Author: Bernardo Reckziegel
Date: 2022-09-29

# Views

#### Bernardo Reckziegel

#### 2022-09-29

This vignette provides a “jump-start” for users that want to quickly understand how to use entropy-pooling (EP) to construct views on the market distribution. The methodology fits well in a variety of scenarios: stress-testing, macro factors, non-linear payoffs, and more.

To demonstrate entropy-pooling’s firepower, the current vignette uses some of the cases Meucci (2008)1 leaves for the reader in the appendix. In particular, the usage of the following functions are covered in depth:

- `view_on_mean()`
- `view_on_volatility()`
- `view_on_correlation()`
- `view_on_rank()`
- `view_on_marginal_distribution()`
- `view_on_copula()`

Along with the examples, the `EuStockMarkets` dataset - that comes with the installation of `R` - is used as a proxy for “the market”:

    library(ffp)
    library(ggplot2)
    set.seed(321)

    # invariance / stationarity
    x <- diff(log(EuStockMarkets))

    dim(x)
    #> [1] 1859    4

    head(x)
    #>               DAX          SMI          CAC         FTSE
    #> [1,] -0.009326550  0.006178360 -0.012658756  0.006770286
    #> [2,] -0.004422175 -0.005880448 -0.018740638 -0.004889587
    #> [3,]  0.009003794  0.003271184 -0.005779182  0.009027020
    #> [4,] -0.001778217  0.001483372  0.008743353  0.005771847
    #> [5,] -0.004676712 -0.008933417 -0.005120160 -0.007230164
    #> [6,]  0.012427042  0.006737244  0.011714353  0.008517217

## Views on Expected Returns

Assume an investor believes the `FTSE` index return will be \(20\%\) above average in the near future. In order to process this view, the investor needs to minimize the relative entropy:

\[ \sum_{j=1}^J x_j(ln(x_j) - ln(p_j)) \]

Subject to the restriction:

\[ \ x_j V_{j, k} = \mu_{k} \]

In which \(x_{j}\) is a yet to be discovered posterior probability; \(V_{j,k}\) is a \(1859 \times 1\) vector with the `FTSE` returns; and \(\mu_{k}\) is a \(1 \times 1\) scalar with the investor projected return for the `FTSE` index. In this case, the \(k\) subscript represents the fourth column in `x`2.

Views on expected returns can be constructed with `view_on_mean`:

    # Returns 20% higher than average
    mu <- mean(x[ , "FTSE"]) * 1.2

    # ffp views constructor
    views <- view_on_mean(x = as.matrix(x[ , "FTSE"]), mean = mu)
    views
    #> # ffp view
    #> Type:  View On Mean
    #> Aeq :  Dim 1 x 1859
    #> beq :  Dim 1 x 1

The `views` object is a `list` with two components, `Aeq` and `beq`, that are equivalent to the elements \(V_{j,k}\) transposed and \(\mu_{k}\), respectively.

The investor also needs to formulate a vector of prior probabilities, \(p_j\), which is usually - but not necessary - a equal-weight vector:

    # Prior probabilities
    p_j <- rep(1 / nrow(x), nrow(x))

Once the prior and the views are established, the optimization can take place with `entropy_pooling`:

    # Solve Minimum Relative Entropy (MRE)
    ep <- entropy_pooling(
      p      = p_j,
      Aeq    = views$Aeq,
      beq    = views$beq,
      solver = "nlminb"
    )
    ep
    #> <ffp[1859]>
    #> 0.0005425631 0.0005340012 0.000544236 0.0005418246 0.0005322989 ... 0.0005451271

What is the interpretation for `ep`? Among all the possible probability vectors, `ep` is the one that can satisfy the views in the “least” evasive way (in which the term “least evasive” is direct reference to the prior). Therefore, `ep` is the best candidate for a posterior distribution.

