Seasonality in Portfolio Risk Calculations

UMEÅUNIVERSITET Department of Physics Master’s Thesis in Engineering Physics HT 2019, 30 Credits

Empirical study on Value at Risk of agricultural commodities using conditional volatility models

v1.0

Author Emil Söderberg

Nasdaq Umeå University Supervisor: Anders Stäring Supervisor: Joakim Ekspong Examiner: Markus Ådahl Seasonality in Portfolio Risk Calculations

Abstract

In this thesis, we examine if the conditional volatility models GARCH(1,1) and eGARCH(1,1) can give an improved Value-at-Risk (VaR) forecast by considering seasonal behaviours. It has been shown that some markets have a shifting volatility that is dependent on the seasons. By including sinusoidal seasonal components to the original conditional models, we try to capture this seasonal dependency. The examined data sets are from the agricultural commodity market, corn and soybean, which are known to have seasonal dependencies. This study validates the volatility performance by in-sample fit and out-of-sample fit. Value-at-Risk is estimated using two methods; quantile estimation (QE) of the return series and filtered historical simulation (FHS) from the standardized return series. The methods are validated by a backtesting process which analyse the number of outliers from each model by the Kupiec’s POF-test (proportional of failures) and the Christoffersen’s IF-test (interval forecast). The results indicates that the seasonal models has an overall better volatility validation than the non-seasonal models but the effect from the distributional assumption of standardized returns varies depending on model. Both VaR methods, for non-seasonal and seasonal models, seems however to benefit from assuming that the standardized return series experience kurtosis. The QE method is greatly dependable on the distributional assumption and almost all models assuming normal distribution fails the POF-test. The seasonal effects on VaR is in general the same for both commodities. Years with more dominant seasonal trends show an increased volatility during the summer season and generally a lower volatility during winter season. It is shown that the seasonal models have fewer violations during summer and is closer to the expected value of violation. It is hard to determine which model has the best overall performance, but the seasonal models have a better VaR estimate during summer.

1 Contents

1 Introduction1

2 Model speciﬁcation3 2.1 GARCH...... 3 2.2 eGARCH ...... 3 2.3 Seasonal components...... 4 2.4 Conditional distributions...... 5 2.4.1 Normal distribution ...... 5 2.4.2 Student’s t ...... 5

3 Value at Risk6 3.1 Historical simulation...... 6 3.2 Quantile estimation...... 7 3.3 Filtered historical simulation ...... 7

4 Estimation8 4.1 Maximum likelihood estimation...... 8 4.1.1 Normal distribution ...... 8 4.1.2 Student’s t distribution ...... 9 4.2 Information criterion...... 9

5 Forecast 10 5.1 Volatility validation ...... 10 5.2 Backtesting Methods...... 11 5.2.1 Kupiec’s POF-Tests ...... 12 5.2.2 Christoﬀersen’s Interval Forecast Test ...... 12

6 Data 14

7 Empirical study 16 7.1 Corn...... 16 7.1.1 Forecast...... 18 7.1.2 Out-of-sample ﬁt...... 20 7.1.3 Backtesting...... 21 7.1.4 Seasonal behaviour of VaR ...... 25 7.2 Soybean...... 28 7.2.1 Forecast...... 29 7.2.2 Out-of-sample ﬁt...... 32 7.2.3 Backtesting...... 32 7.2.4 Seasonal behaviour of VaR ...... 36

8 Conclusions 39

A Thomas Reuters 43 A.1 Data extraction: Settlement prices of Corn ...... 43

B Tables 43 B.1 Corn...... 43

i Seasonality in Portfolio Risk Calculations

B.2 Soybean...... 45

C Figures 47 C.1 Corn...... 47 C.2 Soybean...... 49

ii Seasonality in Portfolio Risk Calculations

1 Introduction

Volatility forecasting has an essential role on the financial market and is a big part of risk management. One method that incorporates volatility to measure the market risk of a portfolio of assets is Value-at-Risk (VaR), which has become one of the most popular risk of loss measurement techniques. VaR can be described as the potential loss in value of a risky asset or portfolio over a defined period for a given confidence interval. Many VaR models use distributional assumptions of the return series which involve numerical calculations of volatility.

One way to calculate volatility is by using conditional volatility models. In 1982 Engle introduced the autoregressive conditional heteroscedasticity model (ARCH) that tries to capture the time-varying volatility and volatility clustering, that is commonly seen in financial time series.[1] The model estimates the variance term by using the past values of the squared return series, which is a normal volatility proxy. This model was further generalized by Bollerslev into the generalized ARCH model (GARCH) which also include the past volatility calculation in the model.[2] The GARCH model has been used to a wide extent and the GARCH(1,1) model is by far the most popular GARCH model.[3] However, the volatility estimate from the family of ARCH-models may not give sufficient results when the volatility of financial time series has seasonal patterns.

The future commodity market, especially the agricultural sector, is well-known for its seasonal behaviour due to harvest seasons and the periods of the year. There are two different kinds of seasonality however, price and volatility. Price tends to change according to the supply which typically leads to higher prices of future contracts with delivery prior to harvest and low prices after harvest when the supply is full.[4] The volatility changes mainly by the effect from information. An example using agricultural commodities is the weather. The months preceding harvest hold a lot of important information since it will have a direct impact on the final yield. This type of information has minor impact on the harvest season and little or no after harvest. Other factors to consider is large information releases like monthly or yearly reports.

Earlier work by Goodwin et al. evaluated determinants of price variability in U.S. corn and wheat futures markets and revealed important seasonal and autoregressive eﬀects in the volatility.[5] They further showed that conditional heteroscedasticity models can estimate the volatility in a eﬀective way. Another study by Wang et al. examined seasonality, long-term memory and structural change of corn future market by using the Fractional Integrated GARCH model (FIGARCH).[6] They concluded that seasonality, structural change and long memory plays an important role in volatility estimation. The importance of long memory was not as important at lower time horizons (1-day ahead) in the presence of seasonality and structural change.

In this study we examine if the 1-day ahead VaR estimate of agricultural futures can further be improved by including seasonal components to the conditional volatility model. The chosen conditional models are the GARCH model and the exponential GARCH model (eGARCH) introduced by Nelson.[7]

The GARCH model is preferred over the FIGARCH model due to simplicity and that Wang et al. showed that the two models had very similar results at the 1-day ahead forecast. The eGARCH model is chosen since it can account for asymmetry in time series.

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It is a popular idea that the return series of commodity futures is positively skewed since commodities have positive exposure to supply shocks.

Regarding the statistical assumption of the return series, it either follows normal distribution or student’s t distribution which has heavier tails than normal distribution. The student’s t distribution is motivated by a study from Kat et al. which concludes that the distribution of commodity futures returns has fat tails and less signiﬁcant skewness.[8]

The VaR estimate will be conﬁrmed by the number of violations and two VaR tests; Kupiec’s proportion of failure test and Christoﬀersen’s interval forecast test.[9, 10] In our analysis, we will also compare how well the seasonal and non-seasonal VaR models are behaving during the year.

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2 Model speciﬁcation

This section introduces the selected models and the continuous distributions used throughout this report. The chosen ﬁnancial time series models are extensions of the auto-regressive conditional heteroscedasticity (ARCH) model.[1] The fundamental 2 idea for the family of ARCH models is that the conditional variance σt (also known as volatility) of the ﬁnancial process rt has a time dependency and a relation to the process {rs, s < t}. The volatility is modelled as,

rt = σtet, {et} ∼ iid N(0, 1) (1) 2 where the volatility is related to the past values of rt according to, p 2 X 2 σt = ω + αirt−i i for some positive integer p and ω, αi > 0 for all i. The model originally assume that the standardized process {et} is independent and identically distributed (iid).

2.1 GARCH

The generalized ARCH or GARCH(p,q) process extends the ARCH model by including a relationship to the past values of the conditional variance according to, p q 2 X 2 X 2 σt = ω + αirt−i + βjσt−j (2) i=1 j=1 for some positive integer p, q and ω, αi, βj > 0 for all i, j.[2] The model has a stationary solution when, p q X X αi + βj < 1 i=1 j=1 One of the models used throughout this report is the stationary GARCH(1,1) which simpliﬁes eq(2) to, 2 2 2 σt = ω + α1rt−1 + β1σt−1 (3)

2.2 eGARCH

The exponential GARCH model or eGARCH was introduced to account for asymmetry with respect to negative and positive disturbances.[7] This attribute is important in portfolio risk calculations since large negative shocks usually has a greater impact on volatiliy than positive ones. This means that we expect negative news to have bigger inﬂuence on the price change then positive news. The model is based on the idea that the volatility model can use a relationship to the standardized process {et} in eq.(1) instead 2 of the {rt } process. The conditional variance is calculated in its natural logarithm form to enable the model to incorporate negative values. The volatility model is written as, p q 2 X X 2 log(σt ) = ω + g(et−i) + βj log(σt−j) i=1 j=1

3 Seasonality in Portfolio Risk Calculations where, g(et−i) = αi |et−i| − E |et−1| + γiet−i for some positive integer p, q and αi, βj > 0 for all i, j. The sign and scale of the γ term determines the asymmetric eﬀect,

If γ = 0, No asymmetric effect. If γ < 0, Volatility is more effected by negative shocks. If γ > 0, Volatility is more effected by positive shocks.

By proposing that et−i ≡ rt−i/σt−i the conditional variance model can further be rewritten into, p q 2 X rt−i rt−i X 2 log(σ ) = ω + αi − |et−i| + γi + βj log(σ ) (5) t σ E σ t−j i=1 t−i t−i j=1

The absolute standardized process |et−i| follows the half distribution function of the assumed conditional density and Wang et al. (2001) shows that the expected value of half normal distribution and half student’s t distribution equals, [11] qπ  2 , when et−i is gaussian distributed.   E |et−i| = q ν Γ[0.5(ν − 1)] when et−i is student’s t distributed with  ,  π Γ[0.5ν] ν degrees of freedom.

