notebooks/sbm/main.ipynb

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   "source": [
    "# Introduction\n",
    "\n",
    "Run this notebook to create an analytical application for the SBM charge - based of the Equity Delta Example. The input data will be stored in-memory and Atoti will perform the computation \"on-the-fly\" based on user query. You can filter, drill down and explore your data and the SBM metrics.\n",
    "\n",
    "<img src=./app-preview.gif/ width = 70%>\n",
    "\n",
    "## About SBM\n",
    "\n",
    "You will find below how the SBM logic can be described in python and injected into Atoti. We'll narrow down the use case to Equity Delta charge for simplicity. The other aggregation chains can be added in a similar manner.\n",
    "\n",
    "Sensitivity-based mathod is one of the parametric market risk methodologies. It can be used to compute a VaR-like metric from sensitivities using pre-calibrated risk weights and correlations, through a sequence of nested variance covariance formulae. Due to its multiple benefits (see [2]), the method is widely used both internally by orgazations to manage market risk, as well as by Regulators for capital requirements purposes (FRTB) as well as margining (SIMM). \n",
    "\n",
    "The notebook assumes that the reader is familiar with the sensitivity-based method and terminology.\n",
    "\n",
    "## What to expect\n",
    "\n",
    "In this notebook, we'll read the required sensitivities and calculation parameters and create a chain of measures aggregating data:\n",
    "\n",
    "sensitivities by trade -> weighted sensitivities by risk factor -> charges by bucket -> equity delta margin\n",
    "\n",
    "\n",
    "## Input data\n",
    "\n",
    "Input data is sourced in a CRIF-like format - see [3].\n",
    "\n",
    "## References \n",
    "\n",
    "- [1]: Consolidated Basel Framework Chapter MAR21: https://www.bis.org/basel_framework/chapter/MAR/21.htm?inforce=20220101. \n",
    "- [2]: ISDA SIMM(TM): From Principles to Model Specification: https://www.isda.org/a/vAiDE/simm-from-principles-to-model-specification-4-mar-2016-v4-public.pdf\n",
    "- [3]: ISDA Risk Data Standard: https://www.isda.org/a/owEDE/risk-data-standards-v1-36-public.pdf\n",
    "\n",
    "# Input data files\n",
    "\n",
    "These are the csv-files that we need to illustrate the Equity Delta aggregation:\n",
    "\n",
    "- \"bigger_data.csv\" - sensitivities in a CRIF-like format\n",
    "- \"eq_delta_gamma.csv\" - cross-bucket correlations, set for each pair of buckets\n",
    "- \"eq_delta_rho.csv\" - risk factor correlations, set per bucket, i.e. all pairs of risk factors in a given bucket have the same correlation\n",
    "- \"eq_delta_rw.csv\" - risk weights, set per bucket and per risk factor type (stored in Label2 crif field)\n",
    "\n",
    "The example data files are stored in s3. "
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"text-align:center\"><a href=\"https://www.atoti.io/?utm_source=gallery&utm_content=sbm\" target=\"_blank\" rel=\"noopener noreferrer\"><img src=\"https://data.atoti.io/notebooks/banners/discover.png\" alt=\"atoti table\" /></a></div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# uncomment to install atoti https://docs.atoti.io/latest/installation.html\n",
    "# !pip install atoti[jupyterlab,aws]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import atoti as tt\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's start by launching atoti cube and web app:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "Open the notebook in JupyterLab with the atoti extension enabled to see this link."
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "session = tt.create_session(config={\"java_options\": [\"-Xmx12G\"]})\n",
    "\n",
    "session.link()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Sensitivities table"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "risks_table = session.read_csv(\n",
    "    \"s3://data.atoti.io/notebooks/sbm/bigger_data.csv\",\n",
    "    keys=[\"TradeID\", \"PortfolioID\", \"RiskType\", \"Qualifier\", \"Label2\"],\n",
    "    table_name=\"Risks\",\n",
    "    hierarchized_columns=[\n",
    "        \"TradeID\",\n",
    "        \"PortfolioID\",\n",
    "        \"RiskType\",\n",
    "        \"Qualifier\",\n",
    "        \"Label2\",\n",
    "        \"Bucket\",\n",
    "    ],\n",
    ")\n",
    "\n",
    "cube = session.create_cube(risks_table)\n",
    "h, l, m = cube.hierarchies, cube.levels, cube.measures"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\n",
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       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>AmountUSD</th>\n",
       "      <th>Bucket</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>TradeID</th>\n",
       "      <th>PortfolioID</th>\n",
       "      <th>Qualifier</th>\n",
       "      <th>Label2</th>\n",
       "      <th>RiskType</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <th>Bigger_Portfolio</th>\n",
       "      <th>Wilmar International</th>\n",
       "      <th>REPO</th>\n",
       "      <th>Risk_Equity</th>\n",
       "      <td>-10332.09</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <th>Bigger_Portfolio</th>\n",
       "      <th>Wilmar International</th>\n",
       "      <th>SPOT</th>\n",
       "      <th>Risk_Equity</th>\n",
       "      <td>-3641606.45</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <th>Bigger_Portfolio</th>\n",
       "      <th>China Minmetals</th>\n",
       "      <th>REPO</th>\n",
       "      <th>Risk_Equity</th>\n",
       "      <td>-2337.90</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <th>Bigger_Portfolio</th>\n",
       "      <th>China Minmetals</th>\n",
       "      <th>SPOT</th>\n",
       "      <th>Risk_Equity</th>\n",
       "      <td>-10549.54</td>\n",
       "      <td>3</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <th>Bigger_Portfolio</th>\n",
       "      <th>China Life Insurance</th>\n",
       "      <th>REPO</th>\n",
       "      <th>Risk_Equity</th>\n",
       "      <td>-7874.94</td>\n",
       "      <td>4</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                                                   AmountUSD  \\\n",
       "TradeID PortfolioID      Qualifier            Label2 RiskType                  \n",
       "0       Bigger_Portfolio Wilmar International REPO   Risk_Equity   -10332.09   \n",
