Valuing stocks: dividend discount and free cash flow models

Owen's colleagues at Prairie Ridge Energy often debate whether their shares are cheap. The stock trades at 38 dollars on the New York Stock Exchange, pays an annual dividend of 2.10 dollars and has raised it every year for six years. Some analysts argue that the shares are undervalued because the dividend yield of 5.5 percent is above the yield on Treasury notes; others say the yield is high because investors doubt the dividend can keep growing. Dr. Varga asks the cohort to turn the debate into numbers: what growth in dividends is the market assuming at 38 dollars, and is that plausible?

A share as a stream of cash flows

A share of common stock entitles its owner to a proportional part of whatever the company distributes, in dividends and buybacks, after all other claims are paid, plus whatever is left if the company is sold or wound up. The value of a share, like any asset, is the present value of the cash it will pay its owner. An investor who buys today and sells in a year receives the dividend plus the sale price. But the sale price in a year depends on dividends after that, and so on. Following the chain forward shows that the value today is the present value of all future dividends.

Here the discount rate is the cost of equity, the return shareholders require given the risk of the shares. Lesson 8 shows how to estimate it with the capital asset pricing model. For now, take Prairie Ridge's cost of equity as 9.5 percent.

The Gordon growth model

If dividends are expected to grow at a constant rate forever, the infinite sum collapses to the growing perpetuity formula from lesson 2. This is the dividend discount model in its simplest form, associated with the economist Myron Gordon.

The model can be turned around. Solving for the cost of equity gives the dividend yield plus growth; solving for growth gives the growth rate the market is implicitly assuming. For Prairie Ridge, next year's dividend at the current rate of increase would be about 2.18 dollars. At a price of 38 dollars and a cost of equity of 9.5 percent, the implied perpetual growth rate is 9.5 percent minus 2.18 divided by 38, or about 3.8 percent. The debate becomes clearer. The bulls are not really arguing that a 5.5 percent yield is high; they are arguing that dividends can grow faster than about 3.8 percent a year forever. For an energy producer whose output and prices are uncertain, that is a strong claim.

The Gordon model is extremely sensitive to its inputs when growth is close to the discount rate. At a cost of equity of 9.5 percent, raising assumed growth from 3.8 to 4.8 percent lifts the value from 38 dollars to about 46 dollars. A growth rate equal to or above the cost of equity gives a meaningless infinite or negative value. Long-run growth for a mature company cannot exceed the growth of the economy for long, which in nominal terms has averaged roughly 4 to 5 percent in Canada and the United States, so perpetual growth rates above that range deserve suspicion.

Where growth comes from

Dividends can grow only if the company's earnings grow, and earnings grow when the company reinvests part of them at a positive return. If a firm retains a fraction b of its earnings, called the plowback or retention ratio, and earns a return on equity of ROE on what it reinvests, its sustainable growth rate is b times ROE. Prairie Ridge pays out about 60 percent of its earnings and earns a return on equity of about 10 percent, so its sustainable growth is 0.4 times 10 percent, or 4 percent, close to what the market is assuming.

This link has an important implication. Growth by itself does not create value. If a firm reinvests at a return on equity equal to its cost of equity, retaining earnings raises future dividends by exactly enough to compensate shareholders for the dividends given up, and the share price is unchanged. Retaining earnings creates value only when the return on reinvestment exceeds the cost of equity. One way to see this is to split a share's value into the value of its current earnings as if they were paid out forever with no growth, EPS divided by the cost of equity, and the present value of growth opportunities.

For a company whose return on new investment equals its cost of capital, PVGO is zero. For a company with valuable new projects, it is positive, sometimes most of the share price. For a company that reinvests in projects earning less than their cost, PVGO is negative, and shareholders would be better off receiving the cash.

Two-stage models

Few companies grow at a constant rate forever. A young company may grow quickly for several years and then slow to a mature rate. A two-stage model handles this by forecasting dividends explicitly for the high-growth years and using the Gordon formula for the value at the end of that period, called the terminal or continuing value, which is then discounted back to today.

