Projecting Mortality Rates and Life Expectancies

From a Forecasted Index to Forecasted Mortality Rates

The entire purpose of forecasting the time index $k_t$ is to allow us to project the full schedule of age-specific mortality rates, $m_{x,t}$. The connection is made through the fundamental equation of the Lee-Carter model.

Recall the core model structure:

Ignoring the error term for our central forecast, we can project the log-mortality rate for age $x$ at a future time $T+h$ using our estimated parameters and the forecast for the index:

Exponentiating both sides gives us the projected central mortality rate:

This single equation is the bridge from our time series forecast to a full mortality forecast. The following video by Ronald Lee, the model's co-creator, provides a concise conceptual summary of this crucial step.

The image below illustrates how the fitted model captures the historical trend in log death rates. The goal of forecasting is to extend the solid "Model" lines into the future, driven by the projected values of $k_t$.

For a more detailed look at the formula and its application, the following paper provides a clear mathematical description.

A Practical Calculation

Let's put this into practice with an exam-style problem. This will test your ability to work with the model's components to find a specific projected rate.

Let's solve this step-by-step:

1. Determine the value of $k_t$ at $t=20$.
   The problem states that $k_t$ decreases linearly from $k_0=1.5$ to $k_{40}=-1.5$. We can define this linear function.

   • The slope (or drift, in this deterministic case) is $\frac{\Delta k}{\Delta t}=\frac{-1.5-1.5}{40-0}=\frac{-3.0}{40}=-0.075$.
   • The equation for $k_t$ is $k_t=k_0+\text{slope}\cdot t=1.5-0.075t$.
   • At $t=20$, we have $k_{20}=1.5-0.075\times 20=1.5-1.5=0$.

2. Calculate the mortality rate $m_{60,20}$.
   The model is given by $m_{60,t}=\exp (a_{60}+b_{60}{k}_t)$. We are given $a_{60}=-4.70$.
   Plugging in our value for $k_{20}$:

   $$m_{60,20}=\exp (-4.70+b_{60}\cdot 0)=\exp (-4.70)$$

   Notice that the value of $b_{60}$ is irrelevant for this specific calculation.

   $$m_{60,20}\approx \mathrm{0.009095...}\approx 0.00910$$

   This matches option D in the exam question.

Projecting Life Expectancy

We have now produced a full table of projected mortality rates, ${\hat{m}}_{x,T+h}$, for all ages $x$ in a given future year $T+h$. This set of rates constitutes a period life table for that future year. From here, calculating the corresponding life expectancy is a standard life table construction exercise.

The process is as follows:

1. Select a forecast year: Choose a future year for which you want to calculate life expectancy, for example, the year 2050.
2. Generate the mortality schedule: Use the method from the previous section to calculate the projected central death rates, ${\hat{m}}_{x,2050}$, for every age $x$.
3. Convert rates to probabilities: Convert the central death rates ($m_x$) into probabilities of death ($q_x$). A common assumption, treating $m_x$ as a constant force of mortality $\mu$ over the year of age, gives the relationship $q_x=1-e^{-m_x}$. Other approximations exist, such as $q_x\approx \frac{m_x}{1+0.5m_x}$.
4. Construct a life table: Starting with a radix $l_0$ (e.g., 100,000), use the calculated $q_x$ values to generate the remaining columns of the life table:
   • $l_{x+1}=l_x(1-q_x)$
   • $d_x=l_x-l_{x+1}$
   • $L_x\approx l_{x+1}+0.5d_x$ (assuming uniform distribution of deaths)
   • $T_x={\sum }_{i=x}^{\omega }{L}_i$ (where $\omega$ is the final age)
5. Calculate life expectancy: Finally, calculate the complete expectation of life for any age $x$ as: $${\hat{e}}_x=\frac{T_x}{l_x}$$

The paper we consulted earlier confirms this final step.

Conclusion

In this lesson, we have closed the loop on the Lee-Carter modeling process. We started with the abstract forecast of the mortality index, ${\hat{k}}_t$, and successfully translated it into concrete, interpretable actuarial quantities.

Key Takeaways:

• Projecting Mortality Rates: The fundamental Lee-Carter equation, ${\hat{m}}_{x,t}=\exp ({\hat{a}}_x+{\hat{b}}_x{\hat{k}}_t)$, is used to convert forecasts of the time index into forecasts of age-specific mortality rates.
• Projecting Life Expectancy: A set of projected mortality rates for a future year forms a period life table for that year. Standard life table calculations can then be applied to these rates to compute the projected life expectancy.
• The Full Workflow: We have now completed the entire central forecasting workflow: from historical data $\to$ parameter estimation (${\hat{a}}_x,{\hat{b}}_x,{\hat{k}}_t$) $\to$ time series forecasting (${\hat{k}}_{T+h}$) $\to$ projecting rates (${\hat{m}}_{x,T+h}$) $\to$ calculating life expectancy (${\hat{e}}_x$).

Preview of the Next Lesson

So far, we have focused exclusively on the central forecast—our single "best guess" for future mortality. However, as actuaries and statisticians, we know that the future is uncertain. Our forecast for ${\hat{k}}_t$ came with prediction intervals, and this uncertainty must be propagated through to our final outputs. In the next lesson, we will address how to quantify the uncertainty in our mortality projections by constructing confidence intervals for both projected mortality rates and life expectancies, a critical step for risk assessment and management.