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Slide 1: Lecture 13Welcome to Lecture 13: The Capital Asset Pricing Model, or what is more commonly referred to as the CAPM.Slide 2: TopicsIn this lecture, we will be covering these five topics:One, we will start by looking at the definition of CAPM and the formula that describes it.Two, we will discuss the interpretation of the model and how it can be used to determine expected asset return and expected portfolio return.Three, we will show a couple of numerical examples that demonstrate how the CAPM can be applied.Four, we will demonstrate how to estimate an asset`s beta by using the market model. Last, as always, we end the lecture with a question to practice your new-found skills.Slide 3: CAPM definedWhat is the Capital Asset Pricing Model? CAPM is a theoretical financial model that says that the expected return on any asset “i" depends on three factors: risk-free rate (represented by Rf), the beta of the asset i (represented by βi), and the expected return on the market portfolio that contains all assets in the market (represented by E(RM)). Note that the difference between the expected market return and the risk-free rate is called the market risk premium, which measures the excess return of the market portfolio over and above the risk-free rate.Slide 4: Determinants of expected returnSo, what do these symbols mean, really?Rf, the risk-free rate, is the pure-time value of money. That is, it is the return that one could earn if one were to forgo all risks, including default risk, and purchase the safest of assets.The expected market risk premium, E(RM) – Rf, represents the reward one would expect for bearing any risk when investing in any asset.The beta of the asset, βi, represents the level of systematic risk on the asset. It is the risk that an average investor in the market will face if he or she purchases the asset i.From the formula E(Ri) = Rf + βi (E(RM) – Rf), we can see that, the higher the risk-free rate, asset beta, and/or market risk premium, the higher will be the expected return on the asset.Slide 5: CAPM applicationsSeeing that the CAPM is a theoretical model, we might say that the value of the model comes from how it can help us understand reality. And so, how can CAPM be applied to this end?CAPM can be used to project the return on any asset or portfolio, as long as we are able to obtain the three relevant and important factors: risk-free rate, expected market return, and the beta on the asset or portfolio.Slide 6: Numerical Example #1 – CAPM with one assetLet‘s go through a numerical example to see how we can apply the capital asset pricing model with one asset.We are given the market data: T-bill rate of 5% and S&P 500 return of 15% and we are also given asset-specific information (i.e., data specific to the asset we are looking at): beta = 2.As usual, the first thing we do is write down the information that we have been given:Rf = 0.05 That is, the T-bill rate is usually used to represent the risk-free rate, because T-bills are issued by governments, and government-issued assets rarely, if ever, have any default risk.The next bit of information we write down is:E(RM) = 0.15That is, the S&P 500 return is used to represent the expected market return, because the S&P 500 index has a sufficiently large number of companies to represent the market.Next, we have asset-specific information relating to the beta of the asset i:i = 2The market risk premium is calculated as:E(RM) – Rf = 0.15 – 0.05 = 0.1Using the CAPM formula,E(Ri) = Rf + Beta[E(RM) – Rf] = 0.05 + 2(0.1) = 0.05 + 0.2 = 0.25 = 25%.That is, the CAPM predicts that, with the current market conditions yielding 15% expected market return and 5% risk-free rate, an asset with a beta of 2 will have a projected return of 25%.That is how we can use the CAPM to find the expected return on an asset, given the three input factors of risk-free rate, expected market return, and asset beta.Slide 7: Numerical Example #2 – CAPM with a portfolio of assetsIn the previous numerical example, we worked on a simple one-asset model. Now, let us move to work on something a bit more complicated. Let us look at a numerical example wherein we use the CAPM to calculate the expected return on a portfolio containing multiple assets.In this example, we are given market data consisting of the risk-free rate of 0.05 and market return of 0.15:Rf = 0.05E(RM) = 0.15We are also given the portfolio data. In this portfolio, we have 5 stocks: Stock A, Stock B, Stock C, Stock D, and Stock E. (If it helps, think of these as abbreviations for stocks of Agrium Inc., Bombardier, CIBC, Dell, and Enbridge.)The betas on Stocks A, B, C, D, and E are, respectively, 0.50.951.522.12.32The dollar amounts invested in Stocks A, B, C, D, and E are, respectively,$100,000$200,000$400,000$200,000$100,000Given