million market value), and a position in the U.S. stock market index ($1 million mar-
ket value). The individual DEARs were:
- Seven-year zero-coupon bonds = $10,770
- Swf spot = $9,320
- U.S. equities = $33,000
However, senior management wants to know the aggregate risk of the entire trad-
ing position. To calculate this, we cannotsimply sum the three DEARs—$10,770 +
$9,320 + $33,000 = $53,090—because this ignores any degree of offsetting covari-
ance or correlation among the fixed-income, FX, and equity trading positions. In par-
ticular, some of these asset shocks (adverse moves) may be negatively correlated. As
is well known from modern portfolio theory, negative correlations among asset
shocks will reduce the degree of portfolio risk.
Exhibit 8.4 shows a hypothetical correlation matrix between daily seven-year
zero-coupon bond yield changes, Swf/$ spot exchange rate changes, and changes in
daily returns on a U.S. stock market index (Wilshire 5000). From the correlation be-
tween the seven-year zero-coupon bonds and Swf/$ exchange rates, z,swf, is negative
(–.2), while the seven-year zero-coupon yield changes with, respectively, U.S. stock
returns,z,U.S.,(.4) and Swf/$ shocks, U.S.,Swf,(.1) are positively correlated.
Using the correlation matrix along with the individual asset DEARs, we can cal-
culate the risk or standard deviation of the whole (three-asset) trading portfolio as:^21
(4)This is a direct application of modern portfolio theory (MPT) since DEARs are di-
rectly similar to standard deviations. Substituting into this equation the calculated in-
12 rU.S.SwfDEARU.S.DEARSwf 2 12 rz,U.S.DEARzDEARU.S. 2 12 rz,SwfDEARzDEARSwf 2DEAR portfolio 3 DEARz 22 1 DEARSwf 22 1 DEARU.S. 228.4 RISKMETRICS MODEL 8 • 11Seven-Year Zero Swf/$1 U.S. Stock IndexSeven-year zero — –.2 .4
Swf/$1 — .1
U.S. stock index —
Exhibit 8.4. Correlations (ij) among Assets.
(^21) This is a standard relationship from modern portfolio theory in which the standard deviation or risk
of a portfolio of three assets is equal to the square root of the sum of the variances of returns on each of
the three assets individually plus two times the covariance among each pair of these assets. With three
assets there are three covariances. Here we use the fact that a correlation coefficient times the standard
deviations on each pair of assets equals the covariance between each pair of assets. Note that DEARis
measured in dollars and has the same dimensions as a standard deviation.
1 > 2