行政院國家科學委員會補助專題研究計畫成果
美國財務會計準則公報 119 號數量性揭露 VAR 預測美國商業銀行利
率風險之有用性
計畫類別：個別型計畫
計畫編號：NSC90-2416-H-002-011-
執行期限：20010801∼20020731
計畫主持人：
劉啟群執行機構：國立臺灣大學會計學系暨研究所
中 華 民 國 91 年 10 月 15 日
Abstract
Jorion (2002) documents that the VAR numbers publicly disclosed by 8 major banks are informative to predict subsequent fluctuations in banks’ trading revenues. Using a relatively larger sample of U.S. commercial banks, this paper investigates (1) whether the quarterly-average VAR disclosures are associated with the variability of current trading revenues and returns (i.e., validation or assessment of VAR statistical accuracy) and (2) whether the end-quarter VAR disclosures are useful to predict the variability of future trading revenues and returns (i.e., predictive value of VAR). The empirical results support that the quarterly average (end-quarter) VAR disclosures provide incremental information for assessing (predicting) the volatility of current (future) trading revenues and returns. Furthermore, since the models estimating VAR numbers involve many different risk measurement methods and assumptions, we argue that larger banks tend to exhibit greater professional expertise and more resources of money, and banks are able to improve their VAR models through learning over time. In other words, we hypothesize that banks size and time horizons significantly affect the accuracy of predicting and assessing power of VAR disclosures. Consistent with the hypotheses, we find that the predictive and assessing powers of the VAR measures are better in the later period than early years, and the VAR disclosures of bigger banks have better predictive and assessing powers than smaller banks.
Keywords: derivatives; market risk disclosures; disclosure regulation; value at risk; volatility; bank size; time horizon; Basel Committee; SEC
Data available: The data used in this study are publicly available from the sources indicated in the text.
I：Introduction
Value at Risk (VAR) was developed in 1993 in response to those famous financial disasters such as Metallgesellschaft, Orange County, and Barings Bank. These problems led to a serious of reports on good risk management practice. The most significant was a report on derivatives risk management published in July 1993 by the Group of Thirty. This report was followed by a report by the US General Accounting Office in May 1994, a joint report issued by the Bank for International Settlements (BIS) and the International Organization of Securities Commissions (IOSCO) in July 1994, and many other reports by the Derivatives Policy Group, the International Swaps and Derivatives Association (ISDA), Moody’s, Standard and Poor’s, and other interested parties. These reports made a number of sensible recommendations that have rapidly become benchmarks for modern financial risk management. They tended to focus particularly on derivatives risks. But their recommendations apply not just to derivatives providers and investments, but to other financial institutions and most corporations as well (Kevin 1998).
VAR measures the potential loss that could occur under normal market conditions at a given confident level. It is an effective tool for describing and communicating risk because it assesses different risks in terms of a common loss relative to a standard unit of likelihood. To calculating VAR, we need to choose a time horizon and a probability. For instance, a bank might say that the daily VAR of its trading portfolio is $1 million at the 99 percent confidence level. In other words, under normal market conditions, only one percent of the time, the daily loss will exceed $1 million. Besides, VAR models typically depend on the past being a good guide to the future. They tend to build around statistical estimates, using probability distributions for the changes in value of a given portfolio over a specific holding
period. Because VAR has considerable intuitive appeal as a condensed big picture of risk exposure, it has gained significant support primarily from the regulatory sectors and has been touted to senior management as a risk control device.
The Securities and Exchange Commission (SEC) required all large U.S publicly traded corporation to report quantitative and qualitative information about market risk in financial reports (SEC, Financial Report Release No. 48, 1997). Three disclosure formats are suggested:
1) Tabular presentation of fair value information and contract terms relevant to determining future cash flows, categorized by expected maturity dates.
2) Sensitivity analysis expressing the potential loss in future earnings, fair values, or cash flows from selected hypothetical changes in market rates and prices.
3) Value at risk disclosures expressing the potential loss in future earnings, fair values, or cash flows from market movements over a selected period of time and with a selected likelihood of occurrence.
Companies may choose one of three alternatives for all of the required quantitative disclosures about market risk. A company may choose one disclosure alternative for market risk sensitive instruments entered into for trading purposes and another alternative for all other market risk sensitive instruments. Also, a company may choose any of the three disclosure alternatives for each risk exposure category within the trading and other than trading portfolios.
Another use of VAR is in the assessment of regulatory capital for market risk. In the Bank of International Settlements (BIS) internal models approach, regulatory capital for market risk is proportional to the VAR that is calculated with a firm’s internal model under certain general parameters, which are specified by regulation.
By making use of VAR, banks can decide how to allocate economic capital and how to trade off risk and return.
Jorion (2002) documents that the VAR numbers publicly disclosed by 8 major banks are informative to predict subsequent fluctuations in banks’ trading revenues. Using a relatively larger sample of U.S. commercial banks, this paper investigates (1) whether the quarterly-average VAR disclosures are associated with the variability of current trading revenues and returns (i.e., validation or assessment of VAR statistical accuracy), and (2) whether the end-quarter VAR disclosures are useful to predict the variability of future trading revenues and returns (i.e., predictive value of VAR). The empirical results support that the quarterly-average (end-quarter) VAR disclosures provide incremental information for assessing (predicting) the volatility of current (future) trading revenues and returns. Furthermore, since the models estimating VAR numbers involve many different risk measurement methods and assumptions, we argue that larger banks tend to exhibit greater professional expertise and more resources of money, and banks are able to improve their VAR models through learning over time. In other words, we hypothesize that banks size and time horizons significantly affect the accuracy of predicting and assessing power of VAR disclosures. Consistent with the hypotheses, we find that the predictive and assessing powers of the VAR measures are better in the later period than early years, and the VAR disclosures of bigger banks have better predictive and assessing powers than smaller banks.
The remainder of the paper is organized as follows. Section II presents the prior literatures about market risk disclosures and hypotheses development. Section III presents the models used to evaluate the predictive and assessing powers of VAR disclosures. Section IV defines the data and samples in the models. Section V shows the results of the regressions and hypotheses. A final section provides some general
conclusions.
II: Prior Literature and Hypothesis Development
Jorion (2002) investigates the informativeness of banks’ publicly available VAR disclosures for the banks’ risk profiles. As in Jorion (2002), we test the informativeness of VAR disclosures to volatility of unexpected trading revenues and holding period returns. In addition, we examine the effect of bank size and learning on assessment and predictive value of VAR numbers.
Schrand (1997) uses regulatory data similar to FFR No. 48 tabular disclosures to examine the association between market interest rate sensitivity and derivatives activities for a sample of S&Ls. The findings suggest that FFR No. 48 disclosures may be useful to investors. Wang (2000) investigates whether the quantitative disclosures about notional amount and fair value of foreign exchange derivatives, required by SFAS No. 119, are associated with the information used by investors to assess the sensitivity of equity returns to currency fluctuations.
