Market risk assessment using VaR and ETL models and its implications from a portfolio management point of view

Transcription

1 Market risk assessment using VaR and ETL models and its implications from a portfolio management point of view Author: Vlădut Nechifor Scientific coordinator: Professor Laura Obreja Brasoveanu, PhD. 1. Introduction The changes in the financial system, post-lehman, have proven how uncertain the evolution of financial markets can be, especially when market shocks are amplified by very volatile emotions and increased leverage. Moreover, the dangerous interplay between the public and the private vulnerabilities induces an even higher level of uncertainty in the financial markets, highlighting the importance of an accurate assessment of market risk. The losses occurred post-lehman on The Bucharest Stock Exchange (BSE) have also a behavioural explanation, as investors focus was dominated by the exuberance of the impressive returns obtained in the period This created the premise of assessing mainly the growth potential of shares and losing sight of the importance of an accurate market risk assessment. Looking back, if investors were interested more about the correct measurement of market risk and its mitigation through hedging with derivative instruments, the major losses post-lehman - that reached even 80% in the case of the shares of the Financial Investment Companies (SIF s) - would have been avoided. This paper addresses the topic of market risk assessment for a portfolio which contains the five SIF shares weighted on a fundamental basis in an effort to obtain a better risk-return trade-off than the BET-FI index (which is tracking the SIF s performance). First, in order to better illustrate this topic, the paper provides a comparison between the methods of computing Value-at-Risk (VaR) and Expected tail loss or Conditional Value-at-Risk (CVaR), illustrating the advantages and disadvantages of the models and testing the effectiveness of these measures of risk. Second, this paper compares the performance in terms of risk and return of an investment in the fundamentally weighted portfolio and in a portfolio tracking the BET-FI index. Third, this paper highlights the implications in terms of market risk and return if a hedging strategy with options (i.e. Protective Put) is in place to limit the market risk. These three concerns raised are presented in this paper as three challenges, respectively: 1

2 1 st challenge: Which model assesses the best the market risk? 2 nd challenge: Is the Portfolio better than BET-FI in terms of risk and returns? 3 rd challenge: Does it worth it to hedge the Portfolio? The in-sample period chosen ( ) and the out-of-sample period ( ) provided an interesting perspective on the shares evolutions because of the exciting international context of events, such as: the beginning of sub-prime crisis, the Lehman Brothers bankruptcy which led to a strong drop in the shares prices, the U.S. government s plan to buy toxic assets from the market (TARP) which reflected in a strong recovery in 2009 and 2010 of the equity markets from the lows of 2008 and also the beginning of the European sovereign debt crisis that led to new turbulences in the financial markets. Also, the local context was full of events which strongly impacted the shares listed on BSE, such as: the programme of the Romanian government with IMF, the uncertainty of continuing it due to the implementation of wage cuts and VAT increases, as well as the legislative initiatives on increasing the maximum percentage of the SIF s allowed to be held by investors. 2. An overview of the market risk models In order to assess the market risk, there are several tools that quantify through a single number the uncertainty in profit/loss of a portfolio, synthesizing the potential deviation from the expected return. The most common and less sophisticated tools are the following: the variance, the standard deviation, the coefficient of variation, the semi variance, but this paper will not focus on these measures of risk as they provide a limited amount of information. As the volatility in the financial markets increased, as the use of derivatives and leverage became more intense, the risk measurement became more and more important. The need for more sophisticated risk models such as Value-at-Risk and Conditional Value-at-Risk (or Expected Tail Loss) has become essential; the historical losses incurred by the Long-Term Capital Management (1998), the Orange County (1994) or the British bank Barings (1995) made these measures even more popular. Value-at-Risk started to become widely used in the mid 90 s due to the JPMorgan "RiskMetrics" service. For a portfolio, given a specific confidence level (1- ) and time horizon (h), VaR is defined as the threshold value such that the mark-to-market loss on the portfolio will not exceed this value. The definition is equivalent to the relationships below: 2

