Relationships between insurance, endowment, and annuity present value random variables

Q: How can the relationship between an insurance PV RV ($Z$) and an annuity PV RV ($Y$) be interpreted conceptually?

It can be viewed as a fixed budget equation: $1 = Z + dY$.

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 4 min read

Welcome to the Bridge: Connecting Insurance and Annuities

Hi there, future actuary! In your studies for Exam FAM, you've likely spent time looking at Life Insurance (which pays when someone dies) and Life Annuities (which pay while someone is alive). At first, they might seem like opposites, but mathematically, they are like two sides of the same coin.

In this chapter, we are going to learn how to build a bridge between them. Why is this important? Because in the exam, you'll often be given information about an annuity and asked to find the value of an insurance policy (or vice-versa). Learning these relationships will save you time and help you avoid complex integrations or summations!

1. The Fundamental Random Variable Relationship

Before we look at averages (expected values), we need to look at the Present Value Random Variables themselves. Let's look at the discrete case (payments at the end of the year) for a Whole Life Insurance and a Whole Life Annuity-Due.

For a person aged $x$:

The Insurance PV RV: $Z = v^{K_x+1}$
The Annuity-Due PV RV: $Y = \ddot{a}_{\overline{K_x+1}|} = \frac{1 - v^{K_x+1}}{d}$

The "Magic" Link:
If you look closely at the formula for $Y$, you'll see $v^{K_x+1}$ (which is $Z$) hidden inside it! This gives us our first major relationship:
$Y = \frac{1 - Z}{d}$

Analogy: The "Fixed Budget"
Imagine you have $1.00 today. You can either keep it as a death benefit (Insurance) or "spend" it to buy a series of annual payments (Annuity). The relationship $1 = Z + dY$ tells us that the initial $1.00 is always accounted for by either the discounted death benefit or the interest lost (the discount $d$) on the annuity payments.

Key Takeaway:

The annuity random variable and the insurance random variable are linearly related. If you know the value of one, you can always find the other using the discount rate $d$.

2. Expected Values: The Actuarial Present Values

Since the random variables are related linearly, their expected values (the prices we charge) follow the exact same pattern. This is one of the most used formulas in all of actuarial science!

For Discrete Whole Life:
$\ddot{a}_x = \frac{1 - A_x}{d}$ or, rearranged: $A_x = 1 - d\ddot{a}_x$

For Continuous Whole Life:
In the continuous world, we swap $d$ (effective discount) for $\delta$ (force of interest):
$\bar{a}_x = \frac{1 - \bar{A}_x}{\delta}$ or, rearranged: $\bar{A}_x = 1 - \delta \bar{a}_x$

Step-by-Step: Converting Insurance to Annuity
1. Identify the interest rate and calculate $d$ using $d = \frac{i}{1+i}$.
2. Subtract the Insurance value ($A_x$) from 1.
3. Divide the result by $d$.
4. You now have the Annuity value ($\ddot{a}_x$)!

Common Mistake to Avoid:
Don't mix your types! Ensure you use $d$ with discrete (annuity-due) formulas and $\delta$ with continuous formulas. If the question asks for an annuity-immediate ($a_x$), remember that $a_x = \ddot{a}_x - 1$.

3. Variance Relationships

Actuaries don't just care about the average; they care about risk (variance). Because $Y = \frac{1 - Z}{d}$, we can use basic probability rules to find the variance.

Remember that $\text{Var}(aX + b) = a^2 \text{Var}(X)$. In our case, $a = -\frac{1}{d}$ and $b = \frac{1}{d}$.

The Variance Formula:
$\text{Var}(Y) = \frac{\text{Var}(Z)}{d^2}$

In continuous notation: $\text{Var}(\bar{Y}) = \frac{\text{Var}(\bar{Z})}{\delta^2}$

Did you know?
The variance of an annuity is usually much "larger" than the insurance variance because of that $d^2$ in the denominator. Since $d$ is a small decimal (like 0.05), dividing by $d^2$ (0.0025) makes the number grow significantly!

