Re: Greenspan's Ponzi Scheme

[Eugene Coyle <eugenecoyle@xxxxxxx>]

The Gordon Growth model, from some ten-year old testimony of mine.  Plus a little more, which I essentially learned from Robin Marris.  Sorry, the drawing doesn't show up here.

Gene Coyle

The Valuation Of A Utility's Common Stock
    In order to consider how a utility's concern with competitiveness is reflected in its behavior, we now briefly turn to finance theory to consider how the utility's rate of growth, and the rate of return it earns, interact to affect the value of the shares.
    The interest of the shareholders of a utility is in maximizing the value of the shares on the stock market.  The value of the common shares as established in the market is dependent in a significant way on policies followed by the utility.  The financial decisions taken by the utility
 and, of particular interest here, the marketing strategy selected by the utility, are taken into consideration by the stock market.  The rate of growth of the utility affects its share price.

Stock Market Value and the Rate of Growth
      To explain how the stock market values growth in profits, as distinct from size of profits, we will use a familiar financial model.  The Discounted Cash Flow (DCF) theory is widely used in CPUC proceedings determining the cost of capital.
  The value of a share is taken to be the Present Value of a stream of future receipts.
  The future receipts are the dividend in each year and the price the stock is expected to sell for at some future time.  The price it is expected to sell for, though, is the Present Value at that time of the stream of future receipts then expected.  Thus we can think of the value now as the Present Value of the expected dividends in each year and a liquidating dividend at the time of the sale of the share.  This can be written in equation form as follows:

(1)    Pø  =    
F(D1,(1+k))   +   
F(D2,(1+k)2)   +   . . .   +  
F(D∞, (1+k)∞)

where Pø is the current price of a share of stock (its Present Value), D1 is the dividend per share expected to be paid in period one, and k is the rate of return which investors require on the stock, i. e. it is the cost of equity capital.  We can write this more generally as
(2)    Pø    =   
SU(t=1,∞,
F(Dt,(1+k)t))
   
where Dt is the dividend per share expected to be paid in period t.
    If dividends are expected to grow at a constant rate, g, in keeping with growth in earnings, equation (2) becomes

(3)    Pø    =   
F(D0(1+g),(1+k)1)  +  
F(D0(1+g)2,(1+k)2)    +  . . . +  
F(D0(1+g)∞, (1+k)∞ )

where D0 is the dividend per share at time 0.
    If dividends are expected to grow at rate "g", and k is greater than g, this equation can be shown to be

(4)    Pø    =      
F(D1,(k - g))

Rearranging terms, the cost of equity capital is then

(5)    k    =    
F(D1,  Pø)   +   g
    This is the familiar form seen in cost of capital proceedings.  In those proceedings, k, the cost of capital, is what the CPUC is attempting to determine.  Here we are more interested in considering g.  In equation (5) the value of the term g is the growth rate of dividends expected by investors.  The higher the rate of growth, the higher would be the required rate of return, k, other things being equal.
    What exactly is "g"?  In the equation it is the expected rate of growth of dividends, but it is more than that.  In the long run earnings must grow at the same rate as dividends, in order to support the dividend.  Productive capacity and sales must grow at the same rate as earnings so that the rate of return will not rise or fall, so that the utility has sufficient capacity to provide the service, that it does not have growing excess capacity, and so on.  Thus "g" is the growth rate of dividends, earnings, capacity, and sales. 
    In the equations we see that the cost of capital, k, depends in part on the rate of growth, i. e. we can say that k is a function of g, which we can write as k = k(g).  Beyond the equations, it is fair to say that it is generally accepted in Finance theory that k is an increasing function of the rate of growth.
  We can make the valuation model richer by explicitly adding a term, r, the rate of return expected to be earned.  The rate of return expected to be earned is dependent both on the rate the Commission allows and also on the expected rate of growth.  This last is because investors may be less confident that a given rate of return can actually be realized at higher and higher rates of growth.  We write r as a function of the rate of growth as r = r(g).  Finally, we can add a term, "v" which is the value of the shares on the stock market, to incorporate all the relationships.  We write, therefore,
    (6)       r    =     r (g)
    (7)    k    =      k (g)
    (8)    v    =    v (r, k, g)
    The value of the shares, v, depends on the rate of growth of earnings, as well as on the rate of return.
    We are now in a position to consider how the rate of return allowed and earned affects the value of the shares.  Recall that maximization of the value of the shares is the goal of the shareholders.
    Figure 1 depicts the relationship between the share price, the rate of growth, and the rate of return.
Figure 1

    The value of a share, v, is a function of the rate of growth, the cost of capital, and the rate of return.  For each given rate of return, the value of a share rises with the growth rate, reaches a maximum, and then declines, as depicted in Figure 1.
    We can see from Figure 1 that, other things being equal, a lower allowed rate of return will lead to the utility choosing a slower rate of growth of investment so as to maximize its value on the stock market.  Utilities can adopt policies to slow growth, for example rate design changes, when it is in their interest to do so.  It is in their interest to do so when doing so will raise the value of the shares on the stock market.
    Regulators have, in cost of capital proceedings, followed the dictates of the Hope and Bluefield cases.
  The US Supreme Court in Bluefield required that the return "… enable [the utility] to raise the money necessary for the proper discharge of its public duties."  In the Hope case the Court said the return "should be sufficient to assure confidence in its credit and to attract capital."  The traditional regulatory approach attempts to provide the utility the opportunity "… to raise the money necessary for the proper discharge of its public duties." by accepting "g" as a given, unaffected by the utility's strategic decisions.  What may appear to be the rate of growth of energy demand that the community wants and is willing to pay for, has been raised by the utility as it reacts to the regulatory constraint by using its monopoly earning power to increase its growth rate.  It does this, as we described earlier, by designing rates which may be unremunerative but which increase sales.




dsquared@xxxxxxxxxxxxxxxx wrote:

On Sat, 29 Nov 2003 03:23:40 -0500, Michael Pollak
wrote:
  

Had there been such a miracle, then some
though not all of the
    

higher valuations would be justified; if the S&P 500
      

Index is selling on
    

20 times earnings, and productivity growth undergoes
      

a secular and
    

permanent increase of 1 percent per annum, without a
      

change in real
    

interest rates, then the S&P 500 should start
      

selling
  

on 25 times
    

earnings (an earnings yield of 4 percent rather than
      

5 percent.)

Could somebody briefly explicate the math in that last
sentence?

    

A somewhat garbled but not absolutely egregious use of
the good old "Gordon Growth Model".

Basically, if you're given the responsibility of
valuing a series of cashflows which stretches off into
perpetuity and grows over time (like a stylised version
of common stock earnings), then the GG model is a
reasonable way of going about it.

Where D is today's earnings, R is the discount rate and
G is the rate at which the cashflows grow, then the
value of a growing series of dividends is given by V=
D/(R-G).  The derivation of the formula shouldn't be
hard to find in a finance textbook or via Google, but I
always contrive to screw it up in some way or other, so
I'm not going to try here.

Hence, if we say that, for a decent average of US
companies, R might be 10% and G might be 5%, then V/D
(the price/earnings ratio) would be 1/(10% - 5%) =
1/0.05 = 20.

Bung an extra 1% onto the growth rate for "productivity
miracles", and you get 1/0.04 = 25.

All the above is of course dependent on the discount
rate being a valid concept, average productivity growth
being a valid concept across sectors, plus the growth
rates being in the "right" sort of order of magnitude
(as you can see, this model will blow up as R-G
approaches zero).  etc etc.  But I'm pretty sure that's
what they mean.

dd



  

Michael