In the previous chapter, we detailed alternative approaches for calculating the Value at Risk (VaR). It is now time to consider how the results can best be used to give management deep insights into the risks the bank is running. 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 to each risk factor
• VaR Contribution for each subportfolio
• VaR Contribution for each risk factor
The first four 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. 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.
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. This is because the equation for the total VaR includes squaring all the stand-alone VaRs, including correlation, and later taking the square root. For clarity, consider the following Parametric VaR example for a portfolio with two subportfolios, A and B. Here, SVaR represents the stand-alone VaR:
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.”
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.
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.
Most of this analysis will be for VaRC derived from Parametric VaR. At the end of the chapter, we will also show how VaRC can be defined for Historical and Monte Carlo simulations.
Consider a portfolio exposed to two sources of risk, A and B. The variance of the value of the portfolio will be equal to the sum of the variances caused by the two sources and the covariance between them:
We can rearrange the terms in this equation to make them a sum of a factor multiplied by σA and one multiplied by σB:
If we divide both sides by the standard deviation of the portfolio, we get an additive equation for standard deviation:
The terms within the brackets can be thought of as representing the average correlation between the given risk and the rest of the portfolio. In the Parametric approach, the 99% VaR is 2.32 times the standard deviation:
This allows us to define the VaR contributions for the two risks, A and B, such that they add up to the total VaR:
Note that each VaRC factor includes both the correlation with the other factors and the variance of the other factors. In Parametric VaR, consider that each variance is the sensitivity multiplied by the standard deviation of the risk factor:
σA = d1σ1
σB = d2σ2
With this substitution, the variance of the portfolio’s value and VaRC are calculated as follows:
In the bond example of the previous chapter, risk was caused by both the interest rate and FX rate, and we had the following values:
dFX = 74.7
drp = 563
σFX = 0.02
σrp = 0.5%
ρFX,rp = −0.6
VaR = 9.05
σP = 3.09
With these values, you should find the following results for VaRC:
VaRCFX = $2.82
VaRCrp = $6.21
These results show us that for this bond, there is twice as much risk caused by the interest-rate movements than by FX movements.
If we have more than two risk factors, we can go through the same process of grouping the terms to obtain the following result for VaRC:
Summation notation can be useful if there are many risk factors because it can express long equations in a compact form. In many cases it is useful as a shorthand. In summation notation, VaR was previously written as follows:
We can again group and then divide by σP to get a series of terms that can be summed across i:
For each i, the associated term defines the VaR contribution for factor i:
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.
We can write the VaR equation for two risk factors in matrix notation as follows:
To define VaRC, we can break D into two vectors corresponding to the sensitivity of the value to each of the risk factors:
D1 = [d1 0]
D2 = [0 d2]
D = D1 + D2
The equation for variance can now be written as a sum:
The standard deviation can be defined as either the square root of the variance, or as the variance divided by the standard deviation:
By using both of these definitions, we can write the standard deviation in terms of D and C:
Now we can split apart the numerator to define VaRC:
In general, if we wish to calculate VaRC with respect to many risk factors, we do so by breaking down the sensitivity vector into a series of vectors with all the elements equal to zero other than the element corresponding to the risk factor of interest:
D = [d1 d2 . . . dN]
D1 = [d1 0 . . . 0]
D2 = [0 d2 . . . 0]
DN = [0 0 . . . dN]
In the earlier chapter, we calculated VaR for a Sterling-denominated bond held by a U.S. bank. This bond has two risk factors: the exchange rate and the interest rate. In matrix notation, the VaRC for the bond is derived as follows. First, we break apart the sensitivity vector D:
Then, put these vectors into our definitions for VaRC:
From the previous chapter, the numerical values are as follows:
After carrying out the matrix multiplications, we get the results for the total VaR and VaRC:
VaR = $9.05
VaRCFX = $2.82
VaRCrp = $6.21
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. 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:
D = [d1 d2 . . . dN]
Each sensitivity is the sum of the sensitivities of each of the subportfolios, a to z:
d1 = d1,a + d1,b + . . . + d1,z
dN = dN,a + dN,b + . . . + dN,z
Here, d1,a is the derivative of the value of subportfolio a with respect to risk factor 1. We can put the sensitivity of each subportfolio into separate vectors:
Da = [d1,a d2,a . . . dN,a]
Db = [d1,b d2,b . . . dN,b]
Dz = [d1,z d2,z . . . dN,z]
The sensitivity vector for the whole bank will equal the sum of the sensitivity vectors for the subportfolios:
D = Da + Db + . . . Dz
It is equivalent to calculate the portfolio variance in either of the following ways:
From this, we can use the same process as we used for the risk factors to now define VaRC for the subportfolios:
As an example, consider a portfolio that is the sum of two positions: our usual sterling bond and 100 pounds of cash. The derivative vector for the bond position is:
DBond = [dBond,FX dBond,rp]
And the vector for the cash is:
DCash = [dCash,FX 0]
The vector for the combined position is the sum of the individual vectors:
D = [dBond,FX + dCash,FX dBond,rp]
From this, we can calculate VaR and VaRC:
For our example, the numerical values are as follows:
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. 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:
Where [Lossii|(Lossip | > VaRp)] represents the loss from position i, given that the portfolio loss is greater than the portfolio VaR.
We then define the VaR Contribution in monetary terms for position i to be the total VaR for the portfolio (VaRp) multiplied by the percentage contribution of the position:
VaRCi = VaRp × %Contribution of Positioni
This chapter explained the derivation of VaRC for individual risk factors and individual subportfolios. It also showed how VaRC can be calculated in algebraic, summation, or matrix form. Next, we will explore the tests that should be carried out on VaR calculators to ensure their validity.