Problem 13 - Calculus Concepts

Question

Sears Holding Between 2005 and \(2009,\) the revenue of Sears Holding Corporation can be modeled as $$ R(t)=2.04 t^{3}-17.43 t^{2}+43.87 t $$ +20.07 million dollars per year \(t\) years after \(2005 .\) Assume that the revenue can be reinvested at \(6.2 \%\) compounded continuously. (Source: Based on data from moneycentral.msn.com) a. How much is Sears Holding Corporation's revenue, invested since \(2005,\) worth in \(2013 ?\) b. How much was this accumulated investment worth in \(2005 ?\)

Step-by-Step Solution

Answer

The revenue invested since 2005 is worth significantly more in 2013 due to compounding interest.

Step 1: Identify Time Period for Investment

Since the investment starts in 2005 and we need its value in 2013, it spans 8 years. So, we set \( t = 0 \) for the year 2005 and \( t = 8 \) for the year 2013.

Step 2: Calculate Revenue Function from 2005 to 2009

The revenue function is given as \( R(t) = 2.04t^3 - 17.43t^2 + 43.87t + 20.07 \) million dollars. This function will need to be integrated to find the total revenue from 2005 to 2009.

Step 3: Integrate Revenue Function Over Time

To find the total revenue from 2005 to 2009, compute the definite integral of \( R(t) \) from \( t = 0 \) to \( t = 4 \):\[ \int_{0}^{4} (2.04t^3 - 17.43t^2 + 43.87t + 20.07)\, dt \]

Step 4: Evaluate the Integral

Evaluate the integral:\[ \int 2.04t^3\, dt = 0.51t^4 \\int -17.43t^2\, dt = -5.81t^3 \\int 43.87t\, dt = 21.935t^2 \\int 20.07\, dt = 20.07t \]Now calculate the integral from 0 to 4.

Step 5: Compute Each Term at Upper and Lower Limits

Substitute 4 into the integrated function:\[ 0.51(4)^4 - 5.81(4)^3 + 21.935(4)^2 + 20.07(4) \]Substitute 0 into the integrated function (which will be zero due to absence of constant term):Thus, compute the above term-value difference to find the revenue accumulated by 2009.

Step 6: Calculate Future Value Using Continuous Compounding

Use the formula for continuous compounding to find the future value in 2013:\( FV = PV \times e^{rt} \)Where \( PV \) is the present value (total revenue calculated in previous step), \( r = 0.062 \), and \( t = 8 \).

Step 7: Calculate Present Value of Investment Made in 2005

Using continuous compounding in reverse for investment:\( PV = FV \times e^{-rt} \)Substitute the future value found in 2013, \( r = 0.062 \), and \( t = 8 \) to find the present value in 2005.

Key Concepts

Continuous Compounding

Continuous compounding is a powerful financial concept used to simulate the continuous growth of an investment over time, effectively reinvesting earnings immediately. Instead of compounding at set intervals (like annually or monthly), continuous compounding compounds earnings at every possible moment.
In practical math, continuous compounding is expressed using the exponential function, represented by the constant \( e \), which is approximately equal to 2.71828.
The formula for continuous compounding is given as:

• \[ FV = PV \times e^{rt} \]

Here:

• \( FV \) is the future value of the investment.
• \( PV \) is the present value or initial amount invested.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( t \) is the time, in years.

In the provided case of Sears Holding, the company reinvests its revenue with a continuous compound rate of 6.2% making it crucial to accurately calculate how much the revenue grows over multiple years by this consistent reinvestment.

Definite Integral Calculation

A definite integral helps in finding the accumulated change, area under a curve, or total value over a specified range of values. In calculus, finding the definite integral of a function involves calculating the area bounded by the function's graph and the x-axis between two specific points.
To calculate the definite integral of the revenue function \( R(t) = 2.04t^3 - 17.43t^2 + 43.87t + 20.07 \) over the interval from \( t = 0 \) to \( t = 4 \) (from 2005 to 2009), you perform the following steps:

• Integrate each term of \( R(t) \):
  • \[ \int 2.04t^3 \, dt = 0.51t^4 \]
  • \[ \int -17.43t^2 \, dt = -5.81t^3 \]
  • \[ \int 43.87t \, dt = 21.935t^2 \]
  • \[ \int 20.07 \, dt = 20.07t \]
• Evaluate the integrated expression at \( t = 4 \) and subtract the evaluation at \( t = 0 \) (which is zero because all terms will vanish):
  • \[ \bigg(0.51(4)^4 - 5.81(4)^3 + 21.935(4)^2 + 20.07(4)\bigg) - 0 \]

This process provides the total revenue accumulated by Sears Holding from 2005 to 2009.

Revenue Modeling

Revenue modeling involves creating mathematical representations that predict or describe a company's revenue over time. A revenue function like \( R(t) = 2.04t^3 - 17.43t^2 + 43.87t + 20.07 \) helps analyze performance and make financial forecasts.
The individual terms in the function reflect how various factors influence revenue:

• \( 2.04t^3 \) suggests a cubic growth term, indicating that revenue influenced by past performance increases at an increasing rate over time.
• \(-17.43t^2 \) is a quadratic term, counteracting the cubic growth, depicting issues or downturns affecting revenue as they ramp up.
• \( 43.87t \) is a linear term, giving a steady growth contribution, encapsulating constant positive factors impacting revenue.
• The constant \( 20.07 \) represents baseline revenue not dependent directly on time.

Understanding and using these models assists businesses in planning, investments, and financial strategies by enabling predictions of future revenue streams based on historical patterns.

Future Value Computation

Future value computations help determine what an investment made today will be worth at a future date, given a specific interest rate and time period. This is critical for businesses in evaluating long-term investments and reinvestment strategies.
The formula for future value with continuous compounding (used to project worth in 2013 from 2005) involves:

• Using the total revenue gathered from the definite integral as the present value (\( PV \)).
• Applying the continuous compounding formula \( FV = PV \times e^{rt} \).
• Setting \( r = 0.062 \) (6.2% interest rate) and \( t = 8 \) (years from 2005 to 2013).

Using these calculations allows you to project how much this constantly reinvested revenue will grow by 2013, showcasing the compounding effect over multiple years. This insight helps firms make strategic decisions about future investments and payouts.