Researching the risk associated with different risk profiles
using VaR and Expected Shortfall
Abstract
In 2008, many portfolio return fell below the return range of their risk profile. This paper examines if the current risk measure, Value‐at‐Risk, is an accurate risk measure for determining the return ranges and what the role is of the risk profiles and the corresponding asset mix in exceeding the return ranges. The performance of the Value‐at‐Risk approach is compared with the Expected Shortfall approach over the past 23 years and under stress situations. The results show that the VaR and ES values largely exceeded the return ranges used by financial institutions, especially in stress situations. Furthermore, results show that the risk profiles overlap each other not only based on the portfolio allocation but also on the measured risk. Moreover, due to the many different portfolio allocation possibilities within a risk profile, there is a large risk spread within a risk profile which makes one return range per risk profile insufficient. To improve the accuracy of the return range calculation per risk profile, the Expected Shortfall method should be used complementary to the Value‐at‐Risk approach and the risk profiles should become isolated categories.
Acknowledgements
This thesis is the final product in completing my MSc of Business Administration, Specialization Finance. Together with writing this thesis, I also completed a five‐month internship at one of the leading financial institutions in the Netherlands at this moment. I would like to thank the company for offering me an internship. Also thanks to my colleagues, of which I learned a lot and made the five months pleasant.
1
Introduction
stage consists of two groups, one that is entering the market and one that is leaving, both groups base the decision on their predictions about asset movements. Figure 1.1 Product life cycle The unpredictability of asset movements creates risk accompanied with investing on a stock market. Besides this equity risk, there are also other types of risk involved with investments, for example interest rate risk, currency risk and commodity risk. The impact of these risk factors can be diminished by creating a well diversified portfolio. To diversify a portfolio, the assets have to be distributed over many different sectors and geographical areas. This way the risk is spread over many different factors, which means that the portfolio is less dependent on one asset and therefore will be less influenced by a large decrease in a certain asset. However, before someone can establish a portfolio, first a persons’ risk profile has to be determined according to the Wet op Financieel Toezicht (Wft)1. This law is introduces to
create a good match between a persons’ investment behaviour and the established portfolio. An investor can be categorized in six different risk profiles, from defensive till offensive. Every risk profile has a range of expected annual returns consisting of a maximum negative expected return and a maximum positive expected return. For example, the Institute for Research and Investment Services, IRIS, calculated that investors with an offensive risk profile have a probability of losing as much as 25 percent and gaining a maximum of 45 percent2. The return achieved in a certain year will lie between these limits given a 95
This thesis will examine if the current risk measure, Value‐at‐Risk (VaR), is an accurate risk measure to determine the return ranges. Furthermore, the role of the risk profiles and the corresponding asset mixes in the exceeding return ranges are examined. To test the performance and accuracy of the VaR approach, I will compare VaR with another risk measure called Expected Shortfall. Both risk measures will be calculated based on historical data and simulated stress scenarios and compared with the current return ranges. For this, the following research question is developed: Are the currently used return ranges accurately calculated based on VaR and what is the role of the risk profiles and the corresponding asset mix in exceeding the return ranges?
The outline of my thesis is as follows. First, a literature review about which factors are considered when creating a portfolio and the different risk measurement methods used in the financial world are presented. I will continue with the data and methodology used in my research, followed by the results. In the last section the conclusion is given, together with the limitations of this paper and recommendations for further research.
2
Literature
In this chapter, several factors that influence a persons’ investment behaviour are discussed. After which various risk measures and the role of risk measures in the realization of risk profiles and the corresponding return ranges will be explained.
2.1 Behavioural Finance
In theory, it is assumed that people make rational investment decisions. Moreover, March (1994) found that although individuals tend to be rational, they are constrained by limited cognitive capabilities and incomplete information. A few examples of these constraints are; heuristics, mood/emotion and a persons psychology (Elton et al, 2007). When looking at the psychology of investors, it is found that investors regard potential losses and gains differently. Many experiments showed that investment decisions are inconsistent with rational decision making. People measure the importance of their gains and losses relative to a certain reference point (Elton et al, 2007). For example, even though an investor achieved a 20 percent return, if a neighbor gained 30 percent on his investment, the investor is not totally satisfied which can persuade him to accept more risk in his next investment decision. Other reference points may be the price at which the asset is purchased or the current wealth of an investor. This can result in irrational investment decision making, depending on whether the outcomes are regarded gains or losses compared to their reference point.
Moreover, the mood or emotions of investors influence their investment decisions. This is seen in the day of the week effect where people feel a certain optimism or pessimism with a certain day. Significant negative stock returns are found on Mondays, which people generally see as a pessimistic day. Researchers even found a significant correlation between stock market movements and the weather, where bad weather leads to negative stock returns (Hull, 2006).
consequently leads to irrational and poor decision making. For example, when a positive news release about a certain company persuades investors to buy that companies’ stock, while ignoring the history of the stock, company or other relevant data. Such decisions are not taken rationally and are often not in the best interest of the investor. To decrease such irrational decision making, an investment advisor could be employed. Investment advisors largely use information calculated by research institutes to base their investment decision on, thereby decreasing the influence of the second and third constraints.
As mentioned earlier, investment decisions can only be made when the risk profile of the investor is established. In the law of Financial Authority, it states that before a financial institution can advise or manage a clients’ assets, they first have to gather information about the financial position of the client, their knowledge and experience, their investment goals and risk acceptance (Loonen & van Raaij, 2008). To gather this information financial institutions created a series of questions, for an example of such a questionnaire see Appendix A.
2.2 Risk tolerance
is of importance. A first‐time investor will respond differently to a large decline in the stock market than a well seasoned investor. Most of these factors can be retrieved in the questionnaire used to determine a persons’ risk profile, see Appendix A.
