VaR gives a solid foundation for assessing the amount of capital that should be held by a bank to protect it in the case of losses arising from market risks. There are three reasons for analyzing the capital consumed by market risks:
• Complying with industry regulations
• Calculating economic capital to control the bank’s default probability
• Measuring Risk-Adjusted Profitability
Each of these measures can be based on VaR, as we will describe below. At the end of this chapter, we also note that VaR can be used to manage the institutional-level risk, not only for banks, but also for asset managers.
In 1996, the Basel Committee on Banking Supervision recommended to national regulators that the minimum capital to be set aside for market risks should be based on VaR.1 Under normal circumstances with a VaR Calculator that is functioning well, the regulatory capital should equal the 99% 1-day VaR, multiplied by 3, times the square root of 10:
The square root of 10 represents the losses that could occur over a 10-day holding period, e.g., if there was a 10-day crisis in which the markets became illiquid. The factor of 3 allows 3 such events to happen each year. If the back-testing shows an unusual number of exceptions, the capital should be increased according to Table 8-2 in the previous chapter.
In calculating the VaR, each bank is allowed to use whatever method it thinks will best pass the back-test. However, the model selected must be used consistently, and it must be the primary model for reporting and managing risks within the bank.
To be allowed to stay in business by the regulators, banks must hold at least the capital discussed above. However, if banks wish to maintain a high credit rating and a low probability of default, they may wish to hold more capital. This amount is calculated using the Economic Capital framework. Chapter 2 discussed the connection between a bank’s economic capital and its default probability. In that chapter, the economic capital for market risks was given as:
Where rf is the risk-free rate, and Wp is the maximum probable loss such that there is only probability p that the profitability over a year will be worse than Wp:
VaR gives us the maximum probable loss that could happen with a 1% probability over one day. To get the economic capital, we need to translate from 1% to the required confidence level and from one day to one year.
If banks only held enough capital to have a 1% chance of avoiding default, they would be rated around BB or BBB. If banks want to be more secure, they must increase the amount of capital.
If we assume that the losses are Normally distributed, then if a bank held capital equal to 2.32 times the standard deviation of the possible annual losses, there would be a 1% chance that the actual loss over 1 year would exceed the capital. If the bank held capital equal to 3.7 times the standard deviation of losses, there would only be a 0.01% chance that the actual loss would exceed the capital. The bank could therefore expect to be rated AA or AAA. Table 9-1 gives the relationship between the amount of capital that a bank holds against market risks and the consequent probability that the actual losses would be greater than the capital.
Typically, A-rated institutions have a default probability of around 0.1%. From Table 9-1, we can say that an A-rated institution should aim to hold capital equal to 3.1 times the standard deviation of the potential annual change in value (assuming a Normal distribution).
Economic CapitalA = 3.1 × σ1 year
TABLE 9-1 The Relationship between the Amount of Capital Held against Market Risks and the Associate Probability of the Banks Defaulting
Now we need to find the standard deviation of the potential annual change in value. We already know that 99% VaR is 2.32 times the standard deviation of the potential daily change in value:
VaR = 2.32σ1 day
Now we need a way to scale from daily standard deviation to annual standard deviation. This can be crudely approximated by assuming that losses are uncorrelated from one day to the next, and that the VaR is constant during the year. With these assumptions, we can use the familiar “square-root-of-T”2 approximation assuming 250 trading days per year:
Bringing these three equations together, we can estimate economic capital as a function of VaR:
This is more than twice as high as the regulatory capital. One reason for the discrepancy is that banks may wish to be more creditworthy than is implied by the minimum regulatory capital. Another reason is that the regulatory capital implicitly includes a reduction due to diversification between market risks and the other risks in the bank, e.g., credit risk (this is covered in Chapter 25). A further reason is that we assumed that the standard deviation of losses scales with the square root of time. This assumption breaks down for two reasons. The first is that over long periods, such as a year, effects such as mean reversions manifest themselves. The second reason is that it assumes that the trader holds a fixed position all year and takes whatever losses the market brings. With the trader’s skill, and with bank’s policies to stop losses, this should not be the case.
The only tractable solution to including the mean reversion and management effects is to use Monte Carlo simulation. This approach is explored in detail in an article entitled. “Changing Regulatory Capital to Include Liquidity and Management Intervention,” Marrison, C.I., Schuermann, T.D., and Stroughair, J., The Journal of Risk Finance, August, 2000. The approach in this article creates many simulations of price changes for a year and the bank’s response to these changes. By combining the market movements and the bank’s expected response, we can relate the one-day VaR to the required economic capital. The scaling between the two is strongly dependent on the bank’s stop-loss policies and the liquidity of the instruments, but in general, the required economic capital for an A rating was found to be between one and two times the minimum regulatory capital.
Once the capital has been established for the portfolio as a whole, it can be allocated to the individual traders, subportfolios, transactions, or desks, using the VaRC methodology. The capital allocated to an individual transaction is the VaRC for the transaction, divided by the VaR for the whole portfolio, and multiplied by the capital for the whole portfolio:
Once we know the capital being consumed by a particular transaction, we can calculate the Risk-Adjusted Return on Capital (RAROC) and the Shareholder Value Added (SVA), as explained in Chapter 2. Knowing the risk-adjusted performance, we can properly decide on whether the profit from a transaction is worth the risk and whether a trader is performing well or just risking a lot of the bank’s capital.
The RAROC is simply the net income from the transaction divided by the capital consumed:
If the transaction only lasts a few days (T) and only requires capital for that time, the calculated RAROC should be annualized so that it can be compared with the RAROC from other transactions:
The Shareholder Value Added is calculated by subtracting the hurdle income from the expected or actual income. The hurdle income is the amount of capital consumed multiplied by the hurdle rate that the shareholders require as a minimum return for risking their capital. The hurdle rate HT, for a short-term transaction lasting T days can be calculated from annual hurdle rate, HAnnual:
The required income is then the capital consumed multiplied by HT:
Required Income = Allocated Capital × HT
The SVA is then the net income minus the required income:
SVA = Net Income - Allocated Capital × HT
Convincing traders that their bonuses should be reduced according to Allocated Capital × HT is left as an exercise for the reader.
Asset managers, such as mutual funds, have two sets of risks to control: risks to their fund holders and risks to their shareholders. Their fund holders suffer losses directly if the value of the funds falls. VaR, as a measure of portfolio volatility, is perfectly suited to measuring this risk and can be used without any modification. The shareholders suffer losses if the net earnings of the asset management company fall. The earnings are typically partially tied to the performance of the funds, but also depend on costs and on the fees charged.
The volatility of the earnings, the source of earnings volatility, and the risk-adjusted profitability can be calculated by using VaR augmented with information on the structure of fees and costs. As this book is focused on risk measurement for banks, we will not spend any more time here discussing risk measurement for asset managers, but simply note that it can be done, and for further information see “Institution-Level Risk Measurement for Asset Managers,” Marrison, C.I., Risk, pp. S26–S28, September, 2001.
In this chapter, we showed how VaR can be used to calculate the amount of capital that should be held by a bank to protect it in the case of losses from market risk. Next, we will explore VaR’s limitations and explain how to minimize them.
1 Amendment to Capital Accord to incorporate market risks, Basel Committee on Banking Supervision, January 1996.