IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Non-Unit Reserves for Unit-Linked Contracts: Zeroisation

Non-unit reserves are amounts held from the non-unit fund so that future negative non-unit cash flows can be paid. Zeroisation works backwards from the end of the term. At each time, you set the reserve as the smallest non-negative amount that makes next year's profit zero if it would otherwise be negative. Where no strain arises, the reserve is zero.

What this chapter covers

Unit-linked contracts split into two parts. The unit fund holds the policyholder's invested premiums. The non-unit fund holds the insurer's own money: charges taken in, expenses and benefits paid out that the unit fund does not cover. This chapter is about the non-unit side.

In early years the non-unit cash flow is often negative. Initial expenses and commission are higher than the charges received. If the insurer does nothing, it must fund that strain from shareholder capital. A non-unit reserve is the amount set aside at the end of a year to cover later negative cash flows. Zeroisation is the method that finds the reserves by working backwards. At each time, you choose the reserve so that the profit in the next year is zero if it would otherwise be negative. You choose the smallest non-negative reserve that does this. Where the profit would already be non-negative with no reserve, the reserve is zero and the profit stays as it is.

This links to the rest of CM1 in three ways. It uses the same profit-testing framework as pricing and reserving: cash flows, survival probabilities, interest and the profit signature. It uses the reserving basis, so you must be clear about which interest rate, mortality and expenses apply. It also uses the profit vector and discounted profit measures such as the net present value at the risk discount rate. Learn the profit test first, then this chapter is a small, mechanical extension.

Pricing and reserving carries the largest syllabus weighting in CM1, and profit testing for unit-linked contracts is a favourite written question. Zeroisation is procedural, so careful students can collect most of the marks. The same short loop of working backwards, checking the sign and adjusting the profit vector appears repeatedly, and an error early in the calculation carries through every later line. Accuracy and a clear layout are worth the effort.

Non-unit reserves for unit-linked contracts (zeroisation): topics in the order to study them

1Unit-Linked Contracts and Non-Unit Cash FlowsYou must be able to build the non-unit cash flow for each year before you can decide whether any reserve is needed.
2Non-Unit Reserves and Negative Cash FlowsOnce you see where negative cash flows come from, the purpose of a reserve and its link to the profit vector become clear.
3Zeroisation Method Step by StepWith the purpose understood, you can learn the backward recursion and practise it on full numerical questions.
4Reserving Basis, Interest and Profit Signature EffectsThis last topic needs the method in hand: you see how the reserve basis and interest rates change the profit vector and the profit signature.

How to prepare Non-unit reserves for unit-linked contracts (zeroisation)

Treat this chapter as a short procedure to practise until it is automatic. Do a little each day, as you can on a phone for theory and on paper for sums.

Revise the profit test for a conventional contract first: cash flows, profit vector, profit signature, NPV. Zeroisation builds directly on it.
Write down the layout of a unit-linked profit test: unit fund roll-forward, then non-unit cash flow, then reserves, then profit vector.
Learn the zeroisation recursion in words first: the reserve at time t must cover the shortfall next year, after interest and survival, taking account of the reserve needed at t+1.
Do one full question slowly, working back from the last year. Check every reserve against the zero condition before moving on.
Recompute the profit vector and profit signature after reserves. Check that the reserve is released in later years and that the sums make sense.
Change one assumption at a time, such as the reserving interest rate or the mortality basis, and note how the reserves and NPV move.
Finish with timed past-paper questions, writing assumptions and each line of the layout as you would in the exam.

Common mistakes in Non-unit reserves for unit-linked contracts (zeroisation)

Working forward through the years instead of backward
Fix: Start at the last year and move back. Each reserve depends on the next one, so you need the later figure first.
Allowing negative reserves
Fix: Set the reserve to zero whenever the calculation gives a negative value, and then continue backwards.
Forgetting the survival probability when carrying reserves back
Fix: Apply the probability of being in force to the reserve needed next year, using the stated mortality basis.
Mixing up the unit fund and the non-unit fund
Fix: Keep separate lines in your layout. Only the non-unit fund enters the profit vector and the reserves.
Using the wrong interest rate for reserves or discounting
Fix: Label each rate before you start. Use the reserving basis for reserves and the risk discount rate only for the NPV.
Not checking the result after the reserves are set
Fix: Recompute the profit vector and check that it is non-negative where expected. Comment on the effect on the NPV in written answers.

Last-day revision: Non-unit reserves for unit-linked contracts (zeroisation)

The unit fund belongs to the policyholder. The non-unit fund holds the insurer's own cash flows.
Non-unit cash flow is charges in, less expenses and any extra benefit costs. It excludes interest on the non-unit fund. Interest is added later through the (1 + i) factor in the profit vector, so it is counted once only.
Negative non-unit cash flows usually arise early because of initial expenses and commission.
A non-unit reserve is set up to meet future negative cash flows, and it is held at the end of the year.
Zeroisation: work backwards from the end of the term. At each time, set the reserve so that the profit in the next year is zero if it would otherwise be negative. Use the smallest non-negative reserve that does this. Where the profit would already be non-negative with no reserve, the reserve is zero and the profit is left as it is.
Each reserve must allow for interest and for the probability of the policy being in force.
Where the next year's profit is already non-negative with no reserve, the reserve is zero, not negative.
The zeroisation recursion uses the reserving basis: the reserving interest rate and the reserving mortality. The profit vector calculation uses the earned interest rate and the pricing or experience basis. Keep the two bases apart.
Profit vector at time t = (reserve at t-1 + non-unit cash flow in year t)(1 + i) − p_{x+t-1} × reserve at t. The reserve at t-1 is brought forward and released. The reserve at t is carried forward. Here i is the earned interest rate, and the non-unit cash flow excludes interest. The reserve brought forward earns interest at this rate, not at the reserving rate.
Profit signature at time t = profit vector at time t × probability of being in force at the start of that policy year, that is, the survival probability from policy inception to the start of year t.
Reserves brought forward earn interest at the earned rate in the profit vector. If a lower reserving interest rate is used to calculate the reserves, the reserve amounts are higher, and profit tends to emerge later. Holding reserves reduces the NPV when the earned rate is below the risk discount rate. This is common but depends on the rates given in the question.
Always state the reserving basis and whether it differs from the pricing basis.

Non-unit reserves for unit-linked contracts (zeroisation) practice questions

Non-unit reserves for unit-linked contracts (zeroisation): frequently asked questions

What does zeroisation mean in unit-linked reserving?

It means choosing reserves, working backwards, so that the profit is zero in the years where it would otherwise be negative. Each reserve is the smallest non-negative amount that does this. Where the profit would already be non-negative with no reserve, the reserve is zero. It removes later strain without holding more capital than necessary.

Why are non-unit cash flows often negative at the start?

Initial expenses and commission are usually paid in the first year, while the charges from the policy are spread over many years. The early charges are not enough to cover those costs, so the non-unit fund is in deficit.

Does a reserve change the NPV of the policy?

Holding reserves delays profit, because money is tied up and earns only the earned rate. If the earned rate is below the risk discount rate, the discounted return on the reserves is less than their cost, so the NPV falls. If the earned rate is higher, the effect can reverse. The size and sign of the effect depend on the rates given in the question.

How should I present a zeroisation answer in the exam?

State your assumptions, then lay out the non-unit cash flows and work back year by year. Show the formula for each reserve and the calculated value. Finish with the profit vector and the profit signature, with a short comment.