This shows that risk gets depressed by the effect of correlation. There is nothing new about this, though the formulation is a fancier way of expressing what is already known by portfolio managers and traders.

Another trivial observation is the market is that profits are perfectly additive. That is, profits from trading asset A and asset B can be linearly added up to give the total profit of the portfolio made up of these two assets. If pi(A) is the profits from trading (or holding) asset A and pi(B) is the profits from trading (or holding) asset B then for the portfolio comprising A and B the total profits is simply given by:

Π_Portfolio = Π_A+B = Π(A) + Π(B)

However, what happens when we try to find out how have these profits have been scaled for the risk undertaken by the trader or the manager. In other words, rather than looking at the Dollar value of profits in trading assets A and B, let us ask the question: what is the risk adjusted return on capital for each asset A and B. Clearly a certain amount of risk is taken in trading (or holding) any asset which either generates a profit or loss. If a profit is generated then it is essential to find out whether it was worthwhile taking on that risk to generate that profit. If too large a risk was taken to generate a small profit then the transaction was not worth it; on the other hand if a small risk was taken to generate a very large profit then one could invoke the "principle of luck". In any case, profits should be always scaled by the risk taken to generate that profit.

This is again nothing new as Sharpe ratio does precisely that. Also, another very similar but perhaps more powerful concept is that of RAROC (risk adjusted return on capital).

RAROC, in its simplest form, assumes that VaR is the risk capital which should be used as the denominator to calculate the risk adjusted return. In fact, for all traders and fund managers it is RAROC which is really pertinent and not the simple return on equity or return on capital. Both ROE and ROC could be grossly misleading in terms of profiling the risk-reward characteristics of the trader and the manager. As a matter of fact, even for financial institutions and banks, it the RAROC that is the ultimate and correct barometer of success and profitability.

Now we come back to our original problem: Is it worth it to combine the treasury divisions of these two banks? As we said the answer lies in the correlation.

Let us once again encapsulate the formulas: