a) A company manufacturing soft drinks is thinking of expanding its plant capacity so as to meet future demand. The monthly sale for the past 5 years are available. State, giving reasons, the type of modelling you will use to obtain good estimates for future demand so as to help the company make the right decisions. Also state four essentials and four non-essentials for the problem.
Answer:
For predicting future demand based on past sales data, a time series analysis model would be an appropriate choice. Time series analysis takes into account the temporal nature of the data, allowing for the identification of trends, seasonal patterns, and other temporal dynamics that are crucial for accurate forecasting.
Reasons for choosing time series analysis:
Trend Analysis: Time series models can capture underlying trends in the sales data, which is essential for long-term planning and forecasting.
Seasonality: Soft drink sales are likely to exhibit seasonal patterns (e.g., higher sales in summer). Time series models can account for this seasonality, providing more accurate estimates for future demand.
Forecasting: Time series models are specifically designed for forecasting future values based on past observations, making them well-suited for this application.
Historical Data: The company has 5 years of monthly sales data, providing a sufficient historical record for modeling and forecasting purposes.
Four essentials for the problem:
Historical Sales Data: Detailed monthly sales data for the past 5 years is essential for building and training the time series model.
Seasonal Factors: Understanding seasonal variations in soft drink sales (e.g., higher sales in summer) is crucial for accurate forecasting.
Trend Analysis: Identifying any long-term trends in the sales data (e.g., increasing or decreasing demand) is important for future planning.
Model Validation: Splitting the data into training and testing sets is essential for validating the accuracy of the time series model.
Four non-essentials for the problem:
Daily Sales Data: While potentially useful, daily data may not be necessary for monthly demand forecasting and could complicate the analysis.
Individual Product Sales: If the company is interested in overall plant capacity, detailed sales data for individual soft drink products may not be essential.
Competitor Sales Data: While competitor information can be useful for strategic planning, it may not be essential for forecasting the company’s own demand.
External Economic Indicators: Economic indicators like GDP or consumer spending indices may not be necessary for the specific task of forecasting soft drink demand, although they could provide additional context.
Overall, a time series analysis model that takes into account trends and seasonality, using the past 5 years of monthly sales data, would be a suitable approach for estimating future demand and assisting the company in making informed expansion decisions.
b) Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?
Portfolio
Expected return
Standard deviation
W
9%9 \%
21%21 \%
X
5%5 \%
7%7 \%
Y
15%15 \%
36%36 \%
Z
12%12 \%
15%15 \%
Portfolio Expected return Standard deviation
W 9% 21%
X 5% 7%
Y 15% 36%
Z 12% 15%| Portfolio | Expected return | Standard deviation |
| :—: | :—: | :—: |
| W | $9 \%$ | $21 \%$ |
| X | $5 \%$ | $7 \%$ |
| Y | $15 \%$ | $36 \%$ |
| Z | $12 \%$ | $15 \%$ |
Answer:
The efficient frontier, as described by Markowitz, is a set of portfolios that offer the highest expected return for a given level of risk (standard deviation) or the lowest risk for a given level of expected return. Portfolios that lie below the efficient frontier are considered suboptimal because they provide a lower return for the same level of risk or a higher risk for the same level of return.
To determine which portfolio cannot lie on the efficient frontier, we can compare the portfolios’ risk-return profiles:
Portfolio W: Expected return = 9%, Standard deviation = 21%
Portfolio X: Expected return = 5%, Standard deviation = 7%
Portfolio Y: Expected return = 15%, Standard deviation = 36%
Portfolio Z: Expected return = 12%, Standard deviation = 15%
Comparing Portfolio W and Z, we see that Portfolio Z has a higher expected return (12% vs. 9%) with a lower standard deviation (15% vs. 21%). This means Portfolio W is dominated by Portfolio Z, as Z offers both higher return and lower risk.
Therefore, Portfolio W cannot lie on the efficient frontier.