University Study Notes: Dividend Discount Models and Stock Valuation

Definition: This model assumes that dividends remain constant forever. It is most appropriate for valuing preference shares, which typically feature a fixed, almost contractual dividend payment.
The Perpetuity Principle: Because a share represents ownership in a company that is expected to continue indefinitely, the value is calculated as a perpetuity.
Core Formula: * $P_0 = \frac{D}{r}$ * Where $P_0$ is the current price, $D$ is the constant dividend, and $r$ is the required rate of return.

Quarterly Dividends: If a company pays dividends quarterly (rather than annually), the time value of money arithmetic must be adjusted, similar to standard compounding periods. * Adjusting Rate ( $r$ ): Divide the annual percentage rate (APR) by the number of periods ( $m$ ). For quarterly, $r_{period} = \frac{APR}{4}$ . * Dividend Frequency: Use the specific dividend payment occurring each quarter. * Infinite Payments: While the number of payments is infinite ( $\infty \times 4$ ), the formula remains a perpetuity based on the periodic rate.
Relative Contribution to Value: * Share value is derived from the present value ( $PV$ ) of all future dividends, not the final principal. * The principle is that the "final value" of a share is so far in the future that its present value is negligible. * Example Proof: If a share is worth $\$100$ in $100$ years and the required return is $10\%$ , its present value is: * $PV = \frac{100}{(1.1)^{100}} = 0.007$ * This represents seven-thousandths of a dollar (less than a cent), demonstrating that distant future values contribute virtually nothing to the price today. Most of a share's value is derived from the first few decades of dividend payments (e.g., the first $10$ payments).

Overview: Named after the economist Gordon, this model assumes dividends grow at a constant rate ( $g$ ) forever.
Rationale for Growth: Marketing and economic principles suggest that if a company is not growing, it is falling behind competitors. Companies aim to at least outgrow inflation to maintain strength and cash flow.
Core Formula (The Gordon Growth Model): * $P_0 = \frac{D_1}{r - g}$ * Crucial Condition: g < r. The growth rate must be lower than the required rate of return for the perpetuity to have a finite value. Logically, if a company grows at an extreme rate, it must be taking on more risk, which in turn increases the required return ( $r$ ).
D1 vs. D0: For all dividend models, the calculation must start with the next dividend due ( $D_1$ ). If you are currently at time period zero ( $t=0$ ), the starting point is the dividend at $t=1$ .

Historical Analysis: Looking back at the company's recent growth to project forward.
The Accounting Perspective (Plowback Ratio): * Companies grow by making profits and reinvesting them into new assets rather than paying them all out as dividends. * Formula: $g = \text{Plowback Ratio} \times \text{Return on Equity (ROE)}$ . * Plowback Ratio: This is the percentage of earnings retained in the company. * $\text{Plowback Ratio} = 1 - \text{Dividend Payout Ratio}$ .
Course Context: While these accounting rituals are useful for estimation, the course focuses primarily on the financial application of the rates rather than complex accounting ratio analysis (with the exception of Topic 6: Working Capital Management).

Scenario Context: A practical problem styled after the TV series The Chase (with a nod to host Bradley Walsh).
Data provided: * Next expected dividend ( $D_1$ ) = $\$4$ * Growth rate ( $g$ ) = $6\%$ * Required return ( $r$ ) = $16\%$
Calculation 1: Current Price ( $P_0$ ): * $P_0 = \frac{4}{0.16 - 0.06} = \frac{4}{0.1} = \$40$ .
Calculation 2: Price in Year 4 ( $P_4$ ): * To find $P_4$ , you need the dividend for the next period, which is $D_5$ . * $D_5 = D_1 \times (1 + g)^4 = 4 \times (1.06)^4$ . * $P_4 = \frac{D_5}{r - g}$ .
Calculation 3: Implied Rate of Return: This involves finding the compound growth return between today and Year 4 by manipulating the present value/future value equation ( $PV = \frac{FV}{(1+r)^n}$ ).

The "Horizon" Metaphor: * Visible Horizon: Short-term period ( $0$ to $t$ ) where we can specifically forecast dividends based on supernormal growth estimates. * Over the Horizon: The infinite future after time $t$ where we assume a stable, constant growth rate or zero growth.
Step-by-Step Calculation Process: 1. Forecast Dividends: Specifically calculate every dividend from period $1$ through the horizon period $t$ (e.g., $D_1, D_2, \dots, D_5$ ). 2. Estimate Terminal Value: Go to the horizon period ( $t$ ) and calculate the price of all future dividends from that point onward ( $P_t$ ). Use the next dividend ( $D_{t+1}$ ) and apply the Gordon Growth Model ( $P_t = \frac{D_{t+1}}{r - g}$ ). 3. Present Value Summation: Discount all the individual visible dividends and the terminal price ( $P_t$ ) back to time zero using the required rate of return ( $r$ ). * $P_0 = \sum_{i=1}^{t} \frac{D_i}{(1+r)^i} + \frac{P_t}{(1+r)^t}$

Inputs: * Last dividend paid ( $D_0$ ) = $\$1$ * Growth Year 1 = $20\%$ * Growth Year 2 = $15\%$ * Growth Year 3+ (indefinite) = $5\%$ * Required Return ( $r$ ) = $20\%$
Dividend Calculations: * $D_1 = 1 \times 1.20 = \$1.20$ * $D_2 = 1.20 \times 1.15 = \$1.38$ * $D_3 = 1.38 \times 1.05 = \$1.449$
Terminal Value at Year 2: * $P_2 = \frac{D_3}{r - g} = \frac{1.449}{0.20 - 0.05} = \$9.66$
Current Price ( $P_0$ ): * $P_0 = \frac{D_1}{(1+r)^1} + \frac{D_2 + P_2}{(1+r)^2} = \frac{1.20}{1.20} + \frac{1.38 + 9.66}{(1.20)^2}$ * $P_0 = 1 + \frac{11.04}{1.44} = 1 + 7.666 = \$8.666$

New Zealand Context: Preference shares in NZ are typically issued with a fixed dividend rate as a percentage of face value and have no maturity date (perpetual).
International Variations: Some jurisdictions (e.g., USA) may issue preference shares with a fixed maturity date. For this course, assume no maturity date unless specified.
Formula: $r = \frac{D}{P_0}$ (used to find the implied rate of return given the market price).

Cheat Sheet: Students are allowed four A4 sides of notes for the exam. This can include formulas, worked examples, and complex derivations.
Workings Importance: In the exam, marks are awarded for the proper use of equations and identifying correct inputs, not just selecting the right formula.
Identifying Variables: Explicitly state your inputs (e.g., $r = 9.5\%$ , $D = \$4.125$ ) to ensure full marks even if the final arithmetic has an error.
CAPM Relationship: The required rate of return ( $r$ ) used in these dividend models is the same $r$ calculated using the Capital Asset Pricing Model ( $CAPM$ ). They are interchangeable approaches to determining the return a shareholder requires for a specific level of risk.