{"id":29576,"date":"2025-01-04T12:00:31","date_gmt":"2025-01-04T12:00:31","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29576"},"modified":"2025-01-05T23:54:04","modified_gmt":"2025-01-05T23:54:04","slug":"simple-market-model-basic-concepts-and-assumptions","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/simple-market-model-basic-concepts-and-assumptions\/","title":{"rendered":"Simple Market Model: Basic Concepts and Assumptions"},"content":{"rendered":"<style>\n  p {\n    text-align: justify;\n  }\n<\/style>\n<p><center><\/p>\n<h1><strong>A Simple Market Model:<\/strong><br \/>Basic Concepts and Assumptions<\/h1>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><em><strong>Abstract:<\/strong><br \/>\nThis lesson introduces the \u00abSimple Market Model,\u00bb 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.<\/em><\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives:<\/strong><br \/>\nBy the end of this lesson, the student will be able to<\/p>\n<ul>\n<li><strong>Identify<\/strong> the characteristics of a Simple Market Model, risk-free assets, and risky assets in investment decision-making.<\/li>\n<li><strong>Understand<\/strong> the difference between risk-free and risky assets, identifying how each affects return and risk in a portfolio.<\/li>\n<li><strong>Apply<\/strong> formulas to calculate the investment return on risky and risk-free assets, using initial and final prices.<\/li>\n<li><strong>Analyze<\/strong> the construction of portfolios that combine risky and risk-free assets to optimize returns while managing risk in a simple market model.<\/li>\n<li><strong>Evaluate<\/strong> the impact of market scenarios on the value and return of a portfolio, considering price variations of assets.<\/li>\n<li><strong>Apply<\/strong> probability to calculate expected return in uncertain market situations, determining potential financial outcomes.<\/li>\n<\/ul>\n<p style=\"text-align:center;\"><strong><u>TABLE OF CONTENTS<\/u><\/strong><br \/>\n<a href=\"#1\"><strong>Introduction<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Definitions and Theoretical Assumptions<\/strong><\/a><br \/>\n<a href=\"#3\">Risky and Risk-Free Assets<\/a><br \/>\n<a href=\"#4\">Timeline in the Model<\/a><br \/>\n<a href=\"#5\">Investment Return<\/a><br \/>\n<a href=\"#6\">Portfolio Construction and Valuation<\/a><br \/>\n<a href=\"#7\">Basic Model Assumptions<\/a><br \/>\n<a href=\"#8\"><strong>Solved Problems<\/strong><\/a><br \/>\n<a href=\"#9\"><strong>Proposed Exercises<\/strong><\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/MK9owXS381U?si=7KmgnPbga5fihMVy\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><br \/>\n<center><\/p>\n<h2>Introduction<\/h2>\n<p><\/center><\/p>\n<p>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.<\/p>\n<p>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.<\/p>\n<p>You decide to investigate and come across an approach called the <strong>\u00abSimple Market Model,\u00bb<\/strong> 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.<\/p>\n<p>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.<\/p>\n<p>Now that you are ready, let&#8217;s dive into the theoretical knowledge you need to understand this market model and apply it to your investment decisions.<\/p>\n<p><center><br \/>\n  <a name=\"2\"><\/a><\/p>\n<h2>Definitions and Theoretical Assumptions<\/h2>\n<p><\/center><br \/>\n<a name=\"3\"><\/a><\/p>\n<h3>Risky and Risk-Free Assets<\/h3>\n<p>To begin understanding the simple market model, we need to familiarize ourselves with the concepts of <strong>risky assets<\/strong> and <strong>risk-free assets<\/strong>. These two types of assets form the basis of most investment strategies.<\/p>\n<p>A <strong>risk-free asset<\/strong> is an investment whose return is known and secure. A classic example of a risk-free asset is a <em>bond<\/em> 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.<\/p>\n<p>On the other hand, a <strong>risky asset<\/strong> is one whose future price is uncertain and can vary, both up and down. A common example of a risky asset is the <em>stock<\/em> of companies traded on the stock exchange. Stocks can be volatile, and their price depends on multiple factors, making their future value unpredictable.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Timeline in the Model<\/h3>\n<p>In the simple market model, we restrict the analysis to only two points in time: the present, called <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span>, and a future moment, such as one year later, called <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span>. This simplified approach allows us to analyze changes in asset value without excessive complexity.<\/p>\n<p>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.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Investment Return<\/h3>\n<p>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.<\/p>\n<p>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 <span class=\"katex-eq\" data-katex-display=\"false\"> t <\/span> is represented by <span class=\"katex-eq\" data-katex-display=\"false\"> S(t) <\/span>, the return of the stock between <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span> and <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> is calculated as follows:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S = \\dfrac{S(1) - S(0)}{S(0)} <\/span>\n<p>This return, represented by <span class=\"katex-eq\" data-katex-display=\"false\"> K_S <\/span>, 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).