Deriving the Finite-Horizon Dividend Discount Equation

Why the Resale Price Must Be Included

A stockholder may receive dividends while holding a share and then sell it at a chosen horizon. Valuing only the dividends received during that period ignores the value of the share at sale. The finite-horizon dividend discount equation accounts for both: it discounts each expected dividend and the expected resale price back to today.

The equation follows by applying the one-period dividend and resale-price relationship repeatedly. Its result is a valuation over a finite holding period; it does not assume that dividends stop after the investor sells. Instead, the expected sale price represents the value of the share, including the cash flows expected after the sale date.

Definition: Let $D_t$ be the cash dividend paid at the end of period $t$, and let $P_t$ be the stock price immediately after that period’s dividend is paid. For a holding horizon of $n$ periods, the finite-horizon dividend discount equation is $$ P_0=\sum_{t=1}^{n}\frac{E_0[D_t]}{(1+k)^t}+\frac{E_0[P_n]}{(1+k)^n}, $$ where $P_0$ is today’s price, $k$ is the required return per period, and $E_0[\cdot]$ is the expectation based on information available at time 0.

The timing convention matters: the investor receives $D_n$ at the end of period $n$, then sells the share for $P_n$, the price just after that dividend. Thus both $D_n$ and $P_n$ are included. If a problem instead defines the sale price immediately before the period-$n$ dividend, its convention must be reconciled before applying the equation.

Deriving the Finite-Horizon Equation

Assume that the one-period valuation relationship applies at each date, that the required return $k$ is constant per period, and that the stock is valued ex-dividend at each date. At time $t$, the relationship is $$ P_t=\frac{E_t[D_{t+1}+P_{t+1}]}{1+k}, $$ where $E_t$ denotes expectation conditional on information available at time $t$. This says that the time-$t$ price equals the expected next-period dividend plus the next-period ex-dividend price, discounted one period.

Substitute the relationship at the next date into the current one. Applying the same relationship repeatedly gives $$ P_0=\frac{E_0[D_1]}{1+k}+\frac{E_0[D_2+P_2]}{(1+k)^2} $$ for a two-period horizon, and the same process continues for longer horizons. The expectation at time 0 of a later conditional expectation reduces to the expectation at time 0 of the underlying cash flow, by the law of iterated expectations. The pattern is therefore $$ P_0=\frac{E_0[D_1]}{1+k}+\frac{E_0[D_2]}{(1+k)^2}+\cdots+\frac{E_0[D_n+P_n]}{(1+k)^n}. $$ By linearity of expectation, the final term separates into the expected period-$n$ dividend and the expected resale price. This yields the stated equation.

Start with one-period value.
At each date, the share is valued as the discounted expected sum of the next dividend and the next ex-dividend price.

Substitute the next price.
Replace $P_{t+1}$ with its own one-period valuation in terms of the following dividend and price.

Continue to the horizon.
Each substitution adds a dividend at its appropriate date and discounts it by the corresponding number of periods.

Keep the terminal price.
After $n$ substitutions, the remaining price is $P_n$. Discount its expectation by $n$ periods and add it to the present values of the dividends.

The derivation relies on the specified one-period valuation relationship and constant required return. If the required return differs by period, each cash flow must instead be discounted by the product of the period-specific discount factors. If risk is modeled with a more general pricing framework, a single constant required return may not be an appropriate discounting rule. The finite-horizon equation should not be used without checking which valuation assumptions the problem provides.

Applying the Equation to Known Forecasts

When dividends and the resale price are forecast as known amounts, the equation is a direct present-value calculation. Each cash flow is discounted from its own payment date. In particular, the horizon dividend and the resale price have the same discount exponent under the ex-dividend timing convention above.

Worked Example: Three Dividends and a Horizon Sale

An investor considers buying a share and holding it for three years. The forecast dividends are $D_1=\$2.00$, $D_2=\$2.50$, and $D_3=\$3.00$, and the expected ex-dividend sale price at the end of year 3 is $\$40.00$. The required return is 10% effective per year. Find the value today.

