Chapter 18 Solutions to exercises

18.1 Chapter 3

For annual values, see 18.1:

data_ml %>%
    group_by(date) %>%                                            
    mutate(growth = Pb > median(Pb)) %>%            # Creates the sort
    ungroup() %>%                                   # Ungroup
    mutate(year = lubridate::year(date)) %>%        # Creates a year variable
    group_by(year, growth) %>%                      # Analyze by year & sort
    summarize(ret = mean(R1M_Usd)) %>%              # Compute average return
    ggplot(aes(x = year, y = ret, fill = growth)) + geom_col(position = "dodge") + # Plot!
    theme(legend.position = c(0.7, 0.8))
The value factor: annual returns.

FIGURE 18.1: The value factor: annual returns.

For monthly values, see 18.2:

returns_m <- data_ml %>%
    group_by(date) %>%                                            
    mutate(growth = Pb > median(Pb)) %>%                         # Creates the sort
    group_by(date, growth) %>%                                   # Analyze by date & sort
    summarize(ret = mean(R1M_Usd)) %>%                           # Compute average return
    spread(key = growth, value = ret) %>%                        # Pivot to wide matrix format
colnames(returns_m)[2:3] <- c("value", "growth")                 # Changing column names
returns_m %>%
    mutate(value = cumprod(1 + value),                           # From returns to portf. values
           growth = cumprod(1 + growth)) %>%
    gather(key = portfolio, value = value, -date) %>%            # Back in tidy format
    ggplot(aes(x = date, y = value, color = portfolio)) + geom_line() +     # Plot!  
    theme(legend.position = c(0.7, 0.8))
The value factor: portfolio values.

FIGURE 18.2: The value factor: portfolio values.

Portfolios based on quartiles, using the tidyverse only. We rely heavily on the fact that features are uniformized, i.e., that their distribution is uniform for each given date. Overall, small firms outperform heavily (see Figure 18.3).

data_ml %>%
    mutate(small = Mkt_Cap_6M_Usd <= 0.25,                        # Small firms...
           medium = Mkt_Cap_6M_Usd > 0.25 & Mkt_Cap_6M_Usd <= 0.5, 
           large = Mkt_Cap_6M_Usd > 0.5 & Mkt_Cap_6M_Usd <= 0.75,
           xl = Mkt_Cap_6M_Usd > 0.75,                            # ...Xlarge firms
           year = year(date)) %>%                        
    group_by(year) %>%
    summarize(small = mean(small * R1M_Usd),                      # Compute avg returns
              medium = mean(medium * R1M_Usd),
              large = mean(large * R1M_Usd),
              xl = mean(xl * R1M_Usd)) %>%
    gather(key = size, value = return, -year) %>%
    ggplot(aes(x = year, y = return, fill = size)) + geom_col(position = "dodge")
The value factor: portfolio values.

FIGURE 18.3: The value factor: portfolio values.

18.2 Chapter 4

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 (4.3) 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.

getSymbols.FRED("BAMLC0A0CM",                                    # Extract data
                env = ".GlobalEnv", 
                return.class = "xts")
## [1] "BAMLC0A0CM"
cred_spread <- fortify(BAMLC0A0CM)                               # Transform to dataframe
colnames(cred_spread) <- c("date", "spread")                     # Change column name
cred_spread <- cred_spread %>%                                   # Take extraction and...
    full_join(data_ml %>% dplyr::select(date), by = "date") %>%  # Join!
    mutate(spread = na.locf(spread))                             # Replace NA by previous
cred_spread <- cred_spread[!duplicated(cred_spread),]            # Remove duplicates

The creation of the augmented dataset requires some manipulation. Features are no longer uniform as is shown in Figure 18.4.

data_cond <- data_ml %>%                                    # Create new dataset
    dplyr::select(c("stock_id", "date", features_short))
names_cred_spread <- paste0(features_short, "_cred_spread") # New column names
feat_cred_spread <- data_cond %>%                           # Old values
cred_spread <- data_ml %>%                                  # Create vector of spreads
    dplyr::select(date) %>%
    left_join(cred_spread, by = "date") 
feat_cred_spread <- feat_cred_spread *                      # This product creates...
    matrix(cred_spread$spread,                              # the new values...
           length(cred_spread$spread),                      # using duplicated...
           length(features_short))                          # columns
colnames(feat_cred_spread) <- names_cred_spread             # New column names
data_cond <- bind_cols(data_cond, feat_cred_spread)         # Aggregate old & new
data_cond %>% ggplot(aes(x = Eps_cred_spread)) + geom_histogram() # Plot example
Distribution of Eps after conditioning.

