Apply formulae for the means and variances of the payments under various assurance and annuity contracts, assuming a constant deterministic interest rate

Welcome to the World of Actuarial Values!

Hello there! In this chapter, we are going to learn how to calculate the Expected Present Value (EPV)—which is basically the "average" cost—and the variance (the risk or "spread") of various insurance and pension contracts.

Think of this as the bridge between probability and finance. We know someone might die or survive (probability), and we know money has a time value (interest). By combining them, we can figure out exactly how much money an insurance company needs to set aside today to pay for future benefits. Don't worry if it seems tricky at first; we will break it down step-by-step!

1. The Core Concept: Random Variables

In CM1, we don't just look at fixed payments. Because we don't know exactly when someone will die, the present value of a payment is a Random Variable.

• For Life Assurances, we usually call the Present Value random variable \(Z\).
• For Life Annuities, we usually call the Present Value random variable \(Y\).

Our goal is to find the Mean (the Expected Value, \(E[Z]\) or \(E[Y]\)) and the Variance (\(Var(Z)\) or \(Var(Y)\)).

Quick Review: Prerequisite Terms

Before we dive in, remember these two important tools:

\(v\): The discount factor, \(v = (1+i)^{-1}\).
\(d\): The discount rate, \(d = i / (1+i) = 1 - v\).
\(\delta\): The force of interest, \(\delta = \ln(1+i)\).

2. Life Assurances: Means and Variances

A life assurance pays out a lump sum when the policyholder dies. This could happen next year, in ten years, or fifty years!

The Mean (Expected Present Value)

The EPV is the sum of (Probability of dying in a specific year) \(\times\) (Present value of the payment if death occurs then).

Common Symbols:
- \(A_x\): Whole Life Assurance (pays at the end of the year of death).
- \(\bar{A}_x\): Whole Life Assurance (pays immediately at the moment of death).
- \(A_{x:\bar{n}|}^1\): Term Assurance (pays only if death occurs within \(n\) years).

The Variance (The "Rule of 2")

To find the variance of an assurance, we use the standard formula: \(Var(Z) = E[Z^2] - (E[Z])^2\).
But how do we find \(E[Z^2]\)?

The Actuarial Trick: To calculate the second moment (\(E[Z^2]\)), you simply take the standard EPV formula and double the force of interest (or use a squared discount factor \(v^2\)). In actuarial notation, we write this as \({}^2A_x\).

The Formula:
\(Var(Z) = {}^2A_x - (A_x)^2\)

Analogy: Imagine a lottery where the prize stays the same, but the "interest rate" of your luck doubles. That’s how we calculate the second moment!

Key Takeaway:

For any assurance contract, the variance is always the "EPV calculated at double the force of interest" minus the "EPV squared."

3. Life Annuities: Means and Variances

Annuities are regular payments made while someone is still alive (like a pension). These are very different from assurances because the payments stop when the person dies.

The Mean (EPV)

We use the symbol \(a_x\) (for payments at the end of the year) or \(\ddot{a}_x\) (for payments at the start of the year).

The Golden Relationship:
There is a famous link between annuities and assurances that you must memorize:
\(\ddot{a}_x = \frac{1 - A_x}{d}\) (Discrete case)
\(\bar{a}_x = \frac{1 - \bar{A}_x}{\delta}\) (Continuous case)

The Variance of Annuities

Calculating the variance of an annuity directly is hard. Instead, we use the relationship above. Since the only "random" part of the annuity is the timing of death (which is what \(A_x\) measures), we can transform the variance of the assurance into the variance of the annuity.

The Formula:
\(Var(Y) = \frac{1}{d^2} [^2A_x - (A_x)^2]\) (Discrete)
\(Var(Y) = \frac{1}{\delta^2} [^2\bar{A}_x - (\bar{A}_x)^2]\) (Continuous)

Did you know? The variance of an annuity is just the variance of the corresponding assurance divided by the discount rate squared! It saves you from doing all the hard math from scratch.

4. Step-by-Step: Solving a Variance Problem

If you are asked to find the variance of a whole life assurance for \( (x) \):

Step 1: Calculate the EPV at the given interest rate \(i\). This is your \(A_x\).
Step 2: Look at the force of interest \(\delta\) (or the discount factor \(v\)). Double the \(\delta\) (or square the \(v\)) to find a "new" interest rate.
Step 3: Calculate the EPV again using this new interest rate. This is your \({}^2A_x\).
Step 4: Plug them into: \(Var(Z) = {}^2A_x - (A_x)^2\).

5. Common Mistakes to Avoid

1. Forgetting to square the EPV: Students often do \({}^2A_x - A_x\). Remember, the formula requires the square of the first moment: \((A_x)^2\).
2. Using the wrong "d": When calculating the variance of an annuity, make sure you use the original \(d\) (based on \(i\)), not the "doubled" version.
3. Mixing up continuous and discrete: Always check if the payment is "immediate" (\(\bar{A}\)) or "at the end of the year" (\(A\)).

6. Summary Table for Quick Revision

Whole Life Assurance:
Mean: \(A_x\)
Variance: \({}^2A_x - (A_x)^2\)

Whole Life Annuity-Due:
Mean: \(\frac{1 - A_x}{d}\)
Variance: \(\frac{1}{d^2} [^2A_x - (A_x)^2]\)

Pure Endowment (\(n\) years):
Mean: \(A_{x:\bar{n}|}^{\:\:\:1} = v^n \cdot {}_np_x\)
Variance: \(v^{2n} \cdot {}_np_x \cdot (1 - {}_np_x)\)
Note: This is just a version of the Binomial Variance \(npq\) because it's a "yes/no" payment!

Key Takeaway:

Don't be intimidated by the symbols. Most of these problems just require you to find two values: the standard EPV and the "doubled interest" EPV. Once you have those, it's just basic subtraction!

Keep going! You're doing great. Mastering these formulas is the key to unlocking the rest of the CM1 syllabus.