Disability Income Premium Calculation Workshop

Why Payment State and Timing Matter

A disability income premium is not generally paid with certainty at every scheduled date. If premiums are due only while the insured is active, the expected present value of premiums depends on the probability of being active on each premium date. Disability benefits depend instead on the probability of being in a benefit-paying state on each benefit date. A level premium must balance these two different streams.

This workshop uses an annual, discrete-time model. It is a convenient way to organize a calculation when the contract specifies annual transition probabilities and payment dates. The transition probabilities, payment dates, interest assumption, and benefit rule must all be stated before applying the equivalence principle.

Definition: Let the states at each annual observation date be active $A$, disabled $D$ (or one of several disabled states), and dead $X$. For an insured active at issue, let $a_k$ be the probability of being active at the start of year $k+1$, and let $d_k$ be the probability of being in a benefit-paying disabled state at the end of year $k+1$. If an annual premium $P$ is paid at the start of each year only when active, and a benefit $B$ is paid at the end of each year only when disabled, then, for a term of $n$ years and effective annual interest rate $i$, $$ \operatorname{EPV}(\text{premiums})=P\sum_{k=0}^{n-1}a_kv^k, \qquad \operatorname{EPV}(\text{benefits})=B\sum_{k=1}^{n}d_kv^k, \qquad v=\frac{1}{1+i}. $$

The indices reflect the payment dates: the first premium is paid at issue, at time 0, while the first benefit is paid at the end of the first year, at time 1. The probabilities are unconditional probabilities from issue, not probabilities conditional on being active or disabled at that date.

Building the Level-Premium Equation

Under the equivalence principle, the level premium is chosen so that the expected present value of premiums equals the expected present value of benefits. This is a pure-premium calculation: expenses, profit margins, and other cash flows are omitted. Those items would have to be included explicitly in a gross-premium calculation.

P\sum_{k=0}^{n-1}a_kv^k = B\sum_{k=1}^{n}d_kv^k \quad\Longrightarrow\quad P = \frac{B\sum_{k=1}^{n}d_kv^k} {\sum_{k=0}^{n-1}a_kv^k}.

The denominator must be positive for a finite level premium to be determined. If the premium is payable only while active, premiums are not due in years in which the insured is not active on the scheduled premium date. Similarly, the benefit term includes only states and dates that satisfy the contract’s benefit rule.

Worked Examples

Worked Example: Three-Year Premium with Changing Transitions

A hypothetical three-year policy is issued to an active insured. At the start of each year, a premium is due if the insured is active. A benefit of $9,000 is paid at each year-end if the insured is disabled then. There is no death state in this simplified example. The effective annual interest rate is 4%.

In year 1, an active insured remains active with probability 0.80 and becomes disabled with probability 0.20. In year 2, the probabilities are 0.80 for an active insured to remain active, 0.20 to become disabled, and 0.35 for a disabled insured to remain disabled. In year 3, a disabled insured remains disabled with probability 0.35; an active insured becomes disabled with probability $72/385$. All remaining probability in each row corresponds to being active at year-end.

Method. First calculate the unconditional state probabilities at each payment date. The year-1 disabled probability is $d_1=0.20$. At the start of year 2, the probabilities are 0.80 active and 0.20 disabled, so $d_2=0.80(0.20)+0.20(0.35)=0.16+0.07=0.23$. Thus, the active probability at the start of year 3 is $1-0.23=0.77$. The year-3 disabled probability is $d_3=0.77(72/385)+0.23(0.35)=0.144+0.0805=0.2245$. The active probabilities on premium dates are $a_0=1$, $a_1=0.80$, and $a_2=0.77$.

Discounting and calculation. Here $v=1/1.04=0.9615384615$, $v^2=0.9245562130$, and $v^3=0.8889963587$. The expected present value of benefits is

9{,}000(0.20v+0.23v^2+0.2245v^3) = 9{,}000(0.19230769+0.21264793+0.19957968) = \$5{,}440.82.

The expected present value factor for premiums is $1+0.80v+0.77v^2 =1+0.76923077+0.71190828 =2.48113905$. Therefore,

P=\frac{5{,}440.82}{2.48113905} =\$2{,}192.87.

The level annual premium is $2,192.87, rounded to the nearest cent. The calculation avoids the mistake of treating every scheduled premium as payable with certainty: the second and third premiums are weighted by the probabilities of being active on those dates.

Worked Example: Recovery and Death in a Two-Year Contract

A hypothetical two-year contract pays $6,000 at each year-end if the insured is disabled and alive at that date. A premium is due at the start of each year only if the insured is active. At each year-end, an active insured transitions to active with probability 0.85, disabled with probability 0.12, or dead with probability 0.03. A disabled insured transitions to active with probability 0.25, disabled with probability 0.70, or dead with probability 0.05. The effective annual interest rate is 5%, and the insured is active at issue.

Method. The first-year disabled probability is $d_1=0.12$, and the active probability at the start of year 2 is $a_1=0.85$. For the end of year 2, add the probabilities of reaching the disabled state from each state occupied at the start of that year: $d_2=0.85(0.12)+0.12(0.70)=0.102+0.084=0.186$. The probability of being dead after year 1 does not contribute to either year-2 occupancy probability.

The premium factor is $1+0.85v$, and the benefit factor is $0.12v+0.186v^2$. With $v=1/1.05=0.9523809524$ and $v^2=1/1.1025=0.9070294785$, these are

$$ 1+0.85v=1+0.80952381=1.80952381, $$

$$ 0.12v+0.186v^2 = 0.11428571+0.16870748 = 0.28299319. $$

Thus the expected present value of benefits is $6{,}000(0.28299319)=\$1{,}697.96$. Equating expected present values gives

P=\frac{1{,}697.96}{1.80952381} =\$938.35.

