Integrated Participating Policy Valuation Workshop

Why an Integrated Valuation Needs a Cash-Flow Map

A participating-policy valuation can involve several benefit components that do not have the same status or trigger. A reversionary bonus may already be declared and vested; a terminal bonus may remain discretionary; premiums and renewal expenses may be due only while the policy is in force. The valuation is reliable only if each cash flow is assigned the correct amount, payment time and probability.

The method in this workshop is an expected-present-value calculation under an explicitly stated mortality and interest model. It is not enough to say that a terminal bonus is “allowed for”: the valuation must specify its amount or scale, what benefit it is based on, and which policy exits receive it. A discretionary benefit included at an assumed scale is a scenario valuation, not a statement that the benefit is guaranteed.

Definition: A declared reversionary bonus is a bonus that has been declared and, under the stated policy terms, vests in the policy benefit. A terminal bonus is an additional benefit payable only under the specified terminal-bonus rules; it may be discretionary. For valuation, state the amount or scale assumed, its calculation base, and whether it applies on death, maturity, or both.

Use annual policy years. Let $q_k$ be the probability of death during year $k$, conditional on being in force at the start of that year, and let $p_k=1-q_k$. Write $s_0=1$ and $s_k=\prod_{j=1}^{k}p_j$, the probability of surviving in force to time $k$. Death during year $k$ then has probability $s_{k-1}q_k$, while maturity at time $n$ has probability $s_n$, assuming the contract remains in force until death or maturity.

Assume an effective annual interest rate $i$, with $v=1/(1+i)$. Benefits for death during year $k$ are paid at the end of that year, at time $k$; a maturity benefit is paid at time $n$. Unless a problem states otherwise, level premiums are paid at the start of each year while the policy is in force, at times $0,1,\ldots,n-1$. Expenses must be assigned their own timing and in-force conditions.

Theorem: Under these assumptions, if the annual premium is $P$, the equivalence principle sets the expected present value of premiums equal to the expected present value of benefits plus expenses. If $B_k^D$ is the death benefit at the end of year $k$, $B_n^M$ is the maturity benefit, and $E$ denotes the present value of expenses, then

P\sum_{k=0}^{n-1}s_kv^k = \sum_{k=1}^{n}s_{k-1}q_kv^kB_k^D +s_nv^nB_n^M+E.

In this formula, death in year $n$ and survival to maturity are different outcomes: their probabilities are $s_{n-1}q_n$ and $s_n$, respectively. The formula also assumes premiums are paid at the start of the year, including at time 0; changing premium timing changes the premium factor.

Why the Expected-Present-Value Formula Holds

For each possible exit, multiply its payment by the probability of that exit and the discount factor to its payment time. The mutually exclusive death outcomes contribute the sum of the death-benefit terms; survival to maturity contributes the maturity-benefit term. The expectation of a sum of cash flows is the sum of their expectations, so each component can be valued separately and then added.

A premium due at time $k$ is paid only if the policy is in force at that time. Its payment probability is $s_k$, and its present value if paid is $Pv^k$. Summing over premium dates gives $P\sum_{k=0}^{n-1}s_kv^k$. Expenses are treated in the same probability-and-discount manner, but are added to the outgo side of the equivalence equation.

Write the exit benefits.
For each death year and for maturity, separate the basic benefit, vested declared bonuses and any assumed terminal bonus.

Assign probabilities and times.
Use the conditional death probability for each year, survival to that year, and the stated payment date.

Value all outgo.
Discount expected benefits and expenses, applying the specified treatment of future discretionary benefits.

Solve and check.
Divide total expected outgo by the expected present value of one unit of premium, then verify the result by substitution.

Worked Valuations

Worked Example: Two-Year Endowment With a Terminal Bonus

A hypothetical two-year endowment has a basic benefit of $20,000. It earns a declared, simple reversionary bonus of $1,000 for each completed policy year. At every exit, a terminal bonus is assumed equal to 10% of the vested reversionary bonuses. Annual mortality is $q=0.10$ in each year, and $i=0$. Death claims are paid at year-end; a surviving policy receives the maturity benefit at time 2. Level premiums are paid at times 0 and 1 while in force, with no expenses. Find the equivalence-principle annual premium, including the stated terminal-bonus scale.

Method and benefit amounts. The vested bonus is $1,000 after year 1 and $2,000 after year 2. The terminal bonus is based on those vested bonuses, not on the basic sum assured. Thus the year-1 death benefit is $20{,}000+1{,}000+0.10(1{,}000)=\$21{,}100$. At year 2, either death or maturity receives $20{,}000+2{,}000+0.10(2{,}000)=\$22{,}200$.

