Last week's Learning Curve covered the fact that volatilities given by some data providers can not be taken at face value but have to be transformed to the required reference or accounting currency. This can be a two-step conversion procedure if the exchange rates, and their volatilities, are not given with respect to the reference currency. The same holds for correlations. The conversion of correlations is more complicated as two market parameters are involved, i.e. the two market parameters to which the correlations refer. Here, two cases have to be distinguished: either one market parameter is given in the reference currency and only the other in a foreign currency, or both market parameters are given in a foreign currency.
In the first case, let the market parameter with price S1 be given in foreign currency, such as a French government bond in euros, the second market parameter S2 in the accounting currency, such as IBM stock in U.S. dollars. The foreign currency has exchange rate D with respect to the accounting currency, so D units of the accounting currency are obtained for one unit of the foreign currency. Then the price of the first market parameter in the accounting currency equals D S1. Making use of the linearity of the covariance we get for the covariance between the logarithmic changes of these prices and those of the second market parameter;
This can be expressed with volatilities and correlations;
Using last week's equation
for the volatility of the product D S1 finally yields after some rearrangements the required conversion;
In the next, more complicated case, let the market parameter S2 also be in a foreign currency. The two currencies may be different. The foreign currency of S1 has exchange rate D1 with respect to the accounting currency, the second foreign currency has exchange rate D2. Then the price of the first market parameter in the accounting currency equals D1 S1, and the price of the second market:
parameter equals D2 S2 correspondingly. Again, consider the covariance of the logarithmic price changes;
And re-write this in terms of volatilities and correlations;
Taking again last week's equation for the volatilities of the products D1 S1 and D2 S2 finally yields the required conversion equation;
A frequent occurance is two identical foreign currencies. For this case just set D1 = D2 = D in equation 6.
The transformations often have to be applied in practice. For example, in J.P. Morgan's Risk-Metrics the volatilities and correlations of market parameters are all given in the original currency and therefore have to be converted to the relevant currency. Even if the reference currency is U.S. dollars, the data of French interest rates or German Bunds has to be converted.
Until now exchange rates were already given with respect to the reference currency. If this is not the case then the exchange rates and their volatilities and correlations must first be converted to the reference currency before the equations are used for the conversion of other market parameters.
For this we need the correlations of cross rates. In last week's example; if D1=USD/EUR and D2=USD/JPY then the correlation of the yen exchange rate for the German investor, i.e. for EUR/JPY= D2/ D1, with a market parameter S (given in the reference currency) can be calculated using equation 3 with the substitutions D=D2, S1=1/D1 and and S2=S. The result is;
Similarly by substituting S1=1/D3 and S2=S in equation 6 the correlation between a cross rate D1/ D3 and a market parameter S not given in the reference currency but in a currency D2 is discovered.
And finally, the correlation between two cross rates D1/ D3 and D1/ D4 is discovered by substituting S1=1/D3 and S2=1/D4 in equation 6.
TO SUMMARIZE:
In a general scenario;
* first convert the volatilities of the cross exchange rates
to the reference currency,
* then the correlations between all cross rates,
* then the correlations between the cross rates and the
other market parameters
* and finally convert the volatilities of the market
parameters and the correlations between them.
Only after all those conversion have been carried out in the appropriate order can value-at-risk calculations or option valuations be started.
This week's Learning Curve was written by Dr. Hans-Peter Deutsch,a partner at Arthur Andersenand head of financial risk management consulting in Germany.
