Welcome to the Future: Projecting Cashflows

Hello there! Welcome to one of the most exciting and practical parts of the CM1 syllabus. So far, you’ve learned about interest rates, life tables, and basic annuities. Now, we are going to put those tools to work by becoming "financial fortune tellers."

In this chapter, we learn how to project expected future cashflows. This is the heart of actuarial work—if we can’t predict what money is coming in and what money is going out, we can't figure out if a product is profitable or if we have enough money saved up (reserves) to pay future claims. Don’t worry if this seems like a lot of moving parts; we will break it down step-by-step!

1. The Big Picture: What are we projecting?

Think of an insurance policy like a "bucket." Money flows in (Premiums) and money flows out (Claims and Expenses). Our job is to estimate how much will be in that bucket at any given time.

The Expected Cashflow at any time \( t \) is simply:
(Amount of Cashflow) \(\times\) (Probability of that Cashflow occurring)

Key Components to Include:

  • Inflows: Premiums (usually at the start of the year).
  • Outflows: Death benefits, maturity benefits, surrender values, and expenses (commissions, administration, etc.).
  • Investment Income: Interest earned on the money currently held in the "bucket."

Quick Review: Remember that in pricing and reserving, we usually work with discrete time steps (like year-by-year) rather than continuous time, to make the calculations manageable.

2. Projecting Conventional Contracts

For standard contracts like Term Assurance, Whole Life, or Endowment, the cashflows depend mainly on whether the policyholder is alive or dead.

How to calculate the expected death benefit for year \( t+1 \):

We look at the probability that the policyholder survived until the start of the year (\( {}_t p_x \)) and then dies during that year (\( q_{x+t} \)).

Expected Outflow = \( S \cdot {}_t p_x \cdot q_{x+t} \)
Where \( S \) is the Sum Assured.

How to calculate expected premiums:

Premiums are only paid if the policyholder is alive at the start of the year.

Expected Inflow = \( P \cdot {}_t p_x \)
Where \( P \) is the annual premium.

Key Takeaway: For conventional products, the cashflow is "fixed" in amount but "uncertain" in timing. We use survival probabilities to find the "mathematical expectation" of these flows.

3. Unit-Linked Contracts: The "Two-Bucket" System

Unit-linked contracts can be tricky because there are two different funds to track. Imagine two separate bank accounts:

  1. The Unit Fund: This belongs to the policyholder. It grows with investment returns and shrinks when we take out "charges."
  2. The Non-Unit Fund: This belongs to the insurance company. This is where the company collects its fees and pays its expenses.

The Unit Fund Projection

The value of the units at the end of the year \( (V_{t+1}) \) is calculated as:
\( V_{t+1} = [V_t + P(1-f) - C] \cdot (1+i_{unit}) \)

  • \( P(1-f) \): The premium after the "allocation charge" or "bid-offer spread" is taken out.
  • \( C \): Risk charges (like the cost of providing a death benefit).
  • \( i_{unit} \): The expected investment growth rate for the unit fund.

The Non-Unit Fund (The Profit/Loss Account)

The company looks at its own cashflows here:
Inflows: Allocation charges, Management fees (often a % of the fund), Risk charges.
Outflows: Expenses (initial and renewal), Commission, and the "Extra" death benefit (the amount by which the death benefit exceeds the unit fund).

Analogy: Think of a Unit-Linked policy like a gym membership where you pay a fee (Unit Fund) that gets invested in equipment, but the gym owner (Non-Unit Fund) takes a monthly "maintenance fee" from your account to pay the electricity bill and keep the profit.

4. With-Profits Contracts and Asset Shares

With-profits contracts are about "smoothing." The policyholder shares in the profits of the company through bonuses.

Asset Share: This is a vital concept. It represents the "fair share" of the company's assets that belongs to a specific policy. It is calculated by accumulating the actual premiums received, adding actual investment returns, and subtracting actual expenses and the cost of cover.

Types of Bonuses to Project:
  • Reversionary Bonuses: Added periodically (usually annually). Once added, they are guaranteed and increase the Sum Assured.
  • Terminal Bonuses: Added only at the very end (death or maturity). These are not guaranteed until the end.

Common Mistake: Students often forget that reversionary bonuses compound. If a bonus is "compound," it applies to the original Sum Assured plus all previous bonuses!

5. Incorporating Multiple Decrement Models

In the real world, a policy doesn't just end because someone dies. It might end because they surrender (cancel) the policy or retire.

When projecting cashflows, we must use Dependent Probabilities. Instead of just using \( q_x \), we use \( (aq)_x^d \) for death and \( (aq)_x^w \) for withdrawal.

The Golden Rule of Multiple Decrements:
The probability of staying in the group (the survival probability \( ap_x \)) is:
\( ap_x = 1 - (aq)_x^d - (aq)_x^w - ... \)

When you project cashflows, you must multiply the cashflow by the probability of that specific event happening. For example:
Expected Surrender Outflow = (Surrender Value) \(\times\) (Probability of Surrender).

Did you know? Actuaries monitor "Withdrawal Rates" (surrenders) very closely. If too many people cancel their policies early, the company might not recover the heavy initial expenses spent on commissions and marketing!

6. Step-by-Step: How to build a Cashflow Projection

If you face a long question on this, follow these steps:

  1. Identify the timing: Are premiums at the start? Are claims at the end? (Usually, yes).
  2. Calculate the "In-force" probability: Determine the probability that the policy is still active at the start of each year.
  3. Project Unit Growth (if applicable): Calculate the fund value year by year.
  4. Calculate "Per-Policy" Cashflows: List all income and outgo for a single policy that survives.
  5. Multiply by Probabilities: Multiply those values by the probability of the policy being in force or the event (death/withdrawal) occurring.
  6. Sum and Discount: Find the Net Cash Flow for each year and discount them back to the present value if you need to find the Profit or Reserve.

7. Summary and Key Takeaways

  • Expected Cashflow = Amount \(\times\) Probability. This is the foundation of everything.
  • Unit-Linked products separate the client's investment (Unit Fund) from the company's operations (Non-Unit Fund).
  • With-Profits products use Asset Shares to track the policy's value and determine bonuses.
  • Multiple Decrements allow us to model realistic scenarios where people leave for various reasons (death, surrender, etc.).
  • Timing is everything: Always check if expenses or premiums happen at the start or end of the year.

Quick Review Box:
- Death Benefit: Usually \( S \cdot {}_t p_x \cdot q_{x+t} \)
- Unit Fund: \( (\text{Old Fund} + \text{New Money} - \text{Charges}) \times (1+i) \)
- Survival Probability: \( ap_x = 1 - \sum (\text{all decrement probabilities}) \)

Don't worry if the math feels heavy at first. The logic is always the same: What is the chance of this money moving, and how much is moving? Master that, and you've mastered the chapter!