6 Survival Models
6.1 Attrition Rates
For most PA professionals, a key requirement is often to determine the attrition patterns that will be exhibited by the staff of a company. Where things sometimes go awry is the treatment of attrition like an interest rate. This can often lead to erroneous results and misinterpretations in applications using attrition rates. Attrition needs to be thought of as a decrement rate and I will explain why in the remainder of this chapter.
A decrement rate refers to the rate at which individuals exit a population or system due to specific causes, such as death, disability, resignation or retirement. In actuarial science, decrement rates are used in models to estimate how many people will leave a group due to these events. Decrement rates measure the probability of events happening within a population over time.
The decrement rate approach would define attrition as:
\[ \begin{aligned} \mathrm{attrition\ rate\ this\ year} &= (1 - (1-\frac{total\ exits\ this\ period}{exposed\ to\ risk\ this\ period}) ^ {12/(number\ of\ periods)})\\ \end{aligned} \]
This can appear to be slightly odd but leads to much more interpretable and accurate results. Traditionally, PA professionals have utilised the formula to calculate attrition:
\[ \begin{aligned} \mathrm{attrition\ rate} &= (\frac{total\ exits}{average\ headcount}) \\ \end{aligned} \]
This rate is then used to estimate attrition over longer or shorter periods using either the simple interest method:
\[ \begin{aligned} \mathrm{attrition\ rate\ this\ year} &= (\frac{total\ exits\ this\ month}{average\ headcount\ this \ month}) * {12}\\ \end{aligned} \] or the compound interest method:
\[ \begin{aligned} \mathrm{attrition\ rate\ this\ year} &= (1 + \frac{total\ exits\ this\ month}{average\ headcount\ this \ month}) ^ {12} - 1\\ \end{aligned} \]
Both of these formulae are inaccurate for reasons I shall explain below.
This approach is not accurate for a few reasons:
- Attrition rates can exceed 100%. This happens when the number of exits is greater than the average population during the period. Example:
- Population at start of month 1 \(P_1\): 100 employees
- Population at end of month 1 \(P_2\): 80 employees
- Number of exits \(exits\): 20 employees
- Average population: \[ \begin{aligned} \mathrm{average\ heacount} &= (\frac{P_1 + P_2}{2})\\ &= (\frac{100 + 90}{2}) = 95 \\ \end{aligned} \] \[ \begin{aligned} \mathrm{attrition\ rate\ per\ month} &= (\frac{total\ exits}{average\ headcount}) \\ &= (\frac{20}{95}) \\ &= 21.05\% \end{aligned} \]
If we now want the annualised attrition rate, it would be either: \[ \begin{aligned} \mathrm{attrition\ rate\ this\ year} &= (1 + \frac{total\ exits\ this\ month}{average\ headcount\ this \ month}) ^ {12} - 1\\ &= (1 + \frac{25}{90}) ^ {12} - 1\\ &= (1.2105) ^ {12} - 1\\ &= 890\% \end{aligned} \]
or
\[ \begin{aligned} \mathrm{attrition\ rate\ this\ year} &= (\frac{total\ exits\ this\ month}{average\ headcount\ this \ month}) * {12}\\ &= (\frac{25}{90}) * 12\\ &= (0.2105) * {12}\\ &= 252\% \end{aligned} \] In this example, you end up with a nonsensical attrition rate over 100%, which is impossible. Attrition rates over 100% suggest that more than the entire population left the organisation, which is not a reasonable interpretation. This happens because treating attrition like an interest rate does not account for situations where a large number of people leave early in the period, causing the population to be much smaller later in the period. As a result, the average population becomes artificially low, while exits remain high, inflating the attrition rate.
- Assumes Linear Population Change If the population size fluctuates significantly during the period, the average population method may provide a misleading measure of attrition. For instance, if an organisation hires or loses employees in bursts, the timing of these changes can distort the attrition rate. The average population formula assumes that population changes linearly over the period. This assumption can lead to inaccurate results, especially when the changes in population are abrupt. Example:
- Population at start of year \(P_1\): 100 employees
- Population at end of year \(P_{12}\): 400 employees
- Number of exits \(exits\): 50 employees
- Assume all exits and new hires occurred in the last month of the year.
- Average population: \[ \begin{aligned} \mathrm{average\ heacount} &= (\frac{P_1 + P_2}{2})\\ &= (\frac{100 + 400}{2})\\ &= 250\\ \end{aligned} \] \[ \begin{aligned} \mathrm{attrition\ rate} &= (\frac{total\ exits}{average\ headcount}) \\ &= (\frac{50}{250}) \\ &= 20\% \end{aligned} \]
However, if the changes were not linear, the real attrition could be very different.
In contrast, a decrement rate formula for the above example would yield:
\[ \begin{aligned} \mathrm{average\ heacount} &= (\frac{P_1 + P_2 + P_3 + P_4 + P_5 + P_6 + P_7 + P_{8} + P_{9} + P_{10} + P_{11} + P_{12}}{12})\\ &= (\frac{100\times 11+400}{12})\\ &= (\frac{1500}{12})\\ &= 125\\ \end{aligned} \] and
\[ \begin{aligned} \mathrm{attrition\ rate\ this\ year} &= (1 - (1-\frac{total\ exits\ this\ year}{average\ headcount\ this \ year})) - 1\\ &= (1 - (1-\frac{50}{125})) \\ &= (1-(1 - 0.4)\\ &= 40\% \end{aligned} \] which is a much more accurate representation of the true situation.
