A Simple Market Model:
Basic Concepts and Assumptions

Abstract:
This lesson introduces the “Simple Market Model,” an approach that facilitates the learning of key investment concepts, combining risk-free assets (bonds, with known return) and risky assets (stocks, with uncertain return). We will see how these assets can be combined in a portfolio that, when managed correctly, allows for superior returns compared to bank interest rates, balancing growth and security. Additionally, we will learn how to calculate the return of these assets over a simplified timeline (present and future) and analyze market assumptions such as price randomness and solvency to make informed decisions about investment and risk.

Learning Objectives:
By the end of this lesson, the student will be able to

• Identify the characteristics of a Simple Market Model, risk-free assets, and risky assets in investment decision-making.
• Understand the difference between risk-free and risky assets, identifying how each affects return and risk in a portfolio.
• Apply formulas to calculate the investment return on risky and risk-free assets, using initial and final prices.
• Analyze the construction of portfolios that combine risky and risk-free assets to optimize returns while managing risk in a simple market model.
• Evaluate the impact of market scenarios on the value and return of a portfolio, considering price variations of assets.
• Apply probability to calculate expected return in uncertain market situations, determining potential financial outcomes.

TABLE OF CONTENTS
Introduction
Definitions and Theoretical Assumptions
Risky and Risk-Free Assets
Timeline in the Model
Investment Return
Portfolio Construction and Valuation
Basic Model Assumptions
Solved Problems
Proposed Exercises

Introduction

Imagine you just received a bonus at work and have saved a considerable amount in the bank. However, observing current interest rates and the impact of inflation, you worry that the purchasing power of your savings will decrease over time. You want your money not only to be preserved but also to grow.

You have heard that investing in stocks and bonds can be a good way to grow your money. You know that some assets, like bonds, are safe, while others, like stocks, offer higher returns but carry more risk. You wonder if you could combine both types of assets in a strategy that allows you to earn more than bank interest without taking excessive risks.

You decide to investigate and come across an approach called the “Simple Market Model,” which facilitates learning the basics of risky and risk-free assets, returns, and portfolio construction. This model is ideal for beginners, as it simplifies financial analysis by focusing on two points in time: the present and a future moment.

With this motivation, you decide to learn more about calculating investment returns and building a portfolio that maximizes your returns. As we progress, we will explore these concepts in-depth so that you can make informed decisions and better manage your personal finances.

Now that you are ready, let’s dive into the theoretical knowledge you need to understand this market model and apply it to your investment decisions.

Definitions and Theoretical Assumptions

Risky and Risk-Free Assets

To begin understanding the simple market model, we need to familiarize ourselves with the concepts of risky assets and risk-free assets. These two types of assets form the basis of most investment strategies.

A risk-free asset is an investment whose return is known and secure. A classic example of a risk-free asset is a bond issued by the government or a stable financial institution, which guarantees a fixed interest payment at the end of a period. These bonds can be viewed as deposits in a bank account or debt instruments that offer a predictable and stable return.

On the other hand, a risky asset is one whose future price is uncertain and can vary, both up and down. A common example of a risky asset is the stock of companies traded on the stock exchange. Stocks can be volatile, and their price depends on multiple factors, making their future value unpredictable.

Timeline in the Model

In the simple market model, we restrict the analysis to only two points in time: the present, called t = 0 , and a future moment, such as one year later, called t = 1 . This simplified approach allows us to analyze changes in asset value without excessive complexity.

This two-point-in-time model is especially useful for beginners, as it helps understand how asset prices change over time and how these changes affect the value of a portfolio.

Investment Return

Return is a measure of how much value an investment has gained or lost over a period of time. Depending on the type of asset, the calculation of return can be uncertain or determined.

For a risky asset, such as a stock, the return is uncertain and calculated using the initial and future price of the asset. If the price of the stock at time t is represented by S(t) , the return of the stock between t = 0 and t = 1 is calculated as follows:

K_S = \dfrac{S(1) - S(0)}{S(0)}

This return, represented by K_S , is a fraction of the initial value of the stock and can be positive (if the price of the stock has risen), negative (if it has fallen), or zero (if the price has not changed).

For a risk-free asset, such as a bond, the return is known in advance. If we represent the price of a bond at time t by A(t) , the return of this bond between t = 0 and t = 1 is calculated as:

K_A = \dfrac{A(1) - A(0)}{A(0)}

This return, K_A , is fixed and guaranteed by the bond issuer. The key difference between K_S and K_A is certainty: while the return of a stock is uncertain, the return of a bond is fixed and known.

