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In this project, the use of the discrete-time Kalman filter is investigated in its application in robust fault detection and diagnosis scenarios in dynamic systems. To achieve the analytical redundancy requirements, the model-based approach is used to design a dedicated-observer scheme consisting of a bank of Kalman filters that generate sets of structured residuals. These residual signals are subjected to statistical analysis and compared against a detection rule to unveil fault signatures. Information from the isolated faults is then determined to reveal its time dependency, fault location and fault time. A proposed fault detection and diagnosis algorithm is proposed and applied through simulations of a linearised discrete-time Simplified Longitudinal Flight Control system and a Gas Turbine Jet Engine system that are both affected by uncertainties.
The modelling of idiosyncratic risk is a frequent and contentious topic especially when related to its effects on future excess returns from a portfolio or an individual stock.
One of the earliest literatures that questions idiosyncratic risk's role is Douglas (1969) that relates it to asset pricing. However, this study was refuted by other findings as significant factors were not considered. Nevertheless, the study by Fama and Macbeth (1973) is often considered the basis of most modern arguments and discussions in relation to the topic of idiosyncratic risk and returns. Hence, the CAPM is introduced as the initial model of asset pricing despite its constraints and the fact that idiosyncratic risk can be diversified away.
Various asset pricing models have been introduced and modified to include factors which affect the security's returns. Examples of models considered are the I-CAPM and the more common Fama-French three-factor model. Such models portray an unambiguous description and understanding of idiosyncratic risk particularly if the traditional CAPM's assumptions cannot be realised.
The relevance of determining a suitable and realistic explanation of whether idiosyncratic risk is a good predictor of stock and portfolio returns is highly beneficial to financial strategies. In addition, it is imperative to establish its behaviour with respect to individual stock or portfolio returns.
The structure of this literature review firstly introduces the mathematical justification of the capital asset pricing model in Section 2. Thereafter, the primary assumptions considered for the idiosyncratic risk to have an effect on portfolios or an individual stock is introduced in Section 3. In Section 4, an overview of the analysis is undertaken to determine whether future excess returns amongst securities, based on their cross-sectional data, can be explained by their idiosyncratic risks. This is followed by the time series analysis in Section 5. Finally, drawn conclusions are detailed in Section 6.
Capital Asset Pricing Model
The asset pricing model that is initially used is the Capital Asset Pricing Model (CAPM) (see expression below). This framework is used based on the relationship of the expected excess market return and the excess return on a security (ER-r) where is the measure of the volatility of a security.
Idiosyncratic risk is synonymous in relation to asset pricing. Unlike market risk that affects the whole market or industry sector, idiosyncratic risk affects specific or a small number of firms or stocks. Additionally, this risk can be diminished through diversification of a portfolio, unlike market risk.
Research over the years has questioned whether idiosyncratic risk can be used to predict expected returns. Hence, the CAPM's presence in explaining the conditions for this to occur regardless of the fact that beta and book-to-market, which are essential in explaining security returns, are missing in the model.
In the traditional CAPM, as idiosyncratic is diversified away, systematic risk is priced at a equilibrium due to homogenous expectations from investors with respect to the market portfolio weights according to Cuthbertson and Nitzsche (2008). However, the assumptions are presumptuous as they disregard reality. In addition, investors are known to include stocks that have idiosyncratic risk so as obtain relatively high returns.
Hence, adjustments of the CAPM to adequately cover these effects have been proposed by Levy (1978) based on the fact that it is suited to test individual securities rather than portfolios as it negatively affects the skewness of the stock returns and residual variance are reduced (Patterson (1995)). In addition, another modified model is the Intertemporal Capital Asset Pricing Model (I-CAPM) which considers risks from both the market returns and changes in future market return forecasts as proposed by Campbell et al (1993).
To determine whether idiosyncratic risk is a good predictor of future excess returns, specific assumptions are generally considered across most literatures.
Studies undertaken by Merton (1987) reflect that a large percentage of investors rarely invest in more than 10 stocks. Thus, under-diversification occurs often as the number of stocks that are randomly chosen to achieve absolute portfolio diversification is approximately 50 as suggested by Campbell et al (2000).
Analysis carried out in relation to idiosyncratic risk and stock returns vary with the data used. Whereas some authors use similar data sources (Center for Research and Security Prices - CRSP) and others have used different sources such as Datastream.
In addition, according to Santa-Clara et al (2003) daily data reflects short-term trends, whereas monthly data reflects long-term trends. This is a possible rationale for the differences observed. Whereby Ang et al (2004) suggests that relationship between idiosyncratic risk is negative, whereas Spiegel and Wang (2006) suggest a positive relationship in regards to idiosyncratic risk and future excess returns.
