A Simple Algorithm for the Valuation ofPreferred Stock

Pradipkumar Ramanlal

Preferred stocks, hoth the straight variety as well as those that are callable and convertible, represent asignificant source of corporate financing. Although several theoretical developments on preferred stockvaluation have been around for some time, they are not amenable to straightforward calculation. Thisarticle provides an algorithm that makes accessible Mcrton's (1974) and IngersoU's (1977a) preferredstocks valuation formulas adapted for dividend paying common stock. The algorithm is simple and requiresevaluation of algebraic expressions. It can be used by practitioners with access to programmable calculatorsand can be incorporated in the finance curricula of advanced students.

• The value of preferred stock, as it is taughl in mostfinance courses, is simply the present value of aperpetuity, i.e.. the promised annual payment dividedby the appropriate discount rate. While the method'ssimplicity is appealing, it provides no practicalguidance to students and practitioners on how todetermine the appropriate discount rate, therebylimiting the method's application. The technique thatis often suggested to overcome this limitation is touse the implied discount rate of another preferred stockwith similar default risk. This approach is reminiscentof the no-arbitrage method of bond pricing using theyield curve implied by prices of bonds with similardefault risk. While the approach may eliminatearbitrage opportunities across securities withsimilar default risk, arbitrage opportunities acrosssecurities with different levels of default risk maynevertheless remain.

Another method of valuing preferred stock isbased on option-pricing theory. Treating thepreferred stock (consol bond) as an option on thevalue of the underlying firm, Merton (1974) derivedthe differential equation for the preferred stock'sprice process accounting for default risk. He thensolved the differential equation to obtain thepreferred stock's valuation formula. Unfortunately,this formula is not readily useable because it isexpressed in terms of the gamma and the incomplete

Pradipkumar Ramanlal is an AssLslanl Professor of Finance althe University of Souih Carolina, Columbia, SC 29208.

gamma functions, which are indeterminate integralsthat must be evaluated by numerical methods. Forthe most part, such methods are not within the reachof either students or practitioners.

The relative merits of the two methods for valuingpreferred stocks, the traditional perpetuity model andthe option-based model, are analyzed by Ferreira,Spivey. and Edwards (1992) for noncotnertihle issues.They find that the perpetuity model more accuratelyprices new issues of preferred securities while theoption-based model more accurately prices seasonedissues. Thus the evidence is mixed. For convertibleissues, the evidence is clearer. Firstly, while the option-based model can be adapted to account for conversionfeatures, the perpetuity model cannot (see Ingersoll,1977a). And secondly, the pricing errors that Ferreira,Spivey, and Edwards (1992) find for nonconvertiblepreferred issues using the perpetuity model are largerthan the pricing errors that Ramanlal, Mann, and Moore(1997) find for convertible issues using the option-based model. Ramanlal, Mann, and Moore furthermoreshow that ignoring the conversion features will resultin gross errors of model prices relative to market prices.Thus using the perpetuity model to price convertibleissues cannot be advised. We propose that the option-based pricing method be used in all cases except fornew nonconvertible issues.

This study provides a methodology that improvesaccessibility to Merton's (1974) pricing formula fornonconvertible preferred stocks. More importantly, the

11

12 FINANCIAL PRACTICE AND EDUCATION — SPRING / SUMMER 1997

methodology can be adapted to Ingersoll's (1977a)valuation results for callable and/or convertiblepreferred slocks. The methodology will be particularlyuseful to practitioners and advanced students offinance. For example, suppose a corporate treasurerconsiders raising capital by issuing, say, callableconvertible preferred stock. How does the treasurerdetermine the offer price for the security? Equallyimportant, how does an investment analyst in advisinga client determine whether a seasoned preferred issueis fairly priced? The apparent choices available to thetreasurer and the iinalyst are to either rely on aninvestment banker or determine the security's valueby solving complex integrals using numericaltechniques. The methodology proposed in this studyprovides an alternative which permits calculatingpreferred stock values using either a hand-heldcalculator or spreadsheet program.

