Why Cause-Specific Intensities Are Needed
In a competing decrement model, a person or policy remains in an active group until an exit occurs. Possible exits might include death, withdrawal, or entry into a care state. For each cause, an actuary needs a measure of how quickly that particular exit is occurring among those who are still in the group. A cause-specific intensity provides that measure at an instant in time.
The intensity is not itself a probability. It is a rate, conditional on being active at the time being considered. Its value can change as duration in the group or attained age changes. Stating the time basis and the conditioning event is therefore essential to interpreting a decrement assumption.
The word first matters: in a competing decrement model, the causes describe alternative ways of leaving the active group. Once an exit occurs, the person is no longer exposed to the active-group exit process being described. The intensity is conditional on being active at the instant \(t\), not on having been active since some earlier time without qualification.
Rate, Time Scale, and Short-Interval Meaning
When the intensity exists and is suitably regular near \(t\), its definition implies the short-interval approximation \(P(\text{exit by }j\text{ in }(t,t+h]\mid\text{active at }t) =\lambda_j(t)h+o(h)\). Here \(o(h)\) denotes a term that becomes negligible relative to \(h\) as \(h\) approaches zero. Thus, for sufficiently small \(h\), the probability is approximately the current intensity multiplied by the interval length.
The time units must match. If \(\lambda_j(t)\) is expressed per year, then \(h\) must be in years in this approximation. An intensity of \(0.04\) per year does not mean a 4% probability over every interval. It gives a first-order probability of about \(0.04h\) over a short interval of \(h\) years, conditional on being active at its start.
An intensity may be greater than 1 without being an invalid probability model. For example, an intensity of \(1.4\) per year is a rate; it is not a probability of 140%. Probabilities concern intervals, while an intensity describes a limiting rate as the interval shrinks.
Worked Example: Reading a Local Probability Expansion
A hypothetical policy group has two possible exits, and the active-group model specifies that, at duration \(t=3\) years, the conditional probability of exit by cause A during the next \(h\) years is \(0.025h+o(h)\) as \(h\downarrow0\). Determine the cause-A intensity and estimate the probability over a short interval of 0.02 years.
Method and assumptions. Use the definition of cause-specific intensity. The interval \(h\) is measured in years, the probability is conditional on being active at duration 3, and the given expansion describes the behavior as \(h\) approaches zero.
Step 1: Identify the coefficient of \(h\). Divide the conditional probability expansion by \(h\): \(0.025h/h+o(h)/h=0.025+o(h)/h\). As \(h\downarrow0\), the second term tends to zero. Therefore, \(\lambda_A(3)=0.025\) per year.
Step 2: Apply the short-interval approximation. For \(h=0.02\) years, the first-order estimate is \(0.025\times0.02=0.0005\). As a percentage, this is \(0.0005\times100=0.05\%\), conditional on being active at duration 3.
Check. The units cancel in the multiplication: \((1/\text{year})(\text{year})\) is dimensionless, as a probability must be. The result is an approximation, not an exact interval probability. The mistake to avoid is reporting \(0.025\) as a 2.5% probability for the 0.02-year interval; the interval length must be included.
Time Dependence: Duration and Attained Age
An intensity may vary with time. The meaning of its argument depends on the model’s time scale. If \(t\) means duration since entry into the group, then \(\lambda_j(t)\) is duration-dependent. If the model is indexed by attained age \(x+t\), an intensity may instead be written as a function of attained age. In some models both attained age and duration matter, so writing \(\lambda_j(x,t)\) makes both inputs explicit.
These time scales cannot be substituted for one another without checking the model definition. A policy issued at age 60 and observed at duration 5 has attained age 65. A duration-dependent rate at 5 and an age-dependent rate at 65 are different model quantities, even though they refer to the same observation time.
Worked Example: Evaluating Two Duration-Dependent Intensities
A hypothetical income-protection portfolio has two competing exits while a policy is active: recovery from a qualifying condition (cause R) and withdrawal (cause W). The model specifies, with duration \(t\) measured in years, \(\lambda_R(t)=0.012+0.003t\) per year and \(\lambda_W(t)=0.024-0.002t\) per year, for the durations considered. Find both intensities at duration 4 and estimate each cause’s conditional exit probability over the next month.
Method and assumptions. Evaluate each specified rate at the stated duration, then use the first-order short-interval approximation separately for each cause. Treat one month as \(1/12\) year. The calculations are local approximations conditional on being active at duration 4.
Step 1: Evaluate recovery intensity. \(\lambda_R(4)=0.012+0.003(4)=0.012+0.012=0.024\) per year.
Step 2: Evaluate withdrawal intensity. \(\lambda_W(4)=0.024-0.002(4)=0.024-0.008=0.016\) per year.
Step 3: Convert the interval to years. The next month has length \(h=1/12\) year. For recovery, the estimate is \(0.024(1/12)=0.002\), or \(0.20\%\). For withdrawal, the estimate is \(0.016(1/12)=0.001333\ldots\), or approximately \(0.1333\%\).
