From Annual Mortality to Multi-Year Probabilities
An annual mortality rate describes one year of risk at a particular age. To find the probability of surviving several years, the calculation must account for the fact that the person reaches each later year only if they survived the earlier ones. This means that annual survival probabilities are multiplied, not added.
The same sequence of rates also gives the probability of death over the full period. The most direct calculation is one minus the probability of surviving the entire period. A second method adds the probabilities of dying in each individual year, with each year’s probability weighted by survival to its start.
The rate \(q_{x+k}\) applies to the year beginning at age \(x+k\). Therefore, an \(n\)-year calculation uses \(q_x,q_{x+1},\ldots,q_{x+n-1}\), or the corresponding \(p\) values. It does not use \(q_{x+n}\), which applies to the year after the \(n\)-year period.
The Product Formula
The probability of surviving every year in a sequence is the product of the probability of surviving each year, conditional on being alive at its start. This follows from the multiplication rule for successive events. In life-table notation, the result is:
Here \({}_np_x\) denotes the probability that a life aged \(x\) survives \(n\) years. The complement is the probability of dying before reaching age \(x+n\):
The notation \({}_nq_x\) in this formula refers to death within \(n\) years, not just death in the first year. In particular, \({}_1q_x=q_x\). When the annual rates differ by age, \({}_nq_x\) generally cannot be calculated by multiplying \(n\) by \(q_x\), or by adding the annual \(q\) values.
Why the Formula Works
Let \(E_k\) be the event that the life survives from age \(x+k\) to age \(x+k+1\). To reach age \(x+n\), the life must experience every event \(E_0,\ldots,E_{n-1}\). Applying the multiplication rule successively gives the derivation:
The probability of surviving from age \(x\) to \(x+1\) is \(P(E_0)=p_x\).
Survival through two years requires \(E_0\) and \(E_1\). The multiplication rule gives \(P(E_0\text{ and }E_1)=P(E_0)P(E_1\mid E_0)=p_xp_{x+1}\), since \(p_{x+1}\) is the one-year survival probability for a life who has reached age \(x+1\).
At each step, the next factor is the probability of surviving the next age interval, given that the life reached its start. Thus \(P(E_0\text{ and }\cdots\text{ and }E_{n-1})=\prod_{k=0}^{n-1}p_{x+k}\).
Within the \(n\)-year period, either the life reaches age \(x+n\) or dies before reaching it. These events are complementary, so the probability of death is \(1-\prod_{k=0}^{n-1}p_{x+k}\).
The same death probability can be expanded by the year in which death occurs. Death in the first year has probability \(q_x\). Death in the second year requires surviving the first year and then dying in the second, with probability \(p_xq_{x+1}\). Continuing in this way gives:
These terms describe mutually exclusive outcomes: death in the first year, death in the second year, and so on. Their sum must equal one minus the probability of surviving all \(n\) years. This is a useful independent check on a calculation.
Worked Calculations
Worked Example: Three Annual Rates
A hypothetical insurance model assigns annual death probabilities \(q_{50}=0.02\), \(q_{51}=0.03\), and \(q_{52}=0.04\). For a life aged exactly 50, calculate the probability of surviving to age 53 and the probability of dying before age 53. Assume the stated rates apply to the respective one-year age intervals.
Method. Convert each annual death probability to a survival probability, multiply the three survival probabilities, and use the complement for death.
The annual survival probabilities are:
Therefore, the probability of surviving all three years is:
The probability of death before age 53 is:
As a check, add the probabilities of death in each year: \(0.02+0.98(0.03)+0.98(0.97)(0.04)=0.02+0.0294+0.038024=0.087424\). The results agree. To four decimal places, survival is \(0.9126\) and death is \(0.0874\). The common mistake avoided here is adding \(0.02+0.03+0.04\); those rates are conditional on reaching different ages, so the later rates must be weighted by prior survival.
Worked Example: Death in a Specified Year
For a hypothetical population model, a life aged 60 has annual death probabilities \(q_{60}=0.04\), \(q_{61}=0.06\), \(q_{62}=0.10\), and \(q_{63}=0.15\). Calculate (a) the probability of surviving to age 64, (b) the probability of dying before age 64, and (c) the probability, from age 60, of dying during the year from age 62 to age 63.
Method. Use the product of the four annual survival probabilities for (a), its complement for (b), and survival through the first two years multiplied by \(q_{62}\) for (c).
