Problem 87 - Brief Applied Calculus

Question

An insurance company keeps reserves (money to pay claims) of \(R(v)=2 v^{0.3}\), where \(v\) is the value of all of its policies, and the value of its policies is predicted to be \(v(t)=60+3 t\), where \(t\) is the number of years from now. (Both \(R\) and \(v\) are in millions of dollars.) Express the reserves \(R\) as a function of \(t\). and evaluate the function at \(t=10\).

Step-by-Step Solution

Answer

Reserves at \(t=10\) are approximately \(8.62\) million dollars.

Step 1: Express Policy Value as a Function of Time

Given that the value of the policies is predicted by the function \(v(t) = 60 + 3t\), where \(t\) is the number of years from now, this equation expresses how the value of policies changes over time.

Step 2: Substitute Policy Value into Reserve Function

The reserve function is given by \(R(v) = 2v^{0.3}\). Since \(v = v(t) = 60 + 3t\), substitute \(v(t)\) into the reserve function: \(R(t) = 2(60 + 3t)^{0.3}\). This expresses the reserves \(R\) as a function of \(t\).

Step 3: Evaluate Reserves Function at t=10

To find the reserves at \(t = 10\), substitute 10 into the function: \[R(10) = 2(60 + 3 \times 10)^{0.3} = 2(90)^{0.3}.\]

Step 4: Simplify and Calculate Numeric Result

Calculate \((90)^{0.3}\) using a calculator: \((90)^{0.3} \approx 4.308869\). Therefore, \(R(10) = 2 \times 4.308869 \approx 8.617738\). Thus, the reserves at \(t = 10\) years is approximately \(8.62\) million dollars.

Key Concepts

Reserves Function

A reserves function is a mathematical expression that helps insurance companies calculate the money they need to keep in reserve to cover potential policy claims.
This function is crucial because it assures that the company can fulfill its promises to policyholders.
In our example, the reserves function is given by \(R(v) = 2v^{0.3}\). This indicates how the reserves depend on the value of policies \(v\). The exponent \(0.3\) shows the rate at which changes in policy values affect the reserves.

• It is important for this function to be accurate, ensuring that reserves are neither inadequate nor excessive.
• An overestimate leads to wasted capital that could be invested elsewhere, while an underestimate risks insolvency.

Understanding the reserves function allows stakeholders to predict future reserves based on changing policy values and helps maintain the financial health of the insurance company.

Policy Value

Policy value is a key metric that reflects the total value of all insurance policies held by an insurance company.
It helps in assessing the company's potential financial liability.
The function given is \(v(t) = 60 + 3t\), where \(t\) is time in years.
This linear function shows how the value of policies is expected to grow over time with a starting value of 60 million dollars.

• The term \(60\) represents the initial total policy value.
• The term \(3t\) represents the annual increase in policy value by 3 million dollars each year.

This understanding allows companies to plan for future changes in policy values and adjust their reserves as necessary.

Exponential Function

Exponential functions, like the one seen in this reserves function, play a significant role in modeling growth patterns that aren't strictly linear.
In the reserves function \(R(v) = 2v^{0.3}\), the exponent \(0.3\) indicates a non-linear relationship between reserves and policy value.
This reflects real-world scenarios where reserves grow at a diminishing rate as policy value increases.

• Exponents less than 1, such as \(0.3\), often represent saturation points.
• This means that as \(v\) grows, each additional unit adds less to the reserves than the one before.

Recognizing the behavior of exponential functions helps in making informed financial and risk management decisions in the insurance industry.

Numeric Evaluation

Numeric evaluation is used to find a specific value of a function at a given point.
It's an important step to gain a concrete understanding of what variables represent in practical terms.
In this example, we evaluate the reserves function \(R(t) = 2(60 + 3t)^{0.3}\) at \(t = 10\).

• Substitute \(t = 10\) into the function: \(R(10) = 2(90)^{0.3}\).
• Calculate \((90)^{0.3} \approx 4.308869\).
• Finally, calculate \(R(10) = 2 \times 4.308869 \approx 8.62\).

This process shows that the reserves needed after 10 years would be approximately 8.62 million dollars.
Numeric evaluation is a practical way to verify analytical results and ensure they align with expectations based on known data.