Integrated Equity-Linked Projection and Valuation Workshop

Why Integrated Projections Need a Consistent Timeline

An equity-linked calculation can involve several moving parts: premium allocation, fund returns, policy deductions, death or lapse, and a maturity guarantee. The central discipline is to follow one timeline throughout. A deduction taken after investment growth does not produce the same account value as one taken before growth, and a guarantee top-up is not the same amount as the full benefit.

This workshop connects account projections to benefit payments, decrement-weighted valuation, reserves, and profit testing. The examples use hypothetical contracts and assumptions. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Let $F_t$ denote the policy account value at time $t$, and let $A_t$ be the premium amount allocated to the account at that time. If the account earns an effective return $j_{t+1}$ during the next year and a flat deduction $C_{t+1}$ is taken at the end of that year, then $$ F_{t+1}=(F_t+A_t)(1+j_{t+1})-C_{t+1}. $$ This timing convention assumes the allocated premium is invested at the start of the year, the return is applied to the account before the deduction, and the deduction cannot exceed the account value. If the contract takes charges at another time or deducts a percentage of account value, the projection must be changed to match those terms.

A death or maturity benefit may be the account value itself, a guaranteed amount, or the greater of the two. For a guaranteed floor $G$, the total benefit is $\max(F,G)$, while the insurer’s guarantee top-up above the fund is $\max(G-F,0)$. These quantities should not be confused: the account assets support the fund portion, while the top-up measures the additional guarantee cost.

Projecting the Account Before Valuing Benefits

The account projection is the common starting point for all later calculations. Record the order of premium allocation, investment return, and deductions explicitly. A negative investment return is applied as a multiplicative factor, not as a fixed dollar subtraction.

Worked Example: Two-Year Account Projection With a Down Market

A hypothetical policy receives a premium of $12,000 at issue. The insurer allocates 96% to the account. The account earns 5% in year 1 and loses 10% in year 2. A flat $120 deduction is taken at each year-end, after the investment return. Find the account value at the end of each year.

Method. First calculate the allocated premium, then apply each year’s return and subtract the year-end deduction. The premium allocation is $12{,}000(0.96)=11{,}520$. Thus $F_0=11{,}520$ for this projection.

At the end of year 1, the account grows by 5%: $11{,}520(1.05)=12{,}096$. After the deduction, $F_1=12{,}096-120=11{,}976$.

At the end of year 2, the account earns $-10\%$, so the growth factor is $1-0.10=0.90$: $11{,}976(0.90)=10{,}778.40$. After the second deduction, $F_2=10{,}778.40-120=\$10{,}658.40$.

Check. The year-2 loss is 10% of the year-2 opening account value, not 10% of the original premium. The final account is therefore $10,658.40. A common mistake is to subtract the charge before applying the return; that would use a different contract timing and produce the wrong result here.

Combining Investment Scenarios and Decrements

To value benefits, project the account under each investment scenario and then attach the appropriate benefit to each possible policy outcome. Suppose annual death and lapse probabilities are specified conditional on being in force at the start of a year. If death and lapse are mutually exclusive, the probability of remaining in force is one minus their sum. These probabilities must be applied to the correct starting population at each duration.

Result: If a scenario $s$ has probability $w_s$, and event $e$ has probability $p_e$ under the stated decrement model, then the expected present value of benefits is the sum over scenarios and events: $$ \operatorname{EPV}=\sum_s w_s\sum_e p_e\,v^{t_e}B_{s,e}. $$ Here $B_{s,e}$ is the payment under scenario $s$ and event $e$, and $t_e$ is its payment time in years. This form assumes the scenario weights are appropriate for the valuation basis and that the event probabilities are conditional as specified. If investment outcomes and decrements are dependent, their joint probabilities must replace the separate weights.

The reason for the formula is that each scenario-event combination describes a mutually exclusive path with probability $w_sp_e$, under the independence assumption. Its present-value contribution is that path’s probability multiplied by its discounted payment. Adding the contributions across all paths gives the expectation by the definition of expected value.

Worked Example: Death, Lapse, and Maturity Under a Benefit Floor

A hypothetical policy receives $10,000 at issue, all allocated to its account. A $100 deduction is taken at each year-end after investment returns. There are two investment scenarios, independent of decrements: a low scenario with probability 0.40 and returns of 0% in year 1 and $-10\%$ in year 2; and a high scenario with probability 0.60 and returns of 8% in year 1 and 5% in year 2. The death benefit and the year-2 maturity benefit are each the greater of the account value and $11,000. At each year-start, conditional on being in force, the annual death probability is 0.02 and the annual lapse probability is 0.05. Death and lapse are mutually exclusive. A lapse receives the account value without a guarantee. Benefits are paid at the end of the year of death or lapse, or at the end of year 2 at maturity. Calculate the expected present value of all benefits at an effective annual interest rate of 4%.

Step 1: Project both accounts. In the low scenario, $F_1=10{,}000(1.00)-100=9{,}900$, and $F_2=9{,}900(0.90)-100=8{,}810$. In the high scenario, $F_1=10{,}000(1.08)-100=10{,}700$, and $F_2=10{,}700(1.05)-100=11{,}135$.

Step 2: Identify the payments. In year 1, death pays $11,000 in either scenario because both account values are below the guarantee. Lapse pays $9,900 in the low scenario or $10,700 in the high scenario. At year 2, death and maturity each pay $11,000 in the low scenario and $11,135 in the high scenario. Lapse pays the account value: $8,810 or $11,135, respectively.

Step 3: Calculate event probabilities. In year 1, death has probability 0.02, lapse has probability 0.05, and persistence has probability $1-0.02-0.05=0.93$. In year 2, the unconditional death and lapse probabilities are $0.93(0.02)=0.0186$ and $0.93(0.05)=0.0465$. The probability of surviving in force to maturity is $0.93(0.93)=0.8649$. The three year-2 terminal probabilities sum to $0.0186+0.0465+0.8649=0.9300$, the probability still in force after year 1.

Step 4: Calculate expected payments by year and scenario. In year 1, the low-scenario expected payment is $0.02(11{,}000)+0.05(9{,}900)=220+495=\$715$. The high-scenario amount is $0.02(11{,}000)+0.05(10{,}700)=220+535=\$755$. Weighting the scenarios gives $0.40(715)+0.60(755)=286+453=\$739$.

In year 2, the low-scenario expected payment is $0.0186(11{,}000)+0.0465(8{,}810)+0.8649(11{,}000)$ $=204.60+409.665+9{,}513.90=\$10{,}128.165$. The high-scenario amount is $0.0186(11{,}135)+0.0465(11{,}135)+0.8649(11{,}135)$ $=207.111+517.7775+9{,}630.6615=\$10{,}355.55$. The weighted year-2 expected payment is $0.40(10{,}128.165)+0.60(10{,}355.55)=4{,}051.266+6{,}213.33=\$10{,}264.596$.

Step 5: Discount and add. The expected present value is $739/1.04+10{,}264.596/(1.04)^2$. The two discounted amounts are approximately $710.5769 and $9,490.1960, giving $10,200.7729. Rounded to the nearest cent, the expected present value is $10,200.77.

Check and trap. Year-2 probabilities are conditional on reaching year 2, so their unconditional weights include the 0.93 probability of persistence through year 1. Applying the year-2 rates to the original policy count would overstate year-2 payments. Also, this is the present value of gross benefit payments; it is not a net reserve after subtracting the fund assets or premiums.

Guarantee Reserves and Profit Testing

A reserve must match the liability being measured. A gross benefit valuation includes the full payment, while a guarantee reserve may measure only the insurer’s top-up beyond the account. When the account is separately backed by fund assets, adding a reserve for the full account-linked payment to those assets can double-count the fund portion.

A profit test makes the timing explicit. Define a profit vector as the sequence of expected net cash flows at each duration, with income positive and outgo negative. Its net present value is the discounted sum of the vector. The assumptions, including whether premiums and expenses are included at issue, must be stated before interpreting the result.

