Price return, total return and the price of a share

Where does an equity return come from?

A share earns a return from two sources only: the cash dividends the company chooses to pay, and the rise or fall in the share price over the period, a capital gain or loss . Unlike a bond, a share promises no payments and never matures.

Total returnprice change plus dividends, over the price paid

• Price return Memorize capital gain or loss; the whole return of a non-payer
• Dividends Memorize
  • Paid at the end add D to the end price
  • Reinvested in shares buy D ÷ payment-date price more shares
  • Reinvested elsewhere use the dividend's value at period end

Use prices adjusted for any split or reverse split. Returns can be measured over any holding period, a day, a week or a year. This lesson ignores taxes.

How do you calculate the return on a share?

Add the price change and the dividend, then divide by the price paid. If the share pays nothing, only the price change is left. That is the price return , and it is the whole return of a share that pays nothing, assuming no restructuring or liquidation during the period.

$$r_E = \frac{P_{t+1} - P_t}{P_t}$$

The total return adds the dividend received. Here $r_E$ is the realized rate of return . For a share that pays nothing, $D_{t+1} = 0$ and the total return reduces to the price return. (Quantitative Methods writes the same return with $r$ and $Inc$ for income. It is one idea in two notations.)

$$r_E = \frac{P_{t+1} - P_t + D_{t+1}}{P_t}$$

Questions often give the return and ask for something else: the end price you must sell at, or the dividend you need. It is the same formula, turned around.

Exam tip Two shares with the same total return can end at different prices. The one that paid a dividend needs less price gain.

The widget below follows the rest of this lesson: its first tab adds a split, then come a mid-year dividend, two years and the price today.

What if the share splits during the period?

Express the purchase price, the end price and the dividend per share on the same basis. The easiest way is to divide the purchase price by the split ratio and use the end price and dividend as quoted after the split. These are split-adjusted prices . A split never changes the return.

A reverse split works the same way and leaves the return unchanged. A 1-for-4 reverse split at year end on a share bought at 2.00 that ends at 2.30 with a 0.03 dividend gives 9.20 and 0.12 per new share against an adjusted cost of 8.00, still (9.20 − 8.00 + 0.12) ÷ 8.00 = 16.50%.

What if the dividend is reinvested?

A dividend received during the period can buy more shares at the price on the dividend payment date , written $P_{PD}$. These are reinvested dividends . You end the period with $1 + D \div P_{PD}$ shares, and the extra fraction rides the rest of the price move.

$$r_E = \frac{P_{t+1}\left(1 + \dfrac{D}{P_{PD}}\right) - P_t}{P_t}$$

Route 2 is not the reinvested-shares formula. When you are reinvesting at the risk-free rate , you grow the dividend to its value at the end of the period and put that value in the ordinary total-return formula. Only reinvestment in the same shares uses $1 + D \div P_{PD}$.

Reinvesting, like a reverse split, can leave you with fractional share ownership (0.024 of a share here), with a proportional claim on dividends and net assets.

How do you get a return over several years?

Use it

Compound the yearly total returns. This is compounding periodic returns , and it is the same as reinvesting each year’s dividend in shares at that year’s closing price. Add the last year’s dividend to the last price; it is not reinvested.

$$r = (1 + r_1)(1 + r_2)\cdots(1 + r_n) - 1$$

The same chaining appears in Quantitative Methods, Learning Module 1.

Price index or total return index?

Know the idea

A price index follows prices only. A total return index also reinvests the dividends. Over long periods the gap can be large, and it is largest where dividends are high.

Take two fictional markets over 15 years, with both indexes starting at 100. In Market A, fast-growing with low dividends, the price index reaches 420 and the total return index 500. Reinvested dividends account for 1 − 420 ÷ 500 = 16% of the ending value. In Market B, mature with high dividends, the price index reaches 180 and the total return index 360. Reinvested dividends account for 1 − 180 ÷ 360 = 50%.

Unlike an unrealized gain, a dividend is usually taxed when received, even if you reinvest it. That trims the total return an investor actually keeps. How indexes report both versions is covered in Quantitative Methods.

What should the share be worth today?

Today’s price should equal the present value (PV) of what you expect to receive: the dividends during the holding period and the price at the end, discounted at the required return on equity , $r_e$. This is the present value of dividends and a future price . The $E(\cdot)$ marks an expected value, because the future price is not known today. The expected price at the end is often an analyst’s price target .

$$P_t = \frac{E(P_{t+1}) + E(D_{t+1})}{1 + r_e}$$

With no dividend, the numerator is $E(P_{t+1})$ alone.

For a longer holding period, discount each dividend and the target separately. With a constant dividend D, the sum of the discounted dividends is an annuity, so the time value of money keys do it in one go.

$$P_t = \sum_{i=1}^{n} \frac{D}{(1 + r_e)^i} + \frac{E(P_{t+n})}{(1 + r_e)^n}$$

If the question gives the dividend just paid and a growth rate, grow it before discounting: the first dividend is $D_0(1 + g)$.

The present value of the expected price usually outweighs that of the dividends, unless the horizon is very long or the discount rate very high. Held indefinitely, a share’s value rests on its dividends. A company that pays nothing is worth only what its future price is worth today, which depends on how fast its net assets grow.

Price todaydiscounted at the required return

• Present value (PV) of expected dividends Memorize the more predictable part
• PV of the expected price at the end Memorize usually the larger part over a short horizon

The dividend part of an expected return is usually the more predictable part, because companies try to keep dividends steady. That is the predictability of dividends . Future prices, and so price targets, are far less certain. Learning Module 6 builds full discounted cash flow models from this idea.