Calculate premiums for state-dependent life insurance, long-term health insurance, and CCRCs using the equivalence principle.

A continuing care retirement community sells a single entry fee and a monthly rent that depends on whether the resident lives in an apartment, an assisted-living unit, or a skilled-nursing bed. You price that bundle of state-contingent cash flows the same way a multi-state disability policy is priced: equate the expected present value of premiums to the expected present value of benefits at issue.

Pricing equates time-zero expectations so the carrier neither siphons capital from policyholders nor under-collects and goes insolvent. The gross premium $P$ solves

$$P \cdot \text{EPV}(\text{premium annuity}) = \text{EPV}(\text{benefits}) + \text{EPV}(\text{expenses})$$

For a state-dependent contract, every EPV is built from transition probabilities ${}_t p_x^{ij}$ and forces $\mu_{x+t}^{jk}$ of a Markov model. Pick the issue state $i$ (almost always state 0, healthy or active), discount at force $\delta$, and integrate.

Two building blocks repeat in every problem. Think of the annuity as renting a meter that ticks only while you sit in a specific state, and the insurance as a smoke alarm that fires the moment you cross from one state...

Common mistakes

• Including disabled-state annuity in the premium-paying denominator. When premiums waive on disability, only $\bar{a}_x^{00}$ belongs in the denominator. Adding $\bar{a}_x^{01}$ underprices a disability income policy by 15 to 30 percent.
• Using $\bar{A}_x$ instead of $\bar{A}_x^{ij\to k}$ for state-contingent death benefits. Single-life $\bar{A}_x$ ignores the path through alive states. The transition-specific insurance $\bar{A}_x^{0j\to k}$ is required when benefits differ by death state.
• Forgetting the deferred period in long-term care. A $5,000-per-month LTC policy with a 90-day waiting period uses $\bar{a}_x^{01,\,0.25}$, not $\bar{a}_x^{01}$. The deferred integral can cut benefit EPV by 10 percent or more.

Bottom line

• Equivalence principle: EPV of premium inflows equals EPV of benefits plus expenses at issue.
• State-contingent flows: premiums collect only in active states, benefits pay in claim states, and state-dependent death benefits use transition-specific $\bar{A}_x^{ij\to k}$ firing the instant a life moves between states.
• Long-term health: when premiums waive on disability only $\bar{a}_x^{00}$ sits in the denominator; deferred periods run from policy issue while elimination periods restart on each new disability spell.
• CCRC: the entry fee $F$ enters equivalence once at $t=0$ and level monthly fees fund housing and care; Type-A holds the fee $M$ level for life while rental contracts let $M_j$ vary by tier.

Exam shortcut

Build a one-line table of all relevant $\bar{a}_x^{ij}$ and $\bar{A}_x^{ij\to k}$ values before writing any equivalence equation. Half of exam errors come from grabbing the wrong superscript pair under time pressure. DECISION: Benefit paid while in a state → annuity $\bar{a}_x^{ij}$. Benefit paid on entering a state → insurance $\bar{A}_x^{ij\to k}$. Lump on death from any alive state → sum the $\bar{A}_x^{0j\to D}$ across alive $j$.

The full lesson (about 9,492 words, 63 min read) adds 7 worked examples, all 14 common mistakes, a self-check, free in the app.

Learning objectives

• 2c

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