Why the Conversion Depends on the Assumption
Mortality data may be reported as an annual death probability, a central death rate, or a force of mortality. These quantities are related, but they are not interchangeable. The conversion depends on how deaths are distributed within the year of age. A force that is constant throughout the year leads to one relationship; the assumption that deaths are uniformly distributed through the year leads to another.
The practical task is therefore twofold: identify the quantity supplied, and identify the mortality assumption that justifies the conversion. If exposure data are supplied, first calculate the central death rate from deaths divided by exposure. Do not treat that rate as an annual probability unless an explicit model supports the conversion.
The distinction between \(q_x\) and \(m_x\) is especially important. The numerator of \(q_x\) is a probability conditional on being alive at the start of the year. The central rate \(m_x\) uses person-years at risk as its denominator. Since some people die during the year, the exposure is generally less than the starting number of lives multiplied by one full year.
Conversions Under a Constant Force
Suppose the force of mortality is constant throughout the age interval, with value \(\mu_x\). Under this assumption, the probability of surviving \(t\) years from age \(x\) is \(e^{-\mu_x t}\) for \(0\leq t\leq1\). In particular, the probability of dying during the year is \(1-e^{-\mu_x}\).
Under the same constant-force assumption, the central death rate equals the force: \(m_x=\mu_x\). To see why, let \(l_x\) be the number alive at exact age \(x\). At time \(t\) within the year, the expected number still alive is \(l_x e^{-\mu_x t}\). The expected deaths during the year are \(l_x(1-e^{-\mu_x})\), and exposure is the integral of the number alive over the year.
Taking the ratio gives \(m_x=d_x/E_x=\mu_x\), as long as \(\mu_x>0\). Thus, when constant force is specified, convert between \(m_x\) and \(q_x\) through the force: \(m_x=\mu_x\) and \(q_x=1-e^{-m_x}\). The equality \(m_x=\mu_x\) is not a general identity; it relies on constant force over the interval.
Worked Example: Converting a Constant Force
A hypothetical life table has a constant force of mortality of \(0.04\) per year between ages 50 and 51. Find \(q_{50}\) and \(m_{50}\). For comparison, calculate the expected deaths among 1,000 lives aged 50 at the start of the year. No interest rate applies. The mortality model is constant force, with \(\mu_{50}=0.04\).
Method. Use the constant-force formula for the annual probability, and use the result \(m_{50}=\mu_{50}\). This avoids confusing the force with the annual probability.
First calculate the one-year survival probability:
Therefore,
The central death rate is \(m_{50}=\mu_{50}=0.04\) per year. For 1,000 starting lives, the expected number of deaths is \(1{,}000q_{50}=39.21056\). To four decimal places, \(q_{50}=0.0392\); to four significant figures, \(m_{50}=0.04000\). The expected deaths are about \(39.21\). As a check, the annual probability is less than the force here, while the central rate equals the force.
Conversions Under Uniform Deaths
A different conversion applies when deaths are assumed to be uniformly distributed through the year of age. Under this assumption, the number alive declines linearly from \(l_x\) at age \(x\) to \(l_{x+1}=l_x-d_x\) at age \(x+1\). The average number alive during the year is the average of these endpoints, so the exposure is \(l_x-d_x/2\), measured in person-years.
The last equality follows because \(d_x=l_xq_x\). Solving for \(q_x\) gives the conversion from central death rate to annual death probability under uniform deaths:
This relationship is not the constant-force formula. Under uniform deaths, the number alive is linear through the year; under constant force, it declines exponentially. Although the resulting conversions may be close for small rates, the assumptions and formulas are distinct.
Worked Example: Converting a Probability Under Uniform Deaths
In a hypothetical cohort, 1,200 lives are observed at age 40. Assume deaths are uniformly distributed between ages 40 and 41, and that \(q_{40}=0.10\). Calculate the deaths, exposure, and \(m_{40}\). There is no interest rate. The mortality assumption is uniform deaths, not constant force.
Method. Find the deaths from the starting lives and the annual death probability. Then use the linear-decline exposure implied by uniform deaths.
The number of deaths is
The number alive at age 41 is \(1{,}200-120=1{,}080\). Under uniform deaths, exposure is the average number alive over the year:
Thus,
To four significant figures, \(m_{40}=0.1053\) per year. Check using the conversion formula: \(q_{40}/(1-q_{40}/2)=0.10/0.95=0.10526316\), which agrees with deaths divided by exposure.
