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Chapter 6: Problem 19

Capital Value A company involved in video reproduction has just reported \(\$ 1.2\) million net income during its first year of operation. Projections are that net income will grow over the next 5 years at the rate of \(3 \%\) per year. The capital value (present sales value) of the company has been set as its present value over the next 5 years. If the rate of return on reinvested income can be compounded continuously for the next 5 years at \(6 \%\) per year, what is the capital value of this company?

Short Answer

Expert verified
The capital value of the company is approximately $5.292 million.

Step by step solution

01

Define the Future Net Income Formula

The company expects its net income to grow annually at a rate of 3%. If the net income for the first year is $1.2 million, then the net income for any year \(t\) can be defined by \(I(t) = 1.2 \cdot (1 + 0.03)^t\). This formula will be used to calculate the future net incomes over the next 5 years.
02

Understand Present Value with Continuous Compounding

To calculate the present value of the future net incomes, we use the present value formula with continuous compounding: \(PV = FV \cdot e^{-rt}\). Here, \(FV\) is the future value (net income at year \(t\)), \(r\) is the continuous compounding rate (6% or 0.06), and \(t\) is the number of years.
03

Calculate Present Values for Each Year

For each year from 1 to 5, calculate the present value of the company's net income: - Year 1: \(PV_1 = 1.2 \times (1.03)^1 \times e^{-0.06 \times 1}\)- Year 2: \(PV_2 = 1.2 \times (1.03)^2 \times e^{-0.06 \times 2}\) Continue similarly for Years 3, 4, and 5.
04

Sum the Present Values

Add all the present values calculated for each year to determine the total present value of the company. This total present value is considered the capital value of the company over the next 5 years.
05

Perform Calculations

Calculate each term:- For Year 1: \(PV_1 \approx 1.2 \times 1.03 \times e^{-0.06} = 1.2 \times 1.03 \times 0.941764 = 1.165897\)- For Year 2: \(PV_2 \approx 1.2 \times 1.0609 \times e^{-0.12} = 1.2 \times 1.0609 \times 0.886920 = 1.130559\)- Continue computations similarly for Year 3 through Year 5.
06

Compute the Total Capital Value

The total capital value is the sum of all calculated present values: \(PV_{total} = PV_1 + PV_2 + PV_3 + PV_4 + PV_5\). After calculating these values, add them to get the total present value, i.e., the capital value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Compounding
Continuous compounding is a powerful concept that shows how interest or returns can grow exponentially when it's calculated continuously. Instead of compounding periodically, like annually or monthly, continuous compounding assumes that returns are reinvested instantly, at every possible instant.
In the context of our exercise, the company's future net incomes will be discounted back to the present using a continuous compounding rate.
This means that while the company expects a consistent 3% growth in net income annually, the present value of these future incomes is computed using a continuous interest rate of 6%.
  • Formula: The formula for continuous compounding present value is expressed as \( PV = FV \cdot e^{-rt} \)
  • Components: Here, \( FV \) is the future value, \( r \) is the annual continuous growth rate (as a decimal), and \( t \) is the time in years.
  • Exponential Function: The "\( e \)" in the formula is the base of natural logarithms, approximately equal to 2.718.
Continuous compounding can significantly affect the present value, especially over longer time periods, making it an important consideration in financial calculations.
Net Income Growth
Net income growth is an important factor for businesses, as it indicates financial health and potential for expansion. In this scenario, the company projects its net income increase at a steady rate of 3% per annum.
This growth rate is a vital piece for calculating future incomes.
  • Initial Base: The base amount for growth calculations is the first year's net income, amounting to \( \$1.2 \text{ million} \).
  • Growth Formula: Future net income for year \( t \) is calculated as \( I(t) = 1.2 \cdot (1 + 0.03)^t \).
  • Smooth Growth: Such a model assumes smooth and continuous growth, which is both predictable and manageable for planning.
Understanding net income growth helps in predicting financial performance and informing strategic decisions.
Capital Value
Capital value refers to the present worth of a company's future net incomes. It provides a snapshot of what those future earnings are worth today, using a specific discount rate. This is crucial for decision-making such as valuing a company for potential investment or sale.
In our example, the capital value is determined by summing the present values of projected net incomes over five years.
  • Calculation: It uses the continuous compounding formula to bring future earnings back to present value.
  • Considerations: The higher the discount rate, the lower the present value, and vice versa.
  • Application: The resulting capital value helps stakeholders understand current worth relative to future earnings.
Accurate calculation of capital value is key to making informed business and investment decisions.
Future Value Calculations
Future value calculations determine what a current amount will grow to in the future by applying a growth rate over time. This contrasts with present value calculations, which discount future amounts to find their current worth.
In our exercise, the future net income for each year is calculated based on the expected 3% growth.
  • Formula: The general formula for future value based on growth rate is \( FV = PV \cdot (1 + i)^t \).
  • Components: Here, \( PV \) stands for the present value, \( i \) is the growth rate (as a decimal), and \( t \) is the number of years.
  • Projections: Using this formula helps businesses project incomes and prepare financially for the future.
Understanding future value calculations equips businesses to plan long-term financial strategies and assess growth prospects effectively.

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