Redeemable Preferred Stock And Required Returns

Why Redemption Changes the Cost Calculation

Preferred stock with no scheduled redemption is often analyzed as a stream of dividends continuing indefinitely. Redeemable preferred stock has a different cash-flow pattern: the issuer pays dividends for a specified number of periods and then pays a redemption amount. The issuer’s cost must reflect both parts of that commitment.

The calculation is a yield calculation: find the rate that makes the present value of the promised cash outflows equal to the issuer’s net proceeds at issue. It is aligned with the SOA VEE Accounting and Finance syllabus. The method is useful when comparing financing offers with different dividends, issue costs, redemption dates, or redemption premiums.

Definition: The cost of redeemable preferred stock is the per-period rate $k_p$ that equates the issuer’s net proceeds at issue with the present value of the scheduled preferred dividends and redemption payment. Net proceeds are the issue price less costs paid to issue the shares.

The Cash-Flow Model

Suppose the issuer receives net proceeds $N$ at time 0. It pays a fixed dividend $D$ at the end of each of $n$ equal periods and pays a redemption amount $R$ at the end of period $n$. Assume the dividends and redemption payment are made as scheduled, all cash flows occur at period ends, and the rate is constant over the term. The final dividend and redemption payment occur at the same time.

N=\sum_{t=1}^{n}\frac{D}{(1+k_p)^t}+\frac{R}{(1+k_p)^n}

Here $k_p$ is the required return per payment period. If payments are annual, it is an annual rate. If payments are quarterly or semiannual, it is the rate for one quarter or one half-year, respectively. Convert it to an effective annual rate when needed; do not label a per-period rate as an annual rate.

Net proceeds are the amount received by the issuer, not necessarily the shares’ stated or redemption value. For example, if shares sell for $100 and issue costs are $3 per share, the issuer receives $N=\$97$. Dividends and redemption are issuer cash outflows, so they appear as positive amounts on the right-hand side of the present-value equation.

The model treats the stated dividend payments as fixed and assumes redemption occurs at the stated date and amount. It does not model a missed dividend, a dividend reset, or a different redemption contingency. Those changes require a different cash-flow specification. Also, do not automatically apply an interest tax shield to preferred dividends: the calculation uses the tax treatment stipulated for the issuer and jurisdiction. In the usual treatment, preferred dividends are not deductible like interest.

Why the Equation Works

The equation follows from the time value of money. At a per-period required return $k_p$, a dollar paid at the end of period $t$ has a present value of $1/(1+k_p)^t$ dollars. Thus each dividend is discounted from its payment date, and the redemption payment is discounted from the final date. Their present values must add to the amount the issuer receives at issue if the return on the financing is $k_p$.

For fixed positive dividends and a positive redemption payment, the present value falls as $k_p$ rises. Differentiating the right-hand side with respect to $k_p$ gives

-\sum_{t=1}^{n}\frac{tD}{(1+k_p)^{t+1}} -\frac{nR}{(1+k_p)^{n+1}}<0

Each term is negative, so there can be at most one positive rate that solves the equation. If the present value at one trial rate is above $N$, increase the trial rate; if it is below $N$, decrease the trial rate. A financial calculator or spreadsheet can solve the equation directly. Bisection or another numerical method can also be used.

Solving for the Required Return

Worked Example: Annual Dividends and a Redemption Discount

A company issues redeemable preferred shares for $98 per share and pays issue costs of $2 per share. Each share pays an annual dividend of $7 for five years and is redeemed for $100 at the end of year 5. Estimate the issuer’s annual cost of the preferred stock, to two decimal places.

Method. First calculate net proceeds, then equate those proceeds to the present value of five annual dividends and the final redemption payment. The final year includes both the fifth dividend and the redemption amount.

Net proceeds are $N=\$98-\$2=\$96$. The equation is

96=\sum_{t=1}^{5}\frac{7}{(1+k_p)^t}+\frac{100}{(1+k_p)^5}

At $k_p=8.00\%$, the discount factor for year 5 is $1/(1.08)^5=0.680583$. The five-year annuity factor is $(1-0.680583)/0.08=3.992710$. Therefore the present value is

7(3.992710)+100(0.680583) =27.9490+68.0583 =\$96.0073

This is slightly above $96, so the exact rate is slightly above 8.00%. Solving the equation gives approximately $k_p=8.0019\%$, which rounds to 8.00% per year to two decimal places. As a check, at 8.01% the present value is approximately $95.97, below $96; the solution therefore lies between 8.00% and 8.01%.

Common mistake avoided: Using the $100 redemption value as the initial investment would ignore the $2 issue cost. The issuer receives $96, not $100, and the correct comparison is between $96 and the present value of its future payments.

Worked Example: Quarterly Dividends and an Effective Annual Return

A hypothetical issuer sells preferred shares for $99.10 each and pays $0.44 per share in issue costs. Each share pays a dividend of $1.50 every quarter for 12 quarters and is redeemed for $105 at the end of quarter 12. Find the quarterly cost and the corresponding effective annual cost, each to two decimal places.