I. The Straight Preferred Stock'sValuation Formula

If the firm's value V follows a lognormal stochasticprocess and the preferred stock's aggregate value f ismodeled as a contingent claim on the value ofthe firm,f(V), the straight-preferred stock's valuation functionsolves the ordinary differential equation (see Merton,1974)

(2)

+(rV-C)f- (1)

with the lower-boundary condition f(0) = 0 and theupper-boundary condition f(°o) = c/r. In Equation (1),the subscripts denote derivatives, o is the annualizedstandard deviation ofthe instantaneous return dV/V. ris the annual riskless rate, C is the aggregate annualcash flow to all equity claims including the preferredstock, and c is the aggregate annual cash flow topreferred stockholders.' The lower-boundary conditionfollows from the firm's limit liability. The upper-boundary condition holds because if the firm's valueincreases without bound, the probability of default isremote and therefore the preferred stock becomesriskless. Accordingly, the preferred stock's value inthat limit is simply a perpetuity with annual cash flowsc discounted at the riskless rate r. The straight-preferred stock's valuation function Is-

'In contrast to Merton's (1974) original formulation for consolbonds as well as Ingersoll's (1977a) adaptation to convertiblesecurities, our forniulalion permits dividend payments on theunderlying common slock (i.e.. C # c).-The fact that Equation (2) solves Equation (1) can be shownby substituting the transformation l'CV) = (c/C)F(V) in (1) andusing Merton's (1974) result that l/2a-V-F^^ + (rV - C)F^ - rF +C = 0 (see Ingersoll, 1977a, for details).

where

P(a,z) =ra

In Equation (2), r(a) is the gamma function, theintegral is the incomplete gamma function usuallydenoted by r(a,z), a = 2r/o' and d = C/rV. FollowingIngersoll (1977a), the firm's value V is interpreted asthe value of all equity securities (i.e., common stockand all convertible securities).

The differential Equation (I) is derived under theassumption of no arbitrage and therefore holdsregardless of the risk preferences of individuals in theeconomy. It follows that the solution ofthe differentialequation, given by Equation (2), which is the straight-preferred stock's valuation function, also holdsregardless of risk preferences.

Note that Equation (2) is simply a perpetuity withannual cash flows c discounted at the riskless rate r,multiplied by the factor in brackets which accountsfor the impact of default risk on price. The default riskinfluences the pricing function via the parameters a, d.It is reasonable to expect that the default risk willincrease as the parameter a decreases, either becauseof a lower riskless rate r (which implies a lower growthrate for the firm),' or because of higher firm volatilityG- (which increases the preferred stock's downsiderisk). It is also reasonable to expect that the defaultrisk will increase as the parameter d increases. To seethis, note that C/r is the upper bound on the presentvalue of cash payments to all equityholders. If C/r islarge relative to V, the likelihood that the firm cansustain such cash payments without facing potentialfinancial distress is low, i.e., the likelihood of defaultincreases as d increases.

II. Methodology to Obtain ApproximatePreferred Stock Values

The valuation formula presented in Equation (2)would be simple to use if the P(a,z) term could beevaluated. To evaluate this term, one has to resort toeither integration by numerical methods, lengthy tablesofthe gamma and incomplete gamma function, seriesapproximation of the functions which have notoriouslypoor convergence properties, or the obscure methodof continued fractions. These methods are neithersimple nor convenient.

To overcome these problems, we provide simplealgebraic expressions that may be used to evaluate

'When the firm's value follows the stated lognormal process,the firtn's mean growth rate is r-o'/2.

RAMANLAL — A SIMPLE ALGORITHM FOR THE VALUATION OF PREFERRED STOCK 13

preferred stock values. For convenience, the straight-preferred stock's value is represented as follows:

f(V) = (3)

where n; is the default risk premium, i.e., the percentagepremium above the riskless rate that compensatespreferred stockholders for the default risk they bear.Equation (3) exactly equals Equation (2) if

n= (4)

To see this, substitute the expression for n given inEquation (4) into the denominator in Equation (3). Tberight-hand side of the resulting expression in (3) exactlyequals the right-hand side of (2). Thus. Equations (3)and (4) together exactly represent Equation (2). Noticethat while the preferred stock valuation formula in (3)appears reminiscent of pricing that assumes riskneutrality, this valuation formula actually holdsregardless of the risk preferences of individuals in theeconomy because, as argued earlier. Equation (2) holdsregardless of preferences.