       "1       Bigger_Portfolio Wilmar International SPOT   Risk_Equity -3641606.45   \n",
       "2       Bigger_Portfolio China Minmetals      REPO   Risk_Equity    -2337.90   \n",
       "3       Bigger_Portfolio China Minmetals      SPOT   Risk_Equity   -10549.54   \n",
       "4       Bigger_Portfolio China Life Insurance REPO   Risk_Equity    -7874.94   \n",
       "\n",
       "                                                                  Bucket  \n",
       "TradeID PortfolioID      Qualifier            Label2 RiskType             \n",
       "0       Bigger_Portfolio Wilmar International REPO   Risk_Equity       1  \n",
       "1       Bigger_Portfolio Wilmar International SPOT   Risk_Equity       1  \n",
       "2       Bigger_Portfolio China Minmetals      REPO   Risk_Equity       3  \n",
       "3       Bigger_Portfolio China Minmetals      SPOT   Risk_Equity       3  \n",
       "4       Bigger_Portfolio China Life Insurance REPO   Risk_Equity       4  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# So far we have loaded data only into the risks_table\n",
    "risks_table.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At this point, a cube has been created, and we can start browsing the sensitivities:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
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        "[Measures].[AmountUSD.SUM]",
        "[Measures].[contributors.COUNT]"
       ],
       "rows": [
        "[Risks].[RiskType].[RiskType]",
        "[Risks].[Bucket].[Bucket]"
       ]
      },
      "query": {
       "mdx": "SELECT NON EMPTY Crossjoin(Hierarchize(Descendants({[Risks].[RiskType].[AllMember]}, 1, SELF_AND_BEFORE)), Hierarchize(Descendants({[Risks].[Bucket].[AllMember]}, 1, SELF_AND_BEFORE))) ON ROWS, NON EMPTY {[Measures].[AmountUSD.SUM], [Measures].[contributors.COUNT]} ON COLUMNS FROM [Risks]",
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      ],
      "text/plain": [
       "Open the notebook in JupyterLab with the atoti extension enabled to build this widget."
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "session.visualize()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Let's create a first measure - to count distinct Qualifiers\n",
    "m[\"Distinct Qualifiers Count\"] = tt.agg.count_distinct(risks_table[\"Qualifier\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "atoti": {
     "widget": {
      "mapping": {
       "columns": [
        "ALL_MEASURES"
       ],
       "measures": [
        "[Measures].[Distinct Qualifiers Count]"
       ],
       "rows": [
        "[Risks].[Bucket].[Bucket]"
       ]
      },
      "name": "",
      "query": {
       "context": {
        "queriesResultLimit.intermediateSize": 1000000,
        "queriesResultLimit.transientSize": 10000000
       },
       "mdx": "SELECT NON EMPTY [Measures].[Distinct Qualifiers Count] ON COLUMNS, NON EMPTY Union(Hierarchize(DrilldownLevel([Risks].[Bucket].[ALL].[AllMember])), Hierarchize(Descendants({[Risks].[Bucket].[AllMember]}, 1, SELF_AND_BEFORE))) ON ROWS FROM [Risks] CELL PROPERTIES VALUE, FORMATTED_VALUE, BACK_COLOR, FORE_COLOR, FONT_FLAGS"
      },
      "serverKey": "default",
      "widgetKey": "pivot-table"
     }
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.atoti.widget.v0+json": {
       "name": null,
       "sessionId": "420477c0-08b8-4a4a-b131-a571557dbc36",
       "sessionLocation": {
        "https": false,
        "port": 59549
       },
       "widgetCreationCode": "session.visualize()"
      },
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      ],
      "text/plain": [
       "Open the notebook in JupyterLab with the atoti extension enabled to build this widget."
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "session.visualize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Risk weights table"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# the risk weights table is created from a csv:\n",
    "eq_delta_risk_weights_table = session.read_csv(\n",
    "    \"s3://data.atoti.io/notebooks/sbm/parameters/eq_delta_rw.csv\",\n",
    "    keys=[\"Bucket\", \"Label2\"],\n",
    "    table_name=\"RiskWeights\",\n",
    ")\n",
    "\n",
    "# risks table is joined with the risk weights table\n",
    "risks_table.join(eq_delta_risk_weights_table)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
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       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th>RW</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bucket</th>\n",
       "      <th>Label2</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <th>SPOT</th>\n",
       "      <td>0.55</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <th>SPOT</th>\n",
       "      <td>0.60</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <th>SPOT</th>\n",
       "      <td>0.45</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <th>SPOT</th>\n",
       "      <td>0.55</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <th>SPOT</th>\n",
       "      <td>0.30</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
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      "text/plain": [
       "                 RW\n",
       "Bucket Label2      \n",
       "1      SPOT    0.55\n",
       "2      SPOT    0.60\n",
       "3      SPOT    0.45\n",
       "4      SPOT    0.55\n",
       "5      SPOT    0.30"
      ]
     },
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   "source": [
    "eq_delta_risk_weights_table.head(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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       "<title>Risks-&gt;RiskWeights</title>\n",
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   ],
   "source": [
    "cube.schema"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can create measures for the numerical columns in the referenced tables such as `RiskWeights` and `RiskFactorCorrelations`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "m[\"RW.VALUE\"] = tt.value(eq_delta_risk_weights_table[\"RW\"])\n",
    "m[\"RW.VALUE\"] = tt.value(eq_delta_risk_weights_table[\"RW\"])\n",
    "# Let's put it into a folder and format as percentages:\n",
    "m[\"RW.VALUE\"].folder = \"Parameters\"\n",
    "m[\"RW.VALUE\"].formatter = \"DOUBLE[#%]\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The risk weights are now available in the cube as a measure.\n",
    "The measure is defined for Bucket and Label2 - these hierarchies need to be present into the view."
