8.6  Valuation using Residual Income

Under clean surplus accounting a₁ = 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 now re-write equation 1) invoking the clean surplus relationship above as:

Intrinsic Value=V₀ = (CI₁ – (BV₁ – BV₀) + V₁)/(1 + k_e)

                                  = (BV₀ + CI₁)/(1+k_e) + (V₁ - BV₁)/(1+k_e)                   4)                                 

Observe that BV₀/(1 + k_e) can be equivalently expressed by adding and subtracting BV₀  as BV₀/(1+k_e) + BV₀  – BV₀ and then re-arranging to yield: 

BV₀ + BV₀/(1+k_e) – BV₀ = BV₀ + (BV₀ – (1+k_e)BV₀)/(1+k_e)  = BV₀ -k_eBV₀/(1+k_e)

Substituting back into 2) yields:

V₀ = BV₀ + (CI₁ – k_eBV₀)/(1+k_e) + (V₁ – BV₁)/(1+k_e)                                 5)                       

Again by substituting for V₁ and re-arranging you can then establish the equivalent form that is a function of V₂ and so on.  We now arrive at the residual income valuation model:

V₀ = BV₀ + RI₁/(1+ k_e) + RI₂/(1+k_e)² + ……..                                            6)                  

This expression starts with the book value per share and adjusts it by the present value of all future residual income.  Residual income is a concept of how the book value of shareholder’s equity grows over time under clean surplus accounting taking into account the opportunity cost of capital.   As a result, the RIV valuation model provides a conceptually coherent framework that ties together clean surplus accounting, book values, comprehensive income and opportunity costs as described above.  It is a technique that applies to any stock irrespective of whether or not it pays an accounting dividend.