Introduction: Connecting the Pieces
Welcome! So far in your CM1 journey, you have looked at assurances (which pay out upon death) and annuities (which pay out while someone is alive) as separate ideas. However, in the world of actuarial modelling, these two are actually two sides of the same coin.
In this chapter, we explore the mathematical "bridge" that connects them. Why does this matter? Because if you know the value of a life annuity, you can often calculate the value of a life assurance policy (and vice versa) without starting your calculations from scratch. This is a massive time-saver in the CM1 exam!
The Intuition: The "Interest vs. Capital" Analogy
To understand the relationship between assurance and annuity functions, think of a simple investment of \(1\).
If you invest \(1\) today at a rate of discount \(d\), you could:
- Keep the interest (discount) for as long as you are alive. This is like a Life Annuity.
- Have the original capital returned when you die. This is like a Life Assurance.
Because the total value must equal the original \(1\) we started with, the Present Value of the interest payments + the Present Value of the capital at death must equal \(1\). This is the foundation of the Equation of Value approach mentioned in the syllabus.
1. The Whole Life Relationship (Discrete)
The most fundamental relationship in this chapter is between a Whole Life Assurance (payable at the end of the year of death) and a Whole Life Annuity Due (payable at the start of each year).
The Formula:
\(A_x = 1 - d\ddot{a}_x\)
Breaking it down:
- \(A_x\): The expected present value (EPV) of \(1\) paid at the end of the year of death.
- \(1\): The initial investment.
- \(d\ddot{a}_x\): The EPV of the "interest" (discount) payments of \(d\) made at the start of every year while the person is alive.
Don't worry if this seems tricky! Just remember that the assurance is what's "left over" after all the annuity interest payments have been made.
2. The Whole Life Relationship (Continuous)
If the assurance is paid immediately upon death (\(\bar{A}_x\)) and the annuity is paid continuously (\(\bar{a}_x\)), we use the force of interest \(\delta\) instead of the discount rate \(d\).
The Formula:
\(\bar{A}_x = 1 - \delta\bar{a}_x\)
Quick Tip: Notice the pattern! Discrete uses \(d\), continuous uses \(\delta\). In both cases, we are subtracting the "interest" portion from the total value of \(1\).
3. Endowment Assurances
An Endowment Assurance (\(A_{x:\overline{n|}}\)) pays out either at the end of the year of death or at the end of \(n\) years if the person is still alive. Because it guarantees a payout of \(1\) at some point (either death or maturity), the relationship is very similar to the whole life version, just limited to \(n\) years.
The Formula:
\(A_{x:\overline{n|}} = 1 - d\ddot{a}_{x:\overline{n|}}\)
Similarly, for the continuous version:
\(\bar{A}_{x:\overline{n|}} = 1 - \delta\bar{a}_{x:\overline{n|}}\)
Common Mistake to Avoid: Students often confuse Term Assurance (\(A^1_{x:\overline{n|}}\)) with Endowment Assurance (\(A_{x:\overline{n|}}\)). This specific relationship formula only works for Endowment Assurances because they represent a certain payment of capital at some point in the future.
4. Variance of Present Value
The syllabus requires you to understand the mean (which we've done above) and the variance of the present value of these contracts.
For a Whole Life Assurance, the variance is:
\(Var(v^K) = {}^2A_x - (A_x)^2\)
Where \({}^2A_x\) is the assurance factor calculated at a rate of interest \(j\) such that \(1+j = (1+i)^2\). In other words, you just double the force of interest or square the discount factor \(v\).
The Annuity Connection:
Because \(\ddot{a}_x = \frac{1 - A_x}{d}\), the variance of the annuity is directly linked to the variance of the assurance:
\(Var(\ddot{a}_{\overline{K+1|}}) = \frac{Var(v^{K+1})}{d^2} = \frac{{}^2A_x - (A_x)^2}{d^2}\)
Memory Aid: If you find the variance of the assurance, you just divide it by \(d^2\) (for discrete) or \(\delta^2\) (for continuous) to get the variance of the annuity!
Summary Table for Quick Review
Here is a snapshot of the relationships you must know for the exam:
| Contract Type | Relationship Formula |
|---|---|
| Whole Life (Discrete) | \(A_x = 1 - d\ddot{a}_x\) |
| Whole Life (Continuous) | \(\bar{A}_x = 1 - \delta\bar{a}_x\) |
| Endowment (Discrete) | \(A_{x:\overline{n|}} = 1 - d\ddot{a}_{x:\overline{n|}}\) |
| Endowment (Continuous) | \(\bar{A}_{x:\overline{n|}} = 1 - \delta\bar{a}_{x:\overline{n|}}\) |
Top Tips for CM1A and CM1B
- Check the timing: If the question asks for an assurance payable immediately, ensure you are using \(\bar{a}_x\) and \(\delta\). If it's end of year, use \(\ddot{a}_x\) and \(d\).
- The "1" Rule: These formulas always relate to a benefit of \(1\). If the benefit is \$50,000, multiply the whole equation by 50,000.
- Excel Efficiency (CM1B): In Excel, you can quickly build a table for \(A_x\) and then create a second column for \(\ddot{a}_x\) using the formula \((1 - A_x) / d\). This is much faster than looking up every value in the life tables!
Key Takeaway
The Equation of Value connects life contingencies. An assurance is simply the "balance" of a fund of \(1\) after all survival-contingent interest payments (annuities) have been stripped away. Mastering these relationships allows you to navigate between different contract types with ease.