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CHAPTER 6

CHAPTER 6. The Three Common Approaches for Calculating Value at Risk. INTRODUCTION. VaR is a good measure of risk To estimate the value's probability distribution, we use two sets of information the current position, or holdings, in the bank's trading portfolio

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CHAPTER 6

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  1. CHAPTER 6 The Three Common Approaches for Calculating Value at Risk

  2. INTRODUCTION • VaR is a good measure of risk • To estimate the value's probability distribution, we use two sets of information • the current position, or holdings, in the bank's trading portfolio • an estimate of the probability distribution of the price changes over the next day.

  3. INTRODUCTION • The estimate of the probability distribution of the price changes is based on the distribution of price changes over the last few weeks or months. • The goal of this chapter is to explain how to calculate VaR using the three methods that are in common use: • Parametric VaR • Historical Simulation • Monte CarloSimulation.

  4. LIMITATIONS SHARED BY ALL THREE METHODS • It is important to note that while the three calculation methods differ, they do share common attributes and limitations. • Each approach uses market-risk factors • Risk factors are fundamental market rates that can be derived from the prices of securities being traded • Typically, the main risk factors used are interest rates, foreign exchange rates, equity indices, commodity prices, forward prices, and implied volatilities • By observing this small number of risk factors, we are able to calculate the price of all the thousands of different securities held by the bank • This risk-factor approach uses less data than would be required if we tried to collect historical price information for every security.

  5. LIMITATIONS SHARED BY ALL THREE METHODS • Each approach uses the distribution of historical price changes to estimate the probability distributions. • This requires a choice of historical horizon for the market data • how far back should we go in using historical data to calculate standard deviations? • This is a trade-off between having large amounts of information or fresh information

  6. LIMITATIONS SHARED BY ALL THREE METHODS • Because VaR attempts to predict the future probability distribution, it should use the latest market data with the latest market structure and sentiment • However, with a limited amount of data, the estimates become less accurate • There is less chance of having data that contains those extreme, rare market movements which are the ones that cause the greatest losses

  7. LIMITATIONS SHARED BY ALL THREE METHODS • Each approach has the disadvantage of assuming that past relationships between the risk factors will be repeated • it assumes that factors that have tended to move together in the past will move together in the future

  8. LIMITATIONS SHARED BY ALL THREE METHODS • Each approach has strengths and weaknesses when compared to the others, as summarized in Figure 6-1 • The degree to which the circles are shaded corresponds to the strength of the approach • The factors evaluated in the table are • the speed of computation • the ability to capture nonlinearity • Nonlinearity refers, to the price change not being at linear function of the change in the risk factors. This is especially important for options • the ability to capture non-Normality • non-Normality refers to the ability to calculate the potential changes in risk factors without assuming that they have a Normal distribution • the independence from historical data

  9. LIMITATIONS SHARED BY ALL THREE METHODS

  10. PARAMETRIC VAR • Parametric VaR is also known as Linear VaR, Variance-Covariance VaR • The approach is parametric in that it assumes that the probability distribution is Normal and then requires calculation of the variance and covariance parameters. • The approach is linear in that changes in instrument values are assumed to be linear with respect to changes in risk factors. • For example, for bonds the sensitivity is described by duration, and for options it is described by the Greeks

  11. PARAMETRIC VAR • The overall Parametric VaR approach is as follows: • Define the set of risk factors that will be sufficient to calculate the value of the bank's portfolio • Find the sensitivity of each instrument in the portfolio to each risk factor • Get historical data on the risk factors to calculate the standard deviation of the changes and the correlations between them • Estimate the standard deviation of the value of the portfolio by multiplying the sensitivities by the standard deviations, taking into account all correlations • Finally, assume that the loss distribution is Normally distributed, and therefore approximate the 99% VaR as 2.32 times the standard deviation of the value of the portfolio

  12. PARAMETRIC VAR • Parametric VaR has two advantages: • It is typically 100 to 1000 times faster to calculate Parametric VaR compared with Monte Carlo or Historical Simulation. • Parametric VaR allows the calculation of VaR contribution, as explained in the next chapter.

  13. PARAMETRIC VAR • Parametric VaR also has significant limitations: • It gives a poor description of nonlinear risks • It gives a poor description of extreme tail events, such as crises, because it assumes that the risk factors have a Normal distribution. In reality, as we found in the statistics chapter, the risk-factor distributions have a high kurtosis with more extreme events than would be predicted by the Normal distribution. • Parametric VaR uses a covariance matrix, and this implicitly assumes that the correlations between risk factors is stable and constant over time

  14. PARAMETRIC VAR • To give an intuitive understanding of Parametric VaR, we have provided three worked-out examples. • The examples are fundamentally quite simple, but they intro-duce the method of calculating Parametric VaR. • There are a lot of equations, but the underlying math is mostly algebra rather than complex statistics or calculus • Three different notations are used in this chapter • Algebraic • Summation • matrix.

  15. PARAMETRIC VAR • If we have a portfolio of two instruments, the loss on the portfolio (Lp) will be the sum of the losses on each instrument: • The standard deviation of loss on the portfolio (σp) will be as follows:

  16. PARAMETRIC VAR Algebraic Notation Summation Notation

  17. PARAMETRIC VAR Matrix Notation

  18. PARAMETRIC VAR

  19. Example One • The first example calculates the stand-alone VaR for a bank holding a long position in an equity. The stand-alone VaR is the VaR for the position on its own without considering correlation and diversification effects from other positions • The present value of the position is simply the number of shares (N) times the value per share, (Vs) • PV$ = N x Vs

  20. Example One • The change in the value of the position is simply the number of shares multiplied by the change in the value of each share: • ΔPV$ = N x ΔVs • The standard deviation of the value is the number of shares multiplied by the standard deviation of the value of each share • σv = N x σs • we have assumed that the value changes are Normally distributed, there will be a 1chance that the loss is more than 2.32 standard deviations; therefore, we can calculatethe 99 VaR as follows • VaR = 2.32 x N xσs

  21. Example Two • As a slightly more complex example, consider a government bond held by a U.K. bank denominated in British pounds with a single payment. • The present value in pounds (PVp) is simply the value of the cash flow in pounds (Cp) at time t discounted according to sterling interest rates for that maturity, rp:

  22. Example Two present value The derivative of PVp with respect to rp

  23. Example Two • To make this example more concrete, consider a bond paying 100 pounds (Cp) in 5 years' time (t), with the 5-year discount rate at 6% (rp), and a standard deviation in the rate of 0.5% (σr). • The present value is then 74 pounds, the sensitivity dr is -352 pounds per 100% increase in rates, and the VaR is 4.1 pounds

  24. Example Two

  25. Example Three • The two examples above were simple because they had only one risk factor • Now let us consider a multidimensional case: the same simple bond as before, but now held by a U.S. bank. • The U.S. bank is exposed to two risks: changes due to sterling interest rates and changes due to the pound-dollar exchange rate. • The value of the bond in dollars is the value in pounds multiplied by the FX rate

  26. Example Three

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  36. Example Three

  37. Homework • Now let us consider a U.S. bond, but now held by a Taiwan’s bank. • Other conditions are consistent with the precious case. • Consider this bond paying 100 U.S. dollars (CUSD) in 5 years' time (t) • Two risks • Changes due to U.S. interest rates (rUS) • changes due to the NTD-USD exchange rate (FX) • Find the current 5-year interest rate (rUS) and the current NTD-USD exchange rate (FX) • Find the standard deviation in the interest rate and the exchange rate • Find the correlation coefficient between rUS and FX

  38. Using Parametric VaR to Calculate Risk Sensitivity for Several Positions • In the example above, we had one security that was sensitive to two different risk factors. • If the portfolio is made up of several securities, each of which is affected by the same risk factor, then the sensitivity of the portfolio to the risk factor is simply the sum of these sensitivities for the individual positions. • For example, consider a portfolio holding our example 100-pound five-year bond and 100 pounds of cash

  39. Using Parametric VaR to Calculate Risk Sensitivity for Several Positions

  40. Using Parametric VaR to Calculate Risk Sensitivity for Several Positions

  41. Using Parametric VaR to Calculate Risk Sensitivity for Several Positions

  42. Using Parametric VaR to Calculate Risk Sensitivity for Several Positions

  43. Homework • Now let us consider a bond portfolio with a U.S. bond and a U.K. bond, • The bond portfolio is held by a Taiwan’s bank. • Other conditions are consistent with the precious case. • Consider the US and UK bonds paying 100 US dollars (CUSD) and 100 British pound (CBP), respectively, in 5 years' time (t) • Four risk factors • Changes due to U.S. interest rates (rUS) • changes due to the NTD-USD exchange rate (FXUSD) • Changes due to U.K. interest rates (rUK) • changes due to the NTD-BP exchange rate (FXBP)

  44. HISTORICAL-SIMULATION VAR • Conceptually, historical simulation is the most simple VaR technique, but it takes significantly more time to run than parametric VaR. • The historical-simulation approach takes the market data for the last 250 days and calculates the percent change for each risk factor on each day • Each percentage change is then multiplied by today's market values to present 250 scenarios for tomorrow's values. • For each of these scenarios, the portfolio is valued using full, nonlinear pricing models. The third-worst day is the selected as being the 99% VaR.

  45. HISTORICAL-SIMULATION VAR • As an example, let's consider calculating the VaR for a five-year, zero-coupon bond paying $100 • We start by looking back at the previous trading days and noting the five year rate on each day. • We then calculate the proportion by which the rate changed from one day to the next

  46. HISTORICAL-SIMULATION VAR The change rate of interest rate for one day

  47. HISTORICAL-SIMULATION VAR

  48. Homework • Consider a Taiwan bond held by a Taiwan’s bank. • Other conditions are consistent with the precious case. • Consider this bond paying 100 NT dollars (CNTD) in 5 years' time (t) • One risk factor • Changes due to Taiwan interest rates (rTAIWAN) • Use the historical simulation approach to calculate the VaR • Use one-year historical data at least

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