Value at risk (VaR) is a risk measure that has been widely implemented by financial institutions. This paper measures the correlation among asset price changes implied from VaR calculation. Empirical results using US and UK equity indexes show that implied correlation is not constant but tends to be higher for events in the left tails (crashes) than in the right tails (booms).Implied Correlation, Value at Risk

Francois Longin

John Cotter

English

Research Papers in Economics

 
 
 
 
UCD GEARY INSTITUTE  
DISCUSSION PAPER SERIES 
 
Implied Correlation from VaR 
 
John Cotter 
Centre for Financial Markets, 
Graduate School of Business, 
University College Dublin 
 
 
Francois Longin 
Department of Finance 
ESSEC Graduate Business School  
 
Geary WP2006/18 
June 2006 
 
 
 
 
UCD  Geary  Institute  Discussion  Papers  often  represent  preliminary  work  and  are  circulated  to 
encourage discussion. Citation of such a paper should account for its provisional character. A revised 
version may be available directly from the author. 
 
Any opinions expressed here are those of the author(s) and not those of UCD Geary Institute. Research 
published in this series may include views on policy, but the institute itself takes no institutional policy 
positions. 
   1 
Implied correlation from VaR
1 
John Cotter
2 and François Longin
3 
 
                                                
1 The first author acknowledges financial support from a Smurfit School of Business research grant and 
was developed whilst he was visiting ESSEC Graduate Business School.  
2 Director of Centre for Financial  Markets,  Department of Banking and  Finance,  Smurfit School of 
Business, University College Dublin, Blackrock, Co. Dublin, Ireland, Tel.: +353-1-7168900. E-mail: 
john.cotter@ucd.ie. 
3 Professor of Finance, Department of Finance - ESSEC Graduate Business School - Avenue Bernard 
Hirsch, B.P. 105 - 95021 Cergy-Pontoise Cedex - France. E-mail: flongin@essec.fr.   2 
Implied correlation from VaR  
 
Abstract 
Value at risk (VaR) is a risk measure that has been widely implemented 
by  financial  institutions.  This  paper  measures  the  correlation  among 
asset  price  changes  implied  from  VaR  calculation.  Empirical  results 
using US and UK equity indexes show that implied correlation is not 
constant but tends to be higher for events in the left tails (crashes) than 
in the right tails (booms). 
 