   ]
  },
  {
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   "execution_count": 13,
   "metadata": {
    "atoti": {
     "widget": {
      "mapping": {
       "columns": [
        "ALL_MEASURES"
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       "measures": [
        "[Measures].[RW.VALUE]"
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       "rows": [
        "[Risks].[Bucket].[Bucket]",
        "[Risks].[Label2].[Label2]"
       ]
      },
      "name": "",
      "query": {
       "context": {
        "queriesResultLimit.intermediateSize": 1000000,
        "queriesResultLimit.transientSize": 10000000
       },
       "mdx": "SELECT NON EMPTY [Measures].[RW.VALUE] ON COLUMNS, NON EMPTY Crossjoin(Union(Hierarchize(DrilldownLevel([Risks].[Bucket].[ALL].[AllMember])), Hierarchize(Descendants({[Risks].[Bucket].[AllMember]}, 1, SELF_AND_BEFORE))), Hierarchize(Descendants({[Risks].[Label2].[AllMember]}, 1, SELF_AND_BEFORE))) ON ROWS FROM [Risks] CELL PROPERTIES VALUE, FORMATTED_VALUE, BACK_COLOR, FORE_COLOR, FONT_FLAGS",
       "updateMode": "once"
      },
      "serverKey": "default",
      "widgetKey": "pivot-table"
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      ],
      "text/plain": [
       "Open the notebook in JupyterLab with the atoti extension enabled to build this widget."
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "session.visualize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Weighted sensitivities\n",
    "\n",
    "In this section, we'll create a measure to compute weighted sensitivities defined in\n",
    "[MAR21.4](https://www.bis.org/basel_framework/chapter/MAR/21.htm?inforce=20220101#paragraph_MAR_21_20220101_21_4):\n",
    "\n",
    "$$WS_k=RW_k \\cdot s_k$$\n",
    "\n",
    "As the risk weights are defined for each Bucket and Label2, this is the level where sensitivities need to be multiplied by the risk weight.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The input sensitivities are multplied by the risk weight for each Bucket and Label2,\n",
    "# and then summed up to obtain weighted sensitivities:\n",
    "m[\"WS\"] = tt.agg.sum(\n",
    "    m[\"AmountUSD.SUM\"] * m[\"RW.VALUE\"],\n",
    "    scope=tt.scope.origin(l[\"Bucket\"], l[\"Label2\"]),\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "atoti": {
     "widget": {
      "mapping": {
       "columns": [
        "ALL_MEASURES"
       ],
       "measures": [
        "[Measures].[WS]"
       ],
       "rows": [
        "[Risks].[Bucket].[Bucket]"
       ]
      },
      "name": "",
      "query": {
       "context": {
        "queriesResultLimit.intermediateSize": 1000000,
        "queriesResultLimit.transientSize": 10000000
       },
       "mdx": "SELECT NON EMPTY [Measures].[WS] ON COLUMNS, NON EMPTY Union(Hierarchize(DrilldownLevel([Risks].[Bucket].[ALL].[AllMember])), Hierarchize(Descendants({[Risks].[Bucket].[AllMember]}, 1, SELF_AND_BEFORE))) ON ROWS FROM [Risks] CELL PROPERTIES VALUE, FORMATTED_VALUE, BACK_COLOR, FORE_COLOR, FONT_FLAGS",
       "updateMode": "once"
      },
      "serverKey": "default",
      "widgetKey": "pivot-table"
     }
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.atoti.widget.v0+json": {
       "name": null,
       "sessionId": "420477c0-08b8-4a4a-b131-a571557dbc36",
       "sessionLocation": {
        "https": false,
        "port": 59549
       },
       "widgetCreationCode": "session.visualize()"
      },
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      ],
      "text/plain": [
       "Open the notebook in JupyterLab with the atoti extension enabled to build this widget."
