Deriving The Perpetual Preferred Cost Formula

Why Derive the Preferred Cost Formula?

A preferred share may promise a fixed dividend, but its cost to the issuer is not automatically the stated dividend rate on the share’s par value. The relevant rate depends on the cash dividend and the amount investors pay for the share. To see why, value the stream of dividends using the same present-value reasoning used for other financial cash flows.

The simple formula applies to a particular case: a fixed dividend paid at regular intervals forever, with the first payment one full interval after valuation and no redemption payment. Changing the dividend amount, payment timing, or end date changes the cash-flow pattern and may change the valuation formula. This derivation is aligned with the SOA VEE Accounting and Finance syllabus.

Definition: Let $D_p$ be the fixed preferred dividend per share for each year, $P_0$ the share price just after a dividend date, and $k_p$ the effective annual return required by investors. For a level annual dividend that continues forever, with the next dividend paid one year from now and no redemption, the perpetual preferred cost is $k_p=D_p/P_0$. The formula assumes $D_p>0$, $P_0>0$, and $k_p>0$.

Here, $k_p$ is an annual rate consistent with annual dividend periods. If a dividend is specified as a percentage of par value, first convert that rate to a dollar dividend. For example, a 6% annual dividend on a $100 par value share means a cash dividend of $0.06(100)=\$6$ per year. It does not mean that the preferred cost is necessarily 6%; the share price also matters.

Deriving the Perpetuity Value

Assume the share pays $D_p$ at the end of each year, indefinitely, and that an investor discounts each future payment at the same effective annual required return $k_p$. The first payment occurs at time 1. The price is the present value of all future dividends.

P_0=\frac{D_p}{1+k_p}+\frac{D_p}{(1+k_p)^2}+\frac{D_p}{(1+k_p)^3}+\cdots

Set $v_p=1/(1+k_p)$. Since $k_p>0$, $0<v_p<1$, so the infinite geometric series converges. Its first term is $D_pv_p$ and its common ratio is $v_p$. The sum of an infinite geometric series with first term $a$ and common ratio $r$, where $|r|<1$, is $a/(1-r)$. Applying that result gives:

P_0=\frac{D_pv_p}{1-v_p} =\frac{D_p/(1+k_p)}{1-1/(1+k_p)} =\frac{D_p/(1+k_p)}{k_p/(1+k_p)} =\frac{D_p}{k_p}.

The cancellation in the last step explains why the share value is dividend divided by required return. Solving this value equation for the rate gives:

k_p=\frac{D_p}{P_0}.

The formula is not a rule that applies to every preferred share. It follows from the specific cash-flow and timing assumptions just stated. In particular, it treats dividends as level and paid as scheduled, starts the stream one full period from now, and includes no principal repayment or redemption amount. If the next dividend is due immediately, or if the share will be redeemed, the cash flows must be valued according to their actual timing and structure.

Identify the cash dividend.
Convert any dividend percentage of par value into the dollar amount paid per share for each payment period.

Check the timing and term.
Confirm that the next payment is one full period away, payments are level and continue forever, and there is no redemption payment.

Match the rate period to the dividend period.
Use an annual rate with annual dividends, or first calculate a periodic rate and convert it if an effective annual rate is needed.

Divide dividend by price.
When all assumptions hold, calculate $k_p=D_p/P_0$, then check that $P_0=D_p/k_p$.

Using the Formula for Annual Dividends

When dividends are annual and the timing assumptions hold, the calculation is direct. The dividend amount is the numerator; the market price is the denominator. The result is the annual return implied by receiving the level dividend for a share purchased at that price.

Worked Example: Dividend Stated as a Percentage of Par

A hypothetical preferred share has a par value of $100 and pays an annual dividend of 7% of par. Its market price is $87.50, and the next dividend is payable one year from now. Assume the dividend continues at the same amount forever and there is no redemption. Find the annual preferred cost.

Method. Convert the stated dividend rate to dollars, then apply the perpetuity formula. This avoids treating the dividend rate on par as the cost when the share is trading away from par.

Step 1: Calculate the annual dividend. The dividend is $D_p=0.07(\$100)=\$7.00$ per share.

Step 2: Divide by the market price.

k_p=\frac{\$7.00}{\$87.50}=0.08=8.00\%.

Step 3: Check by valuing the perpetuity. At an 8.00% annual return, the value of the level dividend stream is $\$7.00/0.08=\$87.50$, equal to the given price. The result is 8.00% to two decimal places.

This result differs from the 7% dividend rate on par because the investor pays $87.50, not $100, for the share. The numerator is the cash dividend actually received; the denominator is the amount paid for the share in this model. A common error is to report the par-based dividend rate as the cost without considering the market price.

Worked Example: Fixed Dollar Dividend and Market Price

A hypothetical preferred share pays $6.40 at the end of each year. Its price is $80 immediately after a dividend date. The dividend is expected to remain level indefinitely, the first future payment is one year away, and the share has no redemption date or redemption payment. Calculate the annual cost.

Method. Use the annual perpetuity yield $D_p/P_0$. The important check is that the quoted price is the price at the start of a full dividend period, not a price that includes an immediate dividend.

The cash dividend is already given, so $D_p=\$6.40$, and $P_0=\$80.00$. Therefore:

k_p=\frac{\$6.40}{\$80.00}=0.08=8.00\%.

To check, discounting the level annual stream at 8.00% gives a value of $\$6.40/0.08=\$80.00$. Thus the calculated rate reproduces the stated share price. The cost is 8.00% to two decimal places.

