Question

Given the following returns and assuming that \(S(0)=45\) dollars, find the possible stock prices in a three-step economy and sketch a tree of price movements: \(\begin{array}{lrrr}\text { Scenario } & K(1) & K(2) & K(3) \\ \omega_{1} & 10 \% & 5 \% & -10 \% \\ \omega_{2} & 5 \% & 10 \% & 10 \% \\ \omega_{3} & 5 \% & -10 \% & 10 \%\end{array}\)

Short answer

At the end of three steps, the stock prices are 46.7775, 57.1725, and 46.7775.

Step-by-step solution

- Understanding Initial Price

The initial stock price is given as \(S(0) = 45\) dollars.

- Calculate Prices After First Step

Using the returns in Scenario 1 for the first step: \(S_1(1) = S(0) \times (1 + K_1(1)) = 45 \times (1 + 0.1) = 45 \times 1.1 = 49.5\)Using Scenario 2 for the first step: \(S_2(1) = 45 \times (1 + 0.05) = 45 \times 1.05 = 47.25\)Using Scenario 3 for the first step: \(S_3(1) = 45 \times (1 + 0.05) = 45 \times 1.05 = 47.25\)

- Calculate Prices After Second Step

For Scenario 1 from Step 1 using returns in Step 2:\(S_{11}(2) = 49.5 \times (1 + 0.05) = 49.5 \times 1.05 = 51.975\)Scenario 2 from Step 1 using returns in Step 2:\(S_{12}(2) = 47.25 \times (1 + 0.10) = 47.25 \times 1.10 = 51.975\)Scenario 3 from Step 1 using returns in Step 2:\(S_{13}(2) = 47.25 \times (1 - 0.10) = 47.25 \times 0.90 = 42.525\)

- Calculate Prices After Third Step

Continuing from Scenario 1 price at second step using returns in Step 3: \(S_{111}(3) = 51.975 \times (1 - 0.10) = 51.975 \times 0.90 = 46.7775\)From Scenario 2 price at second step:\(S_{121}(3) = 51.975 \times (1 + 0.10) = 51.975 \times 1.10 = 57.1725\)For Scenario 3 price at second step:\(S_{132}(3) = 42.525 \times (1 + 0.10) = 42.525 \times 1.10 = 46.7775\)

- Sketch The Tree

Draw the tree with the calculated prices at each step. At Step 0: \(S(0) = 45\) At Step 1: \(S_1(1) = 49.5, S_2(1) = 47.25, S_3(1) = 47.25\) At Step 2: \(S_{11}(2) = 51.975, S_{12}(2) = 51.975, S_{13}(2) = 42.525\) At Step 3: \(S_{111}(3) = 46.7775, S_{121}(3) = 57.1725, S_{132}(3) = 46.7775\)

Key concepts

Three-Step Economy

In financial modeling, a three-step economy is used to predict possible stock prices over three periods. This helps in understanding how an investment might perform under different scenarios.

Each step represents a time period during which the stock price changes based on predefined return rates. The initial stock price is modified at each step according to these rates.

The three-step model is simple but powerful. It teaches foundational concepts for more complex financial models. It's like taking baby steps before running a marathon in finance!

Return Rates

Return rates are crucial in calculating future stock prices. They represent the percentage change in stock prices over a time period.

For instance, a return rate of 5% means the stock price increases by 5%. If the initial stock price is \(45, a 10% return would adjust it to \)49.50 (since 45 * 1.10 = 49.5).

Different scenarios can have different return rates. Understanding these helps in forecasting diverse outcomes.

Higher return rates generally indicate more risk but also higher potential rewards. Hence, it's essential to consider the probabilities associated with each return rate scenario.

Stock Price Tree

A stock price tree is a visual representation of how a stock price can evolve over time considering different return rates.

Start with the initial stock price at step zero.

As you move through each step, use the return rates to calculate possible prices at that step. This creates branches like a tree, showing different possible future prices.

For example, starting from \(45, after the first step, you might have prices like \)49.5, \(47.25, and \)47.25 depending on different return rates. As you further progress, you get new prices derived from these intermediate prices.

It's like creating a roadmap for stock price changes, helping investors visualize potential future prices and make informed decisions.

The tree provides an intuitive way to see the impact of various scenarios on future stock prices.