One-Period Dividend and Resale-Price Valuation

Introduction

An investor who expects to hold a share for one period receives two kinds of cash flow: any dividend paid during the period and the proceeds from selling the share at the end. The share’s value today depends on both. Focusing only on the dividend omits the resale proceeds; focusing only on the future sale price omits the dividend.

The one-period model makes this relationship explicit. It is useful both for valuing a share and for finding the required return implied by a current price. It is a single-period cash-flow model, not a model of an unlimited stream of dividends.

Definition: Let $P_0$ be the share price today, $D_1$ the dividend received at the end of the holding period, $P_1$ the share’s resale price at that time, and $r$ the required effective return for the period. Under the one-period valuation model, the current price is the present value of the expected end-of-period cash flows: $$ P_0=\frac{E[D_1+P_1]}{1+r}. $$ If the dividend and resale price are known with certainty, this becomes $P_0=(D_1+P_1)/(1+r)$. The model assumes the relevant cash flows occur at the period’s end, the return and period use consistent units, and $r$ is appropriate for the investment’s risk. It excludes taxes, transaction costs, and other cash flows unless they are explicitly added.

How the One-Period Valuation Works

The investor’s total end-of-period proceeds are the dividend plus the price received on sale. Discounting those proceeds by one period converts them to a value at time 0. When future proceeds are uncertain, the model uses their expected value, and the required return must reflect the risk of those proceeds. This is a valuation assumption: discounting expected cash flows at an arbitrary rate does not necessarily produce a valid market value.

The model can also be written in terms of the expected holding-period return. For known cash flows, that return is the dividend plus the change in share price, divided by the initial price. For uncertain cash flows, its expected value is calculated using the expected end-of-period cash flows.

\text{Expected holding-period return} =\frac{E[D_1+P_1]-P_0}{P_0}.

Setting this expected return equal to the required return $r$ gives the one-period valuation equation. Rearranging the return expression yields $E[D_1+P_1]=P_0(1+r)$; dividing by $1+r$ gives $P_0=E[D_1+P_1]/(1+r)$. This also explains why the dividend and resale price are added before discounting: both are proceeds received at the same end-of-period date.

For certain cash flows, the return separates into a dividend yield and a price-appreciation component:

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

The first term is the dividend yield on the initial price. The second is the percentage price change over the period. It can be negative if the resale price is below the initial price. Neither component alone is the total return.

Worked Example: Valuing a Share from Its Dividend and Resale Price

A hypothetical investor expects to receive a dividend of $2.00 at the end of one year and sell a share for $53.00 at that time. The required effective annual return is 10%. Assume both cash flows are certain, there are no other cash flows, and the dividend is paid at year-end. Find the share’s value today and verify the expected return.

Method. Add the two end-of-year proceeds, then discount their total by one year at the required annual return.

The total proceeds at year-end are:

D_1+P_1=\$2.00+\$53.00=\$55.00.

Discount these proceeds:

P_0=\frac{\$55.00}{1.10}=\$50.00.

The dividend yield is $\$2.00/\$50.00=0.04$, or 4%. The price appreciation is $\$53.00-\$50.00=\$3.00$, so its return component is $\$3.00/\$50.00=0.06$, or 6%. Together, the components give a 10% holding-period return.

Check. The investor pays $50.00 and receives $55.00 at year-end, so the return is $(\$55.00-\$50.00)/\$50.00=0.10$, or 10%. The calculation is consistent with the required return. The common mistake avoided here is treating the $55.00 future proceeds as if they were already worth $55.00 today.

Solving for a Missing Cash Flow or Price

Once the valuation relationship is stated, it can be rearranged to find an unknown. In particular, the required end-of-period proceeds equal the current price accumulated at the required return. Subtracting a known dividend from those proceeds gives the implied resale price; subtracting a known resale price gives the implied dividend.

$$ E[D_1+P_1]=P_0(1+r). $$

Worked Example: Finding the Implied Resale Price

A hypothetical share is worth $48.00 today. An investor expects a year-end dividend of $1.92, and the required effective annual return is 12%. Under the certain-cash-flow one-period model, what year-end resale price is consistent with that price and return?

Method. First calculate the total year-end proceeds needed to earn 12% on $48.00, then subtract the dividend.

The required total proceeds are:

P_0(1+r)=\$48.00(1.12)=\$53.76.

The dividend is $1.92, so the implied resale price is:

P_1=\$53.76-\$1.92=\$51.84.

Check. The price increase is $\$51.84-\$48.00=\$3.84$. Adding the $1.92 dividend gives a total gain of $5.76; dividing by the initial $48.00 gives $5.76/48.00=0.12$, or 12%. The answer is consistent with the required return. The common mistake avoided is multiplying the resale price alone by $1+r$ and then treating the dividend as an extra return rather than part of total proceeds.

The same rearrangement can find a required return. With known cash flows and a known current price, the return that equates the price to the discounted proceeds is the total gain divided by the initial price. This is the one-period return implied by the inputs; it is not necessarily an appropriate required return for the share’s risk.

