borrow at the same rate as governments. In addition to this, it is important to determine the maturity, whereas, for short-term investments, it is usual to pick a short maturity and, for long term investments, a long-term ma-turity (Berk & DeMarzo, 2020).
There are two approaches to gather the risk-free rate. The fundamental approach by calculating itself by solv-ing for the discount rate that is consistent with the current level of the index. This approach is highly inaccu-rate for an individual firm, and by assuming constant growth then more suitable for the overall market. The second approach is by looking at the historical risk premium. The authors have chosen the historical approach, as this is a historical analysis and not trying to calculate future cash flows. Therefore, assuming these are long-term investments, the historical rate on the ten-year government bond for Denmark, Finland, Norway, and Sweden been located. This rate has been calculated to an average rate where each rate is equally weighted. To make the risk-free rate adjusted to months, it has been divided by 12 on each observation.
Equation 32
𝑟𝑓𝑚= 𝑟𝑓𝑡 12
Equation 34
𝑟𝑖− 𝑟𝑓= 𝛼 + 𝛽𝑖,𝑀𝑘𝑡∗ (𝑟𝑀𝑘𝑡− 𝑟𝑓) + 𝛽𝑖,𝑆𝑀𝐵∗ 𝑆𝑀𝐵 + 𝛽𝑖,𝐻𝑀𝐿∗ 𝐻𝑀𝐿 + 𝛽𝐸𝑆𝐺∗ 𝐸𝑆𝐺
The idea behind this changed factor is to capture the historical effect of the companies ESG score and their excess stock return. In the view of M&As, it should highlight the relationship between the beta and the cost of equity, which further in the valuation will affect the WACC and, therefore also the enterprise value.
Like the calculations for HML, the ESG score being grouped based on a top and bottom percentile score. For each year, a 90th percentile is being calculated as the limit for the ESG Leaders and a 10th percentile for the ESG Laggers. The combined score for each company categories them in either as an ESG Leader ESG Neutral or ESG Lagger. Following Hvidkjær (2017) paper that stocks with high ESG Score have high future returns and Carhart (1997), Jegadeesh and Titman (1993), and Hendriks et al. (1993) findings of stocks continuing their positive/negative returns based on previous year results, then the portfolio is constructed by buying ESG Leaders and selling ESG Lagger. In other words, ESG Leaders Minus ESG Laggers.
As the paper does not analyze momentum strategies regarding the ESG scores, the authors have chosen not to analyze specific holding periods and assume for the paper due to the specific data. The holding period is on a yearly basis.
For formatting the portfolio, the theory of creating the SMB and HML portfolio is used and added with the created ESG portfolio. For a company to access the regression in the given year, it only must fulfil one crite-rion, which is having an ESG that year. As the data did not extract ESG scores of companies who are not pub-licly traded in 2021, then the portfolio will not be affected by companies who were public-traded one year and were not in years later. The value of the portfolio is set by the authors to be 1 DKK in 2011 to give a more vis-ual presentation of increased/decreased value historically.
Portfolio Evaluation
To calculate the performance of the given portfolio then several measures are being calculated in terms of per-formance and risk.
For the expected return (𝐸[𝑅𝑝𝑚]) for each month on a given portfolio (𝑝) where each security is equally weighted, then the average monthly return is calculated as:
𝐸[𝑅𝑝𝑚] = ( ∑ 𝑟𝑖𝑚
𝑛
𝑚=1
) ∗1
𝑛 , ∀ 𝑚 = 1, … , 108
Where 𝑛 is the number of companies in the portfolio at time 𝑚, whereas to calculate the yearly expected re-turn, the assumption of holding the stocks from primo to ultimo is taken into consideration. The yearly ex-pected return is calculated similarly as the formula above:
Equation 36
𝐸[𝑅𝑝𝑡] = ( ∑ 𝑟𝑖𝑚
𝑛
𝑚=1
) ∗1
𝑛 , ∀ 𝑚 = 1, … , 108 ∈ 𝑡
When moving towards the calculations of the risk of the portfolio, a calculation of the portfolio’s volatility is necessary. To derive the volatility of the portfolio, the following formula has been used to calculate the vari-ance of an equally weighted portfolio of 𝑛 stocks:
Equation 37 𝑉𝑎𝑟(𝑅𝑃𝑚) = ∑ ∑ 𝑥𝑖𝑥𝑗(𝑟𝑖, 𝑅𝑝)
𝑗
𝑖 , ∀ 𝑚 = 1, … , 108
This can be simplified further be simplified as an equally weighted portfolio consisting 𝑛 stocks have portfolio weights 𝑥𝑖 of 𝑥𝑖 =1𝑛, the formula will then look as:
Equation 38
𝑉𝑎𝑟(𝑅𝑃𝑚) =1
𝑛∗ (𝐴𝑣𝑔. 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑠𝑡𝑜𝑐𝑘𝑠) + (1 −1 𝑛)
∗ (𝐴𝑣𝑔. 𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑠𝑡𝑜𝑐𝑘𝑠), ∀ 𝑚 = 1, … , 108
Whereas this can be used to calculate the portfolio volatility by:
Equation 39
𝜎𝑃𝑚 = √∑ ∑ 𝑥𝑖𝑥𝑗(𝑟𝑖, 𝑅𝑝)
𝑗
𝑖 , ∀ 𝑚 = 1, … , 108
In one-period models, it is possible to investigate the equilibrium expected returns instead of prices. To inves-tigate whether the gain in return from investing in 𝑖 adequate to compensate for the increase in risk. The
portfolios Sharpe ratio explains how much the return would increase for a given increase in risk (Bodie et al., 2014).
Equation 40
𝑆ℎ𝑎𝑟𝑝𝑒 𝑅𝑎𝑡𝑖𝑜 =𝐸[𝑅𝑃𝑡] − 𝑟𝑓 𝑆𝐷(𝑅𝑃𝑡)
This measure indicates the reward of the taken risk by the investors, which tells the expected excess return in terms of per unit of risk. This reward-to-volatility measure is widely used in practice, and the Sharpe ratio will be higher as when annualized from higher frequency returns. The Sharpe ratio of an optimally constructed portfolio will exceed the Sharpe ratio of the index portfolio, as this is supposed to be the passive strategy. The contribution of the portfolio to the Sharpe ratio is determined by the ratio of its alpha to its residual standard deviation, which is also known as the information ratio. This ratio measures the extra return investors can ob-tain from security analysis compared to the firm-specific risk to the passive market index. This ratio deter-mines how the portfolio will differ from the well-diversified market portfolio (Bodie et al., 2014).
Equation 41
𝐼𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 =𝑟𝑝𝑡 − 𝑟𝑚𝑘𝑡𝑡 𝑇𝐸 Where:
Equation 42
𝑇𝐸 = 𝑇𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝐸𝑟𝑟𝑜𝑟 = √∑𝑛𝑚=𝑖(𝑟𝑝𝑡− 𝑟𝑚𝑘𝑡𝑡 )2
𝑀 − 1 , ∀ 𝑚 = 1, … , 108 ∈ 𝑡
Figure 6: Visual presentation of data selection
(Authors’ own creation, 2021).