Survival and mortality probabilities from a life table with fractional age assumptions

Welcome to Fractional Ages: Filling in the Gaps!

In your actuarial journey so far, you’ve likely worked with life tables. These tables are great, but they usually only give us data for whole years—like age 40, 41, or 42. But what happens if someone dies at age 40 and six months?

In this chapter, we learn how to "fill in the blanks" between the birthdays. Since we don't know exactly when someone dies during the year from the table alone, we make fractional age assumptions. These are mathematical "best guesses" that allow us to calculate probabilities for any point in time. Don't worry if this seems a bit abstract at first—once you see the patterns, it becomes a simple matter of choosing the right formula!

1. Why do we need these assumptions?

Imagine a life table that tells you 1,000 people are alive at age 50 (\(l_{50} = 1000\)) and 900 are alive at age 51 (\(l_{51} = 900\)). We know 100 people died during that year. But when did they die? Did they all die in a winter flu outbreak in the first month? Or did they die gradually throughout the year?

Because we don't know the exact timing, we use two primary models to estimate it:

• Uniform Distribution of Deaths (UDD)
• Constant Force of Mortality (CFM)

Quick Review: Prerequisite Notation

Before we dive in, remember these basics:

• \(q_x\): The probability that a person age \(x\) dies before age \(x+1\).
• \(p_x\): The probability that a person age \(x\) survives to age \(x+1\).
• \(t\): A fraction of a year (where \(0 \le t \le 1\)).

2. Uniform Distribution of Deaths (UDD)

The Uniform Distribution of Deaths (UDD) is the most common assumption you’ll see on Exam FAM. It assumes that deaths are spread evenly throughout the year.

The Analogy: Imagine 120 people are scheduled to leave a building over the course of an hour. Under UDD, exactly 2 people leave every minute. It’s a steady, linear stream.

Key Formulas for UDD:

1. Interpolating Survivors: \(l_{x+t} = (1-t)l_x + t \cdot l_{x+1}\)
(This is just a straight line between the two years!)

2. Probability of Death: \(_t q_x = t \cdot q_x\)
Example: If the probability of dying during the year is 10%, the probability of dying in the first half-year (\(t=0.5\)) is \(0.5 \times 10\% = 5\%\).

3. Force of Mortality: \(\mu_{x+t} = \frac{q_x}{1 - t \cdot q_x}\)
Wait! Notice that even though deaths are "uniform," the force of mortality (the instantaneous risk) actually increases as \(t\) goes from 0 to 1. This is because the "pool" of survivors is getting smaller, but people are still dying at the same rate.

Quick Review: UDD Summary Box

Visual: A straight line (linear interpolation).
Key Logic: \(_t q_x\) is proportional to \(t\).
Common Mistake: Forgetting that \(_t q_{x+s}\) is NOT just \(t \cdot q_x\) if \(s > 0\). Always bring your calculation back to the start of the year (\(x\)) if you get stuck!

3. Constant Force of Mortality (CFM)

The Constant Force of Mortality (CFM) assumes that the "risk" of dying (the hazard rate) stays exactly the same every single second during the year.

The Analogy: Think of a bag of popcorn. If the heat is constant, each kernel has the same probability of popping at any given moment. The number of kernels popping might slow down as fewer are left, but the heat (the force) is constant.

Key Formulas for CFM:

1. Survival Probability: \(_t p_x = (p_x)^t\)
Example: If the probability of surviving the year is 0.90, the probability of surviving the first half-year is \((0.90)^{0.5} = 0.9487\).

2. The Force: \(\mu_{x+t} = \mu = -\ln(p_x)\) for \(0 \le t < 1\).
Notice there is no \(t\) in the force formula! It’s the same regardless of when you measure it during the year.

3. Interpolating Survivors: \(l_{x+t} = l_x \cdot (p_x)^t\)

Quick Review: CFM Summary Box

Visual: An exponential curve (geometric interpolation).
Key Logic: \(_t p_x\) uses an exponent (\(t\)).
Common Mistake: Using the UDD linear formula when the problem specifies Constant Force. Look for the keywords!

4. Comparing UDD and CFM

You might be wondering: "Which one gives a higher value?" This is a classic Exam FAM concept!

Did you know? For the same annual \(q_x\), the UDD assumption will always result in a lower survival probability (\(_t p_x\)) in the middle of the year compared to the CFM assumption.

Memory Trick:
UDD = Under (Lower survival probabilities \(_t p_x\) than CFM).
CFM = Convex/Constant (Higher survival probabilities \(_t p_x\) than UDD).

5. Step-by-Step: Solving a Fractional Age Problem

If you see a question asking for \(_t q_{x+s}\) where \(x\) is an integer but \(s\) is a fraction, follow these steps:

Step 1: Identify the Assumption. Does the problem say "Uniform Distribution" or "Constant Force"?
Step 2: Break it down. Remember the fundamental rule: \(_t p_{x+s} = \frac{_{s+t} p_x}{_s p_x}\). It is almost always easier to calculate the probability from the integer age \(x\) and then divide.
Step 3: Apply the specific formula.
- If UDD: Use \(_k p_x = 1 - k \cdot q_x\).
- If CFM: Use \(_k p_x = (p_x)^k\).
Step 4: Final Calculation. Solve the fraction.

6. Summary and Key Takeaways

- Life Tables provide data at integer ages; Fractional Assumptions bridge the gaps.
- UDD (Uniform Distribution of Deaths): Deaths occur linearly. \(_t q_x = t \cdot q_x\). The force of mortality \(\mu\) increases during the year.
- CFM (Constant Force of Mortality): The risk is constant. \(_t p_x = (p_x)^t\). The force of mortality \(\mu\) is constant during the year.
- Exam Tip: If the question doesn't specify an assumption, read carefully—sometimes it's hidden in the phrasing (e.g., "deaths are distributed linearly" means UDD).

Don't worry if the formulas for \(\mu\) look intimidating! Focus on mastering the survival (\(p\)) and death (\(q\)) formulas first, as they appear most frequently. You've got this!