Valuing Shares Before Dividends Begin

Introduction

A company may retain its earnings during an early stage of growth and make no dividend payments for several years. That delay does not, by itself, prevent a dividend discount valuation. The key is to identify when the first payout is expected, value the later payouts at the appropriate date, and discount that value back to today.

The timing needs careful attention: a dividend paid at the end of year $k$ is received by an investor who owns the share at the end of year $k-1$. If a stable growth model is used for that and all later dividends, its value is first calculated at the end of year $k-1$, then discounted for $k-1$ years. This approach is aligned with the SOA VEE Accounting and Finance syllabus.

Definition: Let $D_t$ be the cash dividend per share paid at the end of year $t$, and let $P_t$ be the share value at the end of year $t$. If the first dividend is paid at the end of year $k$, the dividend-free period is the time from today through the end of year $k-1$. A dividend discount valuation is the present value of forecast future dividends, using the required return on equity $r$ as the annual effective discount rate.

Value the Stream at the Date Before the First Dividend

Suppose the first dividend, $D_k$, is paid at the end of year $k$, and every dividend after it grows at a constant annual rate $g$. Thus, $D_{k+1}=D_k(1+g)$, $D_{k+2}=D_k(1+g)^2$, and so on. Assume the required return $r$ is constant, the dividends are paid at year-end, and $r>g$. The last condition ensures that the infinite present-value series converges.

At the end of year $k-1$, the first dividend is one year away. The value at that date is therefore the value of a growing perpetuity beginning with $D_k$. Once that value is found, discount it to today over $k-1$ years.

P_{k-1}=\frac{D_k}{r-g}, \qquad P_0=\frac{P_{k-1}}{(1+r)^{k-1}} =\frac{D_k}{(r-g)(1+r)^{k-1}}.

These expressions use an annual effective required return and annual dividends. The dividend in the numerator is the first dividend that will actually be paid, not a dividend paid today. The denominator $r-g$ values the stream at the date one year before that dividend.

Why the Formula Holds

At the end of year $k-1$, the first dividend arrives one year later. The dividend received in year $k+j$ is $D_k(1+g)^j$, and it is discounted $j+1$ years to the end of year $k-1$. Therefore, the value there is the following infinite series:

P_{k-1} =\sum_{j=0}^{\infty}\frac{D_k(1+g)^j}{(1+r)^{j+1}} =\frac{D_k}{1+r} \sum_{j=0}^{\infty} \left(\frac{1+g}{1+r}\right)^j.

For $r>g$, the ratio $(1+g)/(1+r)$ is less than 1 under the usual assumption that the annual rates are greater than $-1$. The geometric series then sums to $1/[1-(1+g)/(1+r)]$. Substituting and simplifying gives $P_{k-1}=D_k/(r-g)$. The share value today is this future value discounted $k-1$ years at $r$. This derivation also shows why the formula is not valid when $r\leq g$: the infinite series does not converge to a finite value under these assumptions.

Applying the Model When Dividends Are Delayed

Worked Example: First Dividend in Year Six

A hypothetical share is forecast to pay no dividends through year 5. Its first dividend, $D_6$, is forecast to be $1.50 at the end of year 6. Dividends are expected to grow at 2% annually from then on forever. The required annual effective return is 8%. Find the value at the end of year 5 and the value today. Give dollar amounts to the nearest cent.

Method and assumptions. The first dividend is one year beyond the end of year 5. From year 6 onward, the dividends form a constant-growth perpetuity. The rates are annual effective rates, dividends arrive at year-end, and $r=0.08>g=0.02$.

Step 1: Value at the end of year 5. This is the date immediately before the first dividend, so use $D_6$ in the growing-perpetuity formula:

P_5=\frac{D_6}{r-g} =\frac{\$1.50}{0.08-0.02} =\frac{\$1.50}{0.06} =\$25.00.

Step 2: Discount to today. The value $P_5$ is five years in the future, so discount it for five years:

P_0=\frac{\$25.00}{(1.08)^5} =\frac{\$25.00}{1.4693280768} =\$17.0145809\ldots \approx \$17.01.

Check and trap avoided. The first dividend is paid at the end of year 6, but the perpetuity value is at the end of year 5. Discounting that value five periods, not six, gives a today value of $17.01. The calculation is consistent with discounting the dividend stream beginning at year 6.

Worked Example: A Shorter Dividend-Free Period

A hypothetical company expects to pay its first dividend of $2.40 at the end of year 4. Dividends are forecast to grow by 3% per year thereafter without end. If the required annual effective return is 9%, calculate the share value today, to the nearest cent.

Method and assumptions. The first dividend is at $t=4$, and all later dividends grow at the same rate. The required return is constant, the dividends are annual and paid at year-end, and $0.09>0.03$.

Step 1: Find the value at the end of year 3. The first dividend is one year beyond this date:

P_3=\frac{D_4}{r-g} =\frac{\$2.40}{0.09-0.03} =\frac{\$2.40}{0.06} =\$40.00.

Step 2: Discount three years. Today is three years before the end of year 3:

(1.09)^3=1.295029, \qquad P_0=\frac{\$40.00}{1.295029} =\$30.887\ldots \approx \$30.89.

Check and trap avoided. There are four years until the first dividend, but only three years from today to $P_3$, the date at which the perpetuity value is calculated. Discounting $P_3$ four years would incorrectly move the value one year too far back.

