In the last section we developed the following expression:

P₀/B₀ = (ROE - g) / (r - g) = 1 + (ROE - r) / (r - g)

As P₀ = V₀, this expression is equivalent to:

V₀ = B₀ + B₀ x (ROE - r) / (r - g)

This expression is called a single-stage residual income model. It splits intrinsic value of a stock into two components: the book value per share B₀, and the present value of the expected level stream of residual income, (ROE - r) x B₀.

The second term B₀ x (ROE - r) / (r - g) = extra value from firm's ability to produce returns > cost of equity = PV of firm's expected economic profits.

• If ROE = r, RI = 0 and MV = BV.
• If ROE > r, RI > 0 and MV > BV.

The key point to remember from this discussion is that residual income depends on the present value of the company's future profitability (ROE) and the current book value of equity. These two factors are the fundamental determinants of residual income.

Example: Base Case

All equity firm. Equity capital = $1,000. NI = $250 into perpetuity. r = 15%. long-term growth rate = 5%.

ROE = 250/1000 = 0.25. V B₀ + ((ROE - r)/(r - g)) x B₀ = 1,000 + ((0.25 - 0.15)/(0.15 - 0.05)) x 1,000 = $2,000.

Example: New Investment w/ ROI > r

New investment = $1,000 addition capital. ROI = 20%, which is less than the current ROE, but greater than r which remains at 15%.

Equity capital: 1,000 + 1,000 = $2,000.
NI = 250 + 200 = $450 into perpetuity.
ROE = 450/2,000 = 22.5%.
V = 2,000 + ((0.225 - 0.15)/(0.15 - 0.05)) x 2,000 = $3,500.

Although firm value increased by $1,500 only $500 of value has been created because $1,000 of the growth was additional invested capital.

Example: New Investment w/ ROI < r

New Investment = $500 additional capital. ROI = 10% which is positive, but less than r.

Equity capital: $1,000 + $500 = $1,500.
NI = $250 + $50 = $300 into perpetuity.
ROE = 300/1,500 = 20%.
V = 1,500 + ((0.20 - 0.15)/(0.15 - 0.05)) x 1,500 = $2,250.

Although firm value increased by $225, the initial investment was $500. Therefore, $250 of value was destroyed because ROI < r.

Residual income valuation and justified P/B ratio relationship

A justified price multiple for the stock is the estimated fair value of that multiple. RIMs can be used to establish market multiples such as P/E or P/B. For example, justified P/B is directly linked to expected future RI.

P₀/B₀ = 1 + Present value of expected future RIs/B₀

When the present value of expected future residual income is positive (negative), the justified P/B based on fundamentals is greater (less) than 1.