Ultimate Rates By Attained Age

Why Attained-Age Indexing Matters

An ultimate mortality table gives a one-year mortality rate for each attained age. To calculate survival over several years, the rate changes as the person reaches each successive age. The calculation is therefore not usually one rate raised to the number of years: it is a product of the one-year survival probabilities at the attained ages traversed.

This tutorial considers a life subject to ultimate rates throughout the period being calculated. If a select period applies, first identify which rates govern each year; do not substitute ultimate rates for years that are still governed by select rates. The calculations below assume integer starting ages and complete one-year intervals. A fractional starting age requires an additional fractional-age assumption or rates supplied for that age; the nearest integer-age rate alone does not determine the answer.

Definition: For a life aged \(x\), \(q_x\) is the probability of death before age \(x+1\), conditional on being alive at age \(x\). The corresponding one-year survival probability is \(p_x=1-q_x\). An ultimate table indexes these rates by attained age: a life who survives from age \(x\) to age \(x+1\) uses \(q_x\) for that year, then \(q_{x+1}\) for the next year.

Combining Ultimate Rates Across Years

Suppose a life is alive at integer age \(x\), and ultimate rates apply over the next \(n\) complete years. For the first year, survival to age \(x+1\) has probability \(p_x\). Conditional on reaching age \(x+1\), survival to age \(x+2\) has probability \(p_{x+1}\). Continuing in this way gives the multi-year survival probability.

{}_np_x=p_xp_{x+1}\cdots p_{x+n-1} =\prod_{j=0}^{n-1}p_{x+j} =\prod_{j=0}^{n-1}(1-q_{x+j}).

The product follows from the multiplication rule for conditional probabilities. Each factor is the probability of surviving one year, conditional on having reached that year's starting age. Consequently, the product gives the probability of surviving every interval in sequence. The probability of death before age \(x+n\), given survival to age \(x\), is the complement:

{}_nq_x=1-{}_np_x.

This complement is for death at any point during the \(n\)-year period. To find death within a particular later part of the period, first account for survival to the start of that part, then calculate the conditional probability of death in that part.

List the attained ages.
For a calculation from age \(x\) to age \(x+n\), the one-year intervals start at ages \(x,x+1,\ldots,x+n-1\).

Match each interval to its rate.
Use \(q_{x+j}\) for the year from age \(x+j\) to age \(x+j+1\), and convert it to survival using \(p_{x+j}=1-q_{x+j}\).

Multiply and interpret.
Multiply the one-year survival probabilities. Take the complement only when the requested event is death during the entire stated period.

Worked Example: Survival From an Interior Attained Age

A hypothetical ultimate table gives \(q_{36}=0.012\), \(q_{37}=0.018\), and \(q_{38}=0.025\). For a life alive at age 36, find (a) the probability of surviving to age 39 and (b) the probability of dying before age 39. Give probabilities to four decimal places.

Method. The interval from age 36 to age 39 contains three one-year periods. For part (a), the three one-year survival probabilities needed are \(p_{36}\), \(p_{37}\), and \(p_{38}\).

Convert each mortality rate to a survival probability:

p_{36}=1-0.012=0.988,\qquad p_{37}=1-0.018=0.982,\qquad p_{38}=1-0.025=0.975.

Multiply in attained-age order:

{}_3p_{36}=0.988(0.982)(0.975) =0.970216(0.975) =0.9459606.

Thus the survival probability is \(0.9460\) to four decimal places. The probability of death before age 39 is the complement:

{}_3q_{36}=1-0.9459606=0.0540394,

or \(0.0540\) to four decimal places. As a check, the two event probabilities sum to \(0.9459606+0.0540394=1\). The common mistake is to omit \(p_{38}\): reaching age 39 from age 36 requires surviving all three intervals, not just the first two.

Conditional Probabilities Within a Multi-Year Period

A question may ask about a later interval conditional on being alive at an intermediate age. Once that condition is given, start the calculation at the intermediate age. For example, conditional on being alive at age \(y\), the probability of surviving to age \(z\), for integer \(z>y\), is the product of the rates from ages \(y\) through \(z-1\). The rates before age \(y\) are not part of this conditional calculation.

P(\text{alive at age }z\mid\text{alive at age }y) =\prod_{j=y}^{z-1}p_j,\qquad z>y.

If instead the life starts at an earlier age and the event concerns a later interval, both pieces matter: multiply the probability of reaching the interval by the conditional probability of the event within it. This is another application of the multiplication rule, not a different mortality model.

Worked Example: Death in a Later Interval

A hypothetical ultimate table has \(q_{44}=0.010\), \(q_{45}=0.014\), \(q_{46}=0.020\), and \(q_{47}=0.025\). A life is alive at age 44. Find (a) the probability of death before age 48 and (b) the probability of surviving to age 46 and then dying before age 48. Give probabilities to four decimal places.

Method for (a). First convert all four rates to survival probabilities:

p_{44}=0.990,\qquad p_{45}=0.986,\qquad p_{46}=0.980,\qquad p_{47}=0.975.

Survival from age 44 to age 48 requires survival across all four intervals:

{}_4p_{44} =(0.990)(0.986)(0.980)(0.975) =(0.97614)(0.980)(0.975) =(0.9566172)(0.975) =0.93270177.

Therefore, death before age 48 has probability \(1-0.93270177=0.06729823\), or \(0.0673\) to four decimal places.

Method for (b). Survive from age 44 to age 46, then die during the two-year interval from age 46 to age 48. The probability of the first event is

{}_2p_{44}=(0.990)(0.986)=0.97614.

