- CFA Exams
- 2023 Level II
- Topic 5. Equity Valuation
- Learning Module 26. Residual Income Valuation
- Subject 4. Fundamental Determinants of Residual Income

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##### Subject 4. Fundamental Determinants of Residual Income PDF Download

In the last section we developed the following expression:V

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.

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.P

P

_{0}/B_{0}= (ROE - g) / (r - g) = 1 + (ROE - r) / (r - g)As P

_{0}= V_{0}, this expression is equivalent to:

_{0}= B

_{0}+ B

_{0}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 B_{0}, and the present value of the expected level stream of residual income, (ROE - r) x B_{0}.The second term B

_{0}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 B

_{0}+ ((ROE - r)/(r - g)) x B_{0}= 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.

_{0}/B

_{0}= 1 + Present value of expected future RIs/B

_{0}

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

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**User Contributed Comments**
2

User |
Comment |
---|---|

Allen88 |
(ROE - r + r -g)/(r-g)= (ROE -r)/(r-g) + 1. |

davidt876 |
an implicit assumption is that the retention ratio cannot = 1 forever otherwise g = ROE*1 = ROE and then the starting equation doesn't make sense: P0/B0 = (ROE-g)/(r-g) = 0/(r-g) = 0 |

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