 
JEL Classification: G12 
 
Keywords: Implied Correlation, Value at Risk   3 
1. Introduction 
Value at risk measures the potential loss of a market position over a given time-period 
and for a given confidence level. For example, a 1-day 99% VaR of $ 1 000 000 means that over 
the  next  trading  day,  in  one  case  over  one  hundred,  the  portfolio  loss  will  be  higher  than 
$ 1 000 000. In order to compute the VaR of a portfolio, it is necessary to make assumptions on 
the distribution of asset price changes (see Jorion, 2000; and Dowd, 2002; for a presentation of 
VaR methods).  The Gaussian distribution with a constant correlation matrix is often assumed in 
practice. This paper takes the reverse approach by inferring the correlation implied by VaR 
calculation. 
2. VaR calculation 
In this section we consider a portfolio composed of two assets, asset 1 in proportion x1 
and  asset  2  in  proportion  x2  (x1+  x2=100%).  The  frequency  used  to  measure  the  asset  and 
portfolio price changes, which also corresponds to the holding period used to compute the VaR, 
is denoted by f. The portfolio VaR is computed by two approaches: first, by considering the 
distribution of asset price changes of the whole portfolio and then compute the portfolio VaR as 
a quantile of this distribution; second, by considering the distribution of price changes of each 
asset, then compute the VaR for each asset and finally compute the portfolio VaR by using an 
aggregation formula. 
a) The portfolio approach 
In  the  portfolio  approach,  a  time-series  of  the  portfolio  price  changes  for  a  given 
frequency f is built from the time-series of each asset price:  2 2 1 1 P x P x Pport D × + D × = D . The 
distribution  of  portfolio  price  changes  is  then  built  in  order  to  compute  the  portfolio  VaR,   4 
denoted by  port VaR . In this approach the dependence between asset price changes is implicitly 
taken into account in the creation of the portfolio by building the time-series for  port P D . 
b) The risk factor approach 
In  the  risk  factor  approach,  the  distribution  of  price  changes  of  each  asset  (more 
generally called “risk factors”) is first estimated in order to compute the individual VaR of each 
asset, denoted by VaR1 and VaR2. The portfolio VaR is then computed by using an aggregation 
formula  linking  the  individual  VaR  and  the  portfolio  weights  (more  generally  called  “risk 
sensitivities”).  A  classical  aggregation  formula  used  in  practice  (see  for  example  the 
RiskMetrics software developed by JP Morgan, 1995) computes the portfolio VaR denoted by 
agg
port VaR  for a given f as follows: 
( ) ( ) 2 1 12 2 1
2
2
2
2
2
1
2
1 2 VaR VaR x x VaR x VaR x VaR
agg
port × × × × × + × + × = r      (1) 
This approach uses explicitly the correlation coefficient between asset price changes, 
denoted by  12 r . This approach is also theoretically justified in the case of normality. If asset 
price changes are distributed according to a Gaussian distribution, then the two approaches to 
compute the VaR lead exactly to the same result. 
3. Implied correlation 
In this section we use the VaR calculation derived above in a reverse manner in order to 
estimate the implied correlation between asset price changes. By assuming that both approaches 
lead to the same VaR results, we compute the correlation coefficient in the aggregation formula 
that equates the two VaR. The correlation implied from VaR is given by: 
( ) ( ) ( )
2 2 2 2 2
1 1 2 2
12
1 2 1 2
=
2
port VaR x VaR x VaR
x x VaR VaR
r
- × - ×
× × × ×
            (2)   5 
If asset price changes are distributed according to a Gaussian distribution, then the same 
number  should  be  found  for  the  correlation  coefficient.  Especially,  correlation  should  be 
autonomous from the following parameters: 
-  The probability used to compute the VaR: p, 
-  The weights used to build the portfolio: x1 and x2, 
-  The type of position: long or short, 
-  The frequency used to measure asset price changes: f (under the i.i.d. assumption). 
Under normality, the implied correlation does not depend on the parameters listed above 
and  is  simply  equal  to  the  classical  Pearson  correlation  coefficient.  Whether  these  set  of 
propositions hold true is empirically tested in the next section. 
4. Empirical results 
The correlation implied from VaR is computed for a portfolio comprising the S&P 500 
and FTSE 100 indexes. Data are closing index values over the time-period from January 1, 1995 
to December 31, 2003. Two frequencies are used to measure price changes: daily and weekly. 
Different  probability  levels  are  used  to  compute  the  VaR:  80%,  95.45%,  98.46%,  99.23%, 
99.62% and 99.81% (for daily VaR) and 75%, 92.31%, 96.15% and 98.08% (for weekly VaR). 
These  probability  levels  correspond  to  average  waiting  time-periods  of  1 week,  1 month, 
1 quarter, 1 semester, 1 year and 2 years (for daily VaR) and 1 month, 1 quarter, 1 semester and 
1 year (for weekly VaR). Different weights are chosen to build the portfolios: (25%, 75%), 
(50%, 50%) and (75%, 25%). Two types of position are used: long positions and short positions 
in both indexes. The individual and portfolio VaR are computed with historical distributions. 
Table 1 gives the correlation implied from daily VaR (Panel A) and from weekly VaR (Panel 
B). A graphical representation is also given in Figure 1.   6 
Results indicate that the implied correlation tends to depend on the type of position (long 
or  short),  on  the  probability  level  used  to  compute  the  VaR  and  on  the  frequency  used  to 
measure  price  changes. However,  the  same  pattern  of  implied  correlation  is  obtained  when 
portfolio weights vary. 
Considering first results from daily VaR, implied correlation appears to be higher for 
long  positions  than  for  short  positions  and  particularly  for  more  extreme  probabilities.  For 
example,  for  an  equally-weighted  portfolio  (50%,  50%),  the  implied  correlation  for  long 
positions  is  higher  than  the  Pearson  correlation  coefficient  equal  to  0.42  while  it  is 
systematically lower for short positions. The results indicate that the implied correlation for long 
and short positions tends to diverge as the probability level used to compute the VaR increases 
(that  is  when  we  look  further  in  the  distribution  tails).  For  a  probability  level  of  99.62% 
(corresponding to an average waiting time-period of 1 year), it is equal to 0.516 for the VaR on 
long positions and 0.261 for the VaR on short positions.  
Implied correlation also depends on the frequency used to measure price changes. It is 
higher for weekly price changes than for daily price changes: 0.695 instead of 0.420 on average. 
Due to the lower number of weekly observations (divided by 5 compared to daily observations), 
the pattern of correlation implied from weekly VaR appears more erratic but once again, for 
high probability levels (or equivalently long average waiting time-periods), implied correlation 
is again much higher for long positions than for short positions. For example, for the equally-
weighted portfolio and for a probability level of 99.08% (corresponding to an average waiting 
time-period of 1 year), the implied correlation is equal to 0.744 for the VaR on long positions 
and to 0.358 for the VaR on short positions. As for daily data, at the highest probability level, 
correlation for long positions (crashes) is double that of short positions (booms). 
A similar pattern  of implied correlation is found for different portfolio weights. For 
example, for the highest probability level, implied correlation from daily and weekly VaR of 
long positions is systematically higher than those of short positions.   7 
5. Summary 
The  empirical  results  presented  in  the  previous  section  can  be  summarized  as  the 
following stylized facts:  
1)  Implied correlation is not constant, 
2)  Implied correlation tends to be higher for long positions than for short positions, 
3)  Implied correlation tends to increase (decrease) with the probability level for long 
(short) positions, 
4)  Implied correlation tends to decrease overall with the frequency of price changes. 
5)  Implied correlation behaves in a similar way for different portfolio weights. 
References: 
Dowd K., 2002, Measuring market risk, (John Wiley & Sons, Chichester). 
Jorion P., 2000, Value at Risk: The new benchmark for controlling market risk," 2
nd 
Edition, (McGraw-Hill, Chicago). 
JP  Morgan,  1995,  RiskMetrics
TM  -  Technical  Document,  3
rd  Edition  (see 
www.riskmetrics.com for updated research works). 
Longin F.M. and B. Solnik, 2001, Extreme Correlation of International Equity Markets, 
Journal of Finance, 56, 651-678.   8 
Table 1. Correlation between US and UK index price changes implied from VaR. 
Panel A. Correlation implied from daily VaR. 
Portfolio weights 
(25%, 75%)    (50%, 50%)    (75%, 25%) 
Probability 
(waiting period) 
Long  Short    Long  Short    Long  Short 
80% 
(1 week)  0.407  0.410    0.402  0.374    0.340  0.310 
95.45% 
(1 month)  0.387  0.368    0.451  0.397    0.533  0.425 
98.46% 
(1 quarter) 
0.351  0.403    0.412  0.403    0.430  0.571 
99.23% 
(1 semester)  0.466  0.571    0.597  0.291    0.572  0.158 
99.62% 
(1 year)  0.596  0.441    0.516  0.261    0.568  0.213 
99.81% 
(2 years)  0.511  0.138    0.484  0.209    0.345  0.214 
 
