Presented By:- Abinash Agrawal Times Business School

Presented By:-Abinash Agrawal

Times Business School

Introduction

Quantitative Methods (QM) are used extensively throughout the Finance sector (banking, hedge funds, insurance, risk management, research etc.).

This talk – a brief overview of some of the recent QM developments in three of the main areas in Finance.

1. Passive Fund Management2. Active Fund Management3. Risk Management

1. Passive Fund Management Passive fund managers assume financial markets

are efficient.Thus, no point in trying to use QM to beat the

market (i.e. earn returns in excess of a buy-and-hold strategy).

Passive fund managers working with equities might try to track a stock market index (e.g. have a look at your ISA you might see FTSE trackers, World trackers, etc.).

The aim of tracking an index is to replicate the performance of the market (since there is no point trying to beat it).

1. Passive Fund Management

2000 2001 2002 2003 2004 2005 2006 2007 2008

4.25

4.50

4.75

Figure 1.1. AXA UK trackerLAXA

2000 2001 2002 2003 2004 2005 2006 2007 2008

7.50

7.75

8.00

8.25Figure 1.2. FTSE All ShareLFTSE

2000 2001 2002 2003 2004 2005 2006 2007 2008

0.00

0.05

Figure 1.3. AXA UK tracker - FTSE All SharedmDiff


How have recent developments in QM helped passive fund managers?

Assume the fund manager is tracking a stock market index.

The fund manager has to choose a sub-set of companies to invest in.

They also have to choose how to weight each company in the sub-set.


Traditional approach is quadratic programming (QP) (see e.g. Roll, 1992).

QP gives analytical formulas.

But QP can only give you optimal weights assuming all stocks are held. Not the optimal sub-set and the optimal weights.

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New approaches include:Genetic algorithms and artificial neural networks

(see e.g. Rafaely and Bennell, 2006). Cointegration approaches (see e.g. Alexander and

Dimitriu, 2005). All are computationally expensive, but perform

better than QP.

1. Passive Fund ManagementFor an example of just how computationally

expensive…One cointegration approach: regress the index

against all sub-sets of constituents of size n.If total number of constituents is N, this translates

into N!/n!(N-n)! regressions.Tracking the FTSE 100 with n = 30 requires

29372339821610944823963760 regressions (over 29 septillion!).

World’s fastest computer (Roadrunner) performs 1000 trillion calculations a second. Would take over 8 hours to do these regressions!

2. Active Fund Management

Active fund managers assume financial markets are inefficient.

Using QM to predict the market might be beneficial relative to buy-and-hold.

How have recent developments in QM helped active fund managers?


This topic is enormous!QM (on the maths side) has led to new products: e.g.

asset backed securities, exotic derivatives, swaps…These can be used for risk management and active

trading.QM (on the stats. side) has led to new technical

trading techniques, new arbitrage strategies – and the rise of stat. arb. hedge funds.

QM also means we can now evaluate the historical performance of active fund managers more effectively.


Focusing on the stats. side:1. Technical analysis Technical analysis involves trend identification by

analysing charts (charting) on price and volume information.

Originally this was done by hand! Then by spreadsheet analysis.

Now, genetic algorithms, pattern recognition and artificial neural network algorithms are employed (see e.g. Park and Irwin, 2007).

2. Active Fund ManagementWe now have a better understanding of the

importance of data snooping when analysing technical trading rules, (see e.g. White, 2000; Sullivan et al., 1999).

Data snooping: a problem where data is re-used. If enough different trading rules are considered,

some will have performed well for a particular data series in the past just by chance, even if they have no predictive power.

If when evaluating the effectiveness of trading rules we focus only on a single rule, then the rule may appear to work well.

2. Active Fund ManagementBUT we might have one of the above-mentioned

chance results and not know it. White (2000) has developed a bootstrap method

that helps to correct for this possibility (the “reality check bootstrap”).

Sullivan et al. (1999) apply the methodology to the DJIA.

2. Active Fund Management2. Fundamental analysis and econometric

forecastingFundamental analysis seeks to exploit relationships

between asset returns and fundamental factors (e.g. dividends, earnings).

Fundamental analysis has a long history in Finance research. The same techniques are still used e.g. DDM.

Econometric forecasting seeks to exploit relationships between asset returns and financial and macroeconomic factors – see e.g. Pesaran and Timmermann (1995, 2000).

Lots of developments in this area utilizing increased computational power (e.g. model searches).