FIGURE 18.4: Distribution of Eps after conditioning.

To prevent this issue, uniformization is required and is verified in Figure 18.5.

data_cond <- data_cond %>%                   # From new dataset
    group_by(date) %>%                       # Group by date and...
    mutate_at(names_cred_spread, norm_unif)  # Uniformize the new features
data_cond %>% ggplot(aes(x = Eps_cred_spread)) + geom_histogram(bins = 100) # Verification
Distribution of uniformized conditioned feature values.

FIGURE 18.5: Distribution of uniformized conditioned feature values.

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

getSymbols.FRED("VIXCLS",                           # Extract data
                env = ".GlobalEnv", 
                return.class = "xts")
## [1] "VIXCLS"
vix <- fortify(VIXCLS)                              # Transform to dataframe
colnames(vix) <- c("date", "vix")                   # Change column name
vix <- vix %>%                                      # Take extraction and...
    full_join(data_ml %>% dplyr::select(date), by = "date") %>%    # Join! 
    mutate(vix = na.locf(vix))                      # Replace NA by previous
vix <- vix[!duplicated(vix),]                       # Remove duplicates
vix <- data_ml %>%                                  # Keep original data format
    dplyr::select(date) %>%                         # ...
    left_join(vix, by = "date")                     # Via left_join()

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 (see Figure 18.6).

delta <- 0.5                                       # Magnitude of vix correction
vix_bar <- median(vix$vix)                         # Median of vix
data_vix <- data_ml %>%                            # Smaller dataset
    dplyr::select(stock_id, date, R1M_Usd) %>%
    mutate(r_minus = (-0.02) * exp(-delta*(vix$vix-vix_bar)),  # r_-
           r_plus = 0.02 * exp(delta*(vix$vix-vix_bar)))       # r_+
data_vix <- data_vix %>% 
    mutate(R1M_Usd_Cvix = if_else(R1M_Usd < r_minus, -1,       # New label!
                                  if_else(R1M_Usd > r_plus, 1,0)),
           R1M_Usd_Cvix = as.factor(R1M_Usd_Cvix))
data_vix %>% 
    mutate(year = year(date)) %>%
    group_by(year, R1M_Usd_Cvix) %>%
    summarize(nb = n()) %>%
    ggplot(aes(x = year, y = nb, fill = R1M_Usd_Cvix)) + geom_col()
Evolution of categories through time.

FIGURE 18.6: Evolution of categories through time.

Finally, we switch to the outliers (Figure 18.7).

data_ml %>%
    ggplot(aes(x = R12M_Usd)) + geom_histogram()
Outliers in the dependent variable.

FIGURE 18.7: Outliers in the dependent variable.

Returns above 50 should indeed be rare.

data_ml %>% filter(R12M_Usd > 50) %>% dplyr::select(stock_id, date, R12M_Usd)
## # 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.

data_ml %>% 
    filter(stock_id == 683, year(date) == 2009) %>% 
    dplyr::select(date, R1M_Usd)
## # 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 shows that the stock is the most volatile of the dataset.

18.3 Chapter 5

We recycle the training and testing data variables created in the chapter (coding section notably). In addition, we create a dedicated function and resort to the map2() function from the purrr package.