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "session.visualize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bucket-level aggregation\n",
    "\n",
    "The weighted sensitivities by risk factor are rolled up into charges by bucket (\"bucket-level charges\"), using a variance-covariance-type of formula that can be found in the [MAR21.4](https://www.bis.org/basel_framework/chapter/MAR/21.htm?inforce=20220101#paragraph_MAR_21_20220101_21_4):\n",
    "\n",
    "$$K_{b} =\\sqrt{max \\left( 0, \\sum _{k\\in b} WS_{k}^{2} +\\sum _{k\\in b}\\sum\n",
    "  _{l\\in b, l\\neq k}\\rho_{kl}\\cdot WS_k \\cdot WS_l\\right)}$$\n",
    "  \n",
    "In this section we'll create measures visualizing bucket-level charges. \n",
    "\n",
    "We will be leveraging the fact, that some of the risk factor pairs are correlated with the same correlation parameter, and provide a more computationally efficient calculation. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Risk factor correlations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The parameter $\\rho_{kl}$ denotes correlation between two risk factors $k$ and $l$ in a pair of risk factors. The rules defining the equity delta correlations are set in [MAR21.78](https://www.bis.org/basel_framework/chapter/MAR/21.htm?inforce=20220101#paragraph_MAR_21_20220101_21_78).\n",
    "\n",
    "The rules can be summarised for each pair of risk factors as follows:\n",
    "\n",
    "- Case 1: same name, different type: a single value -> 0.999\n",
    "- Case 2: different name, same type: a single value depending on bucket, for example, 0.15\n",
    "- Case 3: different name, different type: value depending on risk factor multiplier by 0.999, for example, 0.15 x 0.999\n",
    "\n",
    "Stylized example:\n",
    "\n",
    "\n",
    "| risk factors | name1-spot | name1-repo | name2-spot | name2-repo |\n",
    "|------------|-------------|-------------|-------------|-------------|\n",
    "| name1-spot | 1 |  |  |  |\n",
    "| name1-repo | same_name_diff_type | 1 |  |  |\n",
    "| name2-spot | rho_by_name | rho_by_name x type_multiplier | 1 |  |\n",
    "| name2-repo | rho_by_name x type_multiplier | rho_by_name x type_multiplier | same_name_diff_type | 1 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Equity delta risk factor is defined as a combination of fields - \"Qualifier\" and \"Label2\", i.e. equity name and risk factor type.\n",
    "# Creating variables:\n",
    "same_risk_factor = 1.0\n",
    "same_name_diff_type = 0.999\n",
    "diff_type_multiplier = 0.999"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>names_correlation</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bucket</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1.0</th>\n",
       "      <td>0.15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2.0</th>\n",
       "      <td>0.15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3.0</th>\n",
       "      <td>0.15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4.0</th>\n",
       "      <td>0.15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5.0</th>\n",
       "      <td>0.25</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        names_correlation\n",
       "Bucket                   \n",
       "1.0                  0.15\n",
       "2.0                  0.15\n",
       "3.0                  0.15\n",
       "4.0                  0.15\n",
       "5.0                  0.25"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Creating a table holding correlations defined per bucket ([MAR21.78](2)):\n",
    "eq_delta_rho = session.read_csv(\n",
    "    \"s3://data.atoti.io/notebooks/sbm/parameters/eq_delta_rho.csv\",\n",
    "    keys=[\"Bucket\"],\n",
    "    table_name=\"RiskFactorCorrelations\",\n",
    ")\n",
    "risks_table.join(eq_delta_rho)\n",
    "eq_delta_rho.head(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "m[\"names_correlation.VALUE\"] = tt.value(eq_delta_rho[\"names_correlation\"])\n",
    "m[\"names_correlation.VALUE\"] = tt.value(eq_delta_rho[\"names_correlation\"])\n",
    "# Let's put it into a folder and format as percentages:\n",
    "m[\"names_correlation.VALUE\"].folder = \"Parameters\"\n",
    "m[\"names_correlation.VALUE\"].formatter = \"DOUBLE[0.00]\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "atoti": {
     "widget": {
      "mapping": {
       "columns": [
        "ALL_MEASURES"
       ],
       "measures": [
        "[Measures].[names_correlation.VALUE]"
       ],
       "rows": [
        "[Risks].[Bucket].[Bucket]"
       ]
      },
      "name": "",
      "query": {
       "context": {
        "queriesResultLimit.intermediateSize": 1000000,
        "queriesResultLimit.transientSize": 10000000
       },
       "mdx": "SELECT NON EMPTY [Measures].[names_correlation.VALUE] ON COLUMNS, NON EMPTY Union(Hierarchize(DrilldownLevel([Risks].[Bucket].[ALL].[AllMember])), Hierarchize(Descendants({[Risks].[Bucket].[AllMember]}, 1, SELF_AND_BEFORE))) ON ROWS FROM [Risks] CELL PROPERTIES VALUE, FORMATTED_VALUE, BACK_COLOR, FORE_COLOR, FONT_FLAGS",
       "updateMode": "once"
      },
      "serverKey": "default",
      "widgetKey": "pivot-table"
     }
    }
   },
   "outputs": [
    {
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        "https": false,
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      ],
      "text/plain": [
       "Open the notebook in JupyterLab with the atoti extension enabled to build this widget."
