Matching Dividend And Rate Timing Conventions

Why Timing Conventions Matter

A dividend-growth calculation can use the right formula and still produce the wrong answer if its inputs refer to different periods or its share price refers to the wrong date. A quarterly dividend cannot be paired directly with an annual return in the Gordon formula unless the rates are first converted to a common period. Likewise, a value immediately after a dividend payment is not the value immediately before the next payment.

The essential practice is to label the dates first. This tutorial uses a constant-growth dividend model and focuses on aligning dividend frequency, growth period, price date, and required-return period.

Definition: Let $D_0$ be the dividend just paid at time 0, and let $D_1$ be the next dividend, due one payment period later. Let $g$ be the growth rate per payment period and $k$ the required return per payment period. $P_0$ is the ex-dividend share value immediately after the payment of $D_0$. Under constant growth, $D_t=D_0(1+g)^t$, and the ex-dividend value is $P_0=D_1/(k-g)$, provided $k>g$.

The timing assumptions matter. Dividends are paid once at the end of each equal period; the next payment is $D_1$; subsequent dividends grow at the same rate $g$ per period; and investors require the same return $k$ per period. The formula assumes an indefinite stream of dividends and $k>g$. The required return and growth rate must refer to the same period. If any assumption is not appropriate, this constant-growth calculation does not give a valid valuation.

Use Rates for the Same Period

In $P_0=D_1/(k-g)$, $D_1$ is the next payment, while $k$ and $g$ are both rates per interval between payments. If dividends are quarterly, use quarterly rates; if dividends are annual, use annual rates. Do not put a quarterly dividend growth rate beside an annual required return.

For a periodic rate $r_p$ applying $m$ times per year, its equivalent effective annual rate is $(1+r_p)^m-1$. Conversely, an effective annual rate $r_a$ corresponds to periodic rate $(1+r_a)^{1/m}-1$. A nominal annual rate $i^{(m)}$, convertible $m$ times per year, has periodic rate $i^{(m)}/m$, so its effective annual rate is $(1+i^{(m)}/m)^m-1$. These are different conventions: a nominal rate divided by the number of periods is not itself an effective annual rate.

Rate-conversion rule: Convert both the required return and dividend growth to the same period before subtracting them. For $m$ equal periods per year, use $r_a=(1+r_p)^m-1$ for an effective annual rate, or $r_p=(1+r_a)^{1/m}-1$ in the reverse direction. For a nominal annual rate $i^{(m)}$, use $r_p=i^{(m)}/m$.

Ex-Dividend and Cum-Dividend Dates

The ex-dividend value immediately after $D_0$ is $P_0$. At the next payment date, immediately after $D_1$ has been paid, the ex-dividend value is $P_1$. Since the remaining dividends have each grown by one period, $P_1=P_0(1+g)$, or equivalently $P_1=D_2/(k-g)$. Immediately before $D_1$ is paid, the share still carries the right to receive that dividend. Its value at that instant is therefore $D_1+P_1$, not $D_1+P_0$.

P_0=\frac{D_1}{k-g},\qquad P_1=\frac{D_2}{k-g}=P_0(1+g),\qquad \text{value immediately before }D_1=D_1+P_1

Why the Formula and Date Adjustment Work

The Gordon formula follows by discounting the future dividends to the price date. Immediately after $D_0$, the future stream is $D_1,D_2,\ldots$. The value is the sum of each payment discounted at the required return per period:

P_0=\frac{D_1}{1+k}+\frac{D_1(1+g)}{(1+k)^2} +\frac{D_1(1+g)^2}{(1+k)^3}+\cdots

This is an infinite geometric series whose first term is $D_1/(1+k)$ and whose common ratio is $(1+g)/(1+k)$. Because $k>g$, this ratio is less than 1, and the series converges. Its sum is

P_0= \frac{D_1/(1+k)}{1-(1+g)/(1+k)} =\frac{D_1}{k-g}.

