Calculate gross premiums and reserves of assurance and annuity contracts

Welcome to Pricing and Reserving!

In your CM1 journey so far, you have likely looked at net premiums—the "theoretical" price where we only care about the benefits. But in the real world, insurance companies have bills to pay! They have staff salaries, office rent, and commission for agents. This is where Gross Premiums and Reserves come in.

In this chapter, we are going to learn how to bridge the gap between theory and reality. We will look at how to set the actual price a customer pays (the Gross Premium) and how much money the company needs to keep in the bank to ensure they can pay out future claims (the Reserve). Don't worry if this seems like a lot of variables at first; we’ll break it down step-by-step!

1. Understanding Gross Premiums

The Gross Premium is the total amount the policyholder pays. Think of it like buying a loaf of bread: the price doesn't just cover the flour and water (the benefit); it also covers the baker’s time and the electricity for the oven (the expenses).

The Principle of Equivalence

The fundamental rule of actuarial pricing is the Equation of Value. At the start of the policy (time 0), we assume:

\( PV(\text{Gross Premiums}) = PV(\text{Benefits}) + PV(\text{Expenses}) \)

Did you know? Even if a company makes a profit, we often calculate the "break-even" gross premium first. Profit can be added later as an extra "expense" or a loading!

Types of Expenses

In CM1, expenses are usually categorized by when they happen:

• Initial Expenses (I): These happen once, right at the start (e.g., medical checks, underwriting).
• Renewal Expenses (R): These happen every time a premium is paid or every year (e.g., policy administration).
• Termination/Claim Expenses (C): These happen only when the policy ends or a claim is paid (e.g., legal fees for processing a death benefit).

Expenses can also be expressed in different ways:

• Fixed amounts: e.g., £50 per policy.
• Percentage of premium: e.g., 5% of every gross premium paid.
• Percentage of sum assured: e.g., 0.1% of the benefit amount.

Quick Review: Remember that initial expenses only happen at \( t=0 \). Renewal expenses usually happen at the start of every year except the first year, OR every year including the first. Read the question carefully!

2. Calculating the Gross Premium: Step-by-Step

Let's say we want to find the annual gross premium \( G \) for a 20-year term assurance for a person aged \( x \). The benefit is \( S \).

Step 1: Identify the Benefit PV
\( PV(\text{Benefits}) = S \cdot A^1_{x:\overline{n|}} \)

Step 2: Identify the Expense PV
If we have an initial expense \( I \), a renewal expense \( e \) per year (starting in year 2), and a claim expense \( f \):
\( PV(\text{Expenses}) = I + e \cdot a_{x:\overline{n-1|}} + f \cdot A^1_{x:\overline{n|}} \)
Note: We use \( a \) for renewal expenses because they are usually paid if the person is alive at the start of the year.

Step 3: Identify the Premium PV
\( PV(\text{Gross Premiums}) = G \cdot \ddot{a}_{x:\overline{n|}} \)

Step 4: Solve for G
Set \( PV(\text{Premiums}) = PV(\text{Benefits}) + PV(\text{Expenses}) \) and solve the algebra!

Common Mistake: Forgetting to include "percentage of premium" expenses on both sides or failing to group them with the premium term. If the expense is 5% of \( G \), your premium side effectively becomes \( G \cdot 0.95 \cdot \ddot{a}_{x:\overline{n|}} \).

3. Gross Premium Reserves

A Reserve is the amount of money an insurer must hold at a specific time (\( t \)) to meet future obligations. Even if we priced the policy perfectly at the start, as time goes on, the risk of death increases, and we need to make sure we have enough saved up.

Prospective Reserve

The Prospective Reserve is the most common method. It looks forward into the future:

\( \text{Reserve} = PV(\text{Future Benefits}) + PV(\text{Future Expenses}) - PV(\text{Future Gross Premiums}) \)

Analogy: Imagine you are saving for a holiday that costs £1,000. You plan to save £100 a month. After 4 months, your "reserve" should be the cost of the holiday (£1,000) minus what you still expect to save (£600), which equals £400.

Important Note on Expenses in Reserves

When calculating a Prospective Reserve at time \( t \), we ignore initial expenses. Why? Because they happened in the past! We only care about expenses that will occur from time \( t \) until the end of the policy.

Key Takeaway: If the reserve is positive, the company holds an asset. If it's negative (which is rare but possible in some calculations), it suggests the policy is currently "profitable" in a way that offsets future costs.

4. The Recursive Relationship (Thiele's Principle)

Sometimes you don't want to calculate everything from scratch using actuarial symbols. You can calculate the reserve at year \( t+1 \) if you know the reserve at year \( t \). This is like a bank account balance.

The logic:
(Money you had) + (New Premium) - (Expenses) + (Interest) = (Money paid for those who die) + (Money kept for those who survive)

The formula for a year-to-year reserve (assuming death benefits are paid at the end of the year) is:
\( (_tV + G - e)(1+i) = q_{x+t} \cdot S + p_{x+t} \cdot _{t+1}V \)

Where:
- \( _tV \) is the reserve at time \( t \)
- \( G \) is the gross premium
- \( e \) is the renewal expense
- \( S \) is the Sum Assured (benefit)
- \( q_{x+t} \) is the probability of dying in the next year

Memory Trick: Think of this as "Income accumulated" must equal "Expected Outgo."

5. Summary and Tips for Success

Quick Review Box:
1. Gross Premium = Net Premium + Loading for Expenses.
2. Equivalence Principle applies at \( t=0 \).
3. Prospective Reserve = \( PV(\text{Future Outgo}) - PV(\text{Future Income}) \).
4. Ignore Initial Expenses when calculating prospective reserves at \( t > 0 \).
5. Read carefully: Are expenses paid at the start or end of the year? Is the benefit paid immediately on death or at the end of the year?

Don't let the notation scare you. Underneath all the \( A \)'s and \( a \)'s, it's just basic accounting: making sure you have enough money coming in to cover the money going out. Practice setting up the Equation of Value for different products (Whole Life, Endowment, Annuities), and the rest is just arithmetic!