Equity-Linked Asset Shares with Maturity Guarantees

Purpose and Model

An equity-linked policy’s maturity account value depends on the premiums allocated to the account, the charges deducted, and the returns earned. A maturity guarantee adds a further obligation: if the account value is below the guaranteed amount, the insurer must provide a top-up. Assessing that obligation requires two separate calculations: project the asset share for a policy that reaches maturity, then compare it with the guarantee. If persistency is uncertain, estimate how many policies remain in force before aggregating the expected top-up cost.

The calculations below use a deliberately specified annual model. Premiums are paid at the start of each year, allocation and fixed charges are deducted immediately, and the remaining amount earns that year’s investment return. There are no withdrawals, mortality deductions, lapse benefits, taxes, or other cash flows unless stated. The projected asset share is conditional on a policy remaining in force; it is not automatically equal to a policy reserve or the insurer’s total guarantee liability.

Definition: Let $A_{t-1}$ be the asset share per in-force policy at the start of year $t$, $P_t$ the premium paid at that time, $\alpha_t$ the proportion of premium deducted as an allocation charge, $c_t$ a fixed charge in dollars, and $r_t$ the effective investment return for the year. With no other cash flows, the year-end asset share is $$ A_t=\bigl(A_{t-1}+(1-\alpha_t)P_t-c_t\bigr)(1+r_t). $$ For a maturity guarantee $G$, the maturity top-up per surviving policy is $\max(G-A_n,0)$, and the total maturity payment per survivor is $\max(G,A_n)$.

The charge convention matters. Here, the fixed charge $c_t$ is deducted from the account once per year, in dollars, as well as the proportional charge $\alpha_tP_t$. The net amount added before investment is therefore $(1-\alpha_t)P_t-c_t$. If a product deducts charges at a different time, or charges depend on the account value, its projection must reflect that contract instead.

Why the Recursion and Top-Up Formula Hold

The recursion follows directly from the annual cash-flow order. At the start of the year, the policy has $A_{t-1}$. The premium contributes $P_t$, but the allocation charge removes $\alpha_tP_t$, and the fixed charge removes $c_t$. The account immediately before investment is thus $A_{t-1}+P_t-\alpha_tP_t-c_t$. Applying the year’s return multiplies that balance by $1+r_t$, giving the stated formula.

At maturity the insurer pays the larger of the account value and the guarantee. If $A_n\geq G$, the account already funds the guaranteed amount and no top-up is due. If $A_n<G$, the insurer adds $G-A_n$, so the total is exactly $G$. Both cases are represented by $\max(G,A_n)$, while the insurer’s additional payment is $\max(G-A_n,0)$.

Project the account.
For each year, add the net premium to the opening asset share, then apply the specified investment return.

Apply the guarantee to each outcome.
Calculate $\max(G-A_n,0)$ for the maturity asset share in that outcome.

Aggregate over survivors.
Use the appropriate expected number of maturity survivors, provided persistency is independent of the return outcome and policies are otherwise homogeneous.

Step 3 requires care. If lapse is independent of investment returns and all policies have the same premium and charge pattern, persistency changes the expected number of top-ups, not the projected asset share conditional on survival. If lapse rates vary by account performance, or lapse benefits are paid, a joint model of account values and persistency is needed; multiplying a single per-policy top-up by an unconditional persistency rate may then be inappropriate.

Projecting a Maturity Asset Share

Worked Example: Applying the Annual Return at the Correct Time

A hypothetical equity-linked policy receives a premium of $3,000 at the start of each of two years. The allocation charge is 3% of each premium, and the fixed charge is $40 per year. The annual returns are 0% in year 1 and 5% in year 2. There is no opening asset share. Find the maturity asset share and top-up for a $6,000 guarantee, to the nearest cent.

Method. Calculate each year’s net premium, add it before that year’s return, and compare the final asset share with the guarantee.

The allocation charge each year is $0.03(3{,}000)=\$90$. The net amount added at the start of each year is therefore $3{,}000-90-40=\$2{,}870$.

At the end of year 1:

A_1=(0+2{,}870)(1+0)=\$2{,}870.

At the start of year 2, add the second net premium, then apply the 5% return:

A_2=(2{,}870+2{,}870)(1.05)=5{,}740(1.05)=\$6{,}027.

