Why Disability Income Premiums Depend on State Rules
A disability income contract can make benefit payments depend on whether the policyholder is disabled at a payment date, how long a disability has lasted, or whether a waiting period has been satisfied. Premiums may be due only while the policyholder is active, or they may continue while the policyholder is disabled but alive. These rules affect different cash flows, so a level premium cannot be calculated from the benefit amount alone.
The equivalence principle gives the net level premium that balances the expected present value of benefits and the expected present value of premiums under specified assumptions. It is a valuation principle, not a statement that each individual policyholder’s premiums will equal that person’s benefits. The calculations below use hypothetical contracts and clearly state the state and timing conditions for every payment.
State-Contingent Expected Present Values
Consider a discrete-time contract that begins at time 0 in the active state \(A\). Let \(D\) denote the disabled state and \(\dagger\) the dead state. A transition probability describes the chance of occupying a state at a later payment date, given the current state. The state probabilities must follow the transition rules in the contract model, including recovery and death where specified.
Suppose the effective annual interest rate is \(i\), with \(v=1/(1+i)\). A payment at time \(k\) has present value equal to its amount multiplied by \(v^k\). Let \(b_k\) be a benefit payable at time \(k\) if a specified state condition is met, and let \(c_k\) be the amount of a unit premium payable at time \(k\) if its specified condition is met. Then the expected present values can be written as sums over payment dates:
The probability in each term is specific to that payment. For example, a benefit payable only if disabled uses the probability of being disabled at its payment date. A premium payable only while active uses the probability of being active at its due date. If the premium is payable whenever alive, its probability instead includes both active and disabled states.
This result follows directly from the cash-flow definitions. Each premium payment is the level amount \(P\) times an indicator that the premium condition holds. Discounting and taking expectations are linear operations, so \(P\) can be factored out of the expected present value of premiums. Equating that amount to the expected present value of benefits and dividing by the positive unit-premium value gives the formula.
Record when benefits and premiums are due. A premium at the beginning of year 1 is at time 0; a benefit at the end of year 1 is at time 1.
For every payment date, identify the state or event that makes that payment payable.
Use state occupancy probabilities at the specified dates, calculated from the contract’s transition assumptions.
Calculate the expected present value of benefits and of a unit premium stream, then divide.
Worked Examples
Worked Example: Active-Only Premiums and Deferred Benefits
A hypothetical three-year contract starts in state \(A\). At the end of each year, the transitions are \(A\to A,D,\dagger\) with probabilities \(0.80,0.10,0.10\), and \(D\to A,D,\dagger\) with probabilities \(0.30,0.60,0.10\). Death is absorbing. A benefit of $6,000 is paid at the ends of years 2 and 3 if the policyholder is disabled at that date. Level premiums are due at the beginnings of years 1, 2 and 3, only if the policyholder is active. Interest is \(0\%\). Find the net level premium.
Method. Calculate the disabled probabilities at benefit dates and the active probabilities at premium dates. Because interest is \(0\%\), every discount factor is 1.
At time 0, the policyholder is active with probability 1. At time 1, the state probabilities are \(P(A)=0.80\) and \(P(D)=0.10\). At time 2, the disabled probability is \(0.80(0.10)+0.10(0.60)=0.08+0.06=0.14\). The active probability at time 2 is \(0.80(0.80)+0.10(0.30)=0.64+0.03=0.67\).
At time 3, the disabled probability is \(0.67(0.10)+0.14(0.60)=0.067+0.084=0.151\). This value follows from the active and disabled probabilities at time 2; the dead state contributes no transitions back into disability.
The expected present value of benefits is \(6{,}000(0.14+0.151)=6{,}000(0.291)=\$1{,}746\). The premium dates are times 0, 1 and 2. The probabilities of being active at those dates are \(1\), \(0.80\) and \(0.67\), respectively. Therefore, the expected present value of a unit premium is \(1+0.80+0.67=2.47\).
Equating expected present values gives \(P(2.47)=1{,}746\), so \(P=1{,}746/2.47=\$706.88\) to the nearest cent.
Check. The state probabilities used for the benefit payments are at times 2 and 3, while the premium probabilities are at times 0, 1 and 2. The calculated premium produces an expected premium value of \(706.88(2.47)\), approximately $1,746 after rounding.
Worked Example: Discounting Benefits and Active-Only Premiums
A hypothetical two-year contract starts active. At each year-end, an active policyholder remains active with probability \(0.90\) or becomes disabled with probability \(0.10\); a disabled policyholder recovers with probability \(0.20\) or remains disabled with probability \(0.80\). There is no mortality during the term. A benefit of $10,000 is paid at the end of each year if disabled then. Premiums are due at times 0 and 1 only if active at the due date. The effective annual interest rate is \(10\%\). Find the level premium.
Method. Value each benefit at its year-end and each premium at its due date, using the probability of the required state at that date. Here \(v=1/1.10\).
The disabled probability at time 1 is \(0.10\). At time 2, it is \(0.90(0.10)+0.10(0.80)=0.09+0.08=0.17\). The probability of being active at time 1 is \(0.90\); the time-0 active probability is 1.
The expected present value of benefits is \(10{,}000(0.10v+0.17v^2)\). The two discounted terms are \(10{,}000(0.10/1.10)=\$909.09\) and \(10{,}000(0.17/1.10^2)=\$1{,}404.96\). Thus the benefit value is \(\$909.09+\$1{,}404.96=\$2{,}314.05\), rounded to cents.
