Value = Cash Flow / (Cost of Capital - Growth Rate of Cash Flows).
The levers are the cash flows generated by the company, the growth rate of those cash flows, and the opportunity cost of capital, i.e. the minimum return acceptable to investors. The purpose of the formula is to discount future cash flows into a net present value. The formula is not perfect, mainly because it assumes constant growth in cash flows, but it is an excellent rule of thumb. If you're familiar with finance, this is not new and you can probably stop reading. If finance is not your thing, stick with me - you'll soon understand, I hope.
So, what is the key lever being affected by S&P downgrade? It is the cost of capital. The downgrade is a signal that a default is more likely than previously thought. The greater the risk of default, the greater the return required by investors to compensate them for that risk. Of course, this is not cut and dry. US debt investors may have already priced in the riskiness of default, which means treasuries may not swing as widely on the news. Also, if sovereign debt of other nations (like the EU or Japan) is more risky than US debt, than the US is still a "safe haven" and the large pool of buyers will keep prices high and rates low. But for the sake of understanding the fundamentals of the downgrade, lets assume that treasury rates rise. Since the cost of capital sits in the denominator of the value formula, an increase results in a decline in value. Put simply, a rise in the cost of capital with everything else remaining constant results in a lower valuation or stock price.
Let's go one step further and explain how a rise in treasury rates affect the cost of capital of a company. For simplicity purposes, I'm assuming the company is equity-financed, meaning that it does not have any debt.
I need to introduce one more formula: The Capital Asset Pricing Model (CAPM).
The risk free rate is, well, the treasury rate. Beta, without getting too deep in the weeds, is the riskiness of a particular stock. And the market risk premium is the amount above the risk free rate investors expect to earn by investing in the equities market. If beta and the market risk premium remain constant, than a rise in the risk free rate -- i.e. the treasury rate -- results in a higher cost of capital.
DCF = $2,041 = $100 / (7.9% - 3%).
In short, a 0.90% increase in the treasury rate will cause the value of our hypothetical company to fall 18%. To get a better picture, I have published a Google Spreadsheet.