Throughout this report, we use the eGARCH(1,1) model which simplify eq(5) to, 2 rt−1 rt−1 2 log(σt ) = ω + α1 − E |et−1| + γ1 + β1 log(σt−1) (6) σt−1 σt−1

2.3 Seasonal components

The seasonal behaviour of the conditional variance process can be described by adding Fourier pairs to the conditional variance equation.[5] In this model, the seasonal process {st} is written as, p t t s = X a cos 2πi + b sin 2πi (7) t i T i T i where t is the current observation and T is the amount of observations during a year. Since the seasonal model is added to the conditional volatility model the centre of amplitude needs to be increased to avoid a negative volatility. The seasonal model is thereby assumed to have a centre of amplitude corresponding to the lowest numerical value of the seasonal process. The seasonal process can thus be deﬁned as,

St = st − min (st) (8)

Another approach is to consider the exponential-sinusoidal function deﬁned as,

st ESt = c e (9)

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The exponential-sinusoidal function has an extra parameter compared to the regular sinusoidal function and has the advantage of being strictly positive. The exponential function might also prove to be eﬃcient since the function has a more narrow shape around its peak than the regular sinusoidal function. The function thus has lower values for a longer time then the regular seasonal model St.

2.4 Conditional distributions

This section covers the distributions of the standardized process in eq.(1) used in this report. The general assumption is that the ﬁnancial process {rt} follows a normal distribution (gaussian distribution) which the GARCH model was originally based on. The model is symmetric around its mean value and the function gives a bell shaped appearance. However, real data is seldom a perfect normal distribution. Financial data has commonly a distribution with more points further away from the mean value than a normal distribution, this is known as a fat-tailed distribution and is measured by kurtosis. Kurtosis is a statistical measurement of the distribution-tails in relation to the tails of the normal distribution. Kurtosis is also known as the fourth standardized moment and is described by, " # X − µ4 Kurt[X] = . E σ Student’s t distribution is another bell-shaped distribution and it is chosen since it can incorporate the kurtosis eﬀect. 2.4.1 Normal distribution

The ﬁnancial time series rt in eq.(1) follows normality with zero mean and a variance in accordance to the given conditional volatility model. The conditional density function of the ﬁnancial time series can be written as,

r2 − t 1 2σ2 f (rt | ψt−1) = √ e t σt 2π where ψt−1 denotes the σ-field generated by all the available information up through time t − 1. 2.4.2 Student’s t The student’s t distribution for GARCH models was introduced in 1987 by Bollerslev and he showed that the conditional probability density function of the standardized innovation could be written as, ν+1 − ν+1 Γ 2 ! 2 2 rt fν (rt | ψt−1) = p ν 1 + 2 σt (ν − 2)πΓ 2 σt (ν − 2) where ν > 2 is the degrees of freedom parameter which sets the shape (or kurtosis).[12] The excess kurtosis of a student’s t distribution is defined as, 6 Kurt[X] = , ν > 4. ν − 4 Lower values of ν increases the kurtosis effect, which gives the distribution thicker tails. As ν increases it approaches the probability density function of the normal distribution, lim fν = f, which has an excess kurtosis of 0. v→∞

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3 Value at Risk

Value-at-risk is a measurement of the potential risk of loss for an asset or instrument. It is deﬁned as the smallest loss L an asset can take such that the probability of a future loss larger than L is less than or equal to α, where α is the chosen conﬁdence level. The VaR formula can be written as, [13] P r L ≤ −V aR α = α , α ∈ [0, 1] (10)

By assuming that the loss function is dependable on a single asset of one unit,

L =Pt+1 − Pt and the relation of continuous compounded return series, rt+1 = log Pt+1/Pt

rt+1 ⇒ L = Pt (e − 1) (11) the VaR formula can further be simpliﬁed into two cases, ⇒ α =P r L ≤ −V aR α rt+1−1 α =P r Pt e ≤ −V aR

Returns Standardized returns α α V aR rt+1 V aR 1 = P r rt+1 ≤ log − + 1 = P r ≤ log − + 1 Pt σ Pt σ α α which is possible since the −V aR /Pt ≤ 1. The simpliﬁcation shows that V aR can be written as a function of the quantile distribution function F −1(α) of either the return series or the standardized return series,

Returns Standardized returns

−1 −1 α F r (α) α F e (α)σ V aR = −Pt e − 1 (12) V aR = −Pt e − 1 (13) −1 −1 ≈ −Pt F r (α) ≈ −Pt F e (α)σ

The quantile distribution function can further be estimated in two ways, either by a quantile estimation (QE) or by a historical simulation (HS) approach.

3.1 Historical simulation

One of the most widespread approaches to calculate VaR is the HS approach. The HS method is popular since it makes no assumption of the return distribution and is quite easy to implement. One of the shortcomings, however, is that the method applies equal weight when estimating the quantile distribution. The HS formula for the quantile distribution function can be written as,

−1 Fb r (α) = XbαNc where X is the sorted return series, rj−1 < rj for j = 1, ..., N.

6 Seasonality in Portfolio Risk Calculations

3.2 Quantile estimation

Compared to the HS, the QE method assumes that the return series can be modelled according to a chosen distribution, in this case by a conditional volatility model. The QE is easily computed for normal distribution but in the case of student’s t distribution one needs to scale the volatility estimate by the variance implied by ν i.e. σ = v/(v − 2). The QE for each distribution is thus,  σzˆ α , when r is gaussian distributed. −1  Fb (α) = r q  v − 2 σˆ v tα,ν , when r is student’s t distributed. where σˆ is the volatility estimate. zα and tα,ν denotes the quantile function of gaussian distribution and student’s t distribution respectively.

3.3 Filtered historical simulation

An extension of the regular HS approach is the ﬁltered historical simulation (FHS) method. Unlike the regular HS method the FHS method cannot neglect the assumption of no distribution since the VaR estimation includes the estimated standard deviation. The quantile distribution function is however estimated in the same manner as the HS,

−1 Fb e (α) = XbαNc where X is the sorted standardized return series, et−j−1 < et−j for j = 1, ..., N.

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4 Estimation

The parameter estimation of the conditional volatility models uses the solver solnp() which is included in the Rugarch package in R. The rugarch package provides a large set of univariate GARCH processes and will be used in this report to estimate the non-seasonal models, GARCH and eGARCH. The main solver is the solnp() function which solves the general nonlinear programming problem using a augmented Lagrange multiplier method with an SQP interior algorithm. The algorithm was originally written in matlab by Yinyu Ye. The function and algorithm is explained in his user’s guide which is based on his Ph.D. Thesis. [14]

4.1 Maximum likelihood estimation

The maximum likelihood estimation (MLE) is a statistical estimation technique to ﬁnd the optimal parameters of given statistics. As implied by the name, an increasing value from the MLE function indicates that the observed data with given model parameters are more probable. The maximum likelihood of the conditional volatility models is deﬁned as,

n Y L(θ | r) = f(rt | ψt−1) t=1 where f is the conditional density model of given statistics and θ deﬁnes its parameter space. 4.1.1 Normal distribution

The MLE function with normal distribution can be written as,

r2 n − t Y 1 2σ2 Ln(θ | r) = q e t 2 t=1 2πσt which is further simpliﬁed in its logarithmic form,     n 2 X 1 rt LLn(θ | r) = log q  − 2  2 2σt t=1 2πσt " # 1 n n r2 = − n log (2π) + X log σ2 + X t 2 t 2 t=1 t=1 σt

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4.1.2 Student’s t distribution

Following the same reasoning, the student’s t log likelihood function of the MLE can be written as,

 ν+1 − ν+1  n Γ 2 ! 2 X 2 rt LLt(θ | r) = log  1 +  p 2 ν ν t=1 σ (ν − 2)π Γ 2  ν+1  Γ 2 = n log  p ν  σ (ν − 2)π Γ 2 " !# 1 n r2 − X log σ2 + (ν + 1) log 1 + t 2 t 2 t=1 σt (ν − 2)

4.2 Information criterion

Information criteria is a statistical model selection. The diﬀerent approaches aim to ﬁnd the best model (the model with the lowest information loss) while taking into account the simplicity of model. There are multiple tests that incorporates this reasoning and the most common one is the Akaike information criterion (AIC). By assuming a statistical model of k estimated parameters with a maximum value L from a likelihood function, the AIC formula is,

AIC = 2k − 2 ln (Lˆ)

The AIC formula thus punishes the maximum value from the likelihood function by the number of parameters and the best model is given by the lowest AIC-value.

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5 Forecast

The forecasting evaluation can be made in various ways depending on the area of interest. The main goal of this report lies in forecasting Value at Risk and our main focus will be on the out-of-sample backtesting, discussed in section 5.2. The forecasting procedure will also include the overall performance of the conditional volatility model to reassure that the forecasted volatility is near its estimator.

The forecasting process of the volatility uses a rolling window method which means that each h-day ahead forecasted volatility is estimated from a data set of a speciﬁc window size m. The data set contains n samples and the volatility will thus be forecasted n − m 2 times and span the period {σs+h | ψs−1+h, s ∈ [m + 1, n]}.

5.1 Volatility validation

Volatility estimation is a well-known problem since volatility is a latent variable that cannot be observed directly. The estimated volatility will thus have to rely on a proxy that contains the useful information up to a given time. The most common volatility proxy is the daily return squared of the close-to-close price,

2 2 σˆt+1 = rt+1.

The simplicity of this proxy is favourable, but it has been shown that the results can be fairly noisy. The performance of the conditional volatility models was questioned at first due to poor results using this proxy. It was later shown however, that conditional volatility models actually have a high accuracy comparing the results to a more fitting proxy, the cumulative squared intraday returns. By assuming that each intraday has m observations the realized volatility is instead defined as,

m 2 X 2 σˆt+1 = ri,m,t+1 i=1 The cumulative squared intraday returns give a better estimation of the volatility but unfortunately, high frequency data is hard and costly to attain and is hence not an option for this study.

The evaluation of the conditional volatility is validated using the loss functions mean-squared-error (MSE) and the "QLIKE" which, according to Patton (2010), are the most robust and homogeneous loss functions.[15] The loss functions are deﬁned as,

T 1 2 MSE : X σˆ2 − h (14a) T t+1 t+1 t=1 " ! # 1 T σˆ2 σˆ2 QLIKE : X t+1 − log t+1 − 1 (14b) T h h t=1 t+1 t+1 where σˆ is the volatility proxy and the h is the conditional volatility.

