Now,
Π(A+B) = Π(A)+Π(B)
However, if
ρ ≠ +1, then
VaR(A+B)<VaR(A) + VaR(B)
In other words, less than perfect positive correlation will always make the RAROC, the risk adjusted return on capital of the portfolio dominate the RAROC's of the individual assets (or in our problem, the treasuries of each bank).
Given the above the RAROC of treasury A is equal to 60% and the RAROC of treasury B is 50%.
If the correlation between the two respective treasury divisions is 0.6 then the RAROC of the combined operation will be 60.54%. If the correlation is 0.2 then the RAROC will be 69.61% and for correlation equal to 0 (zero) the RAROC of the combined operation will be 76%. For negative correlation between the two treasuries (two assets) the RAROC of the portfolio will improve further reaching a maximum at correlation equal to -1 (minus one) when the RAROC of the combined operation will be a whopping 380%!!
Therefore, for imperfect correlation between two assets and in our case between any to sets of portfolio (two treasury divisions of two banks) will make the combined portfolio RAROC completely dominate the individual RAROCs of the assets (sets of portfolios) and therefore it makes every sense to combine the operations.
Therefore, if you take a Bank (a hedge fund) as a whole portfolio and another Bank (hedge fund) as portfolio the same will apply when we would analyze the RAROC of the merged entity. In most cases underperforming banks (funds) can combine with other banks (funds) with imperfect correlation to improve the overall risk adjusted return of the merged entity. As a matter fact, correlation plays such a central role in the VaR and the RAROC formula that mapping portfolios onto a P&L-correlation map can form the main exercise of strategic planners in banks.
Disclaimer
"Risk Latte uses proprietary and non-proprietary mathematical and empirical models to measure the volatility and estimate the direction of the market. There is no guarantee of any particular outcome happening and readers must exercise caution while interpreting the conclusions of this article. Risk Latte Company is not a registered stock broker or an SFC registered entity and readers must take advise from their financial advisors, stock brokers, research analysts and bankers while making any buying or selling decisions. Risk Latte Company is not in the business of making stock or asset forecasts whether explicitly or implicitly and shall not be responsible for and/or liable for any losses arising out of any trading decisions based on the above article."
  
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