Performance measures

In this thesis performance measures used to evaluate financial sector companies as an invest-ment are introduced in this chapter. Sharpe ratio, Sortino ratio, Treynor ratio and Jensen’s alpha are all risk-adjusted performance measures. In addition to these, traditional return rate and annualized volatility are also used in the performance comparison.

2.5.1 Sharpe ratio

The Sharpe ratio is a risk-adjusted measure used for performance evaluation of an asset and it was presented by William Sharpe in 1966. It measures total return relative to the total risk.

Risk component in the Sharpe ratio is volatility, meaning that it includes the total risk of the investment. (Sharpe, Alexander & Bailey 1999, 844) Because of its simplicity to calculate, it is the most commonly used performance measure (Pätäri 2000, 27). According to Perold (2004), the ratio is also commonly used in portfolio optimization to find the best risk-return tradeoff.

Sharpe ratio, also known as reward-to-variability ratio, measures the returns over the return of risk-free investment per one unit of volatility. The bigger values Sharpe ratio gets, the bet-ter. (Sharpe 1994) Sharpe ratio is

𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 = 𝑅_! − 𝑅_$

𝜎_! , (5)

where si volatility of the excess return of the investment.

At the denominator of the ratio can be used either the volatility of the excess return of the investment or the volatility of the investments return (Vaihekoski 2004, 260-261). In this thesis volatility of the excess returns of the stock is used as a risk component of the Sharpe ratio.

According to McLeod and van Vuuren (2004), the interpretation of negative values of Sharpe ratio is not as simple as it is with positive values; the bigger the better. In a situation were two investments have equal, but negative excess returns and different volatilities, the investment with bigger volatility has bigger value of Sharpe ratio and thus should be considered as a more profitable investment according to the traditional interpretation of Sharpe ratio, as shown in Table 1. (McLeod & van Vuuren, 2004)

Table 1 An example of negative Sharpe ratios

Applying traditional interpretation to negative values leads to a situation where with equal negative excess returns the one with bigger volatility is preferable and the growth of volatility makes the value of Sharpe ratio to approach zero. Due to this problem, inverse ordering of negative values is applied in this thesis. The negative values of Sharpe ratios are interpreted by arranging them from the best performed to the worst performed in the opposite order than positive values: the smaller the negative Sharpe ratio is, the better. As a consequence of this interpretation, it is assumed that a small volatility is preferable also with negative values and negative excess return. However, this choice is not unambiguous, because it depends on the risk preferences of the investor: with stocks with negative returns an investor who is not avoiding risk might see big volatility as an opportunity, because due to the fluctuation the returns might turn positive again. Despite this, the inverse ordering was chosen in this case since with positive values of Sharpe ratio the small volatility is preferable, even though bigger volatility could also enable higher returns. The choice was also made to keep the risk prefer-ences coherent throughout the whole thesis. The same interpretation problem applies with the negative values of Sortino ratio and Treynor ratio, thus the negative values are arranged similarly as with the Sharpe ratio.

Sharpe ratio has been criticized because it penalizes the investment also from fluctuation above the expected or average return, which is mainly positive from the investors point of view. (Pekár 2016) Sortino ratio is a performance measure established as a response to this criticism.

Risk-free rate = 10% Investment A Investment B

Return -10 % -10 %

Volatility 15 % 10 %

Sharpe ratio -1.33 -2.00

2.5.2 Sortino ratio

Sortino ratio is a risk-adjusted performance measure built on the Sharpe ratio. Sortino ratio intends to answer the criticism of Sharpe ratio faced about using volatility as a risk component.

Volatility as a risk component does not separate the fluctuation of returns to positive and negative from investors point of view. “Positive” fluctuation is desired for the investor because the return of the investment goes above the expected return and so is not harmful for the investor. Sharpe ratio penalizes equally from the fluctuation below and over the expected re-turn. This is why the Sortino ratio uses downside deviation as the risk component. (Sortino &

van der Meer 1991)

Downside deviation is the negative fluctuation isolated from the standard deviation, meaning that it takes into account only returns falling below the minimum acceptable return. (Morn-ingstar 2020) Minimum acceptable return, MAR, can be defined by the user of the ratio and it penalizes only from the returns falling below the minimum acceptable return. A difference between Sharpe ratio and Sortino ratio is that the risk-free interest rate used in the Sharpe ratio is replaced with the minimum acceptable return in the Sortino ratio. (Pekár 2016) Equa-tion of the ratio is

𝑆𝑜𝑟𝑡𝑖𝑛𝑜 𝑅𝑎𝑡𝑖𝑜 = 𝑅_! − 𝑀𝐴𝑅

𝐷𝐷 , (6)

where MAR minimal acceptable return of the investment DD downside deviation.