The expected present value of a unit premium is \(1+0.90v=1+0.90/1.10=1.81818\ldots\). Therefore, \(P=2{,}314.049586\ldots/1.81818\ldots=\$1{,}272.73\) to the nearest cent.
Check. The first benefit is discounted for one year, the second for two years, and the time-1 premium for one year. The expected present value of the premium stream at $1,272.73 per payment is approximately $2,314.05.
Worked Example: Premiums Continue While Disabled
A hypothetical three-year contract starts active. At each year-end, \(A\to A,D,\dagger\) with probabilities \(0.80,0.10,0.10\), and \(D\to A,D,\dagger\) with probabilities \(0.30,0.60,0.10\). A benefit of $5,000 is paid at the end of each year if disabled. A level premium is due at times 0, 1 and 2 whenever the policyholder is alive, whether active or disabled. The interest rate is \(0\%\). Find the level premium.
Method. Use disabled-state probabilities for benefits, but use the probability of being alive, not merely active, for premiums.
At time 1, \(P(A)=0.80\) and \(P(D)=0.10\), so the probability of being alive is \(0.80+0.10=0.90\). At time 2, \(P(A)=0.80(0.80)+0.10(0.30)=0.67\) and \(P(D)=0.80(0.10)+0.10(0.60)=0.14\). Thus the probability of being alive at time 2 is \(0.67+0.14=0.81\).
At time 3, the disabled probability is \(0.67(0.10)+0.14(0.60)=0.067+0.084=0.151\). The disabled probabilities at the three benefit dates are \(0.10\), \(0.14\) and \(0.151\). The expected present value of benefits is therefore \(5{,}000(0.10+0.14+0.151)=5{,}000(0.391)=\$1{,}955\).
The premium dates are 0, 1 and 2. The probabilities of being alive are \(1\), \(0.90\) and \(0.81\). Hence the expected present value of a unit premium is \(1+0.90+0.81=2.71\), and \(P=1{,}955/2.71=\$721.40\) to the nearest cent.
Check. Disabled policyholders are included in the premium probabilities because the contract says premiums continue while disabled. The premium value is \(721.40(2.71)\), approximately $1,955 after rounding.
Common Mistakes and Exam Checks
In the first example, a tempting but incorrect calculation is to include only the time-0 and time-1 premiums, giving a unit-premium value of \(1+0.80=1.80\). That omits the premium due at time 2. Since the policyholder is active at time 2 with probability \(0.67\), the correct denominator is \(1+0.80+0.67=2.47\). The omission would produce \(1{,}746/1.80=\$970.00\), not the contract’s equivalence premium of $706.88.
Other checks help prevent errors:
- Match the state to the rule. A benefit payable if disabled does not use the probability of being alive or the probability of ever having become disabled; it uses the probability of disability under the stated payment condition.
- Match dates to discount factors. A beginning-of-year premium and an end-of-year benefit are not at the same time. Use \(v^k\) for a payment at time \(k\).
- Respect waiver rules. If premiums are waived during disability, disabled lives do not contribute to the premium value. If premiums continue while alive, both active and disabled lives may contribute.
- Use the full state model. Recovery can return a disabled policyholder to active status, and death removes the policyholder from later payments. Do not treat disability as necessarily permanent unless the model says so.
- Do not confuse net and gross premiums. The equivalence calculation here excludes expenses and profit margins; adding those amounts requires additional assumptions.
The equivalence principle is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In an exam calculation, a concise schedule of payment dates, conditions, probabilities and discount factors is often the clearest way to demonstrate that the premium rules have been applied correctly.
Check Your Understanding
For each problem, identify the payment conditions before calculating. Amounts are hypothetical.
- A two-year contract starts active. Each year, \(A\to A,D\) with probabilities \(0.85,0.15\), and \(D\to A,D\) with probabilities \(0.25,0.75\). A $4,000 benefit is paid at each year-end if disabled. Premiums are due at times 0 and 1 if active. Interest is \(0\%\). Find the level premium. (Answer: $843.24)
- A two-year contract starts active. Each year, \(A\to A,D\) with probabilities \(0.90,0.10\), and \(D\to A,D\) with probabilities \(0.20,0.80\). A $8,000 benefit is payable only at the end of year 2 if disabled. Premiums are due at times 0 and 1 if active. The effective annual interest rate is \(5\%\). Find the level premium. (Answer: $664.22)
- A two-year contract starts active. Each year, \(A\to A,D,\dagger\) with probabilities \(0.70,0.20,0.10\), and \(D\to A,D,\dagger\) with probabilities \(0.10,0.70,0.20\). A $3,000 benefit is paid at each year-end if disabled. Premiums at times 0 and 1 are payable whenever alive. Interest is \(0\%\). Find the level premium. (Answer: $757.89)
- A two-year contract starts active. Each year, \(A\to A,D\) with probabilities \(0.80,0.20\), and \(D\to A,D\) with probabilities \(0.50,0.50\). Benefits of $1,000 and $2,000 are paid at the ends of years 1 and 2, respectively, if disabled. Premiums at times 0 and 1 are payable only if active. Interest is \(0\%\). Find the level premium. (Answer: $400.00)