Actuarial Mathematics for Modelling · Non-unit reserves for unit-linked contracts (zeroisation)

Zeroisation Method Step by Step for Unit-Linked Reserves

Updated 11 October 2026 · Fact-checked

Zeroisation sets non-unit reserves so that future profits are never negative. Start at the end of the contract with a reserve of zero. Work backwards, setting each reserve to the smallest amount that makes that year's profit zero where a reserve is needed. If that amount is negative, hold zero and accept the positive profit.

Understand Zeroisation Method Step by Step

In a unit-linked contract, the unit fund pays for most benefits. The insurer's own money sits in the non-unit fund. Each year the non-unit fund receives charges and pays expenses and any extra benefits. The result is the non-unit cash flow. Sometimes it is negative in later years, for example when expenses exceed charges.

If you do nothing, a negative cash flow in a later year pulls profit below zero. That means the insurer must find extra capital then. To avoid this, the insurer sets up a non-unit reserve earlier. The reserve is money held back from earlier profits to cover the later shortfall.

Zeroisation is the method for finding the smallest reserve that does the job. You choose reserves so that the profit in each year is zero where a reserve is needed, and never negative. Each reserve is the smallest amount that achieves this. You cannot start at time 0, because each reserve depends on the next one. So you work backwards from the end of the term.

The reserve is the amount that, with a year's interest, plus that year's cash flow, pays for the reserve needed at the end of the year for those still in force. If that amount is negative, no reserve is needed. You hold zero, and the profit that year is positive. You never hold a negative reserve.

Setting reserves costs profit in the years when they are built up. So early profits fall, and later profits are zero or positive. This is why the profit vector changes after zeroisation.

Key rules to remember

Profit in year t (end-of-year cash flows): Pr_t = (t-1)V × (1 + i) + CF_t − p_t × tV; Per policy in force at the start of year t. CF_t is the non-unit cash flow for year t, valued at the end of the year (interest on within-year flows included). i is the interest rate earned on the non-unit fund. p_t is the probability a policy in force at the start of year t is still in force at the end of it.
Zeroising reserve (backward recursion, end-of-year cash flows): (t-1)V = (p_t × tV − CF_t) ÷ (1 + i), if this is positive; Comes from setting Pr_t = 0 and rearranging. Use the value you have already found for tV. This is the smallest reserve that avoids a negative profit in year t.
Zero floor (end-of-year cash flows): (t-1)V = 0 if (p_t × tV − CF_t) ÷ (1 + i) ≤ 0; Never hold a negative reserve. Pr_t is then zero or positive and is calculated from the formula for profit.
Terminal reserve: nV = 0 at the end of the term n; This is the starting point of the recursion.
Profit in year t (start-of-year cash flows): Pr_t = ((t-1)V + CF_t) × (1 + i) − p_t × tV; Use this form when CF_t is the cash flow at the start of the year and has not yet been accumulated with interest. It is the main profit formula with CF_t accumulated for the year.
Zeroising reserve (start-of-year cash flows): (t-1)V = p_t × tV ÷ (1 + i) − CF_t, if this is positive; otherwise (t-1)V = 0; Comes from setting the start-of-year profit formula to zero. Use it, not the end-of-year recursion, when CF_t is a start-of-year flow.

How to solve Zeroisation Method Step by Step questions

This method works for any zeroisation question. Keep the cash flows per policy in force at the start of each year.

1Write out the non-unit cash flow CF_t for each year t = 1, 2, ..., n, the interest rate i and the in-force probabilities p_t. Check the timing of each cash flow.
2Set the reserve at the end of the term, nV = 0. If the question gives a reserve at the end, use that instead.
3Start with t = n. Compute X = (p_t × tV − CF_t) ÷ (1 + i). If the cash flows are at the start of the year, use X = p_t × tV ÷ (1 + i) − CF_t instead.
4If X > 0, set (t-1)V = X. If X ≤ 0, set (t-1)V = 0.
5Move to t = n−1 and repeat, using the reserve you have just found. Continue until you reach time 0.
6Compute the profit vector, using Pr_t = (t-1)V × (1 + i) + CF_t − p_t × tV for every year. For start-of-year cash flows, use Pr_t = ((t-1)V + CF_t) × (1 + i) − p_t × tV.
7Check: years where the reserve held at the start of the year is positive should give profit of exactly zero. Other years should give profit that is zero or positive.

