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

Reserving Basis, Interest and Profit Signature Effects in Unit-Linked Contracts

Updated 11 October 2026 · Fact-checked

Non-unit reserves are set aside from emerging profit to cover future negative non-unit cash flows. They change the timing of profit, not the basic cash flows. Reserves delay profit, reduce NPV when reserve interest is below the risk discount rate, and are released when policies exit. Compute the profit vector, then the signature, then the NPV.

Understand Reserving Basis, Interest and Profit Signature Effects

A unit-linked contract has a unit fund and a non-unit fund. The non-unit cash flow is what is left for the insurer after charges are taken in and expenses and extra benefits are paid out. It is often negative at the start because of the initial expenses. Later it may also turn negative, for example when a guarantee is costly or charges are low.

Timing convention used on this page: CF_0 is at time 0, and CF_t is the cash flow at the end of year t, per policy in force at the start of that year. Reserves are held at the year ends.

If you pay out all of a positive cash flow as profit, a later negative one would need new money from shareholders. A non-unit reserve avoids this. It holds back part of the earlier cash flow so that later negative cash flows can be met. Zeroisation is the method that sets the smallest reserve needed so that no future profit is negative. Reserves are never negative, so any year that would need a negative reserve is set to zero.

The profit vector is the expected profit at each year end per policy in force at the start of that year. It is interest on the opening reserve, plus the non-unit cash flow at the year end, minus the reserve that must be set up at year end for the policies that remain. The profit signature weights each profit vector item by the probability that the policy is still in force at the start of the year. It is the profit per policy issued. The NPV is the signature discounted at the risk discount rate.

The reserving basis may differ from the profit-test basis. The profit test uses best-estimate assumptions to price. The reserving basis is usually more prudent and sets the reserve interest rate, mortality and withdrawals. Prudence depends on the sign of the future cash flows. If future cash flows are positive, higher lapses lose that income, so higher lapses are prudent. If future cash flows are negative, higher lapses remove future costs, so lower lapses are prudent.

Reserves affect profit in three ways. They move profit later, because money is held back first and released later. They earn interest, usually below the risk discount rate, so holding them lowers the NPV. And policies that surrender, lapse or die leave the in-force group, so the reserve held for them is released into profit. The cost of any surrender value or death benefit is already in the non-unit cash flow, so do not count it twice.

Key rules to remember

Profit vector (year t): Pr_t = _{t-1}V × (1 + i) + CF_t − p_{x+t-1}^{(τ)} × _tV; Per policy in force at the start of year t. CF_t is the non-unit cash flow at the end of year t, already allowing for expected death and surrender costs. i is the investment return in the profit test. p^(τ) is the probability of staying in force over the year. Use _tV for survivors only.
Time 0 profit: Pr_0 = CF_0 − _0V; Per policy issued. The reserve at time 0 is often zero under zeroisation, so Pr_0 is often the initial strain.
Profit signature: Π_0 = Pr_0 ; Π_t = _{t-1}p^{(τ)} × Pr_t; Per policy issued. _{t-1}p^{(τ)} is the probability of being in force at the start of year t. Deaths and withdrawals are both included.
Net present value: NPV = Σ Π_t × (1 + r)^(−t); r is the risk discount rate. Use the signature, not the profit vector, unless the question says otherwise.
Zeroisation recursion: _{t-1}V = max(0, (p_{x+t-1}^{(τ)} × _tV − CF_t) ÷ (1 + i_r)); CF_t is at the end of year t. Work backwards from _nV = 0 at the end of the term. i_r is the reserve interest rate. This makes Pr_t ≥ 0 in every year when i_r equals the investment return i, and Pr_t = 0 where a reserve is needed.
In-force probability with two decrements: p^{(τ)} = 1 − q^d − q^w; Use this when q^d and q^w are the probabilities that a life leaves by death and by withdrawal over the year, each measured in the presence of the other decrement (multiple-decrement rates), with exits at the year end. If you are given independent rates for each cause instead, use p^(τ) = (1 − q^d)(1 − q^w) under independence. Follow the question's decrement assumption.

How to solve Reserving Basis, Interest and Profit Signature Effects questions

Use this order for any question on reserves, the profit vector, the signature or the NPV of a unit-linked contract.

