Math Problem Statement
Exercise 4. Suppose that the value of an initial investment of $500 went up 20% the first year, down 12% the second year, and up 15% the third year. (d) Compute the annualized percent change over the three-year period.
Solution
To compute the annualized percent change over the three-year period, we need to determine the equivalent constant percentage increase per year that would result in the same overall change in the investment after three years.
Here’s the step-by-step process:
Step 1: Calculate the overall growth factor.
The investment goes through three percentage changes over three years:
- First year: The investment goes up by 20%, which corresponds to a growth factor of .
- Second year: The investment goes down by 12%, which corresponds to a growth factor of .
- Third year: The investment goes up by 15%, which corresponds to a growth factor of .
To get the total growth factor over the three years, we multiply the individual growth factors:
Step 2: Compute the total growth factor.
Let me calculate that for you.
Step 3: Calculate the annualized percent change.
The annualized percent change is the equivalent constant rate of return that would give the same total growth factor over the 3 years. This is found using the formula:
where (since it's a 3-year period). Substituting in the total growth factor we found:
Now, I'll compute the result.
Final Answer:
The annualized percent change over the three-year period is approximately 6.68%.
Would you like further details on how this is derived? Here are some related questions to explore further:
- What is the difference between nominal and annualized returns?
- How would the annualized rate change if the second year's decrease was larger?
- How does compounding impact the overall return over multiple years?
- Can the annualized return be negative, and under what conditions?
- How is this calculation different from simple average annual returns?
Tip: Always remember to distinguish between the total return and annualized return, especially for investments over multiple periods.
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Math Problem Analysis
Mathematical Concepts
Percentage Growth
Compounding
Annualized Return
Formulas
Total Growth Factor = (1 + Change1) * (1 + Change2) * (1 + Change3)
Annualized Growth Rate = (Total Growth Factor)^(1/n) - 1
Theorems
Compound Growth Theorem
Suitable Grade Level
Grades 10-12
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