Welcome to Unit-Linked Projections!
In the world of actuarial mathematics, unit-linked contracts are a bit like a "choose-your-own-adventure" for policyholders. Unlike conventional contracts where the insurer promises a fixed sum, unit-linked contracts allow the policyholder to see their premiums invested directly into a fund of assets. As an actuary, your job is to model how this fund grows and how the insurance company makes a profit from it.
Don't worry if this seems a bit more complex than standard endowment policies! The secret is that we simply split the contract into two separate "buckets." Once you understand how these two buckets interact, the math becomes much easier to manage.
1. The Two-Bucket System
To project a unit-linked contract, we must maintain two distinct accounts for each policy:
The Unit Fund (The Policyholder’s Bucket)
The Unit Fund represents the value of the units allocated to the policyholder. Think of this like a personal investment account. The insurer manages it, but the value belongs to the customer.
- Money In: The portion of the premium used to buy units (Allocated Premium).
- Money Out: Charges taken by the insurer (to cover costs) and benefit payments.
- Growth: The investment return earned on the underlying assets.
The Non-Unit Fund (The Insurer’s Bucket)
The Non-Unit Fund is where the insurer’s own cashflows live. This is where we calculate whether the company is actually making a profit.
- Money In: Charges taken from the unit fund, the unallocated portion of the premium, and investment income on the non-unit assets.
- Money Out: Expenses (like administration), commissions paid to agents, and the cost of providing the "extra" insurance cover (the death benefit).
Key Takeaway: The "Charges" leaving the Unit Fund are exactly the same as the "Charges" entering the Non-Unit Fund. They are the bridge between the two buckets!
2. Projecting the Unit Fund
To project the unit fund, we follow the units from the start of the year to the end of the year. Let \(U_{t-1}\) be the unit fund at the start of year \(t\).
Step-by-Step Calculation:
- Add Allocated Premium: Not all of the premium \(P\) goes into the fund. If there is a "bid-offer spread" or an allocation rate, only a portion is used. Let \(P_t\) be the premium and \(Alloc_t\) be the allocation rate.
\( \text{Units Added} = P_t \times Alloc_t \). - Add Investment Growth: The fund grows at the unit growth rate \(I_u\).
\( \text{Value before charges} = (U_{t-1} + \text{Allocated Premium}) \times (1 + I_u) \). - Deduct Charges: The insurer usually takes a Fund Management Charge (FMC). This is often a percentage (\(m\)) of the fund value.
\( \text{FMC} = m \times \text{Value before charges} \). - Closing Balance: The fund at the end of the year \(U_t\) is:
\( U_t = (U_{t-1} + P_t \times Alloc_t) \times (1 + I_u) \times (1 - m) \).
Note: The timing of charges (start vs. end of year) depends on the specific contract wording in the exam question. Always read the "fine print"!
3. Projecting the Non-Unit Fund
The Non-Unit Fund projection is essentially a cashflow statement for the insurer. We calculate the cashflow \(CF_t\) for each year.
Common Non-Unit Cashflows:
- Unallocated Premium: \( P_t \times (1 - Alloc_t) \).
- Contract Fees: Fixed policy fees often deducted at the start of the year.
- Bid-Offer Spread: If units are bought at a "bid" price but the premium is paid at an "offer" price (typically a 5% difference).
- Fund Management Charge (FMC): The amount we just calculated in the Unit Fund section.
- Expenses & Commission: The costs the insurer incurs (\(e_t\)).
- Cost of Capital at Risk: If the policyholder dies, the insurer pays a Death Benefit. The Non-Unit fund must cover the "extra" amount: \( \text{Sum Assured} - \text{Unit Fund} \).
Did you know? The "Cost of Capital at Risk" is often calculated as:
\( q_{x+t-1}^{(d)} \times (\text{Death Benefit} - U_t) \).
This represents the expected cost to the insurer for the insurance protection part of the contract.
4. Interaction and Constraints
When projecting these funds, we usually assume the projection is for a single policy in force at the start of the year. This is why we use probabilities of death (\(q_x\)) and surrender (\(q_s\)) to "weight" the cashflows.
The Profit Vector
Once we have the net cashflow for each year from the Non-Unit fund, we have what is called the Profit Vector. For unit-linked business, this usually looks like a large negative value in Year 0 (due to high initial expenses and commission) followed by positive values in later years (as charges from the unit fund roll in).
Quick Review Box:
- Unit Fund: Tracks the customer's investment.
- Non-Unit Fund: Tracks the insurer's profit/loss.
- FMC: The most common "income" for the Non-Unit fund.
- Allocation Rate: The percentage of premium actually invested.
5. Common Pitfalls to Avoid
Students often find this chapter tricky because of the "moving parts." Here are three things to watch out for:
- Interest Rates: Use the unit growth rate for the Unit Fund, but use the insurer’s investment return (earned on their own cash) for the Non-Unit Fund. They are often different!
- Timing: Be careful whether expenses and charges occur at the beginning (t) or end (t+1) of the year.
- Death Benefits: Remember that in a unit-linked contract, the total death benefit is often \( \max(\text{Sum Assured}, \text{Unit Fund}) \) or \( \text{Sum Assured} + \text{Unit Fund} \). Read the question carefully to see how much the Non-Unit fund has to "top up."
Next Steps: Once you have mastered these projections, you will learn how to "zeroise" these cashflows if the Non-Unit fund turns negative in future years. This is covered in the chapter "Zeroising negative cashflows with non-unit reserves."
Summary: Projecting unit-linked contracts is simply a balancing act. Keep the policyholder's money (Unit Fund) separate from the company's money (Non-Unit Fund), and ensure the charges flow correctly from one to the other. Master this, and you've mastered the heart of modern life insurance modelling!