Chapter 2 (1).pdf

Chapter 2 Market Risk* Part I Accompanying presentation: 2. Risk Management — Market Risk (Part I) 2.1. Trading in Financial Markets 2.1.1. Markets Before we can understand the meaning of “market risk”, we must first define the term “market”. A market is a medium which facilitates transac- tions between buyers and sellers. Markets differ from one another in vari- ous aspects, including but not limited to the following: · Location of trade: This might be a physical location (e.g., Venice’s fish market), where buyers meet vendors face to face, or it might be a virtual setting (e.g., the Tokyo Stock Exchange — TSE), where buyers and sellers may be physically situated in different countries, linked by computer programs which allow them to conduct trade without ever meeting in person. · Type of assets and/or services traded: For example, flowers at the Amsterdam Flower Market or options at the Chicago Board Options Exchange. * In this chapter, we will describe some of the advanced methods used to measure loss. Part I and Part II share the same numbering sequence. 37 38 Lecture Notes in Risk Management · Trading methods: Ranging from barter (e.g., swapping tomatoes for eggs) to electronic currency trading. · Target population: Working-class individuals, professional brokers, etc. · Extent of trade: Measured in kilograms of apples per week, hundreds of millions of U.S. Dollars per day, etc. Trade in Financial Markets Stock exchange trade Computerized trade Standardized contracts (derivatives) Over-the-counter (OTC) trade Traders at financial firms, large companies, fund managers Contracts may be non-standard We shall here focus on financial markets, which may be further subdi- vided into two distinct categories: · Stock exchange The stock exchange1 is a well-organized and highly regulated market, in which a particular type of asset is traded (stocks, bonds, currency, indices, futures, options, etc.). These assets are traded according to clearly defined standards which dictate the minimal amounts which may be traded and how precise the prices should be, fix the term to maturity, determine trad- ing priorities, specify the rights and obligations of both buyers and sellers, 1Some interesting facts… Another name for the stock market is “bourse”. Although somewhat less commonly used in English, this word, or variants of it, is the most popular term for the stock exchange in many languages. The term acquired the modern sense of “stock exchange” from the Huis ter Beurze, an inn established in the city of Bruges (in present-day Belgium) in the 13th century. The inn took its name from its owners, the van der Beurze family. Traders from across Europe would meet at the inn to conduct business and receive financial counsel from the owners. This sys- tem was formalized in the 15th century. The idea spread rapidly, and similar institutions were established across Europe, adopting not only the concept but also the name of the original inn. Market Risk: Part I 39 provide collateral, etc. The trade conducted in modern stock exchanges is almost entirely computerized, allowing for countless transactions to be performed simultaneously. Usually, traders who operate in the stock market enjoy greater liquid- ity. In order to purchase a futures contract on gold, for instance, the buyer need only launch the appropriate trading program, select the desired con- tract out of a list and complete the purchase. The deal is typically closed within a fraction of a second with the contract registered in the buyer’s name. The process of selling the contract may be performed just as quickly. On the other hand, the apparent advantage of standardized stock exchange trading can be a double-edged sword in some circumstances. Let us take as an example the case of a Jordanian exporter who wishes to hedge a deal with an Icelandic importer, which would require an immedi- ate payment in the local currency (Jordanian dinars, JOD) and a future receipt in a foreign currency (Icelandic króna, ISK), over a period of 37 days. The exporter will find it difficult to find an appropriate contract at the stock exchange, where such an asset is not tradable. The disadvan- tage of the stock exchange lies in the type of assets traded and in the dura- tion of contracts, which are predefined and cannot be modified to suit every possible demand. The solution to this issue may be found in over- the-counter trading. · Over-the-counter trading Over-the-counter (OTC) trading is conducted directly between the two parties, without the stock market as an intermediary. OTC trading has a number of advantages: it allows greater flexibility in the type of asset, duration, quantity, etc. Thus, using this method, the Jordanian exporter would be able to hedge the aforementioned deal by purchasing a JOD/ISK futures contract for a period of 37 days. OTC trading, however, suffers from a lack of liquidity compared to stock market trading — should the deal be canceled, the Jordanian busi- nessman would be unlikely to find buyers for his futures contract. Furthermore, OTC trade is only partially regulated — although various regulatory measures have been put in place and those who violate them face severe penalties, this system is not nearly as extensive as that which has been established in the stock market. OTC trading is not always con- ducted at market prices — rather, the price is agreed upon during the 40 Lecture Notes in Risk Management negotiations between the two parties. Further disadvantages exist, mostly due to the lack of transparency in OTC trading. 2.1.2. Long and short Traders make use of two basic concepts, “long” and “short”, which are essential to proper management of market risks: Long Profit Examples of long trades: Purchase of 100 IBM shares at the stock exchange Price OTC exchange of 1M GBP for USD OTC purchase of 1,000 ounces of gold Sale of 1M General Motors bonds at the stock exchange Such deals are often referred to as spot trades as the asset is immediately transferred from one party to the other. · Long: The classic type of deal, in which the trader buys the asset and intends to profit from an increase in the value of the asset. For example, a long purchase of Apple shares will return a profit if the share price rises, and a loss if the share price falls.  In a long transaction, the buyer may make a very large profit (potentially infinite, at least in theory) if the price of the shares rises. However, the maximum loss cannot exceed the value of the shares, even if the share price drops all the way to zero.  Having purchased the shares, the buyer now owns the asset. Market Risk: Part I 41 Short Profit In a short sale, the stocks are not owned by the trader. Price The financial intermediary borrows the stocks from another trader and proceeds to sell them on the market like any other stocks. Such deals require collateral. If the asset returns benefits and/or dividends, the trader must pay these to the shareholder. The cash flow may include a fee for lending the share. · Short: A more complex form of transaction in which the trader sells the asset before buying it and intends to make a profit from a decrease in the price of an asset (e.g., if a company’s share price drops to zero). For example, a short sale of Apple shares will return a profit if the share price falls and a loss if it rises.  In a short transaction, the trader can make a profit equal to the value of the asset if the asset value drops to zero. However, the loss potential is infinite if the asset price continues to rise (theoretically, all the way to infinity).  In a short sale, the stock being traded is not owned by the trader. It is equivalent to a loan whose value is linked to the share price. Therefore, in such a deal, the trader does not become the asset owner.  A different trader lends the asset to a financial intermediary, who then proceeds to sell it in the free market.  A short sale requires the employment of collateral by the trader to ensure that the loan will be repaid when it expires.  If the asset returns benefits and/or dividends, the trader must pay these to the shareholder.  