all this information, what is the expected return on this portfolio?Slide 8: Numerical Example #2 – cont.To calculate the expected portfolio return, we follow a three-step process:Step 1: We calculate the weights on each of the five assets in our portfolio by using the dollar amount invested in each asset and the total amount invested in the portfolio (What we will call the ‘total portfolio value’). The portfolio weight on asset i is equal to the dollar amount invested in asset i, divided by the total portfolio value, which means that, since we do not have it yet, we must first calculate the total portfolio value:Total portfolio value = $ invested A + $ invested in B + $ invested in C + $ invested in D + $ invested in E= $100,000 + $200,000 + $400,000 + $200,000 + $100,000 = $1,000,000The weight on each asset is then calculated as:Weight on Asset A = $100,000/$1,000,000 = 0.1Weight on Asset B = $200,000/$1,000,000 = 0.2Weight on Asset C = $400,000/$1,000,000 = 0.4Weight on Asset D = $200,000/$1,000,000 = 0.2Weight on Asset E = $100,000/$1,000,000 = 0.1Note that we must always check that the portfolio weights add up to a sum of 1:0.1 + 0.2 + 0.4 + 0.2 + 0.1 = 1.Slide 9: Numerical Example #2 – cont.Step 2: We calculate the beta of the portfolio. The portfolio beta is calculated as the weighted sum of the betas of the assets in the portfolio, wherein the weights are the portfolio weights we have just calculated in Step 1. The formula for calculating the beta of our portfolio looks like this:Portfolio beta = βP = wA(βA) + wB(βB) + wC(βC) + wD(βD) + wE(βE) Writing out again the information we have on the weights and betas of the assets,wA = 0.1 βA = 0.5wB = 0.2 βB = 0.95wC = 0.4 βC = 1.52wD = 0.2 βD = 2.1wE = 0.1 βE = 2.32We plug these numbers into the portfolio beta formula, and we havePortfolio beta = 0.1(0.5) + 0.2(0.95) + 0.4(1.52) + 0.2(2.1) + 0.1(2.32) = 1.5Therefore, the beta of this portfolio is 1.5. We normally assume that the beta on the market portfolio, M, is equal to 1. As such, we can say that our portfolio is a bit riskier than the market portfolio, and therefore our portfolio should have a higher expected return than the expected market return of 15%. Slide 10: Numerical Example #2 – cont.Step 3: We calculate the expected return on the portfolio using the CAPM formula:E(RP) = Rf + βi (E(RM) – Rf)Plugging the numbers of Rf = 0.05, E(RM) = 0.15, and βP of 1.5 into the CAPM formula, we get E(RP) = 0.05 + 1.5(0.15 – 0.05) = 0.05 + 1.5(0.1) = 0.05 + 0.15 = 0.2 = 20%.The projected return on this portfolio is 20% which, as we have predicted, is higher than the projected market return of 15%.Slide 11: Estimation of BetaYou might ask: How did we get the asset betas in our previous example in the first place? That is a very good question! Most frequently, they are estimated by the market model:Ri = a + bRMWhere Ri = returns on asset iRM = returns on proxy of market portfolio, such as a market indexMarket model is a simple regression model. It allows us to run a regression model to obtain the beta, where the Ri is the y-variable (dependent variable), and RM is the x-variable (independent variable). We regress Ri against RM, and the coefficient b, estimated in the model, is our estimated beta.How is this achieved practically? We collect historical time series data on the asset’s returns, Ri, and the returns on a sufficiently large representation market index, such as the S&P 500 composite index, which we use to represent market returns, RM. Most of the time, researchers use 5 or 10 years of monthly data to estimate beta. They then run an Ordinary Least Squares (OLS) regression to obtain the coefficients a and b, and the coefficient b is then treated as the beta (β) of the asset.Note that in this lecture you are assumed to be already familiar with regression analysis.Slide 12: Data required for Market ModelWhat are the data required to run the market model?Let‘s say that we wish to estimate the beta for the common stock of Telus. What data do we need in order to use the market model to estimate Telus‘s beta?We will need two sets of time-series historical data. First, we will need historical data on Telus‘s common stock. We can find this information from financial websites such as finance.. Let us find the monthly stock price for Telus on finance., and let’s do this for the 10-year period between December 2000 and December 2010. This should give us 121 months of stock price data on Telus.Second, we will need to find the return on a market portfolio. Frequently, we use a market index such as the S&P 500 Composite Index to approximate or represent the market portfolio. We can do this because this index contains quite a lot of the representative companies traded in the stock market. As before, we can find historical data on the S&P 500 composite