Central vs. Initial ETR: Problem Solving

Introduction

In our previous lesson, we established the fundamental concepts of exposure to risk, distinguishing between the duration-based central exposed to risk ( $E_{x}^{c}$ ) and the headcount-based initial exposed to risk ( $E_{x}$ ). You practiced calculating $E_{x}^{c}$ directly from individual data, which is the cornerstone of modern mortality investigation.

This lesson builds directly on that foundation. We will now formalize the mathematical relationship between these two exposure measures. Understanding this connection is not just a theoretical exercise; it provides a powerful tool for converting between the different types of mortality rates derived from these exposures. This is particularly useful when working with historical data or simplified models that might only provide one type of exposure measure. By the end of this lesson, you will be able to apply this relationship to solve practical problems, moving fluently between the two frameworks.

The Relationship Between Central and Initial Exposure

At its heart, the difference between initial and central exposed to risk is how each measure accounts for the individuals who die during the age interval $[x, x + 1)$ .

Initial Exposed to Risk ( $E_{x}$ ): In its simplest form, this is a count of the number of lives that start the year of age $x$ under observation. The associated mortality rate, $q_{x} = θ_{x} / E_{x}$ , represents the probability of dying during the year of age.
Central Exposed to Risk ( $E_{x}^{c}$ ): This is the total time, in person-years, that all individuals were observed at risk while aged $x$ . The associated rate, $m_{x} = θ_{x} / E_{x}^{c}$ , represents the central rate of mortality, which is an estimate of the force of mortality over that year of age.

To connect these two quantities, we need to make an assumption about when deaths occur. The standard assumption is the Uniform Distribution of Deaths (UDD) over the year of age. This means that deaths are assumed to occur evenly throughout the year.

Under UDD, an individual who dies at age $x$ will, on average, have been exposed to the risk of death for half of the year.

Let's derive the relationship:
Suppose $E_{x}$ lives start the year of age $x$ . During that year, $θ_{x}$ of them die and $E_{x} - θ_{x}$ survive to age $x + 1$ .

The $E_{x} - θ_{x}$ survivors each contribute 1 full year to the central exposure.
- Total exposure from survivors = $(E_{x} - θ_{x}) \times 1$ .
The $θ_{x}$ deaths, under the UDD assumption, each contribute an average of $1/2$ a year to the central exposure.
- Total exposure from deaths = $θ_{x} \times \frac{1}{2}$ .

The total central exposed to risk is the sum of these two components:

E_{x}^{c} = (E_{x} - θ_{x}) + \frac{1}{2} θ_{x}

This simplifies to the key relationship:

E_{x}^{c} \approx E_{x} - \frac{1}{2} θ_{x}

This approximation is fundamental. It allows us to estimate the central exposure if we know the initial exposure and the number of deaths.

Connecting Mortality Rates

With the relationship between the exposure measures established, we can now derive a formula to connect the mortality rate $q_{x}$ and the central death rate $m_{x}$ .

Recall the definitions:

q_{x} = \frac{θ _{x}}{E _{x}} and m_{x} = \frac{θ _{x}}{E _{x}^{c}}

We can express $θ_{x}$ as $θ_{x} = q_{x} E_{x}$ . Now, substitute this and the exposure relationship into the formula for $m_{x}$ :

m_{x} = \frac{θ _{x}}{E _{x}^{c}} \approx \frac{θ _{x}}{E _{x} - \frac{1}{2} θ _{x}}

Now, divide the numerator and the denominator by $E_{x}$ :

m_{x} \approx \frac{θ _{x} / E _{x}}{( E _{x} / E _{x} ) - \frac{1}{2} ( θ _{x} / E _{x} )}

Substituting $q_{x} = θ_{x} / E_{x}$ , we get the standard formula connecting the two rates:

m_{x} \approx \frac{q _{x}}{1 - \frac{1}{2} q _{x}}

This formula is invaluable for converting between the two rate types, a common task in actuarial analysis. You can also rearrange it to express $q_{x}$ in terms of $m_{x}$ :

q_{x} \approx \frac{m _{x}}{1 + \frac{1}{2} m _{x}}

Problem-Solving Example

Let's put these formulas into practice. Your problem-solving skills will be key here.

