Present value random variables for life insurance, endowment, and annuities

Q: What are the two fundamental components that define a Present Value Random Variable (PVRV) in life contingencies?

A PVRV is composed of the Time of Death ($T_x$) and the Discount Factor ($v^t$).

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

Welcome to the Core of Actuarial Math!

If you have ever wondered how insurance companies decide how much to charge for a policy, you are in the right place. In this chapter, we are going to look at Present Value Random Variables (PVRVs).

Think of this as the "bridge" between probability (when will someone die?) and finance (how much is a future dollar worth today?). We are combining the Time Value of Money with Life Contingencies. Don't worry if this seems a bit abstract at first—we will break it down into simple building blocks that you can use to master Exam FAM!

1. The Basic Ingredients: Time and Money

Before we dive into the specific types of insurance, let’s look at the two ingredients that make up every Present Value Random Variable:

1. The Time of Death ($T_x$): This is a random variable representing how much longer a person aged $x$ will live. Because we don't know exactly when someone will pass away, the payout time is a "random" variable.
2. The Discount Factor ($v^t$): This represents the "Present Value." A dollar paid 50 years from now is worth much less than a dollar paid today. We use $v^t = e^{-\delta t}$ in continuous models.

Analogy: Imagine you have a coupon that gives you $100, but you can only use it on the day it rains. You don't know when it will rain (that's the random part), and you know that $100 in the future won't buy as much as $100 today (that's the discount part). A PVRV is just the math that combines those two uncertainties.

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2. Life Insurance PVRVs

Life insurance pays a benefit when the insured person dies. We usually denote the Present Value Random Variable as $Z$.

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Whole Life Insurance

This policy pays out whenever death occurs, no matter how far in the future that is.
\nThe Formula: $Z = v^{T_x}$
\nThe Expected Value: We call this the Net Single Premium, denoted by $\bar{A}_x$.
\nQuick Tip: In the continuous case, $\bar{A}_x = E[v^{T_x}] = \int_{0}^{\infty} v^t \cdot {}_tp_x \mu_{x+t} dt$.

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n-Year Term Life Insurance

This only pays if the person dies within a specific window of $n$ years. If they survive past year $n$, the insurance company pays nothing.
\nThe PVRV (Z):
\n• If $T_x \le n$, $Z = v^{T_x}$
\n• If $T_x > n$, $Z = 0$
\nKey Takeaway: Term insurance is "use it or lose it." If you outlive the term, the present value of the benefit is zero.

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n-Year Pure Endowment

This is the opposite of term insurance. It pays only if you are still alive at the end of $n$ years.
\nThe PVRV (Z):
\n• If $T_x \le n$, $Z = 0$
\n• If $T_x > n$, $Z = v^n$
\nMnemonic: Think of a Pure Endowment as a "Survival Bonus." You only get the "E" (Endowment) if you "E"ndure until the end!

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n-Year Endowment Insurance

This is a combination of the two above. You get a payout if you die during the term OR a payout if you survive to the end. It’s a "win-win" for the policyholder.
\nThe Formula: Endowment Insurance = Term Insurance + Pure Endowment
\nSymbol: $\bar{A}_{x:n|} = \bar{A}^1_{x:n|} + A_{x:n|}^{\quad 1}$

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3. Life Annuity PVRVs

While insurance pays a lump sum at death, an annuity pays a stream of income while you are still alive.
\nAnalogy: Insurance is like a single bucket of water dumped on you once. An annuity is like a dripping faucet that keeps running as long as you are there to catch the water.

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Whole Life Annuity

This pays a continuous stream (usually $1 per year) until the person dies.
The PVRV (Y): $Y = \bar{a}_{\overline{T_x}|} = \frac{1 - v^{T_x}}{\delta}$
Expected Value: $\bar{a}_x$

n-Year Temporary Life Annuity

This pays until you die, but for no longer than $n$ years.
The PVRV (Y):
• If $T_x \le n$, $Y = \bar{a}_{\overline{T_x}|}$
• If $T_x > n$, $Y = \bar{a}_{\overline{n}|}$

Quick Review Box: Insurance vs. Annuities

• Insurance (Z): Focuses on the moment of death. High risk for the company if you die early.
• Annuity (Y): Focuses on the duration of life. High risk for the company if you live a very long time.

