Problem 12 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

A tiny financial model. To investigate investment strategies, consider the following: You can choose to invest your money in one particular stock or put it in a savings account. Your initial capital is \(\in 1000\). The interest rate \(r\) is \(0.5 \%\) per month and does not change. The initial stock price is \(\in 100\). Your stochastic model for the stock price is as follows: next month the price is the same as this month with probability \(1 / 2\), with probability \(1 / 4\) it is \(5 \%\) lower, and with probability \(1 / 4\) it is \(5 \%\) higher. This principle applies for every new month. There are no transaction costs when you buy or sell stock. Your investment strategy for the next 5 years is: convert all your money to stock when the price drops below \(\in 95\), and sell all stock and put the money in the bank when the stock price exceeds \(\in 110\). Describe how to simulate the results of this strategy for the model given.

Step-by-Step Solution

Answer

Simulate the stock price changes month by month, applying the strategy rules to switch between stocks and savings.

Step 1: Understand the Simulation Goal

The goal is to simulate your investment strategy over 5 years (60 months) based on the initial conditions and probabilities provided. We'll track the movement between stock and savings account, depending on the stock price changes.

Step 2: Set Initial Conditions

Start with an initial capital of \(1000\) in savings. The initial stock price is \(100\). The interest rate for the savings account is \(0.5\%\) per month.

Step 3: Establish the Stock Price Change Model

Each month, the stock price can either stay the same (50% chance), decrease by 5% (25% chance), or increase by 5% (25% chance). Calculate new prices using the current price and the corresponding percentage changes.

Step 4: Define the Investment Strategy

Convert all money to stock if the price falls below \(95\). Sell all stocks and move money to the savings account if the stock price exceeds \(110\). Adjust the number of stocks you own based on decisions made.

Step 5: Simulation Loop

For each month, use a random number generator to decide the stock price movement according to the model. Update your accounts based on the investment strategy. Repeat this process for 60 months (5 years).

Step 6: Compute Monthly Savings Interest

If money remains in the savings account, compute the new amount by applying \(0.5\%\) monthly interest: New Amount = Old Amount \(\times (1 + 0.005)\). Apply this interest monthly when not fully invested in stocks.

Step 7: Record and Analyze Results

At the end of each month, record the status of your investment (amount in savings vs stocks, current stock price). After 5 years, review the data to assess the performance of the strategy.

Key Concepts

Investment Strategy Simulation

Simulating an investment strategy involves predicting outcomes based on a set of rules and possible future scenarios. In this particular exercise, you're analyzing an investment strategy over a period of 5 years, which translates to 60 months. You decide whether to be invested in stocks or to keep your funds in a savings account, based on stock price movements.
Every month, the stock price can either remain the same, decrease by 5%, or increase by 5%. These movements are probabilistic, with a 50% chance of no change, and a 25% chance for either a decrease or increase.
To run the simulation, you need to decide how your investments will react to these changes under preset rules.

• When the stock price drops below \(95\), all money is converted to stock.
• If the stock price rises above \(110\), stocks are sold and proceeds are moved to a savings account.

The simulation requires you to repeatedly apply these decision rules each month, recalculating and updating your investments accordingly. After running this process for 60 months, you can analyze the results to determine the performance of your strategy.

Financial Modeling

Financial modeling provides a framework to evaluate investment outcomes by simulating different financial scenarios. This exercise uses a simple financial model combining both stocks and savings account elements.
The model is driven by stochastic processes, which introduce randomness in the stock price movements—mirroring real-world uncertainties in financial markets.
Your initial setup includes:

• An initial capital of \(1000\) in savings.
• An initial stock price of \(100\).
• A monthly savings interest rate of \(0.5\%\).

The model requires tracking monthly changes in stock prices using probabilities: a 50% chance of the stock price remaining the same, and equal chances of 25% for increasing or decreasing by 5%. The financial model guides you in deciding when to convert funds between savings and stocks based on stock prices. Throughout the simulation, you apply these financial principles and update your investment accordingly each month.

Probability in Finance

Probability is the backbone of predicting financial outcomes. It allows you to simulate various scenarios and their likelihoods. In this exercise, you rely on probability to determine monthly changes in the stock price.
Every month, there is a:

• 50% chance that the stock price remains unchanged.
• 25% chance that the stock price decreases by 5%.
• 25% chance that the stock price increases by 5%.

These probabilities are used to make investment decisions like buying or selling stocks based on the model’s behavior. This probabilistic approach reflects real-world market randomness, where future prices are uncertain and must be estimated based on historical data and trends. By understanding these probabilities, you can better anticipate market movements and adapt your investment strategy to maximize potential returns while managing risks.
Over the 60-month period, these probabilities are repeatedly used to update your investment strategy and assess performance.