The date adjustment can also be checked from the one-period return. An investor who buys just after $D_0$ for $P_0$ receives $D_1$ and holds a share worth $P_1$ immediately after that payment. The holding-period return is $(D_1+P_1-P_0)/P_0$. Since $D_1=P_0(k-g)$ and $P_1=P_0(1+g)$, the return is

\frac{D_1+P_1-P_0}{P_0} =\frac{P_0(k-g)+P_0(1+g)-P_0}{P_0}=k.

Thus the cum-dividend value just before payment is $D_1+P_1=P_0(1+k)$. This is consistent with earning the required return over the period, including both the dividend and the change in the ex-dividend value.

Worked Valuations with Explicit Dates

Worked Example: Annual Dividends and the Next Payment Date

A hypothetical company has just paid an annual dividend of $1.50. The next annual dividend is forecast to be $1.56, and dividends are expected to grow at a constant 4% per year thereafter. The annual required return is 9%. Find the ex-dividend value just after the payment and the value immediately before the next dividend is paid. State the method and check the result.

Method. Apply the constant-growth formula using annual rates and identify the ex-dividend value at each date. The periodic growth rate is $g=0.04$, and the required return is $k=0.09$, so $k-g=0.05$. Since $k>g$, the series converges.

The next dividend is $D_1=\$1.56$. Immediately after the dividend just paid, the share is ex-dividend:

P_0=\frac{D_1}{k-g} =\frac{\$1.56}{0.09-0.04} =\frac{\$1.56}{0.05} =\$31.20.

To find the ex-dividend value immediately after the next payment, first find $D_2=\$1.56(1.04)=\$1.6224$. Then

P_1=\frac{D_2}{k-g} =\frac{\$1.6224}{0.05} =\$32.448.

Immediately before that payment, the share carries the right to receive $D_1$. Its value is $\$1.56+\$32.448=\$34.008$, or $34.01 to the nearest cent. Check: the one-year return from $P_0$ to the cum-dividend value is $(\$34.008-\$31.20)/\$31.20=0.09$, exactly 9%. A common mistake is to add $D_1$ to $P_0$; $P_0$ is dated one year earlier, so the value after the payment date must be $P_1$.

Worked Example: Quarterly Growth and Quarterly Required Return

A hypothetical stock pays quarterly dividends. The dividend just paid was $0.40. The next dividend is expected to be 1.5% higher, and each later dividend is expected to grow by 1.5% per quarter. The required return is 4.00% per quarter. Find the ex-dividend value just after the payment already made and the value immediately before the next payment.

Method. Use quarterly periods throughout. Here $D_0=\$0.40$, $g=0.015$, and $k=0.04$. The next dividend is $D_1=\$0.40(1.015)=\$0.406$, and $k-g=0.025$.

The ex-dividend value immediately after the dividend just paid is

P_0=\frac{\$0.406}{0.04-0.015} =\frac{\$0.406}{0.025} =\$16.24.

The following dividend is $D_2=\$0.406(1.015)=\$0.41209$. Therefore, immediately after $D_1$ is paid, the ex-dividend value is

P_1=\frac{\$0.41209}{0.025}=\$16.4836.

Immediately before the next payment, the value is $D_1+P_1=\$0.406+\$16.4836=\$16.8896$, or $16.89 per share to the nearest cent. Check: $(\$16.8896-\$16.24)/\$16.24=0.04$, so the one-quarter holding-period return is 4.00%. Adding $D_1$ to $P_0$ would mix a value from the start of the quarter with one from its end.

Worked Example: Converting a Nominal Annual Return for Quarterly Dividends

A hypothetical firm pays dividends quarterly. The dividend just paid is $0.50, and dividends are expected to grow by 1.00% per quarter indefinitely. An analyst estimates the required return as a nominal annual rate of 10.0% convertible quarterly. Find the ex-dividend value just after the latest payment, its equivalent effective annual required return, and the value immediately before the next payment.

