Business — Banking — Management — Marketing & Sales

The goal of risk-based pricing is to ensure a minimum target return on capital, in line with shareholders requirements. A hurdle rate serves as a benchmark for risk-based pricing and for calculating creation or destruction of value with Shareholders Value Added (SVA). It is the target rate of return on existing equity, or the target return on economic capital. Moreover, if the average risk-adjusted return of the portfolio is higher than the minimum, it makes sense to set the hurdle rate at this level.

Risk-based Pricing Requires Marginal Risk Contribution

The existing portfolio is P. Economic capital is a multiple of ct_p ,or m x ct_p ,and m is supposed to be constant. The required pre-tax pre-operating expense return is r%. The target revenue is r % x m x ct_p .A portfolio has a loss volatility of 100. The multiple capital/loss volatility is 2. The hurdle rate is 25% pre-tax and pre-operating expense. The target net revenue is 25% x 2 x 100 = 50. A new transaction increases the portfolio loss volatility to 110. The marginal risk contribution to loss volatility of this additional new transaction is 10, and the approximate new capital of this transaction is 2 x 10 = 20. The target pre-tax net income on the new portfolio becomes 25% x 220 = 55. This is an increase of 55 — 50 = 5. This increase is exactly 25% x (220 — 200) = 5. Pricing on a different basis than marginal risk contribution will not maintain the portfolio return.

General Formulation

We use the previous notation. The initial portfolio is P and it becomes P + f, f being the new transaction. By definition, MRCf = ct_p+f — ct_p . What should be the return required for the additional facility f ? The hurdle rate is k% on the capital of the existing portfolio. We assume that capital K is a constant multiple m of the loss volatility with K = m x ct_p .

We assume that the existing portfolio provides exactly r, and we look for the minimum return rf on the new facility f guaranteeing that the return of the portfolio P + f will remain at (at least) k%. The required income in value is k% x m x ct_p+f. The return without the new facility is k% x m x ct_p . The required return on the new facility rf %is such that the return on the portfolio P + f, summing the income on the existing portfolio plus the income on the new facility, is k% x m x ct_p+f:

Dropping the multiple m implies that r/ % > k%. Consequently, the rule is that the risk-adjusted return on marginal risk contribution to volatility, with a constant multiple m, should be at least k%. The implications are:

• The relevant basis for risk-based pricing is the marginal risk contribution to loss volatility.

• Since the absolute risk contribution ARC to the loss volatility is higher, pricing on ARC would overestimate the required revenue. ARC is not the right risk measure for pricing purposes.

The assumption of a constant multiple m is acceptable subject to the size of the facility and its correlation with the rest of the portfolio. New facilities are generally small compared to the size of the portfolio. However, the assumption collapses when considering adding or removing risk concentrations, such as subportfolios, business lines or even single large concentrations of risk. Then, we need to revert to a direct with and without calculation of the marginal risk contribution to capital.

The Pricing Paradox with Risk Contributions

The usage of MRC lower than ARC for pricing purposes sounds puzzling at first. It means that MRC-based pricing is lower than ARC-based pricing. This does not sound consistent with an overall target return on capital, because capital sums up all ARC and not all MRC! This is the pricing paradox of using marginal, rather than absolute, risk contributions.

An example will help us to understand the paradox. Suppose the marginal risk contribution to capital is 15 and the required rate is 20%, resulting in a target income of 20% x 15 = 3. When the facility enters into the portfolio, it has a new absolute risk contribution, for instance 17, higher than its marginal risk contribution of 15. The revenue of 3 does not provide any more than 20% on 17. How could the revenue of 3 maintain the overall portfolio return?

In fact, 3 is the additional revenue required to compensate the overall portfolio capital after inclusion of the new facility. Using fictitious numbers, suppose that all absolute risk contributions before including the new facility sum to 100. After inclusion of the new facility, all absolute risk contributions sum to 115, since the MRC, 15, is the incremental overall capital. If the new facility now has an ARC of 17, this implies that the sum of all absolute risk contributions over all facilities existing before the inclusion of the new one drops from 100 to 98. This is the only way for the new capital to be 115, given that the incremental capital is 17, since 98 + 17 = 115. The new facility increases the overall risk incrementally, but simultaneously diversifies the existing risks. This solves the pricing puzzle. Pricing according to the absolute risk contribution of the new facility ignores the fact that all other absolute risk contributions of existing facilities decrease. On the other hand, pricing on marginal risk contribution captures both the incremental risk and the increased diversification or, equivalently, the decrease in all existing facility absolute risk contributions.