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5.2 Backtesting Methods

The theory of VaR gives great methods of measuring the potential risk an asset or instrument is carrying but is the result really reliable? Looking back on the previous chapter one can conclude that the VaR estimate carries a lot of assumptions that will greatly aﬀect the accuracy of the result. The backtesting process of the VaR calculation is thus very importance. There are multiple tests to validate the results from a VaR model and our main goal is to verify that eq.(10) is correct. We start by introducing the violation index,

α It+1 = 1 lt+1 ≤ −V aRt+1 where lt+1 is the actual loss of day t + 1. Comparing the loss function in eq.(11) and the VaR estimate in eq.(12,13) one can conclude that the violation index can be rewritten to only consider the daily return series,

α It+1 = 1 lt+1 ≤ −V aRt+1 rt+1 α = 1 Pt(e − 1) ≤ −V aRt+1

Returns Standardized returns

F −1 (α) F −1 (α) σ rt+1 r |ψ rt+1 e |ψ t+1 = 1 Pt (e −1) ≤ −Pt e s s−1 −1 = 1 Pt (e −1) ≤ −Pt e s s−1 −1 n o n o = 1 r ≤ − F −1 (α) (15) = 1 r ≤ − F −1 (α) σ (16) t+1 rs|ψs−1 t+1 es|ψs−1 t+1

−1 where F is the quantile distribution function and ψs denotes the σ ﬁeld generated by the available information up through time s given s < t + 1. This reformulation will make the violation process easier to visualize since the daily return series has its mean close to zero.

The most common and straight-forward backtest is to check if the number of violations is close to the expected number of violations. Looking at the deﬁnition of VaR, eq.(10), the expected violations x from N number of VaR calculations is simply,

N X h i x = E It+i i=1 = αN

By comparing the actual amount of violation with the expected amount of violation, the user gets an guess of how well the VaR model works. But is this enough? To be able to tell if the VaR model really behaves in an acceptable manner two other tests are applied; the POF-test (proportion of failure) by Kupiec, and the IF-test (interval forecast) by Christoﬀersen.

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5.2.1 Kupiec’s POF-Tests

The POF test is widely used as it is the first published backtesting methodology. The test is an unconditional coverage test that measures if the number of violations/failures is within a chosen confidence interval. The test assumes that the VaR model gives the correct violation ratio and that the number of violations follows a binomial distribution. The null hypothesis is thus, xˆ H : α =α ˆ = 0 N To find out if xˆ is significantly different from x, Kupiec suggested that a likelihood-ratio test was the best choice, ! (1 − α)N−x αx LR = −2 ln POF x N−x x x 1 − N N

The probability distribution of the LRPOF -value when the null hypothesis is true, is 2 asymptotically χ (chi-squared) with one degree of freedom. The LRPOF -value will thus be rejected if it exceeds the critical value of χ2 distribution for a certain signiﬁcance level.

The numerical calculation of LRPOF is quite unstable in its current form since larger observations sets N will make the nominator and denominator in the log expression converge to zero. This eﬀect is avoided, or at least postponed, by rewriting the formula into,   N −1 (1 − α) x α LRPOF = −2x ln  N  x x −1 x 1 − N N

5.2.2 Christoﬀersen’s Interval Forecast Test

Christoffersen proposed the first framework to evaluate conditional coverage by interval forecasting. The test aims to determine if the violation index obtained from the VaR model is statistically independent. The test is formulated in a similar manner as the Kupiec’s test but aims to explain if the observed violations depend on the outcome of the previous observation. The procedure starts by defining the number of conditional outcomes nij as the number of outcomes j on the condition that it follows from outcome i the previous observation. The numbers can be visualized in table1.

Table 1 – Visualization of the conditional outcome nij used in the CIF test, i.e. the number of outcomes j given outcome i of the previous observation.

It = 0 It = 1

It+1 = 0 n00 n10 n00 + n10

It+1 = 1 n01 n11 n01 + n11

n00 + n01 n10 + n11 N

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The conditional outcome is then used to deﬁne the probability that a violation is occurring given condition i of the previous observation,

n01 n11 n01 + n11 π0 = , π1 = , and π = n00 + n01 n10 + n11 n00 + n01 + n10 + n11 Finally, the Christoﬀersen’s interval forecast likelihood-test is formulated as, ! (1 − π)n00+n10 πn01+n11 LRIF = −2 ln n00 n01 n10 n11 (1 − π0) π0 (1 − π1) π1

The probability distribution of the LRIF -value is the same as for the LRPOF -value. Under the assumption that the null hypothesis is true it is asymptotically χ2 (chi-squared) with one degree of freedom.

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6 Data

In the empirical study we consider two agricultural commodities: corn and soybean. Most commodities are traded by future contracts (or futures) which is a standardized forward contract; a legal agreement to buy or sell a certain quantity of a commodity at a predetermined price on a future delivery date. The delivery dates are predetermined and diﬀers depending on commodity. The standardized trading months for agricultural commodities are typically reﬂected by the seasonal patterns for planting, harvesting and marketing the underlying crop. The expiration months for corn and soybean can be seen in table2.

Each commodity has a data set containing something called continuous future contracts. A continuous future contract is not a real future contract but several future contracts spliced together to form a single future contract, that spans a longer time period. The continuous future contracts used in the study are labelled by numbers corresponding to the upcoming future contract. The ﬁrst continuous future contract will thus include the ﬁrst upcoming future contract at each date throughout the chosen time period.

Both commodities are traded on CBOT [16] and was extracted from the Reﬁnitiv Data Platform Eikon [17]. The data set descriptions can be seen in table2.

Table 2 – Description of data sets obtained from Eikon. Both commodities are traded on CBOT.

Commodity Dates Start date End date Cont. futures Expiration months

Corn 5037 22/10/1999 21/10/2019 14 Mar, May, Jul, Sep, Dec Soybean 5033 7/12/1999 4/12/2019 17 Jan, Mar, May, Jul, Aug, Sep, Nov

The data can be accessed by using the reuters instrument codes RIC for each continuous commodity contract. The RIC of a continuous commodity contract is a combination of,

h i Root code of commodity, lower-case letter “c”, contract number # , where the root codes for corn and soybean is C and S. The RIC of the second continuous corn contract is thus Cc2. The data can easily be obtained by the Thomson Reuters toolbar in excel and a example cell formula to extract the settlement price for 14 continuous contracts of corn the past 20 years can be seen in appendix A.1.

The data used for the models will have a constant time to maturity (MT ), which means a constant time until the contract expiries. Since the expiration date of the contracts are predetermined the constant-settlement price is determined by interpolating the settlement prices from the active contracts. The settlement price from each active contract forms a curve which is known as a forward curve. The curve can be determined in various ways but the interpolation method used is spline interpolation. The obtained constant-maturity futures prices are denoted by ft(MT ). Some problems might arise when interpolating the future curve since the gap between the contracts might be to large to catch the actual behaviour of the future curve. The chosen MT is six months (MT = 0.5 years) to make sure that there exists at least two control points (two future contracts) before the estimate.

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The data to be used in the conditional volatility model is the daily returns,

f rt+1 ≡ ln ( t+1/ft)

The seasonal conditional models assume a annual time period of 252 observations which is the general assumption of the U.S. market. The actual number of annual observation lies in the time span of T ∈ [250, 253] trading days but the change is neglectable over the whole data sets.

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7 Empirical study

This section covers the empirical study of two data sets, corn and soybean. Each return serie has been extracted according to the procedures in section6. The in/out-sample ﬁtting will compare the seasonal model and its corresponding non-seasonal model. The conditional volatility model abbreviations used in this section is a combination of the abbreviations from Seasonality model, conditional volatility model and the distribution of standardized returns. The abbreviations are,

Model Abbreviations Sinusoidal S Seasonality Exponential sinusoidal ES Generalized ARCH GARCH Volatility Exponential GARCH eGARCH Normal n Distribution Student’s t s

An example is the "ESGARCHs" model which uses the exponential sinusoidal (ES) model along with the GARCH conditional volatility model assuming student’s t distributed standardized returns (s).

7.1 Corn

Future contracts of the agricultural commodity corn are well-known for its seasonal behaviour. The data set contains 5037 observations of constant-maturity futures settlement prices and spans the time period 22/10/1999-21/10/2019. The price curve along with its daily return series is shown in figure1. By inspection one can note that the price changed drastically during the years 2007-2008 and 2010-2013 which also resulted in a more volatile time period. These changes were an effect of the world food crisis which made the prices go to world breaking records. We will later in this section see that the past food crisis will influence the performance of the conditional volatility models.

16 Seasonality in Portfolio Risk Calculations

(a) (b) 800 0.05 700 600 500 0.00 Price: P(t) 400 Daily return: r(t) 300 −0.05 200 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years [t] Years [t] Figure 1 – Time series of a) the price from the transformed continuous future contract of corn and b) its corresponding daily returns.

It is hard, by looking at the daily returns, to see if there exists a seasonal behaviour in the price volatility. To motivate this study of the seasonal volatility properties of corn we check if our volatility proxy, also known as the realized returns in ﬁgure 2a, has a seasonal pattern. A simple approach is to take the average over a set interval of days each year, divide it by the annual average and then take the average of that interval for every year. The annual behaviour of the realized returns can be seen in ﬁgure 2b where the annual interval is 7 trading days, thus giving 36 intervals over a year of ∼252 trading days. The obtained curve seems to indicate that there exists a seasonal behaviour that is behaving in a sinusoidal fashion. This further supports the theory that the seasonal behaviour of the volatility can be described by Fourier pairs, as proposed in section 2.3.

The proposed seasonal component models can also be motivated by inspecting the auto correlation function ACF of the realized returns. Figure 2c illustrates the ACF of realized returns where one lag corresponds to 252 trading days i.e. one year of business days. It is clear that the correlation diﬀers greatly from zero and has a strong sinusoidal correlation with a frequency of one lag. The ACF plot seems to indicate that there is also a long term structural change as concluded in a previous study.[6] The focus of this study lies in the seasonal change however, and the structural change was thus not included in the conditional volatility models used.

