Multiple Decrement Models: Competing Risks for LTAM

Multiple Decrement Tables

A multiple decrement model tracks an individual who is subject to several competing causes of exit from a status. In a pension context, an active employee might leave due to death, disability, voluntary withdrawal, or retirement. Each cause j has a dependent (absolute) probability of decrement q_x^(j) and a force of decrement mu_x^(j). The total force of decrement is mu_x^(tau) = sum of mu_x^(j) over all causes j. The total probability of decrement from any cause is q_x^(tau) = 1 minus the probability of remaining in the status for the entire year.

The key distinction in multiple decrement theory is between dependent and independent rates. The dependent probability q_x^(j) measures the probability of decrement from cause j in the presence of all other causes. The independent (associated single-decrement) probability q_x'^(j) measures what the probability of decrement from cause j would be if it were the only active cause. Because other decrements "compete" by removing individuals before cause j can act, the dependent probability is always less than or equal to the independent probability.

Relationship Between Dependent and Independent Rates

The forces of decrement provide the cleanest connection between the two environments. In the multiple decrement model, the survival probability is t_p_x^(tau) = exp(-integral from 0 to t of mu_x^(tau)(s) ds). Because mu_x^(tau) = sum of mu_x^(j), we can write t_p_x^(tau) = product over j of exp(-integral from 0 to t of mu_x^(j)(s) ds) = product over j of t_p_x'^(j). This multiplicative relationship holds exactly at the level of forces of decrement.

Converting between dependent probabilities q_x^(j) and independent probabilities q_x'^(j) requires additional assumptions because the relationship is not exact at the probability level. The most common assumption for Exam LTAM is UDD (uniform distribution of each decrement in the multiple decrement table), which leads to tractable conversion formulas.

UDD Conversion Formulas

Under the assumption that each decrement is uniformly distributed in the multiple decrement table, the conversion formula for two decrements is: q_x^(1) = q_x'^(1) * (1 minus 0.5 * q_x'^(2)) and q_x^(2) = q_x'^(2) * (1 minus 0.5 * q_x'^(1)). For three decrements, the formula extends to: q_x^(j) = q_x'^(j) * [1 minus 0.5 * (sum of other q_x'^(k)) + (1/3) * (product of other pairs of q_x'^(k))].

For example, suppose independent rates are q_x'^(1) = 0.05 (death) and q_x'^(2) = 0.10 (withdrawal). The dependent rates are q_x^(1) = 0.05 * (1 minus 0.5 * 0.10) = 0.05 * 0.95 = 0.0475, and q_x^(2) = 0.10 * (1 minus 0.5 * 0.05) = 0.10 * 0.975 = 0.0975. The total decrement rate is q_x^(tau) = 0.0475 + 0.0975 = 0.145, which is less than q_x'^(1) + q_x'^(2) = 0.15 because of the competing effects.

Constructing a Multiple Decrement Table

A multiple decrement table starts with an initial cohort l_x^(tau) (often 100,000 at the youngest age) and tracks the number leaving due to each cause at each age. The number of decrements from cause j at age x is d_x^(j) = l_x^(tau) * q_x^(j). The number surviving to age x+1 is l_{x+1}^(tau) = l_x^(tau) minus d_x^(tau), where d_x^(tau) = sum of d_x^(j) over all causes.

To build the table from independent single-decrement studies, you first obtain the independent rates q_x'^(j) from each study, then convert them to dependent rates q_x^(j) using the UDD formulas (or another assumption), and finally compute the decrement counts. This process is essential when combining mortality data from one source with withdrawal data from another, since each source was studied in isolation.

Applications in Pension Valuation

Multiple decrement models are central to pension mathematics. An active employee faces decrements of death, disability, voluntary withdrawal, and retirement. The service table tracks the expected number of employees at each age and the number leaving due to each cause. The expected cost of benefits for each decrement type is computed using the dependent probabilities from the service table.

For example, the expected present value of death benefits for an active employee age x is the sum over future ages of v^(t) * t_p_x^(tau) * q_{x+t}^(death) * b_{x+t}^(death), where b is the death benefit amount. Similarly, retirement benefits use q_{x+t}^(retirement) and the associated pension benefit. Each decrement has its own benefit structure, and the total liability is the sum across all decrements.

Common Pitfalls

The most frequent error is confusing dependent and independent probabilities. When a problem gives you rates from separate single-decrement studies, those are independent rates that must be converted before use in the multiple decrement environment. Conversely, rates extracted directly from a multiple decrement table are dependent rates. Using one type where the other is needed produces incorrect results.

Another common mistake involves the UDD conversion with three or more decrements. Students often apply the two-decrement formula pairwise rather than using the correct three-decrement formula. The interaction term (the product of pairs divided by 3) matters when rates are large. Also, be careful about which UDD assumption is being applied. UDD in the multiple decrement table (used for the conversion formulas above) is different from UDD in each single-decrement table (which produces different conversion formulas).

Study Tips

Exam LTAM tests your ability to convert between dependent and independent rates, construct multiple decrement tables, and apply them to pension and insurance valuations. Practice the conversion formulas until they are automatic, especially the two-decrement case. Work through problems that start with independent rates from separate studies and require building a complete service table. Pay close attention to the problem statement to determine whether given rates are dependent or independent, and which UDD assumption applies. These details determine which formula to use, and getting them wrong leads to entirely different answers.