Quickest way: Backward table with a zero check

When to use it: Use this under time pressure for any term of 3 to 6 years where you are given CF_t, i and p_t.

Draw a table with columns t, CF_t, p_t, tV, Pr_t.
Fill in nV = 0 first.
In each row going upwards, compute (p_t × tV − CF_t) ÷ (1 + i). If it is negative, write 0.
In the final column, check that each profit with a positive reserve in the previous year is exactly zero. A non-zero value signals an arithmetic slip.
Keep at least four decimal places in reserves, and round only the final answers.

Common mistakes in Zeroisation Method Step by Step

Working forwards from time 0 instead of backwards from the end.
Profit vectors and projections are normally done forwards, so students copy that habit.
Fix: The reserve at time t−1 needs the reserve at time t. Always start from nV = 0 and go back.
Holding a negative reserve when the formula gives a negative value.
Students apply the formula mechanically without checking the sign.
Fix: If (p_t × tV − CF_t) ÷ (1 + i) is negative, set the reserve to zero. The profit for that year is then positive.
Forgetting to multiply tV by p_t.
The reserve at the end of the year is held only for policies still in force, and students forget this.
Fix: Write p_t × tV in every line. Per policy in force at the start of the year, only a proportion p_t still need a reserve.
Applying interest to the reserve but not to the cash flow, or twice.
The timing of cash flows is unclear. Some questions give start-of-year flows and others give year-end values.
Fix: Decide first whether CF_t already includes interest. If not, accumulate it with (1 + i) before using it.
Zeroising in a year where the cash flow is positive and the next reserve is zero.
Students think every year needs a zero profit.
Fix: Zero profit is only needed where a reserve is required. If the formula gives a zero floor, leave the positive profit as it is.
Reporting profit as the cash flow without adjusting for the change in reserve.
Students stop after finding the reserves.
Fix: Always compute the full profit vector with the profit formula. Profit in early years is reduced by the reserve set up.

Worked examples

Example 1

A 5-year unit-linked contract has non-unit cash flows per policy in force at the start of each year, valued at the end of the year, of: year 1 ₹120, year 2 ₹80, year 3 −₹50, year 4 −₹30, year 5 ₹40. The probability of a policy staying in force over each year is 0.98. Reserves earn 5% a year. Using zeroisation, find the reserves at times 0 to 4 and the profit vector.

Show the solution

Set 5V = 0. We use CF_t and p_t = 0.98 for all t.
Year 5: (0.98 × 0 − 40) ÷ 1.05 = −38.10. This is negative, so 4V = 0.
Year 4: (0.98 × 0 − (−30)) ÷ 1.05 = 30 ÷ 1.05 = 28.5714. So 3V = ₹28.57.
Year 3: (0.98 × 28.5714 − (−50)) ÷ 1.05 = (28 + 50) ÷ 1.05 = 78 ÷ 1.05 = 74.2857. So 2V = ₹74.29.
Year 2: (0.98 × 74.2857 − 80) ÷ 1.05 = (72.8 − 80) ÷ 1.05 = −6.857. This is negative, so 1V = 0.
Year 1: (0.98 × 0 − 120) ÷ 1.05 is negative, so 0V = 0.
Profit vector: Pr_1 = 0 + 120 − 0.98 × 0 = 120. Pr_2 = 0 × 1.05 + 80 − 0.98 × 74.2857 = 80 − 72.8 = 7.2. Pr_3 = 74.2857 × 1.05 − 50 − 0.98 × 28.5714 = 78 − 50 − 28 = 0. Pr_4 = 28.5714 × 1.05 − 30 − 0 = 30 − 30 = 0. Pr_5 = 0 + 40 − 0 = 40.