17 Seasonality in Portfolio Risk Calculations

(a) 0.006 0.004 0.002 Daily return squared: r(t)^2 0.000 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years [t]

(b) JAN MAR MAY JUL SEP NOV (c)

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● ● ● 10 ● ● ● ● ● ● ● ● −0.05

0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 18 Business days [t] Lag in Years [a.c] Figure 2 – Time series of a) the squared daily return series along with its b) the annual average and c) auto correlation function. The annual average is measured by calculating the average over a 7-day interval each year, divide it by the annual average and then take the average of that interval for every year.

7.1.1 Forecast

The forecast process is a 1-day ahead volatility forecast estimated from a rolling window of 1260 data points (5 years of business days). This gives a forecast period of 3777 data points which spans the time period 2004/09/20 - 2019/09/12. The in-sample fit from each estimation corresponds to the AIC-values obtained from the "optimal" parameters at each forecast. Plotting the AIC-process from each model gives us an opportunity to see how the "goodness-to-fit" ratio changes during the forecast period. It is quite hard to determine how well the different models are performing due to the sheer number of models and since the AIC-value can change quite fast during the forecast period. As such, the AIC-process from each model needs to be plotted in different ways. Figure3 holds four plots,

18 Seasonality in Portfolio Risk Calculations

c-d) The AIC-process of normally and student’s t distributed models, respectively, to see if the distributional behaviour for each model is the same. The GARCH model is set as reference for both cases.

(a) GARCHn SGARCHn ESGARCHn eGARCHn SeGARCHn ESeGARCHn (b) GARCHn GARCHs SGARCHs ESGARCHs eGARCHs SeGARCHs ESeGARCHs GARCHs 0 −20 −6500 −40 AIC AIC −7000 −60 −80 −7500 −100 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years Years

(c) GARCHn ESGARCHn SeGARCHn (d) GARCHs ESGARCHs SeGARCHs SGARCHn eGARCHn ESeGARCHn SGARCHs eGARCHs ESeGARCHs 10 10 0 0 −10 −10 AIC AIC −20 −20 −30 −30 −40 −50 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years Years Figure 3 – AIC value estimated from the log-likelihood function from the optimal parameters for the whole forecasting period. The different plots correspond to the AIC-processes of: a) All models. b) GARCH model of both distributions with normal distribution as reference. c) Models of normal distribution with GARCH model as reference. d) Models of student’s t distribution with GARCH model as reference. The difference between a certain AIC-process and the reference AIC-process is estimated with local regression since the actual difference is quite volatile.

The first thing to note in figure 3a is that the overall performance from the conditional volatility models are decreasing during 2007-2012 (an increasing AIC-value). This is most likely due to the world food price crisis which increased the food price drastically and most likely made the return series more turbulent (less clustered). Another noticeable feature is that the models using student’s-t distribution have better "goodness-to-fit" than the models using normal distribution. This is most prominent when considering plot 3b comparing the performance of the GARCH model of different distributions.

Using the GARCH model of respective distribution as a reference point, plot 3c and plot 3d, show very similar results. The seasonal models seem to have better "goodness-to-ﬁt" during normal market conditions but it is not as distinct for forecasts that includes returns from the two food crisis. It can also be noted that the eGARCH model is performing better than the GARCH model for the majority of the forecast period.

19 Seasonality in Portfolio Risk Calculations

One should keep in mind that the goodness-to-ﬁt estimate by the rugarch package is calculated in a slightly diﬀerent manner. The result is thus not absolute.

The seasonal behaviour is changing to some extent during the forecasted period. An example of how the seasonal component model is complementing the volatility estimate is shown in ﬁgure4 for two cases assuming a student’s t distribution; the last estimate (2019/09/12) where the seasonal models are outperforming the non-seasonal models and one estimate during the food crisis (2008/03/17).

(a) JAN MAR MAY JUL SEP NOV (b) JAN MAR MAY JUL SEP NOV

SGARCH SGARCH ESGARCH ESGARCH 1.2e−04 SeGARCH SeGARCH ESeGARCH ESeGARCH 2.0e−04 8.0e−05 1.0e−04 4.0e−05 Seasonal complement [a.u.] Seasonal complement [a.u.]

0.0e+00 0 50 100 150 200 250 0.0e+00 0 50 100 150 200 250 Business days [t] Business days [t] Figure 4 – Seasonal component model estimate as a function of a business days, day 1 is the ﬁrst annual business day. Plot a) is the seasonal model estimated at the forecast date 2008/03/17 and plot b) is the seasonal model estimated at the forecast date 2019/09/12.

The estimate from the food crisis in figure 4a shows a case of weaker seasonality. The added volatility amplitude is at its peak around 11e-5 and the one can note that the exponential seasonal model differs greatly from the seasonal model. Both models have their maximum in the early summer, around business day 100 to 126. The seasonal model has also an amplified volatility in the beginning of the year which is not as intuitive at the summer peak.

The second plot from the last estimate, figure 4b, shows a more apparent seasonal behaviour then the seasonal models during the financial crisis. The added volatility is almost twice as large and has a more ideal seasonal pattern, with its maximum in the middle of summer, around trading day 126. Both seasonal models show a similar pattern but the exponential model has a higher amplitude then the seasonal model. This is expected since this is one of the attributes the exponential seasonal model carries. 7.1.2 Out-of-sample fit

The volatility validation compares the estimated volatility from each volatility model used with the volatility proxy, the daily return squared. The volatility validation used two loss functions, the MSE and QLIKE. The results from each calculation can be seen in table3.

20 Seasonality in Portfolio Risk Calculations

Table 3 – Volatility validation, using a moving window roll of ﬁve years. * The value is given in the seventh decimal, 10−7.

Model GARCH SGARCH ESGARCH eGARCH SeGARCH ESeGARCH Dist. N t N t N t N t N t N t MSE* 3.032 3.039 3.033 3.031 3.029 3.036 3.045 3.045 3.061 3.047 3.054 3.048 QLIKE 1.477 1.473 1.467 1.461 1.461 1.468 1.476 1.472 1.472 1.464 1.466 1.468

The volatility validation shows that the model with best out-of-sample fit is the seasonal GARCH model with exponential Fourier pairs, from both loss functions. It seems that the seasonal models have a better out-of-sample fit using GARCH conditional volatility but it is harder to determine the best performing eGARCH model. Considering the value from both loss-functions the best eGARCH model seems to be the seasonal model with student’s t. The different GARCH models gives in general better results than the eGARCH models. 7.1.3 Backtesting

This section focus on the behaviour of the violation index for the three VaR estimation techniques considered in section3; historical simulation (HS), Quantile estimation (QE) and ﬁltered historical simulation (FHS). The violation index in this section will only consider the lower VaR boundary which is considered the standard loss. The upper boundary is also interesting however, since the deﬁnition of loss depends on your portfolio position. The results of the upper boundary can be found in appendix B.1.

The number of violations from each VaR model can be visualized in table4.

Table 4 – Number of Violations using three VaR estimation models, HS, QE and FHS. Each estimation model is calculated by different VaR significant levels and conditional volatility models. The best performing model at each VaR significant level is highlighted by a bold style.

Number of Violations x for lower boundary

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% E[x] 19 38 189 378 19 38 189 378 Non-conditional 25 40 180 369 N 37 49 151 311 24 39 179 365 GARCH t 19 38 167 369 20 39 182 365 N 31 51 157 317 18 36 178 366 SGARCH t 20 33 173 366 20 38 180 371 N 33 52 154 312 19 38 178 364 ESGARCH t 22 35 167 360 20 38 173 371 N 34 58 153 310 20 36 191 361 eGARCH t 20 37 171 361 19 34 191 362 N 32 59 159 314 25 39 194 370 SeGARCH t 19 36 167 355 25 37 180 365 N 32 57 157 304 18 40 180 360 ESeGARCH t 21 37 169 346 21 40 183 360

21 Seasonality in Portfolio Risk Calculations

The QE method in table4 is in general providing a better VaR estimate by assuming that the daily returns has a student’s t distribution. This is to be expected since we expect the distribution of the daily returns to have fat tails. This seems to be the case for the FHS method as well, but it is not as easily detectable since the results seems to behave diﬀerently for each model. Considering all VaR signiﬁcant levels, the best performing QE model seems to be the GARCH model with student’s t distribution whilst the best FHS model seems to be the seasonal GARCH model with student’s t distribution.

It further seems like the non-seasonal models perform better than the seasonal models using the QE method. The FHS method does not have the same clear distinction between the seasonal models and the non-seasonal models. It seems to diﬀer from model to model.

To be able to tell if one can reject a VaR model by the number of violations a non-rejection region is calculated from the POF-test deﬁnition given the null-hypothesis signiﬁcant levels of p = 0.1, 0.05, 0.01. The rejection region for a forecast set of 3777 samples is shown in table5.

Table 5 – Nonrejection region of POF-test assuming a signiﬁcance level of p using a forecast set of 3777 samples. The signiﬁcance level of the null hypothesis of the POF-test is denoted by p.

Non-rejection Region for Number of violation N VaR α p = 0.1 p = 0.05 p = 0.01 0.5% 12 < N < 27 11 < N < 28 8 < N < 32 1% 28 < N < 49 26 < N < 51 23 < N < 55 5% 167 < N < 212 163 < N < 216 155 < N < 225 10% 347 < N < 409 342 < N < 415 331 < N < 427

Table5 shows that the non-rejection region is quite large for each signiﬁcance level, and most models should thereby pass the test if they are to be considered reliable. Choosing a signiﬁcance level of 95% and comparing the non-rejection region with the actual number of violation in table4 the following models pass the test,

22 Seasonality in Portfolio Risk Calculations

Table 6 – VaR models passing the POF-test using a null hypothesis signiﬁcant level of 95%. If the the number of violations from a VaR model is within the non-rejection region it is denoted by a check mark, otherwise nothing.

Models passing the POF-test

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% √ √ √ √ Non-conditional √ √ √ √ √ N GARCH √ √ √ √ √ √ √ √ t √ √ √ √ N SGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N ESGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N eGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N SeGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N ESeGARCH √ √ √ √ √ √ √ √ t

It is quite clear from table6 that one can reject all models that are using a QE estimation while assuming a normally distributed daily returns. Only one model passes the test under this assumption, but it should be noted that the number of violations is very close to the rejection region. All other VaR models passes the POF-test.