Method. Apply the finite-horizon equation with $n=3$, discounting each cash flow by the annual required return. The stated forecast amounts are treated as known for this calculation.

Step 1: Write each discounted cash flow. $$ P_0=\frac{2.00}{1.10}+\frac{2.50}{(1.10)^2}+\frac{3.00}{(1.10)^3}+\frac{40.00}{(1.10)^3}. $$ The first dividend is discounted one year, the second two years, and both the third dividend and sale price three years.

Step 2: Evaluate the present values. The terms are $2.00/1.10=1.818181\ldots$, $2.50/1.21=2.066115\ldots$, $3.00/1.331=2.253944\ldots$, and $40.00/1.331=30.052592\ldots$. Carrying unrounded values through the addition gives $$ P_0=36.190833\ldots $$ so the value is $\$36.19$, rounded to the nearest cent.

Check. At year 3 the investor receives $\$3.00$ and sells for $\$40.00$, a combined $\$43.00$. Discounting that combined amount gives $43/1.331=32.306536\ldots$. Adding the present values of the first two dividends gives $32.306536\ldots+1.818181\ldots+2.066115\ldots=36.190833\ldots$, matching the result. The check also confirms that the year-3 dividend is included once.

Common mistake avoided: Discounting the $\$40.00$ sale price but omitting the year-3 dividend understates value by $3/1.331$. Conversely, adding the dividend to the sale price and then adding that dividend again would double-count it.

Expected Cash Flows Under Uncertainty

When outcomes are uncertain, first find the expected cash flow for each date using the stated probabilities. Then discount those expectations. Since addition and expectation are linear, the expected dividend and expected resale price can be calculated separately or as a combined horizon-date cash flow. The probabilities must describe the relevant outcomes, and all scenario cash flows must be measured at the date they occur.

Worked Example: Scenario-Based Horizon Value

A share is expected to pay a certain dividend of $\$1.50$ at the end of year 1. At the end of year 2, there are two possible states: with probability 0.25, the dividend is $\$3.00$ and the ex-dividend sale price is $\$45.00$; with probability 0.75, the dividend is $\$1.00$ and the ex-dividend sale price is $\$25.00$. The annual required return is 10%. Find the time-0 value.

Method. Calculate the expected year-2 dividend and expected year-2 sale price from the two states, then discount the year-1 and year-2 amounts. The scenario probabilities sum to $0.25+0.75=1.00$.

Step 1: Find the expected year-2 dividend. $$ E_0[D_2]=0.25(3.00)+0.75(1.00)=0.75+0.75=\$1.50. $$

Step 2: Find the expected year-2 sale price. $$ E_0[P_2]=0.25(45.00)+0.75(25.00)=11.25+18.75=\$30.00. $$ Thus the expected combined year-2 cash flow is $\$1.50+\$30.00=\$31.50$. Equivalently, the state-based calculation is $0.25(3+45)+0.75(1+25)=12+19.5=\$31.50$.

Step 3: Discount by date. $$ P_0=\frac{1.50}{1.10}+\frac{31.50}{(1.10)^2} =1.363636\ldots+26.033057\ldots =27.396694\ldots $$ The value is $\$27.40$, rounded to the nearest cent.

Check. The expected year-2 amount discounted two years is $31.50/1.21=\$26.033057\ldots$, and the year-1 dividend discounted one year is $\$1.363636\ldots$. Their sum agrees with the reported value. The common mistake avoided is treating the expected year-2 sale price as the only year-2 cash flow and leaving out the year-2 dividend.

Using the Equation to Infer a Terminal Price

The equation can also be rearranged when the current price and interim dividend forecasts are known but the horizon resale price is not. For a constant required return $k$, subtract the present value of the dividends from $P_0$, then multiply by $(1+k)^n$. This gives the expected terminal price implied by the inputs. It is an implied value under the model, not a guarantee of the price that will actually be realized.

Result: If the equation’s assumptions hold and $E_0[P_n]$ is the unknown, then $$ E_0[P_n]=(1+k)^n\left(P_0-\sum_{t=1}^{n}\frac{E_0[D_t]}{(1+k)^t}\right). $$ The dividends are removed from today’s price on a present-value basis before the remaining value is accumulated to the horizon date.