3. Statistical arbitrageUsing stats. to identify arbitrage opportunities. Simple example: consider two companies Ford

and GM. Have a look at their share price and spread since 1998.


1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 20081998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

2.0

2.5

3.0

3.5

4.0

4.5Figure 2. Ford and GMLFord LGM


1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

Figure 3. Ford-GM (de-meaned)dmsspread


Pairs trading involves a long-short arbitrage to exploit this type of relationship.

Lots of research being done on statistical arbitrage - most of it in the private sector (i.e. by Quants at investment banks and hedge funds).

Academic researchers are starting to get interested (e.g. Vidyamurthy, 2004).

3. Risk Management

Risk management: some key topics1. Hedging. 2. Modelling and forecasting market volatility.3. Modelling potential loss.4. Modelling and forecasting credit default risk

(banks have been a bit weak on this recently!).

3. Risk Management

1. Hedging: Developments on the maths side, new products etc.

2. Volatility: Lots of developments on the stats. side using ARCH and GARCH models (see e.g. from Engle, 1995, onwards).

3. Potential loss: VaR developments. 4. Credit default risk: limited dependent variable

models, VaR developments.After Northern Rock/sub-prime, huge demand for

improved risk management models!

4. ConclusionsQM on the maths and stats. side are used

extensively in Finance, both for active trading, passive trading and risk management, and in other areas I haven’t had time to mention.

Look out for many interesting developments, particularly on the stats. side associated with statistical arbitrage.

As QM on the maths side leads to more complex assets, improved QM on the stats. side is needed to fully understand the risks associated with these assets.

Mean:-In statistics, mean has two related meanings:the arithmetic mean (and is distinguished from

the geometric mean or harmonic mean).the expected value of a random variable, which is

also called the population mean.There are other statistical measures that should not

be confused with averages - including 'median' and 'mode'. Other simple statistical analyses use measures of spread, such as range, interquartile range, or standard deviation. For a real-valued random variable X, the mean is the expectation of X. Note that not every probability distribution has a defined mean (or variance)

Arithmetic Mean:-

The arithmetic mean is the "standard" average, often simply called the "mean".

The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions,

the mean is not necessarily the same as the middle value (median), or the most likely (mode).

Geometric Mean:- The geometric mean is an average that is

useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean) e.g. rates of growth.

Standard Deviation:- Standard deviation is a widely used measure

of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" exists from the average (mean, or expected value).

A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data points are spread out over a large range of values.

The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance.

In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions. For example, the margin of error in polling data is determined by calculating the expected standard deviation in the results if the same poll were to be conducted multiple times.

Normal Distribution:- In probability theory,

the normal (or Gaussian) distribution is a continuous probability distribution that has a bell-shaped probability density function, known as the Gaussian function or informally the bell curve.

where parameter μ is the mean or expectation (location of the peak) and is the variance. σ is known as the standard deviation. The distribution withμ = 0 and σ 2 = 1 is called the standard normal. A normal distribution is often used as a first approximation to describe real-valued random variablesthat cluster around a single mean value.

Log-normal distribution:-a log-normal distribution is a

continuous probability distribution of a random variable whose logarithm is normally distributed. If Xis a random variable with a normal distribution, then Y = exp(X) has a log-normal distribution; likewise, if Y is log-normally distributed, then X = log(Y) is normally distributed. (This is true regardless of the base of the logarithmic function: if loga(Y) is normally distributed, then so is logb(Y), for any two positive numbers a, b ≠ 1.)

Corelation:-A correlation function is

the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points. If one considers the correlation function between random variables representing the same quantity measured at two different points then this is often referred to as an autocorrelation function being made up of autocorrelations. Correlation functions of different random variables are sometimes called cross correlation functions to emphasise that different variables are being considered and because they are made up of cross correlations.

For random variables X(s) and X(t) at different points s and t of some space, the correlation function is

C(s,t)=corr(X(s),X(t)) where is described in the article on correlation.

In this definition, it has been assumed that the stochastic variable is scalar-valued.

Time series:-In statistics, signal

processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive time instants spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the Nile River at Aswan. Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. Time series are very frequently plotted via line charts.

Time series data have a natural temporal ordering. This makes time series analysis distinct from other common data analysis problems, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their education level, where the individuals' data could be entered in any order). Time series analysis is also distinct fromspatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses).

A time series model will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility.)

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Presented By:- Abinash Agrawal Times Business School