Problem 33 - Calculus and its applications

Question

An actuary works for an insurance company and calculates insurance premiums. Given an actual mortality rate (probability of death) for a given age, actuaries sometimes need to project future expected mortality rates of people of that age. For example, $$Q(t)=\left(Q_{0}-0.00055\right) e^{0.163 t}+0.00055$$ where \(t\) is the number of years into the future and \(Q_{0}\) is the mortality rate when \(t=0,\) projects future mortality rates. a) Suppose the actual mortality rate of a group of females aged 25 is 0.014 ( 14 deaths per 1000 ). What is the future expected mortality rate of this group of females \(3,5,\) and 10 yr in the future? b) Sketch the graph of the mortality function \(Q(t)\) for the group in part (a) for \(0 \leq t \leq 10\).

Step-by-Step Solution

Answer

Mortality rates: 0.0224 at t=3, 0.0319 at t=5, 0.0692 at t=10.

Step 1: Understand the Given Formula

The formula given is \(Q(t)=\left(Q_{0}-0.00055\right) e^{0.163 t}+0.00055\). It calculates the projected future mortality rate \(Q(t)\) based on the initial mortality rate \(Q_{0}\) and the number of years into the future \(t\).

Step 2: Calculate Mortality Rate at t=3

Substitute \(Q_{0} = 0.014\) and \(t = 3\) into the formula. Calculate as follows:\[ Q(3) = (0.014 - 0.00055) e^{0.163 \times 3} + 0.00055 \]Calculate \(0.014 - 0.00055 = 0.01345\) and \(e^{0.489} \approx 1.630\), then substitute back:\[ Q(3) = 0.01345 \times 1.630 + 0.00055 \approx 0.02242 \]

Step 3: Calculate Mortality Rate at t=5

Substitute \(Q_{0} = 0.014\) and \(t = 5\) into the formula. Calculate as follows:\[ Q(5) = (0.014 - 0.00055) e^{0.163 \times 5} + 0.00055 \]Calculate \(0.014 - 0.00055 = 0.01345\) and \(e^{0.815} \approx 2.260\), then substitute back:\[ Q(5) = 0.01345 \times 2.260 + 0.00055 \approx 0.03190 \]

Step 4: Calculate Mortality Rate at t=10

Substitute \(Q_{0} = 0.014\) and \(t = 10\) into the formula. Calculate as follows:\[ Q(10) = (0.014 - 0.00055) e^{0.163 \times 10} + 0.00055 \]Calculate \(0.014 - 0.00055 = 0.01345\) and \(e^{1.630} \approx 5.104\), then substitute back:\[ Q(10) = 0.01345 \times 5.104 + 0.00055 \approx 0.0692 \]

Step 5: Sketch the Graph for 0 ≤ t ≤ 10

To sketch the graph, plot the calculated mortality rates at \(t=0, 3, 5,\) and \(10\), and draw a smooth curve through these points. The x-axis represents time \(t\) and the y-axis represents the mortality rate \(Q(t)\). The curve should be rising as \(t\) increases, consistent with the calculated points. You can use a tool like Excel or graphing software to plot more precise points for a smoother curve.

Key Concepts

Mortality Rate Projection

Actuarial mathematics involves projecting mortality rates, an essential aspect of insurance mathematics that helps actuaries predict future risks. Projecting mortality rates allows actuaries to estimate the likelihood that a person within a certain group and age will pass away within a given timeframe. This information is vital for calculating insurance premiums and determining liabilities.
Mortality rate projections often use mathematical functions that factor in current mortality rates and anticipated changes over time. For example, the function given in the exercise, \[ Q(t) = \left(Q_{0} - 0.00055\right) e^{0.163 t} + 0.00055 \]averages historic trends and future expectations.
Thus, actuaries, by using such predictive functions, can help insurance companies prepare for future claims and adjust premiums accordingly, ensuring the company remains financially sound.

Exponential Growth

The concept of exponential growth is crucial in understanding how the mortality rate projection evolves over time. In mathematics, exponential growth describes a situation where the rate of increase is proportional to the current value, leading to growth that accelerates over time.
In the provided formula, the expression \[ e^{0.163 t} \] represents exponential growth with respect to time \( t \). - The base \( e \) is a mathematical constant, approximately equal to 2.718, which often appears in growth-related functions due to its unique properties.- The parameter 0.163 indicates how rapidly the mortality rate is expected to grow over time. This function translates to the mortality rate increasing at an accelerating pace as time progresses, consistent with physiological aging processes that increase vulnerability to mortality.
The exponential component in the formula illustrates how even small initial changes in the mortality rate can lead to significant differences in predicted rates over longer periods.

Insurance Mathematics

Insurance mathematics centers around using statistical and mathematical methods to assess risk, which is fundamental to designing and pricing insurance products. Central to this is determining accurate premium rates that balance affordability for customers with the insurer's need for solvency.
Mortality rate projections feed into insurance mathematics by contributing to understanding the risk profiles of different age groups. Calculating projected mortality influences various decisions:

• Setting life insurance premiums based on expected future mortality.
• Determining annuity and pension products pricing, which depend on life expectancy estimates.
• Assessing the overall risk portfolio of an insurance company.

By integrating complex formulas like the one given in the exercise, actuaries can model future outcomes and evaluate financial scenarios for insurance products, thus playing a critical role in securing and maintaining the financial health of insurance companies.