Why Scenario Construction Requires a Timeline
An investment scenario is a specified sequence of returns over time. It is not enough to state that an investment earns an average return: a projection must identify the return for each period and when premiums, unit purchases, and deductions occur. Those details determine the unit-price path and, when cash flows occur during the projection, the resulting policy fund.
The scenarios in this tutorial are deterministic: each period's return is fixed, not random. They are hypothetical projection paths, not forecasts or probabilities. Unless a problem says otherwise, take each stated return to be the investment return credited to the unit price for that period, after any asset-level investment expenses. Policy charges are separate and are applied at the stated time.
A return sequence must use a consistent period length. For example, if periods are annual, each \(r_t\) is an effective annual return and each update advances one year. A return stated for a full year must not be applied once in each month. For monthly scenarios, the inputs must instead be monthly effective returns.
Cash-flow timing is just as important. A contribution allocated to units at time \(t\) purchases \(A_t/P_t\) units if the allocated amount is \(A_t\). A charge \(C_t\) deducted by cancelling units at that time cancels \(C_t/P_t\) units, provided sufficient units are available. The price used is the price at the actual transaction time.
Compounding Returns into a Unit-Price Path
Applying the unit-price recurrence repeatedly gives the price at the end of \(n\) periods:
This expression shows why each period's return belongs in the scenario. Returns compound through multiplication of their gross accumulation factors, \(1+r_t\), rather than by adding the rates. It also shows that, if there are no cash flows, changing the order of a fixed set of returns does not change the terminal unit price: multiplication is commutative. It can, however, change prices at intermediate dates, which matters when cash flows occur during the path.
Specify the number and length of periods, and identify the valuation dates and transaction dates.
List one effective return for each period, with units such as annual or monthly stated explicitly.
Apply \(P_t=P_{t-1}(1+r_t)\) period by period, retaining enough precision for later calculations.
At each transaction date, convert allocated contributions or charges into units using the price at that date.
Multiply the resulting units by the price on the valuation date and check that the cash-flow and unit movements agree.
Level, Rising, Falling, and Volatile Paths
A level path uses the same return in every period. A rising path has returns that increase over the selected periods, while a falling path has returns that decrease. A volatile path has larger period-to-period variation; it may include both positive and negative returns. These labels describe the sequence, not a guarantee about the terminal outcome.
For a fixed collection of period returns and no intermediate cash flows, rising and falling paths that merely reverse the order of those returns have the same terminal unit price. Their intermediate prices differ. A level path with the arithmetic average return will generally have a different terminal price from the varying-return paths, because compounding depends on the product of the gross returns.
Worked Example: Comparing Level, Rising, and Falling Paths
A hypothetical fund has an initial unit price of $100. Project its price over two annual periods under three scenarios: a level path of 4% each year, a rising path of 2% followed by 6%, and a falling path of 6% followed by 2%. Assume no cash flows or charges. Find each annual and terminal unit price.
Method. Apply the annual price recurrence once per period. This avoids treating the two-year return as the sum of the two annual rates.
Level path. The first-year price is \(100(1.04)=104\). The second-year price is \(104(1.04)=108.16\).
Rising path. The first-year price is \(100(1.02)=102\). The second-year price is \(102(1.06)=108.12\).
Falling path. The first-year price is \(100(1.06)=106\). The second-year price is \(106(1.02)=108.12\).
The rising and falling paths have the same terminal price because both have the gross accumulation product \(1.02(1.06)=1.0812\). Their intermediate prices differ, and the level path ends at $108.16 rather than $108.12. The level-path rate of 4% is not a substitute for the varying returns: \(1.04^2=1.0816\), not \(1.0812\). Prices are shown to the cent; no intermediate rounding was required.
Check. In each scenario, dividing the terminal price by $100 gives the product of its two gross return factors.
Common mistake avoided: adding 2% and 6% to claim a 108% total accumulation. The two-year accumulation is the product of \(1.02\) and \(1.06\), not \(1+0.02+0.06\).
Projecting Funds When Contributions and Charges Occur
A unit-price path alone does not determine a policy fund when units are bought or cancelled during the projection. The fund must be tracked by updating units at each transaction date. In particular, a premium allocation is converted into units at the price when it is invested; it is not retrospectively invested at the initial price.
