A New Approach To Allocating Bank Capital

A few months ago, a high Citibank executive was quoted in this newspaper as saying that proper capital allocation is a critical imperative for banking institutions. Actually, it is one of three such imperatives. The other two are improved portfolio diversification and correct pricing.

Although sometimes addressed as if they were independent variables, these three imperatives are in fact closely intertwined. By improving their diversification, banks can reduce the risk of unexpected loss - the chance that losses in any given year will greatly exceed their average, or historical, magnitude. However, most banks remain underdiversified. And even if they greatly improved their diversification. they could not completely neutralize the risk of unexpected loss.

That being the case, banks must allocate enough capital to cover the risks that they either have not diversified away or cannot diversify away. And they must be able to price adequately enough to recoup the cost of this capital, which is equal to the rate of return the marketplace requires banks to earn on capital multiplied by the amount of this capital. Thus, the level of diversification helps to determine the risk of unexpected loss, and the risk of unexpected loss determines the need for capital. In turn, pricing behavior determines whether banks are earning enough on this needed capital to satisfy shareholder requirements.

Since pricing is implemented at the level of individual transactions, there must be a mechanism for allocating institutional capital to each such transaction. Unless this is done appropriately, risk-adjusted profitability cannot be measured, and banks cannot therefore be properly managed. Oliver, Wyman & Co. believes that existing methods of allocating capital are at best rudimentary. This article will present new thinking on how to improve these methods.

Flawed Methodologies

Currently, there are three widespread methods of allocating capital to individual loans and businesses. Some banks allocate a blanket 4%, which is the Tier 1 capital mandated by the regulators. Others vary the allocation based on a subjective determination of differential loan and business risks, resulting in an allocation that is either some fraction or some multiple of 4%. Still others attempt to measure the standard deviation of historical losses - the square root of the variance - by loan or business category and end up allocating capital proportional to one or more of these standard deviations.

None of these methods is especially satisfactory. Although banks cannot ignore the fact that the regulators demand at least 4% equity bankwide, assigning that percentage to every loan or business implies, of course, that they all have the same risk. Varying the allocation according to subjective judgment in effect denies the efficacy of measurement.

The third method - assigning capital according to historical loss volatilities - is also flawed. While it appears sensible to base the allocation on stand-alone loss volatilities, in reality such an approach is superficial. A given loan or line of business is part of the combined bank entity. Therefore, the relevant risk is not a stand-alone magnitude. It depends on the contribution made by that loan or business segment to the bank's overall risk of unexpected loss.

The Portfolio Benefit

Individual loans can be quite risky in isolation, but their addition to a portfolio can serve to reduce aggregate risk if the correlation of bankruptcy losses between these credits and the rest of the portfolio is less than one. (Correlations express the degree of relationship between two random variables. They range from minus one, perfect negative correlation, to plus one, perfect positive correlation.)

The fact that loans in combination may be less risky than the same loans in isolation is called the "portfolio effect." Oliver, Wyman thinks that banks should attempt to take into account this effect by adjusting the capitalization of each loan or business according to a method that measures both relative loss volatilities and loss correlations.

The first step is to determine the amount of capital that would be required on a stand-alone basis. As has been noted, this can be done by computing the standard deviation of losses from their mean. If loan losses are normally distributed (can be described by the traditional bell-shaped curve), a capital allocation equal to one standard deviation will protect the bank against the unexpected-loss risk of the loan, considered in isolation, 84% of the time (provided, of course, the future replicates the past). An allocation equal to three standard deviations will provide almost complete protection.

Unfortunately, the distribution of loan losses is not normal, or bell shaped, because these losses cannot fall below zero.

Given that the actual distribution is skewed to the right, the number of standard deviations needed for complete protection is greater than would be the case for a normal distribution - a circumstance requiring banks to be extremely conservative in allocating enough capital to cover stand-alone loss volatility.

The second step is to adjust the stand-alone requirements for the portfolio effect. Stated in a slightly simplified way, this means multiplying the stand-alone results by the coefficient of correlation between losses in each loan or business segment and losses in the overall portfolio.

In actual fact, the methodology is a bit more complex. The contribution of any given loan segment to the overall variance of the bank portfolio losses depends on (1) the correlation of its losses with those in the portfolio; (2) its size in relation to that of the portfolio; and (3) the raw volatilities (standard deviations of the loan segment and the portfolio).

These elements are linked in a statistical formula for combining the standard deviations of two variables. In principle, all parts of the formula are readily calculable.

A Tale of Two Banks

Consider two hypothetical banks - a typical North Carolina and a typical Arizona institution. Say that each has mining loans equal to 5% of the portfolio. Given that the relative size of the loan segment is the same for both banks, the contribution of mining loans to overall portfolio volatility depends on only two factors - the relative standard deviations of losses and the respective loss correlations.