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "session.visualize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since many of the risk factor pairs share the same correlation value, it is possible to optimize the variance-covariance aggregation. The efficiency of this calculation is critical when the data cardinality along the risk factor is high.\n",
    "\n",
    "We decompose the formula into the three components:\n",
    "1. contribution of pairs with both risk factors being spot - same correlation $\\rho_{names}$ set per bucket,\n",
    "2. contribution of pairs with both risk factors being repo - same correlation $\\rho_{names}$ set per bucket,\n",
    "3. contribution of pairs where one risk factor is spot, another one is repo - same correlation $\\rho_{names} \\cdot 0.999$. We'd need to account that for some of the pairs where one risk factor is spot, one is repo some the equity names will match and need to be correlated at 0.999\n",
    "\n",
    "\n",
    "Let's start with the pairs where both risk factors are either spot or repo - cases 1. and 2. above. Since for any $k$ and $l$ the correlation $\\rho_{kl}$ will be equal to correlation defined per bucket $\\rho_{names}$, their contribution can be rewritten:\n",
    "\n",
    "$$ \\sum_k WS_k^2 +  \\sum_k \\sum_{l \\neq k} \\rho_{kl} WS_k WS_l = \\\\ \\sum_k WS_k^2 + \\rho_{names} \\cdot \\left(\\left( \\sum_k WS_k \\right)^2 - \\sum_k WS_k^2  \\right) = \\\\ (1-\\rho_{names}) \\cdot \\sum_k WS_k^2 + \\rho_{names} \\cdot \\left( \\sum_k WS_k \\right)^2  \\label{reduced_formula} \\tag{reduced_formula}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "# This measure will display sum of WS_k squared by risk factor (Qualifier + Label2):\n",
    "m[\"sum squares\"] = tt.agg.square_sum(\n",
    "    m[\"WS\"], scope=tt.scope.origin(l[\"Qualifier\"], l[\"Label2\"])\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "reduced_formula = (1 - m[\"names_correlation.VALUE\"]) * m[\"sum squares\"] + m[\n",
    "    \"names_correlation.VALUE\"\n",
    "] * m[\"WS\"] ** 2\n",
    "\n",
    "# Total contribution of:\n",
    "# - pairs having only spot risk factors,\n",
    "# - pairs having only repo risk factors,\n",
    "# is the sum of the reduced formula by Label2 members.\n",
    "\n",
    "m[\"spot&repo pairs contribution\"] = tt.agg.sum(\n",
    "    reduced_formula, scope=tt.scope.origin(l[\"Label2\"])\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The contribution of the risk factors where one risk factor belongs to \"SPOT\" and the other belongs to \"REPO\" - case 3 above - can be rewritten:\n",
    "\n",
    "$$ \\sum_k WS_k^2 +  \\sum_k \\sum_{l \\neq k} \\rho_{kl} WS_k WS_l = \\\\ \\vec{WS_{repo}^T} \\cdot J_{n_{repo}, n_{spot}} \\cdot \\vec{WS_{spot}} \\cdot \\rho_{names} \\cdot 0.999 + 0.999 \\cdot (1 - \\rho_{names}) \\sum_{n\\in names}{WS_n^{repo} \\cdot WS_n^{spot}} \\label{spot_vs_repo} $$\n",
    "\n",
    "where:\n",
    "- J - is a matrix of ones,\n",
    "- first term in the above formula performs aggregation of all sensitivities, as if they all are correlated at $\\rho_{names}$,\n",
    "- the second term is to correct the first term and to account for the fact that risk factors, where spot and repo risk factors have the same equity name, must be correlated at 0.999.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**First term**\n",
    "\n",
    "The measure `sum WS_repo |J| WS_spot` will display $\\vec{WS_{repo}^T} \\cdot J_{n_{repo}, n_{spot}} \\cdot \\vec{WS_{spot}} $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "# filtering for repo and spot risk factors:\n",
    "m[\"WS_spot\"] = tt.filter(m[\"WS\"], l[\"Label2\"] == \"SPOT\")\n",
    "m[\"WS_repo\"] = tt.filter(m[\"WS\"], l[\"Label2\"] == \"REPO\")\n",
    "\n",
    "# Collect WS of the spot risk factors in a vector and show against every qualifier in a bucket:\n",
    "m[\"weights_vector\"] = tt.agg._vector(\n",
    "    m[\"WS_spot\"], scope=tt.scope.origin(l[\"Bucket\"], l[\"Qualifier\"])\n",
    ")\n",
    "\n",
    "m[\"spot vector\"] = tt.total(m[\"weights_vector\"], h[\"Qualifier\"])\n",
    "\n",
    "# cross product of weighted sensitivities for the names, having both spot and repo sensitivities:\n",
    "# Multiply vector WS of the spot risk factors by WS of each repo risk factor sum them up\n",
    "repo_scalar_x_spot_vector = m[\"WS_repo\"] * m[\"spot vector\"]\n",
    "m[\"sum WS_repo |J| WS_spot\"] = tt.agg.sum(\n",
    "    tt.array.sum(repo_scalar_x_spot_vector),\n",
    "    scope=tt.scope.origin(l[\"Bucket\"], l[\"Qualifier\"]),\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Second term**\n",
    "\n",
    "The measure `sum WS repo and spot` will display $\\sum_{n\\in names}{WS_n^{repo} \\cdot WS_n^{spot}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "m[\"WS_spot_parent\"] = tt.agg.sum(m[\"WS_spot\"], scope=tt.scope.origin(l[\"Label2\"]))\n",
    "m[\"WS_repo_parent\"] = tt.agg.sum(m[\"WS_repo\"], scope=tt.scope.origin(l[\"Label2\"]))\n",
    "\n",
    "\n",
    "m[\"sum WS repo and spot\"] = tt.agg.sum(\n",
    "    m[\"WS_spot_parent\"] * m[\"WS_repo_parent\"],\n",
    "    scope=tt.scope.origin(l[\"Qualifier\"]),\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Combined result cross spot & repo pairs**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# final contribution of the pairs where one risk factor is spot, one is repo\n",
    "m[\"cross repo spot contribution\"] = (\n",
    "    m[\"names_correlation.VALUE\"] * m[\"sum WS_repo |J| WS_spot\"] * 0.999\n",
    "    + 0.999 * (1 - m[\"names_correlation.VALUE\"]) * m[\"sum WS repo and spot\"]\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Total Kb**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "m[\"Kb\"] = tt.where(\n",
    "    l[\"Bucket\"] == 11,\n",
    "    tt.agg.sum(\n",
    "        tt.math.abs(m[\"WS\"]), scope=tt.scope.origin(l[\"Qualifier\"], l[\"Label2\"])\n",
    "    ),\n",
    "    tt.math.sqrt(\n",
    "        tt.math.max(\n",
    "            0, m[\"spot&repo pairs contribution\"] + 2 * m[\"cross repo spot contribution\"]\n",
    "        )\n",
    "    ),\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.atoti.convert-query-result-to-widget.v0+json": {
       "mdx": "SELECT {[Measures].[Kb]} ON COLUMNS, NON EMPTY [Risks].[Bucket].[Bucket].Members ON ROWS FROM [Risks]",