The cost of a short deal is higher, as the trader must pay a fee to the intermediary from whom the asset was lent. 42 Lecture Notes in Risk Management Short Sale Despite the financial complexity of such transactions, performing a short transaction using a trading platform is quite a simple action. Having selected the type of asset we wish to sell and set the amount, method of execution and other parameters, the only thing which remains is to press the “sell short” button and wait for the system to carry out the transaction. Long/Short Long Short Profit Closed Buy posion Price Profit Closed Sell posion Price Once a long transaction has been performed, it is said that the trader has “opened” a long position or “extended” an existing long position, and Market Risk: Part I 43 from this point forth is taking a long position which exposes him or her to asset price depreciation (the top left in the diagram). Meanwhile, the trader who has “closed” a long position or “reduced” an existing one and therefore is not exposed or at the very least less exposed to asset price depreciation. The chart in the presentation repre- sents a trader who has closed a long position. Once a short transaction has been performed, it is said that the trader has “opened” a short position or “extended” an existing short position, and from this point onwards is taking a short position which exposes him or her to asset price appreciation (the bottom right in the diagram). The trader, by contrast, has “closed” a short position or “reduced” an existing one and therefore is not exposed or at the very least less exposed to asset price appreciation. The diagram represents a trader who has closed a short position. Position is essential to risk management — whether the loss is expected to originate from a decrease or an increase in asset prices will, to a consider- able extent, determine the risk reduction measures which should be adopted. This last sentence may seem obvious, but there have been many instances of companies committing basic errors with regard to the direction of trade: long and short. Instead of hedging the risk of an asset portfolio according to an open position, the company acquired the “wrong” form of protection. The supposed protection, in fact, covered the opposite direction, serving only to increase the exposure to market risks instead of reducing them. 2.1.3. Financial derivatives Financial Derivatives Forwards Futures Swaps Options Exotic options Financial derivatives are a type of security whose price is derived from, or depends on, the price of another asset. Derivatives are often used to 44 Lecture Notes in Risk Management manage market risk. Here, we shall list some of the most widespread types of financial derivatives: · Forwards Forward A contract between two parties which sets the amount and price for the future purchase of an asset, according to terms and conditions agreed upon in the present. Traded OTC. The contract is binding for both sides. However, there is always a certain risk that one of the sides will not honor its commitment. A forward is a contract between two parties, which sets the amount and price for the future purchase or sale of an asset in accordance with the terms of the contract. For example, an Italian energy company predicts a rise in the local demand for oil in the coming months and would like to guarantee the price of the oil it intends to import from Saudi Arabia in two, three and four months. Thus, the company signs a contract with the Saudi oil supplier, which fixes the price per barrel and the number of barrels for the relevant period. Forward contracts are traded over the counter. The contract is binding and compels both sides to fulfill its terms. However, there is always a certain risk that one of the parties will be unable or unwilling to comply with the terms of the agreement. Forward Short Forward Profit Long Forward Profit Profit Profit Price at maturity Price at maturity Market Risk: Part I 45 In a long forward contract, profits increase as the price rises. The Italian energy company, which has purchased a long forward, will make a profit if oil prices go up, since it is still able to purchase oil at the lower price which was set two months before. By contrast, in a short forward deal, profits go down as the price goes up. Thus, for example, the Saudi supplier, who has sold a short forward contract, will lose money if oil prices go up, i.e., the oil could have been sold for a higher price than that agreed upon two months in advance. In forward contracts, reckoning is performed at the end of the contrac- tual period. At its conclusion, rather than paying the entire price, only the difference between the price named in the contract and the actual price is transferred from one party to the other. · Futures Futures Short Future Profit Long Future Profit Profit Profit Price at maturity Price at maturity Future Forward Futures Locaon of trade OTC Stock exchange Financial accounng At maturity Daily Collateral Depends on contract Required A futures contract is a type of forward contract which is traded in the stock market. Futures have standardized features, e.g., asset type, duration and future value. For instance, a chocolate manufacturer, having signed a deal to provide fifty thousand chocolate bars every month for the next year, will have to purchase a precise amount of cocoa every quarter — not so little as to render the manufacturer incapable of meeting the terms of the deal but also not so much that part of the cocoa will be thrown away. To ensure that the deal remains profitable year-round, the chocolate 46 Lecture Notes in Risk Management manufacturer can purchase a futures contract today for each quarter of the year in order to guarantee the price of cocoa for the entire year. Forward and futures contracts differ from each other in the following aspects:  Location of trade: Forward contracts are traded over the counter, whereas futures contracts are traded in the stock market.  Financial accounting: In forward contracts, financial accounting is only performed at the end of the specified period, whereas in futures contracts, financial accounting is conducted on a daily basis, i.e., at the end of each day, the difference between the stated price in the contract and the market price on that day is transferred from one party to the other, according to the position taken by each party.  Collateral: Forward contracts lack a mechanism to guarantee that both sides will fulfill their obligations. It is possible to acquire such a mechanism, but it is not built into the contract by default. For futures contracts, however, the stock exchange guarantees that the contract will be honored by both sides. This is done by continuously tracking market prices as well as the liquid balance of each party. The stock exchange ensures that, at any given moment, the side which is in debt has a sufficient cash balance to honor the agreement (the process of assessing the financial situation of each party is termed “mark to market”). · Swaps Swap A deal in which two sides swap financial assets Example: The ICBC signs a three-year agreement, according to which it will receive a variable interest from JPMorgan Chase, set to match LIBOR* rates. In parallel, the ICBC pays JPMorgan Chase a fixed interest of 5% per year every six months. The extent of this deal is $1M U.S. LIBOR 5% Note: *London Interbank Offered Rate. Market Risk: Part I 47 A swap is a contract in which two parties swap financial assets. For example, the Industrial and Commercial Bank of China (ICBC) signs a three-year agreement with JPMorgan Chase, which stipulates that the lat- ter will pay a variable interest rate to the former, with the interest rate determined according to LIBOR.2 At the same time, ICBC will pay JPMorgan Chase every six months at a fixed annual interest rate of 5%. The scope of the deal is USD 100 million. Swap Potenal cash flow ---------Millions of Dollars--------- according to fluctuaons in LIBOR interest rates. LIBOR FLOATING FIXED Net Date Rate Cash Flow Cash Flow Cash Flow Mar. 5, 2016 4.2% Sept. 5, 2016 4.8% +2.10 –2.50 –0.40 Mar. 5, 2017 5.3% +2.40 –2.50 –0.10 Sept. 5, 2017 5.5% +2.65 –2.50 +0.15 Mar. 5, 2018 5.6% +2.75 –2.50 +0.25 Sept. 5, 2018 5.9% +2.80 –2.50 +0.30 Mar. 5, 2019 6.4% +2.95 –2.50 +0.45 Source: Hull (2014). The fluctuations in LIBOR interest rates are largely responsible for deter- mining the profitability of the swap contract from the perspective of ICBC. For instance, for the first six months, the LIBOR interest rate was 4.2% per year (2.1% for six months) while the fixed interest rate was 5% per year (2.5% for six months). The bank accordingly lost 0.4% during this six-month period. Subsequently, the LIBOR interest rate increased up to 6.4 during the second half of the year, resulting in a profit of 0.45% for ICBC during the whole period. 2The London Interbank Offered Rate (LIBOR) and London Interbank Bid Rate (LIBID) are used in interbank trading in London. Their use has expanded significantly, and they currently serve as a benchmark for the interest rates of several of the largest financial enti- ties in the world. 