Actuarial Mathematics for Modelling · Projecting expected future cashflows and profit testing

Reserves in Profit Testing and Zeroisation

Updated 11 October 2026 · Fact-checked

Reserves in a profit test are amounts held back from each year's cashflow to meet future liabilities. Setting them up costs money and lowers the profit vector. Zeroisation works backwards from the end of the term and holds just enough reserve at each time so that every later negative cashflow becomes exactly zero.

Understand Reserves in Profit Testing and Zeroisation

A profit test projects the cashflows a company expects from a policy. Before reserves, some years may show a profit and others a loss. But the company cannot spend all of an early profit. It must hold reserves to pay future claims and expenses, and sometimes solvency capital on top as a buffer.

Holding a reserve has a cost. At the end of each year you must set up the reserve for the policies still in force. You release the reserve you held at the start of the year, together with the interest it earned. So the reserve changes the profit vector in two ways: a release at the start of the year (with interest) and a set-up cost at the end. The profit vector after reserves is the cashflow that is truly free to be paid out as profit.

The set-up cost is paid per policy in force at the start of the year, so the end-of-year reserve is multiplied by the probability of surviving the year. This is why the term p × tV appears in the formulas. Solvency capital works the same way: it is set up, earns interest, and is released later. Its cost is the gap between the interest earned on it and the return the company demands on that capital, which is the risk discount rate.

Zeroisation deals with negative cashflows in the middle or end of a policy. This is typical for unit-linked contracts, where non-unit cashflows can turn negative later in the term. A negative future cashflow means the company would have to find money later. The company avoids this by setting up a reserve earlier, from the profits of earlier years. You choose each reserve to be just large enough that the next year's cashflow becomes exactly zero. You work backwards from the end of the term, because each reserve depends on the one after it.

Reserves are not allowed to be negative in this process. If a year's cashflow is already positive, you hold a reserve of zero at the start of that year. You only create a reserve where it is needed to remove a loss. Any negative cashflow at time 0 is the new business strain. Zeroisation does not remove it.

Key rules to remember

Profit vector for a traditional contract (year t, per policy in force at start of year): PRT_t = (t-1V + P_t − e_t)(1 + i) − q_{x+t-1}(S_t + E_t) − p_{x+t-1} × tV; Here t-1V is the reserve at the start of the year, P_t the premium, e_t the expenses at the start of the year, S_t the sum assured and E_t any extra death expense. Add other cashflows if the question has them. Check the timing of each item.
Cost of setting up the reserve (year t): Cost = p_{x+t-1} × tV − (t-1V)(1 + i); A positive cost lowers the profit. A negative value means the release is bigger than the cost, which raises the profit.
Profit signature: Π_t = t-1p_x × PRT_t, with Π_0 = PRT_0; Multiply by the probability that the policy is in force at the start of year t, so the result is per policy sold.
Zeroisation reserve (backwards recursion): t-1V = max(0, [p_{x+t-1} × tV − CF_t] ÷ (1 + i)), with nV = 0 at the end of the term; CF_t is the cashflow of year t before the effect of reserves, per policy in force at the start of the year, including interest on cashflows at the start of the year. i is the interest rate earned on the reserve.
Profit with solvency capital: Profit_t = PRT_t + (t-1SC)(1 + i_c) − p_{x+t-1} × tSC; SC is the solvency capital held per policy in force. i_c is the rate earned on it, which may differ from the reserve rate. Discount the result at the risk discount rate.

How to solve Reserves in Profit Testing and Zeroisation questions

Use this method for any question on reserves, solvency capital or zeroisation. Do the work in a table, one column per year.

1Write down the basis: interest rate on reserves, risk discount rate, mortality, expenses, and the reserve or capital rule in the question.
2List the cashflows of each year before reserves, per policy in force at the start of the year. Note the timing of each item and add interest where the item falls at the start of the year.
3For a given reserve basis, work out tV for each year. For zeroisation, set nV = 0 at the end of the term.
4For zeroisation, work backwards from the last year. For each t, compute (p × tV − CF_t) ÷ (1 + i). If the answer is negative, set t-1V = 0.
5Substitute the reserves into the profit formula to get the profit vector. Check that each zeroised year gives a profit of exactly zero.
6Multiply by the probability of being in force at the start of each year to get the profit signature.
7Discount the signature at the risk discount rate to get the NPV, or compute the other profit measure asked for. State clearly what you assumed about time 0.

Quickest way: Backward zeroisation in one table

When to use it: Use this when the question gives the cashflows before reserves and asks for the zeroised reserves and the new profit vector.

Make four rows: CF_t, p, reserve t-1V, and profit.
Put 0 for the reserve at the end of the final year.
For each year going backwards, compute the amount needed as p × tV − CF_t. If it is zero or negative, the reserve at the start of the year is 0.
If it is positive, divide it by (1 + i) to get the reserve at the start of the year.
The profit in a zeroised year must come out as zero. If it does not, you used the wrong timing or the wrong probability.
In all other years the profit is CF_t + (t-1V)(1 + i) − p × tV. Check this once for one year before filling the rest.