Panel B: Correlation implied from weekly VaR. 
Portfolio weights 
(25%, 75%)    (50%, 50%)    (75%, 25%) 
Probability 
(waiting period) 
Long  Short    Long  Short    Long  Short 
75% 
(1 month)  0.564  0.651    0.506  0.675    0.077  0.781 
92.31% 
(1 quarter)  0.643  0.656    0.622  0.737    0.446  0.564 
96.15% 
(1 semester)  0.565  0.714    0.737  0.858    0.794  0.965 
98.08% 
(1 year)  0.684  0.491    0.744  0.358    0.682  0.391 
Note: this table gives the correlation implied from daily VaR (Panel A) and from weekly VaR (Panel B) 
of long and short positions on portfolios composed of the S&P 500 and FTSE 100 indexes. Different 
probability levels (or equivalently average waiting time-periods) are used to compute the VaR. Different 
weights are chosen to build the portfolios.   9 
Figure 1. Correlation between US and UK index price changes implied from VaR. 
 
Note: this figure represents the correlation implied from daily VaR (Figure A) and from weekly VaR 
(Figure B) of long and short positions on an equally-weighted portfolio composed of the S&P 500 and 
FTSE 100 indexes for different average waiting time-periods (or equivalently probability levels) used to 
compute the VaR. 
Figure A. Correlation implied from daily VaR
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Measuring market risk,

RiskMetrics TM -

Implied Correlation from VaR

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Implied Correlation from VaR

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