Allocation of Risk Capital in Financial Institutions

Financial Management (Financial Management Association), Autumn, 1999 by Francesco Saita

B. Accounting for the Diversification Effect: Individual vs. Marginal CaR

A second problem concerning which measure of CaR to use for performance evaluation results from the imperfect correlations among different business unit returns. In fact, total CaR is generally less than the sum of the individual business unit CaRs would suggest. [16] Therefore, a question arises whether the amount of capital charged to each unit should take into account risk-diversification benefits or not.

The first solution that comes to mind is to split the diversification benefits equally over all the business units. We can see in Table 4 that total CaR after diversification, calculated as [17]

Total CaR = [square root of][[[sigma].sup.n].sub.i=1] [[CaR.sup.2].sub.i] [[[sigma].sup.n].sub.i=1] [[sigma].sub.j=1][CaR.sub.i] [CaR.sub.j] [[rho].sub.ij] (5)

where i, j are the business units of the bank and r measures the correlation between business unit returns, is equal to 1134.02. Since the sum of the individual CaRs is equal to 1900, diversification reduces capital-at-risk on average to 59.69% (i.e., 1134.02 divided by 1900) of its initial value. Therefore, individual CaR measures could be reduced accordingly (i.e., the splitting method).

By assuming an equal contribution of all businesses to diversification benefits, the splitting method is easy to apply; however, it is not correct. Merton and Perold (1993) have suggested that marginal risk capital should be used in this context. [18] Marginal risk capital is defined as the reduction in CaR that would be achieved if the unit under evaluation were not part of the firm (see Table 5).

Yet, by using marginal CaR some capital remains unallocated. Merton and Perold (1993) observe that despite this shortcoming, marginal CaR remains the right measure of each unit's contribution to total risk. Consequently, unallocated capital should not be split among the different units. This measure should in fact be considered the right measure to support acquisition or divestiture decisions. [19]

An alternative solution is to assign to each unit an amount of capital equal to total CaR after diversification, multiplied by the unit's "internal beta." The internal beta is defined as the ratio of the covariance between the returns of the individual business units ([R.sub.i]) and those of the bank as a whole ([R.sub.b]), divided by the variance of bank returns ([[[sigma].sup.2].sub.b]). In other words, the internal beta is equal to the beta of the single unit calculated by substituting the bank's portfolio of businesses for the market portfolio:

[[beta].sub.i,b] = Cov ([R.sub.i],[R.sub.b])/[[[sigma].sup.2].sub.b] = [[rho].sub.i,b] [[sigma].sub.i]/[[sigma].sub.b] (6)

where [[rho].sub.i,b] represents the correlation between [R.sub.i] and [R.sub.b] (see James, 1996, and Matten, 1996).

This solution is extremely interesting, both because it is consistent with capital budgeting methodology (using internal betas to rank projects competing for capital) [20] and because it enables the bank to allocate all its existing capital. [21]