.5 yr, 3.0% for 5 yr, 3.5% thereafter
reserve = lt.äx(50, n=10, ts=2.5, ir=ir)
print(reserve)

config.reset()

Note

ir.shifted(ts) is only invoked when ts > 0 and the interest rate is non-constant. A flat rate is translation-invariant and is never shifted.

Growth rates and ts#

When a GrowthRate schedule is combined with ts, the engine calls gr.shifted(ts) internally. Benefit revaluations follow the actuarial convention of whole-year policy anniversaries: shifted() advances the schedule by int(ts) completed anniversaries, not by the raw ts value. This is correct and expected for integer ts:

from lactuca import LifeTable, GrowthRate, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

# Growth schedule: 3% for first 2 years, then 2% indefinitely
gr = GrowthRate(rates=[0.03, 0.02], terms=[2])

# ts=2: exactly 2 anniversaries elapsed → gr.shifted(2) consumes the first segment
# → remaining schedule from valuation date: 2% (constant second segment)
reserve = lt.äx(50, n=8, ts=2, gr=gr)
print(reserve)

config.reset()

With fractional ts and a GrowthRate, the same integer-anniversary convention applies: ts=2.5 advances the schedule by int(2.5) = 2 anniversaries, not 2.5. Because this truncation may be unexpected, Lactuca emits a UserWarning to prompt the user to confirm the convention is appropriate for their calculation:

from lactuca import LifeTable, GrowthRate, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)
gr = GrowthRate(rate=0.02)  # constant 2% growth schedule

reserve = lt.äx(50, n=15, ts=0.5, gr=gr)
# UserWarning: [äx] Fractional 'ts' (shift) detected (got ts=0.5).
#   GrowthRate schedules use whole-year anniversary indices, consistent with
#   actuarial convention.  Set 'Config.force_integer_ts = True' to reject
#   fractional shifts.
print(reserve)

config.reset()

Note

For fractional ts with a GrowthRate:

  • Survival and discounting are computed exactly at the fractional attained age.

  • Growth indexing follows the convention documented in Growth Rate Conventions: the schedule is advanced by int(ts) complete anniversaries. The growth factor in force during the fractional year before the next anniversary is that of the last whole-year milestone — consistent with the actuarial standard that benefit revaluations (CPI indexation, salary scales) operate on complete policy years.

To require strictly integer ts and bypass this convention entirely, set config.force_integer_ts = True.

For further detail on growth shifts see Growth Rate Conventions.

Mode compatibility#

calculation_mode

Integer ts

Fractional ts

discrete_precision

✅

✅

discrete_simplified

✅

✅

continuous_precision

✅

✅

continuous_simplified

✅

✅

All four modes support fractional ts. For highest accuracy at fractional ages use discrete_precision (exact \(\ell_x\) interpolation under the Uniform Distribution of Deaths (UDD) assumption, config.lx_interpolation = "linear") or continuous_precision (numerical integration over the raw \(\ell_x\) curve under the Constant Force of Mortality (CFM) assumption, config.lx_interpolation = "exponential").

Portfolio reserves#

Compute reserves for multiple policies by iterating over a portfolio. For each policy pass the reference age, the full remaining term from that reference, and the elapsed fraction — the engine applies the shift internally. Each policy requires a separate call, so a plain list comprehension is the natural approach:

from lactuca import LifeTable, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

# Reference age (last anniversary), full remaining term, elapsed fraction this year
ages   = [50, 55, 60, 65]
terms  = [20, 15, 10,  5]
shifts = [0.0, 0.25, 0.5, 0.75]

reserves = [
    lt.äx(age, n=term, ts=ts_val)
    for age, term, ts_val in zip(ages, terms, shifts)
]
print(reserves)

config.reset()

Net premium and prospective reserve — worked example#

The prospective reserve formula connects directly to the net level premium — the constant annual benefit payment \(P\) chosen so that the contract is actuarially fair at issue (\({}_{0}V = 0\)). For an \(n\)-year term insurance:

\[ {}_{0}V = A^1_{x:\overline{n}|} - P \cdot \ddot{a}_{x:\overline{n}|} = 0 \quad \Longrightarrow \quad \boxed{P = \frac{A^1_{x:\overline{n}|}}{\ddot{a}_{x:\overline{n}|}}} \]