alpha_seq <- (0:10)/10                     # Sequence of alpha values
lambda_seq <- 0.1^(0:5)                    # Sequence of lambda values
pars <- expand.grid(alpha_seq, lambda_seq) # Exploring all combinations!
alpha_seq <- pars[,1]
lambda_seq <- pars[,2]
lasso_sens <- function(alpha, lambda, x_train, y_train, x_test, y_test){ # Function
    fit_temp <- glmnet(x_train, y_train,                                 # Model
                       alpha = alpha, lambda = lambda)
    return(sqrt(mean((predict(fit_temp, x_test) - y_test)^2)))           # Output
rmse_elas <- map2(alpha_seq, lambda_seq, lasso_sens,                     # Automation
                  x_train = x_penalized_train, y_train = y_penalized_train,
                  x_test = x_penalized_test, y_test = testing_sample$R1M_Usd)

bind_cols(alpha = alpha_seq, lambda = as.factor(lambda_seq), rmse = unlist(rmse_elas)) %>%
    ggplot(aes(x = alpha, y = rmse, fill = lambda)) + geom_col() + facet_grid(lambda ~.) +
    coord_cartesian(ylim = c(0.19,0.193))
Performance of elasticnet across parameter values.

FIGURE 18.8: Performance of elasticnet across parameter values.

As is outlined in Figure 18.8, the parameters have a very marginal impact. Maybe the model is not a good fit for the task.

18.4 Chapter 6

fit1 <- rpart(formula, 
              data = training_sample,     # Data source: full sample
              cp = 0.001)                 # Precision: smaller = more leaves
mean((predict(fit1, testing_sample) - testing_sample$R1M_Usd)^2) 
## [1] 0.04018973
fit2 <- rpart(formula,
              data = training_sample,     # Data source: full sample
              cp = 0.01)                  # Precision: smaller = more leaves
mean((predict(fit2, testing_sample) - testing_sample$R1M_Usd)^2) # Test!
## [1] 0.03699696
rpart.plot(fit1)                         # Plot the first tree
Sample (complex) tree.

FIGURE 18.9: Sample (complex) tree.

The first model (Figure 18.9) 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.

n_trees <- c(10, 20, 40, 80, 160)
mse_RF <- 0
for(j in 1:length(n_trees)){       # No need for functional programming here...
    fit_temp <- randomForest(
        as.formula(paste("R1M_Usd ~", paste(features_short, collapse = " + "))),  # New formula!
        data = training_sample,    # Data source: training sample
        sampsize = 30000,          # Size of (random) sample for each tree
        replace = TRUE,            # Is the sampling done with replacement?
        ntree = n_trees[j],        # Nb of random trees
        mtry = 5)                  # Nb of predictors for each tree
    mse_RF[j] <- mean((predict(fit_temp, testing_sample) - testing_sample$R1M_Usd)^2)
## [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 numbers of trees are preferable and the reason is that each new tree tells 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 by Breiman (2001).

For the last exercises, we recycle the formula used in Chapter 6.

tree_2008 <- rpart(formula,
                   data = data_ml %>% filter(year(date) == 2008), # Data source: 2008
                   cp = 0.001,
                   maxdepth = 2) 
Tree for 2008.

FIGURE 18.10: Tree for 2008.

The first splitting criterion in Figure 18.10 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 poorly were those with the highest EV (i.e., large, robust firms).

tree_2009 <- rpart(formula,
                   data = data_ml %>% filter(year(date) == 2009), # Data source: 2009
                   cp = 0.001,
                   maxdepth = 2) 
Tree for 2009.

FIGURE 18.11: Tree for 2009.

In 2009 (Figure 18.11), 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 K. Daniel and Moskowitz (2016). The rationale is the following: after a market downturn, the stocks 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.

18.5 Chapter 7: the autoencoder model & universal approximation

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.

data_short <- data_ml %>%         # Shorter dataset
    filter(stock_id %in% stock_ids_short) %>%
    dplyr::select(c("stock_id", "date",features_short, "R1M_Usd"))
dates <- unique(data_short$date)  # Vector of dates

N <- length(stock_ids_short)      # Dimension for assets
Tt <- length(dates)               # Dimension for dates
K <- length(features_short)       # Dimension for features

factor_data <- data_short %>%  # Factor side date
    dplyr::select(date, stock_id, R1M_Usd) %>%
    spread(key = stock_id, value = R1M_Usd) %>%
    dplyr::select(-date) %>%

beta_data <- array(unlist(data_short %>%  # Beta side data: beware the permutation below!
                              dplyr::select(-stock_id, -date, -R1M_Usd)), 
                   dim = c(N, Tt, K))
beta_data <- aperm(beta_data, c(2,1,3))   # Permutation