In Equation (3), the value of preferred stock, f,depends on the annual cash flows to preferredstockholders, c, the riskless rate. r. and the default riskpremium, 7U; c and r are constants and therefore onlynneeds to be evaluated according to Equation (4). Ingeneral, evaluating 7i is difficult because it dependson P(»,»), which is an indeterminate integral (seeEquation (2)). To overcome this problem, we providean approximate algebraic expression for TT, whichwe know from Equation (4) depends only on a = 2r/o- and d = C/rV.

For our approximation ofjt to be of practical use, theriskless rate r is assumed to vary between 0.06 andO.I 2,''and the standard deviation of the firm's value isassumed to vary between 0.11 and 0.50.^ Thus, theparameter a = 2r/a- ranges approximately from 0.50 to20.0. The parameter d = C/rV is assumed to range from0 to 1. When d approaches 0, the firm's value V is largerelative to the present value at the riskless rate ofcashflows to all equityholders C/r, thus the security isessentially riskless. In contrast, when d approaches I,V is approximately C/r, thus default on the preferredstock or a reduction in common dividends is imminent.

Two approximations for n in Equation (4) are

•"This is reasonable if Ihe riskless rate is inlerpreied as theyield-Io-maturity of the 30-year US government bond. Theaverage of monthly values of the annualized long-yield of USgovernment bonds from 1988 to 1994 i.s 8.16% (Colenian.Fisher, and Ibbotson, 1995).'These values imply that the value of the firm may vary by aslittle as 11% or as much as 50%i of its initial value over theperiod ol" one year. The average of the standard deviation ofdecile portfolios of New York Stock Exchange firms is 0.30(Ibbotson Associates. 1995),

provided; the first holds when the parameter a exceeds1 and the second holds when the parameter a is lessthan 1:

7r(d) = a(d - 5) + P(d -

where 5 < d< 1 and a > 1

where 0 < d < 1 and a < 1

(5a)

(5b)

In Equation (5a), the values of a, p. ri, and 6 dependon the parameter a, and 7r(d<6) is zero. These valuesare plotted in Exhibit 1 for a between 1 and 20. Noticethat as the parameter a increases from 1 towards 20, ais large while P andr] are small, then p becomes largewhile a andri remain small, and finally r\ becomes largewhile a and p remain small. These suggest that as aincreases, 7E(d) is primarily linear in d, then quadratic,and finally cubic. The value of 6 increasesmonotonically as the parameter a increases. Thisimplies that the default risk premium TC takes on positivevalues that are significantly different from zero forlarger and larger values of d (i.e., smaller and smallervalues of V) as a increases (i.e., as the firm's volatilitya^ decreases). In other words, as expected, the defaultrisk premium is small if the likelihood of default is low.

In Equation (5b), the values of a and P also dependon the parameter a. These values are plotted in Exhibit2 for a between 1/2 and 1. As the parameter a decreasesfrom 1/2 towards I ,a is large while P is small, and thenp increases while a decreases. This suggests that as adecreases, n(d) is primarily linear in d, taking on asquare root dependence for smaller values of theparameter a.

To enable usage of these results to evaluate actualpreferred stock prices, tabulated values are provided.In Exhibit 3, a. p. r|, and 6 are provided for a range ofvalues of the parameter a between 1 and 20. Thesemay be used with Equation (5a). Exhibit 4 provides aandp forvaluesof the parameter a between 1/2 and 1,and these may be used with Equation (5b). If the valueof the parameter a lies between, say, I and 1.2, linearinterpolation may be used to obtain correspondingestimates for a, p. T|, and 5.''