3 The VaR models are divided in two main categories: parametric models in which parameters are estimated and based on them, the distribution of profit/loss is built; this models refer to: Analytical VaR, CAPM VaR, EWMA VaR, GARCH VaR; non-parametric models that reduce the reliance on the distribution assumptions; this models refer to: Historical VaR and Monte Carlo VaR. The advantages of using VaR as a risk measurement tool are the following: It refers to the maximum loss that can be incurred by a portfolio given a particular confidence level; It measures the risk associated with various risk factors and their sensitivity; It can be applied to several different markets and exposures allowing comparisons; It can be broken down in order to isolate the various components of risk related risk factors (eg. systematic risk) or it can indicate the aggregate risk. Disadvantages of using VaR as a risk measurement tool are the following: It indicates the maximum loss of a portfolio given normal market conditions, not extreme conditions; Many VaR models rely on the normal distribution, which is not an accurate reflection of the true behaviour of the markets. The first drawback of VaR measure stated above, is one of a high importance, so it should be treated with special attention. For example, events with low probability, but very high impact on the returns of the portfolio, the so called "black swan events" 1 that have led to the substantial losses during the latest financial crisis, are not captured by VaR. Even if there are restrictions by bank regulators or through Investment Policy Statements, on the maximum amount of risk undertaken, the current possibilities of modelling the payoff and the risks through the use of derivatives allow traders or fund managers to bypass the rules, even if the actual risk taken is much higher. The solution for the above described problem is the latest method to estimate market risk: Expected Tail Loss or Conditional Value-at-Risk. This method essentially measures the magnitude of losses when the portfolio is subject to a VaR 1 The term was first introduced by Nassim Nicholas Taleb in his 2004 book Fooled By Randomness 3

4 exceed, CVaR being simply defined as the answer to the question "When things go wrong, how wrong do they get?". In order to test the performance of the VaR and CVaR models, they need to satisfy the unconditional coverage 2 and independence 3 of exceptions properties (which is equivalent of passing the conditional coverage test). This ensures that the rate of exceptions is not significantly different from the level set through the confidence level and the model is dynamic enough to adapt quickly when exceptions occur. 3 The case study The motivation for this case study built up easily because it addresses legitimate concerns of investors (the three challenges ) and the period analyzed was characterized by high volatility and interesting events, allowing to highlight the differences in the models and the importance of making informed investment decisions. The in-sample period chosen was January December 2010, and the out-of-sample period was January April The portfolio consisted of all the five SIF shares, weighted fundamentally based on PER and VUAN, and was rebalanced annually. As expected, the distribution of the returns of the portfolio is not Gaussian 4, presenting leptokurtosis and being negatively skewed, so the probability of extreme losses is higher than the one supposed by a normal distribution, thus risk assessment must be treated with great caution. 4 Results of the first challenge VaR models characteristics and back-testing results In order to assess risk, and the corresponding CVaR were computed using the six methods: Historical VaR, Analytical VaR, CAPM VaR, EWMA VaR, GARCH VaR, Monte Carlo VaR. For computing the Historical VaR, the empirical distribution of the 10 days returns of the portfolio was determined using 748 observations based on a rolling window. By computing the 1% percentile, the results presented in Annex 1 were obtained. VaR remained at high 2 Verified through the Kupiec test (1995) 3 Verified through the Christoffersen test (2004) 4 Results obtained through the Jarque Bera test 4