Worked Example: Solving for the Implied Expected Sale Price

A share sells for $\$50.00$ today. An investor plans a two-year holding period and expects dividends of $\$1.10$ at the end of year 1 and $\$2.20$ at the end of year 2. The required return is 10% effective per year. Find the expected ex-dividend sale price at the end of year 2 that is consistent with the finite-horizon equation.

Method. Rearrange the equation to solve for $E_0[P_2]$. The year-2 dividend is a separate cash flow and must be subtracted from the horizon-date total after the remaining present value has been accumulated to year 2.

Step 1: Present value the year-1 dividend. $$ \frac{1.10}{1.10}=\$1.00. $$

Step 2: Remove that present value from today’s price and accumulate the residual two years. $$ \left(50.00-1.00\right)(1.10)^2=49.00(1.21)=\$59.29. $$ This is the horizon-date value of the year-2 dividend plus the expected ex-dividend sale price.

Step 3: Subtract the year-2 dividend. $$ E_0[P_2]=59.29-2.20=\$57.09. $$

Check. Discounting the year-1 dividend and the combined year-2 amount gives $1.10/1.10+(2.20+57.09)/1.21=1.00+49.00=50.00$, exactly the observed price. The common mistake avoided is treating $\$59.29$ as the sale price; it includes the year-2 dividend as well.

Interpretation and Common Traps

The finite-horizon equation separates the price today into the present value of cash flows received during the holding period and the present value of the share remaining at the horizon. A short horizon does not make later dividends irrelevant: they are reflected indirectly in the expected resale price, provided that resale price is forecast consistently with the model.

A useful way to check a calculation is to make a date-by-date cash-flow list before discounting. For each date from 1 to $n$, record the dividend; at date $n$, also record the sale price. Then apply the discount exponent corresponding to that date. This simple procedure prevents the most frequent errors:

Omitting the final dividend. Under the ex-dividend sale-price convention, $D_n$ is paid before the share is sold for $P_n$, so both belong in the valuation.
Using the wrong discount exponent. A cash flow at the end of period $t$ is discounted by $(1+k)^t$, not by $(1+k)^n$ unless it occurs at the horizon.
Double-counting cash flows. If the horizon dividend and sale price have already been combined, do not add that dividend again as a separate term.
Assuming the resale price is zero. A finite holding period does not imply that the share has no value at sale. A zero terminal price is appropriate only if the problem explicitly specifies it.
Mixing price conventions. Confirm whether a stated sale price is ex-dividend or includes the dividend. The equation’s cash-flow list must match that convention.

These checks keep the valuation tied to the actual timing of the investor’s receipts. They also clarify the role of the terminal price: it is necessary to close a finite-horizon valuation unless the problem provides a different terminal condition.

Check Your Understanding

For each problem, use the stated end-of-period dividend timing and round dollar values to the nearest cent.

A share will pay $\$1.00$ at the end of year 1 and $\$2.00$ at the end of year 2. Its expected ex-dividend price then is $\$30.00$. At a 5% effective annual required return, find its value today. (Answer: $\$29.98$)
A share pays $\$1.50$ for certain at the end of year 1. At the end of year 2, with probability 0.25 it pays $\$3.00$ and sells ex-dividend for $\$45.00$; with probability 0.75 it pays $\$1.00$ and sells ex-dividend for $\$25.00$. At a 10% annual required return, find today’s value. (Answer: $\$27.40$)
A share is worth $\$60.00$ today. It is expected to pay $\$2.00$ at the end of year 1 and $\$3.00$ at the end of year 2. If the annual required return is 8%, what expected ex-dividend sale price at the end of year 2 is consistent with the equation? (Answer: $\$64.82$)
A share is held for three years. It pays $\$2.00$ at the end of each year, and its expected ex-dividend sale price at the end of year 3 is $\$40.00$. Find today’s value at a 6% effective annual required return. (Answer: $\$38.93$)

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