Beta - Derivation for Conceptual Understanding

A few days back as stock volatility was being discussed in class and Bhatia Sir had mentioned how Beta (hereafter denoted as "B") was the ratio of covariance (stock,market)/variance (market); I tried to search the mathematical relation for the same and wondered how the risk premium formula is deduced as well. I couldn't find it so I tried myself; I think I have got the derivation of B and have linked it to both - regression theory and risk premium approach. I present that derivation here in the hope that it will help clarify the concept of beta for all those who need it as it did for me -

By definition - beta is the % change in price of a stock to the % change in the price of market as whole. Henceforth, let "y" denote price of stock and "x" denote that of market.

Therefore,
                     B = (dy/y)/(dx/x) -----------i.e.  (dy/y) = B*(dx/x)..........Equation A

-- Integreting both sides,
        
1)                              ln y = B*ln x + ln a ................................Equation I

(for non-engineers, ln is log to the base "e"; "ln a" is an arbitary constant)

Rearranging above equation, we have -

                                y = a*(x^B) ..................................Equation II

Above equation shows that by definition of Beta, stock prices have a parabolic relation with market swings.
As beta is a historical concept, a scattered chart for its values is observed relating price of stock and market; to have a better view we make an estimate using regression analysis -
modifying Equation I -
                   y* (y star, i.e estimate of y) = B*x + c.............Equation III

(This is the linear regression model and hence, B = Cov(y*,x)/Var (x)....hence proved)

Above was the first part of the proof; Now to link this concept with the risk premium of the stock I refer to the Dividend growth model where price is the present value of all future benefits.

Therefore,
           P = D/(K-G)             and             p = d/(k-g)

(Capital letters for the market as a whole and small for a certain stock)

K,k is the cost of capital; G,g is the growth rate; P,p is the price and D,d is total dividend of the market and dividend of the stock resp.

For the market, taking ln on both sides of the equation -

                                         ln P = ln D - ln (K-G)

Differentiating both sides and accounting for D as a constant, we arrive at -

                                    dP/P = (-d(K-G))/(K-G).........Equation IV
Similarly for the stock,
                                    dp/p = (-d(k-g))/(k-g).........Equation V

Dividing Eq. V by Eq. IV, we get ---

                     (dp/p)/(dP/P) = {d(k-g)/(k-g)}/{d(K-G)/(K-G)}..............Equation VI

On observation, we realize that the LHS is nothing but B as our stock indices are based on a free-float market capitalization method.
Compare the RHS to Equation III and using the ref of Equation A, we can say that (k-g) is analogous to "y*" and (K-G) is analogous to "x" -

Therefore,
               k-g  =  B*(K-G) + c........Equation VII

Now, let c = m + n*B.....Equation VIII

Reminding all that this is an estimate as we are dealing with historical data and only with great trials can there be increment in accuracy.
Now, put B = 1 in Eq. II, we get  y = a*x (i.e with zero intercept or c = 0)
and when B = 0 in Eq. II, we get y = a (i.e constant or horizontal line where a would be the risk free rate of return or rf)
To re-iterate, while substituting value of B in Eq. II, changes in Eq. III have to be observed simultaneously as well as it is an estimate for Eq. II; infact, from Eq. III we actually get the values of "c"; using the conditions for B and c, we substitute in Eq. VIII,
for B= 1,   m = - n which implies that c = m*(1-B) and
for B = 0, c = rf which implies rf = m finally leading to ----

                           c = rf*(1-B)

substitute above relation into Eq. VIII....

Therefore,

                                          k-g = B*(K-G) + rf*(1-B)
also, g = B*G,
hence,            
                               k - B*G = B*K - B*G + rf - rf*B

Rearranging,
                                  k   =   rf   +   B*(K - rf)           Risk Premium Equation

(Note: I can also upload the derivation of the formula for B in regression analysis in terms of all the summation values; since it was not relevant here plus involving partial differentiation and simultaneous equations, I did not do it. If anyone wants, I can do that too)

I hope it helps all of us to crystallize our concepts further when it comes to volatility and make us great investors in the future.