SOLUTION: If a stock price goes from $10 to $12 from January 1st to January 31, from $12 to $9 from February 1st to February 28th, and from $9 to $15 from March 1st to March 31th is the pric

Question 1164762: If a stock price goes from $10 to $12 from January 1st to January 31, from $12 to $9 from February 1st to February 28th, and from $9 to $15 from March 1st to March 31th is the price change from $10 to $15 a straight line?
It is clear that in each of the three time intervals mentioned there was a complex daily variation of prices as in an electrocardiogram. But what would be a simplified solution for a first naive view of the situation? Would a simple function hold up? What is the simplest function to represent this situation? Does your naïve initial and simplified model allow you to predict the behavior of the stock in the next month?
How can I use three “pieces” of lines to describe the price movements from the beginning of January to the end of March? Show the graph for the price movement.
Answer by CPhill(2189) (Show Source): You can put this solution on YOUR website!
To answer your question directly: **No, the price change from $10 to $15 is not a straight line.** While the net result is an increase of $5, the path taken involves significant fluctuations that a single straight line cannot capture.
Here is a breakdown of the situation from a "naive" perspective to a more structured mathematical model.
---
### 1. The Naive View: A Simple Function?
If you took a "first naive view," you might simply look at the start and end points: **(Jan 1st, $10)** and **(March 31st, $15)**.
* **Would a simple function hold up?** A simple linear function () would connect these two points, suggesting a steady growth of roughly $1.67 per month.
* **The Flaw:** This model fails because it completely ignores the fact that in February, the stock price actually **dropped** below its starting value. It masks the volatility (risk) of the investment.
* **Predictive Power:** This naive model has **zero predictive power** for the next month. Stock prices are "stochastic" (random), and a simple line assumes a trend that doesn't account for market shocks or changes in direction.
---
### 2. The Simplest Multi-Point Model: Piecewise Linear
To represent this more accurately while still keeping it simple, we use a **Piecewise Linear Function**. This is the simplest way to show the "simplified" movement without losing the critical turns in February and March.
By treating time as the independent variable () and price as the dependent variable (), we can define three segments:
| Interval | Start Price | End Price | Trend |
| --- | --- | --- | --- |
| **January** (Month 1) | $10 | $12 | Upward (+2) |
| **February** (Month 2) | $12 | $9 | Downward (-3) |
| **March** (Month 3) | $9 | $15 | Upward (+6) |
#### The Mathematical Function
The function (Price over time in months) would look like this:

---
### 3. The Visual Representation
When you graph these three "pieces," you get a zigzag shape. This is often called a "price chart" in finance, which simplifies daily "noise" into monthly trends.
#### Why this is useful:
* **Volatility:** You can see the "sharpness" of the angles. The March segment is steeper than the January segment, indicating a faster recovery.
* **Support/Resistance:** It shows that the price fell below the initial January level ($9 vs $10) before the final surge.
### 4. Can this predict April?
Mathematically, no. In finance, this is known as the **Random Walk Hypothesis**. Just because the line went "up, down, up" doesn't mean it must go "down" next.
However, technical analysts look at these "pieces" to find **momentum**. Since the March growth (-3$), a momentum trader might guess the trend is now bullish, whereas a mean-reversion trader might guess it's overextended and will drop.
Would you like me to show you how to calculate the **Total Percent Return** versus the **Annualized Return** for this three-month period?
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