Rajgopal (1999) find that disclosures about derivatives presented in a tabular and sensitivity analysis format are associated with firms’ oil and gas price sensitivity. Berkowitz and O’Brien (2002) provide a detailed analysis of the performance of VAR models actually in use and analysis the distribution of historical trading P&L and the daily performance of VAR estimates of 6 large U.S. banks. The data used in this paper is available to bank regulators, and not to the public.
Linsmeier et al. (2002) find that firms’ 10-K market risk disclosures mandated by FFR No. 48, reduce investors’ uncertainty and diversity of opinion about the implications. They use a broad sample of nonfinancial firms and conclude that FFR No.48 market risk disclosures provide useful information to investors.
primarily examines the relation between share prices and changing in a particular financial risk factor. Jorion (2002) is the first paper to investigate the relation between the trading VAR numbers and the subsequent variability of their trading revenues. He concludes that VAR disclosures are informative in that they predict the variability of trading revenues. It is the one using banks’ publicly available data to investigate the informativeness of VAR disclosures. There are only 8 banks in the sample and one model to test the predictive power of VAR measures.
Results in earlier research show that VAR measures are informative for future unexpected trading revenue in predictive model. Since VAR measures the risk of a portfolio, we expect that VAR numbers are also associated with the risk of equity. Therefore, we predict that VAR measures are informative to the volatility of unexpected trading revenues and holding period returns:
H1a: The VAR measures reported by banks are informative to predict future unexpected trading revenues. That is, the end-quarter VAR numbers are positively associated with the subsequent unexpected quarterly trading revenues.
H1b: The VAR measures reported by banks are informative to assess current unexpected trading revenues. That is, the quarter-average VAR numbers are positively associated with the current unexpected quarterly trading revenues.
H1c: The VAR measures reported by banks are informative to predict the volatility of future holding period returns. That is, the end-quarter VAR numbers are positively associated with the subsequent variability of quarterly daily returns.
H1d: The VAR measures reported by banks are informative to assess the volatility of current holding period returns. That is, the quarterly average VAR numbers are
positively associated with the current variability of quarterly daily returns.
As a statistical estimate, VAR is derived from the probability distribution of the portfolio’s potential change in economic value. That probability will be based on some assumptions about and method of simulating the joint probability distribution of changes in market rates and some method for transforming simulated changes in market rates into simulated changes of the portfolio’s value. Models for calculating VAR can and do differ in their broad method, in their assumptions and in the details of their implementation.
We argue that larger banks tend to exhibit greater professional expertise or more resources of money and banks are able to improve their VAR models through learning over time. Therefore, we expect the bank size and time horizon will significantly affect the predictive and assessing power of VAR disclosures. We extend the research by providing empirical evidence about whether the predictive and assessing powers of banks’ VAR disclosures affected by the size and time horizon. As time passing, VAR measures reported by banks should be more informative by the amendment of models and experience accumulation, and many research about VAR increasingly enhance the accuracy of VAR disclosures of banks. We characterize VAR numbers as comprising a “noise” component, which may be introduced by different VAR methods and assumptions. The quality of bank VAR numbers depends on the magnitude of the variability of the “noise” component. We hypothesize that larger banks have better VAR models and learning over time (i.e., experience) increase the quality of earnings. It is well known that the measurement errors in independent variables (e.g., Collins et al. 1994) will underestimate the association of the variables1.
1
Measurement errors in an independent variableX tends to lower coefficient b toward zero, and also to raise its standard error.
H2a: The predictive and assessing powers of the VAR measures for unexpected trading revenues are better in the later period than early years. That is, the end-quarter (quarterly-average) VAR numbers for the later period are more positively associated with the subsequent (current) unexpected quarterly trading revenues.
H2b: The predictive and assessing powers of the VAR measures for the volatility of holding period returns are better in the later period than early years. That is, the end-quarter (quarterly-average) VAR numbers for the latter period are more positively associated with the subsequent (current) variability of quarterly daily returns.
H3a: The VAR disclosures of experienced and bigger banks have better predictive and assessing powers for unexpected trading revenues. That is, the end-quarter (quarterly-average) VAR numbers for the bigger banks are more positively associated with the subsequent (current) unexpected quarterly trading revenues.
H3b: The VAR disclosures of experienced and bigger banks have better predictive and assessing powers for the volatility of holding period returns. That is, the end-quarter (quarterly-average) VAR numbers for the bigger banks are more positively associated with the subsequent (current) variability of quarterly daily returns.
III：Research Design
Model I: Predictive Power of End-Quarter VAR
For a given portfolio, if the portfolio return is normally distributed with mean and standard deviations , the VAR number can be obtained as follows. Denote the t initial portfolio value as W , the cut-off returns as 0 *
R , and the cut-off revenue as * r .
Thus, the VAR in absolute dollar terms is2 0 * * ) (absolute r R W VAR (1) The negative sign arises because we can assume that the extreme tail revenues and returns with which we are dealing are losses, but the VAR is positive. Similarly, if the mean revenue is rand the mean return is , the VAR relative to the mean is 3
0 0 * * ) (relative r r R W W VAR (2) By standard normal table, there is a number corresponding to the confidence level c . For example, if c is chosen to be 95 percent, the corresponding is 1.65,
and if c is 99 percent, is 2.33. If R is normally distributed, we can always
describe the confidence level in terms of a single parameter , which tell us how far away the cut-loss values of the two tails are from the mean , in terms of units of the standard deviation . * R (3) Hence, we have: 0 0 ) (absolute W W VAR (4) 0 ) (relative W VAR (5) In practice, instead of working with two different types of VAR, we would usually work with only one, the relative VAR.4 Thus
t
t s
VAR (6)
2
Jorion, P. (2001) describes the VAR in absolute dollar terms as VAR(zero)
3 Jorion, P. (2001) describes the VAR in relative dollar terms as VAR(mean)
4 The relative VAR is easier to handle because it does not require that we know the mean return, which could otherwise a problem. In any case, if we are dealing with a short time period, the difference between absolute and relative VARs will be fairly small anyway. So we may as well use whichever VAR is more convenient.
With normally distributed market risk factors, the expectation of the absolute valuert1 is5
rt st stE 1 2 0.8
(7)
Thus, the expected absolute value of the trading revenue is linearly increasing in the forecasted volatility of trading revenue, s , which is proportional to VAR. This setup t is valid not only for the normal distribution but also for the conditional distribution, which is fixed and symmetrical.