To test if the violation index is statistically independent, Christoﬀersen’s interval forecast test was used. The obtained value from each model can be seen table7. The rejection region is 3.841 which corresponds to the χ2 value of 95% signiﬁcant level and assuming 1 degree of freedom.

23 Seasonality in Portfolio Risk Calculations

Table 7 – The results from the interval forecast test by Christoffersen. The values in italic font do not pass the IF-test for a null hypothesis significant level of 95%. The χ2 value using a 95% significant level and 1 degree of freedom is 3.841.

Christoﬀersen value for lower bound

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% Non-conditional 0.333 3.182 32.068 21.123 N 0.732 0.183 3.619 2.348 0.307 0.814 2.307 0.744 GARCH t 0.192 0.773 2.693 0.512 0.213 0.814 1.983 0.465 N 0.513 1.397 2.807 0.823 0.172 0.693 1.524 1.852 SGARCH t 0.213 0.582 2.011 1.401 0.213 0.773 0.695 1.379 N 0.582 1.453 1.138 0.46 0.192 0.773 2.421 2.062 ESGARCH t 0.258 0.655 0.919 1.508 0.213 0.773 1.178 1.379 N 1.008 1.06 6.301 0.919 2.783 0.849 8.021 1.418 eGARCH t 2.783 0.776 5.988 0.687 2.974 1.008 8.021 2.282 N 1.186 0.982 5.18 0.66 1.985 0.643 4.642 1.924 SeGARCH t 2.974 0.849 3.877 1.103 1.985 0.776 7.264 1.491 N 1.186 1.141 5.54 2.53 3.178 0.581 4.413 3.761 ESeGARCH t 2.604 0.776 4.895 1.437 2.604 0.581 8.213 2.515

Table7 rejects all eGARCH model estimated with a VaR signiﬁcance level of α = 5% and the HS method calculated with a VaR signiﬁcance level of α = 5%, 10%.

The models that pass both the conditional and unconditional VaR test can be seen in table 8.

Table 8 – VaR models passing the POF-test and IF-test using a null hypothesis signiﬁcant level of 95%. If the the number of violations from a VaR model is within the non-rejection region it is denoted by a check mark, otherwise nothing.

Models passing the POF-test

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% √ √ Non-conditional √ √ √ √ √ N GARCH √ √ √ √ √ √ √ √ t √ √ √ √ N SGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N ESGARCH √ √ √ √ √ √ √ √ t √ √ √ N eGARCH √ √ √ √ √ √ t √ √ √ N SeGARCH √ √ √ √ √ √ t √ √ √ N ESeGARCH √ √ √ √ √ √ t

24 Seasonality in Portfolio Risk Calculations

7.1.4 Seasonal behaviour of VaR

This section aims to investigate how the VaR-process of the seasonal models diﬀers from the non-seasonal models. To reduce the number of results in this section we will only consider the models that are using the GARCH conditional volatility model with an assumed return distribution of student’s t. This selection is mainly based on the fact that all models passed the VaR-tests for both the QE method and the FHS method. The results from both methods, comparing the VaR-process of the non-seasonal model and the seasonal models, are similar. As such, we will only consider one method in this section, the FHS method. The results of the QE method can be found in appendix C.1.

To further reduce the number of presented VaR models we only consider VaR-models using a signiﬁcant level of 1%. This is motivated by the amount of violations and the fact that VaR is usually estimated for extreme cases. A VaR signiﬁcance value of 0.5% does not generate enough of violations (to small data set) to study the seasonal behaviour of the violations.

As such, ﬁgure5 shows the VaR-process with a signiﬁcance level of 1 % for the GARCHs, SGARCHs and ESGARCHs model.

r GARCHs SGARCHs ESGARCHs GARCHs: r < VaR SGARCHs: r < VaR ● ESGARCHs: r < VaR 0.00

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● ● −0.08 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years Figure 5 – 1-day ahead FHS VaR forecast for a signiﬁcance level of 1%, using a 5 year moving data set. The lines corresponds to the VaR estimate for the non-seasonal and seasonal GARCH models whilst the points correspond to violations. The conditional models assume a student’s t distribution.

Comparing the VaR-processes in ﬁgure5 one can see that the seasonal models seems to have a lower VaR value than the non-seasonal model during winter and a higher VaR value during summer time. This is expected since the seasonal models was shown to have an increased volatility during summer, see ﬁgure4. This is more distinct from 2013 when the

25 Seasonality in Portfolio Risk Calculations seasonal models had an better "goodness-to-ﬁt" then the non-seasonal model, see ﬁgure3.

It is hard to determine from ﬁgure5 if the violations from each model is behaving in a seasonal manner. One can assume that a perfect model with no seasonality would have evenly distributed violations throughout the year. The distributions of violations is tested by plotting the violations from each model as a function of business days throughout a year, see ﬁgure6.

GARCHs SGARCHs ● ESGARCHs JAN MAR MAY JUL SEP NOV −0.02 −0.04 Daily return: r(t) −0.06

−0.08 0 50 100 150 200 250 Business days [t] Figure 6 – The returns outside the 1-day ahead FHS VaR boundary of signiﬁcance level 1% as a function of business days throughout a year. The conditional models assume a student’s t distribution.

Figure6 shows that the non-seasonal model has more outliers during the summer, around business day 100-150, than the seasonal models. The seasonal models have almost the same outliers and have more violations than the non-seasonal during wintertime. This is somewhat expected since the seasonal models have an increased volatility during summertime. It is difficult to see how well the violations are distributed throughout the year in figure6. To better visualize this we look at the average ratio of outliers in a set interval throughout the year. The resulting average should be a line corresponding to the VaR significance level. Figure7 shows the average ratio of annual outliers using a 7 trading day interval.

26 Seasonality in Portfolio Risk Calculations

GARCHs SGARCHs ESGARCHs Expected outbounds JAN MAR MAY JUL SEP NOV 0.05 0.04 0.03 Daily return: r(t) 0.02 0.01 0.00 0 50 100 150 200 250 Business days [t] Figure 7 – Average annual outbound ratio from a 1-day ahead FHS Value at Risk forecast at 1% as a function of business days throughout a year. The conditional models assume a student’s t distribution.

It is quite clear from ﬁgure7 that none of the models is close to the expected ratio of outbounds throughout the year. As expected, the seasonal model performs better during the summertime than the non-seasonal model and slightly worse during winter. The seasonal models have however a huge amount of outliers in the beginning of year.

27 Seasonality in Portfolio Risk Calculations

7.2 Soybean

Another agricultural commodity that is known to behave in a seasonal manner is soybean. The data set has 5033 observations of constant-maturity futures settlement prices and spans the time period 07/12/1999-04/12/2019. The price curve along with its daily return is shown in ﬁgure8. The price of soybean is aﬀected by the world food crisis in a similar way as the corn prices with an huge price increase 2007-2008 and 2010-2013.

(a) (b) 0.06 1600 0.02 1200 Price: P(t) −0.02 Daily return: r(t) 800 600 −0.06 400 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years [t] Years [t] Figure 8 – Time series of a) the price from the transformed continuous future contract of soybean and b) its corresponding daily returns.

The volatility properties of the soybean data series are presented in the same manner as the corn data set. Figure9 includes plots of the realized return series, the annual behaviour of the realized returns series (annual average from a 7 trading day interval) and the ACF of the realized returns where one lag corresponds to 252 trading days i.e. one year of business days.

28 Seasonality in Portfolio Risk Calculations

(a) 0.005 0.004 0.003 0.002 0.001 Daily return squared: r(t)^2 0.000 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years [t]

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● [a.c] value Autocorrelation ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

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0 50 100 150 200 250 −0.05 0 2 4 6 8 10 12 14 16 18 Business days [t] Lag in Years [a.c] Figure 9 – Time series of a) the squared daily return series along with its b) the annual average and c) auto correlation function. The annual average is measured by calculating the average over a 7-day interval each year, divide it by the annual average and then take the average of that interval for every year.

The first thing to conclude from the realized return series in figure 9a is that it seems to experience clustering which is a good argument to use conditional volatility models. Secondly the realized returns seems to have similar seasonal trends as the corn volatility, see figure 9b. This is not surprising since corn and soybean has almost the same planting and harvest seasons. The ACF of the realized return series in figure 9c shows a weaker seasonal correlation than the volatility of corn. 7.2.1 Forecast

The forecast process is a 1-day ahead volatility forecast estimated from a rolling window of 1260 data points (5 years of business days). This gives a forecast period of 3773 data points which spans the time period 2004/12/15 - 2019/12/04.

The "goodness-to-fit" ratio or in-sample fit from each forecast tells us how well the chosen volatility model can represent the return series. The in-sample fit from each parameter estimation is given by the AIC-value and by plotting the AIC-process of each model one can see how well the models are performing. Just as for corn, the sheer number of models makes it quite hard to visualize all models at once. The in-sample fit in figure 10 thereby holds four plots,

29 Seasonality in Portfolio Risk Calculations

a) The AIC-process from all models, to see how the AIC-values (the "goodness-to-ﬁt") changes with time. b) The AIC-process from the GARCH model with both distributions to see which distribution has an overall better performance. The GARCHn model is set as reference. c-d) The AIC-process of normally and student’s t distributed models, respectively, to see if the distributional behaviour for each model is the same. The GARCH model is set as reference for both cases.

(a) GARCHn SGARCHn ESGARCHn eGARCHn SeGARCHn ESeGARCHn (b) GARCHn GARCHs SGARCHs ESGARCHs eGARCHs SeGARCHs ESeGARCHs GARCHs 0 −6800 −20 −7200 −40 AIC AIC −60 −7600 −80 −8000 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years Years

(c) GARCHn ESGARCHn SeGARCHn (d) GARCHs ESGARCHs SeGARCHs SGARCHn eGARCHn ESeGARCHn SGARCHs eGARCHs ESeGARCHs 20 0 10 0 −20 AIC AIC −10 −40 −20 −60 −30 −80 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years Years Figure 10 – AIC value estimated from the log-likelihood function from the optimal parameters for the whole forecasting period. The different plots correspond to the AIC-processes of: a) All models. b) GARCH model of both distributions with normal distribution as reference. c) Models of normal distribution with GARCH model as reference. d) Models of student’s t distribution with GARCH model as reference. The difference between a certain AIC-process and the reference AIC-process is estimated with local regression since the actual difference is quite volatile.