To illustrate how well the approximations for the

•̂ Exhibits 3 and 4 are relatively compact, however. u.sage ofour results is enhanced if algebraic expre.ssions for thedependence of a. p. r\. and 8 on the parameter a are available.Exhibits I and 2 show these dependencies are highly nonlinearis some cases, and therefore using the tabulated values isrecommended. Nevertheless, we provide the following algebraicapproximations that may be used instead (these areobtained by nonlinear curve fitting): For use with Equation(5a): a(a) =l/2a'. p(a) = (1.27/a)Expl-2.24a--"]. ii(a) =(0,23a'"-)Ln[0,4la"*''] +0,20a"". 5(a) = Max[O, -0.072 +0.057a - 0.0014a-l. For use with Equation (5b): a(a) = -0-076 + 1.49a - 0.79a=, p(a) = 1.73 - 3..14a + 1.57a^

14 FINANCIAL PRACTICE AND EDUCATION — SPRING / SUMMER 1997

Exhibit 1. Parameters a, p, TI, and 5 as Functions of a for Vaiues of a Between 1 and 20(See Equation (5a))

0 . 6 •

0 . 4 •

0 . 2

20

Exhibit 2. Parameters a and p as Functions of a for Vaiues of a Between 1/2 and 1 (SeeEquation (5b))

0 . 6 •

0 . 4

0 . 2

RAMANLAL — A SIMPLE ALGORITHM FOR THE VALUATION OF PREFERRED STOCK

Exhibit 3. Parameter Vaiues of the Defauit Risi< Premium 7i(d) in Equation (5a).Values of a, p, r\, and 5 are provided for values of a between 1 and 20. • ^ *'

IS

a

I.O

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

4.0

a

0.5000

0.3220

0.2095

0.1380

0.0915

0.0605

0.0395

0.0252

0.0153

0.0084

0.0037

-0.0051

0.0848

0.2846

0.3869

0.4319

0.4459

0.4441

n.4335

0.4186

0.4020

0.3852

0.3683

0.2980

11

-0.0025

-0.0906

-0.1281

-0.1368

-0.1307

-O.I 180

-0.1019

-0.0844

-0.0669

-0.0498

-0.0329

0.0399

5

o.oooo

0.0031

0.0069

0.0123

0.0194

0.0278

0.0371

0.0473

0.0579

0.0689

0.0800

0.1353

a

5.0

6.0

7.0

8.0

9.0

10.0

12.0

14.0

16.0

18.0

20.0

a

-0.0052

-0.0033

-0.0015

-0.0007

0.0025

0.0045

0.0072

0.0092

0.0114

0.0128

0.0144

0.2487

0.2131

0.1884

0.1676

0.1513

0.1362

0.1155

0.1015

0.0867

0.078 i

0.0683

0.0996

0.1518

0.1972

0.2421

0.2845

0.3282

0.4082

0.4841

0.5655

0.6400

0.7191

8

0.1857

0.2303

0.2692

0.3035

0.3338

0.3607

0.4066

0.4444

0.4762

0.5033

0.5269

Exhibit 4. Parameter Values of the Default Risi< Premium 7E(d) in Equation (5b)Values of a and p are provided for values of a between 1/2 and 1.

a

0.50

0.52

0.54

0.56

0.58

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

a

0.4630

0.4813

0.4978

0.5127

0.5261

0.5382

0.5491

0.5589

0.5678

0.5758

0.5831

0.5869

0.5954

P0.4673

0.4239

0.3839

0.3469

0.3128

0.2811

0.2517

0.2244

0.1989

0.1751

0.1529

0.1321

0.1126

a

0.76

0.78

0.80

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

a

0.6006

0.6053

0.6095

0.6133

0.6166

0.6195

0.6221

0.6244

0.6263

0.6280

0.6293

0.6305

0.6315

0.0943

0.0771

0.0610

0.0457

0.0314

0.0179

0.005 1

-0.0069

-0.0183

-0.0291

-0.0393

-0.0489

-0.0581

16 FINANCIAL PRACTICE AND EDUCATION — SPRING / SUMMER 1997

Exhibit 5. The Defauit Risi< Premium (jc) .̂ . ,This exhibit shows exact values (using Equation (4)) and approximate values {using Equations (5a) and (5b)) for valuesof d between 0 and 1 and for a = 0.5,0.7, 1.0,4.0. and 15.0. The exact and approximate values are almost indistinguishable.

3

0.8

0.6

0.4

0.2

a=0.5

a=0.7

a = I.O

0.2 0.4 0.6

default risk premium Ji given by (5a) and {5b) fit theexact relation in Equation (4), Kid) is plotted in Exhibit5 for values of d between 0 and I and for values of aequal to 0.5,0.7, 1.0,4.0, and 15.0, using both the exactand approximate relations. Each line in Exhibit 5 isactually a pair of lines {corresponding to exact andapproximate values) that are almost indistinguishable.We conclude that Equations {5a) and (5b) provide agood approximation of Equation (4) for most practicalpurposes.

