Deterministic profit testing for Universal Life insurance

Exam ALTAM – Advanced Long-Term Actuarial Mathematics · SOA (Society of Actuaries) · 6 min read

Welcome to the World of Universal Life Profit Testing!

Hello, future actuary! If you’ve made it to the ALTAM exam, you already know that life insurance isn't always a simple "pay a premium, get a benefit" transaction. Universal Life (UL) insurance is unique because it’s transparent: the policyholder can see exactly how their money grows, how much they are being charged for expenses, and what the insurance itself costs.

In this chapter, we focus on Deterministic Profit Testing. This is the process where we project the year-by-year cash flows of a UL policy based on a single set of assumptions to see if the product will be profitable for the insurance company. Think of it like building a financial "roadmap" for the policy. Let’s dive in!

1. Understanding the Two "Buckets"

To master UL profit testing, you must visualize two different financial perspectives happening at the same time:

1. The Policyholder’s Bucket (Account Value): This is the "savings" part of the policy. We track how much the customer thinks they have.

2. The Insurer’s Bucket (Profit): This is what the company actually keeps after paying out all benefits, expenses, and setting aside reserves.

Analogy: Imagine you are a personal chef. Your client gives you $100 for groceries (Premium). You take $10 for your time (Expense Charge) and $5 for the risk of burning the roast (Cost of Insurance). The remaining $85 goes into a pot to buy ingredients (Account Value). At the end of the year, you also look at how much you actually spent vs. what you charged to see if you made a profit.

2. The Mechanics of the Account Value (AV)

The Account Value is the engine of a UL policy. In deterministic profit testing, we calculate the AV recursively (year by year). It is crucial to follow the timing of the cash flows exactly as described in the policy contract.

A typical end-of-year Account Value $AV_t$ is calculated as:

$AV_t = [AV_{t-1} + P_t(1 - f_t) - e_t](1 + i^c_t) - CoI_t$

Breaking down the terms:

$AV_{t-1}$: The balance left over from the previous year.
$P_t$: The premium paid by the policyholder at the start of year $t$.
$f_t$: A percentage premium charge (like a sales tax or commission load).
$e_t$: A fixed dollar expense charge (admin fee).
$i^c_t$: The Credited Interest Rate. This is the rate the company promises to the customer.
$CoI_t$: The Cost of Insurance. This is the "price" of the death protection for that year.

How to calculate the Cost of Insurance (CoI)?

The $CoI$ is usually calculated at the end of the period based on the Net Amount at Risk (NAR). The formula usually looks like this:

$CoI_t = v^q \cdot q^{death}_{x+t-1} \cdot (DB_t - AV_t)$

Note: In many exam problems, the formula for $AV_t$ and $CoI_t$ are circular (you need $AV_t$ to find $CoI_t$, but you need $CoI_t$ to find $AV_t$). Don't panic! Usually, the exam will provide a simplified version or a specific formula to solve for $AV_t$ algebraically.

Quick Review: The Account Value is not the company's profit. It is a liability—money the company owes to the policyholder!

3. Projecting the Cash Flows (The Profit Vector)

Now that we know how the policyholder’s account grows, we need to see how much the Insurance Company makes. We calculate the Profit ($Pr_t$) for each year $t$.

The standard deterministic profit formula for year $t$ is:

$Pr_t = (V_{t-1} + P_t - E_t)(1 + i) - [q^{(d)}_{x+t-1} \cdot DB_t + q^{(w)}_{x+t-1} \cdot CV_t + (1 - q^{(d)}_{x+t-1} - q^{(w)}_{x+t-1}) \cdot V_t]$

Key Components:

$V_{t-1}$ and $V_t$: The reserves held by the company. In UL, the reserve is often (but not always) equal to the Account Value ($AV$).
$E_t$: The actual expenses incurred by the company (which might be different from the charges deducted from the policyholder!).
$i$: The Earned Interest Rate. This is the actual rate the company makes on its investments.
$q^{(d)}$: The probability of death.
$q^{(w)}$: The probability of withdrawal (surrender).
$DB_t$: The Death Benefit paid.
$CV_t$: The Cash Value paid on surrender (usually $AV_t$ minus a surrender charge).

Common Mistake Alert!

Students often confuse the Credited Rate ($i^c$) with the Earned Rate ($i$).
- Credited Rate: Used to grow the policyholder’s Account Value.
- Earned Rate: Used to calculate the company’s investment income in the profit formula.
The difference between these two (the "spread") is a major source of profit for the company!