Worked Example: Guarantee Top-Up Reserve and Profit Signature

A hypothetical one-year guarantee rider is attached to an account with opening value $10,000. The account earns either $-20\%$, with probability 0.25, or $10\%$, with probability 0.75. In either scenario, a $100 charge is deducted at year-end after investment return. At the end of the year, the death benefit or maturity benefit is the greater of the account and $10,000. Death has probability 0.02 and survival to maturity has probability 0.98; both outcomes have the same guarantee terms. Assume no lapse, an effective annual interest rate of 4%, a $600 rider charge received at issue, and a $100 issue expense. Find the expected guarantee top-up, its reserve at issue, and the NPV of the stated profit vector.

Method: value only the guarantee top-up. In the low-return scenario, the account is $10{,}000(0.80)-100=7{,}900$, so the top-up is $\max(10{,}000-7{,}900,0)=\$2{,}100$. In the high-return scenario, the account is $10{,}000(1.10)-100=10{,}900$, so the top-up is $\max(10{,}000-10{,}900,0)=\$0$.

The expected year-end top-up, conditional on reaching the terminal event, is $0.25(2{,}100)+0.75(0)=\$525$. Death and maturity probabilities sum to $0.02+0.98=1$, and the top-up is the same for either event in each investment scenario. Therefore the unconditional expected top-up is also $525. The guarantee reserve at issue is $525/1.04=\$504.8077$, or $504.81 to the nearest cent.

The issue-date profit is rider charge less issue expense: $600-100=\$500$. The expected year-1 profit is the negative of the top-up, $-\$525$. Thus the profit vector is $(500,-525)$, at times 0 and 1, and its NPV is $500-525/1.04=500-504.8077=-\$4.8077$. Rounded to the nearest cent, the NPV is $-\$4.81$.

Check. The reserve equals the present value of the expected future guarantee outgo, and the NPV is issue profit minus that same reserve. The calculation excludes the account-backed portion of the benefit from the guarantee reserve. Valuing the full $10,000 benefit as a guarantee claim would confuse the account liability with the insurer’s top-up.

A Practical Integration Sequence

In a longer case, work in the order of the cash flows rather than jumping immediately to a present value. This reduces the chance of mixing account values, benefit amounts, and insurer outgo.

Write the contract timeline.
Specify when premiums are allocated, when returns are credited, when deductions occur, and when benefits are paid.

Project each account scenario.
Use the stated return and charge order to obtain an account value at every relevant duration.

Map each outcome to its payment.
Apply the death, lapse, and maturity terms separately. Distinguish the total benefit from any guarantee top-up.

Weight and discount the paths.
Use unconditional path probabilities when calculating present values from issue, and discount using the payment time.

State the valuation basis.
Identify whether the result is gross benefits, a guarantee reserve, or a net liability after premiums and backing assets.

The same projected cash flows can support several outputs, but the outputs answer different questions. A scenario-weighted value measures expected discounted payments under the stated assumptions. A reserve is a liability measure defined on a particular basis. A profit signature records the timing of expected income and outgo. Keeping those interpretations separate is as important as getting the arithmetic right.

Key takeaway: In an integrated equity-linked calculation, project first, map contract terms to each policy outcome second, then apply probabilities and discounting. State explicitly whether the result includes the fund-backed benefit, only the guarantee cost, or the net profit-test cash flow.

Check Your Understanding

Use the stated timing and assumptions in each problem. Amounts are hypothetical.

A $5,000 premium has 90% allocated to an account. The account earns 4%, then a $50 deduction is taken at year-end. What is the year-end account value? (Answer: $4,630.00)
An account of $8,000 earns $-5\%$, followed by a $100 year-end deduction. A death benefit is the greater of the resulting account and $8,000. Find the account value and guarantee top-up. (Answer: Account $7,500; top-up $500)
Conditional on being in force at the start of a year, death probability is 0.03 and lapse probability is 0.04. What is the probability of remaining in force for the year? (Answer: 0.93)
A one-year guarantee pays a top-up of $1,200 with probability 0.20 and zero otherwise. The effective annual interest rate is 5%. Find the reserve at issue. (Answer: $228.57)
A rider produces an issue-date profit of $300 and an expected year-end guarantee outgo of $420. At an effective annual interest rate of 5%, calculate the two-term NPV, with the outgo at time 1. (Answer: $-\$100.00$)

What You'll Learn