1. The Three Common Approaches for Calculating Value at Risk
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

27. Example Three

28. Example Three

29. Example Three

32. Example Three

33. Example Three

34. Example Three

35. Example Three

36. Example Three

37. Example Three

38. Example Three

39. 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

40. 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

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

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

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

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

45. 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)

46. 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.

47. 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

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

49. HISTORICAL-SIMULATION VAR

50. 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

51. HISTORICAL-SIMULATION VAR
There are two main advantages of using historical simulation:
It is easy to communicate the results throughout the organization because the concepts are easily explained
There is no need to assume that the changes in the risk factors have a structured parametric probability distribution
no need to assume they are Joint-Normal with stable correlation

52. HISTORICAL-SIMULATION VAR
The disadvantages are due to using the historical data in such a raw form:
The result is often dominated by a single, recent, specific crisis, and it is very difficult to test other assumptions.
The effect of this is that Historical VaR is strongly backward-looking, meaning the bank is, in effect, protecting itself from the last crisis, but not necessarily preparing itself for the next

53. There can also be an unpleasant "window effect."
When 250 days have passed since the crisis, the crisis observation drops out of our window for historical data and the reported VaR suddenly drops from one day to the next.
This often causes traders to mistrust the VaR because they know there has been no significant change in the risk of the trading operation, and yet the quantification of risk has changed dramatically

54. MONTE CARLO SIMULATION VAR
Monte Carlo simulation is also known as Monte Carlo evaluation (MCE). It estimates VaR by randomly creating many scenarios for future rates
using nonlinear pricing models to estimate the change in value for each scenario, and then calculating VaR according to the worst losses

55. MONTE CARLO SIMULATION VAR
Monte Carlo simulation has two significant advantages:
Unlike Parametric VaR, it uses full pricing models and can therefore capture the effects of nonlinearities
Unlike Historical VaR, it can generate an infinite number of scenarios and therefore test many possible future outcomes

56. MONTE CARLO SIMULATION VAR
Monte Carlo has two important disadvantages:
The calculation of Monte Carlo VaR can take 1000 times longer than Parametric VaR because the potential price of the portfolio has to be calculated thousands of times
Unlike Historical VaR, it typically requires the assumption that the risk factors have a Normal or Log-Normal distribution.

57. MONTE CARLO SIMULATION VAR
The Monte Carlo approach assumes that there is a known probability distribution for the risk factors.
The usual implementation of Monte Carlo assumes a stable, Joint-Normal distribution for the risk factors.
This is the same assumption used for Parametric VaR.
The analysis calculates the covariance matrix for the risk factors in the same way as Parametric VaR

58. MONTE CARLO SIMULATION VAR
But unlike Parametric VaR
Decomposes the covariance matrix and ensures that the risk factors are correlated in each scenario
The scenarios start from today's market condition and go one day forward to give possible values at the end of the day
Full, nonlinear pricing models are then used to value the portfolio under each of the end-of-day scenarios.
For bonds, nonlinear pricing means using the bond-pricing formula rather than duration
for options, it means using a pricing formula such as Black-Scholes rather than just using the Greeks.

59. MONTE CARLO SIMULATION VAR
From the scenarios, VaR is selected to be the 1-percentile worst loss
For example, if 1,000 scenarios were created, the 99% VaR would be the tenth-worst result
Figure 6-4summarizes the Monte Carlo approach
Most of the Monte Carlo approach is conceptually simple. The one mathematically difficult step is to decompose the covariance matrix in such a way as to allow us to create random scenarios with the same correlation as the historical market data

60. MONTE CARLO SIMULATION VAR

61. MONTE CARLO SIMULATION VAR
For example, in the previous example of a Sterling bond held by a U.S. bank, we assumed a correlation of -0.6 between the interest rate and exchange rate
In other words, when the interest rate increases, we would expect that the exchange rate would tend to decrease.
One way to think of this is that 60% of the change in the exchange rate is driven by changes in the interest rate. The other 40% is driven by independent, random factors.
The trick is to create random scenarios that properly capture such relationships

62. MONTE CARLO SIMULATION VAR
If we just have two factors, we can easily create correlated random numbers in a simple way:
Please refer to P120

67. MONTE CARLO SIMULATION VAR

68. Example Three

69. Example Three

70. MONTE CARLO SIMULATION VAR
For the previous bond example, we would create changes in the risk factors rp and FX by using the following equations: EXCEL

71. Homework
Find the realized data to redo the sample example in the previous homework
Three approaches are asked to apply
Parametric VaR
Historical VaR
Monte Carlo VaR
Compare the differences among them

72. 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 risk factors
Changes due to U.S. interest rates (rUS)
changes due to the NTD-USD exchange rate (FX)

73. Quick Quiz Three approaches for VaR calculation Calculation speed
Non-normality
Nonlinearity
Too heavily dependent on historical data

74. LIMITATIONS SHARED BY ALL THREEMETHODS