In real world, the investor can have many views. Extending the current example, say the investor has a new belief: the `CAC` index returns will be \(10\%\) bellow average3:

    mu <- c(
      mean(x[ , "FTSE"]) * 1.2, # Return 20% higher than average
      mean(x[ , "CAC"]) * 0.9   # Return 10% bellow average
    )

    # ffp views constructor
    views <- view_on_mean(x = as.matrix(x[ , c("FTSE", "CAC")]), mean = mu)
    views
    #> # ffp view
    #> Type:  View On Mean
    #> Aeq :  Dim 2 x 1859
    #> beq :  Dim 2 x 1

In which the elements `Aeq` and `beq` now have \(2\) rows each, one for every view.

The minimum relative entropy problem is solved in the exact same way:

    # Solve MRE
    ep <- entropy_pooling(
      p      = p_j,
      Aeq    = views$Aeq,
      beq    = views$beq,
      solver = "nlminb"
    )
    ep
    #> <ffp[1859]>
    #> 0.0005602371 0.0005473082 0.0005573006 0.0005384979 0.000531066 ... 0.0005434801

It’s easier to analyse the output of `ep` visually.

    class(ep)
    #> [1] "ffp"        "vctrs_vctr"

To that end, the `autoplot` method is available for objects of the `ffp` class:

    autoplot(ep) +
      scale_color_viridis_c(option = "C", end = 0.75) +
      labs(title    = "Posterior Probability Distribution",
           subtitle = "Views on Expected Returns",
           x        = NULL,
           y        = NULL)

The plot reinforces the intuition of \(x_j\) - the posterior probability - as a distribution that distorts the prior in order to accommodate the views.

It’s easy to double-check the investor views by computing the ratio of the conditional vs. unconditional returns:

    conditional   <- ffp_moments(x, ep)$mu
    unconditional <- colMeans(x)

    conditional / unconditional - 1
    #>         DAX         SMI         CAC        FTSE
    #>  0.01335825  0.02078130 -0.09999999  0.20000009

According to expectations, `CAC` outlook is now \(10\%\) bellow average and `FTSE` expected return is \(20\%\) above average. Violà!

## Views on Volatility

Say the investor is confident that markets will be followed by a period of lull and bonanza. Without additional insights about expected returns, the investor anticipates that market volatility for the `CAC` index will drop by \(10\%\).

To impose views on volatility, the investor needs to minimize the expression:

\[ \sum_{j=1}^J x_j(ln(x_j) - ln(p_j)) \] Subject to the constraint:

\[ \sum_{j=1}^{J} x_j V_{j, k}^2 = \mu_{k}^2 + \sigma_{k}^2 \]

In which, \(x_j\) is a vector of yet to be defined posterior probabilities; \(V_{j,k}^2\) is a \(1859 \times 1\) vector with `CAC` squared returns; \(\mu_{k}^2\) and \(\sigma_{k}^2\) are the `CAC` sample mean and variance; and \(k\) is the column position of `CAC` in the panel \(V\).

The `view_on_volatility` function can be used for this purpose:

    # opinion
    vol_cond <- sd(x[ , "CAC"]) * 0.9

    # views
    views <- view_on_volatility(x = as.matrix(x[ , "CAC"]), vol = vol_cond)
    views
    #> # ffp view
    #> Type:  View On Volatility
    #> Aeq :  Dim 1 x 1859
    #> beq :  Dim 1 x 1

The `views` object holds a list with two components - `Aeq` and `beq` - that are equivalent to the elements \(V_{j,k}^2\) transposed and \(\mu_{k}^2 + \sigma_{k}^2\), respectively.

To solve the relative entropy use the `entropy_pooling` function:

    ep <- entropy_pooling(
      p      = p_j,
      Aeq    = views$Aeq,
      beq    = views$beq,
      solver = "nlminb"
    )
    ep
    #> <ffp[1859]>
    #> 0.0005227495 0.0004701207 0.0005609239 0.0005476659 0.0005631671 ... 0.0005349393

Once again, the `ep` vector is what we call posterior: the probability vector that causes the “least” amount of distortion in the prior, but still obeys the constraints (views).