4. Endowment Insurance: The Sum of Parts

An Endowment Insurance is a policy that pays if you die OR if you survive to the end of a term. It is the combination of two simpler pieces:

The Formula:
$A_{x:n|} = A^1_{x:n|} + A_{x:\overline{n}|}^1$

Where:
- $A^1_{x:n|}$ is the Term Insurance (pays only if you die during the $n$ years).
- $A_{x:\overline{n}|}^1$ (also written as ${}_n E_x$) is the Pure Endowment (pays only if you survive the $n$ years).

Quick Review Box:
- Term Insurance: "I pay if you die soon."
- Pure Endowment: "I pay if you live long."
- Endowment Insurance: "I pay no matter what, as long as time passes."

Key Takeaway:

The relationship $A_{x:n|} = 1 - d\ddot{a}_{x:n|}$ also holds true for the n-year term! The math works for both whole life and temporary (term) periods, as long as you are consistent with the timeframe $n$.

5. Summary and Study Tips

Don't worry if these formulas feel like a lot to memorize. Just remember the "Base Bridge": Insurance = 1 - (Discount $\times$ Annuity).

Mnemonic: Think of "AID" — Annuity, Insurance, Discount. They are the three pillars of this chapter.
The "1" Rule: In these relationships, the number "1" represents the total unit of benefit. It is being split between the "living benefit" (annuity) and the "death benefit" (insurance).
Check your work: Annuity values ($\ddot{a}_x$) are usually large numbers (like 12.5 or 15.0), while Insurance values ($A_x$) are small decimals (like 0.2 or 0.4). If you get an Insurance value greater than 1, you've likely flipped a formula!

Keep practicing! These relationships are the "shortcuts" that separate the students who finish the FAM exam from those who run out of time. You've got this!

Quick check

Can you answer these now?

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

What is the mathematical relationship between the present value random variable of a whole life insurance ($Z$) and a whole life annuity-due ($Y$)?

$Y = \frac{1 - Z}{d}$

How can the relationship between an insurance PV RV ($Z$) and an annuity PV RV ($Y$) be interpreted conceptually?

It can be viewed as a fixed budget equation: $1 = Z + dY$.

What is the formula relating the Actuarial Present Values (APVs) of a discrete whole life insurance and a whole life annuity-due?

$A_x = 1 - d\ddot{a}_x$ or $\ddot{a}_x = \frac{1 - A_x}{d}$

What is the formula relating the Actuarial Present Values of a continuous whole life insurance and a continuous whole life annuity?

$\bar{A}_x = 1 - \delta \bar{a}_x$ or $\bar{a}_x = \frac{1 - \bar{A}_x}{\delta}$

When relating insurance and annuities, which interest rate measures must be paired with discrete vs. continuous models?

Use the effective discount rate $d$ for discrete/due models and the force of interest $\delta$ for continuous models.

How do you calculate the variance of an annuity random variable ($Y$) if you know the variance of the insurance random variable ($Z$)?

$\text{Var}(Y) = \frac{\text{Var}(Z)}{d^2}$ (discrete) or $\text{Var}(\bar{Y}) = \frac{\text{Var}(\bar{Z})}{\delta^2}$ (continuous).

What are the two component parts of an endowment insurance ($A_{x:n|}$)?

$A_{x:n|} = A^1_{x:n|} + A_{x:\overline{n}|}^1$

Does the linear relationship between insurance and annuities apply to temporary (n-year) contracts?

Yes: $A_{x:n|} = 1 - d\ddot{a}_{x:n|}$.

What are typical relative magnitudes for insurance and annuity values that can help verify your calculations?

Insurance values ($A_x$) are usually small decimals (e.g., 0.2), while annuity values ($\ddot{a}_x$) are usually much larger (e.g., 15.0).

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