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Investment Management 2FINA8006Kieran O’Reilly

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Introduction

Investing in financial markets is inherently risky. Many investors use the normal probability distribution to quantify it. This report intends to evaluate the quality of this measure by using daily data from the FTSE100 over 20 years and explain the consequences of these findings. Finally alternatives approaches will be evaluated for investors to better quantify investment risk or evaluated risk not take into account by the normal probability distribution.

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The normal distributionThe normal probability distribution in finance aims to forecast the likely gains or losses investors can expect with a confidence level of their choice (standard deviation). For example the normal probability distribution for the FTSE100 over 20 years predict that there is 95.4% probability (or 95.4% of the time) (2 standard deviation) that the return are between -2.35% or +2.35%. However the figures in the excel document demonstrate that the normal probability distribution used in risk management don’t reflects accurately the financial market risk. The number of variations of the FTSE100 which are outside the standard deviation are higher in reality than in theory. Therefore the probability to have between +/-2.35% return is lower than 95.4%. The difference between theory and market reality is particularly significant for 3, 4 and 5 standard deviation.In addition, it is impossible to know that precisely potential losses or gains as it is an estimation. Furthermore, it is based on previous data which may not be a reliable measurement to estimate future losses or gains. The normal distribution underestimates the difficulty to estimate future variations by simplifying the reality of the market.Negative standard deviation events (except for 5 standard deviation) are higher than positive ones; which mean there are more negative variations than positive ones, outside 2 standard deviation.

ImplicationsIn the long term investors won’t attaint the return they expect. As they are under estimating potential losses or gains. There is more often very high or low movements on the marketNormal probability distribution only reflects the risk on the financial market when the market is stable, when there are no non-normal incidents or misbehaviour.Investors are fooled by the model and by people evaluating their risk by using the normal probability distribution. It also means that investors don’t know what there are doing and they don’t know that they don’t know how to quantify risks. They may discover that their model doesn’t correctly quantify risk when they need it the most, when the market doesn’t follow a normal distribution. It might lead to panic if many investors discover it at the same time.It raises several issues: what tools investors should use to quantify investment risk when the market is not stable or add to the model that the market is not always stable, although instability of the market tends to be unpredictable?

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BackgroundBefore suggesting alternative approaches to quantifying investment risk. Investors should have knowledge of this facts:

Although we want to help investors better quantify investment risk, their decision and how they perceive risk also depend of their own risk aversion and time horizon.

The risk is the difference between the actual and expected return or the uncertainty of the returns. High risks are rewarded by potential high returns.

Investors should also consider the fact that risk tend to be lower on the long term than on the short term. Investors may face bad fluctuations on the short term but it will tend to stabilise and growth on the long term as the economy is wealthier.

There are two potential type of risks on the financial market:o Unsystematic risk, also known as specific risk is the type of uncertainty that

comes with the company or industry you invest in. o Systematic risk, also known "un-diversifiable risk", is the uncertainty inherent to

the entire market or entire market segment.

Black swan theoryAs said before the normal probability distribution is not reliable because of non-normal events, an illustration of these events are black swan events.The black swan theory from Taleb concerned events that come as a surprise, unpredictable, have major effect, rare but certain to happen. As rare events have a very low probability, there are underestimate in finance in term of money and people think they won’t happen.This theory come from the fact people generalise events if they happen over and over again because they can’t check every possibilities as it would take too much time or as they don’t know everything. Therefore they accept it as a fact but their conclusion is based on incomplete information and could be wrong. People think there are right until the first contrary event that prove that there are not. If investors can’t predict black swans, the knowledge of its existence mean they won’t base their predication only on usual and predictable facts (normal distribution) but also on unexpected ones (black swans) and therefore temper their expectations. Investors can also try to hedge against bad ones by identify areas of vulnerability and exploit good ones.

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Four momentsAnother way to quantify investment risk are the four moments of return distributions, which can be combine to more accurately describe the risk:

The first moment is the mean. Investors use previous data of the asset to measure the expected return of the investment. Assets which are riskier are expected to have more potential returns to compensate for the risk.

The second moment is the standard deviation. It measures the dispersion of returns around the mean. The higher the dispersion is the higher is the risk and volatility, investors could potentially loss or earn a lot as there are more high variations around the mean. The first and second moment use previous return as the history tends to repeat itself. However it might poorly predict future returns as it is not always the case.

The third moment is the skewness which measure the asymmetry of the distribution.  A neutral skew mean that the distribution is symmetrical. Therefore it is a normal

distribution. A negative skew mean that the left tail is longer; the mass of the distribution is

concentrated on the right of the figure. A positive skew mean that the right tail is longer; the mass of the distribution is

concentrated on the left of the figure. The mean is more on the right, there are higher return possible as there are higher positive variations. As a result the risk is lower for a positive skewness as in average investors will make more money or loss less than in a normal distribution.

Skewness is not a robust statistic because it is greatly affected by outliers (low-frequency, high-impact events) as their values are raised to the power of three. Skewness is highly depend on the measurement period.