The posterior probabilities can be observed with the `autoplot` method:

    autoplot(ep) +
      scale_color_viridis_c(option = "C", end = 0.75) +
      labs(title    = "Posterior Probability Distribution",
           subtitle = "View on Volatility",
           x        = NULL,
           y        = NULL)

To check if the posterior probabilities are valid as a way to display the investor’s view, compare the conditional vs. unconditional volatilities:

    unconditional <- apply(x, 2, sd)
    conditional   <- sqrt(diag(ffp_moments(x, ep)$sigma))

    conditional / unconditional - 1
    #>         DAX         SMI         CAC        FTSE
    #> -0.08377407 -0.06701178 -0.09977906 -0.04552477

The posterior volatility for the `CAC` index is reduced by \(10\%\), in line with the investor’s subjective judgment. However, by emphasizing tranquil periods more often, other assets are also affected, but in smaller magnitudes.

## Views on Correlation

Assume the investor believes the correlation between `FTSE` and `DAX` will increase by \(30\%\), from \(0.64\) to \(0.83\). To construct views on the correlation matrix, the general expression has to be minimized:

\[ \sum_{j=1}^J x_j(ln(x_j) - ln(p_j)) \] Subject to the constraints:

\[ \sum_{j=1}^{J} x_j V_{j, k} V_{j, l} = \mu_{k} \mu_{l} + \sigma_{k} \sigma_{l} C_{k,l} \]

In which, the term \(V_{j, k} V_{j, l}\) on the left hand-side of the restriction consists of cross-correlations among assets; the terms \(\mu_{k} \mu_{l} + \sigma_{k} \sigma_{l} C_{k,l}\) carry the investor target correlation structure; and \(x_j\) is a yet to be defined probability vector.

To formulate this view, first compute the unconditional correlation matrix. Second, add a “perturbation” in the corresponding element that is consistent with the perceived view:

    cor_view <- cor(x) # unconditional correlation matrix
    cor_view["FTSE", "DAX"] <- 0.83 # perturbation (investor belief)
    cor_view["DAX", "FTSE"] <- 0.83 # perturbation (investor belief)

Finally, pass the adjusted correlation matrix into the `view_on_correlation` function:

    views <- view_on_correlation(x = x, cor = cor_view)
    views
    #> # ffp view
    #> Type:  View On Correlation
    #> Aeq :  Dim 10 x 1859
    #> beq :  Dim 10 x 1

In which `Aeq` is equal to \(V_{j, k} V_{j, l}\) transposed and `beq` is a \(10 \times 1\) vector with \(\mu_{k} \mu_{l} + \sigma_{k} \sigma_{l} C_{k,l}\).

Notice that even though the investor has a view in just one parameter4 the constraint vector has \(10\) rows, because every element of the lower/upper diagonal of the correlation matrix has to match.

Once again, the minimum entropy is solved by `entropy_pooling`:

    ep <- entropy_pooling(
      p      = p_j,
      Aeq    = views$Aeq,
      beq    = views$beq,
      solver = "nlminb"
    )
    ep
    #> <ffp[1859]>
    #> 6.709143e-05 0.000659403 0.001085763 0.0003254822 0.0005215729 ... 0.0004748052

And the `autoplot` method is available:

    autoplot(ep) +
      scale_color_viridis_c(option = "C", end = 0.75) +
      labs(title    = "Posterior Probability Distribution",
           subtitle = "View on Correlation",
           x        = NULL,
           y        = NULL)

To fact-check the investor view, compute the posterior correlation matrix:

    cov2cor(ffp_moments(x, p = ep)$sigma)
    #>            DAX       SMI       CAC      FTSE
    #> DAX  1.0000000 0.6932767 0.7317714 0.8346939
    #> SMI  0.6932767 1.0000000 0.6260098 0.6113109
    #> CAC  0.7317714 0.6260098 1.0000000 0.6535585
    #> FTSE 0.8346939 0.6113109 0.6535585 1.0000000

And notice that the linear association between `FTSE` and `DAX` is now \(0.83\)!

