Equivalence Premiums for Multi-State Benefit Packages

Why Equivalence Premiums Need a State Model

A level premium for a multi-state contract is not generally found by dividing the total benefit amounts by the number of years in the contract. The probability of receiving a benefit can depend on the insured’s state, while the probability of paying a premium can depend on a different eligibility condition. A disability contract, for example, might pay while disabled but collect premiums only while active. A pension arrangement might pay a pension after retirement while collecting contributions only before retirement.

The equivalence principle sets the expected present value of premiums equal to the expected present value of benefits, under the same transition, interest, and timing assumptions. The resulting premium is a net level premium: the calculation does not include expenses, profit margins, or risk loads unless these are explicitly added to the model.

Definition: In an annual multi-state model, let $p_t(i)$ be the probability that a covered person is in state $i$ at time $t$, and let $P_{ij}$ be the probability of moving from state $i$ at time $t$ to state $j$ at time $t+1$. If a level premium $G$ is payable at time $t$ only when the person is in an eligible state, its expected present value is $G$ times the expected present value of the eligible premium-payment indicators. The equivalence premium is the value of $G$ that makes this equal to the expected present value of benefits.

Build Both Present Values on the Same Timeline

For the examples below, the model is discrete and annual. State probabilities are evaluated at integer times, transitions occur over each year, and benefits described as year-end payments are paid at the end of that year. A premium due at the beginning of a year is paid at that year’s integer time, provided the person is then in an eligible state. Let $i$ be the effective annual interest rate and $v=1/(1+i)$. The present-value factor for a payment at time $t$ is $v^t$.

Suppose a contract lasts $n$ years. Let $E$ denote its set of premium-eligible states, $B_j$ the benefit paid at year-end when the ending state is $j$, and $L_{ij}$ a lump sum paid at year-end if the transition during that year is from state $i$ to state $j$. The expected benefit in year $t+1$, before discounting, is

C_{t+1}=\sum_i\sum_j p_t(i)P_{ij}(B_j+L_{ij}).

The terms $B_j$ and $L_{ij}$ may both be nonzero: an occupancy benefit and a transition lump sum can be payable in the same year. The expected number of premiums paid at time $t$, per unit premium, is $\sum_{i\in E}p_t(i)$. Consequently, the equivalence equation is

G\sum_{t=0}^{n-1}v^t\sum_{i\in E}p_t(i) = \sum_{t=0}^{n-1}v^{t+1}C_{t+1}, \qquad G= \frac{\displaystyle\sum_{t=0}^{n-1}v^{t+1}C_{t+1}} {\displaystyle\sum_{t=0}^{n-1}v^t\sum_{i\in E}p_t(i)}.

This form assumes premiums are due at the beginning of each covered year and benefits at year-end. If the contract specifies different payment times, the discount exponents must change accordingly. It also assumes the stated transition probabilities apply for the relevant year; a homogeneous transition matrix is not appropriate if probabilities vary by year.

Why the Equivalence Formula Holds

The formula follows by summing expected cash flows at their actual payment times. For a particular year, the probability of being in state $i$ at the start of the year and then moving to $j$ is $p_t(i)P_{ij}$. On that path, the year-end benefit is $B_j+L_{ij}$. Multiplying the payment by its path probability and summing over all possible starting and ending states gives $C_{t+1}$.

Linearity of expectation allows each year’s expected payment to be discounted and added, whether or not benefits in different years are dependent. A premium at time $t$ has amount $G$ if the person is eligible and zero otherwise, so its expected amount is $G\sum_{i\in E}p_t(i)$. Summing discounted premiums and setting them equal to discounted benefits gives the equivalence equation. Dividing by the premium factor gives the stated expression for $G$, provided that factor is positive.

Specify states and timing.
List the states, transition probabilities, premium dates, benefit dates, and contract term. Include any absorbing state, such as death, where relevant.

Find state probabilities.
Starting with the stated initial state, propagate probabilities one year at a time using the transition probabilities.

Value benefits and premiums separately.
Count occupancy benefits in the ending state and transition benefits only on the transitions that trigger them. Count premiums only in eligible states at their due times.

Discount and solve.
Discount each expected cash flow from its own payment time, then divide the benefit present value by the present value of one unit of premium.

Worked Valuations

Worked Example: Disability Benefits with Active-State Premiums

A hypothetical two-year contract starts with the member active. The states are active $A$, disabled $D$, and dead $X$, with death absorbing. In each year, an active member remains active with probability $0.70$, becomes disabled with probability $0.20$, or dies with probability $0.10$. A disabled member recovers to active with probability $0.10$, remains disabled with probability $0.80$, or dies with probability $0.10$. The effective annual interest rate is $5\%$. A year-end benefit of $900 is paid if the member is disabled at year-end, and an additional $300 is paid on an active-to-disabled transition. A level premium is due at times 0 and 1 only if the member is active then. Find the equivalence premium.

