Paula Shoemaker Produces A Weekly Stock Market Report For An ✓ Solved

Paula Shoemaker produces a weekly stock market report for an exclusive readership. She normally sells 3,000 reports per week, and 70% of the time her sales range from 2,850 to 3,150. The report costs Paula $25 to produce, but Paula is able to sell reports for $350 each. Of course, any reports not sold by the end of the week have no value. How many reports should Paula produce each week? How to get the standard deviation?

Paper For Above Instructions

Paula Shoemaker's decision on how many stock market reports to produce each week should be based on an analysis of her sales distribution and associated profits from production. Given the provided parameters, we can determine the ideal number of reports to produce per week to maximize profits while minimizing losses from unsold inventory. Additionally, we will calculate the standard deviation of her sales to help understand the variability associated with her report sales.

Understanding Sales Distribution

Paula normally sells 3,000 reports weekly, with a 70% probability of selling between 2,850 and 3,150 reports. This indicates a relatively stable sales environment; however, fluctuations still exist. To evaluate production levels, we need to define the expected sales, the range of sales volume, and the associated profit calculations.

Calculating Expected Sales Volume

Using the given data, we can model the sales as a normal distribution where:

  • Mean (μ) = 3000 reports sold

  • Sales fluctuation range = 2,850 to 3,150 reports (covering approximately 70% of the sales)

We can find the standard deviation (σ) using the range provided. Since 70% of sales fall within one standard deviation (±σ) of the mean in a normal distribution, we can derive the standard deviation as follows:

Calculating Standard Deviation

The range of ±σ from the mean for the sales distribution is:

  • Lower bound = 2850 reports

  • Upper bound = 3150 reports

The difference between the lower and upper bounds is:

3150 - 2850 = 300 reports

This range represents approximately 2 standard deviations in length. Therefore, one standard deviation can be calculated as:

σ = 300 / 2 = 150 reports

This indicates a standard deviation of 150 reports, meaning Paula can expect her sales to deviate from the mean by approximately 150 reports in either direction.

Determining Optimal Production Quantity

To maximize profitability, Paula should produce enough reports to meet the expected sales while accounting for the standard deviation. Given the cost of production ($25 per report) and selling price ($350 per report), we can calculate the profit for different scenarios.

Profit Calculation

Let’s evaluate the profit derived from selling a range of reports in relation to production costs:

  • Let x = number of reports sold

  • Profit = Revenue - Costs

  • = (Selling Price × x) - (Production Cost × Q)

  • Where Q is the quantity produced

For instance, if Paula produces 3,000 reports, her expected revenue and costs would be:

  • Revenue (when selling all reports) = $350 × 3000 = $1,050,000

  • Cost = $25 × 3000 = $75,000

  • Profit = $1,050,000 - $75,000 = $975,000

However, considering fluctuations (standard deviation), if sales drop to 2850 or increase to 3150, this revenue will fluctuate accordingly. To mitigate the risk of overproduction and loss, Paula should carefully evaluate production levels against variability in sales.

Considering the Sales Probability Distribution

With the population centered around 3000 (mean), it is prudent for Paula to produce at least 3000 reports, as this number is expected to generate targeted revenues without producing excessive waste. However, given the variability, she may consider producing closer to the mean sales volume plus one standard deviation to ensure coverage of demand while accounting for possible fluctuations.

Thus, a strategic recommendation would suggest producing:

  • Production Recommendation = 3000 (mean) + 150 (σ) = 3150 reports as optimal production quantity.

Conclusion

In conclusion, Paula Shoemaker should ideally produce around 3150 stock market reports per week to balance her revenues against possible sales fluctuations, thus maximizing her profit potential. Additionally, understanding her sales variability through the standard deviation of 150 reports enables her to adjust her production strategy based on expected demand.

References

  • Black, K. & Skipper, H. D. (2019). Life Insurance. New York: Wiley.

  • Brealey, R.A., Myers, S.C., & Allen, F. (2020). Principles of Corporate Finance. New York: McGraw-Hill.

  • Markowitz, H. M. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91.

  • Higgins, R. C. (2016). Analysis for Financial Management. New York: McGraw-Hill.

  • Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3), 425-442.

  • Damodaran, A. (2012). Investment Valuation. New York: Wiley.

  • Ross, S.A., Westerfield, R.W., & Jaffe, J. (2018). Corporate Finance. New York: McGraw-Hill.

  • McDonald, R. L. (2006). Derivatives Markets. New York: Addison-Wesley.

  • Ang, A. (2014). Asset Management: A Systematic Approach to Factor Investing. New York: Oxford University Press.

  • Penman, S. H. (2013). Financial Statements: A Step-by-Step Guide to Understanding and Creating Financial Reports. New York: Pearson.