Problem 63 - College Algebra with Corequisite Support

Question

In the year \(1985,\) a house was valued at \(\$ 110,000\). By the year 2005, the value had appreciated to \(\$ 145,000 .\) What was the annual growth rate between 1985 and \(2005 ?\) Assume that the value continued to grow by the same percentage. What was the value of the house in the year \(2010 ?\)

Step-by-Step Solution

Answer

The annual growth rate is 1.375%. The house value in 2010 is approximately $155,276.

Step 1: Understanding the Problem

We need to determine the annual growth rate at which the house value appreciated from \(1985\) to \(2005\), and then use that growth rate to find out what the house value would be in \(2010\).

Step 2: Define the Formula for Growth Rate

The formula for compound annual growth rate (CAGR) is \[ \text{CAGR} = \left( \frac{V_{f}}{V_{i}} \right)^{\frac{1}{n}} - 1 \] where \(V_{f}\) is the final value, \(V_{i}\) is the initial value, and \(n\) is the number of years.

Step 3: Substitute Values into the Formula

Substitute \(V_{f} = 145,000\), \(V_{i} = 110,000\), and \(n = 20\) years (from 1985 to 2005) into the CAGR formula: \[ \text{CAGR} = \left( \frac{145,000}{110,000} \right)^{\frac{1}{20}} - 1 \]

Step 4: Calculate the Growth Rate

Calculate the ratio: \( \frac{145,000}{110,000} = 1.3182 \). Then take the 20th root: \( 1.3182^{\frac{1}{20}} \approx 1.01375 \). Subtract 1 to find the growth rate: \(1.01375 - 1 = 0.01375\), or \(1.375\%\) annual growth rate.

Step 5: Forecasting the Value for 2010

Using the growth rate obtained, calculate the value for 5 more years (from 2005 to 2010): Use the formula \[ V_{2010} = V_{2005} \times (1 + ext{CAGR})^{5} \] where \(V_{2005} = 145,000\) and \(\text{CAGR} = 0.01375\).

Step 6: Calculate the 2010 Value

Substitute into the forecast formula: \[ V_{2010} = 145,000 \times (1 + 0.01375)^{5} \] Calculate \((1.01375)^{5} \approx 1.0708\) and then multiply: \[ V_{2010} = 145,000 \times 1.0708 \approx 155,276 \] Therefore, the value of the house in 2010 is approximately \$155,276.

Key Concepts

Understanding House Value Appreciation

House value appreciation refers to the increase in a home's market value over time. This growth can result from various factors such as inflation, changes in interest rates, neighborhood development, or overall economic conditions. Homeowners and investors consider appreciation important because it can significantly affect the return on investment when selling a property or refinancing.

Watching trends in real estate values gives a good idea about future pricing. When a house appreciates, it simply means it is worth more than it was before. To calculate the appreciation rate, one must understand the total increase in value over a period. For example, in the exercise, the property's value grew from \(\\(110,000\) in 1985 to \(\\)145,000\) in 2005, showing an appreciation. The key is to find the yearly average increase rate using the compound annual growth rate (CAGR) formula.

The Concept of Annual Growth Rate

The annual growth rate is a figure that represents the average yearly increase in value of an investment or property over a specified time. This rate allows investors to compare different assets or evaluate growth trends.

In the exercise, the compound annual growth rate (CAGR) formula is used:

• \( \text{CAGR} = \left( \frac{V_{f}}{V_{i}} \right)^{\frac{1}{n}} - 1 \)

Here, \( V_{f} \) is the final value, \( V_{i} \) is the initial value, and \( n \) is the number of years between these values.

Using the exercise's figures, the substitutions lead to a calculation of \( 1.375\% \) annual growth from 1985 to 2005. This means on average, the house's value increased by \( 1.375\% \) each year over these 20 years. Understanding CAGR helps with making predictions about how a value might grow over time and comparing historical growth rates.

Forecasting Future Value of House

Forecasting future value involves estimating what the house's value could be at a future date, considering the annual growth rate. For this, the previous CAGR is utilized to project future growth over a new period.

To predict the value in 2010, the growth rate calculated is applied over five additional years. The forecast formula is:

• \( V_{2010} = V_{2005} \times (1 + \text{CAGR})^{5} \)

Here, \( V_{2005} \) is the value in 2005, the end year of the calculated annual growth rate, and CAGR continues to estimate future worth.

The exercise concludes with the outcome of \(\$155,276\) as the estimated value in 2010. Forecasting involves assumptions about the future curve of growth. Although it's not certain, it provides a snapshot based on past performance and a tool for strategic decision-making regarding investments and planning.