The UDD Force of Mortality

Why the Force Changes Within a Year

A life table gives the number of survivors at integer ages, but mortality calculations may require a rate at a fractional age, such as age \(x+0.6\). Under the uniform distribution of deaths (UDD) assumption, deaths are spread uniformly across each year of age. The number of deaths per unit of age is therefore constant within that year. The force of mortality, however, is not constant: it is measured relative to the people still alive at each age, and that number decreases as the year progresses.

This distinction matters in fractional-age calculations. The annual probability \(q_x\) is the probability of dying during the whole year for someone alive at exact age \(x\). The force \(\mu_{x+r}\) is an instantaneous rate at a particular age within that year. Under UDD they are related, but they are not interchangeable.

Definition: Under UDD for the year of age from \(x\) to \(x+1\), the time of death within that year, conditional on dying during the year, is uniformly distributed. For a life table with \(\ell_x\) survivors at age \(x\) and \(\ell_{x+1}\) survivors at age \(x+1\), let \(d_x=\ell_x-\ell_{x+1}\) and \(q_x=d_x/\ell_x\). At fractional age \(x+r\), where \(0\le r<1\), the interpolated number alive is \(\ell_{x+r}=\ell_x-rd_x\).

Here \(r\) is the fraction of a year after age \(x\), not a number of months. Thus, for age \(x+0.6\), use \(r=0.6\). The formulas below apply within the specified year of age and assume \(\ell_x>0\). For a finite force throughout the interval, also assume \(\ell_{x+1}>0\), or equivalently \(q_x<1\).

Deriving the UDD Force

The force of mortality at age \(y\) is the instantaneous rate of death among those alive at age \(y\). It can be defined as the limit of the probability of dying in the next small interval, divided by the interval length. Under UDD, the life-table count falls linearly between ages \(x\) and \(x+1\):

\ell_{x+r}=\ell_x-rd_x.

Over a small age interval of length \(h\) that stays within the year, UDD implies a decrease of \(hd_x\) in the life-table count. Among the \(\ell_{x+r}\) people alive at age \(x+r\), the conditional probability of dying before age \(x+r+h\) is therefore \(hd_x/\ell_{x+r}\). Dividing by \(h\) and taking the limit gives the force.

\mu_{x+r} =\lim_{h\to 0^+}\frac{hd_x/\ell_{x+r}}{h} =\frac{d_x}{\ell_{x+r}} =\frac{\ell_x-\ell_{x+1}}{\ell_x-r(\ell_x-\ell_{x+1})}.

Since \(q_x=d_x/\ell_x\), divide the numerator and denominator in the last expression by \(\ell_x\). This gives the form most convenient when the annual mortality probability is supplied.

\mu_{x+r}=\frac{q_x}{1-rq_x}, \qquad 0\le r<1.

Key result: Under UDD, the force at fractional age \(x+r\) is the annual death probability \(q_x\) divided by the fraction of the original age-\(x\) cohort still alive at age \(x+r\). Equivalently, it is the annual number of deaths \(d_x\) divided by the interpolated number alive \(\ell_{x+r}\). The force is measured per year.

Interpreting the Formula and Its Boundaries

At the start of the year, \(r=0\), so \(\mu_x=q_x\) under this year's UDD assumption. As \(r\) increases, the numerator \(d_x\) stays fixed while the denominator \(\ell_x-rd_x\) falls. For \(d_x>0\), the force consequently increases through the year. Differentiating the expression also makes this explicit:

\frac{d}{dr}\left(\frac{q_x}{1-rq_x}\right) =\frac{q_x^2}{(1-rq_x)^2}>0 \quad\text{when }q_x>0.

The limit as \(r\) approaches 1 from below is \(\frac{q_x}{1-q_x}=\frac{d_x}{\ell_{x+1}}\). This is the left-hand limit of the force from the year just considered. At exact integer age \(x+1\), the force for the next instant is calculated from the next year's mortality assumption, using \(q_{x+1}\), if that year is modeled by UDD. There is no requirement that this right-hand value equal the left-hand limit from the previous year.

This endpoint distinction prevents a common indexing error: the formula with \(q_x\) describes the interval from \(x\) to \(x+1\), not the interval beginning at \(x+1\). Also, if \(q_x=1\), there are no survivors at \(x+1\), and the force expression becomes unbounded as \(r\) approaches 1. The finite endpoint limit stated above presumes \(q_x<1\).

Worked Example: Finding the Force from an Annual Probability

A hypothetical mortality model has \(q_{40}=0.04\). Assuming UDD during the year from age 40 to 41, calculate the force at age 40.5. No interest rate is involved. The conditional distribution of death time within the year is uniform, and the annual probability parameter is \(q_{40}=0.04\).