Payment Frequency and Rate Units

The dividend frequency and the return-rate period must match when using a perpetuity formula. With semiannual dividends, first calculate the return per half-year from the price and the dividend per half-year. If the requested answer is an effective annual cost, convert the half-year return by compounding two periods. Simply doubling a periodic rate gives a nominal annual rate, not an effective annual rate.

Worked Example: Semiannual Dividends and Effective Annual Cost

A hypothetical preferred share trades for $90 immediately after a dividend date. It pays $2.25 every six months, with the next payment in six months. The same payment continues every six months forever, and there is no redemption. Find the effective annual cost. Assume the investor’s return is compounded consistently with the six-month dividend periods.

Method. Apply the perpetuity formula using six-month units, then convert the periodic rate to an effective annual rate. This avoids mixing a half-year dividend with an annual rate.

Step 1: Find the six-month return. The price is measured just after a dividend date, so the next $2.25 payment is one full half-year period away:

j=\frac{\$2.25}{\$90.00}=0.025=2.50\% \text{ per half-year}.

Step 2: Convert to an effective annual rate. Two six-month periods compound in one year:

$$ k_p=(1+j)^2-1=(1.025)^2-1=1.050625-1=0.050625. $$

The effective annual cost is 5.0625%, or 5.06% to two decimal places. As a check, the annual cash dividend is $2(\$2.25)=\$4.50$. Dividing it by the price gives 5.00%, which is the nominal annual rate formed by doubling the half-year rate. It is not the effective annual rate, because it excludes the effect of compounding.

Timing Trap: An Immediate Dividend

The formula $k_p=D_p/P_0$ assumes the first payment is one complete period after the valuation date. If a payment is due immediately, it is not discounted. The stream then consists of that immediate payment plus a level perpetuity beginning one period later. The valuation equation must show both parts.

Worked Example: Dividend Payable Immediately

A hypothetical preferred share costs $78 at a valuation date when a $6 dividend is payable immediately to the purchaser. After that payment, it pays $6 at the end of every year forever. There is no redemption. Find the annual return $k_p$, rounded to four decimal places.

Method. Separate the immediate dividend from the future perpetuity. This avoids incorrectly discounting the immediate payment as though it were due in one year.

The immediate payment is worth $6 at the valuation date. The remaining dividends begin one year later, so their value is $6/k_p$. Therefore:

\$78=\$6+\frac{\$6}{k_p}.

Subtract the immediate payment from both sides:

\$72=\frac{\$6}{k_p}, \qquad k_p=\frac{\$6}{\$72}=0.083333\ldots.

Thus $k_p=8.3333\%$ to four decimal places. Check: $\$6+\$6/0.083333\ldots=\$6+\$72=\$78$. By contrast, the incorrect calculation $\$6/\$78=7.6923\%$ applies the ordinary perpetuity formula to a price that includes an immediate payment.

When the Simple Formula Does Not Apply

Before using the perpetual preferred cost formula, check the share’s cash-flow terms rather than relying only on the label “preferred.” The formula is suitable only when the dividend amount and its timing match the model. A finite life, a required redemption payment, a dividend that changes, or an immediate first payment changes the value equation. In those cases, value the stated cash flows with their actual dates and amounts instead of treating them as a level perpetuity.

The formula also presumes the price and dividend are measured on a consistent basis. A price just after a dividend date pairs naturally with the next payment one full period away. A price that includes a dividend payable immediately must account for that payment separately. With dividends less frequent than annually, calculate a rate per payment period first, then convert to the requested annual convention.

A useful exam check is to reconstruct the price from the calculated rate. Under the model, $D_p/k_p$ should equal $P_0$. If it does not, check for a missed conversion from a par-based rate to a dollar dividend, a mismatched time unit, an incorrectly timed first payment, or a cash flow that violates the perpetuity assumptions.

Key takeaway: For a level annual dividend paid forever, with the next payment one year from now and no redemption, the present value is $P_0=D_p/k_p$, so $k_p=D_p/P_0$. The formula’s simplicity depends on its assumptions; when dividend amount, timing, frequency, or term differs, write the cash-flow valuation equation before solving for a rate.

Check Your Understanding

For each problem, identify the relevant timing and apply the matching valuation method.

A hypothetical preferred share has a par value of $100, pays an annual dividend of 5.4% of par, and trades just after a dividend date for $72. Its level dividend continues forever, the next payment is one year away, and there is no redemption. Find the annual cost. (Answer: 7.50%)
A share pays a fixed annual dividend of $5.50 and trades for $88 just after a dividend date. It has no redemption and pays forever, with the next dividend in one year. Calculate the annual cost and check the implied perpetuity value. (Answer: 6.25%; implied value $88.00)
A preferred share costs $64 just after a dividend date and pays $1.60 every six months forever, beginning six months from now. Find the six-month rate and the effective annual cost. (Answer: 2.50% per half-year; 5.0625% effective annually)
A share costs $64 at a date when its $4.80 dividend is payable immediately; it then pays $4.80 at each year-end forever. There is no redemption. Set up the valuation equation and find the annual return. (Answer: $64=4.80+4.80/k_p$, so $k_p=8.1081\%$)
A hypothetical share pays $7.20 annually, with its next dividend one year away, but must be redeemed for $100 at the end of year 10. Explain whether $k_p= D_p/P_0$ is an exact valuation formula for this share. (Answer: No. The dividend stream is finite and a redemption payment is due; the valuation must include the 10 annual dividends and the $100 payment at year 10.)

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