Lets apply the lessons from the book to real world data using the data from the last 936 trading days from Oslo Børs. That means we have 183 instruments
Note: building mlfinlab
$ pip install -r requirements.txt
$ pylint mlfinlab --rcfile=.pylintrc -f text
$ bash coverage
Runs the unit-tests. Alternative:
pip install pytest
pip install coverage
pip install pytest-cov
pytest --cov=foo --cov=bar
S_value contains the time-series of instruments. One instrument per column.
>>> S_value
array([[ 7.3311, 94.83 , 3.4349, ..., 90.03 , 234.85 , 27.6 ],
[ 7.1928, 94.83 , 3.4139, ..., 90.43 , 238.69 , 29.15 ],
[ 7.1005, 94.83 , 3.4069, ..., 89.63 , 239.58 , 29.15 ],
...,
[ 1.36 , 112. , 4.22 , ..., 21.6 , 337.7 , 34. ],
[ 1.465 , 112. , 4.315 , ..., 21.62 , 334. , 33.7 ],
[ 1.38 , 112. , 4.33 , ..., 22.48 , 331.2 , 33.5 ]])
>>> S_value.shape
(936L, 183L)
There is a corresponding list of portfolio names:
>>> portfolio_name[0:3]
['FIVEPG.ol', 'AASB-ME.ol', 'ASC.ol']
>>> portfolio_name[-3:]
['XXL.ol', 'YAR.ol', 'ZAL.ol']
The time-series of values must be converted to returns.
>>> S, instrument_returns = calculate_returns(S_value)
>>> S
array([[-0.01886484, 0. , -0.00611372, ..., 0.00444296,
0.01635086, 0.05615942],
[-0.01283228, 0. , -0.00205044, ..., -0.00884662,
0.00372869, 0. ],
[ 0. , 0. , 0.00616396, ..., 0.00892558,
0.00784707, -0.01989708],
...,
[-0.13375796, -0.00884956, 0.0047619 , ..., -0.02262443,
-0.0088054 , 0. ],
[ 0.07720588, 0. , 0.02251185, ..., 0.00092593,
-0.01095647, -0.00882353],
[-0.05802048, 0. , 0.00347625, ..., 0.03977798,
-0.00838323, -0.00593472]])
Fitting the covariance matrix to the marcenko-pasture distribution assumes the elements of the matrix is normal distribution with mean 0 and variance:
>>>eVal0, eVec0, denoised_eVal, denoised_eVec, denoised_corr, var0 = denoise_OL(S)
>>> var0
0.8858105886313286
Fitting denoised_corr to the marcenko-pasture pdf:
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|---|
| Marcenko-Pasture theoretical probability density function, and empirical density function from 183 instruments the last 936 trading days: |
The largest eigenvalues of 24.02 is the market and can be de-toned.
The eigenvalues before and after denoising:
 |
|---|
| A comparison of eigenvalues before and after applying the residual eigenvalue method: |
The minimum-variance-portfolio with a long/short strategy gives VVL.OL as biggest long, and STB.OL as biggest short:
>>> np.argsort(w)
array([158, 106, 151, 38, 23, 22, 77, 10, 6, 54, 47, 111, 7,
46, 137, 8, 175, 148, 181, 5, 157, 165, 150, 62, 13, 119,
161, 39, 169, 114, 2, 18, 96, 30, 103, 172, 108, 16, 115,
4, 92, 9, 83, 97, 28, 29, 112, 166, 155, 102, 33, 40,
79, 156, 66, 94, 24, 100, 81, 76, 176, 31, 48, 153, 95,
122, 88, 87, 125, 80, 141, 43, 68, 72, 20, 160, 57, 73,
104, 127, 74, 85, 36, 180, 99, 55, 130, 121, 15, 98, 152,
154, 162, 138, 86, 147, 168, 49, 41, 136, 93, 135, 116, 91,
105, 134, 113, 14, 146, 145, 129, 19, 139, 118, 61, 82, 164,
12, 143, 133, 17, 32, 124, 56, 182, 35, 173, 132, 101, 42,
89, 53, 52, 44, 25, 1, 84, 90, 177, 21, 34, 171, 126,
117, 11, 140, 123, 128, 170, 0, 71, 58, 107, 60, 26, 120,
110, 51, 109, 45, 163, 63, 78, 75, 64, 149, 59, 50, 70,
159, 167, 179, 178, 142, 67, 65, 144, 3, 37, 27, 69, 131,
174], dtype=int64)
>>> portfolio_name[174]
'VVL.ol'
>>> portfolio_name[158]
'STB.ol'
Running the ONC algorith on Oslo Børs with 183 instruments and time frame 934:
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|---|
| * There are 2 clusters. A small cluster of 8 instruments which are the salmon farming companies like AUSS.ol, BAKKA.ol, GSF.ol, LSG.ol and SALM.ol: * |
Modern portfolio theory, or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It uses the variance of asset prices as a proxy for risk.