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

main_input <- layer_input(shape = c(N), name = "main_input")  # Main input: returns      
factor_network <- main_input %>%                              # Def of factor side network
    layer_dense(units = 8, activation = "relu", name = "layer_1_r") %>%
    layer_dense(units = 4, activation = "tanh", name = "layer_2_r") 

aux_input <- layer_input(shape = c(N,K), name = "aux_input")  # Aux input: characteristics
beta_network <- aux_input %>%                                 # Def of beta side network
    layer_dense(units = 8, activation = "relu", name = "layer_1_l") %>%
    layer_dense(units = 4, activation = "tanh", name = "layer_2_l") %>%
    layer_permute(dims = c(2,1), name = "layer_3_l")          # Permutation!

main_output <- layer_dot(c(beta_network, factor_network),     # Product of 2 networks
                         axes = 1, name = "main_output") 

model_ae <- keras_model(                                      # AE Model specs
    inputs = c(main_input, aux_input),
    outputs = c(main_output)

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

summary(model_ae)                      # See model details / architecture
## 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
## __________________________________________________________________________________________
model_ae %>% compile(                  # Learning parameters
    optimizer = "rmsprop",
    loss = "mse"

model_ae %>% fit(                      # Learning function
    x = list(main_input = factor_data, aux_input = beta_data),
    y = list(main_output = factor_data),
    epochs = 20,                      # Nb rounds
    batch_size = 49                   # Nb obs. per round

For the second exercise, we use a simple architecture. The activation function, number of epochs and batch size may matter…

model_ua <- keras_model_sequential()
model_ua %>%   # This defines the structure of the network, i.e. how layers are organized
    layer_dense(units = 16, activation = 'sigmoid', input_shape = 1) %>%
    layer_dense(units = 1) # 
model_ua %>% compile(                             # Model specification
    loss = 'mean_squared_error',                  # Loss function
    optimizer = optimizer_rmsprop(),              # Optimisation method (weight updating)
    metrics = c('mean_absolute_error')            # Output metric
summary(model_ua)                                 # A simple model!
## Model: "sequential_15"
## __________________________________________________________________________________________
## Layer (type)                            Output Shape                        Param #       
## ==========================================================================================
## dense_48 (Dense)                        (None, 16)                          32            
## __________________________________________________________________________________________
## dense_47 (Dense)                        (None, 1)                           17            
## ==========================================================================================
## Total params: 49
## Trainable params: 49
## Non-trainable params: 0
## __________________________________________________________________________________________
fit_ua <- model_ua %>% 
    fit(seq(0, 6, by = 0.001) %>% matrix(ncol = 1),              # Training data = x
        sin(seq(0, 6, by = 0.001)) %>% matrix(ncol = 1),         # Training label = y
        epochs = 30, batch_size = 64                             # Training parameters

In full disclosure, to improve the fit, we also increase the sample size. We show the improvement in the figure below.

model_ua2 <- keras_model_sequential()
model_ua2 %>%   # This defines the structure of the network, i.e. how layers are organized
    layer_dense(units = 128, activation = 'sigmoid', input_shape = 1) %>%
    layer_dense(units = 1) # 
model_ua2 %>% compile(                             # Model specification
    loss = 'mean_squared_error',                  # Loss function
    optimizer = optimizer_rmsprop(),              # Optimisation method (weight updating)
    metrics = c('mean_absolute_error')            # Output metric
summary(model_ua2)                                 # A simple model!
## Model: "sequential_8"
## __________________________________________________________________________________________
## Layer (type)                            Output Shape                        Param #       
## ==========================================================================================
## dense_24 (Dense)                        (None, 128)                         256           
## __________________________________________________________________________________________
## dense_23 (Dense)                        (None, 1)                           129           
## ==========================================================================================
## Total params: 385
## Trainable params: 385
## Non-trainable params: 0
## __________________________________________________________________________________________
fit_ua2 <- model_ua2 %>% 
    fit(seq(0, 6, by = 0.0002) %>% matrix(ncol = 1),            # Training data = x
        sin(seq(0, 6, by = 0.0002)) %>% matrix(ncol = 1),       # Training label = y
        epochs = 60, batch_size = 64                            # Training parameters
tibble(x = x) %>%
  ggplot() + 
  geom_line(aes(x = x, y = predict(model_ua, x), color = "Small model")) +
  geom_line(aes(x = x, y = predict(model_ua2, x), color = "Large model")) +
  stat_function(fun = sin, aes(color = "sin(x) function")) + 
  scale_color_manual(values = c("#9999FF", "#333399", "#000000"))
Case with 128 units. The sine function in light blue and approximation in black.

