Equities as discounted outcomes

First, some theoretical mumble jumble to make sure we are all on the same page.

Two main theorems are currently at the core of financial asset valuation:

1. The Dividend Discount model (or the two associated theorems, Modigliani-Miller and Discounted Cash Flow model)
2. The Capital Asset Pricing model (itself part of the Modern Portfolio theory)

In one way or another, market participants are using a combination of the two theoretical frameworks when allocating capital to assets.

The Dividend Discount model (DDM) states the following:

which can be simply re-stated as 

where the discount rate is equal to the cost of capital minus the growth rate of dividends.

Unfortunately, this equation only works for companies paying out earnings as dividends. To generalize it, we need to use a combination of Modigliani-Miller (MM) and the Discounted Cash Flow model (DCF). The main conclusion of MM is that the value of a firm is unaffected by how that firm is financed (between debt and equity). The main conclusion of the DCF states that:

This is where the Capital Asset Pricing model (CAPM) comes in. As is commonly known, the main conclusion from the CAPM states that:

where the important variable is

which represents the extra return market participants require to take on risk above the risk-free rate. In equity markets, we refer to this term as the Equity Risk Premium (ERP). This can be thought of as the discount rate to be used for cash flow valuation.

For reference sake, the S&P500 ERP looks something like the following historically (using the forward P/E as the inverse of the expected return and 10y inflation protected yield as the risk free rate):

Tying in all of the above, we can see that when evaluating a stock, a market participant relies heavily on three variables: the estimation of dividends, the estimation of the terminal value and the discounting factor.

Re-writing the DDM, we can see that:

where the PV factor is driven by 

The main issues with the model are that 1) it assumes perfectly knowable dividends and 2) doesn't take into account the volatility of the terminal value.

To give an example, assume a stock that pays out 1$ in dividend in year 1, has a dividend growth of 2% and an equity risk premium of 5%. Using a DDM, the value of the stock should be:

As we can see in this example, roughly 82% of the value of the stock is derived from the terminal value (40.8 / 49.19).

Now let's look at what a change in the ERP will have as an effect on the stock value:

We could break it down between changes in the terminal value and the discounted dividends, but the general idea is that minute changes in the volatility (as represented by the ERP) of the discounted dividends and the terminal value have a dramatic impact on the value of the stock but the delta between changes in volatility and changes in the stock value will move lower with higher volatility levels. 

As a side note, the above is the reason why I don't like comparing bonds and equities in the first place, something that the CAPM forces you to do. Why would I compare two assets with fundamentally different volatility, especially in regards to the terminal value (remember that a bond terminal value under no default is known in advance).

But the market also has a tautological statement that we can use to value assets. Under the assumption of no arbitrage (a very important assumption which I think "mostly" holds), we know that the put-call parity holds at all times:

which can also be re-written as

again, re-arranging we get

What conclusions can we draw from the above? 

First of all, it becomes quite clear that the relationship between stock and bond yields is "unknowable": Call - Put is positively correlated with interest rates (rho), PV(strike) is clearly negatively correlated with interest rates and finally PV(dividends) has a lets say complicated relationship with interest rates (do dividend grow faster than the reduction in the present value?).

Second, and this is the main conclusion from the long diatribe, it becomes quite clear that the future value of an asset is positively correlated with changes in implied volatility. The distribution of outcomes identified by Call - Put increases with implied volatility (i.e. the distribution is more spread out). 

What I find interesting is that when volatility rises, we generally say that "the market is less certain about the future" when we should actually be saying, the market is discounting more varied outcomes. (and here comes the key sentence) Hence, the potential for the realized volatility to be higher than the implied volatility is dramatically reduced the higher the implied volatility, offering the potential for positive gamma P&L. This is the source of the forward returns and why I think of equities as discounted outcomes and not discounted cash flows.

Of course, this falls apart if you don't agree with the no arbitrage environment (which does sometimes happen).