Method. Convert the nominal annual required return to its quarterly rate before using the quarterly growth rate. With $m=4$, the quarterly required return is $k=0.10/4=0.025$, or 2.50%. The effective annual required return is $(1.025)^4-1=0.103812890625$, or 10.3813% to four decimal places in percent. This is not the same as the nominal 10.0% quote.

The quarterly growth rate is $g=0.01$. The next dividend is $D_1=\$0.50(1.01)=\$0.505$, and the quarterly spread is $k-g=0.025-0.01=0.015$. Thus the ex-dividend value just after the latest payment is

P_0=\frac{\$0.505}{0.015} =\$33.666666\ldots

The next dividend after $D_1$ is $D_2=\$0.505(1.01)=\$0.51005$. The ex-dividend value immediately after $D_1$ is $P_1=\$0.51005/0.015=\$34.003333\ldots$. Therefore, the value immediately before $D_1$ is $\$0.505+\$34.003333\ldots=\$34.508333\ldots$, or $34.51 to the nearest cent. Check: using unrounded values, $(\$34.508333\ldots-\$33.666666\ldots)/\$33.666666\ldots=0.025$, the 2.50% quarterly return. The corresponding annual effective growth rate is $(1.01)^4-1=0.04060401$, or 4.0604%; multiplying quarterly growth by four would not produce the effective annual growth rate.

Common Timing Traps

Mixing annual and periodic rates. If dividends grow quarterly, convert an effective annual required return to a quarterly rate before subtracting $g$, or convert both rates to annual effective rates.
Confusing nominal and effective annual rates. A nominal rate convertible quarterly is divided by four to obtain the quarterly rate. Its effective annual equivalent is then found by compounding that quarterly rate four times.
Using the wrong dividend. $P_0$ uses the next dividend $D_1$, not the dividend just paid $D_0$. If only $D_0$ is given, first calculate $D_1=D_0(1+g)$.
Using the wrong price date. Immediately before $D_1$, the value is $D_1+P_1$. Do not add $D_1$ to $P_0$, which is the ex-dividend value at the previous payment date.
Ignoring the convergence condition. The constant-growth formula requires $k>g$. If $k\leq g$, the infinite discounted dividend series does not converge under these assumptions.

Key takeaway: Label the dividend date and share-price date, express $k$ and $g$ per the same payment period, and use $D_1$ for the ex-dividend value $P_0$. Immediately before the next payment, use $D_1+P_1$, not $D_1+P_0$.

Check Your Understanding

For each problem, keep the dividend interval, rate convention, and valuation date explicit. Round dollar values to the nearest cent and percentages to four decimal places when needed.

A hypothetical company has just paid an annual dividend of $0.80. Its dividends grow 3% per year, and the annual required return is 8%. Find $D_1$ and the ex-dividend value $P_0$. (Answer: $D_1=\$0.824$; $P_0=\$16.48$.)
A hypothetical stock pays quarterly. Its latest dividend was $0.25; quarterly growth is 2%, and the required return is 5% per quarter. Find $P_0$ and the value immediately before the next payment. (Answer: $P_0=\$8.50$; value before the next payment $=\$8.93$.)
A hypothetical company pays quarterly dividends. Its latest dividend was $0.60, quarterly growth is 1%, and the required return is a nominal annual rate of 12% convertible quarterly. Find $P_0$ and the equivalent effective annual required return. (Answer: $P_0=\$30.30$; effective annual required return $=12.5509\%$.)
A hypothetical company’s next annual dividend is $2.40; annual growth is 5%, and the annual required return is 11%. Find the ex-dividend value just after the preceding payment and the value immediately before the $2.40 dividend is paid. (Answer: $P_0=\$40.00$; value before the next payment $=\$44.40$.)
A hypothetical stock pays quarterly dividends with quarterly growth of 1%. The latest dividend was $0.50, and the required return is an effective annual rate of $(1.02)^4-1=8.243216\%$. Convert the required return to a quarterly rate and find the ex-dividend value just after the latest payment. (Answer: quarterly required return $=2.00\%$; $P_0=\$50.50$.)

What You'll Learn