How is PE related to WACC

This post may seem like stating the obvious for seasoned analysts, but how are Price to Earnings ratios related to Weighted Average Cost of Capital (WACC). In CFA prep questions dealing with PE valuation techniques most of the information is given to you to solve a problem. However, in real life, that is hardly ever the case and analysts have to find information out for themselves. For instance the Dividend Discount Model the price is determined by:

P = D / [r - g]

Where:

• P - Current price of equity
• D - Dividends expected in next period
• r - Required rate of return on the market
• g - Divided growth rate

However, consider also:

PE multiplier = P / EPS
P = EPS / PE multiplier

Does something look familiar? Well, substitute dividends for earnings and the PE multiplier for [r - g] and you have the same formula. This should come as no surprise as the underlying math and assumptions are virtually identical for these valuation techniques (infinite geometric series - the discount and growth being the geometric term and earnings being the base case).

What I would like to focus on, is r - the required rate of return. Right away, it should be obvious that this is somehow related to WACC. As r goes up, the divisor goes up and the price goes down. This is a similar relationship with what we should expect from WACC, as the cost of raising capital for a company goes up, on some level its margins on return must also drop.

Looking strictly from a capital markets perspective, it should be clear that if a company were given capital at WACC, it would have to generate returns in excess of WACC (at r) in order to justify the additional risk of the venture. Using a modified version of the Sharpe ratio to decompose what's happening here:

S = [Rp - Rf] / sp

Where

• S - Sharpe ratio
• Rp - Return on portfolio
• Rf - Return on risk free asset
• sp - standard deviation of return (risk)

However instead use:

S = [Rr - Rwacc] / sp'

Where

• Rr - Required rate of return on company's stock
• Rwacc - Weighted average cost of capital
• sp' - change in standard deviation
  

We can see that we can determine the value of management in providing value added by utilizing capital that we provide as investment bankers and sources of capital (which in turn affects the stock price through the PE ratio valuation technique).

Another thought: What is the relationship between required rate of return on the market (r above) and cost of equity component in WACC? Isn't it the same? Isn't the difference in cost of equity and WACC the use of leverage such as debt to provide amplified returns (which also explains why WACC < r)?