First convert the rates:
The four-year survival probability is:
Thus the probability of death before age 64 is \(1-0.690336=0.309664\). For death during the third year, the life must first survive the intervals from age 60 to 61 and from age 61 to 62, then die in the interval from age 62 to 63:
As a check on (b), the probabilities of dying in years one through four are \(0.04\), \(0.96(0.06)=0.0576\), \(0.9024(0.10)=0.09024\), and \(0.81216(0.15)=0.121824\). Their sum is \(0.04+0.0576+0.09024+0.121824=0.309664\). To four decimal places, the answers are \(0.6903\), \(0.3097\), and \(0.0902\), respectively. The common mistake avoided is treating \(q_{62}\) alone as the probability of third-year death from age 60; it applies only after reaching age 62.
Worked Example: Five Years With Changing Rates
A hypothetical projection uses \(q_{70}=0.01\), \(q_{71}=0.02\), \(q_{72}=0.03\), \(q_{73}=0.05\), and \(q_{74}=0.08\). For a life aged 70, find the probability of dying within five years and the probability of dying in the fifth year of the period. Assume each \(q\) applies to the year beginning at its subscript age.
Method. Find the five-year survival product and take its complement for death within five years. For death in the fifth year, multiply survival through the first four years by \(q_{74}\).
The annual survival probabilities are \(0.99,0.98,0.97,0.95,\) and \(0.92\). Multiply them in age order:
Therefore, the probability of death within five years is \(1-0.822516156=0.177483844\). Survival through the first four years is \(0.8940393\), so the probability of dying in the fifth year is:
The fifth-year death probability is one part of the five-year death probability; it is not the whole five-year probability. To four decimal places, the answers are \(0.1775\) and \(0.0715\). As a check, adding the five disjoint annual death probabilities gives \(0.01+0.99(0.02)+0.9702(0.03)+0.941094(0.05)+0.8940393(0.08)=0.177483844\), agreeing with the complement calculation.
Common Exam Traps
The product formula is simple, but errors often come from using the right operation on the wrong interval or from confusing a one-year rate with a multi-year probability. Before calculating, write down the first and last age intervals included. This makes the indexing explicit.
- Adding the annual \(q\) values. The \(q\) values are conditional on being alive at the start of their respective years. Use one minus the product of the \(p\) values, or sum annual death probabilities weighted by survival to each year's start.
- Using the wrong endpoint rate. Survival for \(n\) years from age \(x\) uses \(p_x\) through \(p_{x+n-1}\). The rate \(q_{x+n}\) belongs to the following year.
- Confusing death in a particular year with death by that year. Death during the \(j\)th year from age \(x\) has probability \(\left(\prod_{k=0}^{j-2}p_{x+k}\right)q_{x+j-1}\). Death by the end of that year includes deaths in all earlier years as well.
- Reporting a survival probability as a death probability. These are complements over the same period: \({}_nq_x=1-{}_np_x\).
Check Your Understanding
Use the annual rates as applying to the year beginning at the subscript age. Give probabilities to four decimal places.
- A hypothetical model has \(q_{40}=0.03\), \(q_{41}=0.04\), and \(q_{42}=0.05\). Find the probability that a life aged 40 dies before age 43. (Answer: \(0.1154\))
- For a life aged 55, suppose \(q_{55}=0.02\), \(q_{56}=0.03\), \(q_{57}=0.04\), and \(q_{58}=0.05\). Find the probability of surviving to age 59. (Answer: \(0.8669\))
- A hypothetical population has \(q_{30}=0.10\), \(q_{31}=0.08\), and \(q_{32}=0.06\). Find (a) the probability of dying within three years from age 30 and (b) the probability, from age 30, of dying in the second year. (Answer: (a) \(0.2217\); (b) \(0.0720\))
- For a life aged 65, the annual rates are \(q_{65}=0.025\), \(q_{66}=0.035\), and \(q_{67}=0.045\). Find the probability, from age 65, of dying during the third year. (Answer: \(0.0423\))
- A hypothetical projection gives \(q_{50}=0.04\), \(q_{51}=0.05\), \(q_{52}=0.06\), \(q_{53}=0.08\), and \(q_{54}=0.10\). Find (a) the probability of dying within five years from age 50 and (b) the probability of dying during the fourth year. (Answer: (a) \(0.2902\); (b) \(0.0686\))