<\/p>\n<p>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 <span class=\"katex-eq\" data-katex-display=\"false\"> t <\/span> by <span class=\"katex-eq\" data-katex-display=\"false\"> A(t) <\/span>, the return of this bond between <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span> and <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> is calculated as:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_A = \\dfrac{A(1) - A(0)}{A(0)} <\/span>\n<p>This return, <span class=\"katex-eq\" data-katex-display=\"false\"> K_A <\/span>, is fixed and guaranteed by the bond issuer. The key difference between <span class=\"katex-eq\" data-katex-display=\"false\"> K_S <\/span> and <span class=\"katex-eq\" data-katex-display=\"false\"> K_A <\/span> is certainty: while the return of a stock is uncertain, the return of a bond is fixed and known.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Portfolio Construction and Valuation<\/h3>\n<p>Now that we understand the concept of return, we can combine risky and risk-free assets to form a <strong>portfolio<\/strong>. Suppose you decide to build a portfolio containing <span class=\"katex-eq\" data-katex-display=\"false\"> x <\/span> stocks and <span class=\"katex-eq\" data-katex-display=\"false\"> y <\/span> bonds. The total value of the portfolio at any time <span class=\"katex-eq\" data-katex-display=\"false\"> t <\/span> is:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(t) = xS(t) + yA(t) <\/span>\n<p>Here, <span class=\"katex-eq\" data-katex-display=\"false\"> V(t) <\/span> represents the total value of the portfolio, which is the sum of the value of the stocks (<span class=\"katex-eq\" data-katex-display=\"false\"> xS(t) <\/span>) and the value of the bonds (<span class=\"katex-eq\" data-katex-display=\"false\"> yA(t) <\/span>).<\/p>\n<p>At the initial moment (<span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span>), the value of the portfolio is known if we know the number of stocks and bonds and their respective current prices. However, at time <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span>, the value of the stocks may vary, making the portfolio value uncertain.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Basic Model Assumptions<\/h3>\n<p>To simplify the model, we establish some key assumptions that allow us to perform calculations and analysis more manageable:<\/p>\n<ul>\n<li><strong>Randomness Assumption:<\/strong> The price of a stock in the future (<span class=\"katex-eq\" data-katex-display=\"false\"> S(1) <\/span>) is a <em>random variable<\/em>, meaning it can take different values depending on unpredictable market factors.<\/li>\n<li><strong>Positive Prices:<\/strong> All stock and bond prices are strictly positive, meaning <span class=\"katex-eq\" data-katex-display=\"false\"> S(t) &gt; 0 <\/span> and <span class=\"katex-eq\" data-katex-display=\"false\"> A(t) &gt; 0 <\/span> for <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0, 1 <\/span>. This assumption ensures that asset values are realistic.<\/li>\n<li><strong>Divisibility, Liquidity:<\/strong> 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.<\/li>\n<li><strong>Solvency:<\/strong> The total wealth of an investor must be non-negative at all times, meaning <span class=\"katex-eq\" data-katex-display=\"false\"> V(t) \\geq 0 <\/span>. This means it is not possible to lose more than what has been invested.<\/li>\n<li><strong>Discrete Prices:<\/strong> The future price <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) <\/span> 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.<\/li>\n<\/ul>\n<p>With these assumptions, the model becomes easier to handle, allowing us to analyze portfolio returns and values without additional complexities.<\/p>\n<p>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.<\/p>\n<p><center><br \/>\n  <a name=\"8\"><\/a><\/p>\n<h2>Solved Problems<\/h2>\n<p><\/center><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/BdcylGfSgtA?si=QwVbLqRiAaULJ6v1\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<h3>Exercise 1: Calculating Bond Return (Risk-Free Asset)<\/h3>\n<p>Suppose you have a bond whose price at the initial moment is <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> dollars. At the end of one year, the value of the bond has increased to <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 110 <\/span> dollars.<\/p>\n<p><strong>Question:<\/strong> What is the return on this bond investment?<\/p>\n<p><strong>Solution:<\/strong> Since the bond is a risk-free asset, the return is secure and can be calculated using the return formula for risk-free assets:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_A = \\dfrac{A(1) - A(0)}{A(0)} <\/span>\n<p>Substituting the values:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_A = \\dfrac{110 - 100}{100} = \\dfrac{10}{100} = 0.10 <\/span>\n<p>The return is 10%.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 2: Calculating Stock Return (Risky Asset)<\/h3>\n<p>Suppose you buy a stock at a price of <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span> dollars. At the end of the year, the price of the stock can vary. There are two possible outcomes:<\/p>\n<ul>\n<li>If the market goes up, the stock price will be <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 52 <\/span> dollars, with a probability <span class=\"katex-eq\" data-katex-display=\"false\"> p <\/span>.<\/li>\n<li>If the market goes down, the stock price will be <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 48 <\/span> dollars, with a probability <span class=\"katex-eq\" data-katex-display=\"false\"> 1 - p <\/span>.<\/li>\n<\/ul>\n<p><strong>Question:<\/strong> In a simple market model, what is the return on this investment in each scenario?