# Chapter 19 Solution to exercises

## 19.1 Chapter 4

For annual values:

For monthly values:

Portfolios based on quartiles, using the tidyverse only. We rely heavily on the fact that features are uniforimized, i.e., that their distribution is uniform for each given date. Overall, small firms outperform heavily.

## 19.2 Chapter 5

Below, we import a credit spread supplied by Bank of America. Its symbol/ticker is “BAMLC0A0CM”. We apply the data expansion on the small number of predictors to save memory space. One important trick that should not be overlooked is the uniformization step after the product (5.3) is computed. Indeed, we want the new features to have the same properties as the old ones. If we skip this step, distributions will be altered, as we show in one example below.

We start with the data extraction and joining. It’s important to join early so as to keep the highest data frequency (daily) in order to replace missing points with close values. Joining with monthly data before replacing creates unnecessary lags.

## [1] "BAMLC0A0CM"

The creation of the augmented dataset requires some manipulation.

To prevent this issue, uniformization is required.

The second question naturally requires the downloading of VIX series first and the joining with the original data.

## [1] "VIXCLS"

We can then proceed with the categorization. We create the vector label in a new (smaller) dataset but not attached to the large data_ml variable. Also, we check the balance of labels and its evolution through time.

Finally, we switch to the outliers.

Returns above 50 should indeed be rare.

## # A tibble: 8 x 3
##   stock_id date       R12M_Usd
##      <int> <date>        <dbl>
## 1      212 2000-12-31     53.0
## 2      221 2008-12-31     53.5
## 3      221 2009-01-31     55.2
## 4      221 2009-02-28     54.8
## 5      296 2002-06-30     72.2
## 6      683 2009-02-28     96.0
## 7      683 2009-03-31     64.8
## 8      862 2009-02-28     58.0

The largest return comes from stock #683. Let’s have a look at the stream of monthly returns in 2009.

## # A tibble: 12 x 2
##    date       R1M_Usd
##    <date>       <dbl>
##  1 2009-01-31  -0.625
##  2 2009-02-28   0.472
##  3 2009-03-31   1.44
##  4 2009-04-30   0.139
##  5 2009-05-31   0.086
##  6 2009-06-30   0.185
##  7 2009-07-31   0.363
##  8 2009-08-31   0.103
##  9 2009-09-30   9.91
## 10 2009-10-31   0.101
## 11 2009-11-30   0.202
## 12 2009-12-31  -0.251

The returns are all very high. The annual value is plausible. In addition, a quick glance at the Vol1Y values show that the stock is the most volatile of the dataset.

## 19.4 Chapter 7

## [1] 0.04018973
## [1] 0.03699696

The first model is too precise: going into the details of the training sample does not translate to good performance out-of-sample. The second, simpler model, yields better results.

## [1] 0.03967754 0.03885924 0.03766900 0.03696370 0.03699772

Trees are by definition random so results can vary from test to test. Overall, large number of trees are preferable and the reason is that each new tree tell a new story and diversifies the risk of the whole forest. Some more technical details of why that may be the case are outlined in the original paper Breiman (2001).

For the last exercises, we recycle the formula used in Chapter 7.

The first splitting criterion is enterprise value (EV). EV is an indicator that adjusts market capitalization by substracting debt and adding cash. It is a more faithful account of the true value of a company. In 2008, the companies that fared the least bad where those with the highest EV (i.e., large, robust firms).

In 2009, the firms that recovered the fastest were those that experienced high volatility in the past (likely, downwards volatility). Momentum is also very important: the firms with the lowest past returns are those that rebound the fastest. This is a typical example of the momentum crash phenomenon studied in Barroso and Santa-Clara (2015) and Daniel and Moskowitz (2016). The rationale is the following: after a market downturn, the stock with the most potential for growth are those that have suffered the largest losses. Consequently, the negative (short) leg of the momentum factor performs very well, often better than the long leg. And indeed, being long in the momentum factor in 2009 would have generated negative profits.

## 19.5 Chapter 8: the autoencoder model

First, it is imperative to format the inputs properly. To avoid any issues, we work with perfectly rectangular data and hence restrict the investment set to the stocks with no missing points. Dimensions must also be in the correct order.

Next, we turn to the specification of the network, using a functional API form.

Finally, we ask for the structure of the model, and train it.

