Equity: Dividend Valuation, Growth Opportunities, P/E Ratios, and Index Futures

Equity valuation connects the money a business can distribute to its owners with the price investors pay today. Growth matters because retained earnings can finance future distributions. The important question is whether that reinvestment earns enough to compensate shareholders for its risk.

Learning goals and notation

By the end, you should be able to derive the dividend discount model, distinguish current from next-period dividends, estimate growth under explicit assumptions, calculate PVGO and forward P/E, and explain an index futures hedge.

Assume annual year-end cash dividends, an ex-dividend valuation date, a constant risk-adjusted cost of equity, and no taxes or trading costs unless stated otherwise. All per-share forecasts refer to a consistent share-count basis. Numerical rates are illustrative rather than market estimates.

Symbol	Meaning
$P_t$	Stock price immediately after the date-$t$ dividend
$D_t$	Cash dividend per share paid at date $t$
$r$	Required annual return on equity
$g$	Annual growth rate in the specified per-share quantity
$E_1$ or $\mathrm{EPS}_1$	Expected earnings per share next year
$b$	Retention or plowback ratio
$R$	Return earned on incremental book equity in the simple growth model

Review the Present Value Relations and Fixed-Income Securities lessons for discounting and no-arbitrage foundations.

1. What common stock represents

A common share is an ownership claim. Shareholders are residual claimants: creditors and other senior claims are paid before common shareholders receive the remaining value. Voting rights and other terms depend on the share class. Unlike a plain bond’s contractual schedule, common dividends generally have uncertain amounts and timing.

For a share bought at $P_0$ and sold one year later at ex-dividend price $P_1$, with cash dividend $D_1$, the holding-period return is

$$R_1=\frac{D_1+P_1-P_0}{P_0}.$$

Buying for 50, receiving 2, and selling for 53 gives a 10% return: 4% cash dividend yield plus 6% capital gain. A fall in price can offset or exceed the cash dividend.

Cash dividends, stock dividends, and repurchases

A cash dividend transfers money from the company to its shareholders. A proportional stock dividend instead increases the number of shares held without, by itself, increasing each shareholder’s ownership fraction or the company’s assets.

For example, a 10% stock dividend turns 100 shares into 110. If total equity value is unchanged, the price per share adjusts from 50 to about $50/1.10=45.45$. The investor still owns approximately 5,000 of equity. Do not add the nominal value of the extra shares to cash return as if it were newly created wealth.

Repurchases also distribute cash, but to the shareholders who sell. They change shares outstanding and must be reflected consistently in per-share forecasts or a total-equity payout model. Cash dividends alone can be an incomplete practical measure of shareholder distributions for a company that relies heavily on buybacks.

2. Deriving the dividend discount model

In a simplified one-period required-return model,

$$P_0=\frac{\mathbb{E}_0[D_1+P_1]}{1+r}.$$

Applying the same relationship repeatedly gives, for a horizon $H$,

$$P_0=\sum_{k=1}^{H}\frac{\mathbb{E}_0[D_k]}{(1+r)^k} +\frac{\mathbb{E}_0[P_H]}{(1+r)^H}.$$

The last term is the discounted resale value. To pass to an infinite dividend-only formula, impose a no-bubble or transversality condition:

$$\lim_{H\to\infty}\frac{\mathbb{E}_0[P_H]}{(1+r)^H}=0.$$

Then the dividend discount model (DDM) is

$$P_t=\sum_{k=1}^{\infty}\frac{\mathbb{E}_t[D_{t+k}]}{(1+r)^k}.$$

Capital gains have not been forgotten. The eventual resale price reflects later payouts; repeatedly substituting that price moves those later payouts into the valuation. Adding an independent capital-gain perpetuity to the full DDM would double count value.

A constant discount rate for all risky future dividends is a modeling simplification. More general asset pricing accounts for state-dependent risk and potentially changing required returns. Forecasts are expectations, not promised payments.