48 Lecture Notes in Risk Management A swap deal can make a lot of sense from a risk management perspec- tive. An analysis of the example above from a business viewpoint uncov- ers that, before performing the swap deal, ICBC signed a second deal with one of its clients for a three-year period, during which the bank has prom- ised to pay a variable interest rate according to LIBOR. Meanwhile, JPMorgan Chase has signed another deal in which it pledges to pay a real estate company a fixed annual interest rate of 5% for a period of three years. In signing these deals, each of the banks created exposure which was subsequently completely eliminated through the swap deal between the two banks, as described earlier. · Options Option A contract granting the buyer the right to choose to conduct a future transaction. The transaction is defined in advance according to various parameters, typically quantity, price and timing. By contrast, the seller must conduct the transaction according to the predetermined terms. An option is a contract which grants the buyer the right to perform a transaction at some point in the future. The seller, on the other hand, is obligated to perform the trade according to the predetermined terms. When the transaction is defined in advance according to standard parameters — underlying, quantity, price and duration — the option is termed a vanilla option. For example, an energy company would like to hedge its market risk and ensure that the price of an oil barrel does not exceed USD 50 for the next three months. For this purpose, the company seeks to purchase an Market Risk: Part I 49 option from the oil supplier, predefining the deal as a purchase of 1,000 barrels of oil at USD 50 per barrel for the next three months. Exercising this option will be worthwhile if, during these three months, oil prices exceed USD 50 per barrel. The oil supplier will be obligated to perform the deal according to the terms defined in the con- tract. Conversely, if the price drops below USD 50, the company will choose not to exercise the option, as buying the oil at the market price would be a cheaper and more profitable alternative. Regardless of the circumstances, the company will not have to pay more than USD 50 per barrel. Why would the oil supplier agree to such a deal at all? After all, the market risk is not eliminated but rather transferred in its entirety to the oil supplier. The motive for accepting the deal would be the risk premium charged by the seller. The oil supplier does not offer the option free of charge but demands a fee in return. This fee is called a risk premium (or just a premium for short). The risk premium is intended to serve as recompense to the risk- bearing party (the oil supplier in the example above) for the risks involved in the deal. Such risks depend on market conditions and on various parameters defined in the contract. Thus, in the example above, the primary risk would be fluctuations in oil prices. The greater the fluc- tuations predicted for the next three months, the higher the risk premium. Options Long call option profit Long put option profit Profit Profit Price Price 50 Lecture Notes in Risk Management We distinguish two types of options — call options and put options:  A Call option allows the buyer to perform a purchase in the future.  A Put option allows the buyer to perform a sale in the future. In the example earlier, the energy company has acquired a call option to purchase oil barrels, and the oil supplier has sold (or “written”, to use the professional term) a call option. A vine grower wishes to ensure that the price of grapes during the harvest, in four months’ time, will remain at least GBP 6 per kilogram. The vine grower purchases a put option from the winemaker. If the market price falls below GBP 6, the vine grower will still be able to sell his grapes at GBP 6 per kilogram, and if the price exceeds GBP 6, he will be free to choose not to exercise the option and instead sell his grapes at the higher market price. Options Long call option profit Long put option profit Profit Profit Price Price American opon European opon May be exercised at any point May only be exercised on its during its life me expira on date There are two commonly traded types of options, which differ in the man- ner in which they are exercised:  An American option allows the buyer to perform the transaction at any point during the option’s lifetime.  A European option only allows the buyer to perform the transaction at the end of the option’s lifetime. Market Risk: Part I 51 Options on stocks are usually of the American type, whereas options on indices are usually European. Options Source: Yahoo Finance. In the above table, we can see the quoted price of tradable call options on Apple shares for January 27, 2017, to be exercised on February 3, 2017. The options differ only in the price at which they are exercised (or “strike price”), which is shown in the leftmost column. The options marked in first six rows are “in the money” (ITM) whereas the unmarked options are “out the money” (OTM). The price of an Apple share, as can be seen in the table, is USD 122.02. Supposing we would like to purchase the share on 3 February, 2017, and wish to guarantee its purchase at a price no higher than USD 123, we may buy an option for USD 1.61 (see “ask” column). On February 3, 2017, we would be faced with two possibilities. In the first, the price of an Apple share remains below USD 123, and in the sec- ond, the price of a share has exceeded USD 123. In the first scenario, we will purchase the share directly, losing the option premium (USD 1.61). The second scenario may be further split into two possibilities. If the price of the share is below USD 124.61 (123 + 1.61), we will minimize our loss by purchasing the share at USD 123 while losing the premium. However, if the price is above USD 124.61, we will gain the difference between the market price and the option. 52 Lecture Notes in Risk Management It is evident, therefore, that in the case of a call option, the higher the strike price, the lower the price of the option (or “last price”). · Exotic Options Exotic Options Asian options Barrier options Basket options Binary options Compound options Lookback options An exotic option is a non-vanilla option. It may differ from a vanilla option in the type of underlying, exercise rules, how returns are calcu- lated, etc. Here are a few examples of exotic options:  Asian options yield profits according to the average price of the underlying for the entire lifetime of the option. Asian options were first issued in 1987 by Japanese banks, on futures contracts for crude oil. The main features of this type of option are as follows: – They are less sensitive to market fluctuations and their standard deviation is smaller. They therefore tend to be cheaper than equivalent vanilla options. – The standard deviation of an Asian option decreases over time, as part of the average is already known. – Asian options are less vulnerable to manipulations to the date of exercise (e.g., inside information), thus giving the option writer a significant advantage. – Asian options are well suited to the needs of businesses which trade continuously in a certain type of underlying and wish to hedge the risk involved in this type of trade (for instance, an airline which continuously purchases fuel over the course of a year may hedge itself from fluctuations in jet fuel prices by using an Asian option on fuel prices). Market Risk: Part I 53  Barrier options have barriers for entry or exit: – Knock in: An exotic option with a single barrier, which activates if the price of the underlying hits the barrier price during the option’s lifetime. – Knock out: An exotic option with a single barrier, which becomes invalid if the price of the underlying reaches the barrier price during the option’s lifetime. – Double barrier: An option with two barriers separated by an interval. These barriers may be either both knock ins or both knock outs, or there may be one of each (such an option is termed KIKO — Knock In/Knock Out).  Basket options are options on an average of a basket of assets. This type of option is suitable for hedging the market risks associated with indices.  