FIGURE 18.12: Case with 128 units. The sine function in light blue and approximation in black.

18.6 Chapter 8

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 6.

mse <- function(fit, features, label){             # MSE function
    return(mean((predict(fit, features)-label)^2))
par_list <- list(y = train_label_xgb[1:10000],     # From Tree chapter
                 x = train_features_xgb[1:10000,],
                 type = "eps-regression",
                 epsilon = 0.1,                    # Width of strip for errors
                 gamma = 0.5,                      # Constant in the radial kernel 
                 cost = 0.1)
svm_par <- function(kernel, par_list){             # Function for SVM fit automation
    return(, c(kernel = kernel, par_list))) 
kernels <- c("linear", "radial", "polynomial", "sigmoid") # Kernels
fit_svm_par <- map(kernels, svm_par, par_list = par_list) # SVM models
map(fit_svm_par, mse,                                     # MSEs
    features = test_feat_short,                           # From SVM chapter 
    label = testing_sample$R1M_Usd)
## [[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 that is limited to the 7 major predictors. Even with a smaller number of features, the training is time consuming.

svm_full <- svm(y = train_label_xgb,      # Train label
                x = train_features_xgb,   # Training features
                type = "eps-regression",  # SVM task type (see LIBSVM documentation)
                kernel = "linear",        # SVM kernel 
                epsilon = 0.1,            # Width of strip for errors
                cost = 0.1)               # Slack variable penalisation
test_feat_short <- dplyr::select(testing_sample,features_short)       # Test set
mean(predict(svm_full, test_feat_short) * testing_sample$R1M_Usd > 0) # Hit ratio
## [1] 0.490343

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

xgb_full <- xgb.train(data = train_matrix_xgb,    # Data source 
                      eta = 0.3,                          # Learning rate
                      objective = "reg:linear",           # Objective function
                      max_depth = 4,                      # Maximum depth of trees
                      nrounds = 60                        # Number of trees used (bit low here)
## [14:43:24] WARNING: amalgamation/../src/objective/ reg:linear is now deprecated in favor of reg:squarederror.
mean(predict(xgb_full, xgb_test) * testing_sample$R1M_Usd > 0) # Hit ratio
## [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.

18.7 Chapter 11: ensemble neural network

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

feat_train_1 <- training_sample %>% dplyr::select(features[3*(1:31)]) %>%   # First set of feats
feat_train_2 <- training_sample %>% dplyr::select(features[3*(1:31)-1]) %>% # Second set of feats
feat_train_3 <- training_sample %>% dplyr::select(features[3*(1:31)-2]) %>% # Third set of feats
feat_test_1 <- testing_sample %>% dplyr::select(features[3*(1:31)]) %>%     # Test features 1
feat_test_2 <- testing_sample %>% dplyr::select(features[3*(1:31)-1]) %>%   # Test features 2
feat_test_3 <- testing_sample %>% dplyr::select(features[3*(1:31)-2]) %>%   # Test features 3

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

first_input <- layer_input(shape = c(31), name = "first_input")   # First input      
first_network <- first_input %>%                                  # Def of 1st network
    layer_dense(units = 8, activation = "relu", name = "layer_1") %>%
    layer_dense(units = 2, activation = 'softmax')                # Softmax for categ. output
second_input <- layer_input(shape = c(31), name = "second_input") # Second input      
second_network <- second_input %>%                                # Def of 2nd network
    layer_dense(units = 8, activation = "relu", name = "layer_2") %>%
    layer_dense(units = 2, activation = 'softmax')                # Softmax for categ. output
third_input <- layer_input(shape = c(31), name = "third_input")  # Third input      
third_network <- third_input %>%                                  # Def of 3rd network
    layer_dense(units = 8, activation = "relu", name = "layer_3") %>%
    layer_dense(units = 2, activation = 'softmax')                # Softmax for categ. output

main_output <- layer_concatenate(c(first_network, 
                                   third_network)) %>%            # Combination
    layer_dense(units = 2, activation = 'softmax', name = 'main_output')

model_ens <- keras_model(                                          # Agg. Model specs
    inputs = c(first_input, second_input, third_input),
    outputs = c(main_output)