Why earnings are not automatically distributable cash

Earnings are an accounting measure. Cash may be needed for equipment, working capital, debt payments, or other investment. Discounting all earnings as if paid out while also counting the growth financed by retaining those same earnings would double count resources.

In the simple full-payout model, zero retention means dividends equal earnings. But the outline’s claim that earnings-based valuation is possible only with zero retention is too strong: residual-income models and justified earnings multiples can value firms with retained earnings by accounting for reinvestment and its cost. Free cash flow to equity is another approach when dividend policy is unrepresentative.

A company paying no dividends today can still have value if future shareholder distributions are expected. If a model assumes no dividends, repurchases, liquidation payments, or other shareholder benefits ever, a positive fundamental value cannot come from cash payouts under its no-bubble condition.

3. The Gordon growth model

Suppose expected cash dividends grow forever at constant rate $g$:

$$\mathbb{E}_0[D_k]=D_1(1+g)^{k-1}.$$

Summing the geometric series gives

$$P_0=\frac{D_1}{r-g}=\frac{D_0(1+g)}{r-g}.$$

Under conventional $r>-1$ and $g>-1$, convergence requires $r>g$. The numerator is next year’s dividend, not the dividend just paid. A perpetual growth rate above the discount rate does not produce a sensible negative valuation; it violates the convergence condition.

Worked example: value and sensitivity

A stock just paid $D_0=2$. Assume perpetual growth of 4% and cost of equity of 9%. Then

$$D_1=2(1.04)=2.08,\qquad P_0=\frac{2.08}{0.09-0.04}=41.60.$$

The following table holds $D_1=2.08$ fixed to isolate sensitivity:

Required return	Perpetual growth	Value
9%	3%	34.67
9%	4%	41.60
9%	5%	52.00
10%	4%	34.67

Small changes in $r-g$ can cause large changes in value. A terminal growth assumption should describe a sustainable mature business, rather than extrapolate a temporary high-growth phase forever.

Estimating the cost of equity

Rearrange the model:

$$r=\frac{D_1}{P_0}+g.$$

At price 41.60, expected dividend yield is 5%, and adding 4% growth gives a model-implied required return of 9%. If a quoted yield uses $D_0$, first convert to $D_1$.

This is an implied return conditional on the model and growth forecast, not an independent measurement proving the market price correct. Growth estimates and payout changes can dominate the answer. As a cross-check, an equilibrium model such as CAPM writes

$$r=r_f+\beta\bigl(\mathbb{E}[R_M]-r_f\bigr).$$

For illustrative inputs $r_f=3\%$, $\beta=1.2$, and market risk premium 5%, CAPM gives 9%. Beta and the premium are estimated inputs with uncertainty; matching two estimates does not remove model risk. Discount shareholder payouts at the cost of equity, rather than automatically using a firm’s weighted average cost of capital.

4. Growth from retention and return on equity

Let $B_0$ be beginning-of-year book equity per share, and suppose next year’s earnings satisfy $E_1=RB_0$. Retain fraction $b$ and distribute fraction $1-b$:

$$D_1=(1-b)E_1,\qquad B_1=B_0+bE_1=B_0(1+bR).$$

If the same return and retention policy continue, with no new share issuance or other changes to book equity,

$$g=bR.$$

This is the familiar $g=b\times ROE$ relationship, using a timing-consistent return on beginning book equity. With constant payout, earnings and dividends grow at the same rate. It is often called sustainable growth under these assumptions; terminology such as “internal growth rate” differs across finance contexts and can refer to a distinct asset-based formula.

Historical average ROE need not equal the return available on the next investment. Accounting adjustments, leverage changes, buybacks, and share issuance can also break the simple per-share growth relationship. Use an incremental return forecast when evaluating future projects.

