Surrender Values and Participating Policy Reserves

Introduction

A participating policy can have two different values at the same duration. Its contractual surrender value is the amount payable if the policyholder ends the contract, subject to the policy’s eligibility conditions and surrender schedule. Its prospective reserve is the value, on stated assumptions, of the policy’s future benefits and expenses less future premiums, conditional on the policy continuing in force. One is a payment defined by the contract; the other is a valuation of future cash flows.

The comparison requires careful attention to timing and benefit definitions. A surrender schedule may include declared bonuses, apply a factor to some benefits but not others, or provide no payment until a minimum duration has been reached. The reserve must instead use the stated valuation basis and the benefits and premiums that apply if the policy continues. This distinction is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: At time $t$, the contractual surrender value is the amount payable on surrender at that time, provided the policy satisfies the contract’s eligibility conditions. The prospective reserve immediately before surrender is the conditional expected present value of future benefits and expenses, less the conditional expected present value of future premiums, calculated on the stated valuation basis and assuming the policy remains in force.

Two Values, Two Questions

The reserve is not automatically the amount due on surrender. It answers: what is the value at time $t$ of the future cash flows if this policy continues? The surrender value answers: what does the contract pay if this eligible policy ends now? A surrender payment may be below, equal to, or above the reserve. The relationship depends on the contract and the valuation assumptions; there is no general rule that one must equal the other.

For a discrete annual model, suppose a policy is valued immediately after any premium due at time $t$. Let $i$ be the effective annual valuation interest rate, $v=1/(1+i)$, and let $q$ and $p=1-q$ be the annual mortality and survival probabilities for the relevant policy year. If $B_k$ is the benefit payable at the end of year $k$ on the termination event then occurring, and $P_k$ is a premium payable at time $k$ only if the policy is still in force, the reserve can be written as:

{}_tV = \operatorname{EPV}_t(\text{future benefits and expenses}) - \operatorname{EPV}_t(\text{future premiums}).

The probabilities in each expected present value must reflect the cash-flow conditions. For example, a premium at time $1$ payable only if the insured survives the first year has expected present value $P_1\,{}_1p_x v$. A benefit payable on death in the first year has expected present value $B_1\,{}_1q_x v$. Do not count a cash flow when its stated condition is not met.

Important distinction: The prospective reserve is conditional on the policy continuing. A surrender payment is a cash flow caused by ending the policy. Unless the valuation explicitly models surrender as a decrement, the surrender payment is not included as a future continuation benefit in the reserve.

Worked Examples

Worked Example: Comparing a Surrender Payment With a Reserve

A hypothetical participating endowment is valued immediately after a premium has been paid. It has two years remaining. Its currently guaranteed benefit, including bonuses already declared, is $11,000 on death or maturity. The benefit is payable at the end of the year of death, or at the end of year 2 on maturity if the insured survives. Annual mortality is $q=0.10$ in each year, with independent annual survival, so $p=0.90$. The valuation interest rate is $i=0.05$. One further premium of $500 is due at time 1 if the insured is alive then; no later premium is due. The policy is eligible to surrender now, and its contractual surrender value is $9,000. Calculate the prospective reserve and compare the two amounts.

Method. First calculate the expected present value of benefits over the two years. Since a benefit of $11,000 is paid at the first termination, the probability-weighted discount factors are $qv$ for death in year 1 and $pv^2$ for survival through year 1 followed by termination in year 2. Here $v=1/1.05=20/21$.

\operatorname{EPV}(\text{benefits}) = 11{,}000\left(0.10\frac{20}{21} +0.90\left(\frac{20}{21}\right)^2\right) = 11{,}000\left(\frac{2}{21}+\frac{360}{441}\right) = 11{,}000\left(\frac{134}{147}\right) = \$10{,}027.210884\ldots

The premium is payable at time 1 only if the insured survives the first year. Its expected present value is:

\operatorname{EPV}(\text{premium}) = 500(0.90)\frac{20}{21} = \$428.571428\ldots

Subtract the unrounded expected present values to obtain the reserve:

{}_tV = 10{,}027.210884\ldots-428.571428\ldots = 9{,}598.639455\ldots = \$9{,}598.64.