Lastly, we can train and evaluate (see Figure 18.13).

summary(model_ens)                      # See model details / architecture
## 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_49 (Dense)             (None, 2)           18         layer_1[0][0]                 
## __________________________________________________________________________________________
## dense_50 (Dense)             (None, 2)           18         layer_2[0][0]                 
## __________________________________________________________________________________________
## dense_51 (Dense)             (None, 2)           18         layer_3[0][0]                 
## __________________________________________________________________________________________
## concatenate_1 (Concatenate)  (None, 6)           0          dense_49[0][0]                
##                                                             dense_50[0][0]                
##                                                             dense_51[0][0]                
## __________________________________________________________________________________________
## main_output (Dense)          (None, 2)           14         concatenate_1[0][0]           
## ==========================================================================================
## Total params: 836
## Trainable params: 836
## Non-trainable params: 0
## __________________________________________________________________________________________
model_ens %>% compile(                  # Learning parameters
    optimizer = optimizer_adam(),
    loss = "binary_crossentropy",
    metrics = "categorical_accuracy"

fit_NN_ens <- model_ens %>% fit(               # Learning function
    x = list(first_input = feat_train_1, 
             second_input = feat_train_2,
             third_input = feat_train_3),
    y = list(main_output = NN_train_labels_C), # Recycled from NN Chapter
    epochs = 12,                               # Nb rounds
    batch_size = 512,                          # Nb obs. per round
    validation_data = list(list(feat_test_1, feat_test_2, feat_test_3),
Learning an integrated ensemble.

FIGURE 18.13: Learning an integrated ensemble.

18.8 Chapter 12

18.8.1 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 3.1. The returns are plotted in Figure 18.14.

data_ml %>%
  group_by(date) %>%                     # Group by date
  summarize(return = mean(R1M_Usd)) %>%  # Compute return
  ggplot(aes(x = date, y = return)) + geom_point() + geom_line() # Plot
Time series of returns.

FIGURE 18.14: Time series of returns.

18.8.2 Advanced weighting function

First, we code the function with all inputs.

weights <- function(Sigma, mu, Lambda, lambda, k_D, k_R, w_old){
    N <- nrow(Sigma)
    M <- solve(lambda*Sigma + 2*k_R*Lambda + 2*k_D*diag(N)) # Inverse matrix
    num <- 1-sum(M %*% (mu + 2*k_R*Lambda %*% w_old))       # eta numerator
    den <- sum(M %*% rep(1,N))                              # eta denominator
    eta <- num / den                                        # eta
    vec <- mu + eta * rep(1,N) + 2*k_R*Lambda %*% w_old     # Vector in weight
    return(M %*% vec)

Second, we test it on some random dataset. We use the returns created at the end of Chapter 1 and used for the Lasso allocation in Section 5.2.2. For \(\boldsymbol{\mu}\), we use the sample average, which is rarely a good idea in practice. It serves as illustration only.

Sigma <- returns %>% dplyr::select(-date) %>% as.matrix() %>% cov()  # Covariance matrix
mu <- returns %>% dplyr::select(-date) %>% apply(2,mean)             # Vector of exp. returns
Lambda <- diag(nrow(Sigma))                                          # Trans. Cost matrix
lambda <- 1                                                          # Risk aversion
k_D <- 1
k_R <- 1
w_old <- rep(1, nrow(Sigma)) / nrow(Sigma)                           # Prev. weights: EW
weights(Sigma, mu, Lambda, lambda, k_D, k_R, w_old) %>% head()       # First weights
##             [,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.

sensi <- function(lambda, k_D, Sigma, mu, Lambda, k_R, w_old){
    w <- weights(Sigma, mu, Lambda, lambda, k_D, k_R, w_old)
    out <- c()
    out$div <- 1/sum(w^2)             # Diversification
    out$lev <- sum(abs(w[w<0]))       # Leverage
    out$vol <- t(w) %*% Sigma %*% w   # In-sample vol