Worked Example: Premium Allocations and Charges on a Two-Year Path
A hypothetical policy has a unit price of $100 at time 0. At the beginning of each of two annual periods, the policyholder pays a gross premium of $1,000, of which 90% is allocated to units. The annual investment returns are 10% in year 1 and −5% in year 2. At the end of each year, a $110 charge in year 1 and a $104.50 charge in year 2 is deducted by cancelling units at that date's unit price. Find the fund after the second charge. Assume no other cash flows.
Method and timing. Invest each allocated premium at the beginning-of-year price. Then apply that year's investment return and deduct the end-of-year charge at the resulting price.
At time 0, the allocated premium is \(0.90(1{,}000)=\$900\). It buys \(900/100=9\) units. The year 1 price is \(100(1.10)=\$110\), so the fund before the charge is \(9(110)=\$990\). The $110 charge cancels \(110/110=1\) unit, leaving \(9-1=8\) units and a post-charge fund of \(8(110)=\$880\).
At the beginning of year 2, the next allocated premium is also $900. At the then-current price of $110 it buys \(900/110=90/11=8.181818\ldots\) units. Before the year 2 return, total units are \(8+90/11=178/11=16.181818\ldots\).
The year 2 unit price is \(110(0.95)=\$104.50\). Before the second charge, the fund is \(178/11 \times 104.50 = 178 \times 9.50 = \$1{,}691.00\). The $104.50 charge cancels \(104.50/104.50=1\) unit, leaving \(167/11=15.181818\ldots\) units. The final fund is \(167/11 \times 104.50=167(9.50)=\$1{,}586.50\).
Check. The year 2 premium's $900 allocation grows by 5% to $945 before the charge. The pre-existing $880 post-charge fund also grows by 5% to $924. Their sum is $1,869, which is not the correct check because the earlier fund was $880 at the end of year 1 and the new premium enters at the beginning of year 2: the fund immediately after the premium is $880+$900=$1,780, and after a 5% return it is $1,869. Subtracting the $104.50 charge gives $1,764.50, not the unit-based answer above. This reveals an inconsistency in the stated charge timing or the example arithmetic; therefore the direct unit calculation must be reconciled using the correct beginning-of-year timing. Starting year 2 units are 8 existing units plus 900/110 new units, but the $900 contribution is invested at unit price $110 and then both components earn −5%. Its value at the end is \(8(104.5)+(900/110)(104.5)=836+855=1{,}691\). Subtracting the $104.50 charge leaves $1,586.50. The cash check must instead start from the $880 at year 1 end, add $900 at the beginning of year 2, and apply the return: \((880+900)(0.95)=\$1,691\), then subtract $104.50 to obtain $1,586.50.
Common mistake avoided: cancelling a charge using the previous year's price, or applying the year 2 return to the existing units but not to the newly purchased units. Both the contribution and existing holdings are exposed to the full second-year return under the stated timing.
Why an Average Return Can Mislead
For two returns \(r_1\) and \(r_2\), the compound accumulation factor is \((1+r_1)(1+r_2)\). In general, that is not equal to the square of one plus the arithmetic average return. For example, with returns of 25% and −20%, the arithmetic average is 2.5%, but the varying path's factor is \(1.25(0.80)=1\), whereas a level 2.5% path has factor \(1.025^2=1.050625\).
This is a direct consequence of the price recurrence, not a separate adjustment to apply after projection. Use the actual period returns to build the path. If a problem asks for a level comparison, specify that it is a separate scenario using a constant return; do not replace the original path with its arithmetic average and describe the result as equivalent.
Worked Example: A Volatile Path Versus Its Arithmetic Average
A hypothetical fund starts at $10,000. Compare its terminal value after two years under (a) returns of +25% and −20%, in that order, and (b) a level return equal to the arithmetic average of those two returns. Assume no contributions, charges, or withdrawals.
Method. Compound the two specified returns for the volatile path, then separately calculate the level path from the arithmetic average.
For path (a), the first-year value is \(10{,}000(1.25)=\$12{,}500\). The second-year value is \(12{,}500(0.80)=\$10{,}000\). Equivalently, the two-year factor is \(1.25(0.80)=1.00\).
The arithmetic average return is \((25\%-20\%)/2=2.5\%\). For path (b), the first-year value is \(10{,}000(1.025)=\$10{,}250\), and the second-year value is \(10{,}250(1.025)=\$10{,}506.25\).
The level path ends $506.25 above the volatile path. Both calculations are correct for their stated scenarios; the level path is not the same investment experience as the volatile path. Values are rounded to the cent at the final step.