As a final example, consider the following situation where a company is not hiring any staff but is losing 10% of staff each month.
| month | employees | observed monthly attrition | observed period rate | traditional attrition rate | decrement rate |
|---|---|---|---|---|---|
| 0 | 1000 | ||||
| 1 | 900 | 10.00% | 10.00% | 126.32% | 71.76% |
| 2 | 810 | 10.00% | 19.00% | 125.97% | 71.76% |
| 3 | 729 | 10.00% | 27.10% | 125.39% | 71.76% |
| 4 | 656 | 10.00% | 34.39% | 124.59% | 71.76% |
| 5 | 590 | 10.00% | 40.95% | 123.59% | 71.76% |
| 6 | 531 | 10.00% | 46.86% | 122.38% | 71.76% |
| 7 | 478 | 10.00% | 52.17% | 121.00% | 71.76% |
| 8 | 430 | 10.00% | 56.95% | 119.44% | 71.76% |
| 9 | 387 | 10.00% | 61.26% | 117.74% | 71.76% |
| 10 | 349 | 10.00% | 65.13% | 115.90% | 71.76% |
| 11 | 314 | 10.00% | 68.62% | 113.95% | 71.76% |
| 12 | 282 | 10.00% | 71.76% | 111.91% | 71.76% |
As can be seen from the table above, the decrement rate approach has the correct attrition rate throughout the year. The traditional attrition rate approach is never correct at any point in the year.
6.2 The Kaplan Meier Estimation
The Kaplan-Meier estimator (also known as the product-limit estimator) is a non-parametric statistic used to estimate the survival function from time-to-event data. It’s widely used by actuaries in survival analysis, and also in other fields where it’s important to model the time until a specific event occurs, such as death, failure, or attrition.
The Kaplan-Meier estimator is ideal when you have cases where the event of interest hasn’t occurred for some subjects by the end of the observation period. It allows us to estimate the probability of surviving (or not experiencing the event) beyond a certain point in time, given the data observed up to that time.
The Kaplan-Meier estimator of the survivor function adopts the following conventions.
- The hazard of experiencing the event is zero at all durations except those where an event actually happens in our sample.
- The hazard of experiencing the event at any particular duration, \(t_j\) , when an event takes place is equal to \(\frac{d_j}{n_j}\), where \(d_j\) is the number of individuals experiencing the event at duration \(t_j\) and \(n_j\) is the risk set at that duration (that is, the number of individuals still at risk of experiencing the event just prior to duration \(t_j\)). So if we observed 2 exits out of 10 active employees, the hazard would be equal to \(\frac{2}{10}\).
- People that are censored are removed from observation at the duration at which censoring takes place. People who are censored at a duration where events also take place are assumed to be censored immediately after the events have taken place (so that they are still at risk at that duration). So if we observed 2 exits and 1 retirement occurring simultaneously out of 10 active employees, the hazard would be equal to \(\frac{2}{10}\) but if we observed 2 exits and 1 retirement occurring prior to the exits, the hazard would be equal to \(\frac{2}{9}\). In other words, if any of the individuals are observed to be censored at the same time as one of the exits, the convention is to treat the censoring as if it happened shortly afterwards, ie the resignations are assumed to have occurred first.
6.3 Concepts of Survival
This section may be quite challenging to work through but it has many use cases. Let us start with the observation that the future company employed lifetime of an employee is not known in advance. Further, we observe that lifetimes range from 0 months to in excess of 700 months. A natural consequence therefore is that the future lifetime of a given employee is a random variable.
Let’s denote the future lifetime of a new employee as a random variable, denoted \(T\), which is continuously distributed on an interval \([0,ω]\) where \(0 < ω < ∞\).
The maximum age \(ω\) is called the limiting tenure.
Typical values of \(ω\) for practical work are in the range 480 - 520. The possibility of working beyond tenure \(ω\) is excluded by the model for convenience and simplicity.
The distribution function and survival function of a new employee can be defined as: \(F(t) = P[T <= t]\) is the distribution function of T . \(S(t) = P[T <= t] = 1 - F (t)\) is the survival function of T .
\(S(t)\) is known as the survival function of T because it represents the probability of a new employee working until month t.
The distribution and survival function of an employee with tenure \(x\) for \(0 <= x <= ω\) is: \(F_x(t) = P[T_x <= t]\) is the distribution function of Tx \(S_x(t) = P[T_x > t] = 1 - F_x(t)\) is the survival function of Tx
For example, the probability that an employee who has worked for 10 months resigns before reaching month 25 is given by \(F_{10}(15)\). We can estimate this probability using he Kaplan-Meier hazard rate estimates seen in the previous chapter.
In actuarial notation: \(_tq_x = F_x(t) = P[T_x <= t]\) \(_tp_x = 1 - _tq_x = S_x(t) = 1 - F_x(t) = P[T_x > t]\)
In can then be proved that:
\(_{t+s}p_x = _tp_x\ *\ _sp_{x+t}\ =\ _sp_x\ *\ _tp_{x+s}\)
6.4 Expectation of Tenure
The expected future lifetime after working for time \(t\) can be defined as \(E[T_x]\). Similarly to how demographers denote the expectation of life at age t, we can denote this as \(\mathring{e}_t\) or “e circle t”.
It follows that the complete expectation of tenure can be derived as:
\[ \mathring{e}_t = \sum_{k=1}^{\infty}\:_kp_x \]
i.e. the expected tenure of the employee is the sum of the probabilities that they survive to period \(1, 2, 3, ...k\)
If we are able to calculate decrement rates and survival probabilities for different groups of employees, we will be able to provide many interesting insights which will be discussed in later chapters.