VAR increases with the time horizon over which VAR is calculated because volatility grows approximately in proportion to the square root of time. The time horizon of VAR public-disclosures by banks corresponds to a horizon of one day, which must be extended to match the quarterly horizon of banks’ reported trading revenues. To calculate VAR, the time horizon can be of any length, but it is assumed that the portfolio composition does not change during the holding period. Hence, we can estimate the volatility in quarterly trading revenues from the daily volatility using the square root of time rule
N st
t
(8) where N is the number of trading days in a quarter, taken as N 63, which is the number of average trading days in a quarter from 1995 to 2001.
With distributions other than the normal, the coefficient will in general differ from 2 , but the VAR measures should still be linearly related to the expected absolute value of trading revenues (Jorion, 2002). Thus, we can test the predictive power of bank’s quarterly VAR disclosures by estimating the following equation:
1 1 t t t a b R (9) 5 Let X be ) , ( 2
whereRt1 is the trading revenue for quarter t1, is the forecasted volatility of t quarterly trading revenue inferred from the bank’s public disclosure of its trading VAR as of the end of quarter t , and is random error. The value of t1 b will equal
its theoretical value of 0.80 only if and t Rt1 are measured without error.
In addition, we assume that Rt1 has mean zero and is normally distributed.
Thus, Rt1 must be the unexpected component of trading revenues. To amend for the expected component of the trading revenue, unexpected component is measured as the difference between the quarterly trading revenue and its moving average over the previous four quarters. The unexpected trading revenue is defined as
4 1 1 1 1 1 (1/4) i i t t t t E R R R R (10)This transformation implies that Rt1 E
Rt1 should have an expected value of zero. These estimates of unexpected revenues are statistically indistinguishable from zero for our sample of banks. The trading revenue is replaced by its unexpected component. Besides, in order to know whether quantitative VAR disclosures contain useful information for predicting the volatility of unexpected trading revenues, after controlling for traditional derivative notional amounts. Hence, Equation (9) is amended as the following regression:
1 11
t t t t
t E R a b cNOT
R (11) where NOT is the outstanding notional value of a bank’s derivatives contracts at the t
end of the previous quarter. If VAR provides incremental information for predicting the volatility of unexpected trading revenues, then we would expect the coefficient b
to remain significantly positive. Ideally, the regression R should be high, implying 2
that the regression model has the significant fitness and captures much of the t variation in Rt1E
Rt1 .Model II：Assessment of Quarterly-Average VAR
Quarterly average VAR is reported as a summary of daily VARs in a quarter. Following the Model I, we expect that quarterly average VARs are also correlated with the current unexpected quarterly trading revenues and coefficient b would remain significantly positive after controlling for the derivative notional amount. Thus, we can test the assessing power of bank’s quarterly VAR disclosures by estimating the following equation:
t t t tt E R a b cNOT
R (12) where Rt E
Rt is unexpected trading revenue for quarter t , is the volatility t in unexpected quarterly trading revenue inferred from the bank’s public disclosure of its trading VAR as of the average of quarter t , and NOT is the average of t outstanding notional value of a bank’s derivatives contracts for quarter t .Model III：End-Quarter VAR and Variability of Daily Returns
Denote market value of common stock equity for a bank in quarter t as t
MV , market value of trading assets in quarter t as TAMV , and market value of t
non-trading assets in quarter t as OTAMV . Thus we have t t t
t TAMV OTAMV
MV (13) The equation below is used to find the variance of market value of common stock equity for a bank in quarter t :
OTAMV TAMV OTAMV TAMV t t
t Var TAMV Var OTAMV
MV
Var( ) ( ) ( ) 2 , (14)
where Var(TAMV)t is the variance of market value for trading assets in quarter t , t
OTAMV
Var( ) is the variance of market value for non-trading assets in quarter t , and 2TAMV,OTAMVTAMVOTAMV is the covariance between TAMV and t OTAMV . t
We evaluate Var(OTAMV)t by a proxy which is variance of market returns, and 2TAMV,OTAMVTAMVOTAMV is view as noise to Var(MV)t estimates. Denote
1
) Re
(Daily turn t
Var and Var(Market Return)t1as the variability of daily returns and market returns for the quarter t1.
Our H1c states that we expect to observe the positive association between VAR and quarterly volatility in returns if VAR measures are useful indicators of the risk of equity. Since VAR measures the risk of a portfolio, we consider that VAR numbers are also associated with the risk of equity and should be related to the variability the variability of daily return after controlling for variability of value-weight returns, capturing the effect of systematic risks. In order to test the predictive power of VAR disclosures for the volatility of holding period return, we have the following equation:
1 1
1 ( Re )
) Re
(Daily turn t abVar Market turn t c t dNott t
Var (15)
where is the bank’s public VAR disclosure of its trading portfolio as of the end of t quarter t , and NOT is the outstanding notional value of bank’s derivatives t contracts at the end of the quarter t .
The above equation shows that there is a potential link between VAR and future change in returns after controlling for changes in value-weight returns and notional amounts. Coefficients b,c and dare predicted to be positive. VAR is informative about the risks of next quarter’s returns, if the regression yields a significant estimate for c .
Model IV：Quarterly Average VAR and Variability of Daily Returns
Given that variability of returns is the total risk of a bank, we argue quarterly average VAR disclosures provide incremental information for assessing the volatility of daily return, and coefficient c would remain significantly positive. Similar to the
concept in Model III, Model IV of quarterly average VAR incorporates control variables that capture the impact of systematic risks and changes in notional amount on derivatives. Thus, we have the following equation:
t t t
t
t a bVar Market turn c dNot
turn Daily
Var( Re ) ( Re ) (16) where Var(Daily Return)t is the variability of daily return for quarter t ,
t
turn Market
Var( Re ) is the variability of value-weight returns for quarter t , is t the bank’s public quarterly average VAR disclosure of its trading portfolio in quarter
t , and NOT is the average notional value of bank’s derivatives contracts in quarter t
t . We predict that coefficients of the control variables, NOT and t
t
turn Market
Var( Re ) , are significantly positive.
Subgroups of Time and Bank Size
To evaluate the effects of learning on assessment and predictive power of VAR numbers, the sample is divided into two subgroups. One is all banks’ data in the last 12 quarters (later period) ended in 1st quarter 2002, and the other is the data prior to 12 quarters (early years).
For testing the effect of banks size, we divide the sample into two subgroups according to the paper “Public disclosures by banks: results of the 2000 disclosure survey” (May 2002) of Basel Committee on Banking Supervision. Bank of America, Bank of New York, Bank One, Citigroup, First Union, Fleet Boston and J. P. Morgan Chase in the survey are big-sized and experienced. We choose them in one group except First Union merged with Wachovia in 2001, and the remaining banks are in another group.