# Gordon growth, implied growth, and a two-stage dividend discount model
def gordon(d1, r, g):
    assert g < r, "growth must be below the discount rate"
    return d1 / (r - g)

def implied_growth(price, d1, r):
    return r - d1 / price

def two_stage(d0, g_high, years, g_long, r):
    value, d = 0.0, d0
    for t in range(1, years + 1):
        d *= 1 + g_high
        value += d / (1 + r) ** t
    terminal = d * (1 + g_long) / (r - g_long)
    return value + terminal / (1 + r) ** years, terminal

r = 0.095
print(f"Implied growth at 38.00: {implied_growth(38.00, 2.18, r):.2%}")
for g in (0.030, 0.038, 0.048):
    print(f"Gordon value at g = {g:.1%}: {gordon(2.18, r, g):6.2f}")
v, tv = two_stage(2.10, 0.07, 5, 0.03, r)
print(f"Two-stage: 7% for 5 years then 3%: {v:.2f} (terminal value {tv:.2f} in year 5)")

Implied growth at 38.00: 3.76%
Gordon value at g = 3.0%:  33.54
Gordon value at g = 3.8%:  38.25
Gordon value at g = 4.8%:  46.38
Two-stage: 7% for 5 years then 3%: 39.45 (terminal value 46.67 in year 5)

The two-stage model shows that Prairie Ridge would be worth about 39 dollars, a little above the market price, if dividends grew 7 percent a year for five years and then 3 percent forever. Most of the value comes from the terminal value, a pattern that holds for almost every company, and the reason careful analysts spend so much time on the assumptions behind it.

When dividends are the wrong cash flow

The dividend discount model works well for mature companies with stable payout policies, such as banks, utilities and pipelines. It works badly for companies that pay no dividends, such as Kestrel Analytics, or that return cash mainly through buybacks, or whose dividends are set for reasons unrelated to their cash generation. For these, analysts value the cash the business generates rather than the cash it chooses to distribute.

Free cash flow to the firm, FCFF, is the cash generated by operations after taxes and after the investment needed to sustain and grow the business, but before any payments to lenders or shareholders. MBA-501 introduced the components. FCFF equals operating profit after tax plus depreciation, minus capital expenditure, minus the increase in net working capital. Discounting FCFF at the weighted average cost of capital gives the value of the whole business, its enterprise value. Subtracting net debt gives the value of equity, and dividing by the number of shares gives a value per share.

An alternative is free cash flow to equity, FCFE, which is FCFF minus after-tax interest plus net new borrowing. FCFE is the cash available to shareholders and is discounted at the cost of equity. The two approaches give the same answer if the assumptions about debt are consistent, but FCFF is more common because it separates the value of the operations from how they are financed. Lesson 11 builds a full FCFF valuation for a company, and the capstone applies it to Cedarline Outdoor.

# A simple free cash flow to the firm valuation (millions)
tax, wacc, g_long = 0.25, 0.09, 0.025
years = [1, 2, 3, 4, 5]
ebit  = [40.0, 44.0, 48.0, 51.0, 53.0]
da    = [12.0, 13.0, 14.0, 15.0, 15.5]
capex = [18.0, 19.0, 20.0, 19.0, 18.0]
d_nwc = [5.0, 4.0, 4.0, 3.0, 2.5]
fcff = [e * (1 - tax) + d - c - w for e, d, c, w in zip(ebit, da, capex, d_nwc)]
pv = sum(f / (1 + wacc) ** t for t, f in zip(years, fcff))
tv = fcff[-1] * (1 + g_long) / (wacc - g_long)
pv_tv = tv / (1 + wacc) ** years[-1]
ev = pv + pv_tv
net_debt, shares = 120.0, 20.0
print("FCFF by year:", [round(f, 1) for f in fcff])
print(f"PV of years 1-5 {pv:.1f}, PV of terminal value {pv_tv:.1f}")
print(f"Enterprise value {ev:.1f}, equity {ev - net_debt:.1f}, per share {(ev - net_debt) / shares:.2f}")

FCFF by year: [19.0, 23.0, 26.0, 31.2, 34.8]
PV of years 1-5 101.6, PV of terminal value 356.1
Enterprise value 457.7, equity 337.7, per share 16.89

Multiples as a cross-check

Practitioners also value shares by comparison, using ratios such as price to earnings or enterprise value to EBITDA for similar companies. A multiple is a compressed discounted cash flow model: the Gordon formula implies that the price-to-earnings ratio equals the payout ratio divided by the cost of equity minus growth. High multiples therefore signal some mix of low risk, high growth and high returns on reinvestment. Multiples are quick and grounded in market prices, but they hide their assumptions, and if the whole sector is mispriced, so is the comparison. Lesson 11 treats them in detail.

Dr. Varga notes one more practical point. The dividend discount model values a share from the point of view of a minority investor who receives dividends but cannot control the company. A buyer who acquires control can change the payout, the investment policy and the capital structure, so the value of control can exceed the value of a minority share. That difference is one reason acquirers pay premiums over market prices, a topic lesson 11 takes up.