5 levels (40%) because of the "ghost" effects of negative returns from the in-sample period. Volatility before 2011 made the Historical VaR to overestimate the new market risk. Starting October 2011, the memory of Historical VaR begins to substantially remove the information from In order to compute the Analytical VaR, the standard deviation of the 10-days returns was computed using 748 observations based on a rolling window and as expected standard deviation changes very slowly. In order to fix this problem the number of observations was reduced to 250. By normalizing the values and computing the 1% percentile, the results presented in Annex 2 were obtained. Analytical VaR is lower than the loss suffered by the portfolio within 10 days in 11 of the cases. Risk is being underestimated especially within two periods, August and October 2011, fact that is expected to be severely sanctioned by the independence tests. This is explained by the fact that the assumption of normality of the distribution of the portfolio returns is not realistic as proven by the Jarque-Bera test. For computing the CAPM VaR, the coefficient was computed for the portfolio. After multiplying with the standard deviation of the market (approximated by the BET index) and after normalizing the values, the 1 % percentile was computed as detailed for Analytical VaR. The results obtained were compared with the ones for Analytical VaR with a 748 period in Annex 3. The advantages of the CAPM VaR is the facile way of computation and the capturing of the systematic VaR, but it is based on the unrealistic assumption of normal distribution. EWMA VaR was computed using a coefficient of persistence of the volatility of 0.95 (. Using the past volatility and returns, the updated volatility was determined and consequently the EWMA VaR. In early 2011, the measure tends to overestimate risk VaR because of volatility persistence, causing the EWMA to retain the past high volatility, but compared to the Historical VaR the "ghost" effect is not so significant. After mid-2011 we can see that VaR remains at the same levels due to the low inherited volatility, thus it follows a trend closer to the 10 days losses of the portfolio, but does not react quickly enough to market changes; only starting from the end of 2011 it incorporates these changes. These observations are drawn from the results presented in Annex 4. For computing GARCH VaR, a GARCH model was built for the portfolio returns. The coefficient obtained for the persistence of volatility ( was 0.98 and the coefficient of the 5

6 unexpected return ( was Out of the VaR measures outlined above, GARCH VaR most closely follows the 10 days losses of the portfolio as illustrated in Annex 5. Consequently it does not underestimate or overestimate consistently the market risk; this happens even at the end of 2011, a period during which other VaR measures have not correctly anticipated maximum potential loss. The choice of the volatility persistence coefficient is not made arbitrarily as in the case of EWMA VaR, but by estimating the parameters based on the insample period, this is why GARCH VaR is more accurate than the EWMA VaR. In order to generate the price paths required by the Monte Carlo VaR, the previous GARCH model was used for obtaining the conditional volatility for each scenario. Using the 10 dayforecasted volatility obtained by using the GARCH model and the random generated number, the 10,000 simulated returns were obtained. By computing the 1% percentile of the distributions of 10 days returns, the results in Annex 6 were obtained. This model most accurately assesses the market risk, eliminating the disadvantages of the previous methods, being a forward looking model. The first step in back-testing of the models was to determine the Rate of Failure (the ratio of exceptions out of the total number of observations). The results obtained were the following: Historical VaR (0%), Monte Carlo VaR (0.39%), GARCH VaR (2.14%), Analytic VaR (3.36%), CAPM VaR (2.75%), EWMA VaR (4.28%). The second step was to test the conditional coverage property, thus including the independence property. By comparing with the 1% percentile of the distribution with one degree of freedom (6.6349), as expected during the in-sample period, the VaR measures that didn t pass the test were EWMA VaR and CAPM VaR; during the out-of-sample period these measures were accompanied by the Analytical VaR. It is worth mentioning that the only VaR measure that passed the conditional coverage test, thus both the test of independence, was Monte Carlo VaR, Historical VaR passed the independence test but not the unconditional coverage test. CVaR models characteristics and back-testing results For each of the VaR models, CVaR was computed by computing the surface of the area under the distribution of returns which was bounded to the right by the VaR value. As expected by construction, all CVaR models provided higher values than the VaR models. 6

