Problem 87 - 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 after 10 years: approximately 9.55 million dollars.

Step 1: Insert the given polynomial expression for v(t)

The given function for the policy value is \( v(t) = 60 + 3t \). We will use this expression to substitute into the reserves function \( R(v) = 2v^{0.3} \).

Step 2: Substitute v(t) into R(v)

Substitute \( v = 60 + 3t \) into the reserves function, \( R(v) = 2v^{0.3} \). This gives:\[ R(t) = 2(60 + 3t)^{0.3} \]

Step 3: Evaluate R(t) at t = 10

Now that we have \( R(t) = 2(60 + 3t)^{0.3} \), substitute \( t = 10 \) into this function:\[ R(10) = 2(60 + 3 \times 10)^{0.3} \]Simplify the expression inside the parentheses:\( 60 + 3 \times 10 = 90 \)Then compute \( R(10) = 2(90)^{0.3} \).

Step 4: Calculate the numerical value

Finally, calculate the value of \((90)^{0.3}\) using a calculator, then multiply by 2 to get:\[ R(10) = 2 \times 4.774606 \approx 9.549212 \]This means the reserves after 10 years would be approximately \(9.55\) millions of dollars.

Key Concepts

Insurance Mathematics

Insurance mathematics involves using mathematical models to predict and manage financial risks associated with insurance policies. It is crucial for insurance companies to ensure they have enough reserves to cover potential claims. Reserves are funds set aside from premiums collected to pay for future claims.
In the given exercise, the insurance company's reserve formula is represented as a function of policy value. They use a model to help them decide how much money to keep in reserves. This model takes into account the overall value of the policies they insure.

• Reserves ensure the company can fulfill its promises to policyholders.
• It helps in assessing the company’s financial health and ability to pay claims.
• Models like these use mathematical functions to represent various scenarios and predict future financial needs.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole-number powers. In our example, the policy value is described by the polynomial function \(v(t) = 60 + 3t\). This shows how the value of policies changes over time.
Each term of a polynomial expresses different effects, like initial policy value or growth rate of policy value over time.

• The constant term (60 in this case) represents the initial value of all policies.
• The linear term (3t) represents growth or change in policy value each year.

Polynomial functions uniquely represent certain scenarios, making them useful in financial modeling, like predicting insurance policy values over time.

Substitution Method

The substitution method is a technique used in mathematics to solve and simplify expressions by replacing a variable with its equivalent. This helps to transform functions for easier computations.
In this problem, \(v(t)\) is substituted into \(R(v)\), transforming the function of reserves into one that depends on time \(t\). This substitution allows us to assess the insurance reserves dynamically as time changes.

• Substitute \(v(t) = 60 + 3t\) into \(R(v) = 2v^{0.3}\) to get \(R(t)\).
• Allows for finding the exact reserves at specific future times, such as \(t = 10\) years.
• Facilitates computation by reducing complex problems into straightforward numerical evaluations.

Financial Modeling

Financial modeling uses mathematical constructs to create representations of economic realities. It helps predict future financial performance based on current data.
In this exercise, the financial model helps to predict insurance reserves over time by using functions for policy value growth and reserve needs.

• Financial models translate complex financial data into actionable insights.
• They allow companies to simulate different scenarios for strategic planning.
• In our example, it's used to estimate future reserves required as policy values change.

Overall, financial modeling provides a structured way to tackle financial challenges and prepare for future financial conditions.