Wk 4 CAPM and Single Index Models PDF

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CAPM and Single Index Models CAPM - Historical background and importance The Capital Asset Pricing Model (CAPM) was developed in 1964-1966 by three finance academics, WIlliam Sharpe, Jan Mossin and John Lintner independent of one another. It was, in a sense, a pricing model that was "waiting" to be developed, as after Markowitz' advances in portfolio theory in 1952 how an individual asset contributes to overall portfolio risk was well understood. What remained was to link this risk contribution to reward, that is, expected return. This is what the CAPM does, albeit with a number of constraining assumptions. The model is one of the most utilised ones in finance practice. Its risk metric, called beta, is especially ubiquitous. Firm managers use the CAPM to find the net present value of projects, portfolio managers use it to assess security and portfolio risk and equity analysts rely on CAPM to do equity valuation and come up with fair value estimates for the firms that they follow. In the following chapters of this Moodle book, we will first talk about the CAPM assumptions, how they lead to the risk metric, beta, and its crucial result, the equation that links beta to expected return for any risky security. Then we will talk about applying CAPM's insights in the real world through a statistical tool called a single index model.

Some of these assumptions are mild, for instance, constraining the model to one investment period only can easily be removed and the model's results would continue to hold. Likewise, it is possible to extend the basic results to a more realistic world where there is taxes and transaction costs, where investments

are not infinitely divisible and where borrowing and lending rates are different for investors, the model and its proofs just become more complex in those cases.

The crucial assumption for CAPM to work is the one about homogeneous expectations. This simply means when investors are doing their Markowitz optimisation to find the efficient frontier and then drawing their best CAL to the efficient frontier from the risk-free rate, they are using using the same set of inputs. Recall that what one needs to perform these tasks are estimated average returns, all possible covariances across individual securities and a risk-free rate. By making this crucial assumption, the model makes sure that every investor agrees on the identity of the tangency portfolio. That is, the optimal risky portfolio is the same across all investors, they differ only in how much of their funds they allocate to it, i.e. where they will want to be on the CAL that connects the risk-free rate to the tangency portfolio. This constraint of homogeneous expectations and the result that everyone agrees on the best risky portfolio leads to an interesting implication. If everyone is holding the identical risky portfolio, and since when we aggregate every investor's holdings of risky assets we have to arrive at the total supply of risky assets in the market, each asset in the optimal risky portfolio must have a weight that is identical to its weight in the total market capitalisation of risky assets. To illustrate, suppose that Amazon's market capitalisation makes up 2% of the entire market capitalisation of risky assets in the world. Since every investor is holding the same risky portfolio according to the CAPM, then the weight of Amazon in that portfolio has to be 2% as well. The informal proof of this assertion is as follows: If Amazon's weight in investors' risky portfolios is any number other than 2%, then when we aggregate these identical portfolios across all investors, we obtain the total portfolio held by the entire population of investors having a weight in Amazon stock that is not equal to 2%. We are told that the supply of shares is such that Amazon has a 2% weight there, so this result could not hold in equilibrium, where demand has to equal supply. So Amazon's weight in the tangency portfolio has to be 2% as well. The above reasoning is why the tangency portfolio in CAPM has a special name, the market portfolio. In this model's context, it is the set of all risky securities for which we are given estimated risk-return parameters for performing Markowitz optimisation. As such, this is a theoretical concept. When we try to make use of CAPM in the real world, we typically choose a well-diversified stock index as the market portfolio. CAPM results - the risk-return relationship Armed with the same set of inputs, investors arrive at the same tangency portfolio, called the market portfolio, as discussed in the previous chapter. The CAL that connects the risk-free rate to the market portfolio is known as the Capital Market Line (CML):

As in our previous discussions, investors differ in where they are on the CML. More risk-averse investors would make allocations towards the risk-free rate on this line, while less risk-averse ones, i.e. those who can tolerate higher portfolio volatility, would move towards the right on this line, some of these would even use leverage to give the market portfolio more than 100% allocation. In the market portfolio, all unsystematic risk is diversified away, there is only systematic risk. Recall our portfolio variance formula from previous weeks:

N should be large in the market portfolio, think thousands of stocksm, bonds, etc., and the variance terms in the double summation, there are N of them, are being multiplied by 1/N 2, hence, the contribution of variance terms to portfolio variance goes to zero. The covariance terms are also multiplied on average by 1/N2, but, crucially, there are N2 - N covariance terms in the sum, hence, not only do they not vanish, but their contribution to portfolio variance is the average covariance between all pairs of assets. So the riskiness of the market portfolio is determined by the average covariance that exists between risky market securities. How would we measure the riskiness of individual securities in this setting? Recall that the contribution of an additional security to the risk of a well-diversified portfolio is a function of the covariance of that security's return with the return of what already exists in the portfolio. In this

case, what we have is the market portfolio. So it makes sense that in CAPM the risk of a security is primarily a function of its covariance with the market portfolio. Adding a security that has high covariance with the rest of the market will lead to an increase in the volatility of the market portfolio whereas adding a security that has low covariance with the rest of the market will lead to a decrease in the volatility of the market portfolio. We are not going to go through a rigorous proof of the primary CAPM results that are given below regarding how we measure risk and how risk is related to expected return in equilibrium. Intuitively, think of owning a risky portfolio and consider adding a new stock into that portfolio. If that stock goes up aggressively when your original investments go up in value on average and, conversely, goes down aggressively when your original investments go up in value on average, it is clear that it is increasing the risk of your portfolio. When you need compensating gains from your new stock, it is producing even bigger losses itself instead. From this new stock, you would expect a relatively high return for this reason. Now consider a stock that tends to produce returns that are more or less independent of what is going on with the rest of your portfolio. Sometime, when the rest of the portfolio is down, this stock will produce partially-compensating positive returns due to this nature of it. So it has a hedging effect on your portfolio, i.e. it dampens losses. Hence, for this stock, you would not ask for a high level of return on average. The main CAPM results are as follows: The equilibrium risk-return relationship for any risky asset, i:

You can think of beta as a standardised covariance. The covariance of the asset's return with the market portfolio return is in the numerator and he variance of the market portfolio return is in the denominator. This means the beta of the market portfolio has to be 1, since the covariance term in that case becomes the market portfolio's return variance. A stock that has a beta greater than 1 would be regarded as an aggressive stock, since its addition to the market portfolio would increase its total variance. In contrast, a defensive stock would have a beta that is less than 1 since its addition to the market portfolio would decrease its total variance.

The term (E[rM] - rF ) in the CAPM formula above is known as the market risk premium. This is what bearing one unit of systematic risk gets an investor in terms of excess expected return. Example: Suppose that you estimate Microsoft stock's beta as 0.90, the market risk premium is estimated as 8% per year and the risk-free rate is 2% per year? What is MSFT stock's expected rate of return according to the CAPM? Solution: Plugging in these values into the main CAPM result, we get: E[rMSFT] = 2% + 0.90 * 8% = 9.2% per year Note that the relationship between beta and the expected return is a linear one in CAPM. With the parameters in the example above for example, as stock with a beta of 0.5 would have 6% expected return, a beta of 1 would generate 10% return and a beta of 1.5 for instance would generate 14% expected return, that is, a linear progression. In beta-E[r] space, this plots a line known as the Security Market Line or SML, as shown below. Using CAPM in the real world - single index models Remember that as neat as CAPM results are, the notion of beta relies on finding the covariance of a stock's return with that of the market portfolio. The market portfolio however is something theoretical, unobservable. What we do in practice is to select a well-diversified stock index to act as an admittedly imperfect proxy for the market portfolio and use standard econometrics tools to estimate any security's beta against that proxy using a single-index model.

Working with a single index model, we can obtain historical return data and estimate beta in a practical manner. Typically, to estimate a stock's beta, one would get 24-60 months of return data for the stock and the index from the most recent time period and perform an Ordinary Least Squares (OLS) regression....