2.3 Time Horizon
Besides the factors mentioned above, the investment time horizon also plays an important role in determining the risk profile. If a person wants to invest for two or fifteen years, this certainly affects the chosen asset mix. When investing for longer time horizons, a larger part of the portfolio can be allocated to equity to increase the annual return. According to Sing and Ong (2000) stocks are superior to bonds when held beyond ten years, and are also likely to perform better than other assets for shorter holding periods. However, because stocks are very volatile/risky, stocks can largely decrease in one year. When investing for two years it is likely that this negative return will not be compensated, while in ten years time this is very likely. To achieve the aimed return for a short term investment horizon, often the weight in safer assets is increased and the weight in riskier assets decreased. Thus, when investing with a long term horizon, an investor could allocate a larger part in risky assets to achieve a higher annual expected return.2.4 Portfolio Allocation
Based on the factors mentioned in the previous sections, a persons’ optimal portfolio can be allocated. Optimizing portfolio allocation can be defined as the process of mixing asset weights of a portfolio within the constraints of an investor’s capital resources to yield the most favourable risk‐return trade‐off (Sing and Ong, 2000). A portfolio can be allocated with several different assets; derivatives, stocks, bonds, real estate and liquid assets. The distribution of these different assets is called the asset mix3. Every asset is accompanied with
a certain amount of risk; the more risky the asset, the higher the expected return.
Liquid assets are generally the safest investment because they can quickly be converted into cash, but in return produces the lowest rate of return of the several assets. A few examples of liquid assets are; savings accounts, money market funds and government treasuries. Bonds contain more risk than liquid assets but in general are less risky than stocks and
derivatives. There are many different forms of bonds, from reasonable safe to very risky, respectively; government bonds and junk bonds. Finally, there are the moderate risk investments (stocks, real estate, mutual funds) and speculative investments (options, commodities and derivatives). The first investment category may incur losses but on a long term will achieve a higher level of return than bonds and liquid assets. The latter investment category has the highest risk of all assets, it may yield large gains but also large losses and are therefore called speculative investments4. For the portfolio allocation in this thesis, only
stocks, bonds and liquidities are considered. Most financial institutions also use real estate and a speculative investment for portfolio allocation, but this will largely complicate the necessary calculations and are therefore left out of the created portfolios.
When an asset mix is established, the assets can be further distributed between sectors, countries and certain specific subjects, to diversify the risk as much as possible. When investing globally, the asset mix may be distributed between Emerging markets, Pacific area, North America and Europe. Moreover, a few sectors over which the investment can be spread are: IT‐, financial service‐ and the energy‐sector. Finally, a portfolio can also be distributed as such that it is linked to a certain market movement, for example to inflation or interest rates. Inflation linked bonds are bonds that rise together with the inflation rate. In a situation with a high inflation, an inflation linked bond can diminish or neutralize the impact of the inflation on the portfolio value. When the asset mix and the distribution pattern are determined the individual assets for a portfolio are chosen, this is called stock picking (Elton, et al, 2007). To select individual stocks, the well researched information of the research institutes is used. Based on this information together with the investors’ knowledge of assets or preferences, a portfolio is created. When an investment advisor is employed, it is possible that when two advisors allocate a portfolio for the same person, they will create different portfolios. This is mainly caused by the variety of portfolios that can be established within a risk profile. For each risk profile an advisor/investor is free to allocate the portfolio between a minimum and a maximum weight per asset category (see Table 3.1). This can result in large differences in portfolio allocation within a risk profile and consequently affect the possible risk attached to a risk profile.
2.5 Different approaches of risk measurement
Before different risk measurements methods will be explained, first risk itself has to be defined. How people perceive risk can be divided into two parts. The first part is the personal and social characteristics of an investor (Schwing and Albers, 1980). As is discussed in section 2.2 this can depend on wealth, income, age, gender, education and marital status. The second part of risk perception is the likelihood and consequences of a negative outcome of an event or action; the probability that a loss will occur. In terms of investment decisions, such a negative outcome can be defined as the chance that the actual return of an investment is different than the expected return. The chance that the actual return differs from the expected return can be closely estimated by risk measures. Throughout the years many risk measures have been developed, such as; - Mean‐Variance framework - Value at Risk - Expected Regret - Conditional Value at Risk - Expected Shortfall - Tail Conditional Expectation - Worst Conditional Expectation (Szego, 2005) Even though there are many other risk measures, I will mainly focus on the most commonly used risk measures, which are; the Mean‐Variance framework, the Expected Shortfall approach and Value‐at‐Risk method (Gustafson & Lummer, 1996).
2.5.1 Mean variance framework
drawbacks (Sing and Ong, 2000). First, due to the assumption that returns follow a normal distribution, the use of the mean‐variance model is limited when returns are skewed. Secondly, the standard deviation is not consistent with investors’ actual perception of risk and risk aversion is ignored. In practice, investors are very concerned with the level of risk and most times prefer less to more. Slovic, (1967) examined the behaviour of investors and found that riskiness is more likely to be determined by the probability of loss and the amount of loss, rather than basing their preferences on variance. Consequently, including risk preferences is important when establishing portfolios. Because in this thesis, the risk profile/behaviour of an investor is an important factor in risk measurement, Markowitz’s framework is not the best option for this research.
Harlows’ research (1991) is in agreement with this statement, he concludes that the mean variance framework is not the best approach for portfolio allocation. Harlow used a set of international asset allocation examples to demonstrate the benefits of downside‐risk approaches in comparison with the mean variance framework. By optimizing the different portfolios, he found that the downside measures produces the same or higher returns than the mean variance framework with less downside risk exposure. The downside risk measures thus lower risk while maintaining or improving the level of expected return. This makes downside risk approaches superior to the mean variance framework. When comparing the portfolio allocation of these methods, he found a significantly higher bond allocation in the portfolios based on the downside risk approach. An explanation for this is that bonds have less risk exposure than stocks and because the downside risk approach aims at a low risk exposure, a greater weight in bonds is used to decrease the portfolio risk. Based on Markowitz’ framework, portfolios will be allocated with a larger part in stocks to create the highest return possible for a certain level of standard deviation, while ignoring the risk aversion of investors.
