class: my_title # ESG risk rating of alternative strategies ## Nicolas Gaussel, Metori Capital Management ### <i>Joint work with Laurent Le Saint</i> ## Frontiers in Quantitative Finance Seminar ### February, 4^th, 2021 <div class="logo"></div> --- class: my_title # ESG Investing <img src="esg_cartoon.jpg" width="40%"> <div class="logo"></div> --- # Environmental, Social, and Governance Several providers of company/country/issuer ESG rating - MSCI scale: 0 -> 10 - Morningstar scale: 0 <- 100 <img class = img_right src="sustain_esg_risk_rating.png" width="33%"> <img class = img_left src="msci_esg_risk_rating.png" width="28%"> <div class="logo"></div> --- # Rating agregation: portfolio level - Individual ratings belong to ]0,1[, 0 being the best possible ESG rating and 1 the worst - Widespread industry approach for agregation: Weighted Average `$$rating_{p}=\sum_{i}rating_{i}\times w_{i}$$` - Provide portfolio rating for mutual fund i.e. when `\(0 \leq w_i\leq 1\)` - Equation fails for Hedge Fund strategies i.e. when `\(w\leq0\)` (short position) or `\(w > 1\)` (leveraged position) ? .center[<span style="font-weight:bold;font-size:110%;color:#DF1C41;"> How rating agregation should be done ?</span>] - Hardly found anything in the literature <img style="vertical-align: middle;" src="puzzled.png" width="5%"> - Blocking issue for ESG development for Hedge Funds <div class="logo"></div> --- # Outline <br><br> - Discussion around short, long/short and leveraged ESG ratings <br> - Introduction of ESG risk and interpretation of ESG rating <br> - Calculation of ESG risk and ESG rating <br> - Examples <div class="logo"></div> --- # Leveraged portfolios: introduction of ESG risk <div style="border: solid #DF1C41 thin; border-radius:20px;padding:10px"> 1) Single asset example. Consider a series of product on S&P500 with leverage 50%, 100% and 300%. Should they have the same ESG rating ? <br><br> 2) In a 50% equity - 50% cash fund, does ESG analysis on cash has the same impact as the ESG analysis on equity ? </div> - Insights: - ESG rating shall be related to risk (not notional based) - If ESG is defined as a risk, it should then be related to the total risk - <span style ="color:#DF1C41;"> <b>View ESG risk as a fraction of total risk - <span style ="color:#DF1C41;"> Interpret ESG rating as the ratio between ESG risk and total risk </b> <div class="logo"></div> --- # Short and long/short positions - Shorting raises specific issues: - Is shorting a legitimate action ? - Is shorting a bad rating good ? - Risk factor approach: - Bad (resp. good) rating means high (resp. low) exposure to ESG factors - If factors are priced in, they have no risk neutral bias hence - In a Black & Scholes model, local risk of long and short position is the same - <span style="color:#DF1C41 "> <b> ESG risk of a short shall be similar to the ESG risk of the long </b> </span> - Long/short situation - Difference between long <i>Total</i> short <i>Shell</i> and long <i>Total</i> short <i>Rio Tinto</i> - <span style="color:#DF1C41 "> <b> A good ESG measure shall be able to cope with ESG factor correlation...