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Following the recent financial crises, there has been a proliferation of new risk management and portfolio construction approaches. These approaches all endeavour to better quantify and manage risk by accounting for the stylised facts of financial time series mainly heavy and skewed tails, volatility clustering and converging correlations. Capturing all these stylised facts in a coherent framework has proved to be an elusive and knotty task. We here propose a pure econometric framework that captures all the stylised facts satisfactorily. We use three data sets to show how the approach is implemented in VaR forecasting and correlation analysis. We show how an investment portfolio can be constructed in order to optimise reserve capital holding. The approach employed is linear programming (LP) computable, satisfies second degree stochastic dominance and outperforms the general mean/VaR quadratic optimisation to arrive at efficient asset allocation.

The past decade has seen a number of financial institutions fail worldwide. In Zimbabwe alone, from a peak of 40 banks in 2002, 20 banks remain operational as of January 2014 [

World over emphasis on risk management is quite high as demonstrated by the evolution of the Basel accord. Specifically, it was recognised that some changes were necessary to the computation of capital for market risk in the Basel 2 framework. These changes are referred to as Basel 2.5. There are three changes involving:

・ The calculation of a stressed VaR;

・ A new incremental risk charge; and

・ A comprehensive risk measure for instruments dependent on credit correlation.

These measures all have the effect of greatly increasing the market risk capital that large banks are required to hold. Our interest lies in the approach to VaR and the essence of stressed VaR in order to optimise reserve capital holding.

Studies by [

where

Because stressed VaR is always at least as great as VaR, the formula shows that (assuming

Two main issues are of concern. Fistrly, looking closely we do observe that the stressed VaR period is subjective. In Europe, it was considered that 2008 would constitute a good one-year period for the calculation of stressed VaR. Later it was required that Banks search for a one-year period during which its portfolio would perform very poorly [

1) To develop a practically sound, functional and industry useful framework for market risk management.

2) To demonstrate how serial correlation can be tested and corrected for in financial times series.

3) To show how skewed and leptokurtic tails are accounted for in VaR measurement.

4) To show how GARCH forecasting can be integrated into determining portfolio risk.

5) To test for returns normality in 3 asset classes.

6) To test whether the Skewed t outperforms the normal distribution in explaining asset returns.

Secondly, we also note that randomness and normality generalisation of financial time series may need to be re-thought in order to accurately quantify risk. Studies in [

The risk management approach which we detail is that of a long position in financial instruments, hence great emphasis is put on the left tail of the distribution. Our end goal is to articulate an asset allocation framework that optimises future return expectations with sturdy downside risk management. To comprehensively capture the four stylised facts, a profound knowledge of AR, GARCH processes, and stable Paretian distributions is needed.

From an investment perspective, lower than necessary capital levels increase the risk of failure, whereas higher than required capital levels lower equity rate of return, and locks up vital capital needed for that could be invested elsewhere to maximise value. We test for normality from the three data set, the ZSE industrial index, USD/ZAR exchange rate and Gold which are selected proxies for the equities, foreign exchange markets and commodities markets. The problem of non-normality is addressed in four phases:

1) Serial correlations in returns.

2) Heteroscedasticity in volatility of returns.

3) Asymmetric returns: Negative skewness and leptokurtosis.

4) Correlation convergence.

Let S be a subset of the real numbers. For every

A stochastic process is said to be stationary if its mean and variance are constant over time and the value of the covariance between the two time periods depends only on the lag between the two time periods and not on the actual time at which the covariance are computed; thus

In the autoregressive (AR) time series model, an observation

Here

For each asset class we formally test for serial correlation by calculating Ljung-Box (LB) Q statistic, [

We consider a simple AR smoothing model

where

Assume that the true returns follow a stationary AR(1) process:

where

Combining equations the above, we get;

where

Applying OLS to (8) above we obtain an estimate for

We proceed to test for statistical significance of serial dependence at 5 percent. Where statistical significance is found, we transform our returns by following the relationship in (6):

Financial time series has a tendency to produce returns that are skewed and leptokurtic [

where

where

We estimate the GJR GARCH (1, 1) which is generally sufficient for financial time series

The following restrictions apply:

We apply the Jacque-Bera test for normality of the residuals. The following relationship should hold

otherwise we fit the residuals to some fat tailed distribution.

We here state without proof that the GJR model implies that the forecast of the conditional variance at time T + h is given by:

Our approach to modelling left tail risk is a semi parametric approach. We define left tails as 10 percent of all data to the extreme left. Our choice of extreme value distribution is the Skewed t distribution. We define the loss function as:

Definition 3.1 (Loss Function). The loss function is given by the change in value, V, of the portfolio between time t and

By convention, the loss function is usually expressed as a positive value and we are concerned with the left hand tail, i.e., for long positions. Mathematically, VaR refers to the alpha-quantile of a distribution.