Probabilities. Death in year 1 has probability $0.10$. Survival to year 1 is $0.90$, so death in year 2 has probability $0.90(0.10)=0.09$, and maturity has probability $0.90(0.90)=0.81$. The combined probability of a year-2 payment is $0.09+0.81=0.90$.

Expected present value of benefits. Since $i=0$, there is no discounting:

\operatorname{EPV}(B) =0.10(21{,}100)+0.09(22{,}200)+0.81(22{,}200) =2{,}110+1{,}998+17{,}982 =\$22{,}090.

Expected present value of premiums. The time-0 premium is certain. The time-1 premium is paid with probability $0.90$. Therefore the premium factor is $1+0.90=1.90$, and $1.90P=22{,}090$. Hence

P=\frac{22{,}090}{1.90}=\$11{,}626.32.

Check and trap avoided. Substitution gives $1.90(11{,}626.32)=\$22{,}090.008$, which differs from the target by less than one cent because the premium was rounded to cents. A common error is to omit the terminal bonus or calculate it as 10% of the $20,000 basic benefit. The stated base is vested bonuses, giving terminal additions of $100 and $200, respectively.

Worked Example: Discounting, Renewal Expenses and a Discretionary Scale

A hypothetical two-year policy has a $40,000 basic benefit and a declared simple reversionary bonus of $800 for each completed year. A future terminal bonus is discretionary. For this valuation scenario, include it at 20% of vested reversionary bonuses on both death and maturity. Annual mortality is $q=0.10$ in each year, and $i=0.10$. Death and maturity benefits are paid at the end of the relevant year. A level premium is paid at times 0 and 1 while in force. Expenses are $500 at time 0 and $100 at time 1 if the policy is in force. Find the annual premium under this specified scenario.

Method and exit benefits. At the end of year 1, the vested bonus is $800 and the assumed terminal bonus is $0.20(800)=\$160$, so the death benefit is $40,960. At the end of year 2, vested bonuses total $1,600 and the terminal bonus is $0.20(1{,}600)=\$320$, so death and maturity benefits are each $41,920.

Expected present value of benefits. Year-1 death probability is 0.10. The probability of any year-2 benefit is 0.90: death in year 2 has probability 0.09 and maturity has probability 0.81. The discount factors are $v=1/1.10$ and $v^2=1/1.21$. Therefore

\operatorname{EPV}(B) =0.10(40{,}960)\left(\frac{1}{1.10}\right) +0.90(41{,}920)\left(\frac{1}{1.21}\right) =3{,}723.6364+31{,}180.1653 =\$34{,}903.8017.

Expected present value of expenses. The initial expense is certain. The renewal expense is paid at time 1 only if the policy survives year 1:

E=500+0.90(100)\left(\frac{1}{1.10}\right) =500+81.8182 =\$581.8182.

Premium factor and solution. The time-0 premium is certain and the time-1 premium is paid with probability 0.90:

\operatorname{EPV}(\text{premiums}) =P\left(1+0.90\left(\frac{1}{1.10}\right)\right) =P(1.8181818).

Equating premiums to benefits plus expenses gives

P=\frac{34{,}903.8017+581.8182}{1.8181818} =\$19{,}517.09.

Check and trap avoided. The numerator is total outgo of $35,485.6199; multiplying the rounded premium by 1.8181818 gives approximately that amount. The discretionary terminal bonus is included at the specified 20% scenario scale, not described as guaranteed. Also, the renewal expense is discounted from time 1: omitting that discount would overstate its present value.

Worked Example: Year-Specific Mortality and a Maturity-Only Terminal Bonus

A hypothetical three-year endowment has a $50,000 basic benefit and a declared simple bonus of $500 for each completed year. A terminal bonus is discretionary and is payable only on maturity. For this valuation scenario, include it at 30% of total vested reversionary bonuses. Annual death probabilities are $q_1=0.05$, $q_2=0.10$, and $q_3=0.20$. The effective annual interest rate is $i=0$. Death benefits are paid at each year-end; maturity occurs at time 3. A level premium is paid at times 0, 1 and 2 while in force. Expenses are $200 at time 0 and $50 at times 1 and 2 while in force. Find the annual premium.

Method and probabilities. Survival probabilities to the starts of years 1, 2 and 3 are $s_0=1$, $s_1=0.95$, and $s_2=0.95(0.90)=0.855$. Thus death probabilities in the three years are $0.05$, $0.95(0.10)=0.095$, and $0.855(0.20)=0.171$. Maturity probability is $0.855(0.80)=0.684$.