## Views on Relative Performance

Assume the investor believes the `DAX` index will outperform the `SMI` by some amount, but he doesn’t know by how much. If the investor has only a mild view on the performance of assets, he may want to minimize the following expression:

\[ \sum_{j=1}^J x_j(ln(x_j) - ln(p_j)) \] Subject to the constraint:

\[ \sum_{j=1}^{J} x_j (V_{j, k} - V_{j, l}) \ge 0 \] In which, the \(x_j\) is a yet to be determined probability vector; \(V_{j, k}\) is a \(1859 \times 1\) column vector with `DAX` returns and \(V_{j, l}\) is a \(1859 \times 1\) column vector with the `SMI` returns. In this case, the \(k\) and \(l\) subscripts refers to the column positions of `DAX` and `SMI` in the object `x`, respectively.

Views on relative performance can be imposed with `view_on_rank`:

    views <- view_on_rank(x = x, rank = c(2, 1))
    views
    #> # ffp view
    #> Type:  View On Rank
    #> A :  Dim 1 x 1859
    #> b :  Dim 1 x 1

Assets that are expected to outperform enter in the `rank` argument with their column positions to the right: assets in the left underperform and assets in the right outperform. Because the investor believes that `DAX` \(\geq\) `SMI` (the asset in the first column will outperform the asset in the second column), we fill `rank = c(2, 1)`5.

The optimization is, once again, guided by the `entropy_pooling` function:

    ep <- entropy_pooling(
      p      = p_j,
      A      = views$A,
      b      = views$b,
      solver = "nloptr"
    )
    ep
    #> <ffp[1859]>
    #> 0.0005145646 0.0005403155 0.0005470049 0.0005330238 0.0005446858 ... 0.0005469164

Two important observations:

- Inequalities constraint cannot be handled with the `nlminb` solver. Use `nloptr` or `solnl`, instead;
- Inequalities constraint require the arguments `A` and `b` to be fulfilled rather than `Aeq` and `beq`.

The posterior probabilities are presented bellow:

    autoplot(ep) +
      scale_color_viridis_c(option = "C", end = 0.75) +
      labs(title    = "Posterior Probability Distribution",
           subtitle = "View on Ranking/Momentum",
           x        = NULL,
           y        = NULL)

To fact-check the ongoing view, compare the conditional vs. unconditional returns:

    conditional   <- ffp_moments(x, ep)$mu
    unconditional <- colMeans(x)

    conditional / unconditional - 1
    #>         DAX         SMI         CAC        FTSE
    #>  0.17274275 -0.06507207  0.13576463  0.06261163

Indeed, returns for `DAX` are adjusted upwards by \(17\%\), while returns for `SMI` are revised downwards by \(6.5\%\). Notice that returns for `CAC` and `FTSE` were also affected. This behavior could be tamed by combining the ranking condition with views on expected returns. See the function `bind_views()` for an example.

## Views on Marginal Distribution

Assume the investor has a view on the entire marginal distribution. One way to add views on marginal distributions is by matching the first moments exactly, up to a given order. Mathematically, the investor minimizes the relative entropy:

\[ \sum_{j=1}^J x_j(ln(x_j) - ln(p_j)) \] With, say, two equality constraints (one for \(\mu\) and one for \(\sigma^2\)):

\[ \sum_{j=1}^J x_j V_{j,k} = \sum_{j=z}^Z p_z \hat{V}_{z,k} \] \[ \sum_{j=1}^J x_j (V_{j,k})^2 = \sum_{z=1}^Z p_z (\hat{V}_{z,k})^2 \]

In which \(x_j\) is a yet to be defined probability vector; \(V_{j,k}\) is a matrix with the empirical marginal distributions; \(p_z\) is vector of prior probabilities; and \(\hat{V}_{z,k}\) an exogenous panel with simulations that are consistent with the investor’s views.

When \(j = z\), the panels \(V_{j,k}\) and \(\hat{V}_{z,k}\) have the same number of rows and the dimensions in both sides of the restrictions match. However, it’s possible to set \(z \ge j\) to simulate a larger panel for \(\hat{V}_{z, k}\). Keep in mind though that, if \(z \ne j\), two prior probabilities will have to be specified: one for \(p_j\) (the objective function) and one for \(p_z\) (the views).