       "sessionId": "420477c0-08b8-4a4a-b131-a571557dbc36",
       "widgetCreationCode": "session.visualize()"
      },
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Kb</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bucket</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2,002,940.32</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1,247,249.60</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>4,757.80</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>758,465.49</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>456,861.06</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>932,682.89</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>3,937,788.92</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>2,234,809.96</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>1,819,315.90</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>6,418,348.71</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>286,992.44</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                  Kb\n",
       "Bucket              \n",
       "1       2,002,940.32\n",
       "2       1,247,249.60\n",
       "3           4,757.80\n",
       "4         758,465.49\n",
       "5         456,861.06\n",
       "7         932,682.89\n",
       "8       3,937,788.92\n",
       "9       2,234,809.96\n",
       "10      1,819,315.90\n",
       "11      6,418,348.71\n",
       "12        286,992.44"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cube.query(m[\"Kb\"], levels=[l[\"Bucket\"]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Cross-bucket aggregation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bucket correlations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "eq_delta_buckets_correlations = session.read_csv(\n",
    "    \"s3://data.atoti.io/notebooks/sbm/parameters/eq_delta_gamma.csv\",\n",
    "    keys=[\"Bucket\", \"Other Bucket\"],\n",
    "    table_name=\"eq_delta_corr_outer\",\n",
    ")\n",
    "risks_table.join(eq_delta_buckets_correlations)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
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       "</g>\n",
       "<!-- Risks&#45;&gt;RiskWeights -->\n",
       "<g id=\"edge2\" class=\"edge\">\n",
       "<title>Risks-&gt;RiskWeights</title>\n",
       "<path fill=\"none\" stroke=\"black\" d=\"M173.25,-177C220.77,-177 279.51,-177 326.29,-177\"/>\n",
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       "<!-- eq_delta_corr_outer -->\n",
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       "<title>eq_delta_corr_outer</title>\n",
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       "<IPython.core.display.SVG object>"
      ]
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     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "cube.schema"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, let's create the measure for gamma from the referenced table `eq_delta_corr_outer table`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "m[\"gamma.VALUE\"] = tt.value(eq_delta_buckets_correlations[\"gamma\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "atoti": {
     "widget": {
      "mapping": {
       "columns": [
        "[eq_delta_corr_outer].[Other Bucket].[Other Bucket]",
        "ALL_MEASURES"
       ],
       "measures": [
        "[Measures].[gamma.VALUE]"
       ],
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        "[Risks].[Bucket].[Bucket]"
       ]
      },
      "query": {
       "context": {
        "queriesResultLimit.intermediateSize": 1000000,
        "queriesResultLimit.transientSize": 10000000
       },
       "mdx": "SELECT NON EMPTY Crossjoin(Hierarchize(Descendants({[eq_delta_corr_outer].[Other Bucket].[AllMember]}, 1, SELF_AND_BEFORE)), {[Measures].[gamma.VALUE]}) ON COLUMNS, NON EMPTY Hierarchize(Descendants({[Risks].[Bucket].[AllMember]}, 1, SELF_AND_BEFORE)) ON ROWS FROM [Risks]",
       "updateMode": "once"
      },
      "serverKey": "default",
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    {
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20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20252px%3B%20width%3A%20120px%3B%20background-color%3A%20rgb(250%2C%20250%2C%20250)%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.00%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%201080px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20252px%3B%20width%3A%20120px%3B%20background-color%3A%20rgb(250%2C%20250%2C%20250)%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.00%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%201200px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20252px%3B%20width%3A%20120px%3B%20background-color%3A%20rgb(250%2C%20250%2C%20250)%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.00%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%201320px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20252px%3B%20width%3A%20120px%3B%20background-color%3A%20rgb(250%2C%20250%2C%20250)%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.00%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%200px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20120px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20240px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20360px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20480px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20600px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20720px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20840px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20960px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%201080px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.45%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%201200px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.00%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%201320px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%20280px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E.00%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3Cdiv%20class%3D%22css-1uww2vi-Table%22%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2FforeignObject%3E%3C%2Fsvg%3E\">"
      ],
      "text/plain": [
       "Open the notebook in JupyterLab with the atoti extension enabled to build this widget."
