Problem 13 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A panel of 50 economists was asked to predict the average prime interest rate for the upcoming year. The results of the survey follow: $$\begin{array}{lcccccc}\hline \text { Interest Rate, } \% & 4.9 & 5.0 & 5.1 & 5.2 & 5.3 & 5.4 \\\\\hline \text { Economists } & 3 & 8 & 12 & 14 &8 & 5 \\ \hline\end{array}$$ Based on this survey, what does the panel expect the average prime interest rate to be next year?

Step-by-Step Solution

Answer

The panel expects the average prime interest rate for the upcoming year to be approximately 5.16%.

Step 1: Understand what to calculate.

We need to find the weighted average of the interest rates, with the number of economists predicting each rate serving as the weights. We can find this by multiplying each interest rate by the number of economists predicting that rate, summing these products, and then dividing by the total number of economists.

Step 2: Compute the numerator of the weighted average.

To calculate the numerator, multiply each interest rate by the number of economists predicting it and sum up these values: - 4.9% x 3 economists = 14.7 - 5.0% x 8 economists = 40 - 5.1% x 12 economists = 61.2 - 5.2% x 14 economists = 72.8 - 5.3% x 8 economists = 42.4 - 5.4% x 5 economists = 27 Sum = 14.7 + 40 + 61.2 + 72.8 + 42.4 + 27 = 258.1 The sum of the weighted interest rates is 258.1.

Step 3: Compute the denominator of the weighted average.

The denominator is the total number of economists in the panel, which is 50.

Step 4: Calculate the weighted average.

To find the average prime interest rate, we divide the sum of the weighted interest rates (258.1) by the total number of economists (50): Weighted average = \( \frac{258.1}{50} \) Weighted average ≈ 5.16% Based on the survey, the panel expects that the average prime interest rate for the upcoming year will be approximately 5.16%.

Key Concepts

Weighted Average

The concept of a weighted average is a valuable tool in statistics and economics. It provides a way to calculate an average that takes into account the relative importance or frequency of different values. In this context, the weighted average offers a more nuanced view than a simple average because it considers how often each value occurs.

For our example with the economists predicting interest rates, each predicted rate varies in frequency. Some rates are more commonly predicted than others. By using a weighted average, we ensure that the final average rate reflects the distribution and frequency of the predictions.

Here's how we do it:

• First, multiply each interest rate by the number of economists who predicted it. This gives more weight to rates predicted by more economists.
• Next, sum all these products. This step aggregates the total weighted value.
• Finally, divide this sum by the total number of economists to get the weighted average. This final step normalizes the weighted total, resulting in the weighted average prediction.

Understanding the weighted average involves recognizing that not all numbers are equal; some play a more significant role in the outcome.

Interest Rate Forecasting

Interest rate forecasting is a crucial element in financial and economic planning. It involves predicting future interest rates, which can influence a wide variety of financial decisions. These forecasts help businesses, investors, and policymakers make informed decisions about loans, investments, and economic policies.

There are several methods and models used in interest rate forecasting:

• **Historical Trends:** Examining past interest rate trends to find patterns that might repeat in the future.
• **Economic Indicators:** Using economic data like inflation rates, employment figures, and GDP growth to predict how interest rates might change.
• **Expert Surveys:** As in our example, surveys from knowledgeable individuals in economics can provide valuable insights into future rates.

Forecasts are never certain, but they provide guidance based on the best available data. They help balance risk by preparing individuals and institutions for potential changes in interest rates.

Economics

Economics as a field studies the production, distribution, and consumption of goods and services. It's crucial for understanding markets and how resources are allocated, influencing everything from individual's personal finances to government policy.

In our exercise, the economists' forecast of interest rates is connected to the larger field of economics because:

• **Monetary Policy Impact:** Interest rates are controlled by central banks as part of monetary policy, influencing inflation and economic growth.
• **Investment Decisions:** Changes in interest rates affect the attractiveness of different investments, influencing economic activity.
• **Consumer Behavior:** Interest rates impact borrowing and spending habits, which can drive or slow economic growth.

With interest rate predictions, economists contribute significantly to decision-making in sectors like banking, real estate, and consumer markets.

Surveys

Surveys are a method of gathering information from groups of people by administering a set of questions. In many fields, including economics, surveys are used to collect data and opinions that assist in forecasting and decision-making.

In our exercise, a survey was used to collect predictions from economists regarding future interest rates. This is a common method in data collection because:

• **Diverse Insights:** Surveys can gather a wide range of opinions, providing a comprehensive overview of expectations.
• **Efficiency:** They are a relatively quick and cost-effective way to gather large amounts of data.
• **Trend Analysis:** Subsequent surveys can highlight changes in opinions over time, useful for trend analysis.

Surveys don't just provide data; they can reflect the collective wisdom and judgment of a diverse group, offering insights into likely future trends.