Welcome to Parametric Survival Models!
Hello there! Welcome to one of the most practical parts of Exam FAM. So far, you might have looked at mortality using life tables (those big grids of numbers). But what if we could describe human life using a simple mathematical formula instead? That is exactly what parametric survival models do!
Instead of looking up \(l_x\) values in a table, we use equations to calculate the probability of someone surviving or dying. Think of it like moving from a paper map (the life table) to a GPS (the formula). Both get you to the same place, but the formula is often smoother and easier to work with once you know the "rules" of the road. Let’s dive in!
1. The Foundation: The Force of Mortality (\(\mu_x\))
Before we look at specific models, we need to understand the "engine" that drives them: the force of mortality, denoted as \(\mu_x\). In simple terms, this is the instantaneous risk of death for someone at age \(x\).
To find the probability that a person aged \(x\) survives for another \(t\) years (denoted as \({}_t p_x\)), we use this fundamental relationship:
\({}_t p_x = e^{-\int_0^t \mu_{x+s} ds}\)
Don't worry if this seems tricky at first! You don't always have to do heavy calculus. Many of the models we study have "shortcut" formulas that come from this integral. The key takeaway is: if you know the formula for \(\mu\), you can find the probability of survival.
Quick Review: The Basics
• \({}_t p_x\): Probability a person age \(x\) survives to age \(x+t\).
• \({}_t q_x\): Probability a person age \(x\) dies before age \(x+t\). Remember: \({}_t q_x = 1 - {}_t p_x\).
• \(\mu_x\): The "hazard rate" or force of mortality at exact age \(x\).
Key Takeaway: Parametric models provide a functional form for \(\mu_x\), which we then use to calculate survival probabilities.
2. The Exponential Model (Constant Force)
The Exponential Model is the simplest model you will encounter. It assumes that the risk of death is the same regardless of how old you are. It’s like a lightbulb—it doesn't "age"; it just eventually fails at a constant rate.
The Formula:
\(\mu_x = \mu\) (a constant value for all \(x\))
Survival Probability:
\({}_t p_x = e^{-\mu t}\)
Example: If the constant force of mortality is 0.02, the probability of a 30-year-old surviving 10 years is \(e^{-0.02 \times 10} = e^{-0.2} \approx 0.8187\).
Did you know? In this model, \({}_t p_x\) does not depend on the current age \(x\). A 20-year-old and an 80-year-old both have the same probability of surviving the next 5 years. While unrealistic for humans over long periods, it's very useful for short durations or specific types of insurance risks.
Key Takeaway: Constant force means \({}_t p_x = e^{-\mu t}\). Simple and easy!
3. Gompertz Law
Benjamin Gompertz noticed something in 1825: as humans get older, their risk of dying increases exponentially. This is the Gompertz Law.
The Formula:
\(\mu_x = B \cdot c^x\)
Where \(B > 0\) and \(c > 1\).
Survival Probability:
\({}_t p_x = \exp \left[ -\frac{B \cdot c^x}{\ln c} (c^t - 1) \right] \)
Analogy: Imagine a snowball rolling down a hill. As it goes further (as you get older), it picks up speed and mass faster and faster. That’s how the risk of death behaves in the Gompertz model.
Memory Trick: Gompertz starts with G, like Growth. The risk grows at a geometric rate.
Key Takeaway: Gompertz models aging. If you see a formula where age \(x\) is in the exponent (like \(c^x\)), you are looking at Gompertz!
4. Makeham Law
Makeham decided to improve Gompertz's idea. He realized that people don't just die from "aging." Sometimes, people die from accidents or "bad luck" that has nothing to do with how old they are. He added a constant term to the Gompertz model.
The Formula:
\(\mu_x = A + B \cdot c^x\)
Components:
• \(A\): The "age-independent" component (accidents, lightning strikes, etc.).
• \(B \cdot c^x\): The "age-dependent" component (wear and tear on the body).
Survival Probability:
\({}_t p_x = e^{-At} \cdot \exp \left[ -\frac{B \cdot c^x}{\ln c} (c^t - 1) \right] \)
Common Mistake: Forgetting to multiply \(A\) by \(t\) in the survival formula. Remember, the accident risk (\(A\)) accumulates over the time period \(t\)!
Key Takeaway: Makeham = Gompertz + Accident Constant. It’s the most famous model for human mortality in actuarial science.
5. The Weibull Model
The Weibull Model is another way to model increasing risk, but instead of using an exponential growth (like \(c^x\)), it uses a power function of age.
The Formula:
\(\mu_x = k \cdot x^n\)
(In some textbooks, this is written using different Greek letters like \(\alpha\) and \(\beta\), so always check the parameters provided in the exam question!)
Survival Probability:
If \(\mu_x = k \cdot x^n\), then to find survival, we integrate. A common form used in FAM is:
\({}_t p_x = e^{-[k(x+t)^{n+1} - kx^{n+1}] / (n+1)}\)
Real-World Example: The Weibull distribution is often used in engineering to model the "failure rate" of mechanical parts. Some parts fail early (infant mortality), and some fail more often as they wear out.
Key Takeaway: If you see age \(x\) raised to a power (like \(x^2\) or \(x^3\)), think Weibull!
6. Summary Table for Quick Reference
Use this table to keep the models straight in your head during your final review:
| Model | Force of Mortality (\(\mu_x\)) | Characteristics |
|---|---|---|
| Exponential | \(\mu\) | Constant; age doesn't matter. |
| Gompertz | \(B \cdot c^x\) | Risk grows exponentially with age. |
| Makeham | \(A + B \cdot c^x\) | Gompertz + constant risk of accidents. |
| Weibull | \(k \cdot x^n\) | Risk follows a power function of age. |
7. Final Tips for Success
1. Watch your variables: Be careful with \(x\) (current age) and \(t\) (time into the future). In the exam, they might ask for the probability that a 40-year-old dies before age 50. Here, \(x=40\) and \(t=10\).
2. Integration is your friend: If you forget a specific shortcut formula for \({}_t p_x\), just remember the golden rule: \({}_t p_x = e^{-\int_0^t \mu_{x+s} ds}\). Most of the integrals you'll face for these models are basic power rules or exponential rules.
3. Sanity Check: Survival probabilities (\(p\)) should always be between 0 and 1. If you get a number like 1.2 or -0.5, stop and check your algebra!
You've got this! These models might look intimidating with all the exponents, but once you recognize the patterns (Exponential = Constant, Gompertz = \(c^x\), Weibull = \(x^n\)), you'll be solving these problems in no time.