Worked example: growth changes both payout and future earnings

Let $B_0=20$, $R=15\%$, and $b=40\%$. Then

$$E_1=3,\quad D_1=1.8,\quad g=0.4(0.15)=6\%.$$

With cost of equity $r=10\%$:

$$P_0=\frac{1.8}{0.10-0.06}=45.$$

Retention lowers the immediate dividend from 3 to 1.8, but generates future growth. Whether that trade creates value depends on the return on the retained funds.

5. PVGO: separate existing earnings from future investment

A useful benchmark assumes the firm can sustain next year’s earnings $E_1$ forever while paying all those earnings out. Its no-growth value is $E_1/r$. Define present value of growth opportunities by

$$P_0=\frac{E_1}{r}+PVGO.$$

The benchmark requires an economically sustainable earnings level. Zero growth does not mean a real business needs no replacement investment; earnings and cash-distribution assumptions must remain internally consistent.

Under the constant-retention model,

$$P_0=\frac{E_1(1-b)}{r-bR},\qquad r>bR.$$

Subtracting the no-growth benchmark gives

$$PVGO=\frac{E_1b(R-r)}{r(r-bR)}.$$

For positive $E_1$, positive retention, and a positive denominator, the sign depends on incremental return minus cost of equity:

• $R>r$: reinvestment creates positive PVGO.
• $R=r$: earnings may grow, but PVGO is zero.
• $R<r$: the reinvestment policy destroys value relative to full payout.

If $b=0$, there is no reinvestment and PVGO is zero even if hypothetical opportunities could earn more than $r$. A firm’s ability to change policy matters; negative PVGO here describes the assumed investment policy rather than an unavoidable corporate destiny.

Worked comparison: more growth need not mean more value

Keep $E_1=3$, $r=10\%$, and $b=40\%$, and vary the return on future reinvestment:

Incremental return $R$	Growth $bR$	Dividend $D_1$	Price	PVGO
15%	6%	1.80	45.00	15.00
10%	4%	1.80	30.00	0.00
5%	2%	1.80	22.50	-7.50

The no-growth benchmark is $3/0.10=30$ in every row. In the middle row, the company grows earnings at 4%, but investors receive no extra NPV because new investments merely earn their required return.

In this lecture’s economic usage, valuable growth opportunities support the idea of a growth stock. In market classification, “growth stock” is also a broader label based on expected expansion or valuation characteristics; it does not certify that future investments will earn above the cost of capital.

6. P/E ratios and what they do not tell you

Divide the PVGO decomposition by positive next-year EPS:

$$\frac{P_0}{E_1}=\frac{1}{r}+\frac{PVGO}{E_1}.$$

This is a forward P/E because the denominator is forecast earnings. A trailing P/E uses historical earnings and need not have the same value.

For the positive-PVGO example,

$$P/E=45/3=15,\qquad 1/r=10,\qquad PVGO/E_1=5.$$

The justified P/E in the constant-retention model can also be written

$$\frac{P_0}{E_1}=\frac{1-b}{r-g}.$$

A high P/E can reflect valuable growth opportunities, a lower required return, depressed near-term earnings, or optimistic assumptions. It does not by itself establish overvaluation. A low P/E can reflect risk or poor future prospects rather than a bargain. With negative or near-zero EPS, ordinary P/E comparisons become uninformative or unstable.

The result $P/E=1/r$ when PVGO is zero belongs to this benchmark model, not a universal empirical rule for all firms with low measured growth.

7. Multistage DDM: separate expansion from maturity

A company may grow rapidly for a few years before settling into a slower long-run rate. Forecast the early dividends individually and attach a stable-growth terminal value:

$$P_H=\frac{D_{H+1}}{r-g_{\infty}},\qquad P_0=\sum_{t=1}^{H}\frac{D_t}{(1+r)^t}+\frac{P_H}{(1+r)^H}.$$

$P_H$ is ex-dividend at date $H$: it values dividends starting at $H+1$. Include $D_H$ separately, and discount the terminal value from date $H$.