The Z tests associated with the regression coefficients of variables allow us to test the significance of the effect of being in the designated categories. If b1 and b2
are denoted as VAR measures of slope coefficients for later period (bigger banks) and early years (smaller banks), we test the hypothesis H0 :b1b2 against H1:b1b2, which is equivalent to b1 b20, as both b1 and b2 follow a normal distribution (OSL), ) 1 2 var( 1 2 b b b b Z
should follow the t-distribution of student with )
3
(n1 n2 degrees of freedom, wheren and 1 n are numbers of observations in 2
two subgroups. As the subgroups are independent, we have
) 2 var( ) 1 var( 1 2 b b b b Z .
As the alternative hypothesis is one side inequality, we use one tailed t-test and 00 . 2 025 . 0 t
Zcrit . If Zobs Zcrit and we can conclude that b2b1 with probability of 2.5% of type 1 error.
IV：Data and Descriptive Statistics
The bank samples
The criterion for banks in this sample is the availability of publicly disclosed VAR data of bank holding companies in U.S. We collect annual (10-K) and quarterly (10-Q) reports of top 200 bank holding companies from the Internet, and find only a few companies have disclosed their market risks for VAR measures. Most banks disclose their market risk positions using the cash flow or sensitivity measures, based on static or deterministic interest rate scenarios. Eventually the sample includes 17 banks in top 60 bank holding companies. Because large banks’ investment portfolios become more complex, with holdings of instruments that are in some cases longer-dated or have more options, than smaller bank investments. They have greater incentives to establish risks control system for effective identification, monitoring, and reporting risks. Of these 17 banks in the sample, each one issued VAR disclosures in different ways and began to disclose in different time. We collect all the VAR measures from the quarter in which the banks began to disclose to 1st quarter in 2002. Table 1 shows the disclosures of VAR for 17 U.S. banks. The description of
methods used for VAR measurements are shown in column 2. As is well known, there are three primary VAR calculation methodologies: (1) Historical simulation (2) Variance/Covariance approach (3) Monte Carlo simulation. More than half banks didn’t disclose the methods they used. Some banks even used vague descriptions. The columns 3 and 4 show the confidence level banks used to calculate VAR and where they IPO. The columns 5 to 8 represent the beginnings of the public reporting VAR measures for the 17 banks.
In “The New Basel Capital Accord” the quantitative disclosures for banks using the standardized approach include interest rate risk, equity position risk, foreign exchange risk and commodity risk. It allows a comparison between different types of risks and between different institutions. The column 9 shows the period since which banks began to disclose the VAR for specific risks. Not all the banks disclose their risks by the same taxonomy.
Table 2 shows the details of each bank’s VAR disclosures in each year. Table 3 represents the taxonomy banks use for specific risks. Most banks adopt consistent methods to report VARs over time. But it is hard to find two banks disclose market risks by VAR measures in the same formats, with the same horizons and confidence levels.
Value at Risk
VAR data for trading activities were collected from the banks’ annual (10-K) and quarterly (10-Q) reports. According to the SEC’s market risk disclosure rule FFR No.48, banks must disclose market risk in one of the three methods, that is, tabular presentation, sensitivity analysis and value at risk. It is not effective until January 1,1998. Few banks chose to disclose market risks by VAR after 1998, and fewer banks had disclosed market risks by VAR before 1998. Besides, many banks report VAR
only as an average or an end-of-period value, and the period of the average values are different from 3 months. Hence, we use the following decision rule in collecting VAR measures:
VAR (quarterly-average):
1. Average VAR for the current quarter.
2. Average of the beginning and ending quarterly VAR data. If quarterly data are not available, we turn to annual data. 3. Average VAR for the year
4. Average of the beginning and ending annual VAR data. 5. VAR reported at the end of the year.
VAR (end-quarter):
1. End-of-quarter VAR data for the previous quarter 2. Average VAR for the quarter.
If quarterly data are not available, we turn to annual data. 3. Average VAR for the year.
4. VAR reported at the end of the year.
Using annual VAR data instead of quarter data should reduce the accuracy of the risk forecast and assessment in the regressions (Equations 11, 12, 15 and 16), because four quarterly trading revenue, returns and notional amount observations correspond to the same value of the VAR-based independent variable.
Banks report VAR data in different forms, with different confidence levels and different reporting horizons. We transferred these into a quarterly standard deviation assuming normal distributions and identically and independently distributed returns using Equations (6) and (8).
VAR numbers. For these and other statistics reported below, all numbers are measured in million dollars besides notional amount, measured in billion of dollars.
Trading revenues, trading assets and notional amount
We collect the trading revenues, trading assets and notional amounts of these banks from the database of the Federal Reserve Bank of Chicago named “Commercial Bank and Bank Holding Company Database”. Bank holding companies conform to certain criterion must file the report (FR Y-9C6) required by law. The advantage of using this database is that it covers all bank holding companies. However, not all bank holding companies are publicly traded. Therefore, there is no data for Allfirst, which is a wholly owned subsidiary of Allied Irish Banks (AIB). Trading revenues include revenues from interest rate, foreign exchange, equity security and index, and commodity and other exposures but not include fees and interest revenues earned on assets and liabilities. The notional amount is the sum of interest rate, foreign exchange, equity derivatives and other commodity contracts. Table 5 and 7shows the summary statistics of the trading revenue, trading assets and notional amount for the 17 banks from 1996 through 1st quarter 2002.
Other data
The data employed in this study about total assets, book value, returns and value-weight returns are taken from Bank Compustat and CRSP. The definition of return is the change in the total value of an investment in a common stock a day per dollar of initial investment. Value-weight return indices contain the daily returns, including all distributions, on a value-weighted market portfolio (excluding American Depository Receipts). Summary statistics are reported in Table 6 for the total assets
6
This report collects basic financial data from a domestic bank holding company on a consolidated basis in the form of a balance sheet, an income statement, and detailed supporting schedules, including a schedule of off-balance-sheet items.
and book value of the 17 banks.
V：Results
We test the usefulness and informativeness of VAR measures by estimating the assessing (predicting) power of the volatility of current (future) trading revenues and returns. We adopt White’s [1980] test and reject homoscedasticity for all regressions, so White’s adjusted t statistics are reported. Observations with any variable in the extreme 1% of either tail of the pooled sample distribution of that variable are deleted.
For controlling the scale effect, all variables in Equation (11) and (12) are deflated by previous quarter’s trading assets and current quarter’s trading assets, respectively. Similarly variables ( ,t NOT ) in Equation (15) and (16) are deflated by t previous quarter’s book value and current quarter’s book value, respectively.
Results for H1: Hypotheses for the informativeness of VAR disclosures
Table 8 reports the regression results of absolute value of quarter t+1 unexpected trading revenue on quarter t end-quarter VAR-based volatility and notional derivatives amount. It reports the results of a “pooled sample-OLS” regression with 263 observations. Consistent with H1a, we find that the coefficients on the VAR-based volatility and notional derivatives are positive and statistically significant t = 3.64 and t = 2.98.