Method. Use the UDD force formula with \(r=0.5\):

\mu_{40.5}=\frac{0.04}{1-(0.5)(0.04)} =\frac{0.04}{0.98} =0.0408163265\ldots

Rounded to five decimal places, the force is 0.04082 per year. As a check using life-table counts, choose a cohort of 10,000 people at age 40. The expected count at age 41 is \(10{,}000(1-0.04)=9{,}600\), so \(d_{40}=400\). At age 40.5, UDD gives \(10{,}000-(0.5)(400)=9{,}800\) alive. Then \(400/9{,}800=0.0408163265\ldots\), matching the formula. The common mistake avoided here is using \(0.04\) itself as the force at every fractional age.

Worked Example: Calculating from Life-Table Counts

A hypothetical life table has \(\ell_{25}=10{,}000\) and \(\ell_{26}=9{,}500\). Assuming UDD for the year from age 25 to 26, calculate the force at age 25.6. There is no interest rate; the mortality model is UDD with the stated life-table counts.

Method. First find the deaths during the year and the annual probability:

d_{25}=10{,}000-9{,}500=500, \qquad q_{25}=\frac{500}{10{,}000}=0.05.

The fractional age is \(25+r\), so \(r=0.6\). Interpolate the number alive and then divide the annual deaths by that count:

\ell_{25.6}=10{,}000-(0.6)(500)=9{,}700, \qquad \mu_{25.6}=\frac{500}{9{,}700} =0.0515463917\ldots

To four significant figures, the force is 0.05155 per year. Using \(q_{25}\) gives the same result: \(\frac{0.05}{1-(0.6)(0.05)}=\frac{0.05}{0.97}=0.0515463917\ldots\). The check confirms that the interpolated count is 9,700, not 9,500 or 10,000. The common mistake avoided is dividing by the starting count rather than the number alive at the fractional age.

Worked Example: Recovering the Fractional Age

A hypothetical table has \(\ell_{50}=15{,}000\) and \(\ell_{51}=14{,}100\). Assume UDD during that year and no interest rate. At what fractional age is the force \(0.0625\) per year? The conditional death-time distribution is uniform, and the mortality data are the stated life-table counts.

Method. Calculate \(d_{50}=15{,}000-14{,}100=900\) and \(q_{50}=900/15{,}000=0.06\). Set the UDD force equal to the target and solve for \(r\):

0.0625=\frac{0.06}{1-0.06r}, \qquad 1-0.06r=\frac{0.06}{0.0625}=0.96, \qquad r=\frac{0.04}{0.06}=\frac{2}{3}.

The force is \(0.0625\) per year at age \(50+\frac{2}{3}\), approximately age 50.6667. Check the life-table calculation: \(\ell_{50+2/3}=15{,}000-(2/3)(900)=14{,}400\), and \(900/14{,}400=0.0625\). The solution lies within the year, as required. A common error is to rearrange the formula as if \(q_{50}\) were a force; retaining the denominator \(1-rq_{50}\) is essential.

Common Exam Trap: Annual Probability Is Not a Constant Force

If \(q_x\) is supplied, it may be tempting to report \(q_x\) as the force at every age in the year. That is incorrect under UDD except at the start of the year. For example, with \(q_x=0.05\) at \(r=0.6\), the incorrect calculation would be \(\mu_{x+0.6}=0.05\). The correct calculation is

\mu_{x+0.6}=\frac{0.05}{1-(0.6)(0.05)} =\frac{0.05}{0.97} =0.0515463917\ldots.

The correction reflects the definition of a force: the deaths per year are measured relative to the smaller number still alive at age \(x+0.6\). A second check is to confirm that the calculated age is in the intended interval and that the value is at least \(q_x\) when \(q_x>0\). Under UDD, the force rises from \(q_x\) at the start toward \(q_x/(1-q_x)\) as the year ends.

Key takeaway: For a year with \(q_x<1\), use \(\mu_{x+r}=q_x/(1-rq_x)\), or equivalently \(\mu_{x+r}=(\ell_x-\ell_{x+1})/[\ell_x-r(\ell_x-\ell_{x+1})]\). The fractional-age denominator is the number alive at that age, and \(r\) is measured from the start of the year.

Check Your Understanding

For each problem, assume the stated UDD model applies only to the year specified. No interest rate is involved. Give forces to four significant figures unless an exact answer is requested.

A hypothetical model has \(q_{32}=0.03\). Find the force at age 32.4. (Answer: 0.03036 per year)
A life table has \(\ell_{60}=8{,}000\) and \(\ell_{61}=7{,}600\). Under UDD, find the force at age 60.75 using the counts. (Answer: 0.05195 per year)
For a hypothetical year with \(q_{45}=0.10\), calculate the force at age 45.5. (Answer: 0.1053 per year)
A year has \(q_{70}=0.04\). At what fraction \(r\) of the year does the force reach \(0.041\) per year? (Answer: \(r=0.6098\), approximately)
A hypothetical table has \(\ell_{80}=25{,}000\) and \(\ell_{81}=24{,}000\). Under UDD, find the force at age 80.4. (Answer: 0.04065 per year)
For a year with \(q_x=0.08\), can a force of \(0.09\) per year occur at any \(x+r\) with \(0\le r<1\) under UDD? (Answer: No; the force is less than \(0.08/(1-0.08)=0.08696\) per year throughout the year.)

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