Since $A_2=\$6{,}027$ exceeds $G=\$6{,}000$, the top-up is $\max(6{,}000-6{,}027,0)=\$0$. The maturity payment is the account value, $6,027.00. Rounded to the nearest cent, the asset share is $6,027.00 and the top-up is $0.00.

Common mistake avoided. Applying the 5% return only to the year-1 balance gives $2{,}870(1.05)+2{,}870=\$5{,}883.50$, and would incorrectly suggest a $116.50 top-up. That calculation treats the year-2 premium as if it arrived after the year-2 investment period. Under the stated timing, it is paid at the start of year 2 and earns the year’s return.

Return, Charge, and Persistency Scenarios

The asset share is sensitive to the full sequence of returns, not just a single annual average. Charges affect the amount invested, while persistency determines how many policies are expected to reach maturity and receive a guarantee. Keep these effects separate: a higher premium generally increases the projected asset share, whereas a higher charge generally decreases it, all else equal. The resulting change in top-up depends on whether the asset share is above or below the guarantee.

Worked Example: Comparing Return Paths and Expected Survivors

A hypothetical portfolio has 1,000 identical policies. Each policy pays $2,000 at the start of each of three years. The allocation charge is 2%, the fixed charge is $20 per year, and the maturity guarantee is $6,000. The annual persistency probability is 0.90 in each year, independent of returns. Assume no lapse payment. Compare (a) a 2% return in each year and (b) returns of -10%, 0%, and 2% in years 1, 2, and 3. Calculate the per-survivor top-up and the expected aggregate top-up in each scenario.

Method. First project one policy’s asset share using the stated return sequence. Then apply the guarantee and multiply the per-survivor top-up by the expected number of maturity survivors. The net premium each year is $2{,}000-0.02(2{,}000)-20=\$1{,}940$.

The probability of persisting through all three years is $0.90^3=0.729$. The expected number of maturity survivors is $1{,}000(0.729)=729$.

(a) 2% return each year. The year-by-year asset shares are:

\begin{aligned} A_1&=(0+1{,}940)(1.02)=1{,}978.80,\\ A_2&=(1{,}978.80+1{,}940)(1.02)=3{,}918.80(1.02)=3{,}997.176,\\ A_3&=(3{,}997.176+1{,}940)(1.02)=5{,}937.176(1.02)=6{,}055.91952. \end{aligned}

The unrounded final asset share is $6,055.91952, which exceeds the $6,000 guarantee. The top-up per survivor is zero, so the expected aggregate top-up is $729(0)=\$0$.

(b) Returns of -10%, 0%, and 2%.

\begin{aligned} A_1&=(0+1{,}940)(0.90)=1{,}746,\\ A_2&=(1{,}746+1{,}940)(1)=3{,}686,\\ A_3&=(3{,}686+1{,}940)(1.02)=5{,}626(1.02)=5{,}738.52. \end{aligned}

The top-up per survivor is $6{,}000-5{,}738.52=\$261.48$. The expected aggregate top-up is $729(261.48)=\$190{,}618.92$. Thus, under the lower-return path, the expected guarantee cost is positive even though the expected number of survivors is the same.

Check. In scenario (b), each expected survivor receives an account value of $5,738.52 plus a $261.48 top-up, or $6,000 in total. For 729 expected survivors, the aggregate maturity payment is $729(6{,}000)=\$4{,}374{,}000$, consisting of $729(5{,}738.52)=\$4{,}183{,}381.08$ in account value and $190,618.92 in top-ups. The components sum to $4,374,000.

These aggregate figures are expected amounts, not a statement that a fractional policy survives. For a large homogeneous portfolio, expected counts are useful for projection; an actual outcome has an integer number of survivors. The calculation also assumes persistency is independent of the return path. If poor returns lead to more lapses, the policies reaching maturity may have a different return distribution from the original portfolio.

Why the Top-Up Must Be Calculated by Outcome

When investment returns are uncertain, calculate the guarantee separately in each possible outcome and then take the expectation. The top-up function $\max(G-A_n,0)$ has a kink at $A_n=G$, so in general the expected top-up is not equal to the top-up at the expected asset share. A portfolio that has some outcomes below the guarantee and others above it can have a positive expected top-up even when the mean asset share is at or above the guarantee.

Worked Example: Expected Top-Up Under Two Market Outcomes

A hypothetical one-year policy receives a $5,000 premium at the start of the year. The allocation charge is 4% and the fixed charge is $100. At year-end, the return is either -20% or +20%, each with probability 0.50. The maturity guarantee is $5,000. Find the expected asset share, expected top-up, and expected maturity payment.