Check. The volatile path's gross factors multiply to 1.00, so its terminal value must equal its initial value.
Common mistake avoided: concluding that a 2.5% average return means the volatile path earns 2.5% each year. A constant-return scenario and a varying-return scenario are distinct paths.
Return Order and Intermediate Cash Flows
Although reordering a fixed collection of returns leaves the terminal unit price unchanged when there are no cash flows, it can change a policy fund if contributions or withdrawals occur along the path. The transaction takes place at an intermediate price, and that price depends on which returns have already occurred.
Worked Example: The Same Returns with a Midpoint Contribution
A hypothetical unit account starts with $1,000 invested at a unit price of $100, so it holds 10 units. A $500 contribution is invested at the beginning of the second year. Compare the terminal fund under returns of +20% then −10% with the path −10% then +20%. Assume the contribution is fully allocated and there are no charges.
Method. Project the unit price to the contribution date, purchase units at that date's price, and then apply the remaining return. This exposes the effect of contribution timing.
For the +20%, −10% path, the first-year price is \(100(1.20)=\$120\). The original 10 units are worth $1,200. The contribution buys \(500/120=25/6=4.166666\ldots\) units, giving \(10+25/6=85/6\) units. The terminal unit price is \(120(0.90)=\$108\), and the terminal fund is \(85/6 \times 108=\$1{,}530\).
For the −10%, +20% path, the first-year price is \(100(0.90)=\$90\). The contribution buys \(500/90=50/9=5.555555\ldots\) units, giving \(10+50/9=140/9\) units. The terminal unit price is \(90(1.20)=\$108\), and the terminal fund is \(140/9 \times 108=\$1{,}680\).
Both paths end with the same unit price because \(1.20(0.90)=0.90(1.20)=1.08\). Their policy funds differ by $150 because the contribution buys a different number of units at the two intermediate prices. Values are rounded to the cent where needed.
Check. In each path, the original 10 units are worth \(10(108)=\$1{,}080\) at the end. The contribution's terminal value is $450 in the first path and $600 in the second; adding these to $1,080 gives $1,530 and $1,680, respectively.
Common mistake avoided: assuming that equal terminal unit prices imply equal terminal policy funds. That conclusion ignores the units purchased at the intermediate contribution date.
Exam Checks for a Consistent Projection
A deterministic scenario is useful only if its dates, return units, and cash-flow sequence are internally consistent. Before accepting a result, verify that every return is applied to the intended period, each transaction uses the price on its transaction date, and the terminal fund equals terminal units multiplied by terminal price.
- Do not mix time units. Annual returns require annual updates unless an explicit conversion is supplied.
- Do not confuse gross premium and allocated premium. Only the amount allocated to units purchases units.
- Do not round unit counts early. Keep full precision through the projection and round money at the end.
- Do not treat an average as a path. A constant average-return scenario is a comparison scenario, not a replacement for period-by-period compounding.
- Do not ignore cash-flow timing. Reordered returns can produce different funds when transactions occur between periods.
These projections describe deterministic investment paths and resulting fund mechanics. They do not, by themselves, assign probabilities to scenarios or establish an actuarial valuation basis.
Check Your Understanding
Work each projection using the stated transaction timing and keep full precision until the final value.
- A unit price starts at $80. The annual returns are 5% and then −2%, with no transactions. Find the unit price after each year and at the end. (Answer: $84.00 after year 1 and $82.32 after year 2.)
- An account has 12 units at a price of $50. The price earns 8% over the period, then a $54 charge is deducted by cancelling units at the end of the period. Find the post-charge units and fund value. (Answer: 11 units and $594.00.)
- A fund of $4,000 experiences a 10% return followed by a −10% return, with no cash flows. Find the terminal fund and compare it with the initial amount. (Answer: $3,960.00, which is $40.00 below the initial amount.)
- A policy starts with 5 units at a price of $120. The first-period return is −10%. Immediately afterward, a fully allocated $240 contribution buys units at the current price. The next-period return is 15%. Find the terminal unit price, units, and fund value. (Answer: $124.20, \(65/9\) units, and $897.00.)
- A fund starts at $10,000 and has two annual returns of +30% and −20%, with no cash flows. Compare its terminal value with a level-return scenario using the arithmetic average of the two returns. (Answer: varying path $10,400.00; level 5% path $11,025.00.)