For a life currently aged \(x\), the future lifetime \(T = T(x)\) is the (random) number of additional years the life will survive. \(T\) is a non-negative continuous random variable.
The pdf is \(f_T(t) = -\dfrac{d}{dt} S_T(t)\). When the context is clear we simply write \(s(x) = S_0(x) = P[T(0) > x]\), the survival function from age zero.
From the definition of conditional probability,
\[ {}_tp_x = P[T(0) > x+t \mid T(0) > x] = \frac{s(x+t)}{s(x)}. \]Hence the entire age-\(x\) distribution is encoded in the single function \(s(\cdot)\).
The curtate future lifetime is \(K = K(x) = \lfloor T(x) \rfloor\), the completed number of future years. It is integer-valued and represents the discrete clock used in annuities-due / annuities-immediate at integer durations.
That is: the life survives \(k\) years (probability \({}_kp_x\)) and then dies in the \((k+1)\)-th year (probability \(q_{x+k}\)).
The force of mortality at age \(x\) is the instantaneous death rate:
\[ \mu_x = \lim_{h \to 0^+} \frac{P[T(0) \le x+h \mid T(0) > x]}{h} = -\frac{s'(x)}{s(x)} = -\frac{d}{dx}\ln s(x). \]It is analogous to the hazard rate in survival analysis.
Suppose \(\mu_x = 0.02\) for all \(x\) (constant force). Find \({}_{10}p_{40}\).
\[ {}_{10}p_{40} = \exp\!\left(-\int_0^{10} 0.02\,du\right) = e^{-0.20} = 0.8187. \]
So 81.87% of lives age 40 survive to age 50 under constant-force mortality at level 0.02.
Suppose \(l_{40} = 9{,}313\), \(l_{41} = 9{,}287\), \(l_{42} = 9{,}258\). Estimate \(\mu_{41}\) using the central-difference approximation.
\(\mu_{41} \approx -\dfrac{1}{l_{41}} \cdot \dfrac{l_{42} - l_{40}}{2} = -\dfrac{1}{9287} \cdot \dfrac{9258 - 9313}{2} = -\dfrac{-27.5}{9287} = 0.00296.\)
So mortality intensity at age 41 is roughly 0.296% per year.
Closed-form mortality models are useful for theoretical work, premium derivation, and smoothing of empirical life tables. The five standard laws are:
Assumes a finite limiting age \(\omega\) (commonly \(\omega = 100\) to 110). Mortality increases hyperbolically; mathematically very tractable.
The memoryless property: \({}_tp_x = e^{-\mu t}\) does not depend on \(x\).
Death intensity grows geometrically with age — the empirically observed pattern for adult human mortality. Integrating:
\[ s(x) = \exp\!\left(-\,\frac{B}{\ln c}\,(c^x - 1)\right). \]Adds an age-independent term \(A\) for accidents and infectious causes. The "best-fit" classical law for adult human mortality 30 < \(x\) < 90.
Power-law growth in mortality; widely used in reliability engineering. For \(n = 0\) it reduces to constant force.
| Law | \(\mu_x\) | Parameter Range |
|---|---|---|
| De Moivre | \(1/(\omega - x)\) | \(\omega\) finite limiting age |
| Constant Force | \(\mu\) | \(\mu > 0\) |
| Gompertz | \(Bc^x\) | \(B, c > 0,\ c > 1\) |
| Makeham | \(A + Bc^x\) | \(A \ge -B,\ B > 0,\ c > 1\) |
| Weibull | \(k x^n\) | \(k, n > 0\) |
Assume De Moivre with \(\omega = 100\). Find the probability that a life age 40 survives 20 more years.
\(s(40) = 1 - 40/100 = 0.60\), \(s(60) = 1 - 60/100 = 0.40\).
\({}_{20}p_{40} = \dfrac{s(60)}{s(40)} = \dfrac{0.40}{0.60} = 0.6667\).
So a 40-year-old has a 66.67% chance of reaching age 60 under De Moivre's law.
Gompertz parameters \(B = 0.0001\), \(c = 1.10\). Compute \(\mu_{50}\) and \(\mu_{70}\).
\(\mu_{50} = 0.0001 \times 1.10^{50} = 0.0001 \times 117.39 = 0.01174.\)
\(\mu_{70} = 0.0001 \times 1.10^{70} = 0.0001 \times 789.75 = 0.07898.\)
The force of mortality multiplies by \(1.10^{20} = 6.73\) between ages 50 and 70 — a near-7-fold rise that matches actual human mortality data.
Newly underwritten lives have lower mortality than the general population because they have just passed medical examination. The selection effect wears off over a fixed period (the select period, typically 5 to 15 years), after which mortality merges into the ultimate table.
\(q_{[x]+k}\) denotes the mortality rate for a life selected at age \(x\) and now \(k\) years later. \(q_{[x]+r}\) where \(r \ge\) select period equals the ultimate rate \(q_{x+r}\).
Select period = 2 years.
| \(x\) | \(q_{[x]}\) | \(q_{[x]+1}\) | \(q_{x+2}\) |
|---|---|---|---|
| 50 | 0.00100 | 0.00250 | 0.00420 |
| 51 | 0.00115 | 0.00280 | 0.00470 |
A life selected at age 50 has 2-year survival probability \((1 - 0.00100)(1 - 0.00250) = 0.99650\). After year 2 they revert to the ultimate column starting at \(q_{52} = 0.00420\).
For a 50-year-old just sold a policy (\(q_{[50]} = 0.00100\)), the first-year mortality is 58% lower than that of an "average" 50-year-old (\(q_{50}^{\text{ult}} = 0.00240\), say). Ignoring selection would over-estimate first-year claims by more than half — a serious pricing error for new business.
Life tables give \(l_x\) only at integer ages. To compute probabilities at non-integer durations (\(0 < s < 1\)) we must interpolate. Three standard assumptions are used:
Deaths are spread uniformly across each year of age. The force of mortality varies within the year.
Force is constant within each integer year (jumps at integer ages).
Used in some pension/disability work; gives a force of mortality that decreases within the year — usually undesirable for life work but sometimes mathematically convenient.
\(q_{40} = 0.005\). Find \({}_{0.5}q_{40}\) under each assumption.
All three nearly agree for small \(q\) — but the differences amplify for life-table impairments and old ages.
Under UDD, \({}_sp_x\) is linear in \(s\), so \(\bar A_x\) and similar continuous quantities can be obtained from the discrete \(A_x\) via the simple factor \(i/\delta\). This is the main reason UDD dominates life-insurance textbooks.
Constant force gives clean closed forms for \(\bar a_x\) under multi-state models and is widely used in disability/long-term care work.
Balducci survives mostly because it makes single-decrement to multi-decrement conversions algebraically straightforward.