Definition 3.2 (Value at Risk). The value at risk,

The main assumption in this model is that of conditional normality. The return on day t is normally distributed conditional on the information on day

Once

where

The model parameters are estimated in two steps. In the first step, the parameters of the GARCH process are estimated. In the second step, the standardized residuals

We extend the univariate GARCH models to incorporate the assymetric response of returns to market shocks. For a single asset, conditional variance is the variance of the unpredictable part,

The same is true for the multivariate conditional variance-covariance matrix. We define the conditional variance-covariance matrix for the part of

where

We contrast the estimated v-cov to the industry practice of using the linear correlation coefficient,

As a side note and not to venture far afield we follow [

Let:

The optimisation problem is formulated in the following way:

Objective:

Subject to

We also define the following terms:

where

Our primary performance measure is the ratio of the mean forecast return divided by the forecast

where

We now describe our approach for evaluating the robustness of our findings to alternative performance mea- sures. We define a performance measure as a ratio of reward to risk that is valid with respect to a given utility function. In practice, investor preferences are heterogeneous and there is no single utility function that is valid for all investors. Hence there is no single performance measure that is appropriate for all investors and it makes sense to evaluate performance through the prism of a number of different measures [

Up to now we have considered a regulatory risk measure, VaR, that does not care about losses in excess of VaR. If one looked at the area below the cumulative density function up to a given target payoff, this would be a risk measure which would consider not only the probability, but also the amount of losses. This measure is called Lower Partial Moment One

For all pay-offs above the target, the target is reached and therefore the shortfall is zero: payoffs that are higher than the target cannot compensate payoffs below the target. Then,

Generalised lower partial moments (LPM) provide the basis for our supplementary performance measures. LPM follow directly from the utility function proposed in [

In the same way as in VaR, we measure the mean LPM order 1 and 2 ratios. This gives a complementary view to portfolio performance measurement analysis.

Our approach is made up of two parts. Firstly, we identify the several types of non-normality that are typically not allowed for in traditional asset allocation. Secondly, we then integrate these empirical results in risk mea- surement.

Data for the period March 2009 to April 2014 was used. The Zimbabwe Stock Exchange (ZSE) Industrial index, was used as a proxy for equities, the USD/ZAR exchange rate for currencies and gold for commodities. There is, however, limited exposure by Zimbabwean investors to other asset classes such as bonds and options. A real estate index was constructed but unfortunately its returns differed markedly from those reported in the real estate market hence it was discarded at least for purposes of this study. Figures 1-3 show the returns time plot of the three assets under consideration.

Serial correlation occurs when one period’s return is correlated to the previous period’s return. Figures 4-6 show the time plot of the asset values. Noticeably, the ZSE plot is not stationary i.e. the data does not fluctuate around some common mean or location, this attribute induces dependence in returns over time. However the time/returns of Figures 1-3 does appear to be stationary (the data does fluctuate around a common mean or location). Henceforth, it is not graphically clear to concur on the presence of serial correlation.

When dealing with time series it is desirable to have a stationary data set primarily because the traits of a stationary data set allow one to assume models that are independent of a particular starting point. In essence it becomes unnessecary to compute VaR and Stressed VaR seperately. The two under strict stationarity give similar VaR. Double the VaR computed here is able to satisfy Basel 2.5 requirements.

When there is non-stationary the previous values of the error term will have a non-declining effect on the current value of returns as time progresses. We consolidate our above findings by formally testing for first order serial correlation using the Lung Q-Statistic [

H_{0}: first order serial correlation does not exist in the data.

H_{1}: first order serial correlation does exist in the data.

If the Q-Statistic for a given asset class has significance at a 5 percent, i.e. a p < 0.05 we reject the null hypothesis of no serial correlation and conclude that there is serial correlation in data. In this case we must allow for the effect of serial correlation on future asset class returns. If the p value is higher than 0.05, the null is not rejected and we conclude that there is insufficient evidence to reject the null. We summarise our results in

We conclude that serial correlation is present in equities and the foreign exchange rate returns. We generalise the major drivers of the findings to this case as driven by illiquidity, jumps especially for equities and low frequency of trade. In the case of currencies, the exchange rate is a managed float. This control makes it hard-to- price the true intrinsic value of the asset.

