1. CHAPTER 7 Value-at-Risk Contribution

2. INTRODUCTION The output from a VaR calculation includes the following reports that can be used to identify the magnitude and source of each risk: –Total VaR for the trading operation –Stand-alone VaR for each subportfolio –Stand-alone VaR for each risk factor –Sensitivity (or duration) for each risk factor –Duration matrix for each subportfolio –Duration matrix for Total portfolio –VaR Contribution for each subportfolio –VaR Contribution for each risk factor

3. INTRODUCTION The first six of these reports are generated easily from the analyses we discussed in the previous chapter The total VaR is calculated by including all of the bank's instruments and risk factors. The stand-alone VaR for a subportfolio is the VaR that the portfolio would have if we ignored the rest of the bank.

4. INTRODUCTION Similarly, the stand-alone VaR for each risk factor is calculated by setting the standard deviation on all the other risk factors equal to zero. The sensitivity of the value of the portfolio to changes in risk factors is given by the derivative vector that is used in Parametric VaR.

5. INTRODUCTION The main problem with the stand-alone VaR is that the sum of the stand-alone VaRs does not, in general, equal the total VaR. Also, the stand-alone VaR ignores the correlation with the rest of the portfolio.

6. INTRODUCTION For clarity, consider the following Parametric VaR example for a portfolio with two subportfolios, A and B. Here, SVaR represents the stand-alone VaR: (1)[1,-1] (2)If the value =1, then VAR P =SVAR A +SVAR B

7. INTRODUCTION The VaR Contribution (VaRC) technique is useful because it gives us a measure of risk for each individual subportfolio that includes the interportfolio correlation effects. Furthermore, VaRC is constructed so that the sum of VaRC for all the subportfolios equals the total VaR for the portfolio. This allows us to make straightforward statements such as, "The VaR for the bank is $8 million, caused by contributions of $2 million from the equities desk, $3 million from bonds, $2 million from FX, and $1 million from derivatives."

8. INTRODUCTION As explored in the following chapters, VaRC is also useful for allocating the bank's capital to those units causing the risk and for setting limits on the amount of risk that individual traders may take The process used to define VaRC is the same as the process that is used later in the credit-risk chapters to define ULC, the Unexpected Loss Contribution

9. INTRODUCTION This chapter will show the derivation of VaRC for individual risk factors and for individual subportfolios. We will show how VaRC can be calculated in algebraic, summation, and matrix forms In each case, we will start with a portfolio of just two risks and then generalize to a portfolio of many risks

10. DERIVATION OF VARC IN ALGEBRAIC NOTATION Consider a portfolio exposed to two sources of risk, A and B the average correlation between the given risk and the rest of the portfolio

11. DERIVATION OF VARC IN ALGEBRAIC NOTATION

15. DERIVATION OF VARC IN SUMMATION NOTATION Summation notation can be useful if there are many risk factors because it can express long equations in a compact form.

16. DERIVATION OF VARC IN MATRIX NOTATION Matrix notation also gives a good shorthand way of writing equations. It also allows us to easily show how the VaRC can be calculated either for single risk factors affecting many positions or for single positions affected by many risk factors.

17. DERIVATION OF VARC IN MATRIX NOTATION

21. Example of Calculating VaRC Using Matrix Notation

23. VARC CALCULATED FOR SUBPORTFOLIOS In the derivation above, we showed the VaR Contribution for different risk factors. VaRC can also be calculated for different business units, or subportfolios, each of which may share some risk factors with the other desks. This is most easily shown in matrix notation

24. VARC CALCULATED FOR SUBPORTFOLIOS In this case, we break down the D vector into the sensitivity vector for each subportfolio Consider the following bank consisting of a number of portfolios, a to z The sensitivity vector for the bank as a whole has an element for each of the N risk factors: 1 to N

25. VARC CALCULATED FOR SUBPORTFOLIOS

31. CALCULATING VARC WHEN USING MONTE CARLO ORHISTORICAL SIMULATION With the Monte Carlo and Historical simulation methods, we can calculate a VaRC by going through all the simulation results and examining all the scenarios in which the VaR is exceeded by the experienced losses For example, if we run 5000 scenarios, the 99% VaR would be defined by the 50th- worst result

32. CALCULATING VARC WHEN USING MONTE CARLO ORHISTORICAL SIMULATION VaRC can be calculated using these 50 cases when the losses equal or exceed the VaR On each occasion when the VaR is exceeded, we record the losses from the individual position This gives the percentage contribution of each position to the portfolio's loss in those particularly bad scenarios

33. CALCULATING VARC WHEN USING MONTE CARLO ORHISTORICAL SIMULATION

34. Homework Consider a case of a foreign bond+foreign cash, same with the example in the textbook. Calculate VaRC via the three different VaR approaches