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "session.visualize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Aggregating across buckets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 21.4(5)(a):\n",
    "m[\"WSb\"] = m[\"WS\"]\n",
    "m[\"WSc\"] = tt.at(m[\"WS\"], {l[\"Bucket\"]: l[\"Other Bucket\"]})\n",
    "m[\"sum Kb2 + sum sum WSb WSc gamma\"] = tt.agg.square_sum(\n",
    "    m[\"Kb\"], scope=tt.scope.origin(l[\"Bucket\"])\n",
    ") + tt.agg.sum(\n",
    "    m[\"WSb\"] * m[\"WSc\"] * m[\"gamma.VALUE\"],\n",
    "    scope=tt.scope.origin(l[\"Bucket\"], l[\"Other Bucket\"]),\n",
    ")\n",
    "\n",
    "# 21.4(5)(b):\n",
    "m[\"Sb\"] = tt.math.max(tt.math.min(m[\"WS\"], m[\"Kb\"]), -1.0 * m[\"Kb\"])\n",
    "m[\"Sc\"] = tt.at(m[\"Sb\"], {l[\"Bucket\"]: l[\"Other Bucket\"]})\n",
    "m[\"sum Kb2 + sum sum Sb Sc gamma\"] = tt.agg.square_sum(\n",
    "    m[\"Kb\"], scope=tt.scope.origin(l[\"Bucket\"])\n",
    ") + tt.agg.sum(\n",
    "    m[\"Sb\"] * m[\"Sc\"] * m[\"gamma.VALUE\"],\n",
    "    scope=tt.scope.origin(l[\"Bucket\"], l[\"Other Bucket\"]),\n",
    ")\n",
    "\n",
    "m[\"Risk Charge\"] = tt.where(\n",
    "    m[\"sum Kb2 + sum sum WSb WSc gamma\"] > 0,\n",
    "    tt.math.sqrt(m[\"sum Kb2 + sum sum WSb WSc gamma\"]),\n",
    "    tt.math.sqrt(m[\"sum Kb2 + sum sum Sb Sc gamma\"]),\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.atoti.convert-query-result-to-widget.v0+json": {
       "mdx": "SELECT {[Measures].[Risk Charge]} ON COLUMNS FROM [Risks]",
       "sessionId": "420477c0-08b8-4a4a-b131-a571557dbc36",
       "widgetCreationCode": "session.visualize()"
      },
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Risk Charge</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>9,357,880.51</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    Risk Charge\n",
       "0  9,357,880.51"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cube.query(m[\"Risk Charge\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Enriching the data model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The current data model can be displayed using `cube.schema` command."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"520px\" height=\"338px\" viewBox=\"0.00 0.00 520.00 338.00\">\n",
       "<g id=\"graph0\" class=\"graph\" transform=\"scale(1.0 1.0) rotate(0) translate(4 334)\">\n",
       "<title>Risks schema</title>\n",
       "<polygon fill=\"white\" stroke=\"transparent\" points=\"-4,4 -4,-334 516,-334 516,4 -4,4\"/>\n",
       "<!-- Risks -->\n",
       "<g id=\"node1\" class=\"node\">\n",
       "<title>Risks</title>\n",
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     "execution_count": 33,
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   "source": [
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's group trades into books, desks, business units:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Trade attributes table will hold trade identifiers and trade attributes, in our example, book\n",
    "trade_attributes = session.read_csv(\n",
    "    \"s3://data.atoti.io/notebooks/sbm/trade_attributes.csv\",\n",
    "    keys=[\"TradeID\"],\n",
    "    table_name=\"Trade Attributes\",\n",
    ")\n",
    "\n",
    "# Trade desks table will hold multi-level org structure\n",
    "trading_desks = session.read_csv(\n",
    "    \"s3://data.atoti.io/notebooks/sbm/trading_desk.csv\",\n",
    "    keys=[\"Book\"],\n",
    "    table_name=\"Trading Desk\",\n",
    ")\n",
    "\n",
    "# Linking together the data\n",
    "risks_table.join(trade_attributes)\n",
    "trade_attributes.join(trading_desks)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Setting up a multi-level hierarchy book -> desk -> sub-unit -> unit\n",
    "h[\"Trading Book Hierarchy\"] = {\n",
    "    \"Business Unit\": l[\"business_unit\"],\n",
    "    \"Sub Business Unit\": l[\"sub_business_unit\"],\n",
    "    \"Trading Desk\": l[\"trading_desk\"],\n",
    "    \"BookID\": l[\"Book\"],\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The new schema is as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
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       "<text text-anchor=\"middle\" x=\"245\" y=\"-214.6\" font-family=\"-apple-system, BlinkMacSystemFont, 'Segoe UI', Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol'\" font-size=\"13.00\">Label2 → Label2</text>\n",
       "</g>\n",
       "<!-- eq_delta_corr_outer -->\n",
       "<g id=\"node6\" class=\"node\">\n",
       "<title>eq_delta_corr_outer</title>\n",
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       "<polygon fill=\"none\" stroke=\"black\" points=\"339.5,-4 339.5,-102 501.5,-102 501.5,-4 339.5,-4\"/>\n",
       "</g>\n",
       "<!-- Risks&#45;&gt;eq_delta_corr_outer -->\n",
       "<g id=\"edge1\" class=\"edge\">\n",
       "<title>Risks-&gt;eq_delta_corr_outer</title>\n",
       "<path fill=\"none\" stroke=\"black\" stroke-dasharray=\"5,2\" d=\"M173.03,-185.81C179.11,-182.49 185.14,-179.19 191,-176 234.04,-152.57 281.75,-126.84 322.4,-105\"/>\n",
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       "</g>\n",
       "<!-- Trading Desk -->\n",
       "<g id=\"node4\" class=\"node\">\n",
       "<title>Trading Desk</title>\n",
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       "</g>\n",
       "<!-- Trade Attributes&#45;&gt;Trading Desk -->\n",
       "<g id=\"edge5\" class=\"edge\">\n",
       "<title>Trade Attributes-&gt;Trading Desk</title>\n",
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       "</g>\n",
       "</g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cube.schema"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can compute SBM interactively for various desks and books and drill down to sensitivities by trade and risk factor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "atoti": {