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

lambda <- 10^(-3:2)              # parameter values
k_D <- 2*10^(-3:2)               # parameter values
pars <- expand_grid(lambda, k_D) # parameter grid
lambda <- pars$lambda
k_D <- pars$k_D

res <- map2_dfr(lambda, k_D, sensi, 
                Sigma = Sigma, mu = mu, Lambda = Lambda, k_R = k_R, w_old = w_old)

bind_cols(lambda = as.factor(lambda), k_D = as.factor(k_D), res) %>%
    gather(key = indicator, value = value, -lambda, -k_D) %>%
    ggplot(aes(x = lambda, y = value, fill = k_D)) + geom_col(position = "dodge") +
    facet_grid(indicator ~. , scales = "free")
Indicators related to portfolio weights.

FIGURE 18.15: Indicators related to portfolio weights.

In Figure 18.15, 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.

18.8.3 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 12.

portf_map <- function(t, data_ml, ticks, t_oos, m_offset, train_size, weight_func){
    train_data <- data_ml %>% filter(date < t_oos[t] - m_offset * 30,   # Roll. window w. buffer
                                     date > t_oos[t] - m_offset * 30 - 365 * train_size)
    test_data <- data_ml %>% filter(date == t_oos[t])                   # Test set  
    realized_returns <- test_data %>%                                   # Computing returns via:
        dplyr::select(R1M_Usd)                                          # 1M holding period!
    temp_weights <- weight_func(train_data, test_data, features)        # Weights = > recycled!
    ind <- match(temp_weights$names, ticks) %>% na.omit()               # Index of test assets
    x <- c() 
    x$weights <- rep(0, length(ticks))                           # Empty weights
    x$weights[ind] <- temp_weights$weights                       # Locate weights correctly
    x$returns <- sum(temp_weights$weights * realized_returns)    # Compute returns

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

back_test <- 1:3 %>%             # Test on the first 100 out-of-sample dates
    map(portf_map, data_ml = data_ml, ticks = ticks, t_oos = t_oos,
        m_offset = 1, train_size = 5, weight_func = weights_xgb)
## [14:43:55] WARNING: amalgamation/../src/objective/ reg:linear is now deprecated in favor of reg:squarederror.
## [14:44:04] WARNING: amalgamation/../src/objective/ reg:linear is now deprecated in favor of reg:squarederror.
## [14:44:14] WARNING: amalgamation/../src/objective/ reg:linear is now deprecated in favor of reg:squarederror.
head(back_test[[1]]$weights)     # Sample weights
## [1] 0.001675042 0.000000000 0.000000000 0.001675042 0.000000000 0.001675042
back_test[[1]]$returns           # Return of first period
## [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.

18.9 Chapter 15

We recycle the AE model trained in Chapter 15. 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).

save_model_weights_hdf5(object = ae_model,filepath ="ae_weights.hdf5", overwrite = TRUE)
encoder_model <- keras_model(inputs = input_layer, outputs = encoder)
encoder_model %>% 
    load_model_weights_hdf5(filepath = "ae_weights.hdf5",skip_mismatch = TRUE,by_name = TRUE)
encoder_model %>% compile(
    loss = 'mean_squared_error',
    optimizer = 'adam',
    metrics = c('mean_absolute_error')
encoder_model %>% 
  keras::predict_on_batch(x = training_sample %>% 
                              dplyr::select(features_short) %>% 
                              as.matrix()) %>%
##           [,1]      [,2]      [,3]      [,4]
## [1,] -1.676068 0.2152901 0.1490787 0.8586778
## [2,] -1.684428 0.1919250 0.1985757 0.8494947
## [3,] -1.686019 0.1909507 0.1556696 0.8760668
## [4,] -1.689391 0.1893837 0.1492719 0.8799014
## [5,] -1.694696 0.1924362 0.1314638 0.8783331

18.10 Chapter 16

All we need to do is change the rho coefficient in the code of Chapter 16.

