Welcome to Multiple Decrement Models!
In your actuarial journey so far, you have likely spent a lot of time looking at models where only one thing can happen: death. While "Single Decrement" models are useful, the real world is a bit messier! In a pension fund, for example, a member might leave because they retire, because they pass away, or because they simply quit their job to work elsewhere. Each of these is a different "exit" from the fund.
This chapter is all about tracking those different ways to leave and, more importantly, valuing the cashflows that depend on which "exit" someone takes. Don't worry if this seems a bit more complex than the life tables you've seen before—we will break it down step-by-step!
1. What is a Multiple Decrement Model?
A Multiple Decrement Model describes a situation where an individual is in one "active" state and can leave that state for several different reasons (decrements). Once they leave, they do not come back (this is the key difference between multiple decrement models and multi-state models).
The "Leaky Bucket" Analogy:
Imagine a bucket full of water. This bucket has three different holes at the bottom. One hole represents "Death," another "Retirement," and the third "Resignation." Water (representing people) can leak out of any of these holes. To understand how much water is left, you need to know the flow rate of each hole and how they work together.
Key Terms to Remember:
- Decrement: A specific cause of exit (e.g., death, disability, withdrawal).
- Dependent Probability \( q_x^{(j)} \): The probability that a life aged \( x \) will leave the group within one year specifically due to cause \( j \), in the presence of all other causes.
- Total Probability of Exit \( q_x^{\tau} \): The probability that a life aged \( x \) will leave the group for any reason within one year. This is simply the sum of all individual dependent probabilities: \( q_x^{\tau} = \sum q_x^{(j)} \).
Quick Review: In a Multiple Decrement Table (MDT), we use \( l_x^{\tau} \) to represent the number of active lives at age \( x \). The number of people leaving due to cause \( j \) is denoted as \( d_x^{(j)} \).
2. Forces of Decrement
Just like we have a force of mortality \( \mu_x \), we have a force of decrement for each cause. We denote the force of decrement for cause \( j \) at age \( x \) as \( \mu_x^{(j)} \).
The total force of decrement, \( \mu_x^{\tau} \), is the sum of the forces for each individual cause:
\( \mu_x^{\tau} = \mu_x^{(1)} + \mu_x^{(2)} + ... + \mu_x^{(m)} \)
The Survival Probability:
The probability that a person remains in the active state for \( t \) years is:
\( _{t}p_x^{\tau} = \exp( -\int_{0}^{t} \mu_{x+s}^{\tau} ds ) \)
Wait! Does one cause affect the other?
In these models, we assume the forces are acting simultaneously. If someone dies, they can no longer retire. This is why we call the probabilities "dependent"—the chance of retiring depends on whether the person survives other risks like death!
3. Dependent vs. Independent Rates
This is often where students find things a bit tricky, but here is a simple way to think about it.
Dependent Rates (The Real World)
As discussed, these are the probabilities we observe in a group where all causes of exit are operating at the same time. Symbol: \( q_x^{(j)} \).
Independent Rates (The "What If" World)
These are also called Associated Single Decrement Rates. We imagine a world where only one cause of exit exists. For example, "What is the probability of this person quitting if we assume they literally cannot die or retire?" Symbol: \( q_x^{'(j)} \).
Common Mistake to Avoid:
Never simply add independent rates \( q_x^{'(j)} \) together to get a total. You can only sum dependent rates \( q_x^{(j)} \) to get the total probability of leaving.
Memory Aid:
Think of Dependent rates as "Competing". All the exits are racing to "catch" the person first.
4. Converting Between Rates (The Assumptions)
To move between independent rates \( q_x^{'(j)} \) and dependent rates \( q_x^{(j)} \), we usually need an assumption about how the exits happen during the year.
Assumption 1: Uniform Distribution of Decrements (UDD) in the MDT
If we assume that each decrement \( d_x^{(j)} \) is spread evenly over the year in the multiple decrement table, we use this approximation:
\( q_x^{(j)} \approx q_x^{'(j)} [1 - \frac{1}{2} \sum_{k \neq j} q_x^{'(k)}] \)
Note: This formula is used when you have independent rates and want to find the dependent rate.
Assumption 2: Constant Forces of Decrement
If we assume the force of each decrement is constant over the year of age, the relationship is:
\( q_x^{(j)} = \frac{\mu^{(j)}}{\mu^{\tau}} \cdot q_x^{\tau} \)
Where \( \mu^{(j)} / \mu^{\tau} = \ln(1 - q_x^{'(j)}) / \ln(1 - q_x^{\tau}) \).
Key Takeaway: The dependent rate \( q_x^{(j)} \) is always smaller than the independent rate \( q_x^{'(j)} \). Why? Because in the dependent world, some people who would have left for cause \( j \) are "stolen" by other causes first!
5. Projecting and Valuing Expected Cashflows
Now for the most important part: calculating the value of a benefit. In CM1, you will often be asked to find the Expected Present Value (EPV) of a benefit paid upon a specific exit.
The General Recipe for EPV:
1. Identify the Probability: What is the chance the person is still "active" at time \( t \), and then leaves due to cause \( j \) at exactly that moment? \( _{t}p_x^{\tau} \cdot \mu_{x+t}^{(j)} \)
2. Identify the Cashflow: How much is paid? \( B_{t} \)
3. Identify the Discount Factor: Pull the money back to today's value. \( v^t \)
4. Integrate (or Sum): Add them all up over the possible timeframe.
Formula for a benefit of 1 paid immediately on decrement \( j \):
\( \bar{A}_x^{(j)} = \int_{0}^{\infty} v^t \cdot _{t}p_x^{\tau} \cdot \mu_{x+t}^{(j)} dt \)
Real-World Example:
An employer pays a lump sum of $50,000 if an employee dies while in service, but nothing if they resign. To value this, the actuary only cares about the force of mortality \( \mu^{(death)} \), but must use the total survival probability \( _{t}p_x^{\tau} \) because if the employee resigns, the "death-in-service" benefit is no longer a possibility!
6. Step-by-Step: Solving a Valuation Problem
If you are faced with a calculation question, follow these steps:
Step 1: Identify all the possible decrements (e.g., death, withdrawal).
Step 2: Calculate the Total Force or Total Probability of Survival. You need to know the probability that the person is still "available" to experience the specific decrement you are interested in.
Step 3: Set up the EPV equation for the specific benefit. Only include the "Force" or "Rate" for the cause that triggers the payment.
Step 4: If the benefit is a "survival benefit" (like a pension starting at 65), you only need the total survival probability \( _{t}p_x^{\tau} \) to see if they made it to age 65 without any of the decrements happening.
Did you know?
Pension actuaries spend a huge amount of time on these calculations. They have to assume how many people will stay until retirement age to ensure the company has saved enough money to pay their pensions!
Summary and Key Takeaways
- Multiple Decrements: Used when there is more than one way to leave a state (e.g., Death vs. Retirement).
- Forces: Forces of decrement are additive: \( \mu_x^{\tau} = \sum \mu_x^{(j)} \).
- Dependence: The probability of leaving for one cause depends on the other causes because they compete to "remove" the life from the state.
- EPV: When valuing a benefit for cause \( j \), always use the total survival probability \( _{t}p_x^{\tau} \) combined with the specific rate/force for cause \( j \).
- Accuracy: Always check if you are given independent or dependent rates before you start your calculations!
Don't worry if the notation looks like an alphabet soup at first. Focus on the logic: Survival to time \( t \) (using ALL holes in the bucket) multiplied by the exit rate at time \( t \) (using ONLY the specific hole that triggers the payment). You've got this!