Figure 10a shows that the overall performance of the conditional volatility models are initially decreasing and has the lowest score around 2009-2012 but with an slightly increasing AIC-value during 2010. The poor performance is probably amongst other due to the world food price crisis. As mentioned, this probably made the return series more volatile and volatility less clustered. The distributional assumption of the return series seems to aﬀect the AIC-value quite a lot since the student’s-t distribution has an overall

30 Seasonality in Portfolio Risk Calculations better "goodness-to-ﬁt" ratio. The return series is thus assumed to have fat-tails. This result is most prominent when comparing the performance of the GARCH model of normal distribution and student’s t distribution, see plot 10b.

The performance between the seasonal models and non-seasonal models is best visualized in, plot 10c and plot 10d, where the GARCH model of respective distribution is set as a reference. The result is however, not as intuitive as the results from the corn data set.

Starting with normal distribution, the best seasonal is by far the exponential seasonal model for almost the whole forecast period. The plot seems to indicate that the seasonal behaviour is not distinct until 2015 when all seasonal models have a better "goodness-to-fit" than the non-seasonal models. Before 2015 it seems that the exponential model can give a good estimate of the less clear seasonal effect whilst the regular seasonal model is ineffective. The regular seasonal model has a lower AIC-value then the non-seasonal model for almost all forecasts before 2015.

Assuming student’s t distribution the regular seasonal models still gives a lower "goodness-to-ﬁt" then the respective non-seasonal model until 2015. The exponential seasonal model gives better seasonal estimate, but the AIC-value is still lower than the non-seasonal model for most of forecast until 2015. After 2015 the seasonal models are performing better than the non-seasonal ones and the best seasonal model is the exponential sinusoidal using GARCH model and the sinusoidal using eGARCH.

The seasonal behaviour is changing during the forecast period and it is interesting to see examples of how the added seasonal volatility can look like. Figure 11 shows two cases of added seasonal volatility assuming student’s t distribution; the last estimate (2019/09/12) where the seasonal models are outperforming the non-seasonal models and one estimate when the seasonal models have a lower estimated in-sample ﬁt (2009/01/02).

(a) JAN MAR MAY JUL SEP NOV (b) JAN MAR MAY JUL SEP NOV

SGARCH SGARCH ESGARCH ESGARCH 1.5e−04 SeGARCH SeGARCH

ESeGARCH 1.5e−04 ESeGARCH 1.0e−04 1.0e−04 5.0e−05 5.0e−05 Seasonal complement [a.u.] Seasonal complement [a.u.]

0.0e+00 0 50 100 150 200 250 0.0e+00 0 50 100 150 200 250 Business days [t] Business days [t] Figure 11 – Seasonal component model estimate as a function of a business days, day 1 is the ﬁrst annual business day. Plot a) is the seasonal model estimated at the forecast date 2009/01/02 and plot b) is the seasonal model estimated at the forecast date 2019/12/04.

The first estimate from 2009 shows a case when the in-sample fit is lower from seasonal models, see figure 11a. The shape differs greatly comparing the two seasonality models. The best performing seasonal model for this forecast was the exponential sinusoidal model

31 Seasonality in Portfolio Risk Calculations which suggests that it is closer to the "optimal" solution. The regular seasonal model cannot replicate this narrow shape and ﬁnds another solution which generates a lower in-sample ﬁt.

The last estimate can be seen in figure 11b, which shows the seasonal behaviour during normal market conditions. The seasonal effect is similar for all models, with its maximum in the middle of summer, around trading day 126. 7.2.2 Out-of-sample fit

Table 9 – Volatility validation during the forecast period, using a moving window roll of ﬁve years. * The value is given in the seventh decimal, 10−7.

Model GARCH SGARCH ESGARCH eGARCH SeGARCH ESeGARCH Dist. N t N t N t N t N t N t MSE* 1.704 1.705 1.703 1.704 1.702 1.702 1.710 1.709 1.710 1.709 1.709 1.708 QLIKE 1.493 1.494 1.487 1.484 1.484 1.482 1.494 1.493 1.491 1.487 1.488 1.487

The volatility validation shows that the model with best out-of-sample ﬁt is the seasonal GARCH model with exponential Fourier pairs, from both loss functions. The models that includes seasonality has an overall better out-of-sample ﬁt from both loss functions and the best model for each conditional volatility model is one using exponential seasonality. 7.2.3 Backtesting

The backtesting focus on the behaviour of the violation index and will only consider the lower VaR boundary since it is considered the standard loss. The results of the upper boundary can be found in appendix B.2.

The number of violations from each VaR model can be visualized in table 10.

32 Seasonality in Portfolio Risk Calculations

Table 10 – Number of Violations using three VaR estimation models, HS, QE and FHS. Each estimation model is calculated by different VaR significant levels and conditional volatility models. The best performing model at each VaR significant level is highlighted by a bold style.

Number of Violations x for lower boundary

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% E[x] 19 38 189 378 19 38 189 378 Non-conditional 16 36 152 330 N 32 50 172 314 17 33 179 357 GARCH t 19 37 182 357 16 37 182 361 N 38 51 167 306 20 35 189 355 SGARCH t 22 41 186 356 19 35 185 357 N 35 53 168 317 17 32 184 355 ESGARCH t 20 40 182 367 16 33 179 366 N 37 53 167 310 16 35 184 347 eGARCH t 20 37 179 347 16 35 185 353 N 38 53 166 308 17 33 181 358 SeGARCH t 19 42 178 352 19 38 178 357 N 36 53 163 305 17 35 179 345 ESeGARCH t 20 39 176 354 16 34 178 351

The QE method in table 10 is giving a far better VaR estimate by assuming that the daily returns has a student’s t distribution than normal distribution. As mentioned, this is expected since we expect the daily returns to have fat tails. This is always the case for the FHS method, but it results seems to behave in more reasonable manner (linear) using student’s t distribution. Considering all VaR signiﬁcant levels, the best performing QE model seems to be the GARCH model and the ESGARCH model with student’s t distribution since they have high accuracy while keeping a relatively linear behaviour. The best FHS model is harder to determine but the SGARCH model with student’s t distribution has the best prediction for small signiﬁcance levels.

Table 11 – Nonrejection region of POF-test assuming a signiﬁcance level of p using a forecast set of 3773 samples. The signiﬁcance level of the null hypothesis of the POF-test is denoted by p.

Non-rejection Region for Number of violation N VaR α p = 0.1 p = 0.05 p = 0.01 0.5% 12 < N < 27 11 < N < 28 8 < N < 32 1% 28 < N < 49 26 < N < 51 23 < N < 55 5% 167 < N < 212 162 < N < 216 155 < N < 225 10% 347 < N < 408 341 < N < 414 330 < N < 426

33 Seasonality in Portfolio Risk Calculations

Table 11 shows small changes compared to the non-rejection region of corn since there is only a slight diﬀerence in number of samples. We choose the same signiﬁcance level as before, of 95%, and compares the non-rejection region with the actual number of violation in table 10. The following models pass the test,

Table 12 – VaR models passing the POF-test using a null hypothesis signiﬁcant level of 95%. If the the number of violations from a VaR model is within the non-rejection region it is denoted by a check mark, otherwise nothing.

Models passing the POF-test

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% √ √ Non-conditional √ √ √ √ √ √ N GARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N SGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N ESGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N eGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N SeGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N ESeGARCH √ √ √ √ √ √ √ √ t

Table 12 shows that most models using the QE method and is assuming normally distributed return series is rejected. All models passes however, the VaR signiﬁcance level of 0.5%. It should be noted that the number of violations is gives a bad estimate and is close to the rejection region for most QE models that assumes normality. All other VaR models passes the POF-test with quite a large marginal.

The violation index is further tested by Christoﬀersen’s interval forecast test to see if the violations are statistically independent. The obtained value from each model can be seen table 13. Assuming null-hypothesis with 95% signiﬁcant level and 1 degree of freedom the rejection region corresponds to the χ2 value 3.841.

34 Seasonality in Portfolio Risk Calculations

Table 13 – The results from the interval forecast test by Christoffersen. The values in italic font do not pass the IF-test for a null hypothesis significant level of 95%. The χ2 value using a 95% significant level and 1 degree of freedom is 3.841.

Christoﬀersen value for lower bound

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% Non-conditional 0.136 0.848 12.373 14.465 N 0.548 1.343 2.11 1.477 0.154 0.583 3.347 0.943 GARCH t 0.192 0.733 1.975 0.174 0.136 0.733 2.948 1.011 N 0.773 1.398 1.695 0.465 0.213 0.656 1.321 0.006 SGARCH t 0.258 0.901 1.584 0.013 0.192 0.656 3.642 0.053 N 0.656 1.511 0.847 0.488 0.154 0.548 1.774 0.24 ESGARCH t 0.213 0.857 4.088 0.003 0.136 0.583 3.347 0.009 N 0.733 0.081 0.359 0.91 0.136 0.656 1.029 0.941 eGARCH t 0.213 0.733 1.429 1.338 0.136 0.656 1.678 1.253 N 0.773 0.081 1.792 0.672 0.154 0.583 0.632 0.564 SeGARCH t 0.192 0.469 0.815 0.049 0.192 0.706 0.815 0.023 N 0.694 0.081 2.101 0.085 0.154 0.656 0.751 0.225 ESeGARCH t 0.213 0.642 0.95 0.114 0.136 1.007 1.518 0.004

Table 13 rejects SEGARCH model using student’s t distribution with a VaR signiﬁcance level of α = 5% and the HS method calculated with a VaR signiﬁcance level of α = 5%, 10%.

The models that pass both the conditional and unconditional VaR test can be seen in table 14.

Table 14 – VaR models passing the POF-test and IF-test using a null hypothesis signiﬁcant level of 95%. If the the number of violations from a VaR model is within the non-rejection region it is denoted by a check mark, otherwise nothing.