Answer: Reserves: 0V = 0, 1V = 0, 2V = ₹74.29, 3V = ₹28.57, 4V = 0. Profit vector: (120, 7.2, 0, 0, 40). The year 2 profit is positive because the reserve at time 1 is floored at zero.

Example 2

A 4-year contract has non-unit cash flows per policy in force at the start of the year, valued at the end of the year, of: year 1 ₹45, year 2 −₹20, year 3 −₹10, year 4 ₹15. The in-force probabilities for years 1 to 4 are 0.99, 0.98, 0.97 and 0.96. Interest is 6% a year. Find the zeroised reserves and the profit in year 1.

Show the solution

Set 4V = 0.
Year 4: (0.96 × 0 − 15) ÷ 1.06 is negative, so 3V = 0.
Year 3: (0.97 × 0 − (−10)) ÷ 1.06 = 10 ÷ 1.06 = 9.4340. So 2V = 9.4340.
Year 2: (0.98 × 9.4340 + 20) ÷ 1.06 = (9.2453 + 20) ÷ 1.06 = 29.2453 ÷ 1.06 = 27.5899. So 1V = 27.5899.
Year 1: (0.99 × 27.5899 − 45) ÷ 1.06 = (27.3140 − 45) ÷ 1.06, which is negative. So 0V = 0.
Profit in year 1: Pr_1 = 0 × 1.06 + 45 − 0.99 × 27.5899 = 45 − 27.3140 = 17.686.
Check years 2 and 3: Pr_2 = 27.5899 × 1.06 − 20 − 0.98 × 9.4340 = 29.2453 − 20 − 9.2453 = 0. Pr_3 = 9.4340 × 1.06 − 10 − 0 = 0. Pr_4 = 0 + 15 − 0 = 15.

Answer: Reserves: 0V = 0, 1V = ₹27.59, 2V = ₹9.43, 3V = 0. The year 1 profit is ₹17.69. The profit vector is (17.69, 0, 0, 15).

Exam tips

Show the recursion formula once, then a table of values. Examiners give marks for method even if a figure is wrong.
Check the timing of cash flows in the question before you start. If a flow is at the start of the year, use the start-of-year profit formula and recursion, which accumulate the flow with interest.
After finding reserves, recompute the profit in at least one year with a positive reserve. It should come out as zero. This catches errors fast.
State assumptions clearly, such as reserves earning the interest rate stated and the in-force probabilities used. In a computer-based paper, show the formula in a cell or code comment and check the final reserve is zero.
If a question asks for the effect on the profit signature, or on the NPV or IRR based on it, finish the zeroised profit vector first. Then multiply each year's profit by the probability that a policy is in force at the start of that year, measured from time 0, to get the figures per policy issued.

Practice questions from Non-unit reserves for unit-linked contracts (zeroisation)

Zeroisation Method Step by Step: frequently asked questions

Why do we start zeroisation at the end of the contract?

The reserve needed at the start of a year depends on the reserve needed at its end. Only the final reserve is known without any calculation, because it is zero at the end of the term. So you have to work backwards.

What if the reserve formula gives a negative answer?

Set the reserve to zero. A negative reserve would mean you are counting on future profits to fund today's position, which is not prudent. The profit in that year then comes out positive.

Does zeroisation reduce total profit?

It moves profit from early years to later ones, because reserves are set aside earlier and released when the negative cash flow arrives. Because the reserve earns interest at the reserve rate, the effect on discounted profit depends on how this compares with the risk discount rate.

Is the reserve per policy in force at the start or end of the year?

A reserve (t-1)V is held per policy in force at time t−1. In the profit formula the end-of-year reserve is multiplied by p_t, because only the policies still in force need it.