Profit testing for equity-linked life insurance cash flows

Welcome to Profit Testing for Equity-Linked Insurance!

Welcome! If you’ve made it to Exam ALTAM, you already know that life insurance can be complex. In this chapter, we are zooming in on Equity-Linked Insurance (often called Unit-Linked or Variable Annuity products). These products are exciting because they combine the protection of life insurance with the growth potential of the stock market.

However, because the stock market goes up and down, the insurance company faces unique risks. Profit testing is the tool we use to project cash flows and see if the product is actually going to make money under different scenarios. Don't worry if the formulas look long at first; we’re going to break them down into simple "money in" and "money out" steps!

1. Understanding the Basics: What is Equity-Linked Insurance?

In a standard life insurance policy, the company tells the policyholder exactly what they will get. In an equity-linked policy, the benefit depends on the performance of an investment fund (like the S&P 500).

The Two Accounts:
To understand profit testing, you must think of the money being in two different "buckets":
1. The Unit Fund (Separate Account): This is the policyholder's money. It grows with the market and is used to pay for the benefits.
2. The General Account: This is the insurance company’s "wallet." This is where profit is stored and where the company pays its expenses from.

Analogy: The Managed Investment Account
Imagine you are a financial advisor. Your client gives you $1,000 to invest. You put it in a fund (the Unit Fund). Every month, you take a small fee from that fund to pay for your office rent and your salary (the General Account). If the client dies, you might have to pay their family a guaranteed amount, even if the fund has crashed. Profit testing is simply calculating if those fees you collect are enough to cover your expenses and that "safety net" guarantee.

Key Summary:

Profit testing for equity-linked products involves tracking the Unit Fund (to see what the policy is worth) and the Cash Flows to the company (to see if it’s profitable).

2. The Unit Fund Projection

Before we can figure out the company's profit, we have to know how much is in the policyholder's fund. We project this year by year (or month by month).

The fund at the end of the year \( t \), denoted as \( AV_t \) (Account Value), usually follows this logic:
\( AV_t = [AV_{t-1} + P_t(1 - f_t) - C_t] \times (1 + i^k_t) \)
Where:
- \( AV_{t-1} \): The value at the start of the previous period.
- \( P_t \): The premium paid.
- \( f_t \): The premium front-end load (commission or taxes).
- \( C_t \): Charges taken out (mortality charges or management fees).
- \( i^k_t \): The actual investment return of the fund.

Common Mistake: Students often forget when the charges are taken out. Read the problem carefully! Some charges are taken at the beginning of the year (before growth), and some are taken at the end.

3. Identifying the Cash Flows to the Company

Now we look at the General Account. What makes the company money, and what costs them money?

Income (Money In):
- Front-end Loads: A percentage of the premium kept by the company.
- Management Charges (FMC): Often a percentage of the fund value (e.g., 1% of the assets per year).
- Mortality Charges: Fees taken from the fund to cover the cost of the death benefit.

Outgo (Money Out):
- Expenses: Rent, staff salaries, and administrative costs.
- Commissions: Payments to the agent who sold the policy.
- The "Cost of Insurance" (COI): If the policyholder dies and the fund is worth less than the guarantee, the company must pay the difference out of their own pocket. This is the Embedded Option.

Quick Review: The Embedded Option
The most common guarantee is the GMDB (Guaranteed Minimum Death Benefit).
If the guarantee is \( G \) and the fund is \( AV \), the company pays \( \max(0, G - AV) \). This is essentially a "put option" given to the policyholder!

4. Calculating the Emerging Profit

The Profit Vector is the sequence of profits at the end of each year, usually denoted as \( Pr_t \). This is calculated per policy in force at the start of the year.

Step-by-step for Year \( t \):
1. Start with the Net Cash Flow: (Charges Collected - Expenses - Commissions).
2. Add Investment Income: The company earns interest on its own money in the General Account (use the rate \( i \), not the fund rate \( i^k \)).
3. Subtract the Expected Cost of Guarantees: This is the probability of death \( q_{x+t-1} \) times the excess of the guarantee over the fund value.
4. Subtract the Increase in Reserves: The company must set aside money for future risks. The "profit" is what's left after putting money in the reserve.

\( Pr_t = (\text{Income} - \text{Outgo}) \times (1+i) - \Delta \text{Reserves} \)

Did You Know?

In many equity-linked products, the company actually loses money in Year 0 or Year 1 because of high setup expenses and commissions. This is called "New Business Strain." They hope to recoup this through management fees in later years!

5. Deterministic vs. Stochastic Profit Testing

Because the fund value depends on the stock market, we have two ways to test profit:

1. Deterministic: We assume the market grows at a constant rate (e.g., 6% every year). This is simpler but can be misleading because it ignores market volatility.
2. Stochastic: We run thousands of simulations using different market paths. This is crucial for "Embedded Options" because guarantees usually only "kick in" when the market performs poorly.

Memory Aid: The "Rainy Day" Rule
Deterministic testing is like checking if you have enough money if it's always sunny. Stochastic testing is checking if you have enough money even if it rains 20% of the time. Since guarantees are like umbrellas, you need to test for rain!

6. Profit Measures: Is it a Good Deal?

Once we have our profit vector \( Pr_t \), we use a few standard metrics to judge the product:

1. NPV (Net Present Value):
The sum of all future profits, discounted at the Hurdle Rate (the company's required return).
\( NPV = \sum Pr_t \times (1+r)^{-t} \times {}_tp_x \)

2. Profit Margin:
\( \text{Profit Margin} = \frac{NPV}{\text{Present Value of Premiums}} \)
This tells you how many cents of profit you make for every dollar of premium collected.

3. Internal Rate of Return (IRR):
The interest rate that makes the NPV equal to zero.

Key Takeaway:

The company wants a positive NPV and an IRR that is higher than their cost of capital (the Hurdle Rate).

7. Common Pitfalls to Avoid

- Mixing Interest Rates: Use the Fund Return (\( i^k \)) for the Unit Fund, but use the Company's Earned Rate (\( i \)) or the Hurdle Rate (\( r \)) for discounting profits.
- Ignoring Survivorship: Remember that profit is usually calculated "per policy in force at the start of the year." You must multiply by the probability of the policyholder staying alive (\( {}_tp_x \)) when calculating the total NPV.
- Reserve Changes: Don't just subtract the reserve. Subtract the increase in reserve (\( V_t - V_{t-1} \)). If the reserve goes down, it actually releases profit!

Final Summary Checklist

- [ ] Did I project the Unit Fund (Account Value) correctly?
- [ ] Did I identify all charges taken from the fund?
- [ ] Did I account for the cost of guarantees (the embedded options)?
- [ ] Did I discount the profits using the correct hurdle rate?
- [ ] Did I include the probability of the policy being "in force"?

You've got this! Profit testing is just a detailed accounting exercise. Keep your "buckets" (Unit Fund vs. General Account) separate, and the logic will follow.