The contractual surrender payment is $9,000, which is $598.64 less than the reserve. The difference is not an error: the surrender amount comes from the contract, whereas the reserve values continuation cash flows using the stated assumptions. Common mistake avoided: subtracting rounded displayed values can produce a different cent. For example, subtracting $10,027.21 and $428.57 gives $9,598.64 here, but the reliable method is to retain full precision until the final rounding.

Worked Example: Applying the Eligibility Rule and Bonus Basis

A hypothetical policy permits surrender only after three complete policy years. At duration 4 it is eligible. Its contractual schedule pays 80% of the sum assured plus declared reversionary bonuses, and then adds a fixed $300. The sum assured is $20,000 and declared bonuses total $2,400. For the reserve calculation, the guaranteed benefit, including those declared bonuses, is $22,400 on any termination during the next two years. Annual mortality is $q=0.02$, survival is $p=0.98$, and the valuation interest rate is $i=0.04$. One premium of $1,000 is due at time 1 if the insured is alive; there are no other future premiums. Find the surrender value and the prospective reserve.

Contractual payment. The policy has completed the required three years, so the schedule applies. The stated factor applies to the sum assured and declared bonuses together; the additional $300 is added after applying the factor:

\text{Surrender value} = 0.80(20{,}000+2{,}400)+300 = 0.80(22{,}400)+300 = \$18{,}220.

Prospective reserve. Set $v=1/1.04=25/26$. The benefit is payable at the first termination: death in year 1 with probability $q$, or termination in year 2 after survival through year 1. Therefore:

\operatorname{EPV}(\text{benefits}) = 22{,}400\left(0.02\frac{25}{26} +0.98\left(\frac{25}{26}\right)^2\right) = 22{,}400\left(\frac{1}{52}+\frac{1225}{1352}\right) = 22{,}400\left(\frac{1251}{1352}\right) = \$20{,}726.627218\ldots

The premium is payable at time 1 only if the insured survives the first year:

\operatorname{EPV}(\text{premium}) = 1{,}000(0.98)\frac{25}{26} = \$942.307692\ldots

Thus, using the unrounded values:

{}_tV = 20{,}726.627218\ldots-942.307692\ldots = 19{,}784.319526\ldots = \$19{,}784.32.

The surrender payment is $1,564.32 below the reserve. Common mistake avoided: applying the eligibility rule to the reserve calculation. Eligibility determines whether the contractual payment can be made; it does not change the prospective reserve’s stated benefit and premium cash flows.

Worked Example: A Surrender Payment Above the Reserve

A hypothetical fully paid participating policy has completed its fourth year. Its surrender provision is available after three complete years, and the contract states that the surrender payment now is $8,600. The guaranteed benefit on death or maturity, including bonuses already declared, is $10,000. There are two years remaining to maturity. Annual mortality is $q=0.10$, survival is $p=0.90$, and the valuation interest rate is $i=0.10$. There are no future premiums. Calculate the reserve and state which value is larger.

Method. Since the same $10,000 benefit is paid at the first termination, its expected present value uses the factors $qv$ for death in year 1 and $pv^2$ for termination in year 2. Here $v=1/1.10=10/11$:

{}_tV = 10{,}000\left(0.10\frac{10}{11} +0.90\left(\frac{10}{11}\right)^2\right) = 10{,}000\left(\frac{1}{11}+\frac{90}{121}\right) = 10{,}000\left(\frac{101}{121}\right) = \$8{,}347.107438\ldots = \$8{,}347.11.

The contractual surrender value of $8,600 exceeds the reserve by $252.89. This is possible because the schedule and the reserve are defined differently; neither value is automatically a floor for the other. Common mistake avoided: assuming that surrender value must always be less than the reserve simply because surrender ends future cover.

