It’s going to be tough for the S&P to break above its present range. Last month we saw the equity market show extreme sensitivity to news from the Federal Reserve. It rallied when the Fed indicated that the economy was too weak to permit a cut in interest rates. Why should that be the case? There is actually a simple arithmetic answer to this question (although equity investors may be sorry they asked).

 At extremely low interest rates, stocks are extremely sensitive to changes in the discount rate. If the stock price is equal to e(P)/r, that is, expected earnings over a discount rate, very strange things can happen when both the numerator (expected earnings) and the denominator (the discount rate) are extremely low.

Much as I dislike the old Dividend Discount Model, it is useful for comparative statics (what happens when to one variable, in this case the price of the S&P 500, when we change another variable, namely the risk-free rate). To do this properly we should discount expected future earnings against the yield curve, which now is at record steep levels. That is very good for stocks, because the more valuable cash flows in early years are discounted at much lower rates than the less valuable (and less certain) cash flows for later years. As a rough proxy I simply take the fed funds rate as the risk free rate in the model, assume 3.5% economic growth, and a 5% equity risk premium.

With these admittedly artificial assumptions, we obtain a doozy of a sensitivity graph of equity prices vs. the risk free rate:

The above chart uses the conventional Gordon Dividend Discount Model to value the present earnings of the S&P 500 at different levels of the risk free rate (remembering that we have simplified the calculation by ignoring the yield curve).

The recovery of the S&P 500 since its March 2009 lows reflects an anemic level of earnings as well as a very low discount rate. A rise in the short-term interest rate (in reality, in the whole yield curve) could take a very big bite out of equity prices. I don’t quite believe that a 2% risk free rate implies a drop in the S&P by half — this is a numerical example rather than a realistic model — but it does highlight the sensitivity to watch out for.

Another way to measure the sensitivity is to compare the level of the S&P 500 under different expected growth scenarios at different risk-free rates. This we do in the chart below:

This entry was posted on Monday, March 8th, 2010 at 9:42 pm and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.