7 Risk measured through CVaR reflected the market risk better, including during the periods in which major losses were incurred. The performance of CVaR models depends on the method used for computing VaR, for example Historical and Analytical CVaR overestimate the risk, but changing the number of observations taken into account from 748 to 250 for these two models, the results are improved. The results of the CVaR models are consistently better for each method than those obtained through VaR models; this is due to the fact that the average losses incorporate also the values higher than VaR for each model. The results are presented in Annex 7, Annex 8 and Annex 9. 5 Results of the second challenge Having the confirmation from the previous challenge that VAR and CVaR models through Monte Carlo simulations are the best models for measuring market risk, the risk of BET-FI was quantified by implementing these two models, the results obtained are presented in the Annex 9, Annex 10 and Annex 11. As can be seen in the Annex 11, the risk for the portfolio in 2011 and 2012 is lower than that of BET-FI, except during the period July-August 2011; on an overall basis the risk of the portfolio is less in 64% of cases than that of the BET-FI index. The average CVaR is in all years higher for BET-FI than for the Portfolio, while the returns of BET-FI are lower in all years (except in 2008 when the portfolio and BET-FI have both a return of -83%). In order to assess the trade-off between risk and return, Sharpe and Treynor ratio, as well as and Jensen s Alpha were computed. Taking into account the limits of these indicators when returns are negative, we made the comparison between BET-FI index portfolio only for the years 2009 and The Portfolio presented a superior trade-off between risk and returns in both years. Nevertheless, we must bear in mind the disadvantages of using the standard deviation (used in the above mentioned indicators) as a measure of market risk. Moreover, by computing the Tracking Error and analyzing the values for the Information Ratio, it can be easily observed that the performance of the Portfolio is consistent with an enhanced tracking index strategy. 6 Results of the third challenge 7

8 To reduce the risk of the Portfolio, European put options with one year maturity on the BET- FI were bought. For every year a new number of put options with different strike prices were acquired. The number of puts chosen was based on a Protective Put strategy and the exercise price was chosen to make the option be at the money in case of the last year s scenario. The characteristics of the options are presented in Annex 14. The risk and return of the hedged and unhedged Portfolio are shown in Annex 13 and Annex 15. In 2011, when the BET-FI decreased by 15%, the use of options was reflected not only in the reduction of the average risk with 4% (measured as Monte Carlo CVaR), but ensured a higher return, even if a negative one (-6%) compared with the return of the unprotected portfolio return (-10%). In 2012, the Average CVaR for the protected Portfolio was 7% lower than that of the unprotected Portfolio, justifying clearly in terms of the risk analysis the usage of options. Comparing the effects of using options, it can be observed that the risk of the unprotected Portfolio is higher in 99.86% of cases. In 2012, the risk mitigation was reflected in a return of only 19% compared with the 24% of the unhedged Portfolio. This difference came from purchasing options that were not exercised, while the premium paid for them was quite high due to high levels of volatility. 7 Conclusions In very volatile conditions, the market risk assessment becomes very important because the portfolio losses can be significant and very difficult to anticipate. Knowing the maximum level of losses over a short period of time (10 days) with a confidence level (99%) makes it easier to understand the consequences of being exposed to market risk. The best model that fulfils the unconditional coverage and independence property is Monte Carlo VaR. When computing CVaR the performance of all the models is improved, but the ranking between them remains the same. Using a fundamentally weighted Portfolio of shares a lower risk and a higher return (with only one exception - year 2008) is obtained vis-à-vis the BET-FI indes which is weighted depending on the market capitalization. Using a Protective Put strategy the hedged portfolio reduced its risk, while the impact on returns is asymmetrical, the returns are lower in 2012 and higher in 2011 than those of the unhedged portfolio. 8

9 Bibliography [1] Alexander, C 2008, Market risk analysis - Practical financial econometrics, John Wiley & Sons, West Sussex. [2] Alexander, C 2009, Market risk analysis - Value-at-risk models, John Wiley & Sons, West Sussex. [3] Barone-Adesi, G, Bourgoin, F & Giannopoulos, K 1999, Don t Look Back, Risk, nr.11, p [4] Bodie, Z, Kane, A & Marcus, AJ 2007, Investments, the 7th edition, McGraw-Hill Primis, New York. [5] Bollerslev, T, Chou, RY & Kroner, KF 1992, ARCH Modeling in Finance: A Selective Review of the Theory and Empirical Evidence, Journal of Econometrics, vol. 52, p [6] Boudoukh, J, Richardson, M & Whitelaw, R 1998, The Best of Both Worlds, Risk, no.11, p [7] Brooks, C 2002, Introductory Econometrics for Finance, Cambridge University Press, Cambridge. [8] Campbell, JY, Lo, WA & MacKinlay, AC 1997, The Econometrics of Financial Markets, Princeton University Press, New Jersey. [9] Campbell, SD 2007, A Review of Backtesting and Backtesting Procedures, Journal of Risk, vol. 9, Winter 2007, p [10] Christoffersen, P & Pelletier, D 2004, Backtesting Value-at-Risk: A Duration-Based Approach, Journal of Empirical Finance, no.2, p [11] Codirlau, AD & Chideciuc, NA 2008, Econometrie aplicată utilizând Eviews 5, visited on 10 March 2012, < /econometriebancara2008.pdf>. [12] Glasserman, P 2003, Monte Carlo Methods in Financial Engineering, Springer-Verlag, New York. [13] Graham, B 1949, The Intelligent Investor: The Definitive Book on Value Investing, Collins Business, New York. [14] Greene, WH 2003, Econometric analysis, a V-a ediie, Prentice Hall, New Jersey. [15] Holton, AG 2003, Value-at-risk : theory and practice, Academic Press, Londra. 9

