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Shalit Corporation's 2008 sales were \(\$ 12\) million. Its 2003 sales were \$6 million. a. At what rate have sales been growing? b. Suppose someone made this statement: "Sales doubled in 5 years. This represents a growth of \(100 \%\) in 5 years; so dividing \(100 \%\) by \(5,\) we find the growth rate to be \(20 \%\) per year." Is that statement correct?

Short Answer

Expert verified
Sales grew at 14.87% annually. The 20% rule is incorrect.

Step by step solution

01

Understanding the Problem

We need to find the compound annual growth rate (CAGR) of sales from 2003 to 2008. CAGR is the rate at which sales grow annually, compounded over a number of years.
02

Identify Known Values

We know the sales in 2003 were \( \\(6 \) million and in 2008 were \( \\)12 \) million. The number of years between 2003 and 2008 is \(5\) years.
03

Apply CAGR Formula

The formula for CAGR is \( \left( \frac{{\text{End Value}}}{{\text{Start Value}}} \right)^{\frac{1}{n}} - 1 \), where \( n \) is the number of years. Apply the values: \[ \left( \frac{12}{6} \right)^{\frac{1}{5}} - 1 \].
04

Calculate Exponent

Evaluate \( \frac{12}{6} = 2 \). Therefore, we have to calculate \( 2^{\frac{1}{5}} \).
05

Complete the Calculation

Calculate \( 2^{\frac{1}{5}} \) using a calculator, which is approximately \( 1.1487 \). Subtract \(1\) to find the growth rate: \(1.1487 - 1 = 0.1487\).
06

Convert to Percentage

The growth rate \(0.1487\) as a percentage is \(14.87\%\).
07

Evaluate the Statement

The claim that dividing \(100\%\) by \(5\) gives a \(20\%\) annual growth rate is incorrect. It assumes linear growth, not compound growth, which is inappropriate in this context.

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Key Concepts

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

Sales Growth
Sales growth is an important metric for any business as it shows how much a company's sales increase over time. It helps businesses measure progress and plan for future growth.
Sales growth can provide insights into a company's market position and competitiveness in its industry. When a company experiences consistent sales growth, it implies that more customers are buying its products or services. This can be due to various factors, such as increased market share, new product lines, or expansion into new markets.
Understanding sales growth involves:
  • Monitoring sales numbers over specific periods
  • Identifying trends and patterns
  • Recognizing external factors influencing sales, such as economic changes or competition
These aspects are vital for strategic financial planning and driving the success of a business.
Financial Management
Financial management is the process of planning, organizing, and controlling financial resources within an organization. It plays a crucial role in ensuring financial stability and achieving long-term objectives.
Effective financial management involves several key areas:
  • Budgeting: Allocating resources to various organizational needs
  • Investment: Deciding where to place funds for maximum return
  • Risk Management: Identifying and mitigating financial risks
  • Financial Analysis: Evaluating financial data to inform decision-making
Financial management impacts a business’s ability to grow through managing sales, investments, and ensuring a steady cash flow. It also helps companies adapt to market changes and stay competitive.
CAGR Formula
The Compound Annual Growth Rate, or CAGR, is a useful measure to determine the mean annual growth rate of an investment or metric, over a specified time period. Unlike simple averages, CAGR takes into account the effects of compounding, a key factor in understanding growth rates.
The formula for CAGR is:\[\left( \frac{\text{End Value}}{\text{Start Value}} \right)^{\frac{1}{n}} - 1 \]Where:
  • End Value is the value at the end of the period
  • Start Value is the value at the beginning of the period
  • \(n\) is the number of years
The CAGR gives a clearer picture of growth trends, smoothing out the effects of volatility by focusing solely on the start and end values. This makes it particularly useful in finance and business planning.
Annual Growth Rate
The annual growth rate is a way to measure the yearly increase of some financial variable, such as sales or revenues, usually given as a percentage. It is important to note that there are two common methods to calculate it: linear and compound growth.
Linear growth assumes a fixed increase each year, while compound growth considers the increased base each year, reflecting reality more accurately. The simple calculation provided in the exercise as "20% per year" mistakenly uses linear growth, which can lead to misleading interpretations when analyzing financial data.
Understanding the difference and knowing when to apply each type of growth calculation helps in:
  • Accurately assessing business performance
  • Making informed strategic decisions
  • Communicating financial health to stakeholders
Knowing this distinction is crucial for any business or investor looking to understand financial growth correctly.

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Most popular questions from this chapter

Find the future values of these ordinary annuities. Compounding occurs once a year. a. \(\$ 400\) per year for 10 years at \(10 \%\) b. \(\$ 200\) per year for 5 years at \(5 \%\) c. \(\$ 400\) per year for 5 years at \(0 \%\) d. Rework Parts a, b, and c assuming they are annuities due.

Your firm sells for cash only; but it is thinking of offering credit, allowing customers 90 days to pay. Customers understand the time value of money, so they would all wait and pay on the 90 th day. To carry these receivables, you would have to borrow funds from your bankat a nominal \(12 \%\), daily compounding based on a 360 -day year. You want to increase your base prices by exactly enough to offset your bank interest cost. To the closest whole percentage point, by how much should you raise your product prices?

Find the present value of \(\$ 500\) due in the future under each of these conditions: a. \(12 \%\) nominal rate, semiannual compounding, discounted back 5 years b. \(12 \%\) nominal rate, quarterly compounding, discounted back 5 years c. \(12 \%\) nominal rate, monthly compounding, discounted back 1 year d. Why do the differences in the PVs occur?

Your client is 40 years old; and she wants to begin saving for retirement, with the first payment to come one year from now. She can save \(\$ 5,000\) per year; and you advise her to invest it in the stock market, which you expect to provide an average return of \(9 \%\) in the future. a. If she follows your advice, how much money will she have at \(65 ?\) b. How much will she have at \(70 ?\) c. She expects to live for 20 years if she retires at 65 and for 15 years if she retires at 70 . If her investments continue to earn the same rate, how much will she be able to withdraw at the end of each year after retirement at each retirement age?

You have \(\$ 42,180.53\) in a brokerage account, and you plan to deposit an additional \(\$ 5,000\) at the end of every future year until your account totals \(\$ 250,000 .\) You expect to earn \(12 \%\) annually on the account. How many years will it take to reach your goal?

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