Omitted variables such as book-to-market and size are considered in some studies such as Fama and French (1992) to be primary factors that contribute to the level of idiosyncratic volatility. Whereby, idiosyncratic volatility is used as a measure undiversified idiosyncratic risk.
Fundamentally the average stock variance is the arithmetic average of the monthly variance; therefore it Therefore, it is considered to be the total risk (consisting of both systematic and idiosyncratic risk) of the security.
As (ref....) the return of a security on a particular day is determined by its common factor and firm-specific shock , whereby represents the day of the month and is the return of the security at day . This is shown by the expression below:
Using the above fundamental expression, further assumptions are incorporated to explain the risk measure. The assumptions are:
The portfolio is equally weighted, thereby avoiding the calculation of weighted market capitalisation of various firms with respect to their variance measures.
Factor shocks and firm-specific shocks are not correlated
Firm-specific shocks are independently and identically-distributed across different stocks that have a variance of
Subsequently, with the above assumptions considered the idiosyncratic risk can be shown to constitute a large percentage (over 80%) of the average stock variance as noted by Santa-Clara et al (2003). For this reason, idiosyncratic risk is considered to be a measure of average stock variance and therefore a driving factor of stock returns.
Idiosyncratic risk in different markets
Most studies on idiosyncratic volatilities are based on the US market. Despite this, Malkiel and Xu (2004) undertook an analysis based on an international market. The stock returns of the Japanese market were considered in the study. All other factors such as the sample period are kept constant in the methodology used for analysing the US stock market. The findings indicated that the relationship between future excess returns and idiosyncratic risk remains valid despite for a few country effects.
Further studies have extended the sample size used as suggested by both Santa-Clara et al (2003) and Ang et al (2008). One common practice was including data from the start of the CRSP tapes (January 1926) to the current market period at their time of the study. However, as the daily-data was only available from July 1962, the time-series is split in two and thereby an alternative measure of stock variance is used (ref...). The data prior to July 1962 is considered to be low-frequency as it is based on monthly returns and the data after is considered high-frequency as it is based on daily returns.
The skewness and kurtosis of the measure of variance is observed from the statistical analysis of the data when computed using the standard or modified square of the average stock return for the specific month. The results show that the measures of the volatilities, both high and low frequency, are highly correlated. Therefore, undertaking a regression of a value-weighted portfolio for the whole sample (incorporating both time-series data sets) gives the most significant positive result for the average stock variance in explaining the market returns. However, the high-frequency measure of volatility still gives a significant positive result but not as high when the whole sample is used.
Business Cycle Effects
Does business cycle explain future excess returns? If so how significant is it? A study by Campbell et al (2001) determined that the average stock prices are related to business cycle fluctuations. This finding is further explained by (ref...) whereby it is shown that business cycle variables such as the relative Treasury-bill rate, dividend-price ratio and the default spread amongst others, can be used to predict the stock market (ref...).
Further analysis by (ref...) show the re-sampling (bootstrap) of a regression to determine the significance of these factors in predicting the stock market returns. This consists of adding the business cycle variables in the regression and determining whether the coefficients are significant or not. Thereafter, the measures of volatility are included. The findings show that the average stock variance is still positive in explaining the stock market returns, whereas other variables such as dividend-price ratio are insignificant. However, it is striking that the default spread is significant in explaining the stock market returns, whereas the Treasury-bill rate is negatively significant. From the findings it is established that relationship between returns and the stock risk in terms of idiosyncratic risk still holds when the business cycle variables are controlled.
Further studies have also tried to discount whether it is just idiosyncratic volatilities that could predict the high future excess returns. The considered factors were size, book-to-market and liquidity. To test for robustness, the significance levels of each factor are analysed using the regression model. The findings were as follows:
Size: The size effect is based on the firm's market capitalization. Results from the statistical analysis show that the significance of the size variable is correlated with idiosyncratic risk. Therefore the size effect is an aspect of idiosyncratic risk and it is considered a proxy in its respect.
Book-to-Market value: Regression results indicate that it is not as significant as idiosyncratic risk; however it still shows a relationship with future excess returns.
Liquidity: Using the trading volume based on the bid-ask spread the regression using all variables was statistically insignificant. According to (ref...) this is due to liquidity's low explanatory power in the presence of idiosyncratic risk. Hence, liquidity is a not a major factor in predicting future excess returns.
In conclusion, the regressions results show that idiosyncratic volatilities are more robust than the tested variables.
Studies by Ang et al using the Fama and French (1993) model give contradicting results to Merton (1987) findings in regards to idiosyncratic risk's relationship with excess returns. Results show that the returns have a negative relationship with returns - high (low) idiosyncratic risk will give low (high) excess returns.