4. Measuring Profitability

Once we have the profit vector ($Pr_1, Pr_2, ... Pr_n$), we need to summarize it into numbers that management can understand. We use Net Present Value (NPV) and Profit Margin.

Net Present Value (NPV)

The NPV is the sum of all future profits, discounted back to time 0 using a Risk Discount Rate (r). We also weight each year's profit by the probability that the policy is still in force ($_tp_x^{(\tau)}$).

$NPV = \sum_{t=1}^{n} Pr_t \cdot (1+r)^{-t} \cdot _{t-1}p_x^{(\tau)}$

Profit Margin

The Profit Margin tells us how much profit we make per dollar of premium received. It is calculated as:

$Profit Margin = \frac{NPV}{PV(Premiums)}$

Note: Both the NPV and the PV of Premiums must be discounted at the same Risk Discount Rate ($r$).

5. Step-by-Step: How to Solve a UL Profit Test Problem

If you see a long UL problem on the ALTAM exam, follow these steps to stay organized:

Calculate the Account Value (AV) sequence: Work year-by-year. Calculate $AV_1$, then $AV_2$, etc. Keep track of the $CoI$ charges.
Determine the Cash Values ($CV_t$): Subtract any surrender charges from the $AV_t$.
Calculate the Profit for each year ($Pr_t$): Use the company's earned interest rate and actual expenses. Remember to use the Reserve (often $AV$) in this step.
Apply Probabilities: Calculate the "In-Force" probability ($_{t-1}p_x^{(\tau)}$) for each year.
Discount and Sum: Discount the profits back to time 0 using the Risk Discount Rate to find the NPV.

Did you know? Deterministic profit testing is the foundation for "Stochastic" testing. In stochastic testing, we run this same model thousands of times using random interest rates to see what happens in "worst-case" scenarios!

6. Summary and Key Takeaways

Key Takeaways:

The AV Formula is the "contractual" growth of the policyholder's money.
The Profit Formula reflects the insurer's actual experience (Earned Interest - Benefits - Expenses - Change in Reserves).
Timing is everything: Pay close attention to whether expenses and premiums occur at the beginning of the year and whether benefits are paid at the end of the year.
The Spreadsheet Mindset: Even though you are using a calculator, think of the problem like a table. Columns for Year, AV, CV, Profit, and Discounted Profit will keep you from getting lost.

Don't worry if this seems like a lot of moving parts at first! The math itself is mostly addition, subtraction, and basic interest. The "Advanced" part of ALTAM is simply staying organized and ensuring you use the right rate (Earned vs. Credited) at the right time. Keep practicing those recursive AV tables!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the primary purpose of deterministic profit testing for Universal Life (UL) insurance?

It is the process of projecting year-by-year cash flows based on a single set of fixed assumptions to evaluate the product's profitability.

In UL profit testing, what are the two distinct financial "buckets" that must be tracked?

The Policyholder's Account Value (the liability) and the Insurer's Profit (the surplus generated after benefits and expenses).

What is the recursive formula for calculating the end-of-year Account Value ($AV_t$)?

$AV_t = [AV_{t-1} + P_t(1 - f_t) - e_t](1 + i^c_t) - CoI_t$

How is the Cost of Insurance ($CoI_t$) typically determined in a UL policy?

It is based on the Net Amount at Risk (NAR): $CoI_t = v^q \cdot q^{death}_{x+t-1} \cdot (DB_t - AV_t)$.

What is the standard formula for the annual profit ($Pr_t$) from the insurer's perspective?

$Pr_t = (V_{t-1} + P_t - E_t)(1 + i) - [q^{(d)}_{x+t-1} DB_t + q^{(w)}_{x+t-1} CV_t + (1 - q^{(d)}_{x+t-1} - q^{(w)}_{x+t-1}) V_t]$

What is the critical distinction between the Credited Rate ($i^c$) and the Earned Rate ($i$)?

$i^c$ is the rate used to increase the policyholder's Account Value; $i$ is the actual investment return the insurer makes on its assets.

How is the Net Present Value (NPV) of a profit testing projection calculated?

$NPV = \sum_{t=1}^{n} Pr_t \cdot (1+r)^{-t} \cdot _{t-1}p_x^{(\tau)}$

What is the definition and formula for the Profit Margin?

It is the ratio of the NPV of profits to the PV of premiums: $Profit Margin = \frac{NPV}{PV(Premiums)}$.

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