Method. Find the net invested premium, project the account in each outcome, apply the guarantee separately, and probability-weight the results. The net premium is $5{,}000-0.04(5{,}000)-100=5{,}000-200-100=\$4{,}700$.

With a -20% return, the asset share is $4{,}700(0.80)=\$3{,}760$, so the top-up is $5{,}000-3{,}760=\$1{,}240$, and the maturity payment is $5,000. With a +20% return, the asset share is $4{,}700(1.20)=\$5{,}640$, the top-up is zero, and the maturity payment is $5,640.

The expected asset share is $0.50(3{,}760)+0.50(5{,}640)=\$4{,}700$. The expected top-up is $0.50(1{,}240)+0.50(0)=\$620$. The expected maturity payment is $0.50(5{,}000)+0.50(5{,}640)=\$5{,}320$. This also checks as expected account value plus expected top-up: $4{,}700+620=\$5{,}320$.

Common mistake avoided. Applying the guarantee to the expected asset share gives $\max(5{,}000-4{,}700,0)=\$300$, which is not the expected top-up of $620. The guarantee is applied to each outcome before probability-weighting because the insurer pays a top-up in the low-return outcome but none in the high-return outcome.

Practical Checks and Common Traps

For an exam-style projection, write down the timing and cash-flow assumptions before calculating. Confirm whether premiums are paid at the start or end of each period, whether charges are a percentage of premium or a fixed dollar amount, and whether returns are effective rates for the same period as the cash flows. A return quoted for a different time unit must be converted consistently before it is used.

Input or result	Check	Typical error
Net premium	Subtract both the allocation charge and fixed charge exactly once.	Applying the percentage charge to the premium and then subtracting it a second time.
Asset-share recursion	Add the net premium at the stated payment time, then apply the period’s return.	Leaving a start-of-year premium out of that year’s investment return.
Maturity guarantee	Use $\max(G-A_n,0)$ for the top-up and $\max(G,A_n)$ for the total payment.	Paying a top-up when the asset share already exceeds the guarantee.
Portfolio cost	Apply persistency to the expected number of survivors under the stated assumptions.	Multiplying by the original policy count, or applying persistency twice.
Uncertain returns	Calculate the top-up outcome by outcome before taking an expectation.	Applying the guarantee only to the mean asset share.

The model is intentionally limited to maturity guarantees and the specified account mechanics. It does not value the time value of future guarantee payments, capital requirements, or other benefits. Those require additional assumptions and valuation methods. Within this model, the central discipline is to keep account projection, guarantee application, and persistency aggregation as distinct steps.

Key takeaway: Project the asset share using the contract’s cash-flow timing and return assumptions, calculate the maturity top-up as $\max(G-A_n,0)$ for each relevant outcome, and then aggregate over expected survivors. Higher premiums generally increase asset shares; higher charges generally reduce them, while persistency changes expected portfolio counts when independent of returns.

Check Your Understanding

For each problem, use the stated timing convention: premiums and charges occur at the start of each year, before that year’s investment return.

A hypothetical one-year policy has a $4,000 premium, a 2% allocation charge, a $30 fixed charge, and a 5% return. Its maturity guarantee is $4,200. Find the maturity asset share and top-up, to the nearest cent. (Answer: asset share $4,084.50; top-up $115.50)
A policy receives $2,500 at the start of each of two years. The allocation charge is 4%, the fixed charge is $25 per year, and the returns are 0% and 4%. The maturity guarantee is $5,000. Find the asset share and top-up. (Answer: asset share $4,940.00; top-up $60.00)
A hypothetical block has 800 identical policies, each with a maturity top-up of $150 if it survives two years. Annual persistency is 0.95 in each year, independently of returns. Find the expected number of maturity survivors and expected aggregate top-up. (Answer: 722 survivors; $108,300.00)
A one-year policy invests a $4,000 premium with no charges. The return is -10% or +10%, each with probability 0.50, and the maturity guarantee is $4,000. Find the expected asset share and expected top-up. (Answer: expected asset share $4,000.00; expected top-up $200.00)
A policy receives $3,000 at the start of each of two years. The allocation charge is 2%, the fixed charge is $20 per year, and returns are 0% in year 1 and 5% in year 2. The maturity guarantee is $6,000. Find the asset share and top-up. (Answer: asset share $6,132.00; top-up $0.00)

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