CAPM

Historically, the standard Capital Asset Pricing Model is one of the key models used in studies of ethical fund performance (Bauer, Koedijk, & Otten, 2005). The CAPM model is basically the relationship between risk and return. The appropriate risk premium on an asset will be determined by its contribution to the risk of investors’ overall portfolios.

The CAPM equation states that the risk of the security relative to the market, the assets beta, times the expected excess return relative to the market added to the risk-free rate of return is equal to the expected return of any security (Bodie, Kane & Marcus, 2014).

𝑅_J = 𝑅_K + 𝛽_J(𝑅_C− 𝑅_K)

𝑅_J is expected return on asset i 𝑅K is the risk-free rate

𝛽J is Beta for asset i

𝑅C is expected return of the market portfolio

Furthermore, this relationship is illustrated in Figure 5.1, which is also known as the Security Market Line. The Security Market Line defines the expected return on any security or portfolio in the economy.

Figure 5.1: Security Market Line

The values on the X-axis represent the systematic risk (beta), while the expected return of the portfolio is represented on the Y-axis. The beta measures its contribution to the variance of the market portfolio. Hence, the required risk premium is a function of beta. The market’s beta is 1, we can therefore read off the vertical axis at the point on the X-axis where 𝛽 = 1 to detect the expected return on the market portfolio. The risk of an investment is measured by its beta, and SML specifies the required rate of return needed to compensate investors for risk and the time value of money. The standard form of CAPM has been published in articles by Sharpe, Lintner and Mossin (Bodie, Kane & Marcus, 2014).

5.3.1. Sharpe ratio

The reward-to-volatility ratio was first proposed by William F. Sharpe in 1966 and therefore called the Sharpe ratio. The performance measure is designed to measure how well the return of an asset compensates the investor for the risk taken. Hence, the Sharpe ratio is commonly used to evaluate the performance of investment managers (Bodie, Kane & Marcus, 2014). Graphically, the Sharpe ratio is the slope of the Capital Market Line. It connects the risk-free rate and the portfolio. The underlying assumptions behind the Sharpe ratio state that the investor is able to sell short and could borrow and lend money at the risk-free rate of interest. Hence, the investor will choose the portfolio with the highest Sharpe ratio, while being able to adjust the investment and level of risk by borrowing and lending at the risk-free rate. However, in order for this to be true it is assumed that investors are able to borrow and lend to the same rate, which in most cases is unrealistic. The portfolio with the steepest slope will provide the largest risk-adjusted return. Therefore, the Sharpe ratio provides valuable guidance when ranking portfolio performance.

𝑆_J = 𝑅_J− 𝑅_N 𝑆𝑇𝐷 𝑅_J

SR is the reward − to − volatility ratio on asset i R_R is the return on investment of asset i Re is the risk − free rate of return

STD RR is the standard deviation of the return

The Sharpe ratio measures the excessed average return earned per unit of volatility. The risk-free rate is subtracted from the mean return in order to isolate the performance related to risk-taking activities. Generally, a greater value of the Sharpe ratio implicates a more attractive risk-adjusted return (Elton, Gruber, Brown, & Goetzmann, 2003).

5.3.2. Treynor ratio

The Treynor (1965) ratio is, in similarity to the Sharpe ratio, a reward-to-volatility ratio and is developed to evaluate portfolio performance. The difference between the Sharpe and the Treynor ratio is that the Treynor ratio uses systematic risk (beta) of the portfolio as the measurement of volatility, while Sharpe includes the total risk.

𝑇_J =𝑅_{J i} 𝑅_N 𝛽_J

𝑇J is the reward-to-variability ratio 𝑅J is the return on investment of asset i 𝑅N is the risk-free rate of returns 𝛽J is the beta, or systematic risk of asset i

Graphically, it is a measure of the slope on the Security Market Line. In similarity to the Sharpe ratio, is the Treynor ratio a risk-adjusted measure of return, which is useful when measuring investment performance. The individual shareholder’s aversion of investment risk is taken into account. In similarity to the Sharpe ratio is the assumption behind the model that the investor would be able to adjust the level of risk by borrowing and lending at the risk-free rate.

Furthermore, the return is adjusted for general market fluctuations, since only the systematic risk is taken into consideration (Treynor, 1965).

Figure 5.2: Treynor ratio slopes

Figure 5.2 illustrates the Treynor ratio for three different portfolios. The three colored lines illustrate the optimal combination between the risk-free rate and each of the three portfolios. Those lines symbolize the Security Market Line for each asset. The slope of each line is the Treynor ratio.

The steepness of the different portfolios’ Security Market Line determines the ranking of the investment. The steeper the slope, the higher will the ranking be.

5.3.3. Jensen’s alpha

The main purpose of the Jensen’s alpha is to evaluate the performance of portfolios of risky assets.

The underlying assumption of the model is that investors are risk averse and will only accept additional risk if they are compensated for it in the form of higher expected future returns.

(Jensen, 1969). Furthermore, beta is used as a volatility measurement, which implies that the model analyses portfolios that are diversified. Jensen extended the previous results by (Sharpe, 1966) and (Lintner, 1965). In perfect capital markets should all stock returns lie on the Security Market Line.

Hence, the alpha should be 0. Jensen’s alpha demonstrates the difference between portfolio returns and the market returns when CAPM does not hold.

𝛼_k = 𝑟_k− 𝑟_N+ 𝛽_k 𝑟_l − 𝑟_N

𝛼k Return above or under predicted return by CAPM.

𝑟k Expected total portfolio return

𝑟N Risk-free rate

𝛽k Beta of the portfolio 𝑟l Expected Market return

The formula implies that the portfolios positioned above the SML have a positive alpha and have outperformed the market, while the portfolios positioned below the SML have a negative alpha and have underperformed relative to the market.

Figure 5.3 shows the concept behind Jensen’s alpha, where the different portfolios are located in relation to the Security Market Line.

Figure 5.3: Jensen’s alpha

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Difficulties of using one-factor models

As previously mentioned, CAPM is extensively used when determining performance of a given portfolio of securities. However, the adequacy of single index models to measure fund performance is questioned in recent literature on cross-sectional variations in stock returns Bauer et al. (2005). The assets included in different funds varies, for example certain funds hold only small stocks, while other funds hold a mix of stocks and bonds, or assets from many different countries. The problem arises when comparing the performance of those funds in a single index model as CAPM, since the assets held in the portfolios are not contained in the benchmark.

Consequently, it might perform differently. Single index models are not able to explain if the performance of the fund relative to the benchmarks depends on the fund manager’s stock selection skills or are due to the performance of the security group, which is not included in the benchmark (Elton, 2011).

Further, critics argue that the underlying assumptions to prove CAPM are considered to be unrealistic. Some of the assumptions are that investors have identical preferences, have the same

information, and hold the same portfolio. Furthermore, CAPM only apply the market index when estimating expected return, while multifactor models consider several factors when estimating the appropriate return. The factors, which are thought to influence the return of a specific asset, are included in the multifactor model. The chosen factors are variables that historically seem to predict average returns well. Hence, these factors might capture risk premiums. The different variables are not constant and the factors applied depend on the characteristics of the specific asset. There could be several different macro-economic variables included in the model depending on what the given asset are most sensitive to. Common risk factors could be interest rate fluctuations, inflation or oil price, along with others (Brealey, Allen, & Myers, 2014).