Models passing the POF-test

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% √ √ Non-conditional √ √ √ √ √ N GARCH √ √ √ √ √ √ √ √ t √ √ √ √ N SGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N ESGARCH √ √ √ √ √ √ √ t √ √ √ √ N eGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N SeGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N ESeGARCH √ √ √ √ √ √ √ √ t

35 Seasonality in Portfolio Risk Calculations

7.2.4 Seasonal behaviour of VaR

The seasonal behaviour of the VaR-process is investigated for the same volatility models as in respective corn section. The chosen models are thus the GARCH, SGARCH and ESGARCH with an assumed student’s t distribution. The chosen signiﬁcance level is also the same as in the corn study, 1%. The VaR method represented in this section is the QE method and is mainly motivated by the number of violations from each model. The GARCH model has the same outliers using both VaR methods but the seasonal models have an underestimate using the FHS method and an overestimate using the QE method. Since we want to observe how the seasonal behaviour is for each model, we rather have a model that overestimates than underestimates the expected violation threshold. The plots shown in this section can also be found for the FHS method in appendix C.2.

As such, ﬁgure 12 shows the VaR-process from volatility model GARCH, SGARCH and ESGARCH using a QE method of signiﬁcance level 1%.

r GARCHs SGARCHs ESGARCHs GARCHs: r < VaR SGARCHs: r < VaR ● ESGARCHs: r < VaR 0.00

● ● ● ● ● −0.02 ● ● ● ● ● ● ●

● ● ● ● ● ● ● −0.04 ● ● ●● ● ● ● ● ● ●● ●

Daily return: r(t) ● ● ● ● ● −0.06 ● ● ● ● −0.08 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years Figure 12 – 1-day ahead QE VaR forecast for a signiﬁcance level of 1%, using a 5 year moving data set. The lines corresponds to the VaR estimate for the non-seasonal and seasonal GARCH models whilst the points correspond to violations. The conditional models assume a student’s t distribution.

Comparing the VaR-processes in figure 12 one can see that the seasonal models seems to have a lower VaR value than the non-seasonal model during wintertime and a higher VaR value during summertime. This effect is more distinct from 2013 as the performance of the seasonal models increased in relation to the non-seasonal model, see figure 10.

We further analyse the returns that lies outside of the VaR-boundary to see if they are behaving in an seasonal pattern, this is shown in ﬁgure 13.

36 Seasonality in Portfolio Risk Calculations

GARCHs SGARCHs ● ESGARCHs JAN MAR MAY JUL SEP NOV −0.02 −0.03 −0.04 −0.05 Daily return: r(t) −0.06 −0.07 0 50 100 150 200 250 Business days [t] Figure 13 – The returns outside the 1-day ahead QE VaR boundary of signiﬁcance level 1% as a function of business days throughout a year. The conditional models assume a student’s t distribution.

Figure 13 shows that the non-seasonal model has some isolated outliers during the late summer, around business day 126-160, whilst the seasonal models has more outliers during late winter, around business day 50-60. This is somewhat expected since the seasonal models has, for most forecast an increased volatility during summer time.

The next investigation is how the violations are distributed by looking at the average ratio of outliers in a set interval throughout the year. The resulting average should, as mentioned, be a line corresponding to the VaR signiﬁcance level. Figure 14 shows the average ratio of annual outliers using a 7 trading day interval.

37 Seasonality in Portfolio Risk Calculations

GARCHs SGARCHs ESGARCHs Expected outbounds JAN MAR MAY JUL SEP NOV 0.04 0.03 0.02 Daily return: r(t) 0.01 0.00 0 50 100 150 200 250 Business days [t] Figure 14 – Average annual outbound ratio from a 1-day ahead QE Value at Risk forecast at 1% as a function of business days throughout a year. The conditional models assume a student’s t distribution.

It is quite evident that none of the models from ﬁgure 14 has evenly distributed outliers throughout the year. The seasonal models perform better during summertime than the non-seasonal model but slightly worse during late winter.

38 Seasonality in Portfolio Risk Calculations

8 Conclusions

We have investigated the forecast performance of conditional volatility models with added seasonal dependency. The future contracts of agricultural commodities are known to show seasonal trends and the data sets used are corn and soybean. Both commodities show similar seasonal behaviour, an increased volatility during summer. This is expected since both commodities have their planting season in spring and the harvest in November and December. Both commodities experience a changing in-sample fit during the forecast period and the seasonal correlation seems strongest during last four years, 2016-2019. The corn data set show a better in-sample fit using seasonal models for most of the forecast period while the soybean data set show a lower in-sample for most forecasts before 2016. The in-sample fit is greatly affected by the world food crisis, especially the seasonal models since the regular trends are more vague during this period.

The overall performance can be seen by looking at the volatility validation. Almost all seasonal models show a better result compared to its corresponding conditional model using both MSE and QLIKE loss functions. The GARCH-type models show a higher accuracy than the eGARCH model for both commodities. The best volatility models from the corn data set was the ESGARCH model using normal distribution and SGARCH model using student’s t distribution. The best model for the soybean data set was the ESGARCH model for both distributions. The results from the volatility validation points to the importance of considering seasonality for volatility forecast.

The backtest considers VaR models using both the QE method and the FHS method for all volatility models. The results vary quite a lot and it is hard to determine if a specific model perform better than another. A common factor is that the QE method gives insufficient results when assuming normal distribution. Most models from this assumption, are rejected by the unconditional POF-test or gives very poor results. The non-seasonal models seem to give better VaR forecasts using the QE method especially for lower VaR significance level. The FHS method is not as easily interpreted. The student’s t distributed volatility models seems to give a better result than normal distribution but there is not a clear trend. The results comparing the seasonal and non-seasonal models is inconclusive. There is no model that outperforms the rest. The best model is debatable, but considering FHS models passing both VaR tests, the best model seems to be the SGARCH model for corn and the SeGARCH model for soybean.

The seasonal behaviour from both commodities showed an improved forecast during summer but a declined forecast during certain regions of the year. The seasonal models of corn had a reasonable violation ratio over the year except in the beginning of the year where it was far oﬀ compared to the non-seasonal model. All the violations during this time of the year is almost at the same date and a plausible idea is that it is linked to the annual release of "Crop Production Annual Summary" by the United States Department of Agriculture. This report holds crop production data which includes, amongst other, production data from planted and harvested areas, yield etc. This release seems, to change the price violently certain years when released. It would thus be interesting to add a variable that incorporates this property. Both the non-seasonal and seasonal models show weaker results late in March for reasons unknown.

The soybean data set seems not as eﬀected by the annual crop release as the corn data

39 Seasonality in Portfolio Risk Calculations set. The violation ratio shows that the seasonal models has slightly more outliers during this period. The violation ratio is however higher in late March, both in contrast to the non-seasonal model and the expected value. As mentioned, are the volatility ratio from the seasonal models better during late summer.

Volatility forecasting and VaR calculations is a challenging task and the seasonal models brings some interesting features to this area. The volatility forecast is improved by including seasonal components but more research needs to be done concerning VaR with seasonal volatility. It would be interesting to see how the extreme values of the return series, to some signiﬁcance level, is distributed throughout the year. It could be that values further away from the mean value have a diﬀerent annual distribution then the values closer to mean. With more research, the seasonal models may prove to be an important asset when estimating risk using VaR.

40 Seasonality in Portfolio Risk Calculations

References

[1] Engle, Robert F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inﬂation. Econometrica, vol. 50, no. 4, 1982, pp. 987–1007. Retrieved 22 Oct 2019, JSTOR. [2] Bollerslev, Tim. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, vol. 31, no. 3, 1986, pp. 307–327. Retrieved 22 Oct 2019, JSTOR. [3] Hull, John C. (2015). The GARCH(1,1) Model. In Options, futures, and other derivatives (Ninth edition). Pearson Education, Inc. [4] INTRODUCTION TO AGRICULTURE - Understanding Seasonality in Grains. CME Group. Visited 19 Jan 2020. [5] Goodwin, B. K. & Schnepf, R. (2000). Determinants of endogenous price risk in corn and wheat futures markets. The Journal of Futures Markets, vol. 20, no. 8, 2000, pp. 753-774. Retrieved 22 Oct 2019, Wiley. [6] Wang, Xiaoyang. & Garcia, Philip. (2011). Forecasting Corn Futures Volatility in the Presence of Long Memory, Seasonality and Structural Change. Presentation No 103749, 2011 Annual Meeting, July 24-26, 2011, Pittsburgh, Pennsylvania, Agricultural and Applied Economics Association. Retrieved 11 Nov 2019, EconPapers. [7] Nelson, Daniel B. (1991). Conditional Heteroskedasticity in Asset Returns: A New Approach. Econometrica, vol. 59, no. 2, 1991, pp. 347–370. Retrieved 22 Oct 2019, JSTOR. [8] Kat, Harry M. & Oomen, Roel C.A. (2006). What Every Investor Should Know About Commodities, Part I: Univariate Return Analysis. Cass Business School Research Paper. Retrieved 6 Dec 2019, SSRN. [9] Kupiec, Paul H. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, vol. 3, 1995, pp. 73–84. [10] Christoﬀersen, P. (1998). Evaluating Interval Forecasts. International Economic Review, vol. 39, 1998, pp. 841–862. [11] Wang, Kai-Li., et al. (2001). A Flexible Parametric GARCH Model with an Application to Exchange Rates. Journal of Applied Econometrics, Vol. 16, No. 4 (Jul. - Aug., 2001), pp. 521-536. Retrieved 22 Oct 2019, JSTOR. [12] Bollerslev, Tim. (1987). A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return. The Review of Economics and Statistics, Vol. 69, No. 3 (Aug., 1987), pp. 542-547. Retrieved 12 Nov 2019, JSTOR. [13] Danielsson, Jon. (2011). Risk Measures and Parametric Methods. In Financial Risk Forecasting: The Theory and Practice of Forecasting Market Risk with Implementation in R and Matlab. John Wiley & Sons. Retrieved at ProQuest Ebook Central. [14] Ye, Yinyu (1987). Interior Algorithms for Linear, Quadratic, and Linearly Constrained Non-Linear Programming. Ph.D. Unpublished doctoral dissertation, Department of EES, Stanford University.

41 Seasonality in Portfolio Risk Calculations

[15] Patton, Andrew J. (2010). Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics, Vol. 160, No. 1 (Jan., 2011), pp. 246-256. Retrieved 21 Oct 2019, ScienceDirect.