Different type of skewness

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The fourth moment, kurtosis, measures the extent of the peaks and tail heaviness of a probability distribution.The shape of a probability distribution with a kurtosis of:

3, is the normal distribution. More than 3, leptokurtic, is a fat tail. The mean peaked and is higher than the normal

distribution. The tails approach 0 slowly and are above the normal distribution. There are more extreme events as explain more in details in the fat tails part.

Less than 3, platykurtic, There are more events outside four standard of deviation than a normal distribution. However these events are not as extreme as the fat tails. The distribution is simultaneously less peaked and have thinner tails.

However these statistics are not very reliable as there are raised to the power of four and heavily dependent on the measurement period. Alternative measures of kurtosis are: the L-kurtosis, which is a scaled version of the fourth L-moment; measures based on 4 population or sample quantiles. These correspond to the alternative measures of skewness that are not based on ordinary moments.

Different type of kurtosis

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Different type of return distribution

A view of tails at their extrema, for different kurtosis:

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log-pdf for the Pearson type VII distribution with excess kurtosis of infinity (red); 2 (blue); 1, 1/2, 1/4, 1/8, and 1/16 (gray); and 0 (black)

Fat tailed distributionAs explained previously they were more high loss and gains on the excel calculations than the normal probability distribution forecast. To correct some of the issues of the normal distribution investors may use fat tailed distribution which has the same mean and median. In the normal distribution tail events are very rare however in the stock market there are more frequent, the tails are “fatter”. Fat tail make possible a drop of most shares even if investors had diversified. In fat tailed distribution there are actually more events near the mean than the normal distribution however the a few huge deviations from the mean. Fat tails also traduce excess optimism or pessimism of investors which lead to excessive increase or drop of shares. Fatter tails increase the probability that an investment will move beyond three standard deviations.

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Fat tail (in bold) and normal distribution (bell curve):

Value-at-RiskVaR measures the maximum loss of a portfolio that cannot be exceed at a given confidence. VaR is calculated based on a time period, confidence level and pre-determined loss amount. Investors can compare VaR from different portfolio to reduce the maximum loss they could have, therefore the risk they take.However it is a snapshot which is reliable only for a short period of time. VaR rely on the past, can be complicated and misrepresenting true exposure. It doesn’t work well when markets are highly volatile. VaR was use in 2008 for the mortgage market however it had underestimated the danger of toxic mortgage products. Therefore VaR was part of the factor that lead to the subprime crisis

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Many investors use instead “stressed VaR”, which incorporates periods of market turmoil in its assumptions.

Behavioral riskTo quantify risk investors need to consider human behavior and psychology as a component of risk and not only data and models.If an individual investor correctly forecast an increase of an asset but he is the only one and everybody think it will drop, the asset will drop in value. In addition sometimes shares drop sharply because of panic, people want to get their investment back (or only some of it) although in the long term it will be profitable. People tend to follow the crowd and they might not be rational but normal probability distribution is based on people being rational.Investors can look at some behaviour indicators to quantify these risk:The State street Investor confidence Index measures investor confidence or risk appetite by analyzing the actual buying and selling patterns of institutional investors. Investors can use this index to check how investor confidence fluctuate over time. Therefore investors can estimate the volatility of the market.The trust index from household is also important as it impact asset price of investors and the volatility on the financial market.

Trust index of the University of Michigan

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Trust index of customers in US

Beta

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Beta is a measure of a portfolio volatility in relation to the market. By definition, the market has a beta of 1.0. Individual portfolios are ranked according to how much they deviate from the market. Therefore beta gives a sense of a portfolio’s market risk compared to the market. A portfolio which fluctuate more than the market over time has a beta above 1.0. If a stock moves less than the market, the stock's beta is less than 1.0. High-beta stocks are supposed to be riskier but provide a potential for higher returns (Greece’s bonds) ; low-beta stocks pose less risk but also lower returns (German’s bonds). Beta is calculated by dividing the covariance between the portfolio and market by the variance of the market.

There are some similarities between the beta and standard deviation especially in their limitations: Beta and standard deviation use historical measure of a stock's volatility which may not necessarily predict future beta or future volatility. The betas seem to revert back to the mean. This means that higher betas tend to fall back toward 1 and lower betas tend to rise toward 1.For traders looking to buy and sell stocks within short time periods, beta is a fairly good risk metric. However, for investors with long-term horizons, it's less useful.Beta and standard deviation also doesn’t work for new company because of the lack of data of it. Technically with calculation from beta and standard deviation a portfolio that has fallen sharply in value is riskier than it was before it fell. But it should be the contrary as the portfolio is less likely to further decrease and that it represents now a lower risk-investment. Beta says nothing about the price paid for the portfolio in relation to its future returns.

Finally beta and standard deviation measure only the systematic risk. Therefore investors need to consider specific risks for each of their asset by analyzing each asset: the balance sheet (low ratio of debt to total capital), forecast of growth, earnings or dividends…Investors need to avoid overpay for asset. Investors can look at the price to earnings: Stocks trading at low multiples of their earnings are safer than stocks at high multiples.

