Initial Value and Exponential Factor

Oct 13, 2025 by miodeya.com 99 views

Understanding the Problem

Aaron bought stock worth $125 at the beginning of the day. Every hour, the stock decreases by 3.5%. We need to write an exponential function to model this scenario.

Initial Value and Exponential Factor

The initial value of the stock is $125. The exponential factor, which represents the rate of decline, is $1 - 0.035 = 0.965$ . This means that the stock price decreases by 3.5% every hour.

Equation

To model the stock price decline, we can use the exponential function:

f(t) = 125 * (0.965)^t

where f(t) is the stock price at time t (in hours).

Explanation

In this equation, the initial value of $125 is multiplied by the exponential factor (0.965)^t. The exponent t represents the number of hours that have passed. As t increases, the stock price decreases exponentially.

Discussion

The exponential function f(t) = 125 * (0.965)^t models the stock price decline over time. This function can be used to calculate the stock price at any given time t.

Example Calculations

Let's calculate the stock price at different times:

At t = 0 (beginning of the day), the stock price is f(0) = 125 * (0.965)^0 = $125.
At t = 1 (1 hour later), the stock price is f(1) = 125 * (0.965)^1 ≈ $120.88.
At t = 2 (2 hours later), the stock price is f(2) = 125 * (0.965)^2 ≈ $116.93.

Conclusion

The exponential function f(t) = 125 * (0.965)^t models the stock price decline over time. This function can be used to calculate the stock price at any given time t.

Code Implementation

Here is a Python code implementation of the exponential function:

import math
def stock_price(t):
"""
Calculate the stock price at time t (in hours)
"""
initial_value = 125
exponential_factor = 0.965
return initial_value * (exponential_factor ** t)
print(stock_price(0))  # $125
print(stock_price(1))  # $120.88
print(stock_price(2))  # $116.93

Mathematical Background

Frequently Asked Questions

In this article, we will answer some frequently asked questions about modeling stock price decline with exponential functions.

Q: What is the initial value of the stock price?

A: The initial value of the stock price is $125.

Q: What is the exponential factor?

A: The exponential factor is $1 - 0.035 = 0.965$ . This represents the rate of decline of the stock price, which is 3.5% per hour.

Q: How do I calculate the stock price at a given time t?

A: To calculate the stock price at a given time t, you can use the exponential function f(t) = 125 * (0.965)^t.

Q: What is the stock price at t = 0?

A: At t = 0 (beginning of the day), the stock price is f(0) = 125 * (0.965)^0 = $125.

Q: How does the stock price change over time?

A: The stock price decreases exponentially over time, with a decay rate of 3.5% per hour.

Q: Can I use this model for other types of investments?

A: Yes, this model can be used for other types of investments that exhibit exponential decay, such as bonds or other financial instruments.

Q: How accurate is this model?

A: The accuracy of this model depends on the assumptions made about the stock price decline. In reality, stock prices can be affected by many factors, including market trends, economic conditions, and company performance.

Q: Can I use this model to predict future stock prices?

A: While this model can provide a general idea of how the stock price might change over time, it is not a reliable predictor of future stock prices. Stock prices can be highly volatile and subject to many unpredictable factors.

Q: How can I implement this model in a programming language?

A: You can implement this model in a programming language such as Python using the following code:

import math
def stock_price(t):
"""
Calculate the stock price at time t (in hours)
"""
initial_value = 125
exponential_factor = 0.965
return initial_value * (exponential_factor ** t)
print(stock_price(0))  # $125
print(stock_price(1))  # $120.88
print(stock_price(2))  # $116.93

Conclusion

Modeling stock price decline with exponential functions can be a useful tool for understanding how stock prices change over time. However, it is essential to keep in mind the limitations of this model and not rely on it for making investment decisions.

Additional Resources

For more information on exponential functions and their applications in finance, you can refer to the following resources: