Capital Market Models
RS/GZ, contact: gzakrzewski@wne.uw.edu.pl, rslepaczuk@wne.uw.edu.pl
20 May 2020
Sharpe’s single-index model.
The single-index model is a simplification of the portfolio theory presented in the previous classes. It is based on the assumption that the returns are determined by a factor that reflects changes in the capital market. It was observed, on many stock exchanges, that rates of return from a specific stock are linked to the rates of return of the stock market index (called the market index) and reflecting the overall stock market situation.
I. Security Characteristic Line.
The relationship between the rates of return of the stock and the rates of return of the market index (which is the equivalent of the market portfolio) is determined by the following linear regression equation:
\[R_i = \alpha_i+\beta_i * R_m + e_i\]
where:
\(R_i\) - the rate of return of i-th stock;
\(R_M\) - the rate of return of the market index;
\(\alpha_i, \beta_i\) - equation factors;
\(e_i\) - the random component of the equation.
In the above regression equation, the variable presenting stock’s returns is the explained variable, and the variable presenting the rates of return of the market index is the explanatory variable. The regression equation, after its estimation, is called the characteristic line of the security. The most important part of the equation is the coefficient βi called the beta coefficient. It indicates how much, in the percentage terms, the rate of return of the stock will increase when the rate of return of the market index increases by 1 percent.
By plotting on the graph points indicating the rate of return from a given stock in a given period and the rate of return from the market index in the same period, we can set a hypothetical straight line, which is a regression line or security characteristic line (Figure 1).
Figure 1. Characteristic line of the security
To estimate the characteristic line of the stock, the least squares (OLS) method is used, which is based on minimizing the following expression:
\[\sum_{t = 1}^n(R_{it} - \alpha_i - \beta_i * R_{Mt})^2\]
where:
\(R_{it}\) - the rate of return of the i-th stock in the t-th period;
\(R_{Mt}\) - the rate of return of the market index in the t-th period.
The expression above means that the sum of the squares of differences between the values of the rates of return of the stock and the hypothetical values calculated on the basis of the regression line. After solving the above minimization task, we get the following formulas:
\[\beta_i = \frac{\sum_{t = 1}^n(R_{it} - \bar{R}_i)*(R_{Mt} - \bar{R}_M)}{\sum_{t=1}^n(R_{Mt} - \bar{R}_M)^2}\]
\[\alpha_i = \bar{R}_i - \beta_i * \bar{R}_M\]
where:
\(n\) - the number of periods from which information came;
\(R\) - the arithmetic mean of the rates of return on the i-th stock;
\(\bar{R}_M\) - the arithmetic mean of the return rates of the market stock;
On the basis of the estimated characteristic line of the stock, the variances of the market index and the random component can also be determined:
\[s_M^2 = \frac{\sum_{t=1}^n(R_{Mt} - \bar{R}_M)^2}{n-1}\]
\[se_i^2 = \frac{\sum_{t=1}^n(R_{it} - \alpha_i - \beta_i*R_M)^2}{n-1}\]
where:
\(s_M^2\) - the variance of the market index returns;
\(se_i^2\) - the variance of the random component.
The single-index model is a simplification of the portfolio theory. The simplification is caused by the fact that in order to apply the portfolio theory it is necessary to know the coefficients of correlation of the returns of all stocks pairs, which can be quite time consuming. The single-index model is based on the following relationships:
\[R_i = \alpha_i + \beta_i*\bar{R}_M\]
\[s_i^2 = \beta_i^2 * s_M^2 + se_i^2\]
\[\rho_{}ij = \frac{\beta_i * \beta_j * s_M^2}{s_i*s_j}\]
However, we must remember that the formula (8) is just an approximation that determines the total risk of the stock. Let us also point out that the beta coefficient relates the total risk of stock to the risk of the market portfolio in the following way:
\[\beta_i = \frac{s_i*\rho_{iM}}{s_M}\]
where:
\(\rho_{iM}\) - the coefficient of correlation of the rates of return of the stock and the rate of return of the market portfolio.
II. Beta coefficient.
The beta coefficient indicates the extent to which the rates of return of the stock responds to changes in the rate of return of the market index, i.e. changes in the market. It can take different values:
• \(\beta_i = 0\) - the rate of return of the i-th stock does not react to changes on the market; therefore, the security is free from the market risk.
• \(0 <\beta_i <1\) - the rate of return of i-th stock reacts to a small extent to market changes; this stock is called defensive.
• \(\beta_i = 1\) - the rate of return of the i-th stock varies to the same extent as the rate of return of the market; in particular, the market portfolio has a beta factor of 1.
• \(\beta_i> 1\) - the rate of return of i-th stock to a greater extent reacts to changes taking place on the market; this stock is called aggressive.
• \(\beta_i <0\) - means that the rates of return of the stock reacts opposite to changes of the market; this is a relatively rare case, although very desirable especially, if the yields of most of the stocks are expected to fall in the market.
II. Market risk and specific risk of stocks.
Total equity risk (\(s_i^2\)) is the sum of two components. The first component is a systematic risk, also known as the market risk. It depends on the risk of the market index and the beta factor, which also reflects the market risk as it reflects the link between the rates of return of the stock and the rates of return of the market portfolio. The second component, the variance of the random component, reflects the specific risk associated with the given stock. A high share of systematic risk in the overall risk of the stock indicates that the general risk related to the market situation has a large impact on the stock’s risk.
The concept of the systematic risk and the specific risk is directly related to the diversification of the portfolio. As you know, portfolio diversification can lead to a significant reduction in the risk of this portfolio. However, this risk cannot be completely eliminated. Smart portfolio diversification leads to eliminating the risk of specific stocks included in the portfolio. However, there is still a risk of the market, which occurs to a greater or lesser extent in all shares and which cannot be eliminated. As the number of different companies’ stocks in the portfolio increases, we reduce the share of specific risk in total risk, as illustrated in Figure 2.
Figure 2. Total risk vs. portfolio size.
The beta factor can be referred not only to a single share, but also to the equity portfolio, where the following formula applies:
\[\beta_p = \sum_{i=1}^nw_i*\beta_i\]
Capital market model - CAPM (Capital Asset Pricing Model).
CAPM is the most popular capital market model. The authors of this model were: William Sharpe, John Lintner and Jan Mossin. The basis of this model are two relationships:
• Capital Market Line (CML)
\[R_p = R_f + \frac{R_M - R_f}{s_M}*s_p\]
• Security Market Line (SML)
\[R_i = R_f + \beta_i*(R_M - R_f)\]
where: \(R_M - R_f\) - risk premium.
The basic difference between the two lines is that CML applies only to efficient portfolios, and SML applies to all portfolios, including individual securities, on an equilibrium market.
I. Assumptions of the CAPM model.
The CAPM model is based on the following assumptions.
• Portfolio purchase decisions made by investors refer to one period.
• The investor’s utility is entirely determined by the expected rate of return and the risk of the portfolio held.
• There are a risk-free securities that can be purchased by investors. Its returns is a risk-free rate of return.
• There is an unlimited possibility of granting or taking out a loan with a risk-free rate.
• Equilibrium prices exist only if there are no speculative transactions. It is possible when all market participants have the same information - they have the same expectations as to future values of returns and securities risk.
• There is a short sale of stocks.
• Transaction costs are zero.
• There are no taxes on holding securities.
• Securities can be split in any way.
• Single investor transactions do not affect the price of the security.
II. Characteristics of the model.
In the CAPM model, the market portfolio is of key importance. It is a portfolio that consists of all shares and other risky securities on the market, where the weights of individual stocks in this portfolio are equal to the weights of these stocks in the market.
The first dependency in CAPM is CML, which only applies to effective portfolios. The formula for CML shows that the rate of return of the effective portfolio is the sum of two components:
• Risk-free rate of return, which can be interpreted as the “price of time”, because it is the rate of return required by the investor to compensate for the resignation from current consumption.
• Product of \(\frac{R_M - R_f}{S_M}\) and \(s_p\) where \(\frac{R_M - R_f}{S_M}\) is the risk premium, i.e. an additional component of the rate of return (in percentage terms) that can be obtained for increasing the risk per unit. It is therefore the price of the one unit of risk. In turn, \(s_p\) is the risk of an efficient portfolio. Hence the capital market line can be interpreted as follows:
The return rate of the efficient portfolio = the “price of time” + the price of the risk unit * the size of the risk of the efficient portfolio
The securities market line (SML) concerns any portfolios, and therefore also individual securities. The expected rate of return of such a portfolio is also the sum of two components:
• Risk-free rate of return, that is the “price of time”,
• The second component is the risk price. It is the product of the size of the systematic risk of a given portfolio, as measured by the beta factor, and the risk-free premium.
Let us assume several SML versions depending on the given beta value:
• \(\beta = 1\), then \(R = R_M\) (i.e. SML also includes a market portfolio);
• \(\beta = 0\), then \(R = R_f\) (that is SML is the portfolio containing risk-free instruments);
• \(\beta> 1\) (aggressive portfolio), then \(R> R_M\);
• \(0 <\beta <1\) (defensive portfolio), then \(R_f <R <R_M\);
• \(\beta <0\), then \(R <R_f\);
Based on SML, you can determine the alpha coefficient (it should not be confused with the alpha coefficient of the security characteristic line). It is specified as follows:
\[\alpha = R - (R_f + \beta*(R_M - R_f))\]
where: \(R\) - the expected rate of return of the portfolio (eg. estimated using fundamental analysis).
The alpha coefficient is the surplus of the expected rate of return over the expected rate of return on the market in equilibrium. If the stock is on SML then the alpha coefficient is zero. We can also set it for portfolios of stocks:
\[\alpha_p = \sum_{i=1}^nw_i*\alpha_i\]
The figure below presents the SML graphic interpretation.
Figure 3. Security Market Line.
The beta coefficients of the portfolios are marked on the Figure 3 on the x-axis, and the expected rates of return of the portfolios are presented on the y- axis. Various portfolios and the SML line are presented on the Figure 3. There are six portfolios on SML. Portfolio F contains only risk-free instruments. M portfolio is a market portfolio. Portfolio A is a defensive one, and portfolio D is an aggressive one. All these portfolios belong to SML line, which shows that the market for these portfolios is in equilibrium, and the portfolios themselves are fairly priced. The term “fairly-priced” refers to SML (i.e. to CAPM) and means that the expected rate of return for these portfolios is the same as the most portfolios with the same beta ratio.
Figure 3 also shows two portfolios that do not belong to SML. Portfolio B is above SML. The alpha factor of this portfolio is positive. This means that it corresponds to a higher expected rate of return than the B’ portfolio, which has the same beta coefficient, but lies on the SML (i.e. it is fairly priced). Portfolio B is underestimated or, in other words, undervalued. It becomes attractive to investors, who increase the demand for the B portfolio and increase its price, and therefore they decrease its expected rate of return. These movements will lead to equilibrium and B portfolio will converge to a B’ portfolio, i.e. it will belong to the SML line.
In turn, the C portfolio lies below SML. The alpha ratio of this portfolio is negative. This means that the lower expected rate of return corresponds to C portfolio in comparison to C’ portfolio, which has the same beta coefficient, but lies on the SML (i.e. it is fairly priced). Portfolio C is overestimated or, in other words, overvalued. It becomes unattractive to the investors, so they will try to sell it. This will result in an increased supply of the C portfolio, a decrease in its price and, consequently, an increase in its expected rate of return. These activities will lead to equilibrium and C portfolio will converge to a C’ portfolio, i.e. it will move to the SML line.
In fact, it is never the case that all portfolios are on SML. Achieving equilibrium is a dynamic process and most portfolios are either underestimated or overestimated.
Let’s imagine the following relationship at the end. After substituting equation \(\beta_i = \frac{s_i*\rho_{iM}}{s_M}\) for the SML we get the formula for the portfolio that belongs to SML:
\[R_i = R_f + \frac{R_M - R_f}{s_M}*s_i*\rho_{iM}\]
We see that if \(\rho_{im} = 1\), then the above equation becomes the CML equation. This means that the correlation of the returns of the efficient portfolio in equilibrium and market prtfolio is equal 1. For these two portfolios, changes in the rates of return are proportional.
Model APT (Arbitrage Pricing Theory).
I. The law of one price.
The theory of price arbitrage is based on the law of one price, which says that two identical goods are valued at the same price. If they are sold at different prices then it is possible to use arbitrate by buying one good at a lower price and selling it at a higher price, thus achieving income without risk. Arbitrage concerns mainly financial markets and is present on the market all the time.
The conclusion of the law of one price is as follows: two financial instruments of equal risk must have the same rates of return.
II. APT model
In addition to setting the law of one price, APT also assumes the homogeneity of expectations. It belongs to the so-called factor models that assume that the rates of return of stocks are generated according to the formula:
\[R_i = s_i + b_{i1}F_1 + b_{i2}F2 + ...+b_{im}F_m + e_i\]
where:
\(F_j\) - j-th factor;
\(a_i\) - the constant;
\(b_{ij}\) - thefactor of sensitivity of the i-th stock to the j-th factor;
\(e_i\) - the random component of the equation.
The above equation shows that the rates of return depend on various factors. Sensitivity coefficients play an important role in this equation. Their interpretation is similar to the beta factor interpretation. The sensitivity factor indicates how the rate of return of the stock will react to a unit change in the factor when the other factors remain constant. Sensitivity coefficients are also determined for the portfolio:
\[b_{pj} = \sum_{i=1}^nw_i*b_{ij}\]
where:
\(b_{pj}\) - the portfolio sensitivity factor relative to the j-th factor.
There are many stocks and portfolios on the capital market. It is possible to create portfolios of any sensitivity to particular types of factors. There are many possible ways to construct a portfolio with individual sensitivity to a given factor, as well as a portfolio insensitive to any of the factors. A portfolio whose sensitivity to the jth factor is individual and which is insensitive to other factors is determined by solving the equation:
\[\sum_{i=1}^nw_i*b_{ij} = 1\]
\[\sum_{i=1}^nw_i*b_{ik} = 0,\ \ \ \ k=1, ..., m\ \ \ \ k\ne{j}\]
However, a portfolio that is insensitive to all factors is determined by solving the equation:
\[\sum_{i=1}^nw_i*b_{ij} = 0,\ \ \ \ j=1, ..., m\]
Portfolios that are sensitive to one factor, with the sensitivity factor being unitary have the same expected rates of return (the law of one price and the existence of price arbitrage). Similarly, portfolios that are insensitive to any factor should have the expected rate of return equal to the risk-free rate.
The APT model assumes that there are many stocks and other securities on the market. The basis of this model is the so-called an arbitrage portfolio that meets the following equations:
\[\sum_{i=1}^nx_i=0\]
\[\sum_{i=1}^nx_i*b_{ij} = 0,\ \ \ \ j=1, ..., m\]
\[\sum_{i=1}^nx_i*e_i = 0\]
The first equation means that no expenditures are incurred when creating an arbitrage portfolio. The central equation means that the portfolio is insensitive to risk factors (\(x_ib_{ij}\) expression is the change in the income of the part of the portfolio invested in the i-th stock, which change is caused by the change in the j-th factor). The last equation is an approximate relationship, which means that the portfolio has no specific risk (reflected by the random component).
Based on the above equations, we come to the appropriate APT model, defining the rate of return of the portfolio:
\[R=\lambda_0+\lambda_1b_1+\lambda_2b_2+...+\lambda_mb_m\]
where:
\(b_1, ..., b_m\) - portfolio sensitivity ratios relative to risk factors,
\(\lambda_1, ..., \lambda_m\) - equation factors.
You can also prove that the model’s \(\lambda_i\) coefficients are equal to:
\[\lambda_0=R_f,\ \ \ \ \lambda_j = R_{pj} - R_f \ \ \ \ (j = 1, ..., m)\]
where:
\(R_f\) - the risk-free rate of return;
\(R_{pj}\) - expected rate of return of the portfolio, which is insensitive to all factors except j-th, and whose sensitivity to the j-th factor is unitary.
Let’s also see the graphic interpretation (Figure 4) of the APT one-way model, given by the formula:
\[R = R_f +b\lambda\]
Figure 3. Price arbitrage line.
The figure shows the so-called price arbitrage line. This is the relationship between the expected rate of return of the portfolio and the coefficient of sensitivity to risk. The constant component of this line corresponds to the risk-free rate. In turn, the slope coefficient, i.e. the value of \(\lambda\), is the risk premium. In other words it is the surplus of the expected rate of return of the portfolio over the risk-free rate, where the portfolio has a unit risk-sensitivity. If the market is in equilibrium, the portfolios are fairly priced and belong to the price arbitrage line. Figure 4 also shows examples of underestimated (\(O\)) and overestimated (\(P\)) portfolios. Their interpretation and the process of achieving equilibrium is identical to the CAPM model.
The last issue in the APT model is the answer to the question what the risk factors in it may be. These may include, for example, changes in GDP, changes in the unemployment rate, changes in the interest rate differential between countries, changes in the inflation rate, changes in the industrial production index, etc.
Exercise 1
The Jarvis Corporation returns and market returns are as follows:
| year | R_Jarvis | R_M |
|---|---|---|
| 1991 | -5 | -6 |
| 1992 | 14 | 16 |
| 1993 | 10 | 12 |
| 1994 | 12 | 14 |
| 1995 | 17 | 20 |
Calculate the beta ratio for Jarvis. What percentage of the total risk is a systematic risk?
Exercise 2
Assume that \(R_f = 8\%\), \(R_M = 14\%\) and \(\beta = 1.25\) for the i-th security. Calculate the expected rate of return for a security i-th. How will the expected rate of return change if \(R_M\) increases to 16%? How will this rate change if the beta coefficient falls to 0.75?
Exercise 3
Dividend per share for BMC Corporation increased by 6% during the last 6 years. Calculate the value of BMC shares if this situation continues in the future. We assume that \(D_0 = 3.5\), \(R_f = 9\%\), \(R_M = 16\%\) and \(\beta_{BMC} = 1.3\).
Exercise 4
Jefferson Investment Company manages a fund of five stocks with the following market value and beta indicators.
| stock | mtk_val | beta |
|---|---|---|
| Zell | 100000 | 1.10 |
| Car | 50000 | 1.20 |
| Arms | 75000 | 0.75 |
| Dole | 125000 | 0.80 |
| Ord | 150000 | 1.40 |
Calculate the beta ratio of the portfolio.
Exercise 5
We assume that \(R_f = 9\%\), \(R_M = 15\%\). The expected rates of return and beta factors are given below:
| stock | R_expected | beta |
|---|---|---|
| Hall | 14 | 1.20 |
| Izo | 15 | 0.75 |
| Jenn | 20 | 1.50 |
Which stocks are overvalued and which are undervalued?
Exercise 6
Please define the APT line for the two following portfolios in equilibrium:
| portfolio | R_expected | beta |
|---|---|---|
| A | 15 | 1.5 |
| B | 10 | 0.5 |
Exercise 7
There are three portfolios in equilibrium:
| portfolio | R_expected | b_1 | b_2 |
|---|---|---|---|
| X | 16 | 1.0 | 0.8 |
| Y | 12 | 0.6 | 0.5 |
| Z | 18 | 0.9 | 1.1 |
Determine the APT line. There is \(U\) security available on the market that is not in equilibrium, \(R = 17\%\), \(b_{U1} = 0.8\), \(b_{U2} = 0.65\). What profit can we achieve by creating an arbitrage strategy from X and Y stocks? Please describe individual transactions.