Table 9 presents the results of the regression of quarter t absolute value of unexpected trading revenue on quarter t quarterly average VAR-based volatility and notional derivatives amount. The results, consistent with H1b, shows t-test statistics 4.42 at 0.05 significant level with 262 observations, and the slope coefficient b is
0.6410. The adjusted R-square is 23.22%.
Table 10 reports the relation between end-quarter VAR numbers and the subsequent variability of quarterly daily returns. The slope coefficient on the
VAR-based volatility are 0.2756, which is significant t = 2.74 with 218 observations. The results are consistent with H1c that VAR disclosures are useful in predicting volatility in banks’ holding period return.
Table 11 reports the results of the regression of regressions of variability of quarter t return on variability of quarter t market return, quarter t VAR-based volatility and notional derivatives amounts. The slope coefficient c is 0.2978 (t value=2.97)
with 226 observations. The results are consistent with H1d.
From the results of Table 8, 9, 10 and 11, VAR disclosures provide incremental information for assessing (predicting) the volatility of current (future) trading revenues and returns, after controlling for systematic risk, derivatives notional amounts and scale effects.
Results for H2: Hypotheses for the difference of informativeness of VAR disclosures in the later period and early years.
To obtain additional evidence we run univariate regressions for all banks sample divided according to the time horizon. Table 8 and table 9 indicate that (1) the coefficients of slope b for the earlier stage and later stage in Equation (11) are 0.4383 and 1.0645, with significant t = 3.88 and t = 3.43, (2) In Equation (12) the coefficients are 0.4964 and 1.1248, which are significant t =4.93 and t = 3.32. The Z tests for the difference in coefficients b of these two equations are 1.90 and 1.78. The results are consistent with to the prediction (b2 ), significantly at 0.1 confidence level. b1
As table 10 and table 11 shows, the coefficients of slope c for the earlier stage and later stage in Equation (15) and (16) are 0.1358 and 0.8237, 0.1500 and 0.7926. The Z tests for the difference in coefficients c are 2.60 and 2.54 significantly at 5% confidence level. Results from Z test reveals that the coefficients on earlier stage and later stage are statistically different from each other. The results are consistent with
H2b.
Results for H3: Hypotheses for the difference of informativeness of VAR disclosures in big banks and small banks.
We divide the sample into two subgroups by banks size. The coefficients b in Equation (l1) and (12) are 0.2383 and 0.9004, 0.2701 and 0.8232 respectively. In this case Z = 1.94 and 2.26, we use one tailed t-test that Zcrit t0.05 1.65. Thus we can conclude that b2b1 with probability of 5% of type 1 error. The results are consistent with H3a.
The results of Z test for coefficients c in Equation (15) and (16) are 0.88 and 0.55. The insignificant difference between c and 1 c is not consistent with H3b. 2
Overall, the results indicate that the end-quarter (quarterly-average) VAR numbers for the later period are more positively associated with the subsequent (current) unexpected quarterly trading revenues and returns. Besides, the end-quarter (quarterly-average) VAR numbers for the bigger banks are more positively associated with the subsequent (current) unexpected quarterly trading revenues and returns.
VI: Conclusion
Using a sample of bank holding companies in U.S., the paper provides evidence that (1) whether the quarterly-average VAR disclosures are associated with the variability of current trading revenues and returns (i.e., validation or assessment of VAR statistical accuracy) and (2) whether the end-quarter VAR disclosures are useful to predict the variability of future trading revenues and returns (i.e., predictive value of VAR). The results suggest that the quarterly-average (end-quarter) VAR disclosures provide incremental information for assessing (predicting) the volatility of current (future) trading revenues and returns.
by the amendment of models and experience accumulation, and many research about VAR increasingly enhance the accuracy of VAR disclosures of banks. Hence, we hypothesize that bank size and time horizons significantly affect the predictive and assessing power of VAR disclosures. Consistent with the hypotheses, we find that the predictive and assessing powers of the VAR measures are better in the later period than early years, and the VAR disclosures of bigger banks have better predictive and assessing powers than smaller banks.
We believe that sufficient information to assess banks’ overall market risk exposure is critical to banks, regulator, and financial statement users, and understandability and transparency will only be accomplished with strong disclosure requirements. Thus we expect banks would be forced to disclose market risks in more complete and detailed way.
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TABLE 1
Description of VAR disclosures for the 17 U.S. Banks
Bank Description of method Confidence level IPO
First VAR disclosure (10K) First VAR disclosure (10Q) Reported Average VAR since Reported End-quarter VAR since Disclosed Specific Risks since
BANK ONE various statistical
models and techniques 99% since 1997-Q1 NYSE 1996 1997-Q1 1998-Q4 1996-Q4 1997-Q4 POPULAR numerous assumptions
and estimates 95% since 1999-Q4 Nasdaq 1999 NA NA 1999-Q4 NA(only total amount) PACIFIC CENTURY variance/covariance 95% since 1997-Q4 NYSE 1997 1998-Q1 NA 1997-Q4 NA(only foreign
exchange) BANK OF NEW
YORK monte carlo simulation 99% since 1997-Q4 NYSE 1998 1998-Q1 1997-Q4 1997-Q4 1997-Q4 J P MORGAN CHASE historical simulation 97.5% since 1994-Q4、
99% since 1997-Q4 NYSE 1994 1995-Q1 1994-Q4 1997-Q4 1997-Q4 CITICORP assumptions are based
on historical experience.
97.7% since 1997、
99% since 1997-Q4 NYSE 1997 1998-Q1 1997-Q4 1997-Q4 1997-Q4
ALLFIRST variance/covariance 99% since 1997-Q4 NA 1997 NA NA 1997-Q4 NA(only total amount) FLEETBOSTON industry-standard risk
measurement techniques 99% since 1998-Q4 NYSE 1999 2000-Q1 1998-Q4 1999-Q4 1999-Q4 MELLON Corporation's
methodology 95% since 1996-Q1 NYSE 1996 1996-Q1 1998-Q4 1996-Q1 NA(only total amount) BANK OF AMERICA sophisticated modeling
techniques 99% since 1998-Q3 NYSE 1998 1998-Q3 1998-Q3 NA
1999-Q4(1998-Q4~ 1999-Q3 no total) NATIONAL CITY historical simulation 97.5% since 1997-Q4 NYSE 1997 NA 1998-Q4 NA NA(only total amount) NORTHERN TRUST variance/co-variance
methodology 95% since 1997-Q4 Nasdaq 1997 NA 1998-Q4 1997-Q4 NA(only total amount) PNC NA NA NYSE 1997 1998-Q1 2000-Q4 1997-Q4 NA(only total amount) KEYCORP statistical methods 95% since 1997-Q3 NYSE 1997 1998-Q1 1997-Q4 1997Q4 NA(only total amount) STATE STREET simulation approach 99% since 1997-Q4 NYSE 1997 1998-Q1 1997-Q4 NA NA(no total amount) WACHOVIA variance/covariance 97.5% since 1997-Q4 NYSE 1997 1998-Q1 1998-Q4 1997-Q4 1998-Q1
TABLE 2
Reporting
details
for
the
17
U.S.