Of course, the definitive test is how well theapproximation works in yielding preferred stock prices.For this, we compare predictions from the exact pricingrelation from Equation {2) with those from Equation(3), where n is given by the approximations presentedin Equations (5a) and (5b). Both the exact andapproximate pricing functions, f, are plotted in Exhibit6 for values of d between 0 and 1 and for values of aequal to 0.5,0.7, 1.0,4.0, and 15.0. We take the annualcash flow to preferred stockholders to be c = 10 andthe riskless rate to be r = 0.05. thus the value of thepreferred stock with zero default risk is c/r = 200. InExhibit 6, all valuation functions equal 200 when dequals 0 because the default risk at this point is indeedzero. Again, each line in Exhibit 6 is actually a pair oflines (corresponding to exact and approximate values)that are almost indistinguishable. We conclude thatEquation (3) together with Equations (5a) and (5b)provide a good approximation of the exact preferredstock valuation formula given in Equation (2).

III. Adaptations to Convertible PreferredStock

We now show how the approximate valuationfunction for straight preferred stock given in Equations(3), (5a). and (5b) may also be used to value convertiblepreferred stock. Both the callable and noncallabletypes of preferred stock are examined. The aim is toshow that knowing how to value the straight issue,f(V), is sufficient to value convertible issues.

A. Noncallable Convertible Preferred Stock

Eor a noncallable preferred stock that is convertibleinto a fixed proportion 7 of the firm's value V, thevaluation function, denoted by g(V), satisfiesEquation (1) but with the new upper-boundarycondition g (̂<») =7 . This boundary condition arisesfrom the optimal exercise of the preferred stock'sconversion option (see Brennan and Schwartz, 1977).The convertible-preferred stock's valuation functionI S '

(6)

''The fact that Equation (6) solves (1) can be shown by directsubsiiiution and u.sing the result Ihat f(V) solves Equation (1).The new boundary condition can be shown to hold by notingthat t\,(«>) = 0. The latter equality holds because f(V) approachesthe con.stant value c/r as V approaches infinity.

RAMANLAL — A SIMPLE ALGORITHM FOR THE VALUATION OF PREFERRED STOCK 17

Exhibit 6. The Vaiue of Preferred Stocic (f)This exhibit shows exact values (using Equation {2)) and approximate values (using Equations (3), (5a), and (5b)) forvalues of d between 0 and 1 and for a = 0.5,0.7,1.0,4.0, and 15.0. The exact and approximate values are almost indistinguishable.

200

180

160

140

120

0 . 2 0 . 4

The two terms on the right-hand side of Equation(6) are simply the components of the convertiblepreferred stock's worth in terms of its straight value,f(V). and the value of its conversion option, 7(V -Cf{V)/c). The conversion option vaiue is amenablelo a simple interpretation. Cf(V)/c is simply thepresent value of cash flows, incorporating default risk,to all equityholders including the preferred issue. Thus.(V - Cf{V)/c) can be interpreted as the remaining firmvalue to which preferred stockholders have theproportionate claim 7. i.e.,7(V - Cf(V)/c). This resultsimply states that preferred stockholders are notprotected against equity dilution arising from dividendpayments to other equityholders including commonshareholders."

B. Caiiabie Convertible Preferred Stock

For a callable, convertible preferred stock that isconvertible into a fixed proportion7 of the firm's valueand callable with fixed call price K, the valuationfunction, denoted by h{V), satisfies Equation (1) butwith the new upper-boundary condition h(V=K/7) =7V. This boundary condition arises from the optimal

"II is interesting to note (see Equation (6)) that when cashflows to preferred stockholders is less than what they wouidreceive if they had converted their securities (i.e., c < yC), themarket value of the preferred siock is less than its conversionvalue {i.e., g{V) < yV). Under these conditions, il is better forpreferred stockholders to convert. (See Constantinides andGrundy. 19B7, for discussions on ihis topic.)