[16] CBOT, site: https://www.cmegroup.com/company/cbot.html.

[17] Eikon, site: https://www.refinitiv.com/en/products/eikon-trading-software.

42 Seasonality in Portfolio Risk Calculations Appendix

A Thomas Reuters

A.1 Data extraction: Settlement prices of Corn =_xll.tr("Cc1;Cc2;Cc3;Cc4;Cc5;Cc6;Cc7;Cc8;Cc9;Cc10;Cc11;Cc12;Cc13;Cc14", "TR.SETTLEMENTPRICE","Frq=D SDate=0D EDate=19990913 CH=IN;Fd RH=date",B2)

B Tables

B.1 Corn

Table 15 – Number of Violations using three VaR estimation models, HS, QE and FHS. Each estimation model is calculated by different VaR significant levels and conditional volatility models. The best performing model at each VaR significant level is highlighted by a bold style. The values in italic font do not pass the POF-test for a null hypothesis significant level of 95%.

Number of Violations x for upper boundary

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% E[x] 19 38 189 378 19 38 189 378 Non-conditional 25 40 180 369 N 29 49 175 326 15 34 179 370 GARCH t 12 34 195 384 14 30 179 368 N 33 56 192 344 18 32 193 380 SGARCH t 14 35 205 384 17 35 202 381 N 31 51 190 339 18 30 192 379 ESGARCH t 14 36 202 379 17 30 202 380 N 31 47 180 328 16 31 187 365 eGARCH t 12 33 191 385 14 32 183 378 N 34 57 189 349 15 33 186 394 SeGARCH t 16 38 199 395 15 33 187 386 N 32 51 193 341 11 34 194 390 ESeGARCH t 15 38 196 391 14 33 184 386

43 Seasonality in Portfolio Risk Calculations Appendix

Table 16 – The results from the interval forecast test by Christoffersen. The values in italic font do not pass the IF-test for a null hypothesis significant level of 95%. The χ2 value using a 95% significant level and 1 degree of freedom is 3.841.

Christoﬀersen value for upper bound

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% Non-conditional 0.333 3.182 32.068 21.123 N 0.449 4.815 2.781 0.357 0.12 1.008 2.307 1.977 GARCH t 0.077 0.618 2.364 3.014 0.104 1.384 0.756 4.011 N 0.582 3.54 1.82 1.219 0.172 0.547 1.014 0.109 SGARCH t 0.104 0.655 1.37 0.782 0.154 0.655 1.638 0.222 N 0.513 4.422 2.024 0.513 0.172 1.384 1.088 0.297 ESGARCH t 0.104 0.693 0.971 0.52 0.154 0.481 0.971 0.468 N 0.513 2.174 2.196 0.105 0.136 1.282 1.5 0.777 eGARCH t 0.077 0.582 1.165 1.418 0.104 0.547 0.528 1.226 N 1.008 1.141 2.131 1.668 0.12 0.582 1.591 1.847 SeGARCH t 0.136 0.707 2.886 0.974 0.12 0.582 1.5 0.215 N 1.186 1.71 3.701 0.698 0.064 1.008 2.538 0.695 ESeGARCH t 0.12 0.707 2.307 1.291 0.104 0.582 1.071 0.215

Table 17 – VaR models passing the POF-test and IF-test using a null hypothesis signiﬁcant level of 95%. If the the number of violations from a VaR model is within the non-rejection region it is denoted by a check mark, otherwise nothing.

Models passing the POF- and IF-test

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% √ √ Non-conditional √ √ √ √ √ N GARCH √ √ √ √ √ √ √ t √ √ √ √ √ √ N SGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N ESGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ √ N eGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ √ N SeGARCH √ √ √ √ √ √ √ √ t √ √ √ √ N ESeGARCH √ √ √ √ √ √ √ √ t

44 Seasonality in Portfolio Risk Calculations Appendix

B.2 Soybean

Table 18 – Number of Violations using three VaR estimation models, HS, QE and FHS. Each estimation model is calculated by different VaR significant levels and conditional volatility models. The best performing model at each VaR significant level is highlighted by a bold style. The values in italic font do not pass the POF-test for a null hypothesis significant level of 95%.

Number of Violations x for upper boundary

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% E[x] 19 38 189 378 19 38 189 378 Non-conditional 16 33 176 354 N 39 57 164 323 20 39 186 366 GARCH t 23 37 179 370 20 41 187 368 N 35 58 168 329 21 40 183 365 SGARCH t 24 38 175 366 20 40 186 367 N 39 57 166 320 19 42 189 369 ESGARCH t 20 40 180 359 20 43 187 367 N 35 53 166 320 18 37 182 359 eGARCH t 23 38 180 360 20 38 184 357 N 38 55 175 318 21 40 186 355 SeGARCH t 23 42 186 357 20 40 188 350 N 35 53 173 317 19 39 185 355 ESeGARCH t 21 41 174 351 19 41 182 350

Table 19 – The results from the interval forecast test by Christoffersen. The values in italic font do not pass the IF-test for a null hypothesis significant level of 95%. The χ2 value using a 95% significant level and 1 degree of freedom is 3.841.

Christoﬀersen value for upper bound

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% Non-conditional 0.136 0.583 2.65 2.093 N 0.815 1.749 0.77 0.232 0.213 0.815 0.004 0.673 GARCH t 0.282 0.733 0.307 0.243 0.213 0.901 0.009 0.558 N 0.656 1.812 1.012 0.443 0.235 0.857 0.099 0.097 SGARCH t 0.307 0.773 0.664 0.21 0.213 0.857 0.615 0.364 N 0.815 1.749 0.07 0.629 0.192 0.946 0.026 0.504 ESGARCH t 0.213 0.857 0.046 0.51 0.213 0.992 0.201 0.927 N 0.656 1.511 0.887 0.032 0.173 0.733 0.079 0.031 eGARCH t 0.282 0.773 0.954 0.015 0.213 0.773 0.516 0.189 N 0.773 1.628 0.002 0.072 0.235 0.857 0.887 0.004 SeGARCH t 0.282 0.946 0.381 0.174 0.213 0.857 0.299 0.097 N 0.656 1.511 0.001 0.848 0.192 0.815 0.001 0.017 ESeGARCH t 0.235 0.901 0.125 0.076 0.192 0.901 0.006 0.014

45 Seasonality in Portfolio Risk Calculations Appendix

Table 20 – VaR models passing the POF-test and IF-test using a null hypothesis signiﬁcant level of 95%. If the the number of violations from a VaR model is within the non-rejection region it is denoted by a check mark, otherwise nothing.

Models passing the POF- and IF-test

V aR model QE HS & FHS V aR(α) 0.5% 1% 5% 10% 0.5% 1% 5% 10% √ √ √ √ Non-conditional √ √ √ √ √ N GARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N SGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N ESGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N eGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N SeGARCH √ √ √ √ √ √ √ √ t √ √ √ √ √ N ESeGARCH √ √ √ √ √ √ √ √ t

46 Seasonality in Portfolio Risk Calculations Appendix

C Figures

C.1 Corn

r GARCHs SGARCHs ESGARCHs GARCHs: r < VaR SGARCHs: r < VaR ● ESGARCHs: r < VaR 0.00

●

−0.02 ● ● ●

● ● ● ● ● ● ● ● ● ● ● −0.04 ● ● ● ● ● ● ● ● ●

● ● ● ● ● Daily return: r(t) −0.06 ● ● ● ●

● ● −0.08 −0.10 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years Figure 15 – 1-day ahead QE VaR forecast for a signiﬁcance level of 1%, using a 5 year moving data set. The lines corresponds to the VaR estimate for the non-seasonal and seasonal GARCH models whilst the points correspond to violations. The conditional models assume a student’s t distribution.

47 Seasonality in Portfolio Risk Calculations Appendix

GARCHs SGARCHs ● ESGARCHs JAN MAR MAY JUL SEP NOV −0.02 −0.04 Daily return: r(t) −0.06

−0.08 0 50 100 150 200 250 Business days [t] Figure 16 – The returns outside the 1-day ahead QE VaR boundary of signiﬁcance level 1% as a function of business days throughout a year. The conditional models assume a student’s t distribution.

GARCHs SGARCHs ESGARCHs Expected outbounds JAN MAR MAY JUL SEP NOV 0.05 0.04 0.03 Daily return: r(t) 0.02 0.01 0.00 0 50 100 150 200 250 Business days [t] Figure 17 – Average annual outbound ratio from a 1-day ahead QE Value at Risk forecast at 1% as a function of business days throughout a year. The conditional models assume a student’s t distribution.

48 Seasonality in Portfolio Risk Calculations Appendix

C.2 Soybean

r GARCHs SGARCHs ESGARCHs GARCHs: r < VaR SGARCHs: r < VaR ● ESGARCHs: r < VaR 0.00

● ● ●

−0.02 ● ● ● ● ● ●

● ● ● ● ● −0.04 ● ●●

● ● ● ● ●● ●

Daily return: r(t) ● ● ● ● ● −0.06 ● ● ● ● −0.08 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 Years Figure 18 – 1-day ahead FHS VaR forecast for a signiﬁcance level of 1%, using a 5 year moving data set. The lines corresponds to the VaR estimate for the non-seasonal and seasonal GARCH models whilst the points correspond to violations. The conditional models assume a student’s t distribution.

49 Seasonality in Portfolio Risk Calculations Appendix

GARCHs SGARCHs ● ESGARCHs JAN MAR MAY JUL SEP NOV −0.02 −0.03 −0.04 −0.05 Daily return: r(t) −0.06 −0.07 0 50 100 150 200 250 Business days [t] Figure 19 – The returns outside the 1-day ahead FHS VaR boundary of signiﬁcance level 1% as a function of business days throughout a year. The conditional models assume a student’s t distribution.

GARCHs SGARCHs ESGARCHs Expected outbounds JAN MAR MAY JUL SEP NOV 0.04 0.03 0.02 Daily return: r(t) 0.01 0.00 0 50 100 150 200 250 Business days [t] Figure 20 – Average annual outbound ratio from a 1-day ahead FHS Value at Risk forecast at 1% as a function of business days throughout a year. The conditional models assume a student’s t distribution.