2.5.2 Value at Risk
is the most one can expect to lose, with a 95/99 percent confidence level, in the next week/month/year. Because VaR is always related to some confidence level, mostly between 95 and 99 percent, it does not really predict the maximum loss that may be incurred, but only the worst result than can occur in a certain period of time given the chosen confidence level (Goorbergh & Vlaar, 1999). The definition is consistent with VaR representing the left‐ tail of a one‐sided confidence level. This is shown in the following figure, where VaR is presented as the cutoff point between the colored zone and the rest of the distribution given a 95 percent confidence level. The colored zone represents the values, which only occur in 5 percent of the time, whereas the remaining distribution represents 95 percent of the values. 5 Figure 2.1 VaR given a 95 percent confidence level. There are several approaches to calculate VaR, I will discuss the following three approaches (Gregory & Reeves, 2008); - VaR analytically calculated - VaR based on historical data - VaR based on Monte Carlo Simulation VaR calculated analytically
VaR based on historical data
A second method to calculate VaR is by running hypothetical portfolios based on historical data. This is the most applied method to calculate VaR. Using historical trends, returns and volatilities, an assets’ distribution is produced. Besides the historical data, three important factors have to be determined before VaR can be calculated; the confidence level, the initial value of the portfolio and the time horizon. The choice of the confidence level depends on the risk tolerance of the institution bearing the risk. The confidence level mostly ranges between the 1 and 10 percent (Wharton, 1996). I have chosen to use a confidence level of 5 percent, so it has the same confidence level as is used to create the return ranges given in Table 2.1 The next factor is the initial investment, this is set at € 100.000. Finally, also the VaR time horizon has to be established. The most commonly used time horizons for VaR calculations are day, week or monthly time horizons. But to calculate weekly or monthly VaRs for a dataset of 23 years for five different risk profiles analytically, will create numerical difficulties. Therefore, I have chosen to calculate annual VaRs for the past 23 years for each risk profile. Using all these factors, a distribution can be determined, and VaR can easily be calculated using the distribution table or graph. This approach will be applied in section 3.3
VaR Based on Monte Carlo Simulation
The third method which can be used to calculate VaR is Monte Carlo simulation. The advantage of simulation is that it can simulate the effect of one factor while keeping the environment stable. It gives insight of the real environment but has better control over experimental conditions than in a real environment with many changing factors. Furthermore, when the dataset is large and the distribution is skewed, the required number of quadrature points is likely to increase exponentially with the number of assets. This makes the creation of portfolios exceeding two or three assets using numerical integration very difficult and simulation more attractive (Brooks, 2002).
But a drawback of the simulation process is that the results are often hard to replicate. Due to the randomness of the simulation process, the results are difficult to replicate, unless the sequence of the random draws are known. Moreover, when incorporating unrealistic
assumptions into the simulation process, the final results may not be precise. For example, assuming a normal distribution, while the actual distribution is skewed (Brooks, 2002). How this approach will be executed, is explained in section 3.5. Although, VaR is one of the most applied risk measures, VaR has a few shortcomings. First of all, VaR is not a coherent measure of risk. VaR does not fulfill the axiom of sub‐additivity. This means that the risk of a portfolio made of sub‐portfolios should be, at most, the sum of the separate amounts of risk calculated for the sub‐portfolios (Acerbi & Tasche, 2002). With VaR, it is possible that the portfolio risk is larger than the sum of the standalone risks of is components (Acerbi & Tasche, 2002). Moreover, it is very difficult to translate the VaR of a specific asset into a portfolio VaR. Another main critic is that VaR does not consider any rare, though possible, losses larger than VaR itself, hereby underestimating the real risk (Giannopoulos & Tunaru, 2005).
Despite these disadvantages, VaR is still one of the most applied risk measure by financial institutions. Firstly, because of its simplicity, ease of computation and ready applicability (Yamai & Yoshiba, 2005). Secondly, because the Basle Committee, a committee supervising the banking sector of several countries, permitted banks to use the VaR method to calculate the risk of their capital requirements. However, the credit crisis of 2008 showed that VaR as a risk measure has fallen short. The calculated VaR levels have largely been exceeded, surprising many investors. The question is, if this is caused by badly estimated VaR’s or if the problem lies with the determination of the risk profiles and the corresponding asset mix’s. Therefore, in this thesis, using historical data and Monte Carlo simulation, the accuracy of the VaR method will be tested and the performance of the VaR method is compared with another risk measure. Also, the influence of different asset mixes per risk profile on the measured risk will be examined.
2.5.3 Expected Shortfall/Conditional VaR
measures are; Expected regret, Conditional‐VaR (C‐VaR), Tail conditional expectation and Worst conditional expectation (Szego, 2005). Of these different downside risk measures, ES and C‐VaR are the most frequently used risk measures. When in a research continuous random variables are used, ES and C‐VaR have the same formal definition, which is the case in this thesis. This means that Expected Shortfall is conditional to the VaR method (Szego, 2005). This conditionality represents the probability of one event, Expected Shortfall, given the occurrence of another related event, Value‐at‐Risk (Keller & Warrack, 2003). Thus, the Expected Shortfall approach measures the loss beyond the VaR level. This can be expressed in the question6 ʺwhen things get bad (ie the VaR level is exceeded) what is our expected
loss?ʺ.
Figure 2.2 Expected Shortfall
Figure 2.2 shows how the Expected Shortfall relates to VaR. To calculate VaR the whole distribution is used, see the right picture of figure 2.2, whereas for the ES only the distribution as of the cutoff point of VaR has to be considered, see the left picture in figure 2.2. In several papers, ES is presented as a good alternative for VaR. For example, Yamai and Yoshiba (2002, 2004) have written multiple papers where they compared the performance of VaR and ES, measuring several factors such as; their estimation errors, decomposition into risk factors and optimization. Firstly, they found that rational investors who maximize their expected utility may be misled by the use of VaR as a risk measure. Moreover, they concluded that VaR is less reliable under market stress. This can be solved by using ES, which by definition takes into account losses beyond the VaR level. Therefore, I have chosen to use Expected Shortfall as the risk measure to compare with the VaR method. Expected Shortfall is calculated as the average of a set of VaRs of different confidence levels beyond the cutoff point. This method has also been applied by Giannopoulos and Tunaru (2005).
The left picture of figure 2.2 is the whole ES distribution, where the loss at different confidence levels is calculated using VaR calculations. For example, a 5 percent ES confidence level is the same as calculating VaR at a 0.25 percent confidence level and a 10 percent ES can be calculated by measuring the VaR at 0.5 percent.
I will compare the results of both risk measures based on historical simulation and I will perform a Monte Carlo simulation, to test the performance of both risk measures under stress situations. The risk measured by both methods, will be compared and analyzed with the return ranges given in table 2.1. This will show if the return ranges based on VaR are accurately measured or if ES would be a better risk measure to use for the calculation of the return ranges.