Banks
Bank 2001 2000 1999 1998 1997 1996
BANK ONE Average min max end Average min max end Average min max end Average min max end Average min max end end
POPULAR End End End
PACIFIC CENTURY End End End End End
BANK OF NEW YORK Average min max end Average min max end Average min max end Average min max end Average min max end
J P MORGAN CHASE Average min max end Average min max end Average min max end Average min max end Average end Average CITICORP Average min max end Average min max end Average min max end Average min max end End
ALLFIRST End End End End End
FLEETBOSTON Average min max end Average min max end Average min max end Average
MELLON Average end Average end Average end Average end End End
BANK OF AMERICA Average min max Average min max Average min max Average min max
NATIONAL CITY Average min max Average min max Average min max Average min max Max NORTHERN TRUST Average min max end Average min max end Average min max end Average min max end End
PNC Average end Average end End End End
KEYCORP Average end Average end Average end Average end Average end
STATE STREET Average min max Average min max Average min max Average min max Average min max WACHOVIA Average min max end Average min max end Average min max end Average min max end End
BOK Max end Max end Max end Max end
Note: J P Morgan Chase reported average VAR in 1994 and 1995. We do not present it because of the limitation of the table. Merge information: First Union merged with Wachovia in September 2001. Chase Manhhaton merged with J P Morgan in January 2001 to become J P Morgan Chase; data prior to the merge is that of Chase Manhhaton. Citicorp merged with SSB to become Citigroup in October 1998; data prior to the merge is Citicorp.
TABLE 3
Reporting details for the 17 U.S. Banks
Bank 2001 2000 1999 1998 1997 1996
BANK ONE I, FX, E, Total I, FX, E, Total I, FX, E, Total I, FX, E, C, Total I, FX, E, C, Total Total
POPULAR Total Total Total
PACIFIC CENTURY FX FX FX FX FX
BANK OF NEW
YORK I, FX, Total I, FX, Total I, FX, Total I, FX, Total I, FX, Total
J P MORGAN CHASE I, FX, E, C, HF, Total I, FX, E, C, HF, Total I, FX, E, C, HF, Total I, FX, E, C, Total I, FX, E, C, Total Total CITICORP I, FX, E, C, Total I, FX, E, C, Total I, FX, E, C, Total I, FX, E, C, Total I, FX, E, C, Total
ALLFIRST Total Total Total Total Total
FLEETBOSTON I, FX, E, Total I, FX, E, Total I, FX, E, C, Total I, FX, E, C, Total
MELLON Total Total Total Total Total Total
BANK OF AMERICA I, FX, E, C, FI, Total I, FX, E, C, FI, Total I, FX, E, C, FI, Total I, FX, E, C, FI, Total
NATIONAL CITY Total Total Total Total Total
NORTHERN TRUST Total Total Total Total Total
PNC Total Total Total Total Total
KEYCORP Total Total Total Total Total
STATE STREET I, FX I, FX I, FX I, FX I, FX
WACHOVIA I, FX, E, Total I, FX, E, Total I, FX, E, Total I, Total I, FX
BOK Total Total Total Total
TABLE 4
Panel A: Summary Statistics of Banks’ Quarterly-Average VAR ($ Millions)
Panel B: Summary Statistics of Banks’ End-Quarter VAR ($ Millions)
Sample Mean Max Min Median 1st Qu 3rd Qu
BANK ONE 25 12.14 29 3.9 12 5.1 14
POPULAR 13 0.77 1 0.6 0.8 0.6 1
PACIFIC CENTURY 0 NA NA NA NA NA NA
BANK OF NEW YORK 21 5.18 8.6 2.9 5.2 3.6 6.2 J P MORGAN CHASE 21 35.46 93.3 18.8 24.4 23 28.9 CITICORP 21 29.38 56 15 24 20 32 ALLFIRST 21 0.47 0.7 0.253 0.4265 0.3 0.6 FLEETBOSTON 17 43.59 78 31 43 38 44 MELLON 25 2.68 7 1 2 2 3 BANK OF AMERICA 17 40.15 61.3 21 39.6 31.7 47.9 NATIONAL CITY 17 0.45 0.5 0.3 0.5 0.5 0.5 NORTHERN TRUST 21 0.17 0.285 0.052 0.181 0.091 0.233 PNC 21 0.58 1.2 0.3 0.6 0.3 0.7 KEYCORP 21 0.98 1.6 0.3 1 0.6 1.3 STATE STREET 18 2.89 5.4 0.6 2.1 0.9 5.05 WACHOVIA 21 12.05 19 6 12 11 13 BOK 17 1.16 3.1 0.045 0.539 0.227 2.1 Overall 317 11.52 93.3 0.045 3.1 0.6 16
Note: The VAR measures of Pacific Century are only about foreign exchange.