4. Variance of PVRVs (The "Rule of Two")

Exam FAM loves to ask for the variance of these random variables. To find the variance, you need the "Second Moment."

The Trick: To calculate the second moment of an insurance PVRV, simply double the force of interest ($\delta$).
If the first moment is $\bar{A}_x$ calculated at force $\delta$, the second moment is ${}^2\bar{A}_x$, which is the same formula but using $2\delta$ as the interest rate.

The Variance Formula: $Var(Z) = {}2\bar{A}_x - (\bar{A}_x)^2$

Common Mistake to Avoid:

Students often try to double the interest rate $i$. Don't do that! You must double the force of interest ($\delta$) or square the discount factor ($v^2$). If you are given $i$, convert it to $v$ first, then use $v^2$ for the second moment.

5. Important Relationships (The "Master Equation")

There is a beautiful mathematical link between life insurance and life annuities. You don't need to memorize two separate worlds; you just need to know how they connect.

The Relationship: $1 = \delta \bar{a}_x + \bar{A}_x$

Why this makes sense: Think of it this way: If you have $1 today, you can either keep the interest (the annuity $\delta \bar{a}_x$) or eventually lose the principal when you die (the insurance $\bar{A}_x$). Together, they always equal the original $1.

Summary and Key Takeaways

• PVRVs combine the time of death ($T_x$) with the discount factor ($v^t$).
• Whole Life pays whenever you die; Term pays only if you die soon; Endowment pays no matter what.
• Annuities are the present value of a stream of payments while alive.
• The "Rule of Two": To find the variance of insurance, calculate the expected value at double the force of interest ($2\delta$).
• The Link: Insurance and annuities are two sides of the same coin, connected by the formula $1 = \delta \bar{a}_x + \bar{A}_x$.

Don't worry if the symbols ($\bar{A}, \bar{a}, \delta$) feel like alphabet soup right now. The more you practice setting up the PVRV for each story problem, the more natural it will become. You've got this!

Quick check

Can you answer these now?

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

What are the two fundamental components that define a Present Value Random Variable (PVRV) in life contingencies?

A PVRV is composed of the Time of Death ($T_x$) and the Discount Factor ($v^t$).

What is the mathematical definition of the PVRV for a continuous Whole Life Insurance of 1?

The PVRV is defined as $Z = v^{T_x}$.

Under what circumstances does an n-year term life insurance PVRV result in a value of zero?

The PVRV is zero if the insured person survives beyond the term of $n$ years (i.e., if $T_x > n$).

How is the PVRV for an n-year pure endowment defined?

The PVRV $Z$ is $0$ if $T_x \le n$, and $v^n$ if $T_x > n$.

What is the relationship between Endowment Insurance, Term Insurance, and Pure Endowment?

Endowment Insurance is the sum of Term Insurance and Pure Endowment: $\bar{A}_{x:n|} = \bar{A}^1_{x:n|} + A_{x:n|}^{\quad 1}$.

What is the PVRV for a continuous whole life annuity of 1 per year?

$Y = \bar{a}_{\overline{T_x}|} = \frac{1 - v^{T_x}}{\delta}$.

What is the PVRV for an n-year temporary life annuity?

$Y = \bar{a}_{\overline{T_x}|}$ if $T_x \le n$, and $Y = \bar{a}_{\overline{n}|}$ if $T_x > n$.

How do you calculate the second moment of an insurance PVRV to find its variance?

Apply the "Rule of Two" by calculating the expected value using double the force of interest ($2\delta$).

What formula expresses the relationship between a continuous whole life annuity and a whole life insurance?

$1 = \delta \bar{a}_x + \bar{A}_x$.

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