Non-Unit Reserves and Negative Cash Flows in Unit-Linked Contracts

Understand Non-Unit Reserves and Negative Cash Flows

Key rules to remember

Non-unit cash flow in year t

NUCF_t = charges received − expenses − extra benefit costs + interest on the non-unit money held

Build it from your table: charges less expenses, add interest, then deduct expected death and surrender costs in excess of the unit fund. Check the contract's timing.

Zeroisation condition (cash flow at start of year)

(tV + CF_{t+1}) × (1 + i) − p_{x+t} × t+1V ≥ 0

tV is the reserve held at the start of year t+1, per policy in force at that time. CF_{t+1} is the non-unit cash flow at the start of year t+1, per policy in force at the start of the year. Reserves are set at the smallest values, never below zero, that satisfy this condition. It holds with equality whenever tV is positive. If tV is zero, it can hold with a strict inequality, and the excess is profit.

Zeroisation condition (cash flow at end of year)

tV × (1 + i) + CF_{t+1} − p_{x+t} × t+1V ≥ 0

Use this when CF_{t+1} arises at the end of the year. tV is held at the start of the year, per policy in force at that time. The worked examples on this page use this end-of-year form.

Backward recursion for the reserve (cash flow at start of year)

tV = max(0, p_{x+t} × t+1V ÷ (1 + i) − CF_{t+1})

Use this only when CF_{t+1} is at the start of the year and is per policy in force at that time. Here i is the reserving interest rate and t+1V is the reserve needed at the end of the year. The cash flow already includes the expected cost of decrements. Read the question's timing first.

Backward recursion for the reserve (cash flow at end of year)

tV = max(0, (p_{x+t} × t+1V − CF_{t+1}) ÷ (1 + i))

Use this when CF_{t+1} arises at the end of the year, per policy in force at the start of the year. This is the form used in the worked examples. The reserve is floored at zero. A positive cash flow in a later year cannot reduce a reserve needed earlier. It only reduces the reserve needed at the start of its own year.

Reserve-adjusted cash flow (cash flow at start of year)

Adjusted CF_{t+1} = (tV + CF_{t+1}) × (1 + i) − p_{x+t} × t+1V

Use this for cash flows at the start of the year. Zeroisation makes the adjusted cash flow zero whenever tV is positive. Otherwise it is zero or positive.

Reserve-adjusted cash flow (cash flow at end of year)

Adjusted CF_{t+1} = tV × (1 + i) + CF_{t+1} − p_{x+t} × t+1V

Use this for cash flows at the end of the year. Zeroisation makes the adjusted cash flow zero whenever tV is positive. Otherwise it is zero or positive.

How to solve Non-Unit Reserves and Negative Cash Flows questions

1. 1Write down the non-unit cash flows for each future year, with the timing used in the question: start of year, end of year or mid-year.
2. 2Note the reserving interest rate and the survival probabilities for each year. Check whether the cash flows are already per policy in force or per policy at the start.
3. 3Find the last year with a negative cash flow. Years after the last negative cash flow need no reserve.
4. 4Start at the end of the term. Set the reserve at the end of the final year to zero.
5. 5Work back one year. Reserve at start of year t = (p × reserve at end of year − cash flow at the right date, brought to the same date) ÷ (1 + i), using the question's timing. If the result is negative, set the reserve to zero.
6. 6Repeat for each earlier year until you reach time 0.
7. 7Check by working forward: the reserve-adjusted cash flow must be zero or positive each year, and zero in the years that needed a reserve.
8. 8Form the profit vector from the reserve-adjusted cash flows, then multiply each element by the probability of being in force at the start of that year (from policy inception) to get the profit signature.

Quickest way: Backward table method

Common mistakes in Non-Unit Reserves and Negative Cash Flows

• Working forwards from time 0.

  Valuation questions usually look forward from a date, so students apply the same habit.

  Fix: Start from the last year, where the end reserve is zero, and work back.

• Holding a negative reserve in a year when later cash flows are positive.

  Students see a positive future and let it offset the present.

  Fix: Floor each reserve at zero. A negative reserve would be borrowing against future profit, which zeroisation does not allow.

• Discounting at the wrong date or using the wrong interest rate.

  The reserving rate can differ from the rate in the profit test, and the cash flows may fall at different points in the year.

  Fix: Mark the timing of each cash flow on a time line and use the reserving rate given for the reserve calculation.

• Forgetting to multiply by the survival probability.

  Students treat the reserve as certain, not as held only for survivors.

  Fix: The reserve at the end of year t is held only for policies still in force, so weight it by p_{x+t}, unless the cash flows are already per policy in force.

• Reserving for the whole of future negative cash flows rather than only the strain.

  Students forget that positive cash flows in the same year already cover part of the strain.

  Fix: Reserve only the shortfall after that year's own cash flow and interest.

• Confusing unit reserves with non-unit reserves.

  Both appear in the same contract and both are reserves.

  Fix: The unit reserve equals the unit fund and backs the policyholder's units. The non-unit reserve is the insurer's extra money and is the only one found by zeroisation.

Worked examples