Is The Size Of An Operational Loss Related To Firm Size?

The European Commission recently proposed that capital charges for operational risk might be based on the size and income of an institution (Risk Management Operations, 13 December).

While it seems intuitive that operational risk is to some degree a function of firm size, the nature of this relationship is not straightforward. We therefore conducted a study to empirically test one aspect of this relationship: whether the magnitude of a loss experienced by a firm is correlated with the size of the firm.¹

The study revealed that

 size only accounts for a very small portion (about 5%) of the variability in loss severity;

 the size of a firm is related to the magnitude of its loss, but the relationship between size of loss and size of firm is not linear; and

 there is clear evidence of a diminishing relationship between the size of a firm and loss magnitude.

Identifying the relevant scale variable

We began by calculating level and log correlations between operational losses and three variables associated with the size of a firm.² These "scale variables" were revenue, assets and number of employees.

We found that all three variables were correlated with loss size, with the revenue variable showing the strongest relationship.

We also noted that the logarithm of the scale variables showed a stronger relationship to losses than did the raw scale variables.

This suggested that the relationship between size of firm and loss magnitude was not linear. That is, a firm that is twice as large as another does not, on average, suffer a loss that is twice the size of the loss experienced by the other firm.

We subsequently examined the correlation between the logarithm of the three scale variables and found, not surprisingly, that all three variables were highly correlated.

Because multiple regression in this case would have resulted in little improvement in significance, we chose to specify our regression equation in terms of a single independent variable.

Hypothesizing the scale relationship

In view of the above, we hypothesized the following relationship between size of firm and loss magnitude:

L

= R^axF(q) (1.1)

where:

L

is the actual loss amount;

R

the revenue size of the firm in which the loss took place;

a

is the scaling factor measuring the degree of return to scale; and q is the vector of all the risk factors not explained by revenue R so that F(q) is the multiplicative residual term not explained by any fluctuations in size.

In the above equation a = 1 would represent a linear relationship between R and L, a 1 would represent a diminishing relationship between R and L, while a > 1 would represent an increasing relationship.

Ordinary least squares regression

Taking the logarithm of both sides of equation (1.1) yields the following equation:

l

= axr +b + e(1.2)

where l = lnL,
r = lnR, b = E[lnF(q)] and
e = ln(F(q)) - b

An ordinary least square (OLS) regression was performed on all the data, based upon equation 1.2. The results are shown in Table A. Our initial regression plot showed that the residual data scatter was very symmetric about the horizontal axis, strongly indicating a linear relationship between l and r (the log of loss and the log of revenues).

A. OLS Regression Results

OLS regression results	Coefficients	Standard error	t Statistic	Regression statistics
Intercept (b)	1.275950699	0.121395484	10.51069	R Square	0.054102027
LnR (a)	0.151524065	0.014702495	10.30601	Adjusted R Square	0.053592658

The t-statistics for the coefficient also indicated a very significant linear relationship between these two variables.

We noted that both the R² and adjusted R² were very small - only slightly above 5%. An R² of around 5% suggests that around 95% of the loss variability in the data is attributable to factors other than the independent variable.

Generalized least squares regression

We noted that the residual plot of the results exhibited a funnel shape, indicating a positive linear relationship between loss variability and firm size -- a classic case of heteroskedasticity.3 This may be because large firms suffer large losses, as well as small losses.

B. WLS Regression Results

WLS regression results	Coefficients	Standard error	t Statistic	Regression statistics
Intercept	0.232445601	0.0093497	24.86129	R Square	0.090513977
X variable 1	0.695024654	0.051166479	13.58359	Adjusted R Square	0.090023424

The presence of heteroskedasticity in the error term meant that any OLS estimator for a would be inefficient, ie a reduction in variance of the estimator is no longer guaranteed asymptotically. We therefore ran a weighted least square (WLS) regression, which is a special form of a generalized least square regression.

This was accomplished by dividing both sides of equation 1.2 by r and then regressing the following relationship:

y =

a + bxx + f (1.3)

where y = l/r, x = 1/r are the new dependent and independent variables, respectively, and f is the new error term. (Note that the regression coefficient for l in the OLS equation reappears as the intercept in the WLS equation.)

The GLS regression results in Table B show that both the R² and adjusted R² improved. The WLS estimate of the scaling coefficient was determined to be a = 0.2324 instead of the original 0.1515.

Remarks

If the size of a firm is so weakly related to its size of loss, what are the causes of loss variability? We suspect that that the vast majority of the variability is caused by factors such as inherent differences in risk (based on the type of business conducted), the competence of management, and the quality of the internal control environment.

Note that the results of this study could be applied to the statistical modelling of operational value at risk. The correlation coefficient could be used to scale-adjust external loss data to the size of the firm that is being modelled.

The authors are consultants in the Financial Risk Management practice of PricewaterhouseCoopers in New York.

-- Jimmy Shih, Ali Samad-Khan and Pat Medapa