Worked example: two years of fast growth

Let $D_0=1$, early growth be 20% for two years, mature growth be 4% thereafter, and $r=10\%$:

$$D_1=1.20,\quad D_2=1.44,\quad D_3=1.4976.$$

At date 2,

$$P_2=\frac{1.4976}{0.10-0.04}=24.96.$$

Therefore

$$P_0=\frac{1.20}{1.10}+\frac{1.44+24.96}{1.10^2} \approx22.91.$$

The discounted terminal value is about 20.63, roughly 90% of the total. This is why a detailed short-term forecast cannot compensate for an implausible mature-growth assumption. If risk changes between stages, the discounting framework must change consistently too.

8. Lecture 8 connection: equity forwards and index futures

An equity forward sets a price today for a transaction at a future date. Its delivery price is determined by replication and financing under stated assumptions. It is not obtained by growing the spot price at the stock’s risky expected return.

Let $\tau=T-t$, and assume frictionless trading, feasible short sales, deterministic financing, and known dividends. For discrete cash dividends paid before delivery,

$$F_{t,T}=\left(S_t-PV_t(\text{dividends before delivery})\right)(1+r_f)^\tau,$$

using a flat annual effective financing rate and consistent timing. Owning the share receives those dividends; a forward buyer does not, which is why dividends reduce the delivery price.

With deterministic continuous proportional dividend yield $q$ and continuously compounded risk-free rate $r_f$:

$$F_{t,T}=S_te^{(r_f-q)\tau}.$$

For small annual rates, the outline’s expression

$$F_{t,T}\approx S_t(1+r_f-d)^\tau$$

can be used as a rough carry approximation. It is not a universal exact formula: discrete dividends, effective versus continuous rates, timing, and dividend uncertainty require appropriate treatment.

Worked example: index carry

Take spot index 5,000, continuous financing rate 4%, continuous dividend yield 1.5%, and six months to delivery:

$$F_{0,T}=5{,}000e^{(0.04-0.015)(0.5)}\approx5{,}062.89.$$

If a forward were sufficiently overpriced, a stylized cash-and-carry trade would finance an index basket, receive its dividends, and sell the forward. The basket and forward obligations offset at delivery; financing and dividend assumptions determine the carry cost. Costs and constraints create practical bounds around theoretical parity.

A futures contract is marked to market daily, whereas a standard forward settles later. Their prices coincide under suitable assumptions such as deterministic interest rates; stochastic rates and the correlation of settlement cash flows with rates can create differences. CME’s fair-value explanation connects index futures pricing with financing and dividends.

9. A beta-based portfolio hedge

A long equity portfolio can reduce market exposure by selling index futures. A common approximate hedge size is

$$N_{\mathrm{short}}=\frac{(\beta_P-\beta_{\mathrm{target}})V_P}{M F},$$

where $V_P$ is portfolio market value, $M$ is the contract multiplier, and $F$ is the quoted futures index level. This treats the chosen broad-index futures exposure as approximately beta one relative to the same benchmark.

For a hypothetical portfolio worth 10 million with beta 1.2, target beta zero, futures level 5,000, and an assumed multiplier of 50 per index point:

$$N_{\mathrm{short}}=\frac{1.2(10{,}000{,}000)}{50(5{,}000)}=48.$$

If the index and futures fall 5%, the portfolio’s beta-based estimated loss is 600,000. The futures gain is approximately $48(50)(250)=600{,}000$. This is a local illustration, not a guarantee of exact offset.

The hedge leaves company-specific risk and can suffer from unstable beta, basis changes, mismatched exposures, dividend differences, and contract rounding. Daily variation margin creates cash demands, and an ongoing hedge may require rolling contracts and rebalancing. Hedging reduces participation in market rallies as well as market declines. See CME’s beta-hedging discussion for the sizing framework.

10. Practice questions

Use the simplified annual models unless another convention is stated.