## Model: "model_5"
## __________________________________________________________________________________________
## Layer (type)                 Output Shape        Param #    Connected to
## ==========================================================================================
## aux_input (InputLayer)       [(None, 793, 7)]    0
## __________________________________________________________________________________________
## layer_1_l (Dense)            (None, 793, 8)      64         aux_input[0][0]
## __________________________________________________________________________________________
## main_input (InputLayer)      [(None, 793)]       0
## __________________________________________________________________________________________
## layer_2_l (Dense)            (None, 793, 4)      36         layer_1_l[0][0]
## __________________________________________________________________________________________
## layer_1_r (Dense)            (None, 8)           6352       main_input[0][0]
## __________________________________________________________________________________________
## layer_3_l (Permute)          (None, 4, 793)      0          layer_2_l[0][0]
## __________________________________________________________________________________________
## layer_2_r (Dense)            (None, 4)           36         layer_1_r[0][0]
## __________________________________________________________________________________________
## main_output (Dot)            (None, 793)         0          layer_3_l[0][0]
##                                                             layer_2_r[0][0]
## ==========================================================================================
## Total params: 6,488
## Trainable params: 6,488
## Non-trainable params: 0
## __________________________________________________________________________________________

## 19.6 Chapter 9

Since we are going to reproduce a similar analysis several times, let’s simplify the task with 2 tips. First, by using default parameter values that will be passed as common arguments to the svm function. Second, by creating a custom function that computes the MSE. Third, by resorting to functional calculus via the map function from the purrr package. Below, we recycle datasets created in Chapter 7.

## [[1]]
## [1] 0.03849786
##
## [[2]]
## [1] 0.03924576
##
## [[3]]
## [1] 0.03951328
##
## [[4]]
## [1] 334.8173

The first two kernels yield the best fit while the last one should be avoided. Note that apart from the linear kernel, all other options require parameters. We have used the default ones, which may explain the poor performance of some nonlinear kernels.

Below, we train an SVM model on a training sample with all observations but limited to the 7 major predictors. Even with a smaller number of features, the training is time consuming.

## [1] 0.490343

This figure is very low. Below, we test a very simple form of boosted trees, for comparison purposes.

## [1] 0.5017377

The forecasts are slightly better, but the computation time is lower. Two reasons why the models perform poorly:
1. there are not enough predictors;
2. the models are static: they do not adjust dynamically to macro conditions.

## 19.7 Chapter 12: ensemble neural network

First, we create the three feature sets. The first set gets all multiples of 3 between 3 and 93. The second one, the same indices, minus one and the third one, the initial indices minus two.

Then, we specify the network structure. First, the 3 independent networks, then the aggregation.

Lastly, we can train and evaluate.

## Model: "model_6"
## __________________________________________________________________________________________
## Layer (type)                 Output Shape        Param #    Connected to
## ==========================================================================================
## first_input (InputLayer)     [(None, 31)]        0
## __________________________________________________________________________________________
## second_input (InputLayer)    [(None, 31)]        0
## __________________________________________________________________________________________
## third_input (InputLayer)     [(None, 31)]        0
## __________________________________________________________________________________________
## layer_1 (Dense)              (None, 8)           256        first_input[0][0]
## __________________________________________________________________________________________
## layer_2 (Dense)              (None, 8)           256        second_input[0][0]
## __________________________________________________________________________________________
## layer_3 (Dense)              (None, 8)           256        third_input[0][0]
## __________________________________________________________________________________________
## dense_45 (Dense)             (None, 2)           18         layer_1[0][0]
## __________________________________________________________________________________________
## dense_46 (Dense)             (None, 2)           18         layer_2[0][0]
## __________________________________________________________________________________________
## dense_47 (Dense)             (None, 2)           18         layer_3[0][0]
## __________________________________________________________________________________________
## concatenate_1 (Concatenate)  (None, 6)           0          dense_45[0][0]
##                                                             dense_46[0][0]
##                                                             dense_47[0][0]
## __________________________________________________________________________________________
## main_output (Dense)          (None, 2)           14         concatenate_1[0][0]
## ==========================================================================================
## Total params: 836
## Trainable params: 836
## Non-trainable params: 0
## __________________________________________________________________________________________

## 19.8 Chapter 13

### 19.8.1 EW portfolios with the tidyverse

This one is incredibly easy, it’s simpler and more compact but close in spirit to the code that generates Figure 4.1.

First, we code the function with all inputs.

Second, we test it on some random dataset. We use the returns created at the end of Chapter 2 and used for the Lasso allocation in Section 6.2.2. For $$\boldsymbol{\mu}$$, we use the sample average, which is rarely a good idea in practice. It serves as illustration only.

##             [,1]
## 1   0.0031339308
## 3  -0.0003243527
## 4   0.0011944677
## 7   0.0014194215
## 9   0.0015086240
## 11 -0.0005015207

Some weights can of course be negative. Finally, we use the map2() function to test some sensitivity. We examine 3 key indicators:
- diversification, which we measure via the inverse of the sum of squared weights (inverse Hirschman-Herfindhal index);
- leverage, which we assess via the absolute sum of negative weights;
- in-sample volatility, which we compute as $$\textbf{w}' \boldsymbol{\Sigma} \textbf{x}$$

To do so, we create a dedicated function below.