10 [16] Hull, JC 2002, Options, Futures and other derivatives, The 5th edition, Prentice Hall, New Jersey. [17] J. P. Morgan&Reuters 1996, RiskMetrics Technical Document, The 4th edition, Morgan Guaranty Trust Company, New York. [18] Kuester, K, Mittnik, S & Paolella, 2006, Value at Risk Prediction: A Comparison of Alternative Strategies, Journal of Financial Econometrics, vol.4, no. 1, p [19] Kupiec, P 1995, Techniques for Verifying the Accuracy of Risk Management Models, Journal of Derivatives, no.3, p [20] Mishkin, FS 2003, The economics of Money, Banking and Financial Markets, the 7th edition, Addison Wesley, San Francisco. [21] Necula, C 2009, Evaluarea opiunilor financiare. Volumul I - Modelul Black-Scholes- Merton, Editura ASE, Bucureti. [22] Ozerturk, S 2008, Hedging with an index option, visited on 1st June 2012, < [23] Powell R.J., Allen D.E, 2007, Thoughts on VaR and CVaR, Modelling and Simulation Society of Australia and New Zealand [24] Poonn, SH 2005, A Practical Guide to Forecasting Financial Market Volatility, I ediţie, John Wiley & Sons, West Sussex. [25] Sühan Altay, C. Coşkun Küçüközmen, 2009, An assessment of value-at risk (VaR) and expected tail loss (ETL) under a stress testing framework for Turkish stock market, The Journal of Risk Finance, vol. 11, p [26] Yong Bao, Tae-Hwy Lee, Burak Saltoglu, 2006, Evaluating Predictive Performance of Value-at-Risk Models in Emerging Markets: A Reality Check, The Journal of Forecasting, no. 25, p

11 Annex 1 ( 1) * 10 day return Historical VaR Annex 2 ( 1) * 10 day return Analytical VaR Annex 3 11

12 ( 1) * 10 day return Analytical VaR CAPM VaR Annex 4 ( 1) * 10 day return EWMA VaR Annex 5 ( 1) * 10 day return GARCH VaR 12

13 Annex 6 Monte Carlo VaR ( 1) * 10 day return Annex 7 ( 1) * 10 day return Analytical VaR Analytical CVaR Annex 8 ( 1) * 10 day return Historical VaR Historical CVaR Annex 9 13

14 Monte Carlo VaR Monte Carlo CVaR ( 1) * 10 day return Annex 10 ( 1) * 10 day return Monte Carlo VaR Monte Carlo CVaR Annex 11 Monte Carlo CVaR BET FI Monte Carlo CVaR Portfolio Annex 12 14

15 Year Portofolio BET-FI Return Average CVaR Return Average CvaR % 35% -83% 38% % 39% 90% 42% % 29% -10% 34% % 27% -15% 27% % 21% 22% 26% Annex 13 Monte Carlo CVaR Hedged Portfolio Monte Carlo CVaR Portfolio Annex 14 Year BET-FI before buying the option K No. of puts Cost of puts Value of puts at expiration 5, , RON ,688 31, RON ,140 24, , RON ,79 RON Annex 15 Year Return of hedged Portfolio Average CvaR hedged Portfolio Average CvaR unhedged Portfolio Return of unhedged Portfolio % 22% 27% -10% 15

16 % 14% 21% 24% 16