The interpretation of Sortino ratio is similar to Sharpe ratio (Pekár 2016), also with negative values. In this thesis the downside deviation of the stock return is calculated from the series of excess returns to maintain the comparability between Sharpe ratio and Sortino ratio. Risk-free interest rate is used as minimum acceptable return, because the stock should be able to offer better returns than the risk-free investment in order to be lucrative from the investors point of view. The downside deviation for the Sortino ratio is calculated as a standard devia-tion of excess returns falling below the risk-free interest rate.

When interpreting Sharpe ratio and Sortino ratio together, worth paying attention is the dif-ference between these two ratios, because it implies about the fluctuation of the stock price.

The bigger the Sortino ratio is compared to the Sharpe ratio, the more the fluctuation has

happened above the minimum acceptable return and so is positive from the investors point of view. If the fluctuation happens mostly below the minimum acceptable return, Sharpe ratio and Sortino ratio values are close to each other. In this study the interpretation can be done as explained, because the numerator of both ratios is the same, as the risk-free interest rate is also used as the minimum acceptable return.

2.5.1 Treynor ratio

Treynor ratio was the first risk-adjusted performance measure developed by Jack Treynor in 1965. Treynor Ratio, like the Sharpe ratio, measures the excess returns over the risk-free re-turn. The difference between these two ratios is that when Sharpe ratio takes volatility as a risk measure (total risk), risk component of Treynor ratio is beta and so it considers only sys-tematic part of the risk. (Bodie et al. 2005, 868) The equation is

𝑇𝑟𝑒𝑦𝑛𝑜𝑟 𝑟𝑎𝑡𝑖𝑜 = 𝑅_! − 𝑅_$

𝛽_! . (7)

The bigger values Treynor ratio gets, the better is the risk-adjusted return of the investment.

Similarly, to Sharpe ratio and Sortino ratio, the same problem regarding the interpretation of negative ratio values concerns also Treynor ratio. Because of this, the inverse ordering of neg-ative values is also applied with the negneg-ative values of Treynor ratio.

Most common criticism Treynor ratio has faced is about using beta coefficient as a risk com-ponent. The difficult part is deciding the benchmark used to count it, because the decision will affect to the resulting Treynor ratio value. With different benchmarks, ratio will give different values because the beta coefficient is formed by comparing moves of the investment to the benchmark. (Pätäri 2000, 36)

2.5.2 Jensen’s alpha

The Jensen’s alpha is an investment performance measure that indicates the differences of return compared to the returns CAP-model predicts developed by Michael Jensen 1968. In other words, Jensen’s alpha is a return rate over or below the prediction of CAP-model and it uses portfolio beta as a risk measure. (Pätäri 2000, 40-41) Jensen’s alpha can be counted as follows (Bodie et al. 2005, 868) and the equation is formed from the basic form of CAP-model:

𝑅_! − 𝑅_$= 𝛼_!+ 𝛽_!>𝑅_%− 𝑅_$?,

𝐽𝑒𝑛𝑠𝑒𝑛⁾𝑠 𝑎𝑙𝑝ℎ𝑎, 𝛼_! = 𝑅_! − V𝑅_$+ 𝛽_! >𝑅_%− 𝑅_$?W . (8) Jensen’s alpha can be either positive, zero or negative. A positive alpha implies that the in-vestment is underrated in relation to its risk and it has outperformed the prediction of return of CAP-model. (Nikkinen et al. 2002, 220-221) In this thesis the prediction of CAP-model is based on the benchmark index OMX Nordic 40 and the resulting Jensen’s alpha tells, whether the investment has exceeded the return level relative to its beta coefficient.