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Stocks are financial instruments that represent ownership in a company. They are a key component of equity financing and provide investors with potential returns through capital appreciation and dividends. Stocks are traded on stock exchanges, where their prices fluctuate based on supply and demand, company performance, and broader market conditions.

Stocks

• Share Capital represents the equity claim of shareholders, equal to the stock face value times the number of shares issued.
• In Germany, there’s a distinction of share capital (Grundkapital) and capital reserves (Kapitalrücklage), which is the agio (difference to face value) paid y the investors when the shares were issued. This distinction is not made in the US and is economically irrelevant.

Dividends

Cash Dividends or Stock Dividends (e.g. Deutsche Telekom 2013) are paid yearly (in Germany) or quarterly (in the US). Paying dividends is optional; they are the remaining profit after all expenses, interest, and taxes. Companies with stable cash flows often pay dividends, while high-growth companies may reinvest profits instead.

• Dividend per Share: $DPS=\frac{\text{Dividends}}{\text{Number of Shares}}$
• Dividend Yield: $\text{Dividend Yield}=\frac{DPS}{\text{Share Price}}$
• Payout Ratio: $\text{Payout Ratio}=\frac{\text{Dividends}}{\text{Earnings}}$

Pricing

Law of One Price

Discounted Cashflow Principle: The price of a stock should equal the present value of its future cash flows to shareholders. The price is independent of the company’s financing decisions, and is determined by the expected dividends and the required rate of return (cost of equity).

In reality, however, the value of a stock differs from its price due to factors such as market sentiment, liquidity, and information asymmetry. In a perfect capital market (which this lecture assumes), there’s a large number of investors without market power, equal access to information, rational economic actors, and no transaction costs.

Stock Returns

For an investor with a one year time horizon, for example, they would buy the stock at $P_0$, have the right to receive dividends $D_1$ and sell the stock at $P_1$ after one year. The expected return is then:
$r_e=\frac{D_1+P_1}{P_0}-1=\frac{D_1}{P_0}+\frac{P_1-P_0}{P_0}$
The first term is the dividend yield, and the second term is the capital gain yield.

Dividend Discount Model

For any time horizon, the price of a stock equals the present value of all expected future dividends:
$P_0={\sum }_{t=1}^{\infty }\frac{D_t}{(1+r_e{)}^t}$
Due to the investment horizon, the present value of the future price $P_N$ diverges to zero.

Estimating Future Dividends

Dividend Growth

To achieve earnings growth, additional investments are required, which can be financed through retained earnings or new equity issuance. The growth rate of dividends ($w$) and net investment $I_t$ (capital not paid out as dividends) depend on the dividend ratio $p$.
If a company has a constant payout ratio policy, $w$ can be predicted by the return on equity (ROE) and the payout ratio: $w=ROE\cdot (1-p)$.

To find today’s price, investors need to predict future dividends until infinity. There are multiple common models for this:

Zero Growth Model

$D_t=D$. The company pays a constant dividend $D$ every year, and the stock price is:
$P_0=\frac{D}{r_e}$
This is not very realistic, but it can be used for companies that have stable dividends, such as utilities.

Constant Growth Model

$D_t=D_{t-1}\cdot (1+w)$. The company pays dividends that grow geometrically at a constant rate $w$ every year. The stock price is:
$P_0=\frac{D_1}{r_e-w},D_1=D_0\cdot (1+w)$
This model is more realistic for companies that are expected to grow at a constant rate, such as mature companies in stable industries with small risk (low $r_e$) and slow growth (low $w$).

Time-Varying Growth Model

$D_1=D_0\cdot (1+w_a)$, then $D_{N+1}=D_N\cdot (1+w_b)$. This model assumes constant growth with a factor $w_a$ for the first $N$ years, then either no growth or a different constant growth $w_b$ thereafter.

The stock price in the no-growth case is:
$P_0=\underset{\text{growth }w\text{ (until }N\text{)}}{\underbrace{\sum _{t=1}^N\frac{D_0\cdot (1+w{)}^t}{(1+r_e{)}^t}}}+\underset{\text{perpetuity at time }N}{\underbrace{\frac{D_0\cdot (1+w{)}^N}{r_e}}}\cdot \underset{\text{discount factor}}{\underbrace{\frac{1}{(1+r_e{)}^N}}}$

For the $w_b$ case, the second term changes:
$(1):P_0=\underset{\text{growth }{w}_a\text{ (until }N\text{)}}{\underbrace{\sum _{t=1}^N\frac{D_0\cdot (1+w_a{)}^t}{(1+r_e{)}^t}}}+\underset{\text{perpetuity with growth }{w}_b\text{ at time }N}{\underbrace{\frac{D_0\cdot (1+w_a{)}^N\cdot (1+w_b)}{r_e-w_b}}}\cdot \underset{\text{discount factor}}{\underbrace{\frac{1}{(1+r_e{)}^N}}}$