Check. Each estimate uses its own cause-specific intensity and the same interval length. Neither is an exact one-month probability: the approximation uses the rates at the interval’s starting duration and is justified as a short-interval approximation. The rates are not constant by assumption over all durations; their formulas explicitly vary with \(t\). The common mistake is to use 1 as the interval length because the rates are per year. One month is \(1/12\), not 1 year.
Attained Age and Duration in One Model
A model can make the distinction between age and duration explicit. Let \(x\) be age at entry and \(t\) be duration since entry. For a cause-specific intensity \(\lambda_j(x,t)\), changing \(x\) while holding \(t\) fixed changes attained age; changing \(t\) while holding \(x\) fixed changes both duration and attained age. The formula’s specified arguments determine what effects are included.
Worked Example: Keeping Age and Duration Straight
A hypothetical long-term care model defines an incidence-exit intensity by \(\lambda_I(x,t)=0.004+0.0005(x-60)+0.001t\) per year, where \(x\) is age at entry in years and \(t\) is duration in years. For a person entering at age 64, calculate the intensity at duration 3. Then calculate the intensity at duration 3 for someone entering at age 60, and identify the effect of changing entry age.
Method and assumptions. Substitute entry age and duration into the model as separate inputs. The intensity is conditional on being active at the evaluation time; no interval probability or additional decrement assumption is needed for this calculation.
Step 1: Substitute \(x=64\) and \(t=3\). \(\lambda_I(64,3)=0.004+0.0005(64-60)+0.001(3)\). The age term is \(0.0005(4)=0.002\), and the duration term is \(0.001(3)=0.003\). Thus \(\lambda_I(64,3)=0.004+0.002+0.003=0.009\) per year.
Step 2: Substitute \(x=60\) and \(t=3\). \(\lambda_I(60,3)=0.004+0.0005(60-60)+0.001(3)\) \(=0.004+0+0.003=0.007\) per year.
Step 3: Compare. The difference is \(0.009-0.007=0.002\) per year. In this model, the four-year difference in entry age contributes \(0.0005(4)=0.002\) per year at the same duration.
Check. Both evaluations use duration 3, and the age-at-entry term is the only input that changes. The mistake to avoid is replacing \(x\) with attained age \(x+t\) in the formula: the model explicitly defines its first argument as age at entry. A different model could be specified using attained age, but that is not the formula given here.
Why the Definition Produces a Rate
The intensity’s interpretation follows directly from the probability limit. Let \(p_j(t,h)\) be the conditional probability in the numerator of the definition. For small positive \(h\), the limit says that the ratio \(p_j(t,h)/h\) approaches \(\lambda_j(t)\). Equivalently, the probability has a first-order term proportional to the interval length:
This relation justifies using \(\lambda_j(t)h\) as a short-interval approximation. It also explains the units: \(p_j(t,h)\) is dimensionless and \(h\) has units of time, so \(\lambda_j(t)\) must have units of inverse time. The approximation becomes useful when the interval is sufficiently short and the intensity does not change sharply within it. For a longer interval, one should not assume that multiplying the starting intensity by the full interval gives an exact probability.
The argument does not require that an intensity be constant over time. It describes the local rate at the particular \(t\) being evaluated. If the model specifies a changing intensity, the value at \(t=2\) can differ from the value at \(t=5\). This local definition does not by itself provide an exact finite-interval probability; that requires additional analysis of the model over the interval.
Common Exam Traps
- Treating a rate as a probability. An intensity of \(0.03\) per year is not automatically a 3% probability for any interval. For a short interval of \(h\) years, its first-order probability contribution is about \(0.03h\).
- Using the wrong time unit. If the intensity is per year, a one-month interval must be entered as \(1/12\) year. If the intensity is per month, use an interval measured in months.
- Ignoring the conditioning event. The defined intensity applies among those active at time \(t\), not among the original entry group regardless of what happened before \(t\).
- Confusing attained age with duration. Read the model’s argument definitions before substituting values. Age at entry, attained age, and duration are not interchangeable.
- Assuming the intensity is constant. A time-dependent formula must be evaluated at the time specified. A starting value multiplied by a long interval does not generally give an exact interval probability.
Check Your Understanding
For each question, use the stated time scale and interpret any short-interval calculation as a first-order approximation.
- For a hypothetical cause A, the conditional probability of exit over the next \(h\) years at duration \(t\) is \(0.0009h+o(h)\). What is the cause-A intensity at duration \(t\), including its units? (Answer: \(0.0009\) per year.)
- A duration-dependent intensity is \(\lambda_B(t)=0.010+0.002t\) per year. Evaluate it at duration 6 years. (Answer: \(0.022\) per year.)
- A model specifies a cause-C intensity of \(0.004\) per month. Express the same rate per year. (Answer: \(0.048\) per year.)
- At a specified time, a cause-specific intensity is \(0.03\) per year. Estimate the conditional probability of that cause’s exit over the next \(0.02\) years using the short-interval approximation. (Answer: \(0.0006\), or \(0.06\%\).)
- A model defines \(\lambda_D(x,t)=0.006+0.001(x-50)+0.002t\) per year, where \(x\) is age at entry and \(t\) is duration, both in years. Find the intensity for entry age 55 at duration 2. (Answer: \(0.015\) per year.)