Problem 13 - Applied Calculus

Question

Harley-Davidson Inc. manufactures motorcycles. During the years following 2003 (the company's \(100^{\text {th }}\) anniversary), the company's net revenue can he approximated \(^{3}\) by \(4.6+0.4 t\) billion dollars per year, where \(t\) is time in years since January 1,2003 . Assume this rate holds through January 1,2013, and assume a continuous interest rate of \(3.5 \%\) per year. (a) What was the net revenue of the Harley-Davidson Company in \(2003 ?\) What is the projected net revenue in \(2013 ?\) (b) What was the present value, on January 1,2003 , of Harley-Davidson's net revenue for the ten years from January 1,2003 to January \(1,2013 ?\) (c) What is the future value, on January 1,2013, of net revenue for the preceding 10 years?

Step-by-Step Solution

Answer

(a) $4.6$ billion in 2003; $8.6$ billion in 2013. (b) Present value is about $42.409$ billion. (c) Future value is about $60.851$ billion.

Step 1: Find Net Revenue for 2003

The formula given for net revenue is \( R(t) = 4.6 + 0.4t \). For the year 2003, \( t = 0 \) because it is the starting year. Plugging \( t = 0 \) into the formula, we get: \( R(0) = 4.6 + 0.4 \times 0 = 4.6 \) billion dollars.

Step 2: Find Net Revenue for 2013

For the year 2013, \( t = 10 \) because it is 10 years after 2003. Plugging \( t = 10 \) into the formula, we get: \( R(10) = 4.6 + 0.4 \times 10 = 8.6 \) billion dollars.

Step 3: Determine Present Value Annuity Formula

The present value (PV) of a continuously paid annuity is given by: \( PV = \int_{0}^{10} R(t) e^{-rt} \, dt \), where \( r = 0.035 \). We need to evaluate this integral to find the present value.

Step 4: Evaluate Integral for Present Value

First, split the integral for \( R(t) = 4.6 + 0.4t \): \( PV = \int_{0}^{10} (4.6 + 0.4t) e^{-0.035t} \, dt = \int_{0}^{10} 4.6 e^{-0.035t} \, dt + \int_{0}^{10} 0.4t e^{-0.035t} \, dt \). Evaluate each separately using integration by parts where appropriate.

Step 5: Solve First Integral

The first integral is \( \int_{0}^{10} 4.6 e^{-0.035t} \, dt \). This evaluates to:\( \frac{4.6}{-0.035} e^{-0.035t} \Bigg|_{0}^{10} = -4.6 \times \frac{1}{0.035} (e^{-0.35} - 1) \approx 36.251 \).

Step 6: Solve Second Integral

The second integral \( \int_{0}^{10} 0.4t e^{-0.035t} \, dt \) requires integration by parts. Let \( u = t \) and \( dv = 0.4 e^{-0.035t} \, dt \). This evaluates to: \( \left[ -0.4t \frac{1}{0.035} e^{-0.035t} \right]_{0}^{10} + \int_{0}^{10} 0.4 \frac{1}{0.035} e^{-0.035t} \, dt \approx 6.158 \).

Step 7: Sum Present Value Components

Add both results of the integrals: \( PV_{total} \approx 36.251 + 6.158 \approx 42.409 \). Therefore, the present value of net revenue for the ten years from January 1, 2003, to January 1, 2013, is approximately \( 42.409 \) billion dollars.

Step 8: Determine Future Value Formula

The future value (FV) of a continuously paid annuity is given by: \( FV = PV \times e^{rt} \), where \( r = 0.035 \) and \( t = 10 \). Substituting the known values, compute the future value.

Step 9: Calculate Future Value

Use the formula \( FV = 42.409 \times e^{0.035 \times 10} \). This simplifies to: \( FV \approx 42.409 \times e^{0.35} \approx 60.851 \). Thus, the future value of net revenue for the ten years as of January 1, 2013, is approximately \( 60.851 \) billion dollars.

Key Concepts

Present Value in Financial Analysis

When discussing financial analysis, the present value (PV) is a crucial concept. It helps to determine the current worth of a stream of cash flows that will be received in the future, discounted by a specific interest rate. This could be extremely helpful for projecting or deciding on investments today.
To calculate the present value of Harley-Davidson's revenue from 2003 to 2013 using an interest rate of 3.5%, we apply the formula:

• \( \text{PV} = \int_{0}^{10} R(t) e^{-0.035t} \, dt \),

where \( R(t) = 4.6 + 0.4t \) encapsulates the expected revenue over each year.
This calculation involves integration, which allows us to factor in the continuous growth of revenue over the 10-year period at a decreasing rate influenced by the 3.5% interest rate.Integration helps in accounting for continuous cash flow rates, illustrating the value of future cash flows in today's terms. This method supports effective financial planning and investment decisions by enabling businesses to assess whether future returns justify current spendings.

Future Value and Its Importance

Future Value (FV) refers to the amount of money an investment will grow to over a particular period, given a certain interest rate. In simpler terms, it reflects what the present amount will be worth in the future.
For Harley-Davidson's net revenue, the future value for the revenues accumulated from 2003 to 2013 was computed as:

• \( \text{FV} = \text{PV} \times e^{0.035 \times 10} \),

where 'PV' is the present value calculated previously.
This illustrates how an amount invested or earned today benefits from compound interest over time.
It's important for companies and investors as it gives a clearer picture of potential growth and profitability of revenue streams or investments. Calculating FV provides insights into future financial health and helps in making informed investment choices and financial strategies. Utilizing the continuous compounding interest formula ensures a more realistic growth projection over significant periods like a decade.

Integration by Parts - A Useful Technique

Integration by parts is a fundamental technique in calculus used to find the integral of products of functions. For expressions that are hard to integrate directly, this method can simplify the process substantially.
In our exercise, to solve the integral of Harley-Davidson's revenue formula \( 0.4t e^{-0.035t} \), we use integration by parts.The formula applied here is:

• Create two parts: let \( u = t \) and \( dv = 0.4e^{-0.035t} \, dt \).
• Calculate \( du \) and \( v \) using the differentiation and integration of components.
• Substitute these into integration by parts formula: \( \int u \, dv = uv - \int v \, du \).

Applying this process results in breaking down a complex integral into manageable parts. It is especially beneficial in finance when calculating areas like expected revenue, integrating different growth timetables or calculating financial values over continuous functions. Understanding and applying integration by parts can elevate analytical skills, tackling complex financial computations methodically.