2.6 Risk Profiles and Expected Returns
In the previous sections several factors have been discussed that are necessary to determine a persons’ risk profile and the risk attached to a certain risk profile. Financial institutions have to gather this information according to the law. The Ken‐uw‐Klant beginsel states that before a financial institution may control or advice clients about their assets, they first should gather the necessary information to create a good match between a persons’ risk behaviour and their investment portfolio (Loonen & van Raaij, 2006). Unfortunately, how financial institutions should process this information into a portfolio is not given. As will be explained later in this paragraph, this causes major differences in the measured risk and portfolio allocation per risk profile.
return of three percent, and is achieved by only using savings accounts (IRIS Institute)7.
Since this risk profile does not invest in assets, it will be kept out of this research. Secondly, there is the very defensive investor. This is characterized by people who would like to invest but are very careful. Therefore, the investments will mostly consist of long term bonds and for a small part in stocks. The expected return is slightly higher than that of a no‐risk investor. Furthermore, there is the defensive profile. Investors with a defensive profile will try to limit the risk by using bonds, lodgement/deposits but will have a larger percentage in stocks than in the former risk profile. This profile has an expected return of 5.5 percent. The neutral investor has an expected return of 6 percent. The goal here is to certainly outperform the return of a savings‐account and these investors are willing to take a higher risk for that. An offensive investor has a long‐term strategy and is not worried by a decrease in the portfolio value. It has a higher percentage in stocks than in bonds and has an expected return of 7 percent. Finally, there is the very offensive risk profile. A very offensive investor also has a long‐term strategy and is therefore not concerned by large swings in the stock market, because the goal is to achieve a high average return over ten or more years. A very offensive portfolio will mostly consist of stocks. Although stocks may sometimes yield a greater loss than other assets, the probability of achieving a higher average return is still higher in comparison with other assets (Ho et al, 1994). Given enough years, the average return will converge to the expected return of the risk profile.
Besides the expected return, research institutes also calculated the return range for each risk profile based on the VaR approach. This return range gives the maximum negative return and maximum positive return given a 95 percent certainty. The return ranges per risk profile are presented In Table 2.1. These return ranges show the length in which the returns can vary throughout the years per risk profile. For example, a neutral investor can have a return of ‐10 percent in one year and 25 percent the next. But in the past year these boundaries have been largely exceeded. This is because the boundaries are calculated with a 95 percent certainty rate, which means that there is a 5 percent chance that the boundaries are foregone. The consequences foregone the boundaries are uncertain, which is the weakness of VaR. Because ES does take into account the losses beyond the VaR level, the
7 http://financieel.infonu.nl/beleggen/4298‐waarom‐een‐clientprofiel‐maken‐voordat‐u‐gaat‐beleggen.html
comparison between VaR and ES will show if VaR is an accurate risk measure of if it is better to replace it with ES.
Risk profile Return range 95 % certainty
Very Defensive Between -3 % and 18%
Defensive Between -10 % and 26%
Neutral Between -15% and 33%
Offensive Between -25% and 45%
Very Offensive Between -35% and 57%
Table 2.1 Return ranges per different risk profile
In table 2.1, the different risk profiles with the corresponding return range is given. This table suggests that every risk profile is an isolated profile and that there is no overlap between the profiles. However the determination of the risk profiles and the corresponding portfolio allocation is questionable. Loonen and van Raiij (2008) researched how several financial institutions classify investors in the different risk profiles. They found that although every financial institution uses more or less the same questions to establish a risk profile, the resulting risk profile differs per institution. This means that a person could be classified as a defensive investor at one institution and as a neutral investor at another. Furthermore, they found that where one financial institution would allocate 20 percent in stocks, another advised 100 percent, even though it was both based on the same risk profile. With an initial investment of € 10.000, this resulted in different profit returns per institution. The difference between the highest and the lowest profit of the investments varies from 157 till 6.230 euro, based on an investment horizon of ten years. These numbers show that the determination of the risk profiles and the asset allocation attached to a certain risk profile is subjective to the financial institute chosen.
situation, so people do not take more risk than they want or can afford. Thus, it is questionable to classify a person into one category, when someone can be classified into two risk profiles based on the portfolio. This also affects the return range of the investor. If someone is classified as a defensive investor but has a portfolio that also falls within the neutral risk profile, the return range of the defensive profile underestimates the risk associated with such a portfolio.
Thus, first the accuracy and variability of the current risk measure, VaR, used to determine the return ranges is tested. Different VaR time horizons and confidence levels are applied and the results will be analyzed. After which the effect of different portfolio allocations on the measured risk per profile is examined. Finally, the performance of both risk measures will be tested and compared with the given return ranges, for which historical and Monte Carlo simulation is applied. For this, the following hypotheses are applied; H0: The maximum VaR does not exceed the given negative return for a certain investor type given a 5 percent confidence level H1: The maximum VaR does exceed the given negative return for a certain investor type given a 5 percent confidence level
H0: The maximum ES does not exceed the given negative return for a certain investor
type
H1: The maximum ES does exceed the given negative return for a certain investor type
3
Data and Methodology
In the previous chapter the different risk profiles and the corresponding expected return range were discussed. The expected return is based on the chosen asset mix of a person. In table 3.1, the asset mix of every risk profile is shown. It includes a minimum and maximum advice on how to distribute the different assets. Furthermore, a tactical asset mix advice is given for an investor with a mid‐term horizon of 5 till 10 year and a strategic advice is given for an investor with a long‐term horizon of 10 years and longer. As described in section 2.3 different time horizons can result in a different asset mixes. Therefore, the strategic asset mix contains a larger part in riskier assets, while the tactical asset mix has a larger part in the safer assets.