set.seed(42)                                                 # Fixing the random seed
n_sample <- 10^5                                             # Number of samples generated
rho <- (-0.8)                                                # Autoregressive parameter
sd <- 0.4                                                    # Std. dev. of noise
a <- 0.06 * rho                                              # Scaled mean of returns
data_RL3 <- tibble(returns = a/rho + arima.sim(n = n_sample, # Returns via AR(1) simulation
                                               list(ar = rho),       
                                               sd = sd),
                   action = round(runif(n_sample)*4)/4) %>%   # Random action (portfolio)
    mutate(new_state = if_else(returns < 0, "neg", "pos"),   # Coding of state
           reward = returns * action,                        # Reward = portfolio return
           state = lag(new_state),                           # Next state
           action = as.character(action)) %>% 
    na.omit()                                                # Remove one missing state

The learning can then proceed.

control <- list(alpha = 0.1,                        # Learning rate
                gamma = 0.7,                        # Discount factor for rewards
                epsilon = 0.1)                      # Exploration rate
fit_RL3 <- ReinforcementLearning(data_RL3,          # Main RL function
                                 s = "state", 
                                 a = "action", 
                                 r = "reward", 
                                 s_new = "new_state", 
                                 control = control)
print(fit_RL3)   # Show the output
## 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 16.

pos_3 <- c(-1,0,1)                              # Possible alloc. to asset 1
pos_4 <- c(-1,0,1)                              # Possible alloc. to asset 3
pos <- expand_grid(pos_3, pos_4)                # All combinations
pos <- bind_cols(pos, id = 1:nrow(pos))         # Adding combination id

ret_pb_RL <- bind_cols(r3 = return_3, r4 = return_4, # Returns & P/B dataframe
                       pb3 = pb_3, pb4 = pb_4) 
data_RL4 <- sapply(ret_pb_RL,                        # Combining return & positions
                   times = nrow(pos)) %>%
    data.frame() %>%
    bind_cols(id = rep(1:nrow(pos), 1, each = length(return_3))) %>%
    left_join(pos) %>% dplyr::select(-id) %>%
    mutate(action = paste(pos_3, pos_4),            # Uniting actions
           pb3 = round(5 * pb3),                    # Simplifying states
           pb4 = round(5 * pb4),                    # Simplifying states
           state = paste(pb3, pb4),                 # Uniting states
           reward = pos_3*r3 + pos_4*r4,            # Computing rewards
           new_state = lead(state)) %>%             # Infer new state
    dplyr::select(-pb3, -pb4, -pos_3,          # Remove superfluous vars.
                  -pos_4, -r3, -r4) 

We can the plug this data into the RL function.

fit_RL4 <- ReinforcementLearning(data_RL4,           # Main RL function
                                 s = "state", 
                                 a = "action", 
                                 r = "reward", 
                                 s_new = "new_state", 
                                 control = control)
fit_RL4$Q <- round(fit_RL4$Q, 3) # Round the Q-matrix
print(fit_RL4)                   # Show the output 
## State-Action function Q
##       0 0    0 1  0 -1  -1 -1   -1 0   -1 1   1 -1    1 0    1 1
## 0 2 0.000  0.000 0.002 -0.017 -0.018 -0.020  0.023  0.025  0.024
## 0 3 0.001 -0.005 0.007 -0.013 -0.019 -0.026  0.031  0.027  0.021
## 3 1 0.003  0.003 0.003  0.002  0.002  0.003  0.002  0.002  0.003
## 2 1 0.027  0.038 0.020  0.004  0.015  0.039  0.013  0.021  0.041
## 2 2 0.021  0.014 0.027  0.038  0.047  0.045 -0.004 -0.011 -0.016
## 2 3 0.007  0.006 0.008  0.054  0.057  0.056 -0.041 -0.041 -0.041
## 1 1 0.027  0.054 0.005 -0.031 -0.005  0.041  0.025  0.046  0.072
## 1 2 0.019  0.020 0.020  0.015  0.023  0.029  0.012  0.014  0.023
## 1 3 0.008  0.019 0.000 -0.036 -0.027 -0.016  0.042  0.053  0.060
## Policy
##    0 2    0 3    3 1    2 1    2 2    2 3    1 1    1 2    1 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 16; we have covered much more ground! Some policy recommendations have not changed compared to the smaller sample, but some have! The change occurs 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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