Binary options3 are True/False options. If a certain event occurs, the buyer of the option will receive a payment. If it does not, the buyer 3Some interesting facts… · Binary options have ushered a golden age for online gambling companies, which offer their customers big, quick and, above all, legal profits. · At the same time, illicit gambling, of the sort that tends to cause significant financial losses on most customers, has also seen significant growth. For instance, gambling companies may make it exceedingly difficult for customers to collect their earnings. Another example is the price quote of the underlying asset, the timing of which is often controlled by the gambling company in such a way that being off by just a few milli- seconds could mean the difference between profit and loss for the client. · As a result, over the years, binary options trading has largely become a synonym for gambling, fraud, deceit and deception — both regulators and the legal system have come to view binary options in a negative perception and consider them undesirable. · Let us, therefore, take the opportunity to clarify a number of things regarding binary options, in an attempt to remedy their bad reputation:  Gambling is an act which combines luck with considerations of risk and return. “Luck” here refers to randomly occurring events, whose occurrence cannot be con- trolled and certainly not predicted. This fact is just as true of ordinary stock market trading as it is of filling out a lottery form, both of which are either difficult (stock market) or impossible (lottery) to predict, and yet perfectly legal in many countries.  It turns out that certain companies are indeed capable of predicting the direction in which stock and currency indices are headed, making use of physical and 54 Lecture Notes in Risk Management must pay a certain amount. For example, if in three hours the USD/ EUR exchange rate increases above 1.1, the buyer will receive USD 70, and if it does not the buyer will have to pay USD 90. – Compound options are options which activate other options. The first option may be a binary option, for instance, a type of option well suited for tenders. Let us suppose a company is participating in a tender for building infrastructure and must allocate extensive collateral which it needs to borrow from the bank. If the company wins, it will have no problem to repay the loan. If it does not win, however, it may have to pay back the loan with substantial interest. In order to hedge the risk, the company may choose to purchase a compound option for a cheap price. This compound option will, in turn, activate the option to take a loan only if the company makes the winning bid. – Lookback options are options which are exercised on their expiration date, according to the best price recorded during the lifetime of the option (typically the maximum or minimum price during this period). Despite their exotic name and their implied speculative connotations, exotic options are an effective tool for reducing many forms of market mathematical algorithms (these methods are only partially effective, but they do result in predictions at an accuracy of greater than 50%). This means that it is in fact possible to make a profit from binary options trading, i.e., not all depends on luck.  In between two investment extremes — an investment in AAA-rated bonds on the one hand, where the probability of receiving zero interest is extremely high, and an investment in the lottery on the other, where the probability of winning large sums of money is almost zero — binary options are situated somewhere in the middle, with a probability greater than 50% of making a profit. From a risk/return perspec- tive, binary options are therefore just another legitimate form of investment.  Furthermore, in 2016, the New York Stock Exchange (NYSE) launched a form of binary options called Binary Return Derivatives (ByRDs), in order to offer traders a wider variety of investments.  Those many companies commit fraud, refusing to return interest payments on bonds or tampering with legitimate lottery gambling, which is an issue that the authorities must combat with resolve, regardless of the financial instrument these companies use to commit their crimes. Market Risk: Part I 55 risk which are a part of everyday business activity and in fact anything but speculative (see the infrastructure tender and crude oil examples above). 2.2. Market Risk 2.2.1. Definition Market risk “Market risk” is the possibility of incurring losses in profits or capital as a result of unforeseen changes in market prices. Before we explore deeper into the precise definition of “market risk”, let us first explore the manner in which this risk is defined. The first part of the definition refers to the “possibility of incurring losses” and appears in the definition of any type of risk. In the realm of financial risks, this loss materializes in the form of damage to future prof- its or present capital. However, the loss can certainly vary in nature. It can be a breakup with your great love, the loss of a dear friend, harm to one’s physical well-being, etc. This definition also applies to negative projected profits, i.e., losses. In such a case, the risk is of incurring even greater losses than the already projected loss. Similarly, the definition also applies to negative capital, i.e., an increase in current debt. The second part of the definition, “due to…”, refers to the causes for the potential loss. Such causes are not hard to find. A crisis of trust may lead to a breakup from a great love, moving to another town may result in the loss of a friend and carelessly crossing a road could quite possibly damage one’s health. In the world of financial risks, the cause is typically an unforeseen event which could have an adverse effect on profits or capital. 56 Lecture Notes in Risk Management If the event is predictable, this is then an example of certainty. In this case, assuming proper risk management measures are in place, the risk has already been mitigated. The third and final part of the definition, “in…”, describes the risk factor. Some examples of risk factors are inflation, interest rates, exchange rates, earthquakes and fire. 2.2.2. Risk factors The risk factor “market prices”, which appears in the definition, is a gen- eral term for various prices. The following image gives some examples of such prices: Market Risk Factors Price of goods Price of services Share prices Bond prices Option prices Exchange rates Stock indices For instance, if we were to hold a long position on 100 Coca-Cola shares worth USD 150 each, or USD 15,000 overall, the most significant risk factor on our total Coca-Cola holdings would be the market price of the share. Should the market price per share drop by USD 1, we would lose USD 100 overall. It is important to note that a decline in share prices does not necessar- ily correspond to a loss. Were we to hold a short position on Coca-Cola stock, a fall in share prices would, in fact, represent a profit. For both long and short holdings, the share price is a market risk factor. The two cases differ in the direction the price might go. For long holdings, a decrease in share prices would pose a risk to the value of this investment, whereas for short holdings, an increase in share prices would be a risk to that investment. Market Risk: Part I 57 2.2.3. Measurement Having identified the risk factor, we must now attempt to measure and quantify the scope of risk embodied in this factor. By doing this, it is pos- sible to determine whether a share is a riskier asset than a bond and to what extent holding an option on the stock index of the Shanghai Stock Exchange is a riskier venture than investing in the EUR/GBP exchange rate. However, measuring the scale of risk embodied in market prices is far from simple. Unlike risk, measuring profit and return is relatively straightforward. When we measure profits, we compare the total value of financial assets at the end of a given period of time with their value at its beginning. The profit may then be divided by the value of financial assets at the beginning of the period to calculate the return on investment in percentage. The result of this calculation may be substituted with cash, concrete products, services or other assets. Measuring profit and return is so simple that almost no additional methods have been developed to compare the hold- ings at the end of a given period with the holdings at its start. Measuring risk is a considerably more complex process, in part because what we attempt to measure is the “possibility” of losing profit or capital (see the definition of market risk mentioned earlier). Several meth- ods exist for measuring risk: Measurement Exposure Loss Fluctuation Simulation · Exposure: Measures the potential damage in case of a failure event, regardless of the probability of the damage materializing. For instance, 58 Lecture Notes in Risk Management if we hold a long position worth USD 15,000 on Coca-Cola shares and a USD 30,000 long position on General Electric (GE) bonds, our exposure to Coca-Cola is USD 15,000 and is therefore smaller than our USD 30,000 exposure to GE. This measure makes it very easy to estimate the potential of damage at any given point during a time period and at its end. · Fluctuation: Over a period of time, the value of an asset may increase or decrease. For some assets, these fluctuations are greater and more frequent than those for others. Fluctuations may be calculated using statistical measures (see the following) such as standard deviation, implied volatility, variance, volatility index (VIX) and vega. For example, if the standard deviation of Coca-Cola share prices is greater than that of GE bonds, one may say that the risk embodied in a Coca- Cola share is greater than that of a GE bond. · Loss: Unlike exposure, which only measures the potential damage, loss is a measure which incorporates probability as well. Measuring loss can be done using Value at Risk (VaR), maximum drawdown (MaxDD), etc. For example, at a probability of 99%, loss on Coca- Cola shares over the coming month will not exceed USD 1,300 as a way of loss quantification. · Simulation: Risk assessment using various scenarios in which changes are applied to the risk factor. This may be a simulation of a worst-case scenario, a what-if or Monte Carlo simulation, etc. It is possible to simulate a worst-case scenario in which market prices fall drastically, e.g., by one-third over the next month, and then examine how such a scenario might affect the value of Coca-Cola and GE holdings. In this scenario, the total accumulated value of our hold- ings is USD 45,000 (USD 15,000 Coca-Cola + USD 30,000 GE), and it might drop to USD 30,000 in the worst-case scenario (a loss of USD 15,000). The measurement and assessment of risk feature a wide variety of measures, each one examining a different aspect of the risk. Accurate clas- sification and correct quantification of risk are only possible when all of these are utilized in combination. We have already dedicated a section exposure in the first chapter. In the following sections, we shall continue discussing measurement of loss, as well as expand upon the topic of simulation. Market Risk: Part I 59 2.2.3.1. Preliminary calculations Preliminary Calculations Rather than performing calculations based on the portfolio value, P, we base our calculations on Ln(P) for the following reasons: Simpler calculations (linear rather than exponential structure) Uniform scale Mathematically continuous interest rate The various methods for quantifying risk require preliminary mathemati- cal transformations of the portfolio value. Rather than performing calcula- tions directly upon the portfolio value (which we shall refer to as P), they are performed upon Ln(P). Why is this transform necessary? · This method makes it much easier to perform calculations, as they are performed within a linear framework rather than an exponential one. Thus, for example, the formula for calculating compound interest is not a product, e.g., (1 + r)(1 + r)–1, but a sum such as i + i. · This method ensures that a consistent scale is maintained, even in cases where the portfolio value varies significantly over time. A failure to perform these preliminary calculations may confuse our understanding of how the risk has evolved. Portfolio Value — P 60 Lecture Notes in Risk Management For example, a chart of the S&P 500 Index for the years 1871–2014, drawn without the preliminary calculations, would give the impression that the largest financial crisis of the last 150 years occurred in 2008, with the second-largest occurring in 2000. The various other crises which occurred between these years are simply not represented in the chart. From this chart, we might conclude, by mistake, there were no crises before 2000. This happens because, when the index is worth USD 10, a 10% change will amount to USD 1, whereas if the index is worth USD 1,000, a 10% change will equal USD 100. Therefore, the same crisis event, in which the index lost 10% of its value, would register as a dramatic event for the high index value but would be almost indiscernible at the lower value. If one first performs the preliminary calculation and only then draws the chart, this ensures that such an event is represented in a similar manner in both cases. Portfolio Value Logarithm — Ln(P) This image shows the development of the S&P 500 Index during almost 150 years. It also clearly shows that during the past 150 years, the index experienced its most severe crisis in the 1920s and 1930s. · Mathematical continuity of interest rates ensures that it is possible to derive the portfolio value at any point and, for example, calculate the value of options which assist in determining risk pricing. The Market Risk: Part I 61 conversion from discrete to continuous yearly interest is done in the following manner: Preliminary Calculations Annual interest rate on investment, It is also possible to receive after six months and then reinvest for six additional months: The value of the investment at the end of the period, P2, equals its value at the beginning of the period, P1, multiplied by 1, plus the interest, r. Many banks allow for a semi-annual interest, which equals 2r for half a year, and (1 + 2r ) (1 + 2r ) all together for the entire year. The year may be further divided into smaller segments, e.g., a triannual interest of (1 + 3r ) (1 + 3r ) (1 + 3r ), and so forth. More generally, the value of the invest- ment at the nend of the period may be written in the following way: P2 = P1 (1 + nr ) , with n standing for the number of times per year interest is received. n may, in theory, be increased all the way up to infinity, in which case the coefficient of (1+ nr ) n is reduced to er. Preliminary Calculations 62 Lecture Notes in Risk Management The formula for calculating continuous interest is now very simple: Ln(P2) – Ln(P1) ≅ r In much the same way, it is also possible to calculate the continuous yield of a portfolio: Ln(V2) – Ln(V1) ≅ y where V is the portfolio value. Excel Some useful Excel functions for performing the calculations described above are as follows: (1 + r)n r = 0.02; n = 2 = (1+0.02)^2 er r = 0.03 = EXP(0.03) Ln(P) P = 101 = LN(101) 2.2.3.2. Standard deviation As a general rule of thumb, due to statistical considerations, the various risk calculations are based on the yield rather than on the portfolio value. In the following example, we calculate the daily standard deviation of a portfolio value. In order to calculate the standard deviation, we must first calculate the average. Standard Deviation Portfolio average and standard deviation: = portfolio value on day t = difference in portfolio value between day t and day t-1 Average daily yield: Daily standard deviation: Market Risk: Part I 63 Let us use yt to represent the change in portfolio value between day t and day t–1. The average daily yield is 1 T y= ∑ yt T t =2 And the standard deviation is 1 T Std ( y ) = ∑ ( yt − y ) 2 T − 1 t =2 In order to calculate the average yield and standard deviation for a period of 10 days, we multiply the daily result by the number of days and by the square root of the number of days, respectively: Standard Deviation Conversion to annual average and standard deviation: Average: Standard deviation: The advantage of using standard deviation when measuring risk is that it is very simple to calculate. Almost any data analysis software contains built-in functions for the various calculations required. For example, let us assume the yield on our portfolio is as described in the following image, with a probability of 25% for each possibility: 64 Lecture Notes in Risk Management Standard Deviation 2% 2% 2% -2% Mean = 1% STD = 2% The average yield is 1% and the standard deviation is 2%. Excel Some useful Excel functions for performing the calculations described above are as follows: y 2; 2; 2; –2 = AVERAGE(2,2,2,–2) Std(y) 2; 2; 2; –2 = STDEV(2,2,2,–2) 2.2.3.3. Skewness Skewness, occasionally referred to as “inclination angle of the distribu- tion”, is a measure of the symmetry of a statistical distribution. Skewness Negave Skewness Skewness = 0 Posive Skewness Market Risk: Part I 65 Skewness is calculated according to the following formula: 3 N  y − y  N Skewness ( y ) = ∑  t  ( N − 1)( N − 2) n =1  Std ( y )  Skewness is often used alongside average and standard deviation, as the latter two are not always sufficient to reveal the complete understanding. The following example shows two distributions whose average and standard deviation are identical but whose skewness levels are very different: Skewness The average yield is 1% and the standard deviation is 2% in both exam- ples. However, distribution A is left-skewed whereas distribution B is right-skewed. Were we to compare only the average and the standard deviation of the two investments, we might conclude that both involve similar levels of risk. The difference in skewness, however, indicates that distribution B is less risky than distribution A, containing no possibility of loss at all. Excel This is the Excel function which may be used for performing the calcula- tions above: Skewness(y) 0; 0; 0; 4 = SKEW(0,0,0,4) 66 Lecture Notes in Risk Management 2.2.3.4. Kurtosis Kurtosis is a statistical measure which allows us to determine whether a distribution is higher or flatter than a normal distribution. A positive mea- sure reflects a leptokurtic distribution, i.e., a greater likelihood of outliers; whereas a negative measure represents a platykurtic distribution, meaning outliers are less likely to occur. Kurtosis Leptokurc Mesokurc (normal) Platykurc Kurt > 0 Kurt = 0 Kurt < 0 Kurtosis is calculated in the following way: 4 N  y − y  3( N − 1) 2 N ( N + 1) Kurtosis ( y ) = − + ∑  t  ( N − 2)( N − 3) ( N − 1)( N − 2)( N − 3) n =1  Std( y )  A kurtosis (“Kurt”) measure of zero represents a normal distribution. If Kurt is greater than zero, the distribution has “thick tails”, i.e., a greater probability of outliers than a normal distribution. If Kurt is smaller than zero, most of the observations are concentrated around the center and the distribution therefore has fewer outliers than a normal distribution. The following example shows two distributions with identical aver- age, standard deviation and skewness measures but with different kurtosis measures: Market Risk: Part I 67 Kurtosis The average yield is 1%, the standard deviation is 2% and the skewness is 0% in both cases. However, distribution C has a higher kurtosis than distribution D. Were we to compare average, standard deviation and skewness alone, it might appear as though both investments involve a similar level of risk. However, the positive skewness of distribution C reveals that this is the riskier option, as we should expect more outliers for distribution C than for distribution D. Excel This is the Excel function which may be used for performing the calcula- tions above: Kurtosis(y) 2.45; 0; 0; –2.45 = KURT(2.45,0,0,–2.45) 68 Lecture Notes in Risk Management Histogram of daily Stock Prices Changes since 1928 12.53%, Oct 30, 1929 –20.47% , Oct 19, 1987 Source: finance.yahoo.com/quote/%5EGSPC/history?period1=-1325635200&period2=1667001600&interval=1d&filter=history&frequency=1d&includeAdjustedClose=true. If one constructs a histogram of the daily changes to the U.S. stock index in the 20th century, several interesting phenomena become apparent: First of all, most observations are concentrated within a margin of ±2%. This means that on most days we can expect returns or losses of up to 2%. Second, the two columns at the center of the histogram and the addi- tional two columns flanking them indicate a positive average return over- all. Over time, the returns slightly outweigh the losses, and thus, in the long run, an investor can expect to make a profit by investing in the stock market. Third, outliers are relatively uncommon but not small in number. The most extreme outliers are the worst day of 1987, on which the index fell by almost 20.5% in a single day(!), and the best day, which occurred in 1929 (to be precise, the day after Black Tuesday, during the Great Depression), when the index went up by over 12.5%. When managing market risks, one would be wise to pay attention to the enormous difference between average daily losses and an extreme outlier, which has the potential to eliminate more than 20% of all capital in the market in a single day. Market Risk: Part I 69 2.2.3.5. Maximum drawdown Maximum drawdown (MaxDD) is a financial measure which represents the greatest difference measured between peak prices and low prices for an asset over a period of time. Maximum Drawdown (MaxDD) 210 100 90 80 70 200 60 50 DD Max 40 190 Value 30 20 10 180 0 01-01 15-01 29-01 12-02 26-02 12-03 26-03 09-04 23-04 07-05 21-05 04-06 18-06 02-07 16-07 30-07 13-08 27-08 10-09 24-09 08-10 22-10 This is the formula for calculating maximum drawdown: MaxDD ( P ) = max  Pt , Max ( Pt −1 )  {T } In the example given in the image, the chart shows the value of an asset portfolio increasing from 198,000 Jordanian Dinars (JOD) to 207,000 JOD at the end of the period. In the first quarter, the portfolio quickly soared to 210,000 JOD and subsequently began to descend. It then rose again dramatically at the mid- dle of the period but immediately afterwards fell to the lowest point observed during the entire period. During this time, various losses are shown (in gray arrows). These are called drawdowns (DD). Some of these, where the gray line is steepest, represent severe losses over a short period of time. Out of these, the greatest accumulated loss, MaxDD, is the one marked by the thickest arrow. 70 Lecture Notes in Risk Management Excel Here is a fairly simple method for calculating MaxDD in Excel: The Value column lists the values of the asset portfolio. The Max column contains the formula = MAX( , ) as shown in the following: A B C D 1 DATE Value Max DD 2 01-01-15 188 3 04-01-15 188 = MAX(C2,B3) 0 Result: 188 4 05-01-15 187 = MAX(C3,B4) 1 Result: 188 5 06-01-15 186 = MAX(C4,B5) 2 Result: 188 6 07-01-15 187 = MAX(C5,B6) 1 Result: 188 7 08-01-15 191 = MAX(C6,B7) 0 Result: 191 8 11-01-15 191 = MAX(C7,B8) 0 Result: 191 9 12-01-15 188 = MAX(C8,B9) 3 Result: 191 10 13-01-15 189 = MAX(C9,B10) 2 Result: 191 = MAX(D2:D10) Result: 3 The column DD contains the difference between Max and Value. Finally, we find the highest value in the DD column. This is the MaxDD value, which in this case equals 3, i.e., the greatest loss during this ten-day period is 3,000 JOD. Market Risk Part II Accompanying presentation: 2. Risk Management — Market Risk (Part II) Portfolio Value 1.1.2013–31.12.2013 Which point represents the greatest daily loss measured in 2013? 5.0% 20,00,000 A D 4.0% B C 3.0% 19,00,000 2.0% 1.0% 18,00,000 0.0% -1.0% 17,00,000 -2.0% -3.0% 16,00,000 -4.0% -5.0% 15,00,000 The chart in this image represents the daily value of an investment port- folio during the year 2013. From the chart, we can conclude that during the first few months of 2013, the portfolio value mostly went down. This trend reversed in late April, with the portfolio beginning to rise in value, marking a 10% increase by the end of the year. The description above is short and dull while the real story is more dramatic. This is about an investment house managing a mutual fund for its clients. At the beginning of the year, neither the managers nor their clients knew that the year would end with profit, nor could they know 71 72 Lecture Notes in Risk Management what financial hurdles they would have to overcome. Let us outline the evolves in the order in which they occurred, without assuming prior knowledge of the evolution of the portfolio value during the year. In late January, clients noticed that the fund was losing money and thought of selling their holdings. The fund managers claimed this was only a passing phase and managed to keep most of their clients from leaving. Indeed, in early February, the portfolio value started to go back up, although its value remained low compared to the beginning of the year. With clients now reassured that their investments were safe, the portfolio plodded along for two more months, until, for a short period of time begin- ning in late March, it began to experience sharp drops every day. Many upset clients sold their holdings at this point, incurring a loss of more than 10%. For the investment house, this represented both a loss of income from discontinued management fees and damage to their reputation. In April, the portfolio began to recover. However, those clients who had left — the loss still fresh in their minds — were reluctant to return. Furthermore, new clients did not join up. The fund continuously made a profit until mid-May but failed to recruit new clients. In June, the portfolio lost money once again, which served to further reinforce the conviction of potential clients