Projecting Cashflows with Multiple Decrements

Understand Projecting Cashflows with Multiple Decrements

Key rules to remember

Number in force after one year

(al)_{x+1} = (al)_x − Σ_j (ad)^j_x

Sum over all causes j. Use the same year's decrements for all causes.

Decrements by cause

(ad)^j_x = (al)_x × (aq)^j_x

Use the dependent (all causes acting) rate (aq)^j_x. This gives the expected number leaving by cause j between age x and x+1.

Rate from table

(aq)^j_x = (ad)^j_x ÷ (al)_x

Use this when the question gives a multiple decrement table instead of rates.

Probability of staying in force one year

(ap)^τ_x = 1 − Σ_j (aq)^j_x = (al)_{x+1} ÷ (al)_x

τ means all causes combined.

Probability in force t years ahead

t(ap)^τ_x = (al)_{x+t} ÷ (al)_x = (ap)^τ_x × (ap)^τ_{x+1} × … × (ap)^τ_{x+t−1}

Multiply the one-year survival probabilities of consecutive ages.

Expected cashflow for cause j in year t+1 (single policy)

t(ap)^τ_x × (aq)^j_{x+t} × B_j

B_j is the benefit payable on exit by cause j. Payment is at the time stated, for example the end of the year of exit.

Expected present value

EPV = Σ_t E(CF_t) × v^t

Use v = 1 ÷ (1 + i). If exits happen mid-year, discount from the actual payment time, using the stated timing.

How to solve Projecting Cashflows with Multiple Decrements questions

1. 1Write down the starting group size, or probability 1 for a single policy, and the age or time at which you start.
2. 2List every cause of decrement and the rate for each, for each age. Note which rates are dependent (aq)^j_x. If the question gives independent rates, convert first as the question requires.
3. 3For each year, compute exits by cause: starting number × (aq)^j_x. Subtract all of them to get the starting number for the next year.
4. 4Attach the benefit to each cause. Check the payment timing: start of year, end of year, or on exit. Also work out in-force cashflows such as premiums or salaries, based on the number in force at the right date.
5. 5Form the expected cashflow at each time point by adding all inflows and outflows due then. Keep the sign convention clear.
6. 6If asked, discount each cashflow with v^t from the payment date and add them to get the EPV.
7. 7Check: exits plus survivors equal the starting number each year, and survivors never go up.

Quickest way: Year-by-year table

Common mistakes in Projecting Cashflows with Multiple Decrements

• Applying the rate to the original group every year instead of the survivors.

  It feels simpler to multiply the initial number by each year's rate.

  Fix: Always use the number in force at the start of that year. Update it every row.

• Paying a benefit for every life who leaves, ignoring which cause it was.

  Students total all exits and use one benefit amount.

  Fix: Compute exits separately by cause and multiply each by its own benefit.

• Discounting from the wrong time, for example discounting an end-of-year benefit by v^0.

  Mixing up the time of exit with the time the payment is made.

  Fix: Write the payment time next to each cashflow before discounting. Use the timing given in the question.

• Charging a premium or other in-force payment to those who just left.

  Students forget that in-force cashflows depend on survival to that date.

  Fix: For a start-of-year premium in year t+1, use the number in force at time t, after earlier exits and before this year's exits.

• Mixing dependent and independent decrement rates.

  The notation looks similar, and the question may give one type but the table needs the other.

  Fix: Check whether the given rates are the all-causes-acting (aq)^j_x. Only these can be summed to give the exit probability for the year.

• Rounding expected numbers of lives to whole numbers during the projection.

  Students think half a person cannot exit.

  Fix: Keep fractions throughout. These are expected values. Round only the final answer.

Worked examples