Why Interpolate a Life Table?
A life table commonly gives the number alive at integer ages, such as \(\ell_x\) and \(\ell_{x+1}\). Some calculations require the number alive at an age between them. Under the constant-force assumption within the year, the annual survival ratio determines a specific interpolation rule: the fractional-age value follows an exponential curve, rather than a straight line.
The result is useful only when the assumption matches the calculation. It does not say that a life table must follow this curve at every age; it specifies how to interpolate between a particular pair of consecutive integer ages when mortality force is constant over that year. The resulting value can be fractional because it represents an expected number alive in a life-table cohort, not a count observed in a particular group of people.
The Constant-Force Fractional-Age Formula
Assume that the force of mortality is a finite constant \(\mu_x\) throughout the year of age from \(x\) to \(x+1\). The one-year survival probability then satisfies \(p_x=e^{-\mu_x}\), so \(\mu_x=-\ln(p_x)\) when \(p_x>0\). Survival over \(r\) years is \(e^{-r\mu_x}\). Multiplying that survival probability by the cohort size at age \(x\) gives the fractional-age life-table value.
The formula is valid when the entries are for consecutive integer ages, \(\ell_x>0\), and the annual survival ratio satisfies \(0<\ell_{x+1}/\ell_x\le1\). Equivalently, \(0\le q_x<1\), so the corresponding force is finite and nonnegative. If \(\ell_{x+1}=\ell_x\), then \(p_x=1\), the force is zero, and the formula gives \(\ell_{x+r}=\ell_x\) throughout the year.
If \(\ell_{x+1}=0\) while \(\ell_x>0\), then \(p_x=0\) and \(q_x=1\). No finite constant force gives zero one-year survival, since \(e^{-\mu_x}>0\) for every finite \(\mu_x\). Therefore the finite-force interpolation formula does not apply. The endpoint at \(r=0\) remains \(\ell_x\), and the endpoint at \(r=1\) is the stated \(\ell_{x+1}=0\); the two endpoints alone do not specify a finite-force interpolation between them. In particular, do not substitute \(p_x=0\) into \(p_x^r\), where the expression is undefined at \(r=0\).
Why the Formula Holds
Let \(S(t)\) be the probability of surviving \(t\) years from age \(x\), for \(0\le t\le1\), within the year under consideration. A constant force \(\mu_x\) means that the instantaneous rate of decrease in survival is proportional to current survival: \(S'(t)=-\mu_xS(t)\), with \(S(0)=1\). Separating variables and integrating gives \(\ln S(t)=-\mu_xt\), and hence \(S(t)=e^{-\mu_xt}\). At \(t=1\), this is \(p_x=e^{-\mu_x}\); therefore \(S(r)=p_x^r\). The expected number surviving to age \(x+r\) is \(\ell_x S(r)\), which yields the stated formula.
This also explains why interpolation is exponential. A constant force applies the same proportional rate of mortality at every instant in the year, so survival compounds continuously. The formula agrees with both integer-age entries: setting \(r=0\) gives \(\ell_x\), and setting \(r=1\) gives \(\ell_xp_x=\ell_{x+1}\).
Confirm that the values refer to ages \(x\) and \(x+1\), that \(\ell_x>0\), and that \(p_x=\ell_{x+1}/\ell_x\) lies in the interval \(0<p_x\le1\).
Write the requested age as \(x+r\), with \(0\le r\le1\). Keep \(r\) in years; for example, six months corresponds to \(r=0.5\).
Calculate \(\ell_{x+r}=\ell_xp_x^r\), or use \(\ell_x(1-q_x)^r\) if the annual death probability is supplied.
For \(0\le r\le1\), the interpolated value must lie between \(\ell_x\) and \(\ell_{x+1}\), inclusive. Check the endpoint values and round only as requested.
Worked Applications
Worked Example: Interpolating from an Annual Death Probability
A hypothetical life table has \(\ell_{52}=12{,}000\) and annual death probability \(q_{52}=0.19\). Assume constant force throughout the year of age 52. Find the number alive at age 52.5, and check the result against the integer-age values.
Method. Convert the annual death probability to an annual survival probability, then apply the constant-force fractional-age formula. The fraction of the year is \(r=0.5\).
First, \(p_{52}=1-q_{52}=1-0.19=0.81\). The corresponding force is \(\mu_{52}=-\ln(0.81)\), approximately \(0.2107\) per year. Thus,
The next integer-age value is \(\ell_{53}=12{,}000(0.81)=9{,}720\). The answer, \(10{,}800\), lies between \(12{,}000\) and \(9{,}720\), as it must. The endpoint check also works: \(r=0\) gives \(12{,}000\), and \(r=1\) gives \(9{,}720\).