Continuing on the example, consider the margins of `x` can be approximated by a symmetric multivariate t-distribution. If this is the case, the estimation can be conducted by the amazing `ghyp` package, that covers the entire family of generalized hyperbolic distributions:

    library(ghyp)

    # multivariate t distribution
    dist_fit <- fit.tmv(data = x, symmetric = TRUE, silent = TRUE)
    dist_fit
    #> Symmetric Student-t Distribution:
    #>
    #> Parameters:
    #>       nu
    #> 6.151241
    #>
    #> mu:
    #> [1] 0.0007899665 0.0009594760 0.0004790250 0.0003811669
    #>
    #> sigma:
    #>               DAX          SMI          CAC         FTSE
    #> DAX  1.000013e-04 6.046969e-05 7.933384e-05 5.072439e-05
    #> SMI  6.046969e-05 8.062728e-05 5.869208e-05 4.119624e-05
    #> CAC  7.933384e-05 5.869208e-05 1.216927e-04 5.715865e-05
    #> FTSE 5.072439e-05 4.119624e-05 5.715865e-05 6.398099e-05
    #>
    #> gamma:
    #> [1] 0 0 0 0
    #>
    #> log-likelihood:
    #> 26370.73
    #>
    #>
    #> Call:
    #> fit.tmv(data = x, symmetric = TRUE, silent = TRUE)

The investor then, can construct a large panel with statistical properties similar to the margins he envisions:

    set.seed(123)
    # random numbers from `dist_fit`
    simul  <- rghyp(n = 100000, object = dist_fit)

Where the \(100.000 \times 4\) matrix is used to match the views with `view_on_marginal_distribution`:

    p_z <- rep(1 / 100000, 100000)
    views <- view_on_marginal_distribution(
      x     = x,
      simul = simul,
      p     = p_z
    )
    views
    #> # ffp view
    #> Type:  View On Marginal Distribution
    #> Aeq :  Dim 8 x 1859
    #> beq :  Dim 8 x 1

The objects `simul` and `p_z` corresponds to the terms \(\hat{V}_{z,k}\) transposed and \(p_z\), respectively. Note that \(8\) restrictions are created, four for each moment of the marginal distribution (there are four assets in “the market”).

With the prior and the views at hand, the optimization can be quickly implemented:

    ep <- entropy_pooling(
      p      = p_j,
      Aeq    = views$Aeq,
      beq    = views$beq,
      solver = "nloptr"
    )
    ep
    #> <ffp[1859]>
    #> 0.000528748 0.0005678894 0.000536657 0.0005231905 0.0005349765 ... 0.0005324979

As the visual inspection of the vector `ep`:

    autoplot(ep) +
      scale_color_viridis_c(option = "C", end = 0.75) +
      labs(title    = "Posterior Probability Distribution",
           subtitle = "View on Marginal Distribution",
           x        = NULL,
           y        = NULL)

It’s easy to extend the current example. For instance, the investor could “tweak” some of the fitted parameters for stress-testing (i.e. change the degrees of freedom, increase/decrease expected returns, etc) to verify the immediate impact on the P&L (VaR, CVaR, etc).

Acknowledge that when the restrictions are to harsh, the scenarios will be heavily distorted to satisfy the views. In this case, one may need to compute the effective number of scenarios (ENS). This is very easy to do with `ens()`:

    ens(ep)
    #> [1] 1857.356

Rest on the investor to calibrate this statistic in order to get a desired confidence level6.

## Views on Copulas

Assume the investor would like to simulate a different dependence structure for the market. Views that change the interdependence of assets can be implemented by minimizing the relative entropy:

\[ \sum_{j=1}^J x_j(ln(x_j) - ln(p_j)) \] Subject to the following restrictions:

\[ \sum_{j=1}^J x_i U_{j,k} = 0.5\] \[ \sum_{j=1}^J x_j U_{j,k}U_{j,l} = \sum_{z=1}^Z p_z \hat{U}_{z,k}\hat{U}_{z,l} \]

\[ \sum_{j=1}^J x_j U_{j,k}U_{j,l}U_{j,i} = \sum_{j=1}^J p_j \hat{U}_{j,k}\hat{U}_{j,l}\hat{U}_{j,i} \]

In which, the first restriction matches the first moment of the uniform distribution; the second and third restrictions pair the cross-moments of the empirical copula, \(U\), with the simulated copula, \(\hat{U}\); \(x_j\) is a yet to be discovered posterior distribution; \(p_z\) is a prior probability; When \(j = z\), the dimensions of \(p_j\) and \(p_z\) match.

