Reserves using profit testing

Welcome to Profit Testing for Reserves!

Hello there! If you’ve made it to the Profit Analysis section of Exam ALTAM, give yourself a pat on the back. This is where the theoretical models of actuarial math meet the practical "business" side of insurance. In this chapter, we are going to look at Reserves through the lens of Profit Testing.

Think of it this way: In previous chapters, we calculated reserves using the "Prospective" or "Retrospective" methods. Here, we use a "Cash Flow" approach. Instead of just looking at one big formula, we look at the year-by-year movement of money to see exactly how much we need to hold back to keep the lights on and the promises kept. Don't worry if this seems a bit heavy on the accounting side—at its heart, it’s just making sure your checkbook balances!

1. The Core Concept: Reserves as a Cash Flow Component

In profit testing, we project the Net Cash Flow for each year. A Reserve (\(_tV\)) is simply an amount of money the company sets aside at time \(t\). To understand this, let’s use a simple "Bucket Analogy."

The Bucket Analogy: Imagine each policy has its own bucket. At the start of the year, you have last year's reserve in the bucket. You add premiums and interest, then you take out expenses and claims. Whatever is left in the bucket at the end of the year must be enough to cover the Reserve for next year. If there's extra, that's your Profit.

Key Definition: The Profit in year \(t\) (\(Pr_t\)) is calculated based on the assets at the start of the year, the income earned, and the liabilities (reserves) we must hold at the end of the year.

The Recursive Formula (The "Golden Rule" of Reserves)

The relationship between the reserve at the start of the year and the end of the year is:
\(Pr_t = (_{t-1}V + P_t - E_t)(1+i) - q_{x+t-1} \cdot S - p_{x+t-1} \cdot _tV\)

Where:
- \(_{t-1}V\): Reserve from the previous year.
- \(P_t\): Premium received at the start of year \(t\).
- \(E_t\): Expenses incurred at the start of year \(t\).
- \(i\): The earned interest rate.
- \(q_{x+t-1}\): Probability of death during the year.
- \(S\): The death benefit (Sum Assured) paid at the end of the year.
- \(p_{x+t-1}\): Probability of survival.
- \(_tV\): The reserve we need to have ready at the end of the year.

Quick Summary: The profit is what’s left over after you pay benefits and "fund" the new reserve for survivors.

2. Determining Reserves via Profit Testing

Usually, we are used to being given the reserve and finding the profit. However, in this chapter, we often do the reverse: We decide on a specific Profit Signature (how much profit we want each year) and solve for the Reserve needed to make that happen.

The Zero-Profit Reserve

A common task in ALTAM is calculating the Zero-Profit Reserve. This is the reserve value \(_tV\) that makes the profit in year \(t\) exactly equal to zero (\(Pr_t = 0\)).

Why would we do this?
Companies often want to "smooth" their profits. By setting a higher reserve today, they move profit from today into the future. By setting a lower reserve, they "unlock" profit now. The Zero-Profit Reserve tells us exactly how much we need to hold to "break even" in a specific year.

Step-by-Step: Solving for the Reserve

1. Set the Profit equation to zero: \(0 = (_{t-1}V + P_t - E_t)(1+i) - q_{x+t-1} \cdot S - p_{x+t-1} \cdot _tV\).
2. Rearrange the formula to solve for \(_tV\):
\(_tV = \frac{(_{t-1}V + P_t - E_t)(1+i) - q_{x+t-1} \cdot S}{p_{x+t-1}}\).
3. Check your work: Does the result make sense? If your expenses or claims are very high, your required reserve will likely increase.

Common Mistake to Avoid: Don't forget that premiums and expenses usually happen at the beginning of the year, while benefits and the "end-of-year" reserve happen at the end. The interest rate \((1+i)\) applies to everything that was in the pot at the start!

3. Discounting and the "Hurdle Rate"

In profit testing, we deal with two different interest rates, and it's vital not to mix them up:

1. The Earned Rate (\(i\)): The actual interest the company earns on its investments. We use this to project the cash flows.
2. The Risk Discount Rate (Hurdle Rate, \(r\)): The rate used to calculate the Present Value of the profits. This represents the return the shareholders expect to earn.

Did you know? The "Hurdle Rate" is usually higher than the "Earned Rate." This is because shareholders want a premium for the risk they are taking by investing in an insurance company.

Net Present Value (NPV) of Profits

The NPV is the sum of all future profits, discounted back to the start of the policy using the hurdle rate \(r\).
\(NPV = \sum_{t=1}^{n} Pr_t \cdot (1+r)^{-t} \cdot _{t-1}p_x\)

Note: We multiply by \(_{t-1}p_x\) because the profit \(Pr_t\) only happens if the policyholder is still alive at the start of year \(t\).

4. Modifications: Survival and Surrender

In the real world (and on the ALTAM exam), people don't just die; they also surrender (cancel) their policies. This changes our reserve calculation.

If a policy has a surrender benefit (\(CV_t\)) and a surrender probability (\(w_{x+t-1}\)), our profit formula becomes slightly more complex:
\(Pr_t = (_{t-1}V + P_t - E_t)(1+i) - q_{x+t-1} \cdot S - w_{x+t-1} \cdot CV_t - p^{(total)}_{x+t-1} \cdot _tV\)

Where \(p^{(total)}_{x+t-1}\) is the probability that the policyholder neither dies nor surrenders.

Memory Aid: Think of "Outgos" as three doors. At the end of the year, money leaves through the Death Door (\(q \cdot S\)), the Surrender Door (\(w \cdot CV\)), or the Survival Door (\(p \cdot _tV\)).

Quick Review Box

Important Points:
- Profit Testing is a year-by-year cash flow projection.
- Reserves act as a "liability" in the profit equation—increasing the reserve decreases that year's profit.
- Zero-Profit Reserves are found by setting \(Pr_t = 0\) and solving for \(_tV\).
- Always use the Earned Rate for cash flow growth and the Hurdle Rate for discounting profits.
- Remember to include decrements (death and surrender) when calculating the expected value of end-of-year payments.

Summary of Key Takeaways

Reserves in profit testing are not just static numbers; they are active tools used to manage a company's financial health. By adjusting the reserve levels, an actuary can change the timing of when profits are recognized. When solving exam problems, always start by identifying your cash inflows (premiums, interest) and your outflows (expenses, claims, surrenders, and the end-of-period reserve). If you can master the recursive formula, you can solve almost any problem in this chapter!

Keep practicing! You're doing great. Profit testing is one of the most practical skills an actuary can have, and mastering it for ALTAM will serve you well throughout your career.