<\/p>\n<p><strong>Solution:<\/strong> The return on a stock, being a risky asset, is uncertain and calculated using the return formula for risky assets:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S = \\dfrac{S(1) - S(0)}{S(0)} <\/span>\n<p>We calculate the return in each scenario:<\/p>\n<ul>\n<li><strong>If the price rises to 52 dollars:<\/strong><\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S = \\dfrac{52 - 50}{50} = \\dfrac{2}{50} = 0.04 <\/span>\n<p>The return in this case is 4%.<\/p>\n<li><strong>If the price drops to 48 dollars:<\/strong><\/li>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_S = \\dfrac{48 - 50}{50} = \\dfrac{-2}{50} = -0.04 <\/span>\n<p>The return in this case is -4%.<\/p>\n<\/ul>\n<p>Therefore, depending on market behavior, the return can be positive (4%) or negative (-4%).<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 3: Value of a Portfolio with Risky and Risk-Free Assets<\/h3>\n<p>Suppose you decide to build a portfolio containing 20 stocks and 10 bonds. We know that:<\/p>\n<ul>\n<li>The price of a stock at the beginning is <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span> dollars.<\/li>\n<li>The price of a bond at the beginning is <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> dollars.<\/li>\n<\/ul>\n<p><strong>Question:<\/strong> What is the value of this portfolio at the initial moment <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span>?<\/p>\n<p><strong>Solution:<\/strong> The value of a portfolio at time <span class=\"katex-eq\" data-katex-display=\"false\"> t <\/span> is calculated as:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(t) = xS(t) + yA(t) <\/span>\n<p>Where <span class=\"katex-eq\" data-katex-display=\"false\"> x <\/span> is the number of stocks and <span class=\"katex-eq\" data-katex-display=\"false\"> y <\/span> is the number of bonds.<\/p>\n<p>Substituting the values:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(0) = (20)(50) + (10)(100) <\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(0) = 1000 + 1000 = 2000 <\/span>\n<p>The value of the portfolio at the initial moment <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span> is 2000 dollars.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 4: Calculating Return in a Mixed Portfolio<\/h3>\n<p>Suppose the prices of the assets in the portfolio from Exercise 3 vary at time <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> as follows:<\/p>\n<ul>\n<li>If the market rises, the stock price will be <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 52 <\/span> and the bond will be <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 110 <\/span>.<\/li>\n<li>If the market falls, the stock price will be <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 48 <\/span> and the bond will be <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 110 <\/span>.<\/li>\n<\/ul>\n<p><strong>Question:<\/strong> In a simple market model, what is the value and return of the portfolio in each scenario?<\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>Scenario 1: The market rises<\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = (20)(52) + (10)(110) <\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = 1040 + 1100 = 2140 <\/span>\n<p>The value of the portfolio in this case is 2140 dollars.<\/p>\n<p>The return of the portfolio is:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_V = \\dfrac{V(1) - V(0)}{V(0)} = \\dfrac{2140 - 2000}{2000} = \\dfrac{140}{2000} = 0.07 <\/span>\n<p>The return is 7%.<\/p>\n<p><strong>Scenario 2: The market falls<\/strong><\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = (20)(48) + (10)(110) <\/span>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> V(1) = 960 + 1100 = 2060 <\/span>\n<p>The value of the portfolio in this case is 2060 dollars.<\/p>\n<p>The return of the portfolio is:<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\"> K_V = \\dfrac{V(1) - V(0)}{V(0)} = \\dfrac{2060 - 2000}{2000} = \\dfrac{60}{2000} = 0.03 <\/span>\n<p>The return is 3%.<\/p>\n<p>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%.<\/p>\n<p><center><br \/>\n  <a name=\"9\"><\/a><\/p>\n<h2>Proposed Exercises<\/h2>\n<p><\/center><\/p>\n<h3>Exercise 1: Calculating Return on a Bond<\/h3>\n<p>Suppose you buy a risk-free bond at an initial price of <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 200 <\/span> dollars, and at the end of the year, the bond price increases to <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 220 <\/span> dollars.<\/p>\n<p><strong>Question:<\/strong> What is the return on this bond investment?<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 2: Return on a Risky Stock with Probabilistic Scenarios<\/h3>\n<p>You buy a stock at an initial price of <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 100 <\/span> dollars. At the end of the year, the stock price can be <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 110 <\/span> with a probability of 0.5, or <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 90 <\/span> with a probability of 0.5.<\/p>\n<p><strong>Question:<\/strong> Calculate the return in each scenario and the expected return of this investment in the stock.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 3: Value of a Mixed Portfolio<\/h3>\n<p>You build a portfolio with 15 stocks and 5 bonds. At the beginning, the price of each stock is <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 30 <\/span> dollars, and the price of each bond is <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> dollars.<\/p>\n<p><strong>Question:<\/strong> What is the total value of your portfolio at time <span class=\"katex-eq\" data-katex-display=\"false\"> t = 0 <\/span>?