# LO 48.7: Explain challenges in modeling diversification benefits, including

LO 48.7: Explain challenges in modeling diversification benefits, including aggregating a firms risk capital and allocating economic capital to different business lines.
The overall risk capital for a firm should be less than the total of the individual risk capitals of the underlying business units. That is because the correlation of returns between the business units is likely to be less than +1. Such risk reduction due to diversification effects over risk types and business activities is very difficult to measure in practice. Instead of using an extremely high overall confidence level for the firm, the various business units may use lower confidence levels to avoid an excessively high aggregate risk capital amount.
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2018 Kaplan, Inc.
Topic 48 Cross Reference to GARP Assigned Reading – Crouhy, Galai, and Mark, Chapter 17
For example, assume a firm is subject to only the following four types of risk (risk capital amounts are provided for each risk): Market risk = $400 Credit risk =$300 Liquidity risk = $200 Operational risk =$500 Aggregate risk capital for the firm could be as high as $1,400 assuming a perfect correlation (i.e., sum of the four risk capital amounts). Or it could be as low as$734 assuming zero correlation (square root of the sum of squares of the four risk capital amounts). In taking into account the diversification effects, the firms overall VaR should be computed as some value between $734 and$1,400, which is a very wide range. In addition, there is a lot of subjectivity involved in allocating the diversification benefits back to the business units in a fair manner especially since the allocation will impact the respective business units performance measures (i.e., reduction of risk capital required).
It makes sense that a business unit with earnings or cash flows that are highly correlated to the overall firm would need to be allocated more risk capital than a business unit with earnings or cash flows that are negatively correlated (assuming similar volatility). Having business lines that are countercyclical in nature allows the overall firm to have stable earnings and to attain a given desired credit rating using less risk capital. In practice, the easiest allocation method is a pro-rata allocation based on standalone risk capital amounts.
For example, assume the following information pertaining to a business unit that engages in only two activities, A and B: Activity A alone requires $50 of risk capital Activity B alone requires$60 of risk capital Activities A and B together require a total of $90 of risk capital Stand-alone capital looks at each activity independently and ignores any diversification benefits. Therefore, the stand-alone capital for Activities A and B are$50 and $60, respectively. The stand-alone capital for the business unit is$90. $10.9 is allocated to Activity B. Therefore, Activities A and B have fully diversified capital Fully diversified capital takes into consideration the diversification benefits, which equal$20 ($50 +$60 $90). For simplicity, the diversification benefit can be done on a pro-rata basis as follows: ($20 x $50) /$110 = $9.1 is allocated to Activity A and ($20 x $60) /$ 110 = $10.9 is allocated to Activity B. Therefore, Activities A and B have fully diversified capital of$40.9 and $48.1, respectively. Fully diversified capital should be used to determine a firms solvency and to determine the minimum amount of risk capital required for a given activity. Marginal capital is the extra capital needed as a result of a new activity added to the business unit. Diversification benefits are fully considered. The marginal risk capital for Activity A is$30 ($90 total$60 for Activity B) and the marginal risk capital for Activity B is $40 ($90 total $50 for Activity A). Total marginal risk capital ($70) is below the full risk capital of the business unit (\$90). The general method for computing marginal capital of a new activity is to start with the total risk capital required for the business unit minus all of the risk capital required for the other activities. Marginal capital is useful for making active portfolio management and business mix decisions; such decisions need to fully consider diversification benefits.
2018 Kaplan, Inc.
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Topic 48 Cross Reference to GARP Assigned Reading – Crouhy, Galai, and Mark, Chapter 17
In a performance measurement context, stand-alone risk capital is useful to determine incentive pay and fully diversified risk capital is useful to determine the incremental benefit due to diversification. In allocating the diversification benefits, caution must be taken especially since correlations between the risk factors usually change over time. In a more extreme situation such as a market crisis, correlations could move to 1 or +1, thereby reducing diversification benefits.
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