Among many of the available copulas, say the investor wants to model the dependence of the market as a clayton copula to ensure the lower tail dependency does not go unnoticed. The estimation is simple to implement with the package `copula`:

    library(copula)

    # copula (pseudo-observation)
    u <- pobs(x)

    # copula estimation
    cop <- fitCopula(
      copula = claytonCopula(dim = ncol(x)),
      data   = u,
      method = "ml"
    )

    # simulation
    r_cop <- rCopula(
      n      = 100000,
      copula = claytonCopula(param = cop@estimate, dim = 4)
    )

First, the copulas are computed component-wise by applying the marginal empirical distribution in every \(k\) column of the dataset. Second, the pseudo-observations are used to fit a Clayton copula by the Maximum-Likelihood (ML) method. Finally, a large panel with dimension \(100.000 \times 4\) is constructed to match a Clayton copula with \(\hat \alpha = 1.06\), as in the object `cop`.

The views on the market is constructed with the `view_on_copula`:

    views <- view_on_copula(x = u, simul = r_cop, p = p_z)
    views
    #> # ffp view
    #> Type:  View On Copula
    #> Aeq :  Dim 34 x 1859
    #> beq :  Dim 34 x 1

Once again, the solution of the minimum relative entropy can be found in one line of code:

    ep <- entropy_pooling(
      p      = p_j,
      Aeq    = views$Aeq,
      beq    = views$beq,
      solver = "nloptr"
    )
    ep
    #> <ffp[1859]>
    #> 0.0001990935 0.0004161479 0.0008166059 0.0006512624 0.0003776943 ... 0.0003269579

And the `autoplot` method is available for the objects of the `ffp` class:

    autoplot(ep) +
      scale_color_viridis_c(option = "C", end = 0.75) +
      labs(title    = "Posterior Probability Distribution",
           subtitle = "View on Copulas",
           x        = NULL,
           y        = NULL)

To stretch the current example, assume the investor stresses the parameter \(\hat \alpha = 1.06\)7. Higher \(\alpha\) values are linked to extreme occurrences on the left tails of the distribution. Whence, consider the case where \(\hat \alpha = 5\):

    cop@estimate <- 5

Rerun the previous steps (simulation, views and optimization):

    # simulation
    r_cop <- rCopula(
      n      = 100000,
      copula = claytonCopula(param = cop@estimate, dim = 4)
    )

    # #views
    views <- view_on_copula(x = u, simul = r_cop, p = p_z)

    # relative entropy
    ep <- entropy_pooling(
      p      = p_j,
      Aeq    = views$Aeq,
      beq    = views$beq,
      solver = "nloptr"
    )

To find the new posterior probability vector the satisfy the stress-test condition:

    autoplot(ep) +
      scale_color_viridis_c(option = "C", end = 0.75) +
      labs(title    = "Posterior Probability Distribution",
           subtitle = "View on Copulas",
           x        = NULL,
           y        = NULL)

1. Meucci, Attilio, Fully Flexible Views: Theory and Practice (August 8, 2008). Fully Flexible Views: Theory and Practice, Risk, Vol. 21, No. 10, pp. 97-102, October 2008, Available at SSRN: https://www.ssrn.com/abstract=1213325↩︎

2. The commands `x[ , 4]` or `x[ , "FTSE"]` yield the same results.↩︎

3. An ARIMA, GARCH or VAR model could be used as well. The focus is not on the best forecasting method, but how to input the view once the forecast has already been made.↩︎

4. `cor(ftse, dax) = 0.83`.↩︎

5. In the `x` object, `DAX` it’s in the first column and the `SMI` it’s in the second column.↩︎

6. Meucci, Attilio, Effective Number of Scenarios in Fully Flexible Probabilities (January 30, 2012). GARP Risk Professional, pp. 32-35, February 2012, Available at SSRN: https://www.ssrn.com/abstract=1971808.↩︎

7. `cop@estimate`↩︎