Method. Calculate the expected benefit in each year from the transition probabilities, and calculate the premium factor from active-state probabilities at the two premium dates.

At time 0 the member is certainly active. After year 1, the probabilities are $p_1(A)=0.70$ and $p_1(D)=0.20$. The year-1 benefit is paid only on an active-to-disabled transition, giving

$$ C_1=0.20(900+300)=240. $$

For year 2, the probability of being disabled at year-end is the probability of arriving from active plus the probability of remaining disabled:

$$ p_2(D)=0.70(0.20)+0.20(0.80)=0.14+0.16=0.30. $$

The probability of an active-to-disabled transition in year 2 is $0.70(0.20)=0.14$. Thus the year-2 expected benefit, including both types of payment, is

$$ C_2=0.30(900)+0.14(300)=270+42=312. $$

Here $v=1/1.05=20/21$. The benefit present value is

240v+312v^2 =240\left(\frac{20}{21}\right)+312\left(\frac{400}{441}\right) \approx228.571429+282.993197 =511.564626.

The premium at time 0 is certain. The time-1 premium is payable only if active, which has probability $0.70$. The present value of one unit of premium is therefore

1+0.70v=1+0.70\left(\frac{20}{21}\right)=1+\frac{2}{3}=\frac{5}{3}.

Equating the two present values gives $G=511.5646259/(5/3)\approx306.9387755$. Rounded to the nearest cent, the equivalence premium is $306.94 per year. As a check, using the unrounded premium gives $306.9387755(5/3)\approx511.564626$, matching the benefit present value.

Common mistake avoided: The year-2 disability probability is not just the probability of becoming disabled from active. It also includes members who were already disabled and remain so. Conversely, the transition lump sum is paid only for the active-to-disabled portion, not for every disabled member.

Worked Example: Long-Term Care at Two Benefit Levels

A hypothetical two-year long-term care contract starts in active state $A$. Its other living states are moderate care $C$ and intensive care $I$, with death $X$ absorbing. Annual transition probabilities are as follows: from $A$, probabilities to $A,C,X$ are $0.75,0.20,0.05$; from $C$, probabilities to $A,C,I,X$ are $0.10,0.55,0.30,0.05$. (Transitions from $I$ are not needed to value this two-year term.) The effective annual interest rate is $4\%$. At each year-end, the contract pays $500 in state $C$ or $1,000 in state $I$. It also pays $400 on a $C$-to-$I$ upgrade. Premiums are due at times 0 and 1 only while active. Find the level net premium.

Method. Use the state probabilities at each year-end for care-level payments, and separately count upgrades by their transition probability.

At the end of year 1, the probabilities of being in $A$ and $C$ are $0.75$ and $0.20$, respectively. No upgrade can occur from the initial active state under the specified benefit definition. Therefore,

$$ C_1=0.20(500)=100. $$

For year 2, the probability of being in moderate care is $0.75(0.20)+0.20(0.55)=0.15+0.11=0.26$. The probability of being in intensive care is $0.20(0.30)=0.06$. The $C$-to-$I$ upgrade probability is also $0.20(0.30)=0.06$. Thus,

C_2=0.26(500)+0.06(1{,}000)+0.06(400)=130+60+24=214.

With $v=1/1.04=25/26$, the benefit present value is

100v+214v^2 =100\left(\frac{25}{26}\right)+214\left(\frac{625}{676}\right) \approx96.153846+197.855030 =294.008876.

The time-0 premium is certain. At time 1, the member is active with probability $0.75$, so the unit premium factor is

1+0.75v=1+\frac{75}{104}=\frac{179}{104}\approx1.721154.

The premium is $G=294.0088757/(179/104)\approx170.8207993$, or $170.82 per year to the nearest cent. Multiplying the unrounded premium by $179/104$ recovers approximately $294.008876, which checks against the benefit present value.

Common mistake avoided: The $400 upgrade payment is not paid to every member in intensive care. Only the $C$-to-$I$ transitions trigger it; intensive-care occupancy is already valued by the separate $1,000 benefit.

Worked Example: Pension Benefit with a Retirement Lump Sum

A hypothetical two-year pension contract starts with an active member. In each year, an active member remains active with probability $0.60$, retires with probability $0.30$, or withdraws with probability $0.10$. A retired member remains retired with probability $0.90$ or dies with probability $0.10$; withdrawal and death are absorbing. The effective annual interest rate is $5\%$. A $1,200 pension is paid at year-end if retired, and a $1,500 lump sum is paid on an active-to-retired transition. Level premiums are due at times 0 and 1 only while active. Find the equivalence premium.

