Problem 14 - Calculus Concepts

Question

General Motors The revenue of General Motors Company (GM) in December 2008 was \(\$ 148.98\) billion. Assume that GM continues to invest \(3 \%\) of the 2008 revenue figure continuously throughout each year, beginning at the end of December 2008 , into an account that pays interest at a rate of \(4.8 \%\) compounded continuously. (Source: Hoover's Online Guide) a. Calculate the value of the account in December \(2015 .\) b. How much would GM have had to invest at the end of December 2008 , in one lump sum, into this account to build the same 7 -year furure value as the one found in \(\operatorname{part} a ?\)

Step-by-Step Solution

Answer

The final answer at the end of 7 years is $39,752,190,520.38.

Step 1: Calculate Annual Investment

The annual investment amount is 3% of $148.98 billion. So, calculate the annual investment using the formula:\[\text{Annual Investment} = 0.03 \times 148.98\]This results in an annual investment of \(4.4694\) billion dollars.

Step 2: Continuous Compound Interest Formula

Use the formula for continuous compounding to calculate the future value of continuous investments:\[FV = P \cdot \frac{e^{rt} - 1}{r} \cdot e^{rt}\]where\(P\) is the annual investment,\(r\) is the interest rate,\(t\) is the time in years.Here, \(P = 4.4694\), \(r = 0.048\), and \(t = 7\).

Key Concepts

Annual Investment Calculation

To determine how much money General Motors would invest yearly, we start by looking at the given percentage of revenue allocated for investment. General Motors decided to reinvest 3% of their 2008 revenue of \(\$148.98\) billion annually. To calculate this, multiply the revenue by the percentage: \[\text{Annual Investment} = 0.03 \times 148.98\] This calculation results in an annual investment of \(4.4694\) billion dollars. It's essential to ensure that the annual investment is correctly calculated as it forms the base for future calculations. This yearly reinvestment helps build wealth over time by consistently contributing to the investment account.

Future Value Calculation

After determining the annual investment, the next step is calculating the future value in 2015. Future value refers to what an investment will grow to over time when left in an account with a fixed interest rate. We use the continuous compounding formula to find this value given that General Motors' investments earn interest compounded continuously. For each dollar of annual investment, the continuous compounding formula expands as one invests: \[FV = P \cdot \frac{e^{rt} - 1}{r} \cdot e^{rt}\], where \(P\) is the annual investment, \(r\) is the interest rate, and \(t\) is the time in years. Substituting \(P = 4.4694\), \(r = 0.048\), and \(t = 7\) gives us the future value General Motors can expect by 2015. By understanding future value, investors can anticipate how their investments will perform across specific timeframes, taking into account compound interest.

Continuous Compounding Formula

Continuous compounding is a concept that represents the ideal scenario of earning interest. Unlike regular compound interest that compounds at specific intervals, continuous compounding is all about earning interest on top of already earned interest, at every possible moment. The continuous compounding formula is written as: \[FV = Pe^{rt}\]. Here, \(FV\) is the future value, \(P\) is the present or initial amount, \(r\) is the annual interest rate, and \(t\) is the number of years. In this example, continuous compounding allows for precise calculation over extended periods, making it an attractive investment consideration. This formula is essential for determining how an investment account grows over time when continuously compounded fees or interest rates apply.