Presence of Heteroscedasticity makes it difficult to gauge the true standard deviation of the forecast errors, usually resulting in confidence intervals that are too wide or too narrow. In particular, if the variance of the errors is increasing over time, confidence intervals for out-of-sample predictions will tend to be unrealistically narrow. In Figures 1-3 we observed volatility clustering. We will estimate the GJR GARCH model to capture Heteroscedasticity in returns, generating heavy tails in the unconditional distribution of returns. Through modelling Heteroscedasticity we show simple, yet effective approaches to forecasting future volatility and VaR computation.

We summarise the returns data with summary statistics in

The table below shows data is non-normal. In practise, when normality is imposed, risk is understated. To further stress the fact that data is not normal, we show in Figures 7-9 the empirical histogram superimposed with the normal distribution of equal mean and standard deviation. The graphs clearly show the existence of stylised fact; fat tails.

From the diagrams, it is visibly shown that the third stylised fact, fat tails are real. Negative returns are observed in greater magnitude and with higher probability than implied by the normal distribution. Neglect of this non-normality leads to underestimation of risk. On all the two asset classes we reject normality and also conclude that the left tail is indeed heavier than predicted by the normal distribution.

Benchmark index | Test statistic | p value | Reject null | |
---|---|---|---|---|

Equities | ZSE industrial | 169.18 | 0.00 | Yes |

Currencies | USD/ZAR | 5.34 | 0.021 | Yes |

Commodities | Gold | 1.032 | 0.310 | No |

Skewness | Kurtosis | J-B | Reject normality | Fat left tail | |
---|---|---|---|---|---|

Equities | 0.417 | 14.4 | 11,294 | Yes | Yes |

Currencies | −0.372 | 0.598 | 49.7 | Yes | Yes |

Commodities | −0.9367 | 0.250 | 2325 | Yes | Yes |

It is common observation in financial literature that correlations tend to be unstable over time and converge in times of economic turmoil. We investigate whether correlations between asset classes tend to increase during periods of high market volatility compared to periods of relative calm. We compare the correlations during the first two years after dollarisation^{1} with correlations spanning the whole period. We attempt portfolio con- struction using GARCH DCC analysis.

Shockingly, the results show that correlations not only defy stationarity but they do converge during times of high market volatility. This simply means that the benefits of diversification are overestimated. Frameworks which assume normality and linear co-movement of asset returns lead to significant underestimation of joint negative returns during bearish markets.

In this section, we offer statistical methodologies for incorporating the four categories of non-normality dis- cussed above. Our belief is that achieving optimal portfolio efficiency should be based on a more precise estimation of risk and advertently requires embracing non-normality in financial time series.

Existence of serial correlation conceals the true risk characteristics of an asset. If ignored, this will reduce risk estimates from a time series by smoothing true asset volatility. Our task is to compute the unsmoothed more volatile return series for both equities and currencies. Industry convention is to use the partial autocorrelation function (PACF) as a guide to determine the appropriate lag length. The PACF of our data is shown in Figures 10-12. The order is determined by viewing the lines that fall outside the confidence bounds (the blue lines) and

Gold | USD/ZAR | ZSEI | |
---|---|---|---|

Gold | 1.000 | −0.0467 | 0.0358 |

USD/ZAR | 0.0198 | 1.000 | 0.0287 |

ZSEI | 0.0432 | 0.0536 | 1.000 |

counting how many lags it takes for the data to fall inside the confidence bounds. By viewing the PACF, the evidence is weak towards finding a good fitting AR model for the data. According to the PACF the data looks random and certainly shows no easily discernible pattern. Our thrust is for correcting for first order serial correlation. This would support the appearance of the time series plot since it looks a lot like white noise except for the change in spread (variation) of observations. Such Heteroscedasticity would most likely not be evident in a truly random data set. This, however, does not mean we rule out the possibility that the data fits an AR model with weak autocorrelation.

In order to correct for serial correlation we apply [

Step 1. We estimate

The USD/ZAR returns had jumps. This was largely attributed to the exchange rate regime which are managed floats. Also, the slow decay of its ACF may imply the presence of Heteroscedasticity, hence we attempt to capture the stylised fact in the next section.

Step 2. We produce our unsmoothed return series in

A simple observation can be made that unsmoothed data even deviates more from normality than the un- smoothed returns data. We invoke the Jacque-Bera test to verify this. The data speaks for its self.

The increase in the series volatility as a result of removing first order serial correlation coupled with strong evidence of non-normality are compelling findings for use of more robust risk measuring tools.