Method. Calculate each year’s retirement occupancy probability and retirement-transition probability separately. Then include both premium dates in the denominator.

In year 1, the retirement-transition probability is $0.30$, so the expected benefit is

C_1=0.30(1{,}200+1{,}500)=810.

At time 1, the probabilities of being active and retired are $0.60$ and $0.30$. In year 2, the probability of being retired at year-end is $0.60(0.30)+0.30(0.90)=0.18+0.27=0.45$. The active-to-retired transition probability is $0.60(0.30)=0.18$. Therefore,

C_2=0.45(1{,}200)+0.18(1{,}500)=540+270=810.

With $v=20/21$, the benefit present value is

810v+810v^2 =810\left(\frac{20}{21}\right)+810\left(\frac{400}{441}\right) \approx771.428571+734.693878 =1{,}506.122449.

The unit premium factor includes the certain payment at time 0 and the time-1 payment with probability $0.60$:

1+0.60v=1+\frac{12}{21}=\frac{11}{7}\approx1.571429.

Thus $G=1{,}506.122449/(11/7)\approx958.441558$, giving a premium of $958.44 per year to the nearest cent. The check $958.441558(11/7)\approx1{,}506.122449$ agrees with the benefit present value.

Common mistake avoided: Dividing by only one premium would give $1,506.12, incorrectly ignoring the possible premium at time 1. The time-1 premium is not certain, either: it is paid only with probability $0.60$.

Common Traps and Scope of the Result

A reliable calculation keeps the probability model, cash-flow definitions, and payment times aligned. The equivalence principle does not itself decide whether a benefit is payable on entry to a state, while occupying a state, or on both. Those terms must come from the contract specification.

Using benefit probabilities for premiums. A person can be eligible to pay a premium in a state that does not trigger benefits. Construct the premium factor from the premium-eligibility rule, not from the benefit-state probabilities.
Counting a transition payment as an occupancy payment. A lump sum attached to a transition is multiplied by the probability of that transition. It is not automatically payable to everyone who ends the year in the destination state.
Discounting from the wrong date. A beginning-of-year premium at time $t$ has discount factor $v^t$; a year-end benefit for that year has factor $v^{t+1}$.
Omitting persistent states. A benefit-state probability at a later time can include both new entrants and people who remain in that state. Propagate all relevant state probabilities before valuing the payment.
Applying the formula outside its assumptions. If rates vary by year, use the appropriate year-specific transition probabilities. If premiums or benefits are paid continuously or at other specified times, use cash flows at those times rather than the annual timing used here.

The same method applies to disability income, long-term care, and pension packages, but the state definitions and payment conditions differ. It is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus to identify those conditions, construct the correct expected cash flows, and solve the equivalence equation without mixing premium and benefit timing.

Check Your Understanding

For each problem, use the stated annual transition model and payment timing to find the level net premium.

A one-year disability contract starts active. During the year, the member remains active with probability $0.85$, becomes disabled with probability $0.10$, or dies with probability $0.05$. At year-end, $800 is paid if disabled and an additional $200 is paid on an active-to-disabled transition. A premium is due at time 0 while active. Interest is $0\%$. (Answer: $100.00)
A two-year disability contract starts active. Each year, active members remain active with probability $0.50$, become disabled with probability $0.30$, or die with probability $0.20$. Disabled members recover with probability $0.20$, remain disabled with probability $0.70$, or die with probability $0.10$. At each year-end, $400 is paid if disabled and $100 on an active-to-disabled transition. Premiums are due at times 0 and 1 while active. Interest is $0\%$. (Answer: $206.00)
A two-year pension contract starts active. Each year, active members remain active with probability $0.80$ or retire with probability $0.20$; retired members remain retired with probability $0.90$ or die with probability $0.10$. At year-end, $600 is paid if retired and an additional $200 on an active-to-retired transition. Premiums are due at times 0 and 1 while active. Interest is $0\%$. (Answer: $220.00)
A two-year care contract starts active. Each year, active members remain active with probability $0.70$ or enter moderate care with probability $0.30$. From moderate care, members move to active with probability $0.10$, remain in moderate care with probability $0.60$, or enter intensive care with probability $0.30$. At year-end, benefits are $300 in moderate care and $700 in intensive care, plus $200 on a moderate-to-intensive transition. Premiums are due at times 0 and 1 while active. Interest is $0\%$. (Answer: $169.41)
A two-year pension contract starts active. Each year, active members remain active with probability $0.65$, retire with probability $0.25$, or withdraw with probability $0.10$. Retired members remain retired with probability $0.90$ or die with probability $0.10$. The effective annual interest rate is $4\%$. At each year-end, $900 is paid if retired and $600 on an active-to-retired transition. Premiums are due at times 0 and 1 while active. (Answer: $475.79)

What You'll Learn