Calculate policy values for state-dependent life insurance, long-term health insurance, and CCRCs.

A disability income policy in force five years carries one premium stream, two benefit streams, and several possible state transitions each instant. Valuing it means tracking a different reserve for every state the insured could currently occupy.

In a single-state whole life policy, age and duration pin down the reserve. In a multi-state world, two policyholders the same age and duration can carry vastly different liabilities depending on whether they are healthy, disabled, or in skilled nursing. Each occupied state has its own expected future cash flows. The notation ${}_t{V}^{(i)}$ reads "policy value at time t given the life occupies state i."

A model is Markov when, given the current state, the future evolution of the process is independent of the path taken to reach that state. Time since entry, transitions visited, and prior premiums paid are all forgotten the instant you specify the current state. This memorylessness is what lets you build a transition matrix $[p^{ij}]$ and reuse it for every life in state $i$.

Common mistakes

• Using a single reserve for all states. Computing one "policy value" for a DI insured ignores the ALR/DLR distinction. The exam expects two separate values; mixing them produces wrong reserves by a factor of 5 to 10.
• Counting premium in the disabled state. DI and LTC almost never collect premium during a claim. Including $2,400 of phantom premium in the DLR recursion understates the reserve. (Critical-illness riders that do collect in state 1 are the exception; read the question.)
• Wrong sign on sum at risk. Sum at risk on i-to-j transition is $S^{(ij)}+{}_t{V}^{(j)}-{}_t{V}^{(i)}$, not ${}_t{V}^{(i)}-{}_t{V}^{(j)}$. Reversed sign flips Thiele's equation and gives a reserve growing the wrong direction.

Bottom line

• State-conditioned reserves via Thiele: compute one ${}_t{V}^{(i)}$ per occupied state, equal to EPV future benefits and expenses minus EPV future premiums from $i$. Same age and duration, different state, different liability.
• Premium-paying and benefit-receiving states are disjoint in DI and LTC: premium collected only in active or IL; benefit paid in disabled, AL, or SN.
• DLR exceeds ALR mid-claim (ALR hump-shaped, DLR monotonically decreasing); LTC caps cumulative benefit via a pool while DI pays an uncapped lifetime annuity.
• Sum at risk on transition $i\to j$ equals $S^{(ij)}+{}_t{V}^{(j)}-{}_t{V}^{(i)}$: lump-sum benefit plus reserve assumed minus reserve released.

Exam shortcut

When the prompt names ALR or DLR, read the state immediately. ALR uses both premium and benefit terms in the recursion. DLR drops the premium term entirely. DECISION: Multi-state product. Build the transition matrix first, then plug into the recursion. CCRC. Identify the contract type before computing FSO; Type A produces the largest reserve. DECISION: Term-limited EPV with only whole-life table values?

The full lesson (about 12,703 words, 85 min read) adds 15 worked examples, all 16 common mistakes, a self-check, free in the app.

Learning objectives

• 2d

Browse all free Exam ALTAM lessons or jump into free Exam ALTAM practice questions.