Common mistake avoided: treating the half-year survival probability as \(1-0.5q_{52}=0.905\). That is a linear interpolation of survival, not the constant-force model. Here the model gives \((1-q_{52})^{0.5}=0.9\), so the fractional-age value is \(12{,}000(0.9)=10{,}800\), not \(12{,}000(0.905)=10{,}860\).
Worked Example: Recovering the Fractional Age
A hypothetical table has \(\ell_{68}=9{,}600\) and \(\ell_{69}=6{,}144\). Under constant force within the year, at what fractional age between 68 and 69 does the table have \(7{,}680\) lives?
Method. Find the one-year survival ratio, form the fractional-age equation, and solve for \(r\) using logarithms. The survival ratio is
Set the interpolation formula equal to the stated fractional-age value:
Taking natural logarithms of both sides gives \(r=\ln(0.8)/\ln(0.64)=0.5\), since \(0.64^{0.5}=0.8\). The requested age is therefore \(68.5\). As a check, \(\ell_{68.5}=9{,}600(0.8)=7{,}680\), the specified value, and \(r=0.5\) is within the required interval.
Common mistake avoided: using the ratio \(7{,}680/9{,}600\) as the fractional age itself. That ratio is the survival factor over the fractional interval, not the time in years; the exponent \(r\) must be solved from \(0.64^r=0.8\).
Worked Example: Using the Life-Table Ratio and Checking a Midyear Value
For a hypothetical cohort, \(\ell_{41}=25{,}000\) and \(\ell_{42}=10{,}240\). Assume constant force during this year. Calculate \(\ell_{41.5}\), state the annual death probability, and compare the result with the straight-line midpoint.
Method. Use the consecutive-age entries to obtain the annual survival ratio, take its square root for half a year, and then calculate the expected number alive.
For \(r=0.5\), the survival factor is \((0.4096)^{0.5}=0.64\), because \(0.64^2=0.4096\). Therefore,
The straight-line midpoint between the table entries would be \((25{,}000+10{,}240)/2=17{,}620\). That is not the constant-force result. The constant-force value \(16{,}000\) is between the endpoint values \(25{,}000\) and \(10{,}240\), and substituting \(r=1\) into the formula returns \(10{,}240\).
Common mistake avoided: confusing the constant-force assumption with linear interpolation. A straight-line midpoint is appropriate only under a different interpolation assumption; it cannot replace the exponential survival factor in this calculation.
Interpretation and Boundary Checks
The value \(\ell_{x+r}\) is a life-table survivor count on the same scale as \(\ell_x\). If the table is based on a radix of, for example, \(100{,}000\), the formula estimates how many of that original cohort remain alive at age \(x+r\). It need not be an integer. Rounding is a reporting choice, not part of the model.
For a valid finite-force interpolation, \(p_x^r\) decreases from 1 to \(p_x\) as \(r\) moves from 0 to 1 when \(0<p_x<1\). This establishes the useful check
When \(p_x=1\), the two bounds are equal and the interpolated value is constant. When \(p_x=0\), the displayed inequality still describes the endpoint range, but it does not make the finite-force formula valid: the annual survival ratio required to define \(\mu_x=-\ln(p_x)\) is zero. Check this case before taking a logarithm or raising the ratio to a fractional power.
Check Your Understanding
For each problem, use the stated within-year assumption and check whether the inputs permit finite constant-force interpolation.
- A hypothetical table has \(\ell_{55}=15{,}000\) and \(q_{55}=0.19\). Under constant force, find \(\ell_{55.5}\). (Answer: \(13{,}500\))
- A table has \(\ell_{62}=9{,}600\) and \(\ell_{63}=6{,}000\). Under constant force, find \(\ell_{62.5}\), rounded to two decimal places. (Answer: \(7{,}589.47\))
- A hypothetical table has \(\ell_{70}=12{,}500\), \(\ell_{71}=8{,}000\), and constant force between these ages. At what fractional age is \(\ell_{70+r}=10{,}000\)? (Answer: \(r=0.5\), so age \(70.5\))
- Under constant force, a table gives \(\ell_{47}=10{,}000\) and \(\ell_{47.5}=9{,}000\). Find the annual survival probability \(p_{47}\) and annual death probability \(q_{47}\). (Answer: \(p_{47}=0.81\), \(q_{47}=0.19\))
- A hypothetical table has \(\ell_{80}=7{,}200\) and \(\ell_{81}=0\). Can the finite constant-force formula determine \(\ell_{80.4}\)? State what can be concluded from these data under the stated model. (Answer: No. The ratio is zero, so no finite constant force applies; the endpoints give \(\ell_{80}\) and \(\ell_{81}\), but do not determine a finite-force interpolation between them.)