that it would be unwise to invest in this “failed” mutual fund. In August, the investment house managed to recoup all of its losses. At this point, its clients were divided into two groups: those who had stayed true to the fund, who had recovered all the money they had lost, and those who had abandoned and lost money. The fund experienced additional substantial losses in September but had recovered and even turned a profit by October. By December, those clients who had remained loyal had made a profit of almost 10%. At this point, some of those clients who had left sought to reinvest in the portfolio and reap the rewards of the fund’s upswing. Let us here comment on the conduct of the clients and the investment house in the case study earlier: (i) Inexperienced traders tend to act rashly, selling at cheap prices and buying at higher prices. At the core of this financial behavior often lie irrationality, fears and concerns rather than a genuine understanding of the financial risks involved and how to manage them. In our example, this caused Market Risk: Part II 73 clients to sell their holdings when the portfolio value was at its lowest and buy them only after the value had gone back up. (ii) Loyalty to an investment house is no guarantee of return on an invest- ment, however. In this example, we used the term “loyal clients” and showed how it was this group that had made a profit. This fact in and of itself does not mean that remaining loyal to a particular investment house will necessarily earn us money or, conversely, that not remain- ing loyal will cause us to lose money. Investment decisions are usu- ally more complex than being loyal or not. (iii) Many financial entities tend to mischaracterize their clients’ risk profiles, and thus they might match a risky portfolio to a risk-averse client. Although regulations stipulate that the risk profile of a client should be mapped in an accurate and reliable manner, many financial entities continue to ask the wrong questions despite the significant progress which has been made in this area. As a result, the investment brokers in this case had to deal with an outflow of clients whose risk profile did not match the largely foreseeable jolts experienced by the investment portfolio. If we were asked to identify the worst day of the entire year, i.e., on which day during 2013 the portfolio suffered its greatest losses, we would concentrate on the four options marked by arrows in the below image. Whenever this question is posed to students, along with the accompa- nying image, they tend to point to B and C as showing the greatest daily losses. Occasionally, option D is selected as well. Only once have we been told that A was the correct answer, and this was from a student who had arrived late to class and had not even heard the question. Daily Change in Portfolio Value 1.1.2013–31.12.2013 Which point represents the greatest daily loss measured in 2013? 5.0% 20,00,000 A D 4.0% B C 3.0% 19,00,000 2.0% 1.0% 18,00,000 0.0% –1.0% 17,00,000 –2.0% –3.0% 16,00,000 –4.0% –5.0% 15,00,000 74 Lecture Notes in Risk Management The presentation of data affects the way in which the brain inter- prets the facts and draws conclusions from them. In order to come up with a precise answer, we must use suitable measurement methods. In this case, all that means is calculating the daily changes in portfolio value and displaying them in a separate graph. In this way, it is evident that option A is the correct one, as on that day the greatest loss was measured. Daily Change in Portfolio Value 1.1.2013–31.12.2013 5.0% 20,00,000 A 4.0% 3.0% 19,00,000 2.0% 1.0% 18,00,000 0.0% tomorrow –1.0% 17,00,000 –2.0% –3.0% The greatest daily loss measured was just over 4% 16,00,000 - The extent of loss is known –4.0% - The duraon is known - Missing informaon: what is the probability that the porolio will incur such a loss tomorrow? –5.0% 15,00,000 The greatest loss measured in a single day was 4%, making A the correct answer. However, this is not sufficient for the purpose of risk management. There is a significant difference between a scenario where such a loss may be expected once a year and one where it is likely to occur on a weekly basis. We therefore wish to measure the probability that such a loss will be incurred tomorrow. In the example above, the calculation is simple. The probability is one out of 260 trading days (0.3846%). Market Risk: Part II 75 Distribution of Daily Returns 1.1.2013–31.12.2013 5.0% 20,00,000 4.0% Frequency 3.0% 19,00,000 2.0% 90 1.0% 18,00,000 80 0.0% 70 -1.0% 17,00,000 -2.0% 60 -3.0% 16,00,000 50 -4.0% -5.0% 15,00,000 40 30 20 10 0 –4.5% –4.0% –3.5% –3.0% –2.5% –2.0% –1.5% –1.0% –0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% Daily Returns We have seen again that a different presentation of the data can improve our understanding of the risks of a portfolio. Now that we have at our disposal all the daily value changes for 2013, we may proceed to arrange them in a different format. For instance, we may display the data as a histogram showing the number of days (y-axis) on which the change in portfolio value (x-axis) was within the bounds of 0.5% intervals. For instance, the number of days on which the change was between 0% and 0.5% is 77. The number of days on which the change was between –0.5% and 0% is 69. Distribution of Daily Returns 1.1.2013–31.12.2013 Frequency 90 80 70 Frequency 60 Normal Distribuon 50 40 30 20 10 0 –4.5% –4.0% –3.5% –3.0% –2.5% –2.0% –1.5% –1.0% –0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% –10 Daily Returns 76 Lecture Notes in Risk Management On top of our histogram, we may superimpose the normal distribution of value changes, based upon the average (0.037%) and the standard devia- tion (0.79%) of the portfolio’s value changes. This method of displaying the data reflects the distribution of daily changes for 2013 both as an empirical distribution and as a constrained normal distribution. This visual representation of the distribution reveals that it is more or less symmetrical, with most days seeing changes of ±1%, and that it has “thick tails”, i.e., a higher probability of outliers than a normal distribution. Now that we have examined the distribution of daily changes, we may proceed to discuss the concept of “Value at Risk”. 2.2.3.6. Value at risk Value at risk is a statistical method for assigning a monetary value to port- folio risk, reflecting the greatest potential loss which could, at a given probability, materialize during a given period. The Value at Risk is marked as VaR.1 A higher value represents a higher risk level. 1Some interesting facts… · In 1997, many financial institutions began to publish the results of their VaR calcula- tions in order to more accurately disclose the levels of risk to which their portfolios were exposed. · In 1999, banks worldwide began to employ VaR calculations in accordance with the recommendations put forward in Basel II. · Many financial professionals erroneously use the initials VAR (with an uppercase A) in their reports on value at risk in investment portfolios. This three-letter combination can be rather confusing, as it represents three different concepts which differ only in letter case: VaR Value at Risk VAR Vector An econometric model for capturing linear interdependencies Autoregression in time series Var Variance Statistical variance Market Risk: Part II 77 VaR VaR is the greatest potential loss which could occur at a given probability over a given period of time. Value at risk is a measure which answers the following question: “What is the potential loss of an investment portfolio, at its current composition, in the near future?” Loss is typically measured in terms of money or percentage out of the total portfolio value. Potential refers to the probability that the loss will materialize (typi- cally 95% or 99%). The portfolio includes all existing assets and liabilities, including derivatives such as forwards and options. The current composition is a combination of the assets and liabilities which make up the portfolio at the moment of the value at risk calculation, in terms of the number of securities held in each asset (as opposed to a portfolio composition which also takes into account the price of the assets). The near future is typically the next day, week, month, quarter or year. VaR — Definition Frequency 1st percenle VaR(Daily, 99%) = P(1-0.99)*V 90 P(0.01)= 1st percenle 80 V = porolio value 70 60 50 40 30 20 10 0 –4.5% –4.0% –3.5% –3.0% –2.5% –2.0% –1.5% –1.0% –0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% 78 