Scenario:
An actuarial mortality study for a large pension plan provides the following summary statistics for members aged 65 last birthday:

Initial exposed to risk, $E_{65}$ = 25,000
Number of deaths observed during the year of age 65, $θ_{65}$ = 300

Tasks:

Calculate the mortality rate, $q_{65}$ .
Estimate the central exposed to risk, $E_{65}^{c}$ , using the relationship derived above.
Calculate the central death rate, $m_{65}$ .
Verify that your calculated $q_{65}$ and $m_{65}$ satisfy the formula $m_{x} \approx \frac{q _{x}}{1 - 0.5 q _{x}}$ .

Solution:

Calculate $q_{65}$ :
$q_{65} = \frac{θ _{65}}{E _{65}} = \frac{300}{25 , 000} = 0.012$
Estimate $E_{65}^{c}$ :
$E_{65}^{c} \approx E_{65} - \frac{1}{2} θ_{65} = 25, 000 - \frac{1}{2} (300) = 25, 000 - 150 = 24, 850$
Calculate $m_{65}$ :
$m_{65} = \frac{θ _{65}}{E _{65}^{c}} \approx \frac{300}{24 , 850} \approx 0.0120724$
Verify the relationship:
Let's plug our value of $q_{65}$ into the conversion formula:
$m_{65} \approx \frac{q _{65}}{1 - 0.5 q _{65}} = \frac{0.012}{1 - 0.5 ( 0.012 )} = \frac{0.012}{1 - 0.006} = \frac{0.012}{0.994} \approx 0.0120724$
The result matches our direct calculation of $m_{65}$ , confirming the relationship holds.

Practical Ambiguities of Initial Exposure

The relationship we've used is clean and exact under its assumptions. However, in real-world applications, especially with historical data collected over calendar years, the calculation of "initial exposed to risk" itself can be ambiguous. Different actuaries, using different (but reasonable) rules, could arrive at different values for $E_{x}$ from the same raw data.

This ambiguity was a major motivation for the actuarial profession to standardize on the direct calculation of central exposed to risk ( $E_{x}^{c}$ ), which is less open to interpretation. The following reading illustrates this point perfectly.

[PDF] Comparison of approaches for calculating initial exposure

This CMI technical note demonstrates why the seemingly simple concept of initial exposed to risk can be problematic in practice. It compares two historical approaches and shows they produce different results.

Please read the section titled Comparison of the two approaches. Focus on the example provided, where the authors calculate the mortality rate q_x using two different methods for determining the initial exposed to risk. First, take a look at the example setup. Then, see how Approach 1 results in a mortality rate of q_x = 0.100. Following this, observe how Approach 2 yields a different rate of q_x = 0.103 from the same underlying data.

You don't need to master the details of the exposure calculations themselves right now (they involve calendar-year effects we will discuss later). The crucial insight is that the term "initial exposed to risk" is not monolithic. Different valid calculation conventions exist, leading to different results and underscoring the superiority of the direct, unambiguous calculation of central exposed to risk.

Conclusion

In this lesson, we have forged the critical link between the two primary measures of exposure. You've learned how a simple assumption about the timing of deaths allows us to move between the initial and central frameworks with confidence.

Key Takeaways:

The fundamental relationship connecting the two exposure measures is $E_{x}^{c} \approx E_{x} - \frac{1}{2} θ_{x}$ , which relies on the Uniform Distribution of Deaths (UDD) assumption.

This relationship enables the conversion between the mortality probability $q_{x}$ and the central death rate $m_{x}$ via the formula $m_{x} \approx \frac{q _{x}}{1 - 0.5 q _{x}}$ .
While these formulas provide a powerful theoretical bridge, the practical calculation of initial exposed to risk can be ambiguous. This is a key reason why modern actuarial practice prefers the direct and unambiguous calculation of central exposed to risk.

You are now equipped not just to calculate exposure, but to understand and navigate the relationship between the different ways it can be measured.

In our next lesson, we will address a common real-world challenge: what to do when you don't have exact entry and exit dates for every individual. We will explore methods for approximating central exposed to risk using only census data from the start and end of an observation period.