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) 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.
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 average:
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
parameter 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).
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():
Rest on the investor to calibrate this statistic in order to get a
desired confidence level.
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 \(\hat \alpha = 1.06\). Higher
\(\alpha\) values are linked to extreme
occurrences on the left tails of the distribution. Whence, consider the
case where \(\hat \alpha = 5\):
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)