Lecture Notes in Risk Management Value at risk is calculated using the following formula: VaRa(y) = Percentile(y, 1 – a) × Value where y represents all the changes in portfolio value at its current composition. In the example earlier, over the entire period, the composition of the portfolio did not change in terms of the number of securities held. The value at risk for the portfolio at the end of 2013 was USD 42.7 thousand, or 2.23% of the portfolio value. There are three widely used methods for calculating value at risk. 2.2.3.6.1. Historical method Calculation Methods Historical method (based on past data) Advantage: easy to implement Disadvantage: sensitive to the length of the time windows selected The historical method is based on past data. The advantage method is that it is fairly easy to implement. It is therefore frequently used in business risk management and integrated into many information systems. The weakness of the method lies in the fact that specific time win- dows must be selected in order to perform the calculation. One window refers to the frequency of changes (daily, weekly, monthly, quarterly, yearly, etc.). The other refers to the historical time period we wish to examine for the sake of our calculation. Having set the frequency and historical depth, the calculation accord- ing to the historical method is performed as follows: (1) The composition of the portfolio is determined. For instance, 100 Apple shares and 150 Google shares. (2) The units of time for which the calculation is performed are set (day, week, month, quarter, year, etc.). (3) The reference period is set (last 360 days, last 206 weeks, last 60 months, etc.). In order to calculate the changes in the value, we need Market Risk: Part II 79 to increase the number of observations by one (i.e., 361 days, 207 weeks and 61 months). (4) The portfolio value is calculated at the end of each interval, according to the historical prices recorded for that point in time. (5) The changes in price2 between the end points of the intervals are measured (hence the additional interval added to the number). (6) The 1st and 5th percentiles of all the changes are calculated. (7) The result is multiplied by (–1) and by the portfolio value at the end of the period. The following table gives an example of the data needed to calculate value at risk using the historical method: Historical Method 1 –0.475% 21 –0.237% 41 –0.025% 61 0.154% 81 0.310% 2 –0.470% 22 –0.227% 42 –0.020% 62 0.154% 82 0.356% 3 –0.465% 23 –0.219% 43 0.000% 63 0.162% 83 0.361% 4 –0.456% 24 –0.206% 44 0.015% 64 0.173% 84 0.364% 5 –0.415% 25 –0.206% 45 0.025% 65 0.173% 85 0.364% 6 7 –0.405% –0.387% 26 27 –0.173% –0.144% 46 47 0.025% 0.026% 66 67 0.186% 0.187% 86 87 0.366% 0.382% P(0.01) = –0.470% 8 –0.385% 28 –0.128% 48 0.035% 68 0.200% 88 0.406% 9 –0.371% 29 –0.122% 49 0.042% 69 0.210% 89 0.416% 10 –0.368% 30 –0.119% 50 0.050% 70 0.230% 90 0.421% Value = 100,000$ 11 –0.340% 31 –0.118% 51 0.060% 71 0.258% 91 0.432% 12 –0.326% 32 –0.102% 52 0.105% 72 0.262% 92 0.433% 13 –0.321% 33 –0.095% 53 0.106% 73 0.267% 93 0.459% VaR = 470$ 14 –0.279% 34 –0.093% 54 0.112% 74 0.272% 94 0.461% 15 –0.271% 35 –0.089% 55 0.118% 75 0.277% 95 0.481% 16 –0.270% 36 –0.084% 56 0.131% 76 0.278% 96 0.491% 17 –0.265% 37 –0.057% 57 0.133% 77 0.293% 97 0.494% 18 –0.261% 38 –0.034% 58 0.146% 78 0.298% 98 0.495% 19 –0.256% 39 –0.034% 59 0.149% 79 0.308% 99 0.495% 20 –0.238% 40 –0.030% 60 0.152% 80 0.310% 100 0.497% 2This refers to changes in market price which are not the result of technical reasons such as a stock split or devaluation resulting from a dividend payout. There are two possible courses of action in such a case: The first is to remove the problem, i.e., omit the day on which it is known that the tech- nical change will occur. The advantage of this is the ease with which this may be done. The disadvantage is that data are omitted. The second method is to “string” the data to create a consistent asset price index over the entire period: For instance, if the share price was USD 250 on 2.3.2013 and at the end of the day a 1:2 stock split was performed, on the following day (3.3.2013), the share price would be USD 125 and the change would register as 50% due to a technicality. In such instances, all that needs to be done is to double the share price starting from 3.3.2013. The advantage of the second method is that it preserves all the available data. The dis- advantage is that operating errors may occur in its application. 80 Lecture Notes in Risk Management The values table contains 100 observations of returns, sorted in ascending order according to size. The first percentile equals –0.47%. If the total portfolio value is USD 100,000, the VaR will equal USD 470. At the start of this chapter, we showed a chart representing the value of an investment portfolio over the course of 2013. According to the his- torical method, the value at risk at a 99% probability is USD 42,699, which means that in 99%, the maximum loss of this portfolio during the next day will not exceed USD 42,699. Excel This Excel function may be used for performing the calculations above: P (y,a) y = 1,2,3,4,5; x = 0.05 = PERCENTILE({1,2,3,4,5},0.05) 2.2.3.6.2. Analytical method Calculation Methods Historical method (based on past data) Advantage: easy to implement. Disadvantage: sensitive to the length of the time windows selected. Analytical method (based parametric structure of distribution) Advantage: maintains consistency with the selected time windows and with other risk measures (e.g. standard deviation). Disadvantage: requires an a priori assumption regarding the form of distribution (difficult to apply to derivatives). The analytical method for calculating value at risk requires an a priori assumption regarding the breakdown of the distribution of portfolio value changes. Generally, a normal distribution is assumed for the changes in value of the assets composing the portfolio. This approach aids greatly in maintaining consistency in VaR calculations. For instance, there is no need to set the time windows in advance, as the calculations already take them into account (naturally, when reporting VaR, it is necessary to men- tion the time horizon to which it pertains). Furthermore, it ensures Market Risk: Part II 81 consistency with other risk measures, such as standard deviation. Thus, if one measure indicates an increase in risk, the other will necessarily do so as well, and vice versa. The analytical method is widely used in academic research. The disadvantage of this method is that it requires an a priori assump- tion regarding the distribution breakdown. In many cases, the theoretical distribution model does not match the empirical (historical) distribution and may even cause businesses to reach the wrong conclusions. For exam- ple, if there is a correlation between the prices of several assets in a port- folio, we must learn the nature of this correlation and incorporate it into our VaR calculations. In addition, if the portfolio contains non-standard assets, e.g., derivatives, it becomes apparent that they do not conform to an analytically structured distribution which is dependent on the distribu- tions of the other assets contained in the portfolio. The most common form of analytical distribution is normal distribu- tion, which is defined only by average and standard deviation. Therefore, with this approach, the differences between various assets are simply the result of the differences in average and standard deviation between them. The values of different assets may show a positive correlation, nega- tive correlation or zero correlation. For the sake of simplicity, let us here assume zero correlation. VaR — Definition Frequency 1st percenle VaR(Daily, 99%) = P(1-0.99)*V 90 P(0.01)= 1st percenle 80 V = porolio value 70 60 50 40 30 20 10 0 –4.5% –4.0% –3.5% –3.0% –2.5% –2.0% –1.5% –1.0% –0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% 4.0% 4.5% 5.0% The calculation of value at risk according to the analytical method is done in the following manner: 82 Lecture Notes in Risk Management (1) The composition of the portfolio is determined. For instance, 100 Apple shares and 150 Google shares. (2) The units of time for which the calculation is performed are set (day, week, month, quarter, year, etc.). (3) The reference period is set (last 360 days, last 206 weeks, last 60 months, etc.). In order to calculate the changes in the value, we need to increase the number of observations by one (i.e., 361 days, 207 weeks and 6

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