Once \(P\) is known, the prospective reserve at any elapsed duration \(t\) is:

\[ {}_{t}V = A^1_{x+t:\overline{n-t}|} - P \cdot \ddot{a}_{x+t:\overline{n-t}|} \]

where \(A^1_{x+t:\overline{n-t}|}\) and \(\ddot{a}_{x+t:\overline{n-t}|}\) are the actuarial present values of the future benefit and the premium annuity-due for the remaining term \(n - t\), evaluated at the attained age \(x + t\). In Lactuca, both are obtained from the same table instance via the ts=t parameter:

from lactuca import LifeTable, config

config.decimals.annuities = 6
config.decimals.insurances = 6

lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

x = 40    # issue age (integer years)
n = 25    # term in years

# ── Step 1: compute net level premium at issue (ts = 0) ──────────────────────
A0   = lt.Ax(x, n=n)      # APV of term insurance benefit at inception
a0   = lt.äx(x, n=n)      # APV of premium annuity-due at inception
P    = A0 / a0             # net level premium per unit of sum insured
print(f"Net level premium P = {P:.6f}")

# ── Step 2: prospective reserve at each anniversary ──────────────────────────
print(f"\n{'t':>4}  {'A(x+t)':>12}  {'äx(x+t)':>12}  {'tV':>12}")
for t in range(0, n + 1, 5):
    At  = lt.Ax(x, n=n, ts=t)    # APV of benefits from attained age x+t
    at  = lt.äx(x, n=n, ts=t)    # APV of premiums from attained age x+t
    tV  = At - P * at             # prospective reserve
    print(f"{t:>4}  {At:>12.6f}  {at:>12.6f}  {tV:>12.6f}")

config.reset()

Note

The reserve at ts=25 (end of term) is zero because the contract has fully elapsed: both At and at equal zero, so tV = 0. The reserve starts at zero at issue, rises through mid-term, and returns to zero at expiry — the typical shape for a term policy under net premium valuation.

Whole-life variant#

For a whole-life insurance (n=None) the same pattern applies; the premium annuity runs for the rest of life:

\[ P_x^{\infty} = \frac{A_x}{\ddot{a}_x} \qquad {}_{t}V_x^{\infty} = A_{x+t} - P_x^{\infty} \cdot \ddot{a}_{x+t} \]
from lactuca import LifeTable, config

config.decimals.annuities = 6
config.decimals.insurances = 6

lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

x = 40

# Net level premium for a whole-life policy
A_wl  = lt.Ax(x)       # whole-life insurance at issue
a_wl  = lt.äx(x)       # whole-life annuity-due at issue
P_wl  = A_wl / a_wl
print(f"Whole-life premium P = {P_wl:.6f}")

# Reserve at selected policy years
for t in [0, 5, 10, 20, 30]:
    tV = lt.Ax(x, ts=t) - P_wl * lt.äx(x, ts=t)
    print(f"  t={t:2d}: tV = {tV:.6f}")

config.reset()

Note

For a whole-life insurance, \({}_{t}V\) increases monotonically and approaches 1 as \(t \to \omega - x\). This is the classic “savings” accumulation in whole-life contracts. The reserve is identically zero at \(t = 0\) (the actuarial equivalence principle), which is a useful numerical self-check: verify that lt.Ax(x) - P_wl * lt.äx(x) returns a value indistinguishable from zero before proceeding to production calculations.

Methods that support ts#

ts is available on all annuity, insurance, and pure-endowment methods:

Category

Methods

Annuities (single-life)

ax, äx

Annuities (joint-life)

axy, äxy, axyz, äxyz, ajoint, äjoint

Insurances

Ax, Axy, Axyz, Afirst

Pure endowments

nEx, nExy, nExyz, nEjoint

Probability, commutation, and life-expectancy methods do not accept ts and always operate at the age passed directly. To evaluate the \(t\)-year survival probability at the attained age after a shift, pass x + ts explicitly and use t as the forward duration:

# t-year survival from the shifted attained age (t here is a duration, not ts)
survival = lt.tpx(x + ts, t=n)

Here t=n is the future duration (years to survive), wholly distinct from ts (years already elapsed).

See also#