Unit-Linked and Multiple-State Profit Tests Explained Step by Step

Understand Unit-Linked and Multiple-State Profit Tests

Key rules to remember

Unit fund roll-forward

UF(t) = (UF(t−1) + A(t)) × (1 + i) − MC(t)

A(t) is the amount of the premium allocated to units. It is the premium P(t) less the part not allocated, such as the allocation charge and any bid-offer spread. MC(t) is the management charge, here taken at year end on the grown fund. UF is per policy in force, so decrements are not applied here. They enter later through the probabilities. Follow the question's timing for charges and use the order of events given.

Unit-linked non-unit cashflow

NUCF(t) = (P(t) − A(t) − E(t)) × (1 + i) + MC(t) − q_{x+t−1} × (D(t) − F(t))

P(t) − A(t) is the unallocated premium, including any bid-offer spread. E(t) is the expense at the start of the year. Interest is earned on the unallocated premium less expenses for the year. MC(t) is taken at year end here, so it earns no interest in the year. D(t) is the death benefit and F(t) is the fund value paid on death, so D(t) − F(t) is the extra cost the insurer funds. It is weighted by q_{x+t−1}, the probability of dying in the year. NUCF is per policy in force at the start of year t. NUCF excludes interest on the opening reserve. That goes into the profit vector. Always build from the pieces; do not memorise one string.

Profit vector with reserves

Profit(t) = NUCF(t) + {}_{t−1}V × (1 + i) − p_{x+t−1} × {}_tV

This is per policy in force at the start of year t. {}_{t−1}V is the opening reserve and {}_tV is the closing reserve. p_{x+t−1} is the probability of surviving the year, because the closing reserve is held only for survivors. The extra death cost in NUCF is already weighted by q_{x+t−1}, so do not weight it again. Check the exact form against the timing of the reserve in your question.

Profit signature

Π(t) = Profit(t) × {}_{t−1}p_x

Use the probability of being in force at the start of year t. Weights each year's profit per policy issued.

Net present value

NPV = Σ Π(t) × v(t) at the risk discount rate

Use the risk discount rate, not the investment return, unless told otherwise.

Profit margin

Profit margin = NPV ÷ PV of premiums

The PV of premiums uses the same risk discount rate and the same in-force probabilities.

Multiple-state expected cashflow

E[CF(t)] = Σ over states j of {}_{t−1}p(0→j) × CF_j(t)

Use the probability of being in state j at the start of the year, then apply the state's cashflow. Transition benefits are weighted by the transition probability.

How to solve Unit-Linked and Multiple-State Profit Tests questions

1. 1Read the basis: allocation percentages, charges, bid-offer spread, fund growth, expense inflation, interest on the non-unit fund, mortality or transition rates and the risk discount rate.
2. 2Project the unit fund year by year, following the stated order of events: premium in, allocation, growth, management charge out.
3. 3Project the non-unit cashflow: unallocated premium, plus charges, minus expenses, with interest on the start-of-year amounts, minus any extra cost of death, surrender or other benefits above the fund.
4. 4Decide the reserves. If negative non-unit cashflows occur, set the reserve by zeroisation, working backwards from the latest year.
5. 5Form the profit vector per policy in force: cashflow plus interest on the opening reserve, minus the cost of setting up the closing reserve, using the survival or transition probabilities.
6. 6Convert to the profit signature by multiplying by the probability of being in force at the start of each year. For multiple-state policies, use the probability of being in each state and weight state cashflows.
7. 7Discount at the risk discount rate to get the NPV. Compute the profit margin or IRR if asked.
8. 8State the result, check the sign and size, and comment if the question asks for it.

Quickest way: Table method with a fixed row order

Common mistakes in Unit-Linked and Multiple-State Profit Tests

• Charging the whole death benefit to the non-unit account.

  Students forget that the unit fund is paid out on death.

  Fix: Only the excess of the death benefit over the fund value is a non-unit cost. The fund value itself comes from the unit account.

• Ignoring or misordering timing of charges and interest.

  Questions state when charges are taken, and students apply a default order.

  Fix: Follow the order of events stated. Apply interest only to amounts held at the start of the year, unless the question says otherwise.

• Using the investment return as the discount rate.

  Both are interest rates in the basis.

  Fix: Discount profit signatures at the risk discount rate. Use the investment return only for fund growth and interest on reserves.

• Multiplying by the wrong in-force probability.

  The profit vector is per policy in force, and the signature is per policy issued.

  Fix: Multiply by the probability of being in force at the start of year t. Use that probability consistently when you compute PV of premiums.

• Forgetting reserves or setting them with the wrong sign.

  Negative non-unit cashflows look like losses and are left alone.

  Fix: Use zeroisation to set reserves, work backwards, and remember a reserve is released when the following year's cashflow is positive.

• In a multiple-state test, using total survival probability instead of state probabilities.

  Students carry over the single-decrement setup.

  Fix: Use the probability of being in each state at the start of the year, then apply that state's premium, benefit and expense.

Worked examples