# LO 67.2: Explain the role of correlation on portfolio risk.

LO 67.2: Explain the role of correlation on portfolio risk.
In a two-asset portfolio, the equation for the standard deviation is:
P = VW12(J12 + w 22(J22 + 2 wlw2Pl,2CTl2
and the VaR is:
VaRp Z cP Jw ^aj2 + w 2 2 c t 2 2 + 2 w1w2p12a 1a 2
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2018 Kaplan, Inc.
Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
We can square Z and P and put them under the square-root sign. This allows us to express VaRp as a function of the VaRs of the individual positions, which we express as VaRj for each position i. For a two-asset portfolio we will have VaRj and VaR2. If the correlation is zero, p 2 = 0, then the third term under the radical is zero and:
VaR for uncorrelated positions: VaRp = ^VaRj2 + VaR22
The other extreme is when the correlation is equal to unity, p 1 2 = 1. With perfect correlation, there is no benefit from diversification. For the two-asset portfolio, we find:
Undiversified VaR = VaRp = ^VaR,2 + VaR22 + 2 VaR,VaR2 = VaR! + VaR2
In general, undiversified VaR is the sum of all the VaRs of the individual positions in the portfolio when none of those positions are short positions. Notice how evaluating VaR using both uncorrelated positions and perfectly correlated positions will place a lower and upper bound on the total (or portfolio) VaR. Total VaR will be less if the positions are uncorrelated and greater if the positions are correlated. The greatest risk is a correlation o f1 where one asset magnifies the loss of the other asset. The following examples illustrate this point.
Example: Computing portfolio VaR (part 1)
An analyst computes the VaR for the two positions in her portfolio. The VaRs: VaRj = $2.4 million and VaR2 =$1.6 million. Compute VaRp if the returns of the two assets are uncorrelated. Answer:
For uncorrelated assets:
VaRp = ^VaRj2 +VaR22 = ^ 2 .4 2 + 1.62 j($millions)2 = ^8.32($millions)2
VaRp = $2.8844 million Example: Computing portfolio VaR (part 2) An analyst computes the VaR for the two positions in her portfolio. The VaRs: VaRj =$2.4 million and VaR2 = $1.6 million. Compute VaRp if the returns of the two assets are perfectly correlated. Answer: For perfectly correlated assets: VaRp = VaRj + VaR2 =$2.4 million + $1.6 million =$4 million
2018 Kaplan, Inc.
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Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7
Under certain assumptions, the portfolio standard deviation of returns for a portfolio with more than two assets has a very concise formula. The assumptions are: The portfolio is equally weighted. All the individual positions have the same standard deviation of returns. The correlations between each pair of returns are the same. The formula is then:
where: N = the number of positions a = the standard deviation that is equal for all Appositions p = the correlation between the returns of each pair of positions
Professors Note: This formula greatly simplifies the process o f having to calculate portfolio standard deviation with a covariance matrix.
To demonstrate the benefits of diversification, we can simply set up a 2 x 2 table where there is a small and large correlation (p) column and a small and large sample size {N) row. Assuming that the standard deviation of returns is 20% for both assets, we see how the portfolio variance is affected by the different inputs.
Figure 1: Portfolio Standard Deviation
Sample size/correlation N = 4 N = 10
p = 0.1
o p = 11.40% cjp = 8.72%
ia
i
.
Op = 15.81% Op = 14.83%
Example: Computing portfolio VaR (part 3)
A portfolio has five positions of $2 million each. The standard deviation of the returns is 30% for each position. The correlations between each pair of returns is 0.2. Calculate the VaR using a Z-value of 2.33. Page 76 2018 Kaplan, Inc. Topic 67 Cross Reference to GARP Assigned Reading – Jorion, Chapter 7 Answer: The standard deviation of the portfolio returns is: ii 1 – + 1- – 5 5 0.2 Op = 30%V036 aP = 18% The VaR in nominal terms is: VaRp = Zc x a p x V = (2.33)(18%)($10 million)
VaRp = \$4,194,000
M a r g in a l Va R
Marginal VaR applies to a particular position in a portfolio, and it is the per unit change in a portfolio VaR that occurs from an additional investment in that position. Mathematically speaking, it is the partial derivative of the portfolio VaR with respect to the position:
Marginal VaR = MVaRj
________ <9 VaRp________ ^(monetary investment in i) cov(Rj,Rp) Op Using CAPM methodology, we know a regression of the returns of a single asset i in a portfolio on the returns of the entire portfolio gives a beta, denoted pj, which is a concise measure that includes the covariance of the positions returns with the total portfolio: cov(Ri5 Rp) Using the concept of beta gives another expression for marginal VaR: Marginal VaR = MVaR; =------------------x ft portfolio value 2018 Kaplan, Inc. Page 77 Topic 67 Cross Reference to GARP Assigned Reading - Jorion, Chapter 7 Example: Computing marginal VaR Assume Portfolio X has a VaR of 400,000. The portfolio is made up of four assets: Asset A, Asset B, Asset C, and Asset D. These assets are equally weighted within the portfolio and are each valued at 1,000,000. Asset A has a beta of 1.2 . Calculate the marginal VaR of Asset A. Answer Marginal VaRA = (VaRp / portfolio value) x (3A Marginal VaRA = (400,000 / 4,000,000) x 1.2 = 0.12 Thus, portfolio VaR will change by 0.12 for each euro change in Asset A. In c r e m e n t a l Va R