Sample Mean Max Min Median 1st Qu 3rd Qu
BANK ONE 25 12.04 27.5 4 12.5 6.65 15
POPULAR 13 0.78 1 0.6 0.8 0.6 1
PACIFIC CENTURY 0 NA NA NA NA NA NA
BANK OF NEW YORK 21 5.13 6.7 3.4 5.4 3.7 6.3
J P MORGAN CHASE 21 34.11 73.7 19.75 24.5 23 26.8 CITICORP 21 29.29 56 15 24 19 28 ALLFIRST 21 0.46 0.7 0.2765 0.4265 0.3 0.6 FLEETBOSTON 17 40.41 45 32 40 38 44 MELLON 25 2.60 6.5 1 2 1.5 3.5 BANK OF AMERICA 17 41.68 61.3 25.5 40.98 32.2 47.9 NATIONAL CITY 17 0.45 0.5 0.3 0.5 0.5 0.5 NORTHERN TRUST 21 0.20 0.285 0.091 0.209 0.181 0.233 PNC 21 0.57 1.05 0.3 0.55 0.3 0.775 KEYCORP 21 1.04 2 0.4 1 0.6 1.3 STATE STREET 18 2.89 5.4 0.6 2.1 0.9 5.05 WACHOVIA 21 11.90 17.5 7 12 11 12 BOK 17 1.24 3.1 0.101 0.5865 0.3315 2.15 Overall 317 11.32 73.7 0.091 3.1 0.6 17
TABLE 5
Panel A: Summary Statistics of Banks’ Quarterly Trading Revenue ($ Millions) Sample Mean Max Min Median 1st Qu 3rd Qu BANK ONE 25 82.2 277.0 5.4 64.0 20.5 130.0
POPULAR 25 0.6 4.0 -2.0 1.0 0.0 2.0
PACIFIC CENTURY 19 1.3 21.2 -18.2 0.5 -0.1 2.6 BANK OF NEW YORK 24 119.7 338.3 10.1 88.3 50.8 172.6 J P MORGAN CHASE 24 ##### ##### 62.5 1223.0 789.3 ##### CITICORP 24 ##### ##### 295.0 1725.0 ##### ##### ALLFIRST 0 NA NA NA NA NA NA FLEETBOSTON 20 215.0 873.0 7.5 100.5 46.9 259.3 MELLON 24 166.0 859.0 21.2 89.5 50.7 166.0 BANK OF AMERICA 24 386.1 ##### 38.9 192.3 132.0 511.2 NATIONAL CITY 24 373.7 ##### -0.2 8.8 0.9 433.5 NORTHERN TRUST 24 90.6 264.0 -38.7 67.1 31.3 129.8 PNC 24 44.2 176.2 -34.3 18.3 -0.3 86.1 KEYCORP 24 36.8 117.4 -37.7 29.1 20.0 50.4 STATE STREET 24 126.8 293.8 35.1 116.5 73.2 166.8 WACHOVIA 24 121.0 386.9 3.4 56.5 16.9 211.5 BOK 24 54.6 345.0 1.0 21.1 3.0 72.6 Overall 377 389.0 ##### -38.7 73.9 12.5 224.8
Panel B: Summary Statistics of Banks’ Quarterly Trading Assets ($ Millions)
Sample Mean Max Min Median 1st Qu 3rd Qu
BANK ONE 25 9120 92890 196 8397 874 10040
POPULAR 25 265 372 153 270 225 299
PACIFIC CENTURY 19 12 30 2 10 7 14
BANK OF NEW YORK 24 4459 12051 586 2170 1644 8734 J P MORGAN CHASE 24 86634 250550 8357 63343 57567 71532 CITICORP 24 88592 208216 28432 107920 34872 124039 ALLFIRST 0 NA NA NA NA NA NA FLEETBOSTON 20 2080 7819 156 494 230 2097 MELLON 24 1839 7312 300 668 547 971 BANK OF AMERICA 24 28660 66311 773 23714 14318 52366 NATIONAL CITY 24 19730 77287 12 28 19 59378 NORTHERN TRUST 24 227 597 22 204 163 275 PNC 24 199 476 9 191 123 265 KEYCORP 24 525 1521 92 321 188 657 STATE STREET 24 1332 2058 702 1357 949 1559 WACHOVIA 24 1694 3762 665 1420 1055 2148 BOK 24 3810 28227 3 800 6 1038 Overall 377 16910 250550 2 877 220 8680
TABLE 6
Panel A: Summary Statistics of Banks’ Quarterly Total Assets ($ Millions)
Sample Mean Max Min Median 1st Qu 3rd Qu BANK ONE 25 2E+05 3E+05 95708 3E+05 1E+05 269425 POPULAR 25 22974 30745 15805 23174 19301 27230 PACIFIC CENTURY 21 14295 15017 12905 14295 13987 14731 BANK OF NEW YORK 24 66664 89677 51499 63658 59823 75567 J P MORGAN CHASE 24 4E+05 8E+05 79607 4E+05 3E+05 398560 CITICORP 23 4E+05 7E+05 3E+05 3E+05 3E+05 411939 ALLFIRST 21 15782 18551 9799 17507 11246 17991 FLEETBOSTON 24 99219 2E+05 17713 92708 82974 107311 MELLON 24 78418 2E+05 41582 47398 43703 49732 BANK OF AMERICA 24 3E+05 7E+05 32798 2E+05 2E+05 614138 NATIONAL CITY 24 2E+05 7E+05 35110 84058 52510 612297 NORTHERN TRUST 24 48424 1E+05 20302 28394 25050 85918 PNC 24 59615 77207 33219 71895 37683 73662 KEYCORP 24 71940 80889 64764 70762 69149 75494 STATE STREET 24 59124 87270 27229 50179 36366 83733 WACHOVIA 24 59890 73298 45425 64376 48302 66082 BOK 22 37941 76853 4240 35627 5314 70312 Overall 401 ##### ##### 4240 69660 31520 113100
Panel B: Summary Statistics of Banks’ Quarterly Book Value ($ Millions)
Sample Mean Max Min Median 1st Qu 3rd Qu BANK ONE 25 15300 20870 8185 18630 9916 19900
POPULAR 25 1585 2199 1060 1577 1350 1736
PACIFIC CENTURY 21 1165 1372 1047 1167 1082 1212 BANK OF NEW YORK 24 5354 6467 4804 5137 4964 5609 J P MORGAN CHASE 24 23796 42004 6178 21561 18732 23298 CITICORP 23 25949 47912 17684 22569 19400 28867 ALLFIRST 21 1684 1981 1138 1823 1232 1906 FLEETBOSTON 24 8050 15606 1656 7636 6168 9112 MELLON 24 6898 19454 3332 4059 3565 4507 BANK OF AMERICA 24 24018 47223 3443 20378 10886 45304 NATIONAL CITY 24 17893 50084 2790 6057 4342 46978 NORTHERN TRUST 24 3410 7672 1349 1866 1565 6509 PNC 24 4471 6036 2132 5470 2590 5789 KEYCORP 24 5740 6822 4674 5881 5056 6184 STATE STREET 24 4017 6702 1560 2353 1895 6474 WACHOVIA 24 3780 5376 2441 3690 3219 3882 BOK 22 3159 6865 306 2961 396 5799 Overall 401 9293 50080 306 5338 2046 9665
TABLE 7
Panel A: Summary Statistics of Banks’ Quarterly Notional Amount ($ Billions) Sample Mean Max Min Median 1st Qu
BANK ONE 25 490.0 1332.5 0.1 729.6 0.8
POPULAR 25 0.1 1.8 0.0 0.0 0.0
PACIFIC CENTURY 19 0.6 0.9 0.0 0.6 0.4
BANK OF NEW YORK 24 7006.5 162229.3 58.1 295.9 195.9 J P MORGAN CHASE 24 11042.7 24199.1 413.9 9612.6 6711.4 CITICORP 24 6362.8 23520.4 1883.8 5340.6 2566.7 ALLFIRST 0 NA NA NA NA NA FLEETBOSTON 20 74.2 308.7 7.8 16.7 11.5 MELLON 24 92.8 287.6 32.4 52.2 40.9 BANK OF AMERICA 24 2274.6 6059.9 41.9 1509.0 829.9 NATIONAL CITY 24 2414.0 9633.7 1.9 10.3 5.8 NORTHERN TRUST 24 18.0 47.5 1.6 14.4 13.2 PNC 24 64.1 1212.8 2.7 15.3 10.1 KEYCORP 24 30.5 56.9 8.2 30.8 21.7 STATE STREET 24 80.9 138.6 42.4 74.8 61.0 WACHOVIA 24 88.8 201.2 7.9 73.0 11.5 BOK 24 310.4 2104.7 0.0 23.2 0.0 Overall 377 1945.0 ######## 0.0 41.4 10.6
Note: The period for the summary statistics of Table 5, 6 and 7 is from 1996 through 1st quarter 2002. Allfirst is a wholly owned subsidiary of Allied Irish Banks, (AIB), an Irish banking corporation. It is a bank holding company in U.S. but not publicly traded.
