# Ortec Finance – Forecasting Financial Scenarios using Expert Opinions #SWI2016

**1 About Ortec Finance**

Ortec Finance is a company of about 200 employees and its two main offices are located in Rotterdam and Amsterdam. The central mission of Ortec Finance is *improving financial decision-making and monitoring by providing understanding, transparency and control through a combination of market knowledge, mathematical models, and Information Technology (IT)*.

Pension funds form a significant part of our clients. The way in which we use mathematical models for pension funds can be summarized as follows. Given historical data, we use time-series models, i.e., discrete time models of the form \( x_{t+1} = f (x_t ) + \) stochastic error term, to calculate a density forecast (multivariate and very high dimensional) for the future evolution of finance and economic variables like economic growth, interest rates, equity indices, etc. If a pension fund tells us how they are invested and what the structure of their liabilities is (i.e., the pensions they have to pay in the future) we can use the models to answer questions like: “what is the probability that you have insufficient money over five years”.

**2 Background problem description**

A key component of our services is the ability to forecast financial and economic variables for the next decades, on monthly time steps, in a stochastic manner. For this, we use a proprietary model: the Dynamic Scenario Generator. Multi-variate aspect modeling, i.e., modeling multiple interest rates, stock indices, macroeconomic variables, and so on at the same time, is very important to us.

The scope for the problem we would like to address is initially univariate, though we are highly interested in a multivariate solution to the problem. For the sake of simplicity, we will formulate the problem in a univariate manner. Consider an interest rate \(r_t\)* *, \(t\)* *denoting time on a monthly basis, and suppose we know from historic data the value of \(r_t\)* *up to now: \(t_0\) . We would like to make a stochastic/density/scenario forecast for \(t > t_0\) using both historic data and expert opinion. Typically, for short-term dynamics, we rely (fit model) on historic data, but for longer-term behavior (e.g., 10-year changes, the long term mean) we want to be able to adjust the model based on expert opinion. So in particular we are *not *just looking for models that can best fit a set of data, but models that also provide dials we can use to adjust the model to our views.

**3 Problem description**

Our model to forecast interest rates is of the form

$$r_t = r_t^{\text{Trend}} + r_t^{\text{Business cycle}} + r_t^{\text{Month}}$$

where \(r_t^{\text{Month}}\) models the short-term movements, \(r_t^{\text{Business cycle}}\) the medium-term (business cycles) fluctuations and \(r_t^{\text{Trend}}\) long-term (trend) fluctuations. This enables have explicit control over different horizons and use different samples of historic data for different models. The idea is more or less as follows: short-term fluctuations are significant changes on a monthly basis but do not accumulate to annual changes. Medium-term fluctuations are significant changes year-on-year that do not cumulate to decade-on-decade changes, but also have little intra-year changes. Long-term fluctuations are significant decade-on-decade changes, but vary little year-on year, let alone intra-year. Together \(r_t^{\text{Trend}}\) , \(r_t^{\text{Business cycle}}\) , \(r_t^{\text{Month}}\) should capture all dynamics, but they should not interfere (too much). This three-layer approach is something we want to keep, our question is in the modeling of these three components. Let us first discuss how this is done at the moment: band-pass filtered AR models.

A very simple approximate band pass filter consists of a Discrete Fourier Transform (DFT) of the input data, deletion of all the frequency components in the stop-band, and applying an inverse DFT to obtain filtered series in the time domain. The transformation of a data (column-)vector \(y\) to the frequency domain and the selection of frequencies in the pass band can be written in as the matrix multiplication \(WSy\), where \(S\) is a binary selection matrix that selects the columns from the DFT matrix \(W\) , which has entries \(w_{jk}\) given by

$$w_{jk} = \omega^{jk}/\sqrt{N},\quad \omega = e^{-\frac{2\pi i}{N}}, \quad i^2=-1, \quad N=\dim(y)$$

The inverse DFT can likewise be written as a matrix multiplication. We simulate our interest rates using AR models, but let us assume for simplicity that we use AR(1) with filtered error terms:

$$r_t = ar_{t-1} + \widetilde{\varepsilon_t}, \quad t-t_0 = 1, \ldots, N$$

where the error terms \(\widetilde{\varepsilon}_t\) have been filtered to the appropriate band-pass using the simple Fourier transformation filter described above. More explicitly:

$$\widetilde{\varepsilon}_t = \left(WS \varepsilon\right)_t$$

For the short-horizon component, this is essentially the model we use, where in particular \(t\) indeed is specified in monthly time steps. For the medium-term model we use the form, but in this case, let \(t\) take yearly steps. For the long-term model, decade steps are used. The medium and long-term models are then interpolated to a monthly basis.

Our question boils down to

- Can you design a (VAR?) model that automatically splits scenarios into three distinct/non-overlapping/non- interfering frequency bands?

- Would a single model (that has the pass bands as input parameter) suffice or should we consider three distinct models for each frequency scale?

- How does your solution extend to the multivariate situation?

- Do you have other insights or knowledge that fit with this theme and could be useful for making a frequency restricted model?

For the purpose of the study group, we would like to stick with the DFT as a means to measure and split frequencies unless it is clear the filtering methodology plays an important role.

We will provide you with real financial long-term data.