Expected present value and variance of benefit payments

Introduction: Why "Expected" Values Matter

In your previous CM1 chapters, you learned how to calculate the present value of certain cashflows. But in life insurance, nothing is certain! We don’t know exactly when a policyholder will die, which means we don’t know when we will pay a death benefit or for how long we will pay an annuity.

To handle this uncertainty, we treat the Present Value (PV) of a benefit as a random variable. In this chapter, we will learn how to find the "average" of this random variable (the Expected Present Value or EPV) and how much it might vary (the Variance). This is the foundation for pricing every insurance product you see in the real world.

1. The Random Variables: \(T_x\) and \(K_x\)

Before we look at money, we need to look at time. There are two ways to measure how much longer a person aged \(x\) will live:

• Continuous Future Lifetime \(T_x\): This is the exact time until death. It can be any value (e.g., 12.432 years). We use this for benefits paid immediately on death.
• Curtate Future Lifetime \(K_x\): This is the number of complete years lived before death. It is always a whole number (e.g., if someone lives 12.7 years, \(K_x = 12\)). We use this for benefits paid at the end of the year of death.

2. Expected Present Value (EPV) of Assurances

The EPV of an assurance is the amount an insurer needs to set aside today to cover a future death benefit, on average. We often call these Assurance Factors.

Whole Life Assurance

This pays a benefit whenever the person dies.
If paid at the end of the year of death, the PV random variable is \(Z = v^{K_x+1}\).
The EPV is denoted as \(A_x\):
\(A_x = E[v^{K_x+1}] = \sum_{k=0}^{\infty} v^{k+1} \cdot P(K_x = k) = \sum_{k=0}^{\infty} v^{k+1} \cdot {}_k|q_x\)

If paid immediately on death, the PV random variable is \(Z = v^{T_x}\).
The EPV is denoted as \(\bar{A}_x\):
\(\bar{A}_x = E[v^{T_x}] = \int_{0}^{\infty} v^t \cdot f_x(t) dt\)

Term Assurance

This only pays if death occurs within a fixed term of \(n\) years. If the person survives the term, the benefit is zero.
Symbol: \(A^1_{x:\overline{n|}}\) (The "1" is over the \(x\), meaning death must happen first).
\(A^1_{x:\overline{n|}} = \sum_{k=0}^{n-1} v^{k+1} \cdot {}_k|q_x\)

Pure Endowment

This is the opposite of term assurance: it pays only if the person survives \(n\) years.
Symbol: \(A_{x:\overline{n|}}^{\ \ \ 1}\) (The "1" is over the \(n\), meaning survival to time \(n\) is required).
\(A_{x:\overline{n|}}^{\ \ \ 1} = v^n \cdot {}_n p_x\)

Quick Tip: You can combine these! An Endowment Assurance (\(A_{x:\overline{n|}}\)) is just a Term Assurance + a Pure Endowment.
\(A_{x:\overline{n|}} = A^1_{x:\overline{n|}} + A_{x:\overline{n|}}^{\ \ \ 1}\)

3. Expected Present Value (EPV) of Annuities

Annuities are regular payments made while the policyholder is still alive. These are Annuity Factors.

Whole Life Annuity Due

Payments of 1 are made at the start of every year while the person lives.
Symbol: \(\ddot{a}_x\)
\(\ddot{a}_x = \sum_{k=0}^{\infty} v^k \cdot {}_k p_x\)

The Crucial Relationship

One of the most important formulas in CM1 links assurances and annuities. Think of it this way: if you have 1 unit of money today, you can either keep it or trade it for an annuity of the interest (\(d\)) and the unit itself when you die (\(A_x\)).
\(A_x = 1 - d\ddot{a}_x\)
Similarly, for continuous functions: \(\bar{A}_x = 1 - \delta \bar{a}_x\)

Key Takeaway: If you know the annuity value, you can easily find the assurance value (and vice versa) without doing long summations!

4. Calculating Variance: The "Rule of Moments"

The variance of the present value tells us about the risk. If the variance is high, the actual cost could be much higher (or lower) than the EPV.

The formula for variance of a random variable \(Z\) is:
\(Var(Z) = E[Z^2] - (E[Z])^2\)

How to find \(E[Z^2]\) for Assurances

There is a fantastic shortcut! To find the "second moment" (\(E[Z^2]\)) for an assurance, you simply take the standard EPV formula and double the force of interest (\(\delta\)). This is equivalent to squaring the discount factor \(v\).
Symbol: \({}^2 A_x\)

Steps to calculate Variance of a Whole Life Assurance:
1. Calculate \(A_x\) at interest rate \(i\).
2. Calculate \({}^2 A_x\) by using the same formula but with a new interest rate \(i^*\) such that \((1+i^*) = (1+i)^2\).
3. \(Var(Z) = {}^2 A_x - (A_x)^2\)

Note: This shortcut only works for Assurances, not directly for Annuities. To find the variance of an annuity, we convert it into an assurance using the relationship \(Z_{annuity} = \frac{1-Z_{assurance}}{d}\).

5. Using the Life Table: \(l_x\) and \(d_x\)

In the exam, you won't always be given probabilities like \({}_k p_x\). You will need to extract them from a Life Table.

• \(l_x\): The number of people alive at age \(x\).
• \(d_x\): The number of people who die between age \(x\) and \(x+1\). (\(d_x = l_x - l_{x+1}\)).
• \({}_n p_x = \frac{l_{x+n}}{l_x}\) (Probability of surviving \(n\) years).
• \({}_n q_x = \frac{l_x - l_{x+n}}{l_x}\) (Probability of dying within \(n\) years).

Common Mistakes to Avoid

• End of year vs. Start of year: Remember that \(\ddot{a}_x\) (annuity due) starts at time 0, while \(A_x\) (assurance) pays at time \(K_x + 1\).
• Squaring the wrong thing: When calculating variance, students often square the interest rate (\(i^2\)). Don't do this! You must square the discount factor (\(v^2\)), which means using a new rate \(i^* = (1+i)^2 - 1\).
• Forgetting the "1": In the relationship \(A_x = 1 - d\ddot{a}_x\), don't forget the "1"! It represents the present value of a guaranteed unit payment.

Quick Review

EPV: The actuarial "average" cost of a benefit.
Assurance Factor: \(A_x\) (End of year) or \(\bar{A}_x\) (Continuous).
Annuity Factor: \(\ddot{a}_x\) (In advance) or \(a_x\) (In arrear).
Variance Trick: Use \({}^2 A_x\) (calculated at double the force of interest) to find the second moment of an assurance.