Instead of using the baseline map2 function, we rely on a version thereof that concatenates results into a dataframe directly.

Each panel displays an indicator. In the first panel, we see that diversification increases with $$k_D$$: indeed, as this number increases, the portfolio converges to uniform (EW) values. The parameter $$\lambda$$ has a minor impact. The second panel naturally shows the inverse effect for leverage: as diversification increases with $$k_D$$, leverage (i.e., total negative positions - shortsales) decreases. Finally, the last panel shows that in-sample volatility is however largely driven by the risk aversion parameter. As $$\lambda$$ increases, volatility logically decreases. For small values of $$\lambda$$, $$k_D$$ is negatively related to volatility but the pattern reverses for large values of $$\lambda$$. This is because the equally-weighted portfolio is less risky than very leveraged mean-variance policies, but more risky than the minimum-variance portfolio.

### 19.8.3 Functional programming in the backtest

Often, programmers prefer to avoid loops. In order to avoid a loop in the backtest, we need to code what happens for one given date. This is encapsulated in the following function. For simplicity, we code it for only one strategy. Also, the function will assume the structure of the data is known, but the columns (features & labels) could also be passed as arguments. We recycle the function weights_xgb from Chapter 13.

Next, we combine this function to map(). We only test the first 6 dates: this reduces the computation times.

## [1] 0.001675042 0.000000000 0.000000000 0.001675042 0.000000000 0.001675042
## [1] 0.0189129

Each element of backtest is a list with two components: the portfolio weights and the returns. To access the data easily, functions like melt from the package reshape2 are useful.

## 19.9 Chapter 16

We recycle the AE model trained in Chapter 16. Strangely, building smaller models (encoder) from larger ones (AE) requires to save and then reload the weights. This creates an external file, which we call “ae_weights”. We can check that the output does have 4 columns (compressed) instead of 7 (original data).

## tf.Tensor(
## [[-0.9493618  -0.27035978  0.19658312 -0.8683738 ]
##  [-0.9410028  -0.279021    0.19459371 -0.88132095]
##  [-0.9250736  -0.27920857  0.15385853 -0.85924625]
##  [-0.92141783 -0.28385705  0.14627664 -0.85949975]
##  [-0.9247371  -0.29159465  0.14994244 -0.8663718 ]], shape=(5, 4), dtype=float32)

## 19.10 Chapter 17

All we need to do is change the rho coefficient in the code of Chapter 17.

The learning can then proceed.

## State-Action function Q
##          0.25         0         1      0.75       0.5
## neg 0.7107268 0.5971710 1.4662416 0.9535698 0.8069591
## pos 0.7730842 0.7869229 0.4734467 0.4258593 0.6257039
##
## Policy
## neg pos
## "1" "0"
##
## Reward (last iteration)
## [1] 3013.162

In this case, the constantly switching feature of the return process changes the outcome. The negative state is associated with large profits when the portfolio is fully invested while the positive state has the best average reward when the agent refrains from investing.

For the second exercise, the trick is to define all possible actions, that is all combinations (+1,0-1) for the two assets on all dates. We recycle the data from Chapter 17.

We can the plug this data into the RL function.

## State-Action function Q
##        0 0    0 1  0 -1  -1 -1   -1 0   -1 1   1 -1    1 0    1 1
## X0.2 0.000  0.000 0.002 -0.017 -0.018 -0.020  0.023  0.025  0.024
## X0.3 0.001 -0.005 0.007 -0.013 -0.019 -0.026  0.031  0.027  0.021
## X3.1 0.003  0.003 0.003  0.002  0.002  0.003  0.002  0.002  0.003
## X2.1 0.027  0.038 0.020  0.004  0.015  0.039  0.013  0.021  0.041
## X2.2 0.021  0.014 0.027  0.038  0.047  0.045 -0.004 -0.011 -0.016
## X2.3 0.007  0.006 0.008  0.054  0.057  0.056 -0.041 -0.041 -0.041
## X1.1 0.027  0.054 0.005 -0.031 -0.005  0.041  0.025  0.046  0.072
## X1.2 0.019  0.020 0.020  0.015  0.023  0.029  0.012  0.014  0.023
## X1.3 0.008  0.019 0.000 -0.036 -0.027 -0.016  0.042  0.053  0.060
##
## Policy
##   X0.2   X0.3   X3.1   X2.1   X2.2   X2.3   X1.1   X1.2   X1.3
##  "1 0" "1 -1" "0 -1"  "1 1" "-1 0" "-1 0"  "1 1" "-1 1"  "1 1"
##
## Reward (last iteration)
## [1] 0

The matrix is less sparse compared to the one of Chapter 17: we have covered much more ground! Some policy recommendations have not changed compared to the smaller sample, but some have! The change occur for the states for which only a few points were available in the first trial. With more data, the decision is altered.

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