     "height": 229,
     "widget": {
      "mapping": {
       "columns": [
        "ALL_MEASURES"
       ],
       "measures": [
        "[Measures].[Risk Charge]",
        "[Measures].[WS]",
        "[Measures].[AmountUSD.MEAN]"
       ],
       "rows": [
        "[Trading Desk].[Trading Book Hierarchy].[Business Unit] => [Trading Desk].[Trading Book Hierarchy].[Sub Business Unit]"
       ]
      },
      "query": {
       "mdx": "SELECT NON EMPTY {[Measures].[Risk Charge], [Measures].[WS], [Measures].[AmountUSD.MEAN]} ON COLUMNS, NON EMPTY Hierarchize(Union(Descendants({[Trading Desk].[Trading Book Hierarchy].[AllMember]}, 1, SELF_AND_BEFORE), Descendants({[Trading Desk].[Trading Book Hierarchy].[AllMember].[Equity]}, [Trading Desk].[Trading Book Hierarchy].[Sub Business Unit]))) ON ROWS FROM [Risks]",
       "updateMode": "once"
      },
      "serverKey": "default",
      "widgetKey": "pivot-table"
     }
    }
   },
   "outputs": [
    {
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       "widgetCreationCode": "session.visualize()"
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ell%22%3E%3Cdiv%20style%3D%22overflow%3A%20hidden%3B%20display%3A%20flex%3B%20align-items%3A%20center%3B%20font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%200px%3B%20padding%3A%204px%204px%203px%3B%20position%3A%20absolute%3B%20text-align%3A%20left%3B%20top%3A%200px%3B%20width%3A%20115px%3B%20font-weight%3A%20600%3B%22%3E%3Cspan%20style%3D%22text-overflow%3A%20ellipsis%3B%20overflow%3A%20hidden%3B%20white-space%3A%20nowrap%3B%20max-width%3A%20calc((100%25%20-%200px)%20-%200px)%3B%22%3EAmountUSD.MEAN%3C%2Fspan%3E%3C%2Fdiv%3E%3Cdiv%20class%3D%22css-xxcrt1-ResizableCell%22%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3C%2Fdiv%3E%3Cdiv%20class%3D%22%20css-11w18m2%22%3E%3Cdiv%20class%3D%22css-14n5d26%22%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%200px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%200px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E9%2C357%2C880.51%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20120px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%200px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E-5%2C090%2C583.39%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20240px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%200px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E-8%2C334.92%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%200px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%2028px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%20border-top%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E5%2C215%2C021.72%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20120px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%2028px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%20border-top%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E-1%2C068%2C726.24%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20240px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%2028px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%20border-top%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E-6%2C850.56%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%200px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%2056px%3B%20width%3A%20120px%3B%20background-color%3A%20rgb(250%2C%20250%2C%20250)%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E3%2C481%2C758.56%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20120px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%2056px%3B%20width%3A%20120px%3B%20background-color%3A%20rgb(250%2C%20250%2C%20250)%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E-2%2C896%2C373.87%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%20240px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%2056px%3B%20width%3A%20120px%3B%20background-color%3A%20rgb(250%2C%20250%2C%20250)%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E-18%2C641.52%3C%2Fdiv%3E%3Cdiv%20class%3D%22aui-table-cell%22%20style%3D%22font-size%3A%2012px%3B%20height%3A%2028px%3B%20left%3A%200px%3B%20padding%3A%204px%204px%203px%3B%20overflow%3A%20hidden%3B%20position%3A%20absolute%3B%20text-align%3A%20right%3B%20top%3A%2084px%3B%20width%3A%20120px%3B%20background-color%3A%20transparent%3B%20border-right%3A%201px%20solid%20rgb(233%2C%20233%2C%20233)%3B%22%3E2%2C302%2C194.33%3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      ],
      "text/plain": [
       "Open the notebook in JupyterLab with the atoti extension enabled to build this widget."
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "session.visualize()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "atoti": {
     "widget": {
      "mapping": {
       "horizontalSubplots": [],
       "splitBy": [
        "ALL_MEASURES"
       ],
       "values": [
        "[Measures].[Risk Charge]"
       ],
       "verticalSubplots": [],
       "yAxis": [
        "[Risks].[Bucket].[Bucket]"
       ]
      },
      "query": {
       "mdx": "SELECT NON EMPTY {[Measures].[Risk Charge]} ON COLUMNS, NON EMPTY Hierarchize(Descendants({[Risks].[Bucket].[AllMember]}, 1, SELF_AND_BEFORE)) ON ROWS FROM [Risks]",
       "updateMode": "once"
      },
      "serverKey": "default",
      "widgetKey": "plotly-clustered-bar-chart"
     }
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.atoti.widget.v0+json": {
       "name": "SBM charge by Bucket",
       "sessionId": "420477c0-08b8-4a4a-b131-a571557dbc36",
       "sessionLocation": {
        "https": false,
        "port": 59549
       },
       "widgetCreationCode": "session.visualize()"
      },
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