A Question of Balance

Brainstorms by Stephen W. Philbrick

I was recently in a session where we were discussing the pricing of a sizable piece of business. The speaker discussed various aspects of the potential transaction, and remarked that one of the concerns is the impact it would have on our line of business "balance." As part of our strategic plan, we had goals relating to the proportion of business in various segments. This piece of business would add to a category that was "overrepresented."

He remarked, however, that if the price were sufficiently high, it could make up for the lack of balance. This conclusion seemed reasonable, but it wasn't immediately obvious how to quantify the trade-off. I decided to give some thought as to how to calculate the additional price needed to make up for a reduction in a balance goal.

I thought this would be a quick exercise—finding some way to quantify the concept of "balance," then calculating how much additional premium a particular risk would require if it were adversely affecting balance. It turned out to be more complicated than I imagined. I'll share the progress I made, and ask for additional thoughts.

A portfolio (in our hypothetical case) is considered balanced if the proportions of premium by line of business (LOB) match the business plan goals. For simplicity, assume that the company only writes two lines of business, Line A and Line B. The business plan includes a total premium goal, to be equally written in each line. Near the end of the year, the company has met its premium goals in Line B, but is short in Line A, by an amount equal to 10 percent of total planned premium. Underwriters are discussing two potential accounts, each representing 10 percent of the total business for the year. One account is in Line A and the other in Line B. Obviously, selecting the account in Line A will lead to a balanced book, while selecting the other account will lead to an unbalanced book. If the pricing of the two accounts is identical, then the choice is obvious. However, suppose the underwriters believe that more favorable pricing is possible for the account in Line B. The question is how much higher the premium would have to be to make the Line B account equivalent to Line A. In other words, what additional premium would induce the company to accept an unbalanced book?

I assumed perfect correlation between risks within a line, and complete independence between the lines. I also assumed that the capital needs of the company were proportional to the standard deviation of the portfolio results. The capital needs of the unbalanced portfolio will be higher than the balanced portfolio, so the company will need a higher profit load to compensate the increased need for capital.

I'll spare the math, but my first calculation indicated that I needed roughly 1/4 percent more premium to cover the additional risk. My first reaction was that this amount seemed miniscule—far smaller than the margin of error in the pricing. If this was the right amount, it seemed not worth worrying about. After reflection, I realized I had blundered. The entire portfolio requires only a modest amount of additional premium, but I must collect all of it from this particular account. Given that this account represents 10 percent of the book, I need 2.6 percent more premium just to break even.

However, as I reviewed my calculations, I realized they were dependent on the relative mix of the LOBs, on the distribution assumptions for each LOB, on the correlation assumptions, as well as the risk measure/capital allocation assumptions. I had hoped to come up with a rough rule of thumb, or some formula to evaluate the trade-off between price and balance, but it now appears to be more complicated than I originally thought. Has anyone else tried to quantify this?