We now shift gears in an attempt to find some sort of pattern in the data that would suggest the use of a different model. The result of the ACF plot in Figures 9-12 does suggest that a pattern exist in the unconditional distribu- tion of the mean equation. This is noticed from the slow decay of the ACF lag plots. This indicates there is correlation between the magnitude of change in the returns. In other words, there is serial dependence in the variance of the data. Our proposed risk measure, VaR, is a prediction concerning possible loss of a portfolio in a given time horizon. Following the Basel recommendations, it should be computed using the predictive dis- tribution of future returns and volatility. We estimate the conditional mean and variance equations.

Using Gretl on quasi maximum likelihood (QML) we simultaneously estimate the parameters

Prediction

If

Thus

(a) We assume

(b) We assume the errors follow a Skewed t distribution, our mean and variance equation are:

ZSE | Unsmoothed ZSE | USD/ZAR | Unsmoothed USD/ZAR | |
---|---|---|---|---|

Standard dev | 0.0130 | 0.0306 | 0.0333 | 0.116 |

Minimum | −0.11754 | −0.363 | −0.224 | −0.389 |

Maximum | 0.0944 | 0.299 | 0.190 | 0.2498 |

Mean | 0.000898 | 0.000897 | 0.000213 | 0.000230 |

When we use the GJR GARCH and distributions that allow for leptokurtic returns we enhance risk reporting. It can also be shown as is generally misconstrued that VaR is not a function of time but rather of the returns conditional distribution. In Figures 12-15 we plot the quantile plots for the fit data. It can be shown that the skewed t distribution is a better fit. It might not be perfect but it does have a fair track of the left tail better.

Outliers still persist as in Gold returns and USD/ZAR exchange rate returns, however the Skewed t distribu- tion manages to tracks the tails fairly well.

A multivariate GARCH model of the diagonal VECH type is employed. The coefficient estimates are easiest presented in the following equations for the conditional variance or covariance:

where

The unconditional covariance between the assets are positive. It is however interesting to note that there is a very strong positive correlation between gold and the USD/ZAR exchange rate. This inadvertently implies that it is unwise for a trader to overweight long positions on both gold and USD.

Asset class | Gaussian | Skewed t | |
---|---|---|---|

Equities | 1.662 percent | 3.683 percent | |

Equities | 5.385 percent | 8.077 percent | |

Currencies | 19.88 percent | 26.989 percent | |

Currencies | 9.953 percent | 14.086 percent | |

Commodities | 1.975 percent | 3.039 percent | |

Commodities | 2.582 percent | 3.624 percent |

is contrasted to the normal distribution case.

In this section we have conducted a statistical analysis in a stepwise framework. We first removed the autocorrelation component to unmask true asset volatility, then the GJR-GARCH model is estimated assuming normal errors and finally the skewed t-distribution is fit to the errors. The fitted model is used as a basis to estimate VaR and the correlation matrix in the case of a portfolio.

Econometric approaches have been used extensively in risk measurement to address stylised facts in financial time series. In recent years, a number of people have proposed various models with diverse transformations and adaptations. These models endeavour to better quantify risk. Unfortunately, in practice, usefulness of these models could be associated with unintended consequences especially as their level of complexity increases with every step taken to enhance accuracy. In this paper a stepwise model that captures the stylised facts in a simple, coherent and user friendly method is presented.

The main thrust of the model is in accounting for heavy tails in returns data. Incorporating these fat tails generally increases capital requirements, and thus effectively reduces chances of failure though inadvertently return on capital is reduced. Contrary to what literature suggests, VaR is a function of the returns distribution for a given asset and not of time. The study proposes a stepwise AR/GJR-GARCH Skewed-t distribution to incor- porate deviations from normality. The first step involves unsmoothing returns using the AR to unmasks auto- correlation and bring out the true volatility. This results in a more jerked returns time plot. The GJR-GARCH captures heteroscedasticity and the leverage effect. The Skewed-t captures asymptotic behaviour of the tails. The choice of innovations distribution structure is a purely statistic one. The GPD, Multivariate Student t, EVT, Skewed Normal distribution, the Frechet and Gumbul distributions may all be used. We use three data sets to show how the approach is implemented in VaR forecasting and correlation analysis. We show how an invest- ment portfolio can be constructed in order to optimise reserve capital holding. The approach employed is linear programming (LP) computable, satisfies second degree stochastic dominance and outperforms the general mean/ VaR quadratic optimisation to arrive at efficient asset allocation.

VictorGumbo,SimisoSiziba, (2016) An Econometric Approach to Incorporating Non-Normality in VaR Measurement. Journal of Mathematical Finance,06,82-98. doi: 10.4236/jmf.2016.61010