DISCOUNTED CASH FLOW MODEL DPU EXHIBIT 6.1 DOCKET N UMBER 02-057-02
The Discounted Cash Flow (DCF) model is based on the theory that the current price of a stock embodies all future income generated by the stock discounted at an appropriate rate. Algebraically, assuming the stock is held indefinitely, the current price of a stock can be represented as,
DD D D 3 2 P1 2 3. . . 0 ( )1 ( ) 1 () 1 k k k () 1 k
where P0 is the stock’s current price, Di is the expected dividend to be paid in the future period i, and k is the discount rate. The discount rate k is also the investor’s opportunity cost of investing in the stock and, thus, is the investor’s required rate of return on equity. The key to estimating the required return is to solve Equation 1, under various assumptions, for k. To solve for k, define P n as,
D DD D 3 n 2 P 1 2 3. n . . n ( )( ) ( ) 1 1 1 k k k () 1 k
Equation 1 and Equation 2 are equivalent as n approaches infinity. Equation 2 will serve as the basis for the following derivations.
CONSTANT GROWTH DCF MODEL If dividends grow at a constant rate, g, then Equation 2 can be rewritten as,
2 n 1 D ) () 1 11 g g ()1 g 1 ( D D1D 1 P 2 3. . . n n ()() () 1 1 1 k k k () 1 k
Now define P n+1 as (1 + g)*P n /(1 + k); that is,
1 ( DD1 11 1 (2 1 g ) g ) (n (n g ) DD g ) 1 1 P 2 1 3 . . . n n 1 1 ( 1 ( 1 k ) k ) ( 1 (n 1 k ) k )
Thus,
n (g 1D ) ( g D 1 k ) P1 P 1 n nP n n 1 (k (k (k 1 1 1 ) ) )
Discounted Cash Flow Model Docket No. 02-057-02
DPU Exhibit 6.1 DPU Witness: Artie Powell Page 22
Solving for P n, we get,
D ( gD 1) 1 n P 1 n ( (1) kg kg ) ) k ( n
If the discount rate k is strictly greater than the expected growth rate g, then,
D lm 0 1 i P P n n ( ) kg
Solving Equation 7 for k yields the familiar constant growth DCF model:
k D 1 g P 0
where D1 is defined as the dividend to be paid in the next period, P 0 is the current price, and g is the dividend growth rate.