Probabilities, means, variances, and covariances of the present value random variables

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 5 min read

Welcome to the World of Present Value Random Variables!

Hello there! If you’ve made it this far in your actuarial journey, you’ve already mastered the basics of interest rates and survival models. Now, we’re going to combine them! In this chapter, we explore Present Value (PV) Random Variables. Think of this as the "heart" of life insurance math. We are trying to figure out: "If I promise to pay someone $1 when they die, how much is that promise worth today?"

Because we don’t know when someone will die, the present value is a random variable. Don't worry if this seems tricky at first—we’re going to break it down into simple pieces, from finding the average (mean) to measuring the risk (variance).

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1. Understanding the Random Variable $Z$

In life insurance, we usually call the present value random variable $Z$. It represents the value today of a future benefit payout.

For a Whole Life Insurance policy of $1 paid at the moment of death:

$Z = v^{T_x}$

Where:

$v$ is the discount factor $\frac{1}{1+i}$.
$T_x$ is the future lifetime of a person aged $x$.

Analogy: Imagine you have a coupon for a free pizza, but you can only use it on the day it rains. Since you don't know when it will rain, the "value" of that coupon today depends on the probability of rain each day and how much you'll care about a pizza in the future.

Key Types of $Z$:

Continuous Insurance: Benefit is paid at the exact moment of death ($Z = v^{T_x}$).
Discrete Insurance: Benefit is paid at the end of the year of death ($Z = v^{K_x+1}$).

2. The Mean: Actuarial Present Value (APV)

The mean of $Z$ is the average amount we expect to pay, discounted to today. In actuarial terms, we call this the Net Single Premium or Actuarial Present Value (APV).

For continuous whole life insurance, the symbol is $\bar{A}_x$:

$E[Z] = \bar{A}_x = \int_{0}^{\infty} v^t \cdot f_x(t) \, dt$

For discrete whole life insurance, the symbol is $A_x$:

$E[Z] = A_x = \sum_{k=0}^{\infty} v^{k+1} \cdot {}_k p_x \cdot q_{x+k}$

Quick Review: Remember that $E[Z]$ is just a weighted average. We take every possible payout time, calculate its present value, and multiply it by the probability of dying at that exact time.

Key Takeaway: The mean tells us the "fair price" to charge for the insurance if we weren't worried about risk or profit.

3. The Variance: Measuring the Risk

Insurance companies care about more than just the average; they need to know how much the actual results might differ from that average. This is where variance comes in.

The formula for variance is the same one you learned in statistics:

$Var(Z) = E[Z^2] - (E[Z])^2$

The "Rule of Moments" Trick:

Calculating $E[Z^2]$ might look scary, but there is a famous actuarial shortcut! To find the second moment ($E[Z^2]$), you simply calculate the APV using a new interest rate.

If the original interest rate is $i$, the "squared" version uses a force of interest that is double the original ($2\delta$), or a discount factor that is squared ($v^2$).

We denote this as ${}^2 A_x$.

$Var(Z) = {}^2 A_x - (A_x)^2$

Common Mistake to Avoid: A very common error is thinking that $E[Z^2]$ is the same as $(E[Z])^2$. It is NOT! Always calculate the "Second Moment" (${}^2 A_x$) separately by squaring the discount factor inside the expected value calculation.

4. Probabilities of $Z$

Sometimes the exam will ask: "What is the probability that the present value of the benefit is greater than 0.5?"

To solve this, you need to "un-discount" the value to find the time $t$.

Step-by-Step Process:

Set up the inequality: $P(Z > k)$.
Substitute the definition of $Z$: $P(v^T > k)$.
Take the natural log of both sides to solve for $T$.
Because $v$ is less than 1, the inequality sign will flip!
Convert the statement into a survival probability like ${}_t p_x$ or a death probability ${}_t q_x$.

Did you know? As time goes on, the present value $Z$ gets smaller. This is because a dollar paid 50 years from now is worth much less than a dollar paid tomorrow!

5. Covariance of Present Value Random Variables

In some cases, you might have two different insurance benefits for the same person (like a whole life insurance policy and an annuity). Covariance measures how these two random variables move together.

$Cov(Z_1, Z_2) = E[Z_1 Z_2] - E[Z_1]E[Z_2]$

If the two benefits are paid based on the same death time, they are highly correlated. For example, if someone dies early, the Life Insurance payout is worth a lot (high PV), but the Life Annuity payments stop early (low PV). This usually results in a negative covariance!

Key Takeaway: Covariance is essential for understanding "hedging." If one value goes up while the other goes down, the insurance company's total risk is reduced.

6. Summary and Quick Tips

Summary Box:

$Z$ is the random variable for the Present Value of a benefit.
$E[Z]$ is the actuarial symbol (like $A_x$ or $\bar{A}_x$).
$Var(Z) = {}^2 A - (A)^2$. Use the "double the force of interest" rule for the first term.
To find Probabilities, solve the inequality for $T$ or $K$.

Memory Aid: Think of the "Two-Two Rule" for variance. The Second moment uses Two times the force of interest.

Don't worry if this seems tricky at first! The most important thing is to get comfortable switching between the random variable $Z$ and the time of death $T$. Once you can "see" that $Z$ is just a function of time, the probabilities and means will start to make perfect sense. Keep practicing those integrals and summations!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What does the random variable $Z$ represent in the context of life insurance?

$Z$ is the Present Value (PV) random variable representing the value today of a future benefit payout.

What are the definitions of $Z$ for continuous versus discrete life insurance?

Continuous insurance: $Z = v^{T_x}$. Discrete insurance: $Z = v^{K_x+1}$.

What is the Actuarial Present Value (APV)?

The APV is the mean or expected value of the present value random variable, denoted as $E[Z]$.

What is the integral formula for the APV of a continuous whole life insurance?

$E[Z] = \bar{A}_x = \int_{0}^{\infty} v^t \cdot f_x(t) \, dt$

What is the summation formula for the APV of a discrete whole life insurance?

$E[Z] = A_x = \sum_{k=0}^{\infty} v^{k+1} \cdot {}_k p_x \cdot q_{x+k}$

How is the variance of the present value random variable $Z$ calculated?

$Var(Z) = E[Z^2] - (E[Z])^2$

What is the 'Rule of Moments' trick for calculating the second moment $E[Z^2]$?

To find $E[Z^2]$, calculate the APV (${}^2 A_x$) using a squared discount factor $v^2$ or double the force of interest ($2\delta$).

Why does the inequality sign flip when solving for time $T$ in probability expressions like $P(Z > k)$?

Because the present value $Z$ is a decreasing function of time $T$.

How is the covariance between two different present value random variables calculated?

$Cov(Z_1, Z_2) = E[Z_1 Z_2] - E[Z_1]E[Z_2]$

Why do life insurance and life annuities typically have a negative covariance?

They react oppositely to the time of death: early death increases insurance PV but decreases annuity PV.

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