Conditional on reaching age 46, survival to age 48 is \((0.980)(0.975)=0.9555\). Hence conditional death before age 48 is \(1-0.9555=0.0445\). Multiply the probability of reaching age 46 by this conditional probability:

$$ (0.97614)(0.0445)=0.04343823. $$

The requested probability is \(0.0434\) to four decimal places. As a check, the probability of dying before age 46 is \(1-0.97614=0.02386\), and \(0.02386+0.04343823=0.06729823\), agreeing with part (a). A common error is to report \(0.0445\) as the answer to part (b); that is conditional on already being alive at age 46 and does not include the probability of reaching age 46 from age 44.

Using Rates When the Starting Age Is Not a Table Boundary

A starting age does not have to be the first age shown in a table or the boundary of a requested multi-year period. What matters is that the table supplies the one-year ultimate rates for every attained age whose interval is traversed. Begin at the life’s actual integer attained age and use the rates from that age onward. Do not restart the rate sequence at the first row displayed, and do not use the rate at the ending age for an interval that has already ended.

For example, survival from age 53 to age 57 uses \(p_{53}\), \(p_{54}\), \(p_{55}\), and \(p_{56}\). It does not use \(p_{52}\), because that interval ends at age 53, or \(p_{57}\), because that interval begins at age 57. The age index names the age at the beginning of the one-year interval.

Worked Example: Starting Within the Table’s Age Range

A hypothetical ultimate table gives the following rates: \(q_{52}=0.009\), \(q_{53}=0.011\), \(q_{54}=0.015\), \(q_{55}=0.019\), \(q_{56}=0.024\), and \(q_{57}=0.030\). A life is alive at age 53, an interior age in the supplied range. Find (a) the probability of surviving to age 57 and (b) the probability of surviving to age 55 and then dying before age 57. Give probabilities to four decimal places.

Method for (a). The age-53-to-57 period has four intervals, so it uses \(q_{53}\) through \(q_{56}\). The rates at ages 52 and 57 are not used for these intervals. Convert the required rates:

p_{53}=0.989,\qquad p_{54}=0.985,\qquad p_{55}=0.981,\qquad p_{56}=0.976.

Multiply the four survival probabilities:

{}_4p_{53} =(0.989)(0.985)(0.981)(0.976) =(0.974165)(0.981)(0.976) =(0.955655865)(0.976) =0.93272012424.

Thus the probability of surviving to age 57 is \(0.9327\) to four decimal places.

Method for (b). First find the probability of surviving from age 53 to age 55:

{}_2p_{53}=(0.989)(0.985)=0.974165.

Given survival to age 55, the probability of dying before age 57 is one minus survival through the age-55 and age-56 intervals:

1-(p_{55})(p_{56}) =1-(0.981)(0.976) =1-0.957456 =0.042544.

Combine the two stages:

$$ (0.974165)(0.042544)=0.04144487576. $$

The requested probability is \(0.0414\) to four decimal places. As a check, this event is contained within survival-to-age-55, whose probability is \(0.974165\); the result is smaller, as it must be. It is also not the total probability of death before age 57, because it excludes deaths before age 55.

Data Limits and Common Exam Traps

The product method is exact when the applicable one-year rates are known for every interval, and the life is subject to those rates for the period in question. It does not supply missing rates, justify extrapolation beyond the table, or determine survival from a fractional age using integer-age rates. If the question gives only an overall rate or omits a rate needed for an interval, additional information or a stated mortality assumption is required.

Using duration instead of attained age. The second year after age 40 uses \(q_{41}\), not \(q_{40}\) again, under an ultimate table indexed by attained age.
Including an extra rate. Survival from age \(x\) to age \(x+n\) uses the intervals starting at \(x\) through \(x+n-1\); the rate \(q_{x+n}\) is not part of that product.
Taking a complement too early. For death during a multi-year span, first calculate survival across the full span, then subtract from one. For death in a later subinterval, also account for reaching that subinterval.
Assuming fractional-age rates are automatic. Integer-age ultimate rates alone do not specify how mortality is distributed within a year. Use a fractional-age assumption only when it is supplied or established.

Attained-age indexing is the central discipline in these calculations: write down the one-year intervals before selecting rates. Once every interval is paired with the correct \(q_x\), the conversion to \(p_x\), multiplication, and any required complement follow directly. This topic is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the relevant attained-age intervals before calculating. Give each final probability to four decimal places.

A hypothetical ultimate table has \(q_{30}=0.020\) and \(q_{31}=0.030\). Find the probability that a life alive at age 30 survives to age 32. (Answer: \(0.9506\))
A life alive at age 40 faces ultimate rates \(q_{40}=0.008\), \(q_{41}=0.012\), and \(q_{42}=0.016\). Find the probability of death before age 43. (Answer: \(0.0356\))
Given that a life is alive at age 61, the ultimate rates are \(q_{61}=0.025\) and \(q_{62}=0.040\). Find the probability of dying before age 63, conditional on being alive at age 61. (Answer: \(0.0640\))
A life is alive at age 70. The ultimate rates are \(q_{70}=0.030\), \(q_{71}=0.040\), \(q_{72}=0.050\), and \(q_{73}=0.060\). Find the probability of surviving to age 74. (Answer: \(0.8316\))
A life is alive at age 50, and the ultimate rates are \(q_{50}=0.010\), \(q_{51}=0.020\), \(q_{52}=0.030\), and \(q_{53}=0.040\). Find the probability of surviving to age 52 and then dying before age 54. (Answer: \(0.0667\))

What You'll Learn