Dividend and Residual

Contents - Index

Dividend and Residual Income Valuation: Derivations
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If stock markets are efficient then the intrinsic value equals the spot stock price. The spot stock price can in turn be expressed as the present value of next period's expected dividend plus the end of period stock price discounted by the cost of equity capital.
Intrinsic Value = P0 = (d1 + P1)/(1 + ke)                                                                                1)
where d1 is next period's dividend, P1 is the end of period price and ke is the stock's cost of equity capital.
By recursively re-applying the definition of P for P1, P2 etc., yields the fundamental relationship that the intrinsic value of a stock equals the present value of future dividends discounted at the stock's cost of equity capital.
P0 = d1/(1+ke) + d2/(1+ke)2 + d2/(1+ke)2 + ЕЕЕЕЕЕ.                                                       2)
Equivalently equation 1) can be rewritten by invoking the clean surplus relationship that underlies the change in the fundamental accounting identity (A - L) = OE. Clean surplus accounting captures the idea that all changes in shareholder equity, (OE1 - OE0), not involving shareholders such as dividends, Treasury stock or new issues) pass through the income statement. This is the case for the accounting concept of "Comprehensive Income"as opposed to "Net Income" primarily because of foreign currency translation adjustments, derivative accounting and certain pension liability adjustments. Further if we divide the accounting identity by shares outstanding then this is the Book Value (BV) per share.
Under clean surplus accounting d1 = CI1 - (BV1 - BV0), where CI1 is comprehensive income for the next period, BV1 is end of period book value per share and 0 denotes the present.
We can then re-write equation 1) invoking the clean surplus relationship above as:
Intrinsic Value (P0) = (CI1 - (BV1 - BV0) + P1)/(1 + ke)
                                  = (BV0 + CI1)/(1+ke) + (P1 - BV1)/(1+ke)                                             3)
Observe that BV0/(1 + ke) can be equivalently expressed by adding and subtracting BV0 as BV0/(1+ke) + BV0 - BV0 and then re-arranging to yield:
BV0 + BV0/(1+ke) - BV0 => BV0 + (BV0 - (1+ke)BV0)/(1+ke) = BV0 -keBV0/(1+ke)
Substituting back into 2) yields:
P0 = BV0 + (CI1 - keBV0)/(1+ke) + (P1 - BV1)/(1+ke)                                                            4)
Again by recursively substituting for P results in the RIV model of intrinsic value:
P0 = BV0 + Residual Income1/(1+ ke) + Residual income2/(1+ke)2 + ЕЕ..                       5)
That is the above expression starts or anchors on the spot book value per share and then adjusts the present value of all future residual income. Residual income is a concept of ongoing income that continues to be earned over time net of the required rate of return that investors expect from the beginning period book value per share.   As a result, RIV as depicted in 5) is equivalent to the dividend model and provides a conceptually coherent framework that ties together clean surplus accounting, book values and income as derived above.
Finally, observe that the equality above to the spot stock price is under the assumption that markets are ex ante efficient. If this assumption fails to hold then the RHS can be different from the LHS and the intrinsic value can be assessed by working directly with the RHS.
Summary: RIV Approach to Estimating Intrinsic Value
Intrinsic value = Book value per share + Present value of Residual Earnings discounted back at the stock's cost of equity capital (i.e., investors' required rate of return from investing in stocks).