Investors should also calculate the correlation between the assets of their portfolio. The higher the number of assets are positively correlated the higher is the risk. Additionally investors need to measure how much correlated are their assets. They can use regression analysis to do so. They take data from two assets and add a trend line, then determine how closely all of the data points fall to the trend line. The closer the data points fall to the trend line, the stronger is the effect, or correlation, between the two variables.

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Conclusion

The finding from the excel sheet on the FTSE100 and additional research clearly indicate that the normal distribution doesn’t adequately reflects financial market risk, especially in volatile times. The calculations don’t take into account (or badly), non-normal or extreme events, the severity of the risk impact, behavior, correlation between assets and specific risk. It is also a limited measure of the volatility. Investors under estimating risk by using this measure.

To better quantify these risks investors can use: VaR, trust index, fat tailed distribution, correlation measurement, assets price to earning…

Investors should also rely on their own financial education to take the final decision to invest in an asset or not. Investors need to rely on more than one measurement as none is perfect.

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References:

Black swan theory, wikipedia

https://fr.wikipedia.org/wiki/Th%C3%A9orie_du_cygne_noir

https://en.wikipedia.org/wiki/Black_swan_theory

Forbes, Uncertainty and risk management: What to do about black swans? Bill Conerly, Febuary 20 2013.

http://www.forbes.com/sites/billconerly/2013/02/20/uncertainty-and-risk-management-what-to-do-about-black-swans/#25ab59262c1a

Investopedia:

Measuring and managing investment risk, by Katrina Lamb, July 24, 2007

http://www.investopedia.com/articles/08/risk.asp

Complete guide to corporate finance. Return, risk and the security market line-systematic and unsystematic risk

http://www.investopedia.com/walkthrough/corporate-finance/4/return-risk/systematic-risk.aspx

Capital asset pricing model CAPM

http://www.investopedia.com/terms/c/capm.asp

Beta: Know the risk, by Ben McClure

http://www.investopedia.com/articles/stocks/04/113004.asp

Market risk:

http://www.investopedia.com/video/play/market-risk/

How investment risk is quantify, by Trevir Nath

http://www.investopedia.com/articles/investing/032415/how-investment-risk-quantified.asp

Risk

http://www.investopedia.com/terms/r/risk.asp

Beta definition

http://www.investopedia.com/terms/b/beta.asp

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Fat- tailed and skewed asset return distribution, by Svetlozar T.Rachev

http://www.worldcat.org/wcpa/servlet/DCARead?standardNo=0471718866&standardNoType=1&excerpt=true

Financial Times. Risk management. Modelling: Normal distribution is not always the norm, by Tracy Alloway, April 13, 2012.

http://www.ft.com/intl/cms/s/0/67d05d30-7e88-11e1-b7e7-00144feab49a.html#axzz44WH47t9g

Wikipedia:

68-95-99.7 rule

https://en.wikipedia.org/wiki/68%E2%80%9395%E2%80%9399.7_rule

Skewness:

https://en.wikipedia.org/wiki/Skewness

Kurtosis:

https://en.wikipedia.org/wiki/Kurtosis

L-moment:

https://en.wikipedia.org/wiki/L-moment

Lévy distribution:

https://en.wikipedia.org/wiki/L%C3%A9vy_distribution

Stable distribution:

https://en.wikipedia.org/wiki/Stable_distribution#Applications

Cauchy distribution :

https://en.wikipedia.org/wiki/Cauchy_distribution

Financial risk

https://en.wikipedia.org/wiki/Financial_risk#Asset-backed_risk

Regression analysis

https://en.wikipedia.org/wiki/Regression_analysis

Normal vs. Fat tailed distribution, by Lewis Lehe and Victor Powell

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http://vudlab.com/fat-tails.html

Video:

The Economist, Fat tails, illustrated, February 16 2010

http://www.economist.com/blogs/freeexchange/2010/02/financial_risk

Report

Non-normality of Market Returns

A framework for asset allocation decision making, JPMorgan

Book

Multi-Asset Investing: A practical guide to modern portfolio management, by Yoram Lustig, 2013

Image/ pictures:

https://www.bogleheads.org/wiki/Excess_kurtosis

Distribution of S&P500 daily returns vs normal distribution

http://managed-futures-blog.attaincapital.com/2013/02/04/fat-tails-and-tall-heads/

Can the “black swan” rescue investors from market declines?

http://www.feg.com/research/focus-topic/?nID=157&issue=2012_02

Financial Times, definition

http://lexicon.ft.com/Term?term=value-at-risk-_-VaR

http://lexicon.ft.com/Term?term=long-tail

http://lexicon.ft.com/Term?term=fat-tails

State Street index

http://www.statestreet.com/ideas/investor-confidence-index.html

Michgan index and household trust in USA:

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https://www.abcbourse.com/marches/economie_indice_confiance_michigan-17

Quantifying the volatility of your portfolio

http://tradingcommonsense.com/?p=531

Defining risk management -Part 4 : Risk quantification, Brad Egeland, 14 September 2009

http://pmtips.net/Blog/defining-risk-management-part-4-risk-quantification

Trading multiple definition

http://www.wallstreetoasis.com/finance-dictionary/what-is-a-trading-multiple

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