Portfolio Construction and Valuation

Now that we understand the concept of return, we can combine risky and risk-free assets to form a portfolio. Suppose you decide to build a portfolio containing x stocks and y bonds. The total value of the portfolio at any time t is:

V(t) = xS(t) + yA(t)

Here, V(t) represents the total value of the portfolio, which is the sum of the value of the stocks ( xS(t) ) and the value of the bonds ( yA(t) ).

At the initial moment ( t = 0 ), the value of the portfolio is known if we know the number of stocks and bonds and their respective current prices. However, at time t = 1 , the value of the stocks may vary, making the portfolio value uncertain.

Basic Model Assumptions

To simplify the model, we establish some key assumptions that allow us to perform calculations and analysis more manageable:

• Randomness Assumption: The price of a stock in the future ( S(1) ) is a random variable, meaning it can take different values depending on unpredictable market factors.
• Positive Prices: All stock and bond prices are strictly positive, meaning S(t) > 0 and A(t) > 0 for t = 0, 1 . This assumption ensures that asset values are realistic.
• Divisibility, Liquidity: Assets can be purchased in fractional quantities, allowing investors to adjust their portfolios without restrictions. Additionally, assets are assumed to be bought or sold in any quantity.
• Solvency: The total wealth of an investor must be non-negative at all times, meaning V(t) \geq 0 . This means it is not possible to lose more than what has been invested.
• Discrete Prices: The future price S(1) of a stock is a random variable that can take only a finite number of possible values. This simplifies the analysis and modeling of the market.

With these assumptions, the model becomes easier to handle, allowing us to analyze portfolio returns and values without additional complexities.

So far, we have covered the fundamental theoretical concepts to understand the simple market model. In the next section, we will apply this knowledge in practical exercises to see how to calculate portfolio value and return in different scenarios.

Solved Problems

Exercise 1: Calculating Bond Return (Risk-Free Asset)

Suppose you have a bond whose price at the initial moment is A(0) = 100 dollars. At the end of one year, the value of the bond has increased to A(1) = 110 dollars.

Question: What is the return on this bond investment?

Solution: Since the bond is a risk-free asset, the return is secure and can be calculated using the return formula for risk-free assets:

K_A = \dfrac{A(1) - A(0)}{A(0)}

Substituting the values:

K_A = \dfrac{110 - 100}{100} = \dfrac{10}{100} = 0.10

The return is 10%.

—

Exercise 2: Calculating Stock Return (Risky Asset)

Suppose you buy a stock at a price of S(0) = 50 dollars. At the end of the year, the price of the stock can vary. There are two possible outcomes:

• If the market goes up, the stock price will be S(1) = 52 dollars, with a probability p .
• If the market goes down, the stock price will be S(1) = 48 dollars, with a probability 1 - p .

Question: In a simple market model, what is the return on this investment in each scenario?

Solution: The return on a stock, being a risky asset, is uncertain and calculated using the return formula for risky assets:

K_S = \dfrac{S(1) - S(0)}{S(0)}

We calculate the return in each scenario:

• If the price rises to 52 dollars:

K_S = \dfrac{52 - 50}{50} = \dfrac{2}{50} = 0.04

The return in this case is 4%.

• If the price drops to 48 dollars:

K_S = \dfrac{48 - 50}{50} = \dfrac{-2}{50} = -0.04

The return in this case is -4%.

Therefore, depending on market behavior, the return can be positive (4%) or negative (-4%).

—

Exercise 3: Value of a Portfolio with Risky and Risk-Free Assets

Suppose you decide to build a portfolio containing 20 stocks and 10 bonds. We know that:

• The price of a stock at the beginning is S(0) = 50 dollars.
• The price of a bond at the beginning is A(0) = 100 dollars.

Question: What is the value of this portfolio at the initial moment t = 0 ?

Solution: The value of a portfolio at time t is calculated as:

V(t) = xS(t) + yA(t)

Where x is the number of stocks and y is the number of bonds.

Substituting the values:

V(0) = (20)(50) + (10)(100)

V(0) = 1000 + 1000 = 2000

The value of the portfolio at the initial moment t = 0 is 2000 dollars.

—

Exercise 4: Calculating Return in a Mixed Portfolio

Suppose the prices of the assets in the portfolio from Exercise 3 vary at time t = 1 as follows:

• If the market rises, the stock price will be S(1) = 52 and the bond will be A(1) = 110 .
• If the market falls, the stock price will be S(1) = 48 and the bond will be A(1) = 110 .

Question: In a simple market model, what is the value and return of the portfolio in each scenario?

Solution:

Scenario 1: The market rises

V(1) = (20)(52) + (10)(110)

V(1) = 1040 + 1100 = 2140

The value of the portfolio in this case is 2140 dollars.

