Combining Portfolios to Increase RAROC - Is that Why Banks Always Merge?
Team Latte
Mar 07, 2005

Take two very ordinary, underperforming bank treasury & capital markets division. Suppose the treasury of bank A, which specializes in and trades G7 FX options generates US$9 million in annual profits on a capital of $100 million and has a volatility (of the G7 currencies in general and on a basket basis) of 15%. The treasury of another bank B, which specializes in and trades equity options and swaps generates US$10 million in profits on a capital of $80 million and has a volatility (of the basket of the instruments) of 25%.

Let us ask a purely theoretical question: Does it make sense to combine the two treasury divisions of these banks and one unit as a whole?

In practice this is ridiculous and non-workable but a far more practical and workable question is what if the treasury divisions were the entire banks themselves then would it be worthwhile to combine the two banks?

What are the benefits of combing these two operations? And why should it be done? The answer lies in the correlation and cuts through the heart of portfolio theory. Understanding how returns are measured against the risks taken to generate them is quintessential to not only understanding asset allocation models at the trader or fund manager level but also unraveling how banks, hedge funds and fund of funds operate in terms of capital allocation and trading strategies.

Let us approach the problem from a microcosm of two assets, A and B and a simple portfolio with a combination of A and B. Remember, A and B could eventually be two funds, two portfolios themselves or two banks.

It is a trivial observation in the markets that if two assets which are not perfectly positively correlated are combined then the non-systemic risk is diversified away. Under constant volatility assumption, if the two assets have perfect negative correlation then maximum risk diversification is achieved. Further, as long as the two (or more than two, in a general case) assets have less than perfect positive correlation risk will always be diversified away if they are combined in a portfolio.

Risk, which is defined and measure by the volatility of the asset or the asset return, is diversified away because of the mathematical property of non-additivity. Volatility, and therefore, risk, cannot be linearly added together. You cannot add the risk of an asset A and asset B simply as the sum of the two risks.

Risk(A + B) ≠ Risk(A) + Risk(B)

In fact, if asset A and asset B are added together then the sum of the risks would be given by:

Risk(Portfolio)=Risk(A + B) =

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:

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.

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