<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 4: Portfolio Return in Different Market Scenarios<\/h3>\n<p>For the portfolio from the previous exercise, at the end of the year, the stock price can be <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 35 <\/span> if the market rises, or <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 25 <\/span> if the market falls. The risk-free bond will have a price of <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 105 <\/span> in both cases.<\/p>\n<p><strong>Question:<\/strong> Calculate the value and return of the portfolio in each market scenario.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 5: Impact of Changes in Stock Price on the Portfolio<\/h3>\n<p>Suppose you have a portfolio consisting of 10 bonds and 40 stocks. The price of each bond is <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 90 <\/span> dollars, and each stock is <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 20 <\/span> dollars initially. At the end of the year, the stock price increases to <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 30 <\/span>, and the bond price to <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 95 <\/span>.<\/p>\n<p><strong>Question:<\/strong> Calculate the initial and final value of the portfolio and determine the return of the portfolio.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 6: Calculating Weighted Return in a Diversified Portfolio in a Simple Market Model<\/h3>\n<p>You invest 60% of your portfolio in risk-free bonds and 40% in stocks. The initial price of the bonds is <span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 200 <\/span> dollars, and their final price is <span class=\"katex-eq\" data-katex-display=\"false\"> A(1) = 210 <\/span> dollars. The initial price of the stocks is <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span>, and their final price depends on whether the market rises (<span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 55 <\/span> with a probability of 0.6) or falls (<span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 45 <\/span> with a probability of 0.4).<\/p>\n<p><strong>Question:<\/strong> Calculate the total expected return of the portfolio.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 7: Risk Assessment via Standard Deviation<\/h3>\n<p>In a short-selling strategy, you borrow 500 dollars to short-sell stocks that are initially priced at <span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 25 <\/span> dollars. At the end of the year, the stock price can be <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 20 <\/span> (probability of 0.7) or <span class=\"katex-eq\" data-katex-display=\"false\"> S(1) = 30 <\/span> (probability of 0.3).<\/p>\n<p><strong>Question:<\/strong> Calculate the expected return and the standard deviation of this investment in the short sale.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 8: Creating a Portfolio to Guarantee a Specific Return<\/h3>\n<p>You have 2000 dollars and want to build a portfolio with risk-free bonds (<span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 100 <\/span> dollars, with a return of 5%) and stocks (<span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 50 <\/span> dollars) with an expected return of 8%.<\/p>\n<p><strong>Question:<\/strong> How many bonds and stocks should you buy so that the total expected return of the portfolio is 6%?<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 9: Impact of Diversification on Portfolio Risk<\/h3>\n<p>You invest 3000 dollars in a portfolio consisting of bonds and stocks. Half of your investment is in bonds (<span class=\"katex-eq\" data-katex-display=\"false\"> A(0) = 150 <\/span> dollars, with a guaranteed return of 4%), and the other half in stocks (<span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 75 <\/span> dollars), whose price at <span class=\"katex-eq\" data-katex-display=\"false\"> t = 1 <\/span> can be 90 (probability of 0.5) or 60 (probability of 0.5).<\/p>\n<p><strong>Question:<\/strong> Calculate the expected return and the standard deviation of the portfolio.<\/p>\n<p>&#8212;<\/p>\n<h3>Exercise 10: Effect of Price Changes on Portfolio and Solvency<\/h3>\n<p>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 (<span class=\"katex-eq\" data-katex-display=\"false\"> S(0) = 35 <\/span>) can fall to 25 or rise to 45 with equal probabilities.<\/p>\n<p><strong>Question:<\/strong> What is the value of the portfolio in each scenario in the simple market model? Evaluate if the portfolio meets the solvency assumption (<span class=\"katex-eq\" data-katex-display=\"false\"> V(t) \\geq 0 <\/span>).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A Simple Market Model:Basic Concepts and Assumptions Abstract: This lesson introduces the \u00abSimple Market Model,\u00bb 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, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29422,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":67,"footnotes":""},"categories":[749,903],"tags":[],"class_list":["post-29576","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-economics-and-finance","category-financial-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Simple Market Model: Basic Concepts and Assumptions<\/title>\n<meta name=\"description\" content=\"Fundamentals of the Simple Market Model: learn how to combine risk-free and risky assets in a portfolio to calculate returns and risks.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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