Problem 81 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

MORTALITY RATES It is sometimes useful for actuaries to be able to project mortality rates within a given population. A formula sometimes used for computing the mortality rate \(D(t)\) for women in the age group \(25-29\) is $$ D(t)=\left(D_{0}-0.00046\right) e^{-0.162 t}+0.00046 $$ where t is the number of years after a fixed base year and D0 is the mortality rate when t 0. a. Suppose the initial mortality rate of a particular group is 0.008 (8 deaths per 1,000 women). What is the mortality rate of this group 10 years later? What is the rate 25 years later? b. Sketch the graph of the mortality function D(t) for the group in part (a) for 0 t 25.

Step-by-Step Solution

Answer

The mortality rate after 10 years is approximately 0.001945. After 25 years, it is approximately 0.000592.

Step 1: Understanding the Problem

Given the mortality rate formula: \[ D(t)= (D_{0} - 0.00046) e^{-0.162 t} + 0.00046 \] where \( t \) is the number of years after a fixed base year and \( D_0 \) is the initial mortality rate. The task is to find the mortality rates after 10 and 25 years when the initial rate \( D_0 \) is 0.008.

Step 1: Substitute the Initial Rate

Substitute \( D_0 = 0.008 \) into the formula. \[ D(t) = (0.008 - 0.00046)e^{-0.162t} + 0.00046. \]

Step 2: Calculate the Mortality Rate at t = 10

Substitute \( t = 10 \) into the equation. \[ D(10) = (0.008 - 0.00046)e^{-0.162 \times 10} + 0.00046 \] Calculate the exponent: \[ -0.162 \times 10 = -1.62. \] Then compute: \[ D(10) = (0.00754)e^{-1.62} + 0.00046 \] Using the value of \( e^{-1.62} \approx 0.197 \), \[ D(10) = 0.00754 \times 0.197 + 0.00046 \] \[ D(10) \approx 0.001485 + 0.00046 = 0.001945. \]

Step 3: Calculate the Mortality Rate at t = 25

Substitute \( t = 25 \) into the equation. \[ D(25) = (0.008 - 0.00046)e^{-0.162 \times 25} + 0.00046 \] Calculate the exponent: \[ -0.162 \times 25 = -4.05. \] Then compute: \[ D(25) = (0.00754)e^{-4.05} + 0.00046 \] Using the value of \( e^{-4.05} \approx 0.0175 \), \[ D(25) = 0.00754 \times 0.0175 + 0.00046 \] \[ D(25) \approx 0.000132 + 0.00046 = 0.000592. \]

Step 4: Sketch the Graph

To sketch the graph of \( D(t) \) for \( 0 \leq t \leq 25 \), plot the values of \( D(t) \) at various points between \( t=0 \) and \( t=25 \). Use important calculated points like \( t=10 \) and \( t=25 \) and other intermediate values to perceive the decay trend of the mortality rate.

Key Concepts

actuarial science

Actuarial science is a discipline that assesses the financial risk in insurance, finance, and other sectors by using mathematical and statistical methods. Actuaries are professionals who use this science to calculate metrics like mortality rates, which are crucial for setting premiums and reserves for insurance contracts. They analyze historical data and apply mathematical models to predict future events. Understanding mortality rates allows actuaries to estimate how long policyholders are likely to live, thereby helping in the accurate calculation of life insurance premiums. It's important to understand these mortality calculations as they impact both individual financial planning and the financial stability of insurance companies.

exponential decay

Exponential decay is a key concept in the mortality rate formula given in the exercise. It describes the process of decreasing quantities that decelerate over time. Mathematically, it is expressed as the function:




f(t) = A e^{-kt}, where A and k are constants, and t is time.
In this exercise, the mortality rate function D(t) = (D_0 - 0.00046) e^{-0.162t} + 0.00046
demonstrates exponential decay with a decay rate of 0.162, as indicated by the exponent in the formula. As time increases, the value of the exponential term decreases rapidly, causing the mortality rate to approach a certain baseline value over time. Exponential decay is seen in various real-world applications, such as radioactive decay, cooling of objects, and depreciation of assets. By understanding this concept, one can better grasp how rapidly or slowly rates like mortality might change over time.

mortality computation

Mortality computation helps us determine how the probability of death within a specific group changes over time. In the given exercise, the formula for mortality rate is: D(t) = (D_0 - 0.00046) e^{-0.162t} + 0.00046.

















This formula predicts the mortality rate D(t) at any time t given the initial mortality rate D_0. Actuaries use such formulas to forecast future mortality rates, adjust premiums, and manage risk. Let's see an example from the exercise:

• 
  Step 1: Set initial mortality rate
  D_0 = 0.008 (8 deaths per 1,000 women).
• 
  Step 2: Calculate mortality rate after 10 years
  D(10) = (0.00754) e^{-1.62} + 0.00046 ≈ 0.001945.
• 
  Step 3: Calculate mortality rate after 25 years
  D(25) = (0.00754) e^{-4.05} + 0.00046 ≈ 0.000592.

Using these steps, one can understand how mortality computations are conducted using exponential decay models and other mathematical tools.

graphing functions

Graphing functions is an essential skill to visually interpret and understand mathematical relationships. In our case, plotting the mortality rate function will help us visualize how the mortality rate changes over time. With the formula: D(t) = (D_0 - 0.00046) e^{-0.162t} + 0.00046,
given an initial mortality rate (D_0) of 0.008, let's plot it for t ranging from 0 to 25.
Here’s how to do it:

• Calculate D(t) for different values of t (e.g., t=0, t=5, t=10, t=15, t=20, t=25).
• Plot these points on a graph where the x-axis represents time (t) and the y-axis represents mortality rate (D).
• Connect these points to see the curve.

As derived in the steps, important points are:

• D(10) ≈ 0.001945
• D(25) ≈ 0.000592

This graph will show an exponential decay curve, illustrating that the mortality rate decreases over time and stabilizes at a lower value. Understand the shape of such curves is crucial for many scientific, financial, and actuarial forecasts.