Lets compare markowitz minimum variance portfolio with minimum variance portfolio from NCO algorithm on a small worked example of 5 stock time series.
| time-step/stock | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| t.1 | 1 | 1 | 3 | 4 | 5 |
| t.2 | 1.1 | 1 | 2 | 3 | 5 |
| t.3 | 1.2 | 1 | 1.3 | 4 | 5 |
| t.4 | 1.3 | 1 | 1 | 3 | 5 |
| t.5 | 1.4 | 1 | 1 | 4 | 5.5 |
| t.6 | 1.5 | 1.1 | 1 | 3 | 5.5 |
Covariance matrixes are generally evalueted on returns and not price as - even random walks has spurious correlations. While returns are stationary (mean 0). Therefore correlation is calculated on returns:
| time-step/stock | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| t.1 | - | - | - | - | - |
| t.2 | .1 | 0 | -.33 | -.25 | 0 |
| t.3 | .09 | 0 | -.35 | .33 | 0 |
| t.4 | .08 | 0 | 0 | -.25 | 0 |
| t.5 | .08 | 0 | 0 | .33 | 0.1 |
| t.6 | .07 | .1 | 0 | -.25 | 0 |
Return Covariance matrix:
| stock/stock | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 0 | 0.00012774 | -0.00032726 | -0.00178085 | -0.0001758 | -0.00018989 |
| 1 | -0.00032726 | 0.002 | 0.00457051 | -0.00583333 | -0.0005 |
| 2 | -0.00178085 | 0.00457051 | 0.02993721 | 0.00228098 | 0.00457051 |
| 3 | -0.0001758 | -0.00583333 | 0.00228098 | 0.10208333 | 0.00875 |
| 4 | -0.00018989 | -0.0005 | 0.00457051 | 0.00875 | 0.002 |
Return on investment and volatility (std)
| function/stock | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| ROI | .5 | .1 | -.67 | -.25 | .1 |
| Volatility | .17 | .04 | .74 | .5 | .24 |
| NCO allocations | 0.92342949 | 0.04993295 | 0.04785576 | 0.00165416 | -0.02287237 |
| Markowitz alloc | 2.00849581 | -0.51407928 | 0.32638057 | 0.04062639 | -0.86142349 |
| M Expected ret | -0.0250346 | 0.111758771 | 0.009465804 | -0.01020999 | -0.009415984 |
| M Expected sum | 0.0765640 | - | - | - | - |
| M volatility | 0.020303852 | -0.02056317 | 0.050509720 | 0.011609958 | -0.034456939 |
| M sum vol | 0.02740342 | ||||
| NCO Expected ret | 0.00173010 | 0.124913494 | -0.03988751 | -0.14265565 | 0.0003460207 |
| NCO Expected sum | -0.05555356 | - | - | - | - |
| NCO volatility | 0.009334934 | 0.001997318 | 0.007406020 | 0.000472716 | -0.000914895 |
| NCO sum vol | 0.018296094 |
Expected return of NCO (-0.06) is lower than markowitz (0.08), as expected. Minimum variance Markowitz volatility (0.02) is higher than NCO volatility(0.01), which is unexpected.