Everyone is familiar with the idea of return on capital (ROC). If an investor makes an investment, he wishes to be sure of a good return on the capital invested. When the capital is being used, for example, to buy shares, then calculating the return is a simple business. Once we get on to derivatives, however, the situation is more complex. The amount of capital 'invested' in a derivatives portfolio is that amount considered necessary to support the market and credit risks.
To understand how to calculate the amount of capital which needs to be allocated to support these risks, we can consider a hedged derivatives portfolio. Such a portfolio is immune to market risk, unless a default occurs on one of the deals. Let us assume, somewhat simplistically, that all the deals are in matched pairs, hedging each other with a small but safe profit margin in between. At the time the deal and its hedge were put on, both had a market value of roughly zero. However, as time goes on, a deal like a swap may radically change its value. If, for example, I had agreed to borrow at 5% for 5 years, and the borrowing rates for that period fell to 4%, my deal would have decreased in value considerably. Thus, if one half of a matched pair of deals is suddenly eliminated by a default, the holder of the portfolio is still obliged to maintain the opposing cashflows, which may be very expensive to do. Thus market risk enters via default risk.
The risk of loss depends upon
(1) the probability of default (credit risk)
(2) the loss in the event of default (market risk)
For the manager of a derivatives portfolio, the capital question is critical. It is necessary to allocate sufficient capital to the portfolio to absorb losses when they do occur, but allocating too much capital will mean investors receive a poor ROC. Thus calculation of the 'right' amount of capital is extremely important. It is worth noting that poor calculations lead to over-capitalization, as approximations must always err on the conservative side.
In the youth of the derivatives industry, deal spreads were wide, and only crude calculations were necessary to show that a given deal was generating a satisfactory return. However, in the current mature market, deal spreads are much tighter. Only the most accurate calculations will now suffice to give investors an idea of the true ROC of a deal. Moreover, a single derivatives deal at market spreads will never give an adequate ROC. This is nothing to do with fixed costs like communications and computers. Unless the deal is considered in the context of a large portfolio, the capital necessary to support the risk is always unreasonably large. Once the deal is considered as being added to a large portfolio, however, it becomes possible to look at the incremental capital needed which will be added to the capital already allocated to the portfolio. This is inevitably much less than that needed to support the deal on its own, due to the netting diversification effects within the portfolio.
MARKET AND CREDIT RISK CALCULATIONS
How would we go about calculating the amount of capital which we need to allocate? We want to combine the market and credit risk distributions to find the 'expected' or average loss due to default on a single deal, and allocate that much capital to the deal. However, what if the average loss in any one year is $1,000,000, but there is a 30% chance that there will be a loss of $10,000,000? Obviously, unless we know very accurately what the expected loss will be, we need to allocate additional capital to cover our uncertainty. This additional capital (sometimes called economic capital) will depend upon the loss distribution obtained from combining the market and default risks.
The market risk distribution is similar to the normal distribution, but with fatter tails. The default risk distribution is skewed, with non-zero probabilities of very high losses, shown in Figure 1(a). The combination of the two has the worst of both worlds, as illustrated in Figure 1(b), which also shows the relationship of economic capital to the risk of loss distribution. Several standard deviations of the risk of loss distribution are used to determine the amount of capital allocated to the portfolio, depending upon the safety level desired. For example, if the safety level desired was 99.97% (equivalent to an AA rating) then capital equal to about 3.5 standard deviations would be necessary, assuming the risk of loss distribution is normal. In fact, more (6-7) may be needed due to the fat tails. A safety level of 99.97% means that in any one year there is only a 0.03% chance losses will exceed the total capital held.
How is this calculation done in practice? While a Monte Carlo simulation would be ideal, lack of default data and computing power mean it is not an automatic choice for all but very large institutions. What tends to happen is that market risks are calculated separately and then multiplied by a factor dependent upon the credit status of the counterparty. These methodologies, however, will be addressed more fully in a subsequent article.
This week's Learning Curve was written by Jessica James, from First Chicago NBD's strategic risk management group in London.