Minimum Maximum Tactical Strategic
Stocks 0% 15% 10% 10%
Very Defensive Bonds 40% 60% 45% 50%
Liquidity 25% 60% 45% 40% Stocks 20% 40% 30% 30% Defensive Bonds 30% 50% 35% 40% Liquidity 10% 50% 35% 30% Stocks 35% 55% 45% 45% Neutral Bonds 20% 60% 30% 40% Liquidity 0% 30% 25% 15% Stocks 50% 90% 70% 70% Offensive Bonds 0% 40% 10% 20% Liquidity 0% 30% 20% 10% Stocks 80% 100% 90% 90%
Very Offensive Bonds 0% 10% 0% 0%
Liquidity 0% 20% 10% 10%
Table 3.1 The different weight allocation per risk profile
3.1 Dataset
one that is used most as a benchmark for other assets. For stocks, this is the MSCI World Index. The MSCI World index consists of stock indices all over the world, this makes the index a good representation of the world economy and thus for the standard portfolio. Since people cannot invest in the index itself, I would have to use the MSCI World index tracker. Such an index tracker is constructed to follow the exact value of the index itself. The problem with using index trackers is that trackers only exist since a few years and are therefore not useful for a research with a long term horizon. Therefore, I will use the MSCI world index itself, which will not make a significant difference, because the tracker is developed as such that it precisely follows the index itself and that makes the two interchangeable.
For bonds, it is more difficult to find one specific fund that is best to use to fulfill the bond allocation. There are several global bonds that are frequently used as benchmarks, which all have the same characteristics. Therefore, I select the five global bonds that are most used as benchmark for bonds. Of the five bonds, there are only two bonds for which the necessary data is available. The final choice was based on which bond had the longest time horizon availability, which was the JP Morgan Government Bond Index.
Finally, a liquid asset has to be chosen. One important characteristic of liquid assets is that it can be converted into cash, be made liquid, in a short period of time. Therefore, I choose to use a liquid asset published by the US government, the US Treasury‐bill (T‐bill) or the US Treasury‐note (T‐note). T‐bills are short‐term debt obligations, while T‐notes are intermediate‐long term debt obligations with several maturities from 2 years till 10. Both are backed by the US government. I choose to use US Treasury bills, since T‐bills fulfill investment needs similar to savings accounts8 and are often referred to as the least risky
investment. In Datastream, two T‐bills are available, one with a 3‐month and one with a 6‐ month maturity. I will use the 3‐month T‐bill because it is more liquid than the 6‐month T‐ bill, hereby satisfying the most important feature of a liquid asset.
Monthly data of all three assets is gathered using Datastream. I will use the total return index of the assets, because this index calculates the performance of an asset assuming that all dividends, interest and distributions are reinvested. Originally, the time horizon was set on 25 years. Unfortunately, the data of the global bond was only available since 1986,
therefore the time horizon is adjusted to 23 years; 1986 ‐ 2008. This time span is used to determine the asset mix for the risk profiles. The strategic weights are chosen for the portfolio allocation, because the time horizon of 23 years fulfills the condition for the strategic advice; 10 year horizon and longer. When allocating the portfolios according to the strategic weights, it is important that the portfolios are rebalanced every once in a while. Through the years the portfolio weights can drift away from their original weights due to value increase or decrease of the different assets, resulting in a different asset mix. For example, due to a large value increase of stocks, the asset mix changes from 10 percent in stocks to 30 percent, resulting in an increased risk exposure due to the higher volatility of stocks, which may not be acceptable to the investor. To make sure that the portfolio continues to satisfy the demands of the investor, it is necessary to periodically rebalance the portfolio back to the original strategic weights. The drawback of rebalancing is that it may reduce the optimal return.9 For example, imagine that two subsequently years have a stock return of 10 percent while bonds and liquid assets have a return of 5 percent, for a defensive investor the asset weight for stocks will be rebalanced back to 10 percent at the end of year 1, while a higher weight in stocks can result in a higher portfolio return the following year if it was not rebalanced. Normally, rebalancing should increase the transaction costs, but these are not incorporated in this research and therefore have no impact on the performance of the portfolios. I choose to rebalance the portfolios every year, so the created portfolios and the accompanied risk remain to be a good representation of the risk profiles. Besides the assumption that transaction costs do not exist, other assumptions also have to be taken into account such as; riskless assets do not exist and portfolio weights should be positive and sum to one, which means that short sales are not allowed.
Furthermore, I assume that stock prices are distributed lognormally. Researchers found strong empirical evidence that returns are non‐normal and are driven by asymmetric and/or fat‐tailed distributions (Jondeau & Rockinger, 2006). Moreover, a normal distribution assumes that values can be minus infinity and thus have large short/long positions, which is not in congruence with the earlier made assumption that short sales are not allowed. Also, when the distribution is normal, VaR and Expected Shortfall essentially give the same results (Yamai & Yoshiba, 2004). Because both risk measures are multiples of the standard
deviation, both measures give the same information under a normal distribution. Therefore, I assume a lognormal distribution. If the stock prices follow a lognormal distribution, the return distribution is normally distributed with a mean μΔt and variance σ2Δt (Benninga,
2006).
3.2 Portfolio calculation
Annual return MSCI World JP Morgan Glob. bond US T-bill 1986 30,13% 17,16% 6,76% 1987 1,56% 10,67% 6,38% 1988 21,50% 9,26% 6,58% 1989 22,01% 4,72% 8,70% 1990 -23,81% 10,91% 7,97% 1991 14,23% 11,18% 6,45% 1992 2,48% 8,52% 4,02% 1993 15,00% 11,12% 3,16% 1994 4,15% 1,85% 3,90% 1995 16,36% 16,86% 5,82% 1996 19,98% 6,10% 5,29% 1997 17,12% 0,41% 5,17% 1998 17,91% 13,23% 5,16% 1999 20,56% -3,48% 4,68% 2000 -1,97% -1,64% 5,89% 2001 -17,32% 5,11% 4,71% 2002 -18,61% 10,01% 1,79% 2003 13,04% 14,74% 1,16% 2004 12,00% 12,15% 1,20% 2005 16,55% -6,56% 2,91% 2006 12,81% 9,74% 4,64% 2007 8,40% 8,65% 5,07% 2008 -56,59% 4,14% 2,26%
Mean Annual Return 6,41% 7,60% 4,77%
Standard Dev. 15,01% 6,62% 0,61%
Table 3.2 The annual lognormal returns and standard deviations per asset
It might seem remarkable that the mean annual return of the bond is higher than that of the stock, while many researchers concluded that stocks always perform better than bonds over a long term horizon. This can be explained by the credit crisis and its huge impact on the stock market, and consequently on the MSCI World Index. The negative return of ‐56.59 percent of 2008 drastically lowers the mean annual return of the MSCI World Index. When 2008 is left out of the sample, the mean annual return of the stock, bond and liquid asset are, respectively, 9.28, 7.76 and 4.88 percent. The differences in the returns calculated with and without the returns of 2008 clearly show the exposure of the assets to equity risk.
between this range, but the distribution of the MSCI World index appears to have a very high kurtosis and a very negative skewness. The high kurtosis value means that the return distribution contains a few outliers, both positive as negative, and the negative skewness means that the chance of a negative return is higher than the chance on a positive return. To see how these values manifest in the distribution, I made a frequency distribution of MSCI World index returns. When looking at the distribution in a histogram, it has the shape of a normal distribution with a few abnormalities.(see Appendix B) This distribution is only based on 276 monthly returns, by increasing the dataset the distribution will become more accurate and the shape of the normal distribution may become more clear. Looking at the frequency distribution, it is reasonable to assume that stock prices are lognormally distributed.