</b></span> <div class="logo"></div> --- class: my_title # A framework for computing portfolio ESG risk and ESG rating <div class="logo"></div> --- # Our target Set out definitions of ESG risk and ESG rating consistent with market returns. The set of all factors impacting asset prices can be split into two separate sets: `\(F=F_{ESG}\cup F_{!ESG}\)` where `\(F_{ESG}\)` and `\(F_{!ESG}\)` stand for the set of ESG and non ESG factors. Let `\(X\)` be a `\(F\)` meas. random value. In this framework we can envisage: - total risk = `\(\sqrt{Var\left(X\right)}\)` - ESG risk = `\(\sqrt{Var\left(X|F_{!ESG}\right)}\)` and from the above: <span syle="color:#DF1C41"> `$$\text{ESG rating }= \sqrt{\frac{Var\left(X|F_{!ESG}\right)}{Var\left(X\right)}}$$` </span> How can this be put into practice? --- # A model for ESG risk Factor model where local log-price variations `\(dX\)` are defined as `$$dX=(...)dt+\beta_{ESG} dW_{ESG}+\beta_{!ESG}dW_{!ESG}$$` `\(dW_{ESG}\)`, `\(dW_{!ESG}\)`: ESG and non ESG underlying independant brownian factors `\(\beta_{ESG}\)`, `\(\beta_{!ESG}\)`: the two corresponding rectangular exposure matrices. - Returns covariance matrix `\(\Sigma\)` `\begin{align} \Sigma & =\Sigma_{ESG}+\beta_{!ESG}\beta_{!ESG}' \\ where\quad\quad \Sigma_{ESG} & \equiv \beta_{ESG}\beta_{ESG}' \end{align}` - Without further specification this model is equivalent to impose two conditions: `\begin{align} \Sigma_{ESG} &\succeq 0 \quad\quad\quad (SDP1)\\ \Sigma - \Sigma_{ESG} & \succeq 0 \quad\quad\quad (SDP2) \end{align}` <div class="logo"></div> --- #Model specification and related constraints Let `\(\sigma\)` be the diagonal matrix of volatilities and `\(R\)` the diagonal matrix of ratings We have `\(\Sigma=\sigma\Gamma\sigma\)` where `\(\Gamma\)` is the correlation matrix of asset returns. Asset by asset, ratings represent a fraction of risk. Hence we specify: `$$\Sigma_{ESG}:=R\sigma\Gamma_{ESG}\sigma R$$` with `\(\Gamma_{ESG}\)` the ESG correlation matrix. Previous conditions become: `\begin{align} \Gamma_{ESG} \text{ is a correlation matrix} \quad\quad\quad(SDP1-c) \\ \Gamma- R\Gamma_{ESG}R \succeq 0 \quad\quad\quad(SDP2-c) \end{align}` <div class="logo"></div> --- # Infering ESG correlation from returns correlation There are many correlation matrix `\(\Gamma_{ESG}\)` possibly complying with the two conditions. How to make a choice ? Without further information we would consider the case where `$$\Gamma_{ESG}=\Gamma_{emp},$$` where `\(\Gamma_{emp}\)` is an empirical correlation matrix. This can only work if `$$\Gamma_{emp}- R\Gamma_{emp} R \succeq 0 \quad\quad\quad(SDP2-cemp)$$` which in general does not hold... Note that `\((SDP2-cemp)\)` holds for `\(\Gamma_{emp}=Id.\)` <div class="logo"></div> --- # ESG shrinkage procedure Question: find `\(\Gamma\)` such that `\begin{align} \Gamma- R\Gamma R \succeq 0 \quad\quad\quad(SDP2-cemp)\\ \\ \Gamma \text{ "as close as possible to" }\Gamma_{emp} \end{align}` With `\(\epsilon\in[0,1]\)`, define `$$\Gamma(\epsilon)=(1-\epsilon)\Gamma_{emp}+\epsilon Id \\f(\epsilon)=det(\Gamma(\epsilon)-R\Gamma(\epsilon)R).$$` `\(f\)` is a continouous, strictly increasing function such that `\(f(1)=\prod_{i\leq n}\left(1-r_{i}^2\right)>0\)`. So there exists a unique `\(\epsilon_{opt}\)` defined by `$$\epsilon_{opt}=argmin\{\epsilon\in[0,1]\:|\:f(\epsilon)\geq0.