exercise of the preferred's call and conversion optionsassuming perfect capital markets (see Ingersoll. 1977a).^The corresponding valuation function is"'

h(V) = g(V) - f{V) (7)

V^ = K/7 is the firm's value when the preferred stock iscalled. The two terms on tiie right-hand side ofEquation (7) are the components of the callablepreferred stock's worth in terms of its noncallablecounterpart's value, g(V), and the vaiue of the firm'scall option, A '(c/C - 7)(V - Cf(V)/c). It can be shown

•"The preferred stock should be called when the firm's valueincreases such that the conversion value equals the call price{i.e., y\ - K); preferred stockholders convert at Ihat pointand so the preferred stock's value just prior equals itsconversion value {i.e., h(V= y/ic) = yV). While early studies ofcalls of convertible securities found deviations from thisoptimal behavior (see. e.g., Ingersoll, 1977b, and Mikkelson.1985), more recent work finds either insignificiint violations{Asquith, 199.'i) or that !he violalions are such ihal the statedupper-boundary condition holds nevertheless {Byrd, Mann,Moore, and Ramanlal. 1997).

'"Il is straightforward lo show that Equation {7) solves {I) bydirect substitution and using the result that g{V) solves (I) .The new upper-houndary condition can be verified by sellingV - K/y in Equation (7).

18 FINANCrAL PRACTICE AND EDUCATION — SPRING / SUMMER 1997

that the firm's option to call the preferred stock lowersits market value relative to an otherwise identicalnoncallable issue.

C. Comparison of Callable and NoncallableIssues

Rewriting the expression for h(V) in Equation (7) bysubstituting the expression for g(V) given in Equation(6) yields an interesting result:

(8)

where

Equation (8) suggests that a callable preferred stockwith conversion ratio 7 can be thought of as anoncallable preferred stock with a smaller effectiveconversion ratiov ' {compare Equations (6) and (8)).

IV. Numerical Example

A numerical example for a callable and convertiblepreferred issue will illustrate how the previous resultsmay be used in practice. Several parameters valuesmust be specified. To ensure that the illustration isrepresentative, parameter values based on actualmarket data obtained from Ibbotson Associates (1995)and Coleman, Fisher, and Ibbotson (1995) are used.

A. Raw Data

Number of common shares outstanding (N ):"50 million shares

Market price of common stock (P^):$40 per share

Annual common dividend (D ):$1.60 per share

Number of preferred shares outstanding (N );5 million shares

Price of preferred stock (P,):dollars per share {to be determined)

Annual preferred dividend {D ):$5.00 per share

Nominal call price {CALL):'̂120

Nominal conversion ratio {CONV):1.2 common per preferred converted

"The number of shares outstanding, the market price, andannual dividends for common and preferred slocks areobtainable from the NYSE Daily Stock Price Record,"The nominal call price and conversion ratio are obtainablefrom Moody's manuals.

B. Input Values for the Valuation Function

Aggregate value of preferred stock (h{P )):PN=5P

p p pValue of all equity securities (V(P )):'^

PN + P N =2000 + 5Pc c p p p

Aggregate dividends to all equityholders (C):DN +DN =105

t 1; p p

Aggregate dividends on preferred issue (c):DN =25

p p

Call price (K):

Conversion ratio (y):CONV(Np/(CONV(Np + NJ) = 0.1071

Annualized riskless rate (r): '̂8.16%

Annualized standard deviation of dV/V (o)30.12%

C. Parameter Values Required for Valuation

The parameter a:The parameter d(Pp):

= 1.8p C/rV = 1286/(2000 + 5Pp)

Firm's value when call is announced (V^):

Coefficient a:'"Coefficient p:Coefficient Tl:The parameters:Effective conversion ratio (7'):"

0.09150.4457

-0.13070.01940.0699

The numerical values in sections B and C above areused to set up the valuation function (8):

5P = (9)

In Equation {9), V(P ) is defined in Section B {item2), d{P ) is defined in Section C {item 2), and K(d) isgiven by Equation {5a). It is straightforward to solveEquation {9} iteratively for P using either spreadsheetsoftware or a programmable calculator. An initial value