Common mistakes in Reserves in Profit Testing and Zeroisation

Working forwards instead of backwards when zeroising
Students follow the order of the time axis. But the reserve at t-1 depends on the reserve at t.
Fix: Always start from nV = 0 at the end of the term and move back one year at a time.
Forgetting the survival probability on the end-of-year reserve
Students treat the reserve like a cashflow per policy at the end of the year. But only the survivors need it.
Fix: Use p_{x+t-1} × tV for the set-up cost, per policy in force at the start of the year. Write the factor in the formula before putting in numbers.
Holding a negative reserve
The recursion gives a negative number when the cashflow is already positive, and the student keeps it.
Fix: If the result is negative, set the reserve to zero. Do not count on future profits to fund the present.
Thinking zeroisation removes the negative cashflow at time 0
Students apply the same idea to every negative value in the profit vector.
Fix: Zeroisation removes later negatives by holding reserves from earlier profits. The time-0 outflow is the new business strain and stays in the vector.
Discounting at the wrong rate
There are two rates: the rate earned on the reserve and the risk discount rate. Students mix them up.
Fix: Use the earned rate when working out reserves and the interest on them. Use the risk discount rate only when discounting the profit signature.
Applying the profit signature before zeroising or applying it twice
Students multiply by the in-force probability at the wrong stage.
Fix: Zeroise first on a per-policy-in-force basis. Then multiply the final profit vector by the in-force probabilities once.

Worked examples

Example 1

A 4-year contract has non-unit cashflows before reserves, per policy in force at the start of each year, of 40, 25, −30 and 20 at the ends of years 1 to 4. There is an initial outflow of 100 at time 0. The probability of surviving each year is 0.98. Reserves earn 5% a year. Zeroise the negative cashflow and find the profit vector.

Show the solution

Set 4V = 0 at the end of the term.
Year 4: the needed amount is 0.98 × 0 − 20 = −20, which is negative, so 3V = 0. The profit in year 4 is 20 + 0 − 0 = 20.
Year 3: the needed amount is 0.98 × 3V − CF_3 = 0 − (−30) = 30. So 2V = 30 ÷ 1.05 = 28.5714.
Year 2: the needed amount is 0.98 × 28.5714 − 25 = 28 − 25 = 3. So 1V = 3 ÷ 1.05 = 2.8571.
Year 1: the needed amount is 0.98 × 2.8571 − 40 = 2.8 − 40 = −37.2, which is negative, so 0V = 0.
Profit year 1: 40 + 0 − 0.98 × 2.8571 = 40 − 2.8 = 37.2.
Profit year 2: 25 + 1.05 × 2.8571 − 0.98 × 28.5714 = 25 + 3 − 28 = 0.
Profit year 3: −30 + 1.05 × 28.5714 − 0.98 × 0 = −30 + 30 = 0.

Answer: Reserves: 0V = 0, 1V = 2.8571, 2V = 28.5714, 3V = 0, 4V = 0. The profit vector is (−100, 37.2, 0, 0, 20).

Example 2

For a traditional contract, the annual premium is ₹1,000, paid at the start of each year. The expense at the start of each year is ₹100. The interest rate is 6% a year. The mortality rate is q = 0.01 in each year. The death benefit is ₹20,000, paid at the end of the year of death. The reserves per policy in force are 0V = 0, 1V = ₹500 and 2V = ₹1,200. Find the profit vector for years 1 and 2, the cost of setting up the reserve in year 1, and the profit signature for year 2.

Show the solution

Year 1, before reserves: (0 + 1,000 − 100) × 1.06 − 0.01 × 20,000 = 954 − 200 = ₹754.
Cost of the reserve at the end of year 1: 0.99 × 500 − 0 = ₹495.
PRT_1 = 754 − 495 = ₹259.
Year 2: (1V + P − e)(1.06) = (500 + 1,000 − 100) × 1.06 = 1,400 × 1.06 = ₹1,484.
Subtract expected death claims: 1,484 − 0.01 × 20,000 = 1,484 − 200 = ₹1,284.
Subtract the cost of the end-of-year reserve: 0.99 × 1,200 = ₹1,188. So PRT_2 = 1,284 − 1,188 = ₹96.
Profit signature for year 2: 1p_x × PRT_2 = 0.99 × 96 = ₹95.04.

Answer: PRT_1 = ₹259 and PRT_2 = ₹96. The cost of the year-1 reserve is ₹495. The year-2 profit signature is ₹95.04.

Exam tips

Always state your timing assumptions. Say whether the cashflow is at the start or the end of the year and whether the reserve is per policy in force at the start of the year.
Show the recursion for at least one year in full, then fill in the table. Examiners give method marks even if one number is off.
After zeroising, check that each year that was negative now shows a profit of exactly zero. This is the fastest way to catch an arithmetic slip.
In written answers, explain why reserves are needed and why a negative reserve is not held. Comment questions often ask for this.
In the computer-based paper, build the backward recursion as a single formula copied across the columns. Use MAX(0, …) so you do not hold negative reserves.

Practice questions from Projecting expected future cashflows and profit testing

Reserves in Profit Testing and Zeroisation: frequently asked questions

What is zeroisation in profit testing?

It is a method of removing negative cashflows in later years of a policy. You hold a reserve earlier, just big enough that the later cashflow becomes zero. You work backwards from the end of the term.

How do reserves affect the profit vector?

A reserve lowers the profit in the year it is set up and raises it in the year it is released, together with the interest it earned. The overall effect is to move profit from early years to later ones. It does not create profit.

Can a reserve be negative in a profit test?

In zeroisation you set the reserve to zero rather than hold a negative one. A negative reserve would mean counting on future profits, which is not prudent. Always check the basis given in the question.

Does zeroisation change the NPV?

Yes, usually. Reserves are held from profits in earlier years, and those profits are then released later. The NPV falls if the risk discount rate is higher than the interest earned on the reserve, which is the usual case.