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. 

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 $1 + 0.20 = 1.20$.
- **Second year**: The investment goes down by 12%, which corresponds to a growth factor of $1 - 0.12 = 0.88$.
- **Third year**: The investment goes up by 15%, which corresponds to a growth factor of $1 + 0.15 = 1.15$.

To get the total growth factor over the three years, we multiply the individual growth factors:

\[
\text{Total Growth Factor} = 1.20 \times 0.88 \times 1.15
\]

### Step 2: Compute the total growth factor.
Let me calculate that for you.

\[
\text{Total Growth Factor} = 1.20 \times 0.88 \times 1.15 = 1.2144
\]

### 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:

\[
\text{Annualized Growth Rate} = \left( \text{Total Growth Factor} \right)^{1/n} - 1
\]

where $n = 3$ (since it's a 3-year period). Substituting in the total growth factor we found:

\[
\text{Annualized Growth Rate} = \left( 1.2144 \right)^{1/3} - 1
\]

Now, I'll compute the result.

\[
\text{Annualized Growth Rate} = \left( 1.2144 \right)^{1/3} - 1 \approx 0.0668 = 6.68\%
\]

### Final Answer:
The annualized percent change over the three-year period is approximately **6.68%**.

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Would you like further details on how this is derived? Here are some related questions to explore further:

1. What is the difference between nominal and annualized returns?
2. How would the annualized rate change if the second year's decrease was larger?
3. How does compounding impact the overall return over multiple years?
4. Can the annualized return be negative, and under what conditions?
5. 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.

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.

Learn how to compute the annualized percent change for an investment over three years. This problem demonstrates percentage growth, compounding, and the formula for annualized returns. It is suitable for high school students in Grades 10-12.

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