When Payouts Begin Before Growth Stabilizes

The first dividend does not have to be the start of a constant-growth sequence. If one or more dividends are forecast separately before a stable growth phase, value those dividends individually. Then calculate a terminal value at the date immediately before the first dividend in the stable phase, and discount both the separately forecast dividends and the terminal value to today.

A terminal value is not an additional dividend. It represents the value, at its stated date, of dividends from the following period onward. The date attached to it determines which dividends it includes; adding a dividend already included in that terminal value would count it twice.

Worked Example: Two Initial Dividends Before Stable Growth

A hypothetical share is expected to pay $1.20 at the end of year 3 and $1.50 at the end of year 4. Starting in year 5, dividends are forecast to grow at 2% annually forever. The required annual effective return is 8%. Find the value today to the nearest cent.

Method and assumptions. Discount the year 3 and year 4 dividends separately. Value the stable-growth stream at the end of year 4, immediately before the year 5 dividend. Assume all dividends arrive at year-end and the required return remains constant. Because $0.08>0.02$, the terminal value is finite.

Step 1: Find the first stable-growth dividend and terminal value. The year 5 dividend is 2% greater than the year 4 dividend:

D_5=\$1.50(1.02)=\$1.53, \qquad P_4=\frac{D_5}{r-g} =\frac{\$1.53}{0.08-0.02} =\frac{\$1.53}{0.06} =\$25.50.

Step 2: Discount each cash flow to today. The year 3 dividend is discounted three years. The year 4 dividend and the terminal value at year 4 are discounted four years:

\begin{aligned} P_0 &=\frac{\$1.20}{(1.08)^3} +\frac{\$1.50+\$25.50}{(1.08)^4}\\ &=\frac{\$1.20}{1.259712} +\frac{\$27.00}{1.36048896}\\ &=\$0.952598\ldots+\$19.845806\ldots\\ &=\$20.798404\ldots \approx \$20.80. \end{aligned}

Check and trap avoided. $P_4$ includes the year 5 dividend and all later stable-growth dividends, but not the year 4 dividend. Therefore, the year 4 cash flow is added separately. Discounting the combined year 4 amount and terminal value by four years, while discounting the year 3 dividend by three years, respects the payment dates.

Model Limits and a Common Timing Error

A dividend discount model values the cash distributions forecast to reach shareholders. Its result depends on there being a defensible forecast of eventual payouts. If a company is expected never to pay dividends and the model includes no other shareholder distributions or proceeds, a dividend-only model assigns no value to its forecast cash flows. That conclusion is about the model and its inputs; it is not a claim that every share with no near-term dividend has no value. A company may begin distributions later, and a dividend-free period can be valued if eventual payouts can be forecast credibly.

Do not use a constant-growth perpetuity merely because a first dividend has been estimated. The later dividends must actually be modeled as growing at $g$ indefinitely, with a required return greater than $g$. If only a finite set of dividends is forecast, or the long-run growth assumption is not suitable, the constant-growth formula does not by itself value the full share. Any separate forecast cash flows and the date of any terminal value must be stated explicitly.

A frequent exam error is to discount the perpetuity value by the number of years until the first dividend. For a first dividend at the end of year $k$, the perpetuity value is at the end of year $k-1$. Thus the value is discounted $k-1$ years to today, not $k$. The first dividend is already one period beyond the valuation date used for the perpetuity formula.

Locate the first dividend.
Record its payment year and distinguish the payment date from the date immediately before it.

Choose the appropriate payout model.
Use constant growth only for the portion of the dividend forecast that satisfies its assumptions, including $r>g$.

Value at the correct date.
For a stable stream beginning with $D_k$, calculate its value at the end of year $k-1$.

Discount every amount to today.
Discount interim dividends from their own payment dates and the future share value from the date at which it is calculated.

Key takeaway: When the first dividend is paid at the end of year $k$ and all later dividends grow at a constant rate $g$, the value at the end of year $k-1$ is $D_k/(r-g)$, provided $r>g$. Discount that value $k-1$ years to obtain today’s value.

Check Your Understanding

For each problem, use the stated dividend timing and annual effective required return. Round dollar answers to the nearest cent.

A hypothetical share pays its first dividend of $1.80 at the end of year 5. Dividends then grow at 2% forever, and the required return is 8%. Find the value at the end of year 4 and today. (Answer: value at year 4 = $30.00; value today = $22.05)
A share’s first dividend of $0.90 is paid at the end of year 3. Dividends grow at 4% annually from then on forever, and the required return is 10%. Find today’s value. (Answer: $12.40)
A company pays no dividends through year 2. Its first dividend, of $2.00, is paid at the end of year 3; thereafter dividends grow at 3% forever. If the required return is 9%, find today’s value. (Answer: $28.06)
A share pays $1.00 at the end of year 2 and $1.10 at the end of year 3. Dividends grow at 2% annually starting with the year 4 dividend. If the required return is 8%, find today’s value. (Answer: $16.58)
A share is forecast to make no dividends, repurchases, liquidation payments, or other distributions to its owners at any future date. What value does a dividend-only model assign to the forecast stream, and what limitation should be stated? (Answer: $0 for the forecast distribution stream; this does not establish that the share has no value under a model that includes other shareholder benefits or proceeds.)

What You'll Learn