1. A share costs 40, pays a cash dividend of 1.50, and is sold a year later for 42. Find the holding-period return.
2. A company just paid a dividend of 3. Growth is 3% forever and the cost of equity is 9%. Find price and expected dividend yield.
3. A stock trades at 60 and next year’s dividend is 2.40. If sustainable growth is 5%, find the Gordon-implied required return.
4. Next-year EPS is 4, retention is 25%, return on reinvested equity is 16%, and the cost of equity is 10%. Find growth, next dividend, price, PVGO, and forward P/E.
5. Repeat question 4 with a reinvestment return of 8%. Does positive earnings growth create positive PVGO?
6. A company will pay 1 next year and 1.20 the year after. Thereafter dividends grow at 3%. At a 10% cost of equity, find today’s value.
7. An index is at 4,000. Continuous financing is 5%, continuous dividend yield is 2%, and delivery is one year away. Find the theoretical forward price.
8. A portfolio worth 5 million has beta 0.9. To target beta 0.3, how many futures should be sold if each has notional value 200,000?
9. Explain why a proportional stock dividend does not create wealth by itself, and why future earnings cannot all be counted as distributions while also counting the reinvestment growth they finance.

11. Answers and interpretation

1. $(1.50+42-40)/40=8.75\%$: a 3.75% dividend yield plus a 5% capital gain.
2. $D_1=3(1.03)=3.09$. Price is $3.09/(0.09-0.03)=51.50$; next-year dividend yield is 6%.
3. $r=2.40/60+0.05=9\%$. This estimate depends on the assumed growth and model.
4. $g=0.25(0.16)=4\%$, $D_1=4(0.75)=3$, and $P_0=3/(0.10-0.04)=50$. No-growth value is $4/0.10=40$, so PVGO is 10 and forward P/E is 12.5.
5. Growth is 2%; the dividend is still 3. Price is $3/(0.10-0.02)=37.50$, PVGO is -2.50, and forward P/E is 9.375. Growth destroys value here because the incremental return is below the required return.
6. $D_3=1.20(1.03)=1.236$ and $P_2=1.236/(0.10-0.03)\approx17.6571$. Thus $P_0=1/1.10+(1.20+17.6571\ldots)/1.10^2\approx16.49$.
7. $F=4{,}000e^{0.05-0.02}\approx4{,}121.82$. Use the risk-free carry rate with the dividend adjustment, rather than the equity cost of capital.
8. $N=(0.9-0.3)(5{,}000{,}000)/200{,}000=15$ short contracts under the stated beta-one futures approximation.
9. A proportional stock dividend divides the same ownership into more shares. Retained earnings finance future investment rather than an immediate cash payout; counting both as available shareholder cash would double count the same resources.

Further reading

• Andrew W. Lo, MIT OpenCourseWare, Equities lecture and slides: Lecture 7 context for DDM, growth, and valuation multiples.
• MIT OpenCourseWare, Forward and Futures Contracts: the next stage in the course’s no-arbitrage framework.
• CME Group, Calculating Fair Value: financing and dividend inputs to equity-index carry.
• CME Group, Hedging with E-mini S&P 500 Futures: beta-based hedge sizing and portfolio exposure.

Feynman study lab

Explain it simply

A share is a residual claim: its value depends on cash that can ultimately reach shareholders.

Concrete example. A constant-growth dividend model with next dividend 3, required return 8% and growth 2% gives 3 / (0.08 − 0.02) = 50.

Teach it back without looking

Close the lesson and explain the idea aloud to a beginner in two minutes. Define each technical term you use, walk through the example, and state one assumption or limitation. Then answer:

Why does increasing assumed growth near the required return make this estimate fragile?

Check your explanation after you try

The denominator becomes small and value becomes highly sensitive. The perpetual-growth model requires growth below the required return.

Find the gap and try again

If you hesitated, relied on a formula you could not explain, or missed the self-check, write down the exact gap. Revisit the relevant section of this lesson and work through its example. Close the notes, explain it again in simpler language, then change one number or assumption and predict what changes. Recalculate or check the lesson to test your prediction. Try once more tomorrow from memory.