Choosing a Conversion From Exposure Data
When a problem provides deaths and exposure, the direct calculation is \(m_x=d_x/E_x\). This is a rate formed from the observed quantities; it does not by itself identify \(q_x\). To infer an annual death probability, the problem must also specify, or reasonably imply, a within-year mortality assumption. If it specifies constant force, set \(\mu_x=m_x\) and use \(q_x=1-e^{-m_x}\). If it specifies uniform deaths, use \(q_x=2m_x/(2+m_x)\).
A probability \(q_x\), a force \(\mu_x\), or deaths and exposure determine different starting calculations.
For death count \(d_x\) and exposure \(E_x\), compute \(m_x=d_x/E_x\). Keep the exposure in person-years for the same age interval.
Use the exponential relationship under constant force and the linear-exposure relationship under uniform deaths.
An annual probability must lie between 0 and 1. Confirm that the interval, time unit, and assumed model agree with the data.
Worked Example: Choosing Between Two Conversions
A hypothetical age interval has 72 deaths and 360 person-years of exposure. Find the central death rate, then calculate the implied annual death probability under (a) constant force and (b) uniform deaths. Treat the exposure as applying to this one-year age interval. No interest rate applies.
Method. First compute the observed central rate. Then apply each assumption separately; the two probabilities are alternatives, not quantities to average.
The central death rate is
(a) Under constant force, \(\mu_x=m_x=0.20\). Therefore,
(b) Under uniform deaths,
To four decimal places, the respective probabilities are 0.1813 and 0.1818. Both are plausible probabilities, but they differ because the assumptions describe different patterns of deaths within the year. As a check, substituting the uniform-deaths value into \(q_x/(1-q_x/2)\) returns \(0.20\); substituting the constant-force value into \(1-e^{-0.20}\) returns the same probability calculated for that model.
Common Mistakes and Practical Checks
A frequent error is to set \(q_x=m_x\). This confuses a probability with a central rate. For example, with \(m_x=0.20\), the incorrect calculation \(q_x=0.20\) ignores the specified mortality pattern. Under constant force, the correct value is \(1-e^{-0.20}=0.1813\); under uniform deaths, it is \(0.40/2.20=0.1818\). The correction is not to memorize one universal conversion, but to identify the model first.
- Do not equate force and annual probability. Under constant force, \(q_x=1-e^{-\mu_x}\), not \(q_x=\mu_x\). They are approximately equal only when the force is small.
- Do not use the uniform-deaths formula for constant force. The uniform-deaths exposure is based on a linear decline in lives, whereas constant force implies exponential decline.
- Check the exposure period. Dividing deaths by exposure in months or days without converting to consistent time units produces a rate on the wrong scale.
- Do not infer an assumption from the word “rate” alone. If the problem gives \(m_x\) but no mortality pattern, state that an additional assumption is needed to determine \(q_x\).
For small mortality rates, the two conversions can be numerically close. That similarity is not evidence that the formulas are interchangeable. A reliable solution labels each rate, states the assumption, uses the matching relationship, and checks the units and probability range. These distinctions are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, state the mortality assumption before converting. Round probabilities to four decimal places and rates to four significant figures where needed.
- A hypothetical life has a constant force of \(0.06\) per year from age 30 to 31. Find \(q_{30}\) and \(m_{30}\). (Answer: \(q_{30}=0.0582\), \(m_{30}=0.06000\) per year.)
- A cohort has 1,200 lives at age 70 and \(q_{70}=0.15\). Under uniform deaths, find the deaths, exposure, and central death rate. (Answer: 180 deaths, 1,110 person-years, \(m_{70}=0.1622\) per year.)
- Exposure data give \(m_x=0.10\) per year. Under uniform deaths, find \(q_x\). (Answer: \(q_x=0.0952\).)
- A hypothetical study records 45 deaths over 500 person-years in one age interval. Calculate \(m_x\), then find \(q_x\) under constant force and under uniform deaths. (Answer: \(m_x=0.09000\) per year; constant-force \(q_x=0.0861\); uniform-deaths \(q_x=0.0861\), more precisely \(0.0861\) under both to four decimals.)
- For a one-year interval, \(q_x=0.20\). Find the constant force and central rate under constant force, then find the central rate under uniform deaths. (Answer: constant-force \(\mu_x=m_x=0.2231\) per year; uniform-deaths \(m_x=0.2222\) per year.)