TABLE 8
Regressions of Absolute Value of Q t+1 Unexpected Trading Revenue on Q t
VAR-Based Volatility and Notionals
, 1 , , , , , , 1 1 ,t it it i i( it it) i( it it) it i E R A a b A c NOT A R OLS Constant t Slope b t NOT Slope c R-square Z Samples Obs (t – statistics) 0.0339 0.6398* 0.2675* 16.02% Overall 263 (3.53) (3.64) (2.98) 0.0006 0.4383* 0.6567* 42.30% Prior to 12 quarters 100 (0.06) (3.88) (8.65) 0.0474 1.0645* 0.0654 15.04% Last 12 quarters 163 (4.42) (3.43) (1.08) 1.90 0.0694 0.2383 0.3249* 13.12% Small 156 (5.55) (1.29) (3.01) 0.0062 0.9004* 0.0250 54.39% Big 107 (0.78) (3.15) (0.38) 1.94Note: Ri,t1 is the quarterly trading revenue,E[Ri,t1] is the expected trading revenue in quarter
t+1 based on a moving average of the last four quarters, is the ending of VAR-based i,t unexpected trading revenue for banks i in quarter t, and NOTi,t1 is the ending of notional
derivatives amount for banks i in quarter t. The variables are scaled by the previous quarter’s trading assetsAi,t. The Z value is defined as
) 2 ( ) 1 ( 1 2 b Var b Var b b
. Significantly at the 0.01 level denoted by *.
TABLE 9
Regressions of Absolute Value of Q t Unexpected Trading Revenue on Q t
VAR-Based Volatility and Notionals
it it i i it it i it it it t i E R A a b A c NOT A R, , , ( , , ) ( , , ), OLS Constant t Slope b t NOT Slope c R-square Z Samples Obs (t – statistics) 0.0251 0.6410* 0.3730* 23.22% Overall 262 (2.64) (4.42) (3.86) -0.0016 0.4964* 0.6795* 54.01% Prior to 12 quarters 101 (-0.14) (4.93) (6.03) 0.0437 1.1248* 0.0696 12.51% Last 12 quarters 161 (3.95) (3.32) (0.93) 1.78 0.0599 0.2701 0.4271* 22.29% Small 156 (4.98) (1.94) (3.96) 0.0084 0.8232* 0.0076 57.59% Big 106 (1.32) (4.10) (0.13) 2.26Note: Ri,t is the quarterly trading revenue,E[Ri,t] is the expected trading revenue in quarter t
based on a moving average of the last four quarters, is the average of VAR-based unexpected i,t trading revenue for banks i in quarter t, and NOTi,t1 is the average of notional derivatives
amount for banks i in quarter t. The variables are scaled by the quarter’s average trading assetsAi,t. The Z value is defined as
) 2 ( ) 1 ( 1 2 b Var b Var b b
. Significantly at the 0.01 level denoted by *.
TABLE 10
Regressions of Variance of Q t+1 Return on Variance of Q t+1 Market Return,
Q t VAR-Based Volatility and Notionals
1 , , , , 1 , 1 , ( Re ) ( ) ( ) ) Re
(Daily turn it ai biVar Market turn it ci it Ait di NOTit Ait it
Var OLS 1 ) (MR t Var t t A t t A NOT Z Constant Slope b Slope c Slope d R-square
Samples Obs (t – statistics) 0.0078 1.2004* 0.2756* 0.0001* 36.03% Overall 218 (6.26) (12.09) (2.74) (2.21) 0.0062 1.3578* 0.1358 0.0001* 47.88% Prior to 12 quarters 106 (4.26) (10.26) (1.34) (3.50) 0.0090 0.9981* 0.8237* 0.0029 25.25% Last 12 quarters 112 (3.94) (6.59) (3.37) (1.72) 2.60* 0.0079 1.1873* 0.2783* 0.0020 37.13% Small 130 (5.14) (9.23) (2.57) (1.15) 0.0063 1.2193* 0.5474 0.0001* 35.96% Big 88 (2.95) (8.04) (1.91) (2.46) 0.88
Note: Var(Daily Return)i,t1is the variance of daily return of bank i for the quarter t1,
1
) (MR t
Var is the variance of daily return on a value-weighted market portfolio for the quarter
1
t , is the end-quarter VAR-based volatility in unexpected trading revenue for banks i,t i in quarter t, and NOTi,t1 is the notional amount of derivatives for banks i in quarter t+1. The
variables ( ,i,t NOTi,t1) are scaled by the book value Ai,t . The Z value is defined as
) 2 ( ) 1 ( 1 2 c Var c Var c c
TABLE 11
Regressions of Variance of Q t Return on Variance of Q t Market Return, Q t
VAR-Based Volatility and Notionals
t i t i t i i t i t i i t i i i t
i a bVar Market turn c A d NOT A
turn Daily Var( Re ), ( Re ), ( , , ) ( , , ) OLS t MR Var( ) t t A t t A NOT Z Constant Slope b Slope c Slope d R-square
Samples Obs (t – statistics) 0.0074 1.2118* 0.2978* 0.0001* 38.14% Overall 226 (6.25) (12.71) (2.97) (2.44) 0.0061 1.3624* 0.1500 0.0001* 50.52% Prior to 12 quarters 103 (4.38) (10.69) (1.38) (3.59) 0.0086 1.0196* 0.7926* 0.0025 27.30% Last 12 quarters 123 (4.09) (6.97) (3.47) (1.62) 2.54* 0.0073 1.2322* 0.3309* 0.0018 39.35% Small 135 (4.83) (9.77) (2.95) (1.07) 0.0067 1.1773* 0.4892 0.0001* 37.81% Big 91 (3.22) (8.38) (1.85) (3.27) 0.55
Note: Var(Daily Return)i,t is the variance of daily return of bank i for the quarter t ,
t
MR
Var( ) is the variance of daily return on a value-weighted market portfolio for the quarter t , t
i,
is the average of VAR-based volatility in unexpected trading revenue for banks i in quarter t, and NOTi,t1 is the average notional amounts of derivatives amount for banks i in quarter t. The
variables ( ,i,t NOTi,t) are scaled by the quarter’s average book valueAi,t. The Z value is defined as ) 2 ( ) 1 ( 1 2 c Var c Var c c
