# LO 6.4: Estim ate the im pact o f different correlations between assets in the trading

LO 6.4: Estim ate the im pact o f different correlations between assets in the trading book on the VaR capital charge.
The primary goal of risk management is to mitigate financial risk in the form of market risk, credit risk, and operational risk. A common risk management tool used to measure market risk is value at risk (VaR). VaR for a portfolio measures the potential loss in value
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Topic 6 Cross Reference to GARP Assigned Reading – Meissner, Chapter 1
for a specific time period for a given confidence level. The formula for calculating VaR using the variance-covariance method (a.k.a. delta-normal method) is shown as follows:
VaRp = crpaVx
In this equation, crp is the daily volatility of the portfolio, a is the .z-value from the standard normal distribution for a specific confidence level, and x is the number of trading days. The volatility of the portfolio, a p, includes a measurement of correlation for assets within the portfolio defined as:
a P = ^/(3h xCx(3v
where:
= horizontal (3 vector of investment amount
C = covariance matrix of returns 3 = vertical /3 vector of investment amount
Example: Computing VaR with the variance-covariance method
Assume you have a two-asset portfolio with \$8 million in asset A and \$4 million in asset B. The portfolio correlation is 0.6, and the daily standard deviation of returns for assets A and B are 1.5% and 2%, respectively. What is the 10-day VaR of this portfolio at a 99% confidence level (i.e., a = 2.33)?
The first step in solving for the 10-day VaR requires constructing the covariance matrix,
0.00018 C N C
== 0.0152 == 0.000225 covn == c o v 22 = 11 C N C
I cov12 == Pl2XCTl XCT2 == 0.6×0.015×0.02
I = 0.022 == 0.0004
Nb
Thus, the covariance matrix, C, can be represented as:
COVn cv2i
c o v 12n
c o v 22,
0.000225 0.00018′ 0.0004 0.00018
/
k
Next, the standard deviation of the portfolio, op, is determined by first solving for (3h x C, then solving for ((3^ x C) x |3p and finally taking the square root of the second step.
Step 1: Compute (3^ x C:
/
[8 4] 0.000225 0.00018
\ 0.00018 0.0004
[(8 x 0.000225) + (4 x 0.00018) [0.00252 0.00304] (8 x 0.00018) + (4 x 0.0004)] Page 72
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Step 2: Compute ((3^ x C) x (3 :
Topic 6 Cross Reference to GARP Assigned Reading – Meissner, Chapter 1
(0.00252 x 8) + (0.00304 x 4) = 0.03232
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[0.00252 0.00304] Step 3: Compute a p:
ctp = 7(3hxCx|3v = VO03232 = 0.1798 or 17.98%
The 10-day portfolio VaR (in millions) at the 99% confidence level is then computed as:
VaRp = a Pa sfe = 0.1798 x 2.33 x VlO =1.3248
This suggests that the loss will only exceed \$1,324,800 once every 100 10-day periods. This is approximately once every 1,000 trading days or once every four years assuming there are 250 trading days in a year.
Figure 7 illustrates the relationship between correlation and VaR for the previous two-asset portfolio example. The VaR for the portfolio increases as the correlation between the two assets increases.
Figure 7: Relationship Between VaR and Correlation for Two-Asset Portfolio
The Basel Committee on Banking Supervision (BCBS) requires banks to hold capital based on the VaR for their portfolios. The BCBS requires banks to hold capital for assets in the trading book of at least three times greater than 10-day VaR. The trading book includes assets that are marked-to-market, such as stocks, futures, options, and swaps. The bank in the previous example would be required by the Basel Committee to hold capital of:
\$1,324,800 x 3 = \$3,974,400
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Topic 6 Cross Reference to GARP Assigned Reading – Meissner, Chapter 1
C o r r e l a t i o n s D u r i n g t h e R e c e n t F i n a n c i a l C r i s i s