1Write down the non-unit cash flows CF_0, CF_1, ... per policy in force at the start of each year. State that CF_t falls at the end of year t. Check they already include the cost of death and surrender benefits.
2Write the in-force probabilities for each year, using deaths and withdrawals. Also note the investment return i, the reserve interest rate i_r and the risk discount rate r.
3Find the reserves. For zeroisation, start with _nV = 0 and work backwards using _{t-1}V = max(0, (p × _tV − CF_t) ÷ (1 + i_r)). If reserves are given, use them.
4Compute the profit vector year by year with Pr_t = _{t-1}V × (1 + i) + CF_t − p × _tV. Check that no entry is negative after time 0 when zeroised.
5Compute the in-force probabilities at the start of each year and build the profit signature Π_t = _{t-1}p × Pr_t.
6Discount the signature at the risk discount rate to get the NPV. Add other measures only if asked.
7For 'effect of' questions, give the direction of the change and the reason: timing, interest on reserves, or release of reserve on exit. Include a number if the data allow.
8Check the result. The reserve should release as policies leave, and a reserve at interest below r should reduce NPV.

Quickest way: Backward reserve, forward profit

When to use it: Use this for timed questions where you must find zeroised reserves, the profit vector and the NPV for a short-term contract.

Make one table with columns for t, CF_t, p, _{t-1}V and _tV.
Fill the reserve column backwards, row by row, from the end of term. Floor each entry at zero.
Then compute Pr_t from the table in one pass. Do not recompute the reserve.
Multiply by the cumulative in-force probability to get the signature.
Discount only the nonzero items. Zero entries need no working.
If the question asks for the effect of a change, recompute only the rows that change.

Common mistakes in Reserving Basis, Interest and Profit Signature Effects

Holding the reserve for all policies at the end of the year instead of only those still in force.
Students forget that the reserve is per policy in force, and that policies leaving release their reserves.
Fix: Always multiply _tV by the probability of staying in force, p, in the profit vector.
Counting the surrender or death cost twice, in the cash flow and again as a reserve release or extra deduction.
The decrements appear in two places and students apply them to both.
Fix: The expected cost of death and surrender is already in CF_t. Use p only for the reserve and the in-force probabilities.
Using the profit vector instead of the profit signature to find NPV.
Both are rows of profits and the names are similar.
Fix: The vector is per policy in force at the start of the year. Multiply by the start-of-year in-force probability and discount the signature.
Allowing a negative reserve, or forgetting to floor it at zero.
The backward recursion gives a negative number when the cash flow is large and positive.
Fix: Take max(0, ...) in every step. A negative result means no reserve is needed that year.
Assuming higher lapses are always prudent for the reserving basis.
Students memorise a rule from non-linked contracts.
Fix: Check the sign of the future non-unit cash flows. Higher lapses are prudent when future cash flows are positive and imprudent when they are negative.
Saying reserves always reduce total profit.
Reserves reduce early profit, so students extend this to all profit.
Fix: Reserves mainly change timing. The undiscounted total can change slightly through interest and release on exit. The NPV falls when the reserve earns less than the risk discount rate.

Worked examples

Example 1

A 3-year unit-linked contract has non-unit cash flows per policy in force at the start of each year: CF_0 = −20 at time 0, then CF_1 = 50, CF_2 = −40, CF_3 = 60, where CF_t is received at the end of year t. The probability of staying in force over each year is 0.9. The investment return and the reserve interest rate are both 4% a year. The risk discount rate is 10% a year. Find (a) the zeroised reserves, (b) the profit vector, (c) the profit signature, and (d) the NPV. Then (e) show, as an illustration only, what happens to the NPV if no reserves are held.

Show the solution

Reserves use _{t-1}V = max(0, (0.9 × _tV − CF_t) ÷ 1.04). _3V = 0. _2V = max(0, (0.9 × 0 − 60) ÷ 1.04) = max(0, −57.69) = 0.
_1V = max(0, (0.9 × 0 − (−40)) ÷ 1.04) = 40 ÷ 1.04 = 38.462.
_0V = max(0, (0.9 × 38.462 − 50) ÷ 1.04) = max(0, (34.615 − 50) ÷ 1.04) = max(0, −14.79) = 0.
Profit vector: Pr_0 = −20 − 0 = −20.
Pr_1 = 0 × 1.04 + 50 − 0.9 × 38.462 = 50 − 34.615 = 15.385.
Pr_2 = 38.462 × 1.04 + (−40) − 0.9 × 0 = 40 − 40 = 0.
Pr_3 = 0 × 1.04 + 60 − 0 = 60.
Signature: Π_0 = −20. Π_1 = 1 × 15.385 = 15.385. Π_2 = 0.9 × 0 = 0. Π_3 = 0.81 × 60 = 48.6.
NPV = −20 + 15.385 ÷ 1.1 + 0 + 48.6 ÷ 1.331 = −20 + 13.986 + 36.514 = 30.50.
Illustration with no reserves: Pr_1 = 50, Pr_2 = −40, Pr_3 = 60. Signature: −20, 50, −36, 48.6.
NPV = −20 + 50 ÷ 1.1 − 36 ÷ 1.21 + 48.6 ÷ 1.331 = −20 + 45.455 − 29.752 + 36.514 = 32.22.
This no-reserve case is not a valid benchmark. Its year 2 profit of −36 is negative, so shareholders would have to put in new capital. Zeroisation removes that, and the NPV of 30.50 is the figure to use.
The zeroised NPV is about 1.72 lower than the illustration. The reserve of 38.46 earns 4% while the risk discount rate is 10%, and it delays some profit.

Answer: Reserves: _0V = 0, _1V ≈ 38.46, _2V = 0, _3V = 0. Profit vector: −20, 15.385, 0, 60. Signature: −20, 15.385, 0, 48.6. NPV ≈ 30.50. The no-reserve illustration gives ≈ 32.22 but has a negative profit of −36 in year 2, so it is not a valid comparison.

Example 2

In one year of a unit-linked contract, per policy in force at the start of the year, the opening non-unit reserve is 100, the non-unit cash flow received at the end of the year is 15, the closing reserve required is 120, and investment earns 5%. The probability of staying in force over the year is 0.95. (a) Find the profit vector item. (b) Find it if the probability of staying in force is 0.90, with the cash flow and reserves unchanged. (c) If 0.8 of policies issued are in force at the start of the year, find the signature item in each case and comment.

Show the solution

(a) Pr = 100 × 1.05 + 15 − 0.95 × 120 = 105 + 15 − 114 = 6.
(b) Pr = 105 + 15 − 0.90 × 120 = 120 − 108 = 12.
The extra 6 comes from the reserve released for 5% more of the policies that leave: 0.05 × 120 = 6.
(c) With 0.8 in force at the start, the signature item is 0.8 × 6 = 4.80 in case (a) and 0.8 × 12 = 9.60 in case (b).
Comment: higher exits release more reserve and raise this year's profit. This holds only because the cash flow is unchanged. In practice higher lapses also change the cash flow and remove future charges, so the whole projection must be re-run.

Answer: (a) 6. (b) 12. (c) Signature items are 4.80 and 9.60. Higher exits release more reserve in this year, but the full effect needs the whole projection to be recalculated.

Exam tips

Show a table with CF, p, opening and closing reserve and profit vector. Marks are given for method even when the arithmetic slips.
State your assumptions: the decrement timing, the in-force probabilities, and the interest rates for reserves and for investment.
For comment questions, say what changes (timing, interest, release on exit) and in which direction. Then give a number if you can.
Do not assume the reserving basis equals the pricing basis. Read which basis the question uses for the reserve and which for the profit test.
In MCQs, check whether the answer needs the vector, the signature or the NPV. The wrong one is usually among the options.

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

Reserving Basis, Interest and Profit Signature Effects: frequently asked questions

How do non-unit reserves affect the profit signature?

They move profit later. Early profit is held back to cover later negative cash flows, and it is released in later years. The signature shows zero or lower values in the years when reserves build up and higher values when they release.

Why does a reserve usually reduce the NPV?

The reserve earns interest at the investment or reserve rate, which is usually below the risk discount rate. The shareholder therefore gets money back later and at a lower rate than they require. The NPV falls as a result.

What is a prudent lapse assumption for the reserving basis?

It depends on the future cash flows. If they are positive, lapses lose future income, so higher lapses are prudent. If they are negative, lapses remove future costs, so lower lapses are prudent.

What is the difference between the profit vector and the profit signature?

The profit vector is the profit per policy in force at the start of each year. The signature is the profit per policy issued, found by multiplying the vector by the probability of being in force at the start of the year. Use the signature for the NPV.