# 1 Executive Summary.

# 2 Company Background

Marks and Spencer plc is a major British multinational retailer that specialises in selling of clothing, home products and luxury food products. The company was established in 1884 by Michael Marks and Thomas Spencer in Leeds and is currently headquartered in Westminster, London. It is listed on the London Stock Exchange and is a constituent of the FTSE 100 Index.

# 3 Model Specification

## Definition

Given that is a sequence of independent and identically distributed random variables such that then () is called the generalized autoregressive conditionally heteroskedastic or process if

The strong positivity in the conditional variance described in equation 2.2 is maintained by the conditional parameters. If equation 2.2 is written in term of the lag operator B we obtain

If the roots of the characteristic equation, i.e. lie outside the unit circle and the process is stationary, then equation 2.2 can also be written

where and are coefficients of in the expansion of It should be noted that expression (2.6) implies that process is an ARCH process

of infinite order with a fractional structure of the coefficients.

It is evident from equation 2.1 that the process is stationary if the process.

() is stationary.

# 4 Advantages and Disadvantages

## Advantages

The main advantages of this model include weak stationarity, the fact that volatility clusters are modelled because a large or a large will give rise to a large, the fact that distribution of the series have fat tails if which implies that

and the use of simple parametric representation to describe the volatility evolution.

## Disadvantages

On the other hand models are symmetric to both positive and negative prior returns and restrictive on and otherwise they results in an infinite fourth moment. Additionally, they do not provide any explanation as to what causes the variation in volatility and are not sufficiently adaptive in prediction – because they react slowly to large isolated shocks. Finally, the tail behaviour of GARCH (1,1) models is relatively short even with standardized Student-t innovations.

The model will be used to forecast the volatility of the stock price for Air BP Plc. The data under investigation relates to weekly stock prices. It should be noted that oil prices and consequently oil returns are relatively more volatile that stock prices because they have a greater correlation to macroeconomic factors compared to the latter. This is evidenced by the fact that historical oil prices have a higher standard deviation than historical stock prices.

The natural frequency of data used in GARCH models is weekly. However, daily and monthly data are used. Nevertheless, the lower frequencies such as monthly prices reduce the GARCH-iness, which is smoothened out of the data. A GARCH with an intraday frequency can also be used but it is relatively complicated owing to the seasonality of volatility throughout a single day. Such seasonality is mainly influenced by the particular market where trading is taking place. From the graphs illustrated in section 4 it is evident that volatility spikes upwards and decays away until the next spike. This is however not the case in real data because the latter is characterised by shocks of all sizes. It should however be noted that volatility from announcement is slightly similar only it goes in the opposite direction because volatility builds up towards the announcement time and disappears when announcement results are made public. GARCH models essentially estimate the rate at which volatility decays.

# 5 Results

Weekly

# 6 Summary Statistics

Daily Prices | Weekly Prices | |||

Oil | Air France | Oil | Air France | |

Mean | 67.22501 | 14.13725 | 14.12865482 | 67.21013 |

Standard Error | 0.537096 | 0.111915 | 0.249725688 | 1.198912 |

Median | 65.31 | 13.03 | 13 | 65.54 |

Mode | 27.3 | 20 | 19 | 27.3 |

Standard Deviation | 33.66613 | 7.015025 | 7.010134125 | 33.65506 |

Sample Variance | 1133.409 | 49.21057 | 49.14198045 | 1132.663 |

Kurtosis | -1.3426 | 1.441065 | 1.487041243 | -1.326 |

Skewness | 0.186369 | 1.130753 | 1.140284562 | 0.196352 |

Range | 128.46 | 35.245 | 35.1 | 127.96 |

Minimum | 17.15 | 3.055 | 3.2 | 17.65 |

Maximum | 145.61 | 38.3 | 38.3 | 145.61 |

Sum | 264127.1 | 55545.25 | 11133.38 | 52961.58 |

# 7 Conclusion

# 8 References

Ardia, D., 2008. *Financial risk management with Bayesian estimation of GARCH models : theory and applications. *Berlin : Springer.

Brooks, C., 2008. *Introductory econometrics for finance. *2 ed. Cambridge : Cambridge University Press.

Francq, C. & Zakoian, J.-M., 2013. *Garch models : structure, statistical inference and financial applications. *Hoboken, N.J. : Wiley.

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