Why the Reserve Formula Holds

The prospective reserve is a net value: the expected present value of future outgo less the expected present value of future income. This follows from the policy’s future cash-flow balance, conditional on the policy still being in force at the valuation time. For a two-year contract with a benefit at the first termination and a premium at time 1 only if alive, the calculation can be separated by mutually exclusive events.

List the cash flows.
Identify each future benefit, expense, and premium, and state exactly when it is payable and what condition triggers payment.

Weight each cash flow by its condition.
For example, a first-year death benefit is weighted by $q$, while a premium at time 1 conditional on survival is weighted by $p$.

Discount to the valuation time.
A cash flow at time $k$ is discounted by $v^k$, using the stated valuation interest rate.

Take net prospective value.
Subtract the expected present value of future premiums from the expected present value of future benefits and expenses. Round only after this subtraction.

The surrender value is not obtained by repeating this calculation with a different probability. It is obtained by applying the contract’s surrender provision at the current duration. A useful check is to keep the two calculations visibly separate: first determine whether surrender is allowed and calculate its payment; then calculate the reserve from continuation cash flows on the specified basis.

Assumptions and Common Traps

A comparison is meaningful only when the assumptions and timing are explicit. In particular, state whether the reserve is immediately before or after a premium due at the valuation time. A reserve just before a premium may differ from one just after it because the premium is a cash flow at that instant. Similarly, identify whether a declared bonus is already included in the stated benefit or must be added separately. Counting it twice overstates the benefit value.

Ignoring surrender eligibility. A quoted schedule may apply only after a specified duration, after a required number of premiums, or subject to another contractual condition. Check that condition before reporting a payment.
Applying the wrong bonus basis. If the schedule applies a factor to the sum assured and declared bonuses together, do not apply it to the sum assured alone. Conversely, do not add bonuses again if the stated benefit already includes them.
Using the surrender payment as a reserve benefit. The continuation reserve does not normally include a payment that arises only if the policy terminates by surrender. Include surrender only if the valuation model explicitly includes it.
Using inconsistent timing or assumptions. Premium payment dates, benefit dates, mortality probabilities, and interest must all match the problem statement. A premium payable only on survival is not certain.
Rounding too early. Retain unrounded expected present values through the subtraction, then round the final money amount to the cent.

Key takeaway: The surrender value is a contractual payment, subject to eligibility and the specified bonus basis. The prospective reserve is a valuation of future continuation cash flows on stated assumptions. Calculate each independently, then compare them.

Check Your Understanding

For each problem, determine the requested contractual payment or reserve from the information given. Round monetary answers to the nearest cent.

A policy has one year remaining, no future premiums, a $5,000 benefit payable at year-end on death or survival, and a contractual surrender payment of $4,300. The annual effective interest rate is 5%. Find the reserve and the difference between the reserve and surrender payment. (Answer: reserve $4,761.90; reserve exceeds surrender payment by $461.90)
A policy has two years remaining and pays $8,000 at the first termination. Annual mortality is $q=0.10$, survival is $p=0.90$, and the annual effective valuation interest rate is 10%. A $220 premium is due at time 1 only if the insured survives year 1. Find the prospective reserve immediately after the current premium. (Answer: $6,497.69)
A contract allows surrender after two complete policy years. At duration 3, the sum assured is $12,000, declared bonuses are $1,500, and the surrender provision pays 75% of their combined amount plus $200. Find the surrender value and state whether the eligibility condition is met. (Answer: eligible; $10,325.00)
A policy has two years remaining. Its $11,000 benefit, including declared bonuses, is payable at the first termination. Annual mortality is $q=0.10$, the annual effective valuation interest rate is 5%, and a $500 premium is due at time 1 only if the insured survives the first year. The eligible surrender payment is $9,000. Find the reserve and compare it with the surrender payment. (Answer: reserve $9,598.64; it exceeds the surrender payment by $598.64)
A fully paid policy has two years remaining and pays $10,000 at the first termination. Annual mortality is $q=0.05$, survival is $p=0.95$, and the annual effective valuation interest rate is 4%. There are no future premiums. Find the reserve. (Answer: $9,264.05)

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