index from websites such as finance.. Let us do this now. Let’s obtain the monthly S&P 500 index value for the 10-year period between December 2000 and December 2010. This should give us 121 data points of S&P 500 index values.Since these data are stock prices and index values, we will need to convert them to stock and index returns using these two formulas shown here:Return on Telus stock over time t = (Price at time t – Price at time t-1) / Price at time t-1Return on market portfolio = (Index value at time t – Index value at time t-1) / Index value at time t-1This is the basic formula for calculating any return over a single period. We do this for all 121 months in our raw data sets, and we should obtain 120 returns for Telus stock and 120 returns for the market portfolio (S&P 500 index).Slide 13: Data SampleOur data sets should look something like this table. Month Ri Rm Jan-01 0.010539 0.034637 Feb-01 -0.13006 -0.09229 Mar-01 -0.10226 -0.0642 Apr-01 -0.00079 0.076814 May-01 0.052673 0.00509 Jun-01 0.005975 -0.02504 Jul-01 -0.21232 -0.01074 Aug-01 -0.15174 -0.06411 Sep-01 -0.17444 -0.08172 Oct-01 0.273217 0.018099 Nov-01 0.096195 0.075176 Dec-01 -0.07425 0.007574 andJan-10 -0.04093 -0.03697 Feb-10 0.051141 0.028514 Mar-10 0.156108 0.058796 Apr-10 -0.01423 0.014759 May-10 -0.02181 -0.08198 Jun-10 0.062166 -0.05388 Jul-10 0.038427 0.068778 Aug-10 0.04583 -0.04745 Sep-10 0.090093 0.087551 Oct-10 -0.00449 0.036856 Nov-10 0.032606 -0.00229 Dec-10 0.012388 0.0653 For each month, there are two data points: one for the return on Telus shares, and one for the return on the S&P 500 Index.Note that I have only included the first 12 and last 12 data points for each data set here.Slide 14: Estimating Telus Beta using Market ModelSo, we regress the data series for Return on Telus stock against the data series for Market Return, and what we get is:RTelus = 0.0129 + (1.11 x RM)WhereRTelus = monthly return on TelusRM = monthly return on S&P 500 IndexThe coefficient beside the Market Return is what we want: our estimated beta for Telus. Therefore, the beta on Telus stock is estimated to be 1.11 with our 10-year monthly data sets.Slide 15: Using estimated BetaSo, now that we have the estimated beta for Telus, we can use it, as in our Numerical Example #1 from before, to calculate a projected return on the Telus stock using the CAPM formula: E(RTelus) = Rf + βTelus (E(RM) – Rf). That is, provided that we know the expected return on the market portfolio and the risk-free rate. Let’s say that we use a similar method to the one from before to collect data on the last twelve months of returns on the S&P 500 index (proxying as the market portfolio) and the 1 month t-bill rate (proxying as the risk-free rate). Using these data sets, we calculate the average return on the S&P 500 index to be 11% and the average return on the t-bill to be 8%. We then take a giant leap of faith and assume that the expected return on the market over the next 12 months will be the same as our average S&P return of 11%. Now we are driving with gas!!So, we now haveE(RM) = 0.11Rf = 0.08Estimated Beta of Telus = 1.11We plug these numbers into the CAPM formula, and we getE(RTelus) = Rf + βTelus (E(RM) – Rf) Expected return on Telus stock = 0.08 + 1.11(0.11 – 0.08) = 0.1133 = 11.33%.That is, given a beta of 1.11, Telus stock is expected to return 11.33% over the next twelve months.Slide 16: Practice QuestionSince practice is more fun than skimming textbook, or for the love of all that is sacred in knowledge, listening to audio lectures, here is a practice question for you to try your newly-learned skills. Press the pause button to pause this lecture, and take down the information shown on this slide, try to solve the problem, and then move on to the next slide to see the solution provided.Given the following information, calculate the expected return on this portfolio:S&P 500 expected return = 13%3-month T-bill rate = 6%There are three assets, Stock X, Stock Y, and Stock Z, in this portfolio.Dollars invested in Stock X, Y, and Z are, respectively, $5000, $6000, and $7000.The betas of Stock X, Y, and Z are, respectively, 1.1, 2.1 and 2.6.Solve for the expected return on this portfolio.Slide 17: Solution to Practice QuestionThe solution to the practice question:Step 1: Calculate portfolio weights:Total portfolio value = $5000 + $6000 + $7000 = $18000Weight on X = wx = 5/18Weight on Y = wy = 6/18Weight on Z = wz = 7/18Step 2: Calculate portfolio beta:βP = (5/18)(1.1) + (6/18)(2.1) + (7/18)(2.6) = 2.016666667Step 3: Calculate expected portfolio return:E(RP) = Rf + βP(E(RM) – Rf)= 0.06 + (2.016666667)(0.13 – 0.06) = 0.201166667= 20.12% (approximately)Happy practicing!Slide 18: End of Lecture 13Here ends Lecture 13 on the Capital Asset Pricing Model. ................
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