To calculate the risk attached to the risk profiles, first the portfolio returns and standard deviations have to be calculated. The returns of the portfolios can be calculated, using the annual data presented in Table 3.2. This is simple done by applying matrix multiplication, the first matrix contains the strategic weights, XN, given in Table 3.1 multiplied with the
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
−
NN N N N N N Nx
ianceMatri
Co
Variance
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
....
....
....
....
var
3 2 1 3 33 32 31 2 23 22 21 1 13 12 11 [3]The portfolio standard deviation can be calculated by multiplying the variance‐covariance matrix, given above, with the strategic weights of Table 3.1, according to the following equation (Benninga, 2006);
[
]
⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ × ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ × = 3 2 1 33 32 31 23 22 21 13 12 11 3 2 1 X X X X X X pσ
σ
σ
σ
σ
σ
σ
σ
σ
σ
[4] In Appendix C, the standard deviations per risk profile are given. In the following table the annual returns of the portfolios and the average return and standard deviation over the past 23 years are given. Also, the mean annual return without 2008 has been calculated, to show the large impact of the crisis on the average portfolio returns over 23 years. The data given in table 3.3 will be used to calculate VaR and Expected Shortfall in the following sections.Portfolio returns Very Defensive Defensive Neutral Offensive Very Offensive
1986 14,30% 17,93% 21,44% 25,20% 27,80% 1987 8,04% 6,65% 5,93% 3,87% 2,05% 1988 9,41% 12,13% 14,37% 17,56% 20,01% 1989 8,04% 11,10% 13,10% 17,22% 20,68% 1990 6,26% -0,39% -5,16% -13,69% -20,64% 1991 9,59% 10,67% 11,84% 12,84% 13,45% 1992 6,11% 5,36% 5,13% 3,84% 2,63% 1993 8,33% 9,90% 11,67% 13,04% 13,82% 1994 2,90% 3,16% 3,19% 3,67% 4,13% 1995 12,39% 13,40% 14,98% 15,41% 15,31% 1996 7,16% 10,02% 12,23% 15,74% 18,51% 1997 3,99% 6,85% 8,64% 12,58% 15,92% 1998 10,47% 12,21% 14,13% 15,70% 16,64% 1999 2,19% 6,18% 8,56% 14,16% 18,97% 2000 1,34% 0,52% -0,66% -1,12% -1,19% 2001 2,71% -1,74% -5,04% -10,63% -15,12% 2002 3,86% -1,04% -4,10% -10,85% -16,57% 2003 9,14% 10,16% 11,94% 12,19% 11,85% 2004 7,76% 8,82% 10,44% 10,95% 10,92% 2005 -0,46% 3,21% 5,26% 10,57% 15,19% 2006 8,01% 9,13% 10,36% 11,38% 12,00% 2007 7,19% 7,50% 8,00% 8,12% 8,07% 2008 -2,68% -14,64% -23,47% -38,56% -50,70%
Mean Annual Return 6,35% 6,40% 6,64% 6,49% 6,25% Mean without 2008 6,76% 7,35% 8,01% 8,53% 8,84%
Standard Dev. 0,99% 1,35% 1,87% 2,74% 3,51%
3.3 Value‐at‐Risk
The first step in determining the VaR is to calculate the returns and volatilities for each portfolio over the past 23 year. The annual portfolio returns are found in table 3.3, whereas the annual standard deviations can be found in Appendix C. The three other factors that are necessary to calculate VaR are; the confidence level, the initial value of the portfolio and the time horizon. These factors have already been determined in section 2.5.2 as following; a confidence level of 5 percent, an initial portfolio value of €100.000 and a time horizon of 1. Now that all variables are known, the portfolio distribution for each risk profile can be established. After which, the percentile calculation at the confidence level is used to determine VaR. Using an initial investment value of €100.000, the portfolio distribution of a certain year per risk profile can be calculated using the following formulas (Benninga, 2006):
[
LN
(
V
)
(
/
2
)
T
]
2 0 lnμ
σ
μ
=
+
−
[5] And[
σ
ln=
σ
×
T
]
[6] Where; μln = The logarithmic portfolio mean LN(V0)= Logarithmic portfolio value μ = annual return σ = annual standard deviation T = time period Since, the time horizon is 1, this has no impact on the results and therefore can be left out of the equations. Following these formulas the logarithmic portfolio mean and standard deviation can be calculated. These values are parameters of a normal distribution of Ln(V0). For example, the logarithmic mean of a very defensive portfolio in 1986 is:656
.
11
1
)
2
/
016
.
0
143
.
0
(
)
000
.
100
(
+
−
2×
=
LN
which the value falls below the 95 percent certainty demand, which is known as the quantile of the distribution (Benninga, 2006). Excel has a built‐in function to find this quantile, using the inverse function of the lognormal distribution, log.inv. This function finds the cutoff point at the chosen 5 percent confidence level of the distribution with a mean of x and a standard deviation of y. Again the variables of a very defensive portfolio of 1986 are used to explain the function; log.inv.(0.05; 11.656; 0.016) which results in a cutoff point of €112.363,40. Subtracting this value from the initial investment and VaR is found. In this calculation, subtracting the cutoff point from the initial investment, results in a negative value; €100.000 ‐ € 112.363,40 = € ‐12.363,40. A negative VaR implies that the portfolio has a high probability of making a profit. In this case, it has a 95 percent probability that at least €112.363,40 is achieved, which is a profit of €12.363,40. Thus a negative VaR actually says there is ´no risk´ because there is a 95 percent chance a profit is achieved, but what could happen in the remaining 5 percent is uncertain. Applying the log.inv. function for all years, results in the VaR outcomes shown in table 3.4.