The level annual premium is $938.35, rounded to the nearest cent. The key step is to include both persistence in disability and new disability from the active state when finding $d_2$, while excluding the dead state from future payments.

Worked Example: Different Benefits for Partial and Total Disability

A hypothetical two-year contract has three living states: active $A$, partially disabled $P$, and totally disabled $T$, as well as death. A premium is paid at the start of each year only if active. At year-end, the policy pays $3,000 in state $P$, $7,000 in state $T$, and nothing in states $A$ or death. The effective annual interest rate is 5%.

At the end of year 1, the probabilities are 0.80 active, 0.12 partially disabled, 0.06 totally disabled, and 0.02 dead. For year 2, the transition probabilities from $A$ to $A,P,T,\text{death}$ are $0.80,0.12,0.06,0.02$; from $P$, they are $0.25,0.65,0.08,0.02$; and from $T$, they are $0.10,0.05,0.80,0.05$.

Method. The year-2 start active probability is $a_1=0.80$. The end-of-year-2 partial disability probability is $0.80(0.12)+0.12(0.65)+0.06(0.05)=0.096+0.078+0.003=0.177$. The total disability probability is $0.80(0.06)+0.12(0.08)+0.06(0.80)=0.048+0.0096+0.048=0.1056$. These are unconditional probabilities from issue.

Expected benefits at the end of year 1 are $0.12(3{,}000)+0.06(7{,}000)=\$780$. At the end of year 2 they are $0.177(3{,}000)+0.1056(7{,}000)=\$531+\$739.20=\$1{,}270.20$. With $v=1/1.05=0.9523809524$ and $v^2=0.9070294785$, the expected present value of benefits is

780v+1{,}270.20v^2 = 742.85714+1{,}152.10884 = \$1{,}894.97.

The premium factor is $1+0.80v=1+0.76190476=1.76190476$. Therefore,

P=\frac{1{,}894.97}{1.76190476} =\$1{,}075.52.

The level annual premium is $1,075.52, rounded to the nearest cent. The calculation avoids treating the disabled states as having one common benefit: each state’s occupancy probability is multiplied by its own benefit amount before discounting.

Why the Equivalence Equation Holds

The formula follows from linearity of expectation. For each premium date, define the payment as $P$ if the insured is active and zero otherwise. Its expected value at that date is $P$ multiplied by the probability of being active. Discounting that expected payment gives the corresponding term in the premium sum. For each benefit date, the payment is $B$ if the insured is in a qualifying disabled state and zero otherwise, so its expected present value is the discounted benefit multiplied by the probability of that state.

Adding the date-specific expected present values gives the expected present value of the whole stream. This does not require payments at different dates to be independent. The equivalence principle sets the two expected present values equal, and solving the resulting linear equation for $P$ gives the level-premium formula.

For multiple disabled states with benefit $B_j$ in state $j$, let $p_{j,k}$ denote the unconditional probability of being in that benefit-paying state at the end of year $k$. The benefit present value becomes $\sum_{k=1}^{n}\sum_j B_jp_{j,k}v^k$. This is the structure used in the partial- and total-disability example.

Common Exam Trap: Certain Premiums Versus Conditional Premiums

In the three-year example, an incorrect calculation would discount three certain premiums, using $1+v+v^2=2.88609467$ as the premium factor. That would give $5{,}440.82/2.88609467=\$1{,}885.18$, rather than the correct $2,192.87. The incorrect denominator assumes premiums are paid even when the insured is disabled. The contract instead requires each premium to be weighted by the probability of being active at its due date, giving the factor $1+0.80v+0.77v^2$.

Before calculating, write down the payment dates and the state condition for each cash flow. In particular, do not confuse the probability of becoming disabled during a year with the probability of being disabled at the specified benefit date. In an annual model with a year-end benefit, the payment probability is based on the state at year-end under the stated transition rules.

Key takeaway: A level disability income premium is a ratio of expected present values: the discounted expected benefits in the numerator, and the discounted probability-weighted premium payments in the denominator. Correct state occupancy and payment timing are essential to both.

Check Your Understanding

For each problem, use the stated annual payment timing and round the final premium or benefit to the nearest cent.

A two-year policy pays $5,000 at each year-end if disabled then. Premiums are due at the start of each year if active. The end-of-year disabled probabilities are 0.10 and 0.18, and the active probability at the start of year 2 is 0.90. The effective annual interest rate is 5%. Find the level premium. (Answer: $695.97)
A three-year policy has annual transitions from active to disabled with probability 0.10, and from disabled to disabled with probability 0.60. There is no death. The insured starts active. A $4,000 benefit is paid at each year-end if disabled; premiums are paid at each year-start if active. At 4% effective annual interest, find the level premium. (Answer: $589.02)
A two-year policy has a $700 annual premium at the start of each year if active. At 5% effective annual interest, the active probabilities at premium dates are 1 and 0.90, while the disabled probabilities at benefit dates are 0.10 and 0.15. Find the level annual benefit paid at each year-end if disabled. (Answer: $5,620.59)
A two-year policy pays $2,500 at each year-end if disabled. Premiums are paid at the start of each year if active. The active probabilities at premium dates are 1 and 0.92, and the disabled probabilities at benefit dates are 0.08 and 0.14. At zero interest, find the level premium. (Answer: $286.46)
A two-year policy pays $10,000 at each year-end if disabled and alive. At each year-end, active transitions to active, disabled, or dead with probabilities 0.90, 0.08, and 0.02; disabled transitions to active, disabled, or dead with probabilities 0.10, 0.80, and 0.10. The insured starts active, premiums are due at each year-start if active, and the effective annual interest rate is 3%. Find the level premium. (Answer: $1,098.65)

What You'll Learn