The Leverage Equation Insights

Last time at the corporate finance course we were learning about the leverage effect. It is a very nice concept showing that the return on shareholders��� equity can be maximized by financing investments partially through banks, rather than solely by shareholders. The equation looks like this: Return on Equity (ROE) = Return on Capital Employed (ROCE) + (ROCE - Cost of Debt)*(Net Debt/Shareholders Equity). In a very simplified scenario, a shareholder would seek to maximize the Net Debt in order to maximize ROE. Obviously, that is not a realistic result since cost of debt increases as net debt rises. In order, to reflect this relationship I decided to find the optimal level of shareholder���s equity in respect to debt. The solution would be as following:

Let ROE=y(x) where x=Net Debt. Also let ROCE=b and Sh.Equity=c, where b is a constant and c (c>=0) is a variable of interest. Also let the r=interest rate (on debt). Assuming that the cost of debt consists only of interest payments, we then have the leverage equation: y(x,c)=b-(b+xr)*(x/c). This is just a simple quadratic equation y(x,c)=b+bx-(x^2r/c) with two unknowns. Our goal is to maximize y(x,c) in respect to x, therefore, we impose y'(x,c)=0 condition (partial derivative in respect to x). We get: x-2xr/c=0 which can be expressed as cx=2xr, therefore, c=2r. It is interesting to notice that even though x's simplify in our equation, there is an interesting condition when x=0 in the original leverage equation. When that happens (x=0) we get that ROE=ROCE, which means that shareholders receive full benefits of their investments. However, that is usually a corner solution, since very few firms would have net debt=0. But conventional solution is rather interesting as well, since it says that ROE is maximized when shareholders��� equity does not exceed (or is no less) twice the interest rate on debt. Since at this level the ROE is stable (i.e. no further change can cause ROE to increase) the objective of shareholders is to stay at this level of equity financing.

There are some practical applications of this result as well. This finding could partially explain why, for example inflation, is so crucial for economic growth. To explain this statement better, imagine that inflation soars and the central bank decides to increase the interest rate. Then private banks will also increase their lending rates (r), and shareholders will have to provide more of their own funds in order to stay at the optimal ROE level. Moreover, each increase in the interest rate increase will a double effect on businesses, since 1% increase in ���r��� will require shareholders to raise their equity by 2%. This effect would be further amplified by the fact that during the inflationary periods, banks become more risk averse and therefore, might respond by increasing required debt repayment rates more than proportionately to the central bank���s rate. That said, his short analysis shows why today���s credit crunch might have even stronger negative effect on businesses. As the credit provided by banks became more expensive, companies��� shareholders now have to provide bigger ���chunk��� of the investments in their own businesses. But that���s obviously not that easy to do, since shareholders face the full risk of investment, and therefore, require higher returns. Then obviously more risky projects will be abandoned and overall investment levels will fall, leaving people with fewer jobs, lower benefits and overall more depressed future perspective���