Benefits by exit. Death benefits are $50,500, $51,000 and $51,500 in years 1, 2 and 3. The maturity benefit includes $1,500 of vested bonuses and a maturity-only terminal bonus of $0.30(1{,}500)=\$450$, for a total of $51,950. No terminal bonus is added to death benefits under the stated terms.

Expected present value of benefits. With $i=0$, all year-end discount factors equal 1:

\begin{aligned} \operatorname{EPV}(B) &=0.05(50{,}500)+0.095(51{,}000)+0.171(51{,}500)+0.684(51{,}950)\\ &=2{,}525+4{,}845+8{,}806.50+35{,}533.80\\ &=\$51{,}710.30. \end{aligned}

Expected present value of expenses and premiums. Expenses have value $200+0.95(50)+0.855(50)=\$290.25$. The premium factor is $1+0.95+0.855=2.805$. The equivalence equation and premium are therefore

2.805P=51{,}710.30+290.25=52{,}000.55, \qquad P=\frac{52{,}000.55}{2.805}=\$18{,}538.52.

Check and trap avoided. The year-3 death probability and maturity probability sum to $0.171+0.684=0.855$, the probability of being in force at the start of year 3. The premium factor likewise includes a time-2 premium only for the 0.855 policies still in force. Applying the maturity-only terminal bonus to year-3 death claims would contradict the specified benefit rule.

Interpreting the Valuation and Avoiding Common Errors

A participating valuation is conditional on its benefit and scenario assumptions. Declared bonuses are included in benefits once vested under the stated contract terms. For future discretionary benefits, a sound calculation states whether they are excluded, included at an assumed scale, or evaluated under alternative specified scales. The numerical result should not be presented as though a discretionary amount were certain merely because it was included in one scenario.

The same framework can value a premium or a present-value liability: list the contingent cash flows, determine their probabilities and payment times, and discount them consistently. If solving for a premium, the expected present value of one unit of premium is the denominator. If valuing benefits net of premiums, subtract the premium present value from benefit and expense outgo. In either case, keep the assumed policy terms distinct from the valuation treatment of future discretionary benefits.

Do not confuse death and maturity. For the final policy year, death and survival to maturity have separate probabilities and may have different benefits.
Do not apply bonuses to an unstated base. A percentage of vested bonuses is not a percentage of the basic sum assured.
Do not use unconditional mortality probabilities as conditional rates. Year-$k$ death probability is $s_{k-1}q_k$, not $q_k$ alone.
Do not ignore payment timing. A benefit paid at the end of year $k$ is discounted by $v^k$; a premium at time $k$ is discounted by $v^k$ and conditional on being in force then.
Do not mix scenarios. If a terminal bonus is included at a stated scale in benefit outgo, use that same scale consistently wherever the terms say it applies.

Key takeaway: A complete participating-policy valuation makes the exit benefits, bonus bases, discretionary-benefit treatment, mortality probabilities, payment dates and premium conditions explicit before calculating present values.

Check Your Understanding

For each problem, use the stated annual timing and assumptions; round monetary answers to the nearest cent.

A two-year endowment has a $10,000 basic benefit, a declared simple bonus of $300 per completed year, and a terminal bonus equal to 10% of vested bonuses on death or maturity. Annual mortality is $q=0.20$, $i=0$, and premiums are payable at times 0 and 1 while in force. There are no expenses. Find the equivalence-principle annual premium. (Answer: $5,885.56)
A two-year policy has a $25,000 basic benefit and a declared simple bonus of $500 per completed year. A terminal bonus equal to 20% of vested bonuses applies on death and maturity. Let $q=0.10$ each year and $i=0.05$. Benefits are paid at year-end. Find the expected present value of benefits, without premiums or expenses. (Answer: $23,825.85)
A two-year endowment has a $30,000 basic benefit and a declared simple bonus of $300 per completed year. A discretionary terminal bonus equal to 50% of vested bonuses is included at the stated scale on maturity only. Let $q_1=0.05$, $q_2=0.15$, and $i=0$. Death benefits are paid at year-end. Premiums are payable at times 0 and 1 while in force; expenses are $100 at time 0 and $40 at time 1 while in force. Find the annual premium. (Answer: $15,879.62)
A three-year policy pays a maturity-only discretionary terminal bonus equal to 25% of vested bonuses. The declared simple bonus is $600 per completed year, annual mortality rates are $q_1=0.10$, $q_2=0.20$, and $q_3=0.25$, and $i=0.04$. The terminal bonus is valued at the stated scale and is paid at time 3 only if the policy matures. Find its expected present value. (Answer: $216.03)

What You'll Learn