Worked Example: Finding the Required Return

A hypothetical share costs $64.00 today. Its expected year-end dividend is $2.56, and its expected resale price is $66.56. Assume the dividend and resale price are certain for this calculation. Find the one-year return implied by these figures, to two decimal places, and check the valuation.

Method. Add the dividend to the resale price to find total proceeds, then compare the proceeds with the initial price.

D_1+P_1=\$2.56+\$66.56=\$69.12, $$ $$ r=\frac{\$69.12-\$64.00}{\$64.00} =\frac{\$5.12}{\$64.00}=0.08.

The implied required return is 8.00%. Its components are a dividend yield of $\$2.56/\$64.00=0.04$, or 4%, and a price-appreciation return of $(\$66.56-\$64.00)/\$64.00=\$2.56/\$64.00=0.04$, also 4%.

Check. Discounting the $69.12 total proceeds at 8% gives $\$69.12/1.08=\$64.00$, the stated share price. The common mistake avoided is calculating only the price-appreciation return and omitting the dividend yield.

When the Resale Price Is Uncertain

A future sale price may depend on events that are unknown today. In that case, calculate its expected value using the stated probabilities, then combine that expectation with any known dividend. The probabilities must cover all stated outcomes and sum to 1. If the dividend is also uncertain, the expectation must include it as well; the expected value of the total cash flow is $E[D_1+P_1]$.

The expected cash flow is not a promise that the share will sell for that amount. Actual proceeds will depend on the realized outcome, and the realized holding-period return may differ from its expected value. Risk matters: the required return must be suitable for the uncertainty of the share’s cash flows.

Worked Example: Using an Expected Resale Price

A hypothetical share will pay a certain $2.00 dividend in one year. Its resale price is forecast to be $45.00 with probability 0.25, $55.00 with probability 0.50, or $65.00 with probability 0.25. Assume these are the only possible resale outcomes and the probabilities represent the investor’s valuation assumptions. The required one-year return is 14%. Find the model value today.

Method. Calculate the expected resale price, add the certain dividend, and discount the expected total proceeds at the stated required return.

E[P_1]=0.25(\$45.00)+0.50(\$55.00)+0.25(\$65.00) $$ $$ =\$11.25+\$27.50+\$16.25=\$55.00.

The probabilities sum to $0.25+0.50+0.25=1.00$, as required. Expected total proceeds are $\$2.00+\$55.00=\$57.00$, so:

P_0=\frac{\$57.00}{1.14}=\$50.00.

Check. At a price of $50.00, expected proceeds of $57.00 imply an expected return of $(\$57.00-\$50.00)/\$50.00=0.14$, or 14%. The result is the value under the stated expected-cash-flow and required-return assumptions; it does not mean that the investor is certain to receive $57.00.

Common Exam Traps and Model Limits

A useful way to avoid mistakes is to identify the date and units of every quantity before calculating. $P_0$ is a present price; $D_1$ and $P_1$ are end-of-period cash flows or values. If the period is one year, use an annual effective return. If a different period is intended, the rate must be appropriate to that period rather than used as though it were automatically annual.

Leaving proceeds undiscounted. Adding $D_1+P_1$ gives the end-of-period proceeds, not the current value. Divide by $1+r$ to value those proceeds today.
Omitting one of the cash flows. The total includes both the dividend and the resale price. A dividend-only calculation ignores the sale proceeds; a price-only calculation ignores the dividend.
Confusing expected with certain proceeds. If $P_1$ is uncertain, an expected price does not guarantee the realized sale price. State the outcome assumptions and use probabilities consistently.
Using an unsuitable rate. The required return must correspond to the holding period and be appropriate for the risk being valued. A return inferred from a price is not automatically a suitable required return.

The one-period model is most useful when the investor’s horizon and end-of-period cash flows are clearly specified. It does not, by itself, describe how to value dividends received across several future periods or justify a particular required return. For those questions, additional assumptions and valuation relationships are needed.

Key takeaway: Under the one-period model, current share value equals expected dividend plus expected resale proceeds, discounted by one period at the appropriate required return. Check that the cash flows, rate, and holding period match, and include both components of the investor’s payoff.

Check Your Understanding

For each hypothetical share, use the one-period model and the stated assumptions. Answers are rounded to cents where they are dollar amounts.

A share is expected to pay a certain $1.50 dividend and sell for $32.50 at the end of one year. If the required effective annual return is 10%, what is its value today? (Answer: $30.91)
A share costs $72.00 today and is expected to sell for $75.60 in one year. If the required effective annual return is 8%, what year-end dividend is consistent with the current price? Assume certain cash flows. (Answer: $2.16)
A share costs $40.00 today, pays a certain $1.60 year-end dividend, and is expected to sell for $42.40. What one-year return is implied by these cash flows? (Answer: 10.00%)
A share will pay a certain $1.50 dividend in one year. Its resale price is expected to be $38.00 with probability 0.20, $44.00 with probability 0.50, or $53.00 with probability 0.30. If the required one-year return is 10%, what is its value today? (Answer: $42.73)
A share costs $90.00 today and is expected to pay a certain $3.60 dividend in one year. If its expected one-year return is 12%, what expected resale price is implied? (Answer: $97.20)

What You'll Learn