The return of the portfolio is:

K_V = \dfrac{V(1) - V(0)}{V(0)} = \dfrac{2140 - 2000}{2000} = \dfrac{140}{2000} = 0.07

The return is 7%.

Scenario 2: The market falls

V(1) = (20)(48) + (10)(110)

V(1) = 960 + 1100 = 2060

The value of the portfolio in this case is 2060 dollars.

The return of the portfolio is:

K_V = \dfrac{V(1) - V(0)}{V(0)} = \dfrac{2060 - 2000}{2000} = \dfrac{60}{2000} = 0.03

The return is 3%.

In summary, the return of the portfolio depends on market behavior. If the market rises, the return is 7%; if the market falls, the return is 3%.

Proposed Exercises

Exercise 1: Calculating Return on a Bond

Suppose you buy a risk-free bond at an initial price of A(0) = 200 dollars, and at the end of the year, the bond price increases to A(1) = 220 dollars.

Question: What is the return on this bond investment?

—

Exercise 2: Return on a Risky Stock with Probabilistic Scenarios

You buy a stock at an initial price of S(0) = 100 dollars. At the end of the year, the stock price can be S(1) = 110 with a probability of 0.5, or S(1) = 90 with a probability of 0.5.

Question: Calculate the return in each scenario and the expected return of this investment in the stock.

—

Exercise 3: Value of a Mixed Portfolio

You build a portfolio with 15 stocks and 5 bonds. At the beginning, the price of each stock is S(0) = 30 dollars, and the price of each bond is A(0) = 100 dollars.

Question: What is the total value of your portfolio at time t = 0 ?

—

Exercise 4: Portfolio Return in Different Market Scenarios

For the portfolio from the previous exercise, at the end of the year, the stock price can be S(1) = 35 if the market rises, or S(1) = 25 if the market falls. The risk-free bond will have a price of A(1) = 105 in both cases.

Question: Calculate the value and return of the portfolio in each market scenario.

—

Exercise 5: Impact of Changes in Stock Price on the Portfolio

Suppose you have a portfolio consisting of 10 bonds and 40 stocks. The price of each bond is A(0) = 90 dollars, and each stock is S(0) = 20 dollars initially. At the end of the year, the stock price increases to S(1) = 30 , and the bond price to A(1) = 95 .

Question: Calculate the initial and final value of the portfolio and determine the return of the portfolio.

—

Exercise 6: Calculating Weighted Return in a Diversified Portfolio in a Simple Market Model

You invest 60% of your portfolio in risk-free bonds and 40% in stocks. The initial price of the bonds is A(0) = 200 dollars, and their final price is A(1) = 210 dollars. The initial price of the stocks is S(0) = 50 , and their final price depends on whether the market rises ( S(1) = 55 with a probability of 0.6) or falls ( S(1) = 45 with a probability of 0.4).

Question: Calculate the total expected return of the portfolio.

—

Exercise 7: Risk Assessment via Standard Deviation

In a short-selling strategy, you borrow 500 dollars to short-sell stocks that are initially priced at S(0) = 25 dollars. At the end of the year, the stock price can be S(1) = 20 (probability of 0.7) or S(1) = 30 (probability of 0.3).

Question: Calculate the expected return and the standard deviation of this investment in the short sale.

—

Exercise 8: Creating a Portfolio to Guarantee a Specific Return

You have 2000 dollars and want to build a portfolio with risk-free bonds ( A(0) = 100 dollars, with a return of 5%) and stocks ( S(0) = 50 dollars) with an expected return of 8%.

Question: How many bonds and stocks should you buy so that the total expected return of the portfolio is 6%?

—

Exercise 9: Impact of Diversification on Portfolio Risk

You invest 3000 dollars in a portfolio consisting of bonds and stocks. Half of your investment is in bonds ( A(0) = 150 dollars, with a guaranteed return of 4%), and the other half in stocks ( S(0) = 75 dollars), whose price at t = 1 can be 90 (probability of 0.5) or 60 (probability of 0.5).

Question: Calculate the expected return and the standard deviation of the portfolio.

—

Exercise 10: Effect of Price Changes on Portfolio and Solvency

You build a portfolio with 1000 dollars, investing 300 in bonds and 700 in stocks. For bonds, the return is fixed at 3%, while the stock price ( S(0) = 35 ) can fall to 25 or rise to 45 with equal probabilities.

Question: What is the value of the portfolio in each scenario in the simple market model? Evaluate if the portfolio meets the solvency assumption ( V(t) \geq 0 ).