Suppose that the standard deviation of mining losses were equal to the standard deviations of losses in both the typical North Carolina and the typical Arizona bank. And suppose further that the correlation between mining losses and losses in the portfolios of both banks amounted to plus one. Then there would be no portfolio effect - neither benefit nor loss - from adding the mining loans to either portfolio.

Actually, the raw volatility of mining is much greater than that of either bank (standard deviations of loss of 0.533% vs. 0.095% in North Carolina and 0.533% vs. 0.165% in Arizona). By itself, this would suggest that mining loans increase bankwide loss volatilities.

But the correlation of losses between mining and each bank portfolio is considerably less than plus one. The loss correlation with the typical Arizona portfolio is 0.54, while that with the North Carolina bank is even less, 0.35. This means that the factors contributing to defaults in each portfolio are not likely to operate with the same intensity on the mining segments of these portfolios.

Reducing Simultaneity

This correlation effect swamps the relative volatility effect and contributes to a very substantial portfolio benefit. That is, adding mining loans to portfolios with which they are much less than perfectly correlated greatly reduces the likelihood of simultaneous defaults and thus lowers bankwide portfolio volatility.

Applying the formula yields the following numbers. The North Carolina bank would have to allocate capital to its mining segment equal to only 47% of what would be needed if these loans were capitalized on a stand-alone basis. Since mining loan losses are somewhat more strongly correlated with losses in its portfolio, the Arizona bank would allocate capital equal to 59% of what these loans would require in isolation. The capital allocations of all industries in the two portfolios should be adjusted according to the same formula.

Unfortunately, the formula cannot be mechanically applied. That's because one has to correlate mining losses with those in a preexisting portfolio. And if that preexisting portfolio is assumed to be one currently held by a typical North Carolina or a typical Arizona bank, this implies that the mining loans were the last to enter these portfolios, which may not necessarily be the case.

In other words, all correlations are order-dependent. They are a function of the particular mix of the portfolio at the time any given loan is added. If one assumes that the mining loans were indeed the last to enter the portoflio, the numbers churned out by the formula are correct. But if one assumes that the mining loans were the first to enter, there is nothing else with which to correlate mining losses, and the contribution of mining loans to the bank's risk of unexpected loss is self-evidently equal to their stand-alone loss volatility.

The way to solve this portfolio sequencing problem is to allocate capital to each loan segment 50% according to our formula and 50% according to the segment's stand-alone loss volatility. That is, capital should be based on the equal probability of last-in and first-in entry. This wrinkle has the added benefit of solving the problem of what to do when there is a negative correlation between a loan segment and the portfolio. Negative correlations suggest negative capital, a result that may be theoretically sound but practically unacceptable.

The economic justification for this 50-50 weighting is that the sum of the incremental volatilities of each industry segment in the economy, if it is assumed to be the last to enter a portfolio composed of all industries, weighted according to their respective importance, is equal to about half the observed volatility of the total portfolio.

Using the Loan-Sales Market

It can be readily seen that the determination of correct capital allocations has profound implications for loan pricing and competitive advantage. If the North Carolina bank needs less capital to backstop mining loans than the Arizona bank, it can either offer lower prices on these loans or extract a higher rate of return at prevailing prices. This circumstance suggests that the Arizona bank should not seek to originate or purchase mining loans. Rather, it should gravitate to those loan segments whose mesh with its portfolio is such as to permit lower capital allocations and more competitive pricing.

In practice, banks rarely evaluate their overall portfolio position before originating and/or purchasing loans. Decisions are instead based on origination expertise and loan availabilities. But even if the Arizona bank has a comparative advantage in originating available mining loans, the data suggest that it does not have a similar advantage in funding these loans.

It should therefore use the loan-sales market to offload its mining loans to those institutions that can fund the loans more advantageously because of lower capital costs. By the same token, it should arrange to purchase loans that require less capital support, presumably because the correlation between their losses and those of the overall Arizona portfolio is weaker than is the case for mining loans.

Key to Proper Pricing

The above approach should enable banks to price loans according to true economic risks as well as to calculate valid returns in all component businesses. To be sure, given the glut of existing lending capacity, it is probably a fallacy of composition to argue that the industry as a whole can raise risk-adjusted loan compensation to adequate levels.

Nevertheless, individual banks may succeed in doing so, especially since the market for many types of bank loans remains reasonably inefficient - that is, there is a wide range of market-clearing loan prices. But no individual bank can achieve its pricing goals unless it has a pricing model, an indispensable component of which is a capital allocation methodology similar to the one we have just reviewed.