The model used by AHXZ was the Farma-French three-factor model as shown below. The CAPM model is not used as it does not take into account idiosyncratic volatility when its assumptions are not satisfied.
where is the excess market return MKT, SMB and HML represent the portfolio's market, size and value factors respectively. Furthermore idiosyncratic risk is represented by . According to Fama and French (1993) SMB and HML represent the risk factor of the returns. SMB is the difference in returns between the large and small portfolios, whereas HML is the difference in returns between the large and small book-to-market equities portfolios. In both these cases the weighted-average book-to-market equity should be similar. The use of this model is based on the fact that it incorporates additional factors such as size and book-to-market equity which are common drivers of stock returns in the US market (Fama and French (1996)).
Despite the various models that have been used in previous findings such as CAPM and behavioural models (Barberis and Huang (2001)) which both predict a positive relationship, findings by Ang et al give strong contradicting and significant results. Possible reason to this is the methodology in the examination of the relationship. Unlike previous studies, Ang et al focuses on the firm-level when determining the idiosyncratic volatility. In addition, the study classes the value-weighted portfolios based on their measure of idiosyncratic volatility (ref..).
Return reversal theory
Findings by Huang et al (2009) have shown that return reversal of stocks with high idiosyncratic volatilities tend to increase over next month when the future excess returns are driven by small-sized firms. The firms are considered small with respect to the total market capitalization. Hence, the reversal of the high future excess returns over the following month results in the negative relationship.
Effects of lagged idiosyncratic volatilities
A study by Fu (2008) disputes the negative relationship between stock returns and idiosyncratic risk due to fact that lagged idiosyncratic volatilities is used. The exponential generalised autoregressive conditional heteroskedasticity (EGARCH) is used to counter the findings. This is corroborated when lagged idiosyncratic volatilities are included in a statistical regression of the Fama-French three-factor model that show time-varying properties such as the periodical release of a firm's earnings. Therefore idiosyncratic risk and future excess returns are contemporaneous and the use of lagged variables in the analysis is not justified. However, this contradicts the method carried out by Fama and French (1992) whereby the current market-book values were used to describe next year's future excess returns.
This is due to the fact that hypothese of the presenc of random walk is rjeceted at the 10% level
Lack of information
Firms that show expected reduced earnings tend to limit the disclosure of information according (Jiang Xu 2006...see boyer mitton 207 p1). Therefore, the firm's idiosyncratic risk will increase as its prospects in regards to future excess returns decreases.
Tests are carried out to determine whether the negative relationship findings are robust by controlling certain factors that could contribute to the observed effect. Some of these factors are size (market capitalisation), liquidity, book-to-market, bid-ask spread and leverage amongst others. To control these factors, each variable is included in a regression of the Farma-French three-factor model and its significance level is observed. At the 5% level all the variables are statistically insignificant therefore it can be concluded that the low (high) average stock returns with high (low) idiosyncratic volatilities are not due to the stated factors (ref...?)
However, AHXZ findings were bolstered by AHXZ 2009 findings further undertaking the analysis in international markets and not only the US market. In addition, further economic determinants are considered such as whether past idiosyncratic volatilities affect future expected returns and the effects of private information as mentioned by Easley, Hvdkjaer and O'Hara 2002). More importantly, the examination methodology is also kept in alignment with the analysis from the findings that gave a positive relationship between the idiosyncratic risk and stock returns.
From the study it was deduced that the stock returns from the international countries (G7 countries excluding the US) give similar results to those found with the US market. However, the US market has a stronger negative relationship as its idiosyncratic volatility has a comprehensive range with respect with its stocks compared to the other countries.
Robustness -value weighting
Findings from Farma-Macbeth used equally-weighted (stated in section) firms rather than value-weighted as used by AHXZ (mention other journal). Results
The negative relationship effect observed in the US market is one of the primary drivers for the low stock returns around the world in the presence of high idiosyncratic volatilities. This indicates that the international markets face the same idiosyncratic volatilities as the US market.
Most studies focus on cross-section returns rather than time-series based returns. Despite this, findings undertaken by (ref Malkiel 2007) have shown there are number of similarities in the case time-series returns. The results show that idiosyncratic risk is a good predictor in showing returns over time. The statistical results from the regression were similar in significance before and after controlling both size-effects and book-to-market value.
It is evident idiosyncratic risk is highly useful in predicting excess returns using either model - CAPM or Fama-French three-factor model. Furthermore, other significant factors such as book-to-market, size, liquidity, private information and size amongst others are controlled to corroborate the findings of the relationship. Hence, idiosyncratic risk shows its predictive power in relation to stock returns. Furthermore, studies show that international (non-US) markets also portray a similar relationship of idiosyncratic risk and future excess returns.
Despite the encouraging outcome in predicting future excess returns, serious differences are observed when determining whether the relationship of idiosyncratic risk with individual stock or portfolio returns is positive or negative. Theories and studies criticise each of the proposed outcomes, however an in-depth analysis of this is beyond the scope of this literature, but nevertheless it forms the basis of future investigation.