Two-life survival probability for independent lives

Computes the survival probability for two independent lives with actuarial ages x and y at time 0 under a joint-life or last-survivor status.

Usage

t_pxy(lt, x, y, t, frac, status = c("joint", "last"))

Arguments

lt

Either:

a single life table data frame, used for both lives; or
a list of two life tables list(lt_x, lt_y), one for each life.

Each life table must contain columns x and lx.

x

Integer actuarial age of the first life at time 0.

y

Integer actuarial age of the second life at time 0.

t

Nonnegative time (may be fractional).

frac

Fractional-age assumption: "UDD", "CF", "CML" (alias of CF), or "Balducci". If not specified and the supplied life table(s) carry a frac attribute, that value is used. If two tables are supplied and their frac attributes differ, frac must be supplied explicitly. Uses the same assumptions as t_px. Inherited "CML" is normalized to "CF" before comparing the two tables' assumptions.

status

Two-life status: "joint" or "last".

Value

A single numeric value.

Details

Independence is assumed throughout (Finan, Sections 56–57).

**Joint-life status** (Finan, Section 56): the status survives as long as both lives are alive. $${}_tp_{xy} = {}_tp_x \cdot {}_tp_y.$$

**Last-survivor status** (Finan, Section 57): the status survives as long as at least one life is alive. $${}_tp_{\overline{xy}} = {}_tp_x + {}_tp_y - {}_tp_x \cdot {}_tp_y.$$

Individual survival probabilities ${}_tp_x$ and ${}_tp_y$ use observed lx ratios and fractional survival under UDD, constant force, or Balducci (Finan, Section 24). Logarithmic calculations retain small annual survival without subtracting a rounded death probability from one. Last-survivor survival is evaluated as $\max(p_x,p_y)+(1-\max(p_x,p_y))\min(p_x,p_y)$. Tables must have unique consecutive annual integer ages and finite, nonnegative, nonincreasing lx. Supplied px/qx must lie in [0,1] and agree with each other and observed annual lx ratios within 1e-10; unknown probabilities may be NA. Ages must fit R's integer range and durations must be finite. The established t_px horizon and zero-duration conventions apply; survival beyond the last recorded age is a truncation, not extrapolation.

Examples

lt <- data.frame(
  x  = 60:66,
  lx = c(100000, 99000, 97500, 95500, 93000, 90000, 86000)
)

# Joint life, 2.5 years, UDD (Finan, Sec. 56)
t_pxy(lt, x = 60, y = 62, t = 2.5, frac = "UDD", status = "joint")
#> [1] 0.9056154

# Verify: joint = product of marginals
t_px(lt, x = 60, t = 2.5, frac = "UDD") *
  t_px(lt, x = 62, t = 2.5, frac = "UDD")
#> [1] 0.9056154

# Last survivor, 2.5 years, constant force (Finan, Sec. 57)
t_pxy(lt, x = 60, y = 62, t = 2.5, frac = "CF", status = "last")
#> [1] 0.9978385

# Same result if the same table is supplied explicitly for both lives
t_pxy(list(lt, lt), x = 60, y = 62, t = 2.5, frac = "UDD", status = "joint")
#> [1] 0.9056154

# Different life tables for the two lives
lt_m <- data.frame(
  x  = 60:66,
  lx = c(100000, 98500, 96800, 94800, 92400, 89500, 86000)
)
lt_f <- data.frame(
  x  = 60:66,
  lx = c(100000, 99000, 97800, 96400, 94700, 92700, 90300)
)
t_pxy(list(lt_m, lt_f), x = 60, y = 62, t = 2.5, frac = "UDD", status = "joint")
#> [1] 0.9178384

# Finan Example 56.2 style: integer survival
# 10_p_{50:60} = 10_p_50 * 10_p_60
lt_ilt <- data.frame(
  x  = 50:70,
  lx = c(8950901, 8879913, 8804189, 8723382, 8637048,
         8544731, 8445974, 8340310, 8227261, 8106334,
         7977338, 7839775, 7693040, 7536522, 7369603,
         7191658, 7002051, 6800139, 6585264, 6356752,
         6114913)
)
t_pxy(lt_ilt, x = 50, y = 60, t = 10, status = "joint")
#> [1] 0.6831617

# Finan Problem 56.1: t_q_xy = t_q_x + t_q_y - t_q_x * t_q_y
p_joint <- t_pxy(lt, x = 60, y = 62, t = 3, status = "joint")
q_joint <- 1 - p_joint
qx <- 1 - t_px(lt, x = 60, t = 3)
qy <- 1 - t_px(lt, x = 62, t = 3)
c(q_joint = q_joint, q_sum = qx + qy - qx * qy)  # should match
#>   q_joint     q_sum 
#> 0.1184615 0.1184615