Var at a 5% level Very Defensive Defensive Neutral Offensive Very Offensive
1986 -€ 12.363,40 -€ 15.820,15 -€ 18.905,36 -€ 22.284,80 -€ 24.458,05 1987 -€ 6.722,67 -€ 3.496,38 -€ 868,24 € 4.607,50 € 9.103,03 1988 -€ 8.093,14 -€ 10.672,31 -€ 12.503,36 -€ 15.184,19 -€ 17.120,18 1989 -€ 6.552,74 -€ 9.465,24 -€ 10.973,53 -€ 14.597,11 -€ 17.615,69 1990 -€ 4.096,49 € 4.132,21 € 9.930,31 € 19.019,18 € 25.702,85 1991 -€ 8.382,78 -€ 8.901,03 -€ 9.236,94 -€ 8.671,14 -€ 7.856,33 1992 -€ 4.115,63 -€ 3.244,55 -€ 2.508,43 -€ 607,22 € 1.285,94 1993 -€ 7.501,84 -€ 8.539,30 -€ 9.775,79 -€ 10.240,23 -€ 10.235,05 1994 -€ 1.392,49 -€ 792,31 € 32,61 € 751,85 € 1.246,57 1995 -€ 11.599,62 -€ 12.822,97 -€ 14.236,14 -€ 13.965,03 -€ 12.950,41 1996 -€ 6.528,84 -€ 9.105,58 -€ 10.866,18 -€ 13.549,12 -€ 15.607,28 1997 -€ 2.623,81 -€ 4.606,29 -€ 5.397,75 -€ 7.694,77 -€ 9.671,35 1998 -€ 9.663,38 -€ 10.274,41 -€ 10.864,13 -€ 9.654,79 -€ 8.176,34 1999 -€ 705,07 -€ 4.338,84 -€ 6.134,09 -€ 11.032,08 -€ 15.348,63 2000 € 424,87 € 2.246,75 € 4.408,54 € 6.318,32 € 7.570,00 2001 -€ 660,91 € 4.721,09 € 8.987,02 € 15.759,83 € 20.946,44 2002 -€ 2.414,66 € 3.318,14 € 7.360,12 € 15.430,21 € 21.783,82 2003 -€ 7.104,34 -€ 8.381,32 -€ 9.648,43 -€ 8.395,91 -€ 6.339,04 2004 -€ 5.848,63 -€ 6.938,68 -€ 8.280,80 -€ 8.689,51 -€ 8.422,39 2005 € 1.833,39 -€ 1.866,17 -€ 3.547,79 -€ 8.259,29 -€ 12.328,32 2006 -€ 7.178,22 -€ 8.249,75 -€ 9.174,73 -€ 9.600,40 -€ 9.571,77 2007 -€ 6.125,73 -€ 6.612,22 -€ 6.589,59 -€ 5.269,29 -€ 3.819,22 2008 € 4.348,41 € 16.339,22 € 24.628,91 € 37.078,34 € 45.676,31 Average VaR -€ 4.915,99 -€ 4.494,35 -€ 4.094,08 -€ 2.988,25 -€ 2.008,92
Average VaR without 2008 -€ 5.337,10 -€ 5.441,33 -€ 5.399,67 -€ 4.809,45 -€ 4.176,43
Table 3.4 Annual VaR calculation per risk profile
same year would lose around €46.000 with the same probability and initial investment. Thus, the higher the VaR, the higher the risk of that particular portfolio. These portfolios should be adjusted to be less risky, whereas the portfolios with negative VaRs should be adjusted to be more risky to obtain a better risk‐return tradeoff. When comparing the VaRs of the risk profiles, it clearly shows that all VaRs of 2008 are much higher than the other VaRs within each risk profile, which is the result of the credit crisis of 2008. The impact of the crisis is also reflected in the average VaR with and without 2008 over the past 23 years. Especially, the average VaRs of the more risky profiles are largely affected by the crisis. Still, all average VaRs are negative, which is logical because if investment portfolios would have a higher probability of loss than of profit, nobody would invest anymore but rather place their money on a savings account.
To see how the calculated VaRs correspond to the return ranges given in Table 2.1, the maximum VaR of each risk profile over the past 23 years is compared with the maximum negative returns of table 2.1. Furthermore, both values will be compared with the maximum VaR without 2008, to show the difference between the maximum negative returns if the crisis had not occurred. To make the comparison easier, the maximum VaRs are transformed in percentages by dividing the maximum VaRs by the initial investment. This results in the maximum percentages that can be lost of the initial investment per risk profile. These values are showed in the following table.
Maximum negative return Very Defensive Defensive Neutral Offensive Very Offensive
Given in tabel 2.1 -3,00% -10,00% -15,00% -25,00% -45,00%
Of the past 23 years -4,35% -16,34% -24,63% -37,08% -45,68%
Without 2008 -1,83% -4,72% -9,93% -19,02% -25,70%
3.4 Expected Shortfall
As is said in section 2.5.3, Expected Shortfall is calculated using VaR calculations at different confidence levels. The confidence levels are chosen based on the VaR at a 5 percent level being 100 percent. To calculate the left tail of the ES distribution, I will take the following percentages of the 5 percent VaR; 1, 5, 10 and 25. This can be calculated by transforming these percentages in VaR confidence levels. For example, an ES of 5 percent is the same as calculating VaR at a 0.25 percent confidence level, which is 5 percent of the 5 percent VaR level. Transforming all ES percentages results into the following confidence levels; 0.05, 0.25, 0.5 and 1.25. By simple adjusting the confidence level in the function log.inv in excel, a different cut‐off point is calculated, which then is subtracted from the initial investment to obtain VaR at a different confidence level. See appendix E for an overview of the ES at all the confidence levels. To obtain ES per year and risk profile, I took the average of the four different VaR calculations of a certain year and risk profile. For example, the ES of a neutral investor in 2008 is the average of the following VaRs; €28.137,75; €27.124,16; €26.634,40 and €25.920,22, which is €26.954,14. This value, and all the other ES values, can be found in the table given below.