$$` `\(\Gamma_S\equiv\Gamma(\epsilon_{opt})\)` can be seen as the closest matrix to `\(\Gamma_{emp}\)` that complies with both condition `\((SDP1-cemp)\)` and `\((SDP2-cemp)\)`. <div class="logo"></div> --- # ESG risk and ESG rating definitions We can now define both total and ESG risks with the same correlation matrix `\(\Gamma_S\)` <p style="border: solid #DF1C41 thin; border-radius:20px;height:170px"> `\begin{align} risk_{tot}^{2}\left(w\right) & = w'\sigma\Gamma_S\sigma w \\ risk_{ESG}^{2}\left(w\right) & = w'\sigma R \Gamma_S R \sigma w \\ rating_{ESG}\left(w\right) & = \frac {risk_{ESG}\left(w\right)}{risk_{tot}\left(w\right)} \end{align}` </p> and be sure that `\(\forall w:0<rating_{ESG}<1\)`. Single asset case: `\begin{align} risk_{tot}\left(w\right) & =\left|w\right|\sigma\\ risk_{ESG}\left(w\right) & =\left|w\right|\sigma r \\ rating_{ESG}\left(w\right) &= \left|w\right|\sigma r / \left|w\right|\sigma = r \end{align}` <div class="logo"></div> --- #Some properties 1) ESG rating independant of leverage `\(\Rightarrow\)` .center[<span style="color:#DF1C41"><b>ESG rating of derivative = ESG rating of underlying</b></span>] <br><br> 2) When all assets have the same ESG rating `\(r\)`, `\(R=rId\)` `\begin{align} risk_{tot}^{2}\left(w\right) & = w'\sigma\Gamma_S\sigma w \\ risk_{ESG}^{2}\left(w\right) & = w'R\sigma\Gamma_S\sigma Rw=r^2 risk_{tot}^{2}\\ rating_{ESG}\left(w\right) & = r \end{align}` Hence: .center[<span style="color:#DF1C41"><b>ESG rating of any strategy <=> ESG rating of its investment universe</b></span>] <div class="logo"></div> --- # Shrinking: the two asset case In the two assets case with empirical correlation `\(\rho\)` and ratings `\(r_1\)` and `\(r_2\)`. Maximum acceptable value, when `\(\rho>0\)` can be calculated: `\(\rho_S=min\left(\rho,\frac{\sqrt{\left(1-r_{1}^{2}\right)\left(1-r_{2}^{2}\right)}}{\left(1-r_{1}r_{2}\right)}\right)\)`. <div id="htmlwidget-d75a2c6d1ad3dd7a9918" style="width:648px;height:360px;" class="plotly html-widget"></div> <script type="application/json" data-for="htmlwidget-d75a2c6d1ad3dd7a9918">{"x":{"visdat":{"2d3d7b9f857d":["function () ","plotlyVisDat"]},"cur_data":"2d3d7b9f857d","attrs":{"2d3d7b9f857d":{"x":{},"y":{},"color":{},"alpha_stroke":1,"sizes":[10,100],"spans":[1,20],"type":"scatter","mode":"lines","inherit":true}},"layout":{"margin":{"b":40,"l":60,"t":25,"r":10},"xaxis":{"domain":[0,1],"automargin":true,"title":"r2"},"yaxis":{"domain":[0,1],"automargin":true,"title":""},"legend":{"orientation":"h","xanchor":"center","x":0.5,"y":0.4},"title":{"text":"Max correlation 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Its investment universe includes financial futures on stock indices, government bonds, currencies and short-term rates. - The strategy can be long, short and leveraged. - Futures are ESG-proxied by their equivalent financial position: long underlying, short 1/3 month financing - https://www.msci.com/esg-fund-ratings/funds/ <div class="logo"></div> --- # Individual ratings <div class=box_table> <div id="htmlwidget-e8d6e6611dde7f7109d8" style="width:100%;height:75%;" class="datatables html-widget"></div> <script type="application/json" data-for="htmlwidget-e8d6e6611dde7f7109d8">{"x":{"filter":"top","filterHTML":"<tr>\n <td><\/td>\n <td data-type=\"character\" style=\"vertical-align: top;\">\n <div class=\"form-group has-feedback\" style=\"margin-bottom: auto;\">\n <input type=\"search\" placeholder=\"All\" class=\"form-control\" style=\"width: 100%;\"/>\n <span class=\"glyphicon glyphicon-remove-circle form-control-feedback\"><\/span>\n <\/div>\n <\/td>\n <td data-type=\"character\" style=\"vertical-align: top;\">\n <div class=\"form-group has-feedback\" style=\"margin-bottom: auto;\">\n <input type=\"search\" placeholder=\"All\" class=\"form-control\" style=\"width: 100%;\"/>\n <span class=\"glyphicon glyphicon-remove-circle form-control-feedback\"><\/span>\n <\/div>\n <\/td>\n <td data-type=\"number\" style=\"vertical-align: top;\">\n <div class=\"form-group has-feedback\" style=\"margin-bottom: auto;\">\n <input type=\"search\" placeholder=\"All\" class=\"form-control\" style=\"width: 100%;\"/>\n <span class=\"glyphicon glyphicon-remove-circle form-control-feedback\"><\/span>\n <\/div>\n <div style=\"display: none; position: absolute; width: 200px;\">\n <div data-min=\"3.6\" data-max=\"9.3\" data-scale=\"1\"><\/div>\n <span style=\"float: left;\"><\/span>\n <span style=\"float: right;\"><\/span>\n <\/div>\n <\/td>\n <td data-type=\"number\" style=\"vertical-align: top;\">\n <div class=\"form-group has-feedback\" style=\"margin-bottom: auto;\">\n <input type=\"search\" placeholder=\"All\" class=\"form-control\" style=\"width: 100%;\"/>\n <span class=\"glyphicon glyphicon-remove-circle form-control-feedback\"><\/span>\n <\/div>\n <div style=\"display: none; position: absolute; width: 200px;\">\n <div data-min=\"0.1\" data-max=\"0.6\" data-scale=\"1\"><\/div>\n <span style=\"float: left;\"><\/span>\n <span style=\"float: right;\"><\/span>\n <\/div>\n <\/td>\n <td data-type=\"character\" style=\"vertical-align: top;\">\n <div class=\"form-group has-feedback\" style=\"margin-bottom: auto;\">\n <input type=\"search\" placeholder=\"All\" class=\"form-control\" style=\"width: 100%;\"/>\n <span class=\"glyphicon glyphicon-remove-circle form-control-feedback\"><\/span>\n <\/div>\n 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£"],[6.4,6.4,6.4,6.4,6.4,6.4,6.4,6.4,7.8,7.8,7.8,5,5,6.4,6.4,6.4,6.4,6.4,6.4,6.4,7.2,5.2,5.2,5.2,5.2,6.4,6.4,6.4,6.4,7.8,7.8,7.8,7.8,7.8,7.8,7.8,7.8,7.8,9.3,3.6,5,5,6.4,6.4,6.4,6.4,6.4],[0.4,0.4,0.4,0.4,0.4,0.4,0.4,0.4,0.2,0.2,0.2,0.5,0.5,0.4,0.4,0.4,0.4,0.4,0.4,0.4,0.3,0.5,0.5,0.5,0.5,0.4,0.4,0.4,0.4,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.2,0.1,0.6,0.5,0.5,0.4,0.4,0.4,0.4,0.4],["A","A","A","A","A","A","A","A","AA","AA","AA","BBB","BBB","A","A","A","A","A","A","A","AA","BBB","BBB","BBB","BBB","A","A","A","A","AA","AA","AA","AA","AA","AA","AA","AA","AA","AAA","BB","BBB","BBB","A","A","A","A","A"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th> <\/th>\n <th>Sector<\/th>\n <th>Name<\/th>\n <th>MSCI Scale<\/th>\n <th>Metori Scale<\/th>\n <th>Proxy Rating<\/th>\n <\/tr>\n 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<div class="logo"></div> --- # Conclusion <br><br> - Portfolio level ESG rating an important question - We propose a risk-based methodology to compute, for any alternative portfolio, an ESG risk rating based on the ESG ratings of its constituents - What's next? 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