Method. Solve first for the quarterly rate because the cash flows occur quarterly. Then convert that periodic rate to an effective annual rate. Net proceeds are $N=\$99.10-\$0.44=\$98.66$, and the equation is

98.66=\sum_{t=1}^{12}\frac{1.50}{(1+k_p)^t} +\frac{105}{(1+k_p)^{12}}

At a trial quarterly rate of 2.00%, $1.02^{12}=1.268242$, so $1/(1.02)^{12}=0.788493$. The annuity factor for 12 quarters is $(1-0.788493)/0.02=10.57535$. The present value of the dividends is $1.50(10.57535)=\$15.8630$, and the present value of redemption is $105(0.788493)=\$82.7918$. The total is $\$98.6548$, just below the $98.66 received.

Numerical solution gives a quarterly rate of approximately 1.9995%, which rounds to 2.00% per quarter. The effective annual rate is

(1+0.019995)^4-1\approx0.08241

Thus the effective annual cost is 8.24% to two decimal places. This is not the same as multiplying the quarterly rate by four, which would give a nominal annual rate; the effective annual conversion accounts for quarterly compounding.

Common mistake avoided: Applying an annual rate directly to 12 quarterly periods would mismatch the rate unit and the cash-flow dates. Use a quarterly rate in the equation, then convert the result.

Worked Example: Redemption Premium and Numerical Bracketing

A company receives net proceeds of $96 per share from a redeemable preferred issue. It pays an annual dividend of $5.50 for four years and redeems each share for $110 at the end of year 4. Estimate the annual cost to two decimal places.

Method. Include the redemption premium over a $100 reference value in the final cash flow, and find the rate that matches the present value to $96:

96=\sum_{t=1}^{4}\frac{5.50}{(1+k_p)^t} +\frac{110}{(1+k_p)^4}

At 8.92%, $(1.0892)^4\approx1.407442$, so the year-4 discount factor is approximately 0.710508. The four-year annuity factor is $(1-0.710508)/0.0892\approx3.24543$. Therefore, the dividend present value is $5.50(3.24543)=\$17.8499$, and the redemption present value is $110(0.710508)=\$78.1559$. Their sum is approximately $\$96.0058$.

The present value at 8.92% is slightly above $96, so the solving rate is slightly higher. At 8.93%, the present value is approximately $95.97, below $96. Numerical solution gives approximately 8.9218%; the cost is therefore 8.92% per year to two decimal places. The bracket confirms that the rate is consistent with the cash flows.

Common mistake avoided: Omitting the redemption premium would discount $100 rather than $110 in year 4 and understate the issuer’s promised outflow. Use the actual contractual redemption payment.

Interpreting and Using the Result

The cost is not simply the stated dividend rate. The dividend rate describes the dividend relative to a stated share value; the cost calculation also reflects the price actually received, issue costs, the timing and duration of dividends, and the redemption amount. A redemption premium increases the final outflow, while issue costs reduce proceeds. Either change generally raises the rate needed to equate present value with net proceeds, holding other cash flows fixed.

The result is a return implied by the specified cash flows under the stated assumptions. It is not a guarantee that every investor will realize that return: the model assumes payments are made as scheduled. It is also important to distinguish the periodic rate from the effective annual rate, especially when comparing financing with different payment frequencies.

Before accepting a result, check that the model uses the issuer’s net proceeds at time 0; includes every scheduled dividend; includes the redemption payment in the correct period; uses one consistent period for rates and cash flows; and converts the result correctly if an annual figure is required. These checks prevent the most common errors in estimating the cost of redeemable preferred stock.

Key takeaway: For fixed, scheduled dividends and redemption, solve for the rate that equates net proceeds with the present value of all promised payments. Use the actual redemption amount, align the rate period with the payment period, and report the effective annual rate when an annual comparison requires it.

Check Your Understanding

For each original problem, set up the issuer cash flows before solving. Answers are rounded as stated.

A company sells preferred shares for $92 and pays $4.68 per share in issue costs. It pays a $6 annual dividend for four years and redeems each share for $100 at the end of year 4. Estimate the annual cost to two decimal places. (Answer: 10.00% per year)
An issuer receives net proceeds of $102.68 per share. It pays annual dividends of $7.50 for three years and redeems the shares for $105 at the end of year 3. Estimate the annual cost to two decimal places. (Answer: 8.00% per year)
A hypothetical issuer sells preferred shares for $93 and pays $1.53 per share in issue costs. It pays $2 every six months for 10 half-year periods and redeems each share for $100 with the tenth dividend. Find the effective annual cost to two decimal places. (Answer: 6.09% effective per year)
An issuer receives net proceeds of $110.80 per share. It pays annual dividends of $9 for five years and redeems each share for $110 at the end of year 5. Estimate the annual cost to two decimal places. (Answer: 8.00% per year)

What You'll Learn