The Expected Shortfall values presented above show the possible values that can be achieved when the VaR level is exceeded. A negative ES value has the same definition as a negative VaR, namely that is has a high probability of profit, whereas a positive ES gives the possible amount that is lost when the VaR level is exceeded. In Appendix F an overview of the difference in values between the ES and the VaR is given. For example, where for a defensive investor the VaR is calculated at €16.000 in 2008, Expected Shortfall states that when this loss is exceeded, the amount of loss will reach around €18.000 of every €100.000 invested. The differences between the VaR and the Expected Shortfall values do not seem very large. This can be explained, since the assets used for the portfolio allocation in this thesis, are all very well diversified assets with decreased risk exposure. When a portfolio contains, for example, 12 individual stocks instead of an index fund, the risk exposure and the difference between the VaR and ES values are likely to increase. Another example is, when instead of the reasonably safe government bond, high yield/high risk bonds are used for the portfolio allocation. This will create more risk exposure and increases the volatility of the portfolio, which causes the tails of a return distribution to become more fat tailed. Looking at the kurtosis of the assets used, it is clear that the riskier an asset becomes, the higher the kurtosis. Therefore, an increase in the kurtosis will increase the differences between the VaR and ES values.
3.5 VaR and ES based on Monte Carlo simulation
So far the historical approach has been used to calculate VaR and ES over the past 23 years, now the Monte Carlo simulation is applied to simulate future portfolio returns of which VaR and ES is calculated. Monte Carlo simulation is similar to the historical approach, except that the distribution is selected instead of based on the historical data (Gregory & Reeves, 2008). I can perform the simulation based on the return distribution of the three assets or the return distribution per risk profile. I have chosen to simulate the returns of the three assets, because then the influence of different asset allocations can also be examined. For each asset a few variables are selected to perform a Monte Carlo simulation. The simulation is performed using the following equation (Benninga, 2006);
This equation describes a lognormal price process, where the future stock price can be calculated using the lognormal mean and standard deviation, the stock price at t=0 and Z, which is a standard normal variable.
The first two necessary variables that are selected for the simulation are a return and volatility per asset. Assuming that the past gives information about how the returns of the assets will behave in the future, the selection of returns and volatilities are based on the historical information. Since the interest in this thesis lies in the performance of risk measures in unlikely financial situations, the largest standard deviation of each asset is selected. A large standard deviation means large profits but also large losses, the latter is of interest for this research and will be used for the VaR and ES calculations. In appendix G all the standard deviations are given, whereby the largest standard deviations of each asset are highlighted.
Having determined all the necessary variables, a simulation can be performed. For example, a possible simulated MSCI World index stock price with the parameters;
%
41
.
6
=
μ
,σ
=25.46%, t=1/12 and Z =−0.2582 can be calculated as follows;[
0.0641*0.08333 0.2546*( 0.2582)* 0.0833]
1958.95 exp 85 , 1985 × + − = = Δt SThis calculation is executed manually, to perform 1000 of such calculations is time consuming and difficult. Therefore, a macro is applied, which is very useful when a calculation has to be performed repeatedly. This macro performs exactly the same calculations, but then in a few seconds, depending on the number of runs. In appendix H, the program that runs the macro is described. Using this macro, a thousand trails per asset of one year are performed. This means 1000 yearly asset price movements are simulated, resulting in 12.000 monthly asset prices. For all asset prices, the lognormal annual return is calculated using equation [1]. After which the dataset is ranked from lowest to highest return per asset. The ten worst annual returns of each asset are put together into the five portfolios. I do not take into account the correlations between the asset, and in order to create a worst case scenario I assume that when it is a bad year for stocks it also is a bad year for the other assets. This means that the worst annual return of the stock is put together with the worst annual return of the bond and the liquid asset and the second worst return of the stock with the second worst return of the bond and the liquid asset, and so on. The creation of the portfolios will be according to the equations [2] and [4]. An overview of the asset returns and the portfolio returns and standard deviations of the ten scenarios can be found in Appendix I. Next, VaR and ES can be calculated as is described in section 3.3 and 3.4, the results will be used to test the following hypotheses; H0: The maximum VaR does not exceed the given negative return for a certain investor type given a 5 percent confidence level H1: The maximum VaR does exceed the given negative return for a certain investor type given a 5 percent confidence level
H0: The maximum ES does not exceed the given negative return for a certain investor type
3.6 Test the impact of different VaR time horizons and confidence levels
The time horizon of one year used for the VaR calculations is very unusual. Most VaR calculations are based on day, week or monthly time horizons. The larger the time horizon, the less accurate the VaR calculations become. Therefore, I will test if a larger time horizon causes VaR underestimate the portfolio risk. I will compare one VaR calculation based on the whole 23 years with a VaR based on 23 separate annual VaR calculation for each risk profile. The annual VaRs already have been averaged into one VaR over the 23 years, as is seen in Table 3.4.
For the VaR based on the whole 23 years, the mean annual return and standard deviation over the past 23 years of each risk profile are necessary for the formulas [5] and [6], which can be found in table 3.3. After which, for both calculations the Excel function log.inv is used, applying the same confidence level as the annual VaR calculation 5 percent results in the cutoff points for each risk profile. Subtracting the cutoff points from the initial investment of €100.000 results in the VaR calculations. Furthermore, the impact of a different confidence level on both risk measures will be tested by comparing the results of using a 5 percent with a 1 percent confidence level. Both VaR and ES will be recalculated for the ten worst case scenarios with a confidence level of 1 percent. ES will again be determined by taking the 1, 5, 10 and 25 percent of the 1 percent VaR. This means averaging the VaRs at a 0.01; 0.05; 0.1 and 0.25 percent confidence level. Since, the portfolio parameters,
μ
andσ
, are already known, only new cutoff points have to be established to determine the VaR and ES values.3.7 Sensitivity test
To research if the time horizon of an investor has an impact on the VaR and ES values, I will replace the strategic weights with the tactical weights, seen in Table 3.1. Hereby the portfolios are allocated based on a 5 till 10 year time horizon instead of the 10 years and longer time horizon, which results in the portfolios being allocated less risky. This test is an example of a sensitivity test, which is used to research the variability of returns in response to the change of a key variable, in this case the asset mix10. First, I will show if an investor
could have achieved a better portfolio performance of the past 23 years according to the
