Stock valuation

A share of common stock is a claim on whatever is left of a company's cash flows after it pays its suppliers, employees, lenders and taxes. Unlike a bond, a share has no promised payments and no maturity; its value depends entirely on expectations about an uncertain future. Yet the same valuation principle applies: a share is worth the present value of the cash it is expected to deliver to its owner, discounted at a rate that reflects its risk. This lesson develops the dividend discount model, connects growth to reinvestment, values firms from their free cash flows, and explains how analysts use valuation multiples. It is not a guide to picking investments; it shows how financial managers and investors think about what equity is worth, which in turn tells a firm what its decisions do to its share price and what return its shareholders require.

Common and preferred shares

Common shareholders own the residual claim: they receive dividends if the board declares them, vote on directors and major transactions, and receive whatever remains after all other claims if the firm is liquidated. Their liability is limited to their investment. Some companies issue several classes of common shares with different voting rights. Dual-class structures, where founders or families keep control through multiple-voting shares, are relatively common among Canadian listed companies and increasingly common among US technology listings; they protect long-term strategies but weaken the accountability mechanisms described in lesson 1, as BUS-310 discusses.

Preferred shares sit between debt and common equity. They usually pay a fixed dividend that must be paid before any common dividend, and unpaid dividends on cumulative preferred shares accumulate. But preferred dividends are not a legal obligation like interest, and preferred holders usually cannot vote. A perpetual preferred share paying a fixed dividend is valued as a perpetuity, \( P = D/r \), so its required return is simply \( D/P \).

The dividend discount model

An investor who buys a share and holds it for a year receives a dividend and the price at the end of the year. That future price, in turn, depends on dividends after that year and the price after that, and so on. Carrying the argument forward, the price today equals the present value of all expected future dividends. If dividends are expected to grow at a constant rate \( g \) forever, the growing perpetuity formula from lesson 4 gives the constant-growth dividend discount model, often called the Gordon growth model.

Rearranging the Gordon formula shows that the required return is the dividend yield plus the growth rate: \( r = D_1/P_0 + g \). That decomposition is useful in reverse: given a share price, a forecast dividend and a growth rate, it estimates the return investors expect, one way of estimating the cost of equity in lesson 11.

Where growth comes from

Dividends can only grow if the firm reinvests some of its earnings in projects that earn a return. If a firm retains a fraction \( b \) of its earnings (the plowback or retention ratio) and earns a return on equity of ROE on new investment, earnings and dividends grow at the sustainable growth rate \( g = b \times ROE \). Growth is therefore not free: it is bought by paying out less today.

Whether that trade is worth it depends on whether the new investments earn more than the cost of equity. Split the share price into two parts: the value of the current earnings stream if the firm paid out everything and stopped growing, \( EPS_1/r \), and the present value of growth opportunities (PVGO). If reinvested earnings earn exactly \( r \), PVGO is zero and retention neither creates nor destroys value. If they earn more, PVGO is positive; if less, growth destroys value even though earnings rise.

Non-constant growth

Young firms often grow quickly for some years before settling to a steady rate, and many pay no dividends at first. A two-stage (or multi-stage) model forecasts dividends year by year during the fast-growth period, then uses the Gordon formula to value all dividends after that as a terminal value at the end of the period, and discounts everything back to today.

# Two-stage dividend discount model
def two_stage(d0, g_fast, years, g_long, r):
    pv, d = 0.0, d0
    for t in range(1, years + 1):
        d *= 1 + g_fast
        pv += d / (1 + r) ** t
    terminal = d * (1 + g_long) / (r - g_long)
    return pv, terminal / (1 + r) ** years

pv_div, pv_tv = two_stage(1.00, 0.15, 4, 0.04, 0.11)
print(f"PV of high-growth dividends {pv_div:.3f}  PV of terminal value {pv_tv:.3f}  price {pv_div + pv_tv:.2f}")
for g in (0.03, 0.04, 0.05):
    a, b = two_stage(1.00, 0.15, 4, g, 0.11)
    print(f"long-run growth {g:.0%}: price {a + b:.2f}")

PV of high-growth dividends 4.374  PV of terminal value 17.117  price 21.49
long-run growth 3%: price 19.21
long-run growth 4%: price 21.49
long-run growth 5%: price 24.54

Valuing the whole firm from free cash flow

Dividends are set by the board and many firms pay none, so analysts often value the whole business from its free cash flow instead, the measure defined in lesson 2. The discounted cash flow (DCF) approach forecasts free cash flow for several years, adds a terminal value, and discounts at the weighted average cost of capital (WACC, lesson 11), because free cash flow belongs to all investors, lenders and shareholders alike. The result is enterprise value. Subtracting the market value of debt and adding cash gives equity value; dividing by shares outstanding gives value per share.

Valuation multiples

In practice many valuations rely on multiples: the price-earnings (P/E) ratio, enterprise value to EBITDA, price to book, or price to sales of comparable companies. If similar firms trade at 15 times earnings and the firm earns $3 per share, a first estimate of value is $45. Multiples are quick, use market information and are easy to explain. But they hide assumptions rather than avoiding them: a multiple is just a compressed DCF, and the Gordon model shows that a P/E ratio depends on payout, growth and risk, since \( P_0/EPS_1 = (1-b)/(r-g) \). Comparable firms are never identical, and if the whole market is mispriced, multiples carry the mispricing over.

Market efficiency

If shares are valued by thousands of informed investors competing to find mispricing, prices should quickly reflect available information. This is the efficient market hypothesis. In its weak form, past prices cannot be used to predict future returns; in its semi-strong form, prices reflect all public information; in its strong form, even private information. The evidence suggests that large, liquid markets are reasonably efficient with respect to public information, which is why most actively managed funds fail to beat low-cost index funds after fees, but there are well-documented anomalies, bubbles and periods of mispricing. For a financial manager, efficiency has a practical message: the market price is usually a better guide to value than a single analyst's model, and a firm cannot easily create value by timing share issues or by accounting changes that do not affect cash flow.