Non-unit reserves for unit-linked contracts (zeroisation): formula sheet

Key formulas

Allocated premium

A_t = α_t × P_t × (1 − s)

α_t is the allocation percentage, P_t the premium, s the bid-offer spread as a proportion of the offer price (s = 0 if there is none). Check how the question defines the spread.

Unit fund roll-forward (charge at start of year)

U_t = (U_(t−1) + A_t) × (1 − m) × (1 + g)

m is the management charge as a proportion of the fund after allocation, g is the net unit fund growth rate. This is the fund per policy in force at start of year t, before deaths and withdrawals.

Management charge

AMC_t = m × (U_(t−1) + A_t)

Charged at the start of the year in this layout. Use the timing stated in the question.

Non-unit cash flow at start of year

P_t − A_t − e_t + AMC_t (+ fixed fee if any)

e_t is the expense at start of year t. The unallocated premium and charges come in. Expenses go out.

Non-unit cash flow at end of year, per policy in force at start

NUCF_t = (P_t − A_t − e_t + AMC_t) × (1 + j) − q_(x+t−1) × max(0, S − U_t)

j is the non-unit (reserve or earned) interest rate. q is the death probability in the year. S is the guaranteed death benefit. Add other items such as surrender strain if the question gives them.

Expected cash flow per policy in force at time 0

Expected NUCF_t = (t−1)p_x × NUCF_t

Weight by the probability of being in force at the start of year t. Include lapses in the survival probability if given.

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.

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.

Quick revision

• 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.

Common mistakes

• Treating the whole premium as income to the insurer. Fix: Split the premium first. Only the unallocated part and the charges reach the non-unit fund.
• Deducting the full death benefit in the non-unit cash flow. Fix: The unit fund pays its own value. The non-unit fund pays only the excess of the guarantee over the fund value, and zero if the fund is higher.
• Working forwards from time 0 instead of backwards from the end. 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. Fix: If (p_t × tV − CF_t) ÷ (1 + i) is negative, set the reserve to zero. The profit for that year is then positive.
• Holding the reserve for all policies at the end of the year instead of only those still in force. 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. Fix: The expected cost of death and surrender is already in CF_t. Use p only for the reserve and the in-force probabilities.

Exam tips

• Write the timing of every item (start or end of year) next to it before you calculate. Many marks are lost on timing.
• Always state whether your cash flow is per policy in force at the start of the year or an expected value from time 0.
• Show the unit fund projection even if the question asks only for the non-unit flow. The death cost depends on it, and method marks are given for it.
• In MCQs, check the sign and the guarantee condition first. Often the fund is above the guarantee, so the extra death cost is zero.
• If you are asked to comment, say what a negative flow means and link it to the need for non-unit reserves.
• 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.