"Assuming there are no other convertible preferred stock,convertible debt or warrants outstanding.'••The yield-to-maturity of the US government long bond istaken as the riskless rate. These rates are provided by Coleman,Fisher, and Ibbotson (1995), For the illustration, the averageof these monthly values from 1988 to 1994 is used.'^This can be estimated using tbe sample variance of log(VyV̂ i) using daily data for V̂ available in tbe NYSE Daily StockPrice Record and then annualizing this variance. For thevariance calculation, the market price of the preferred issuemay be used to evaluate V .̂ For the example at hand, theaverage of the standard deviation of decile portfolios of NYSEfirms obtained from Ibbotson Associates (1995) is used."The four parameters a, p, r). 8 are from Table I for a = 1.8."The effective conversion ratio is defined following Equation(8). It is evaluated using the parameter after Equation (7).Equation (3), and Equation {5a).

RAMANLAL — A SIMPLE ALGORITHM FOR THE VALUATION OF PREFERRED STOCK 19

for P ,̂ say, P̂ " = {c/r)/N^ = $61.24 {i.e., the per share presentvalue of cash flows to the preferred issue discounted atthe riskless rate), is substituted on the right-hand side ofEquation (9). The first iteration yields P ' = $69.69, andsubstituting this back into the rigbt-hand side of {9) forPp yields the second iterated value P - = $70.34. Thethird iteration yields $70.3989 while an infinite manymore yields $70.4048. The exact value for P obtainedby evaluating the gamma and incomplete gammafunctions is $70.3866. Thus, the suggested numericalprocedure has desirable properties of simplicity andrapid convergance to a result that is very close to theexact value. The component values of the callable,convertible preferred stock in terms of its straight valueand the value of its call and conversion options arestraightforward to obtain.

V. ConclusionTo help bridge financial theory and practice, this

study develops an algorithm that practitioners,including corporate treasurers and financialanalysts, will find useful when calculating the valueof preferred stocks, both the straight variety andthe callable convertible type. Using a hand-heldcalculator, corporate treasurers will be able todetermine the offer price of preferred issues whenmaking capital financing decisions. Moreover,financial analysts will be able to advise tbeir clientswhether such issues are fairly priced in thesecondary market. The proposed valuationalgorithm makes the theoretical pricing results ofMerton (1974) and Ingersoll {1977a) available topractitioners by removing the need to evaluatecomplex integrals numerically. Only algebraicexpressions must be evaluated. The study alsoprovides some intuition on what influences thepreferred stock's call and conversion option valuesand how these option values may be determined. •

References

Asquitb, Paul. 1995, "Convertible Bonds are not Called Late,"Journal of Finance 50 (No. 4. September). 1275-1290.

Brennan, Michael J. and Eduardo Schwartz, 1977, "ConvertibleBonds: Valuation and Optimal Strategies for Call andConversion," Journal of Finance 32 (No- 5. December),1699-1715.

Byrd, Anthony K., Steven V. Mann. William T. Moore, andPradipkumar Ramanlal. 1997, "Rational Timing of Calls ofConvertible Preferred Stock." University of Soutb CarolinaWorking Paper.

Coleman, Thomas S.. Lawrence Fisber. and Roger G. Ibbotson,1995. U.S- Treasury Yield Curves J926-i994. New York,NY, Moody's Investors Service, Inc.

Constantinides, George M. and Bruce Grundy, 1987, "Call andConversion of Corporate Bonds: Theory and Evidence,"tJniversity of Chicago Working Paper.

Ferreira, Eurico. J., Michael F. Spivey, and Charles E. Edwards,1992, "Pricing New-Issue and Seasoned Preferred Stocks."Financial Management 21 (No. 2, Summer), 52-62.

Ibbotson Associates. 1995, Stocks. Bonds. Bills, and Inflation1995 Yearbook. Cbicago. IL. Ibbotson Associates,

Ingersoll, Jonathan, 1977a. "A Contingent-Claims Valuationof Convertible Securities." Journal of Financial Economics4 (No. 3, May), 289-322.

Ingersoll, Jonathan. 1977b. "An Examination of CorporateCall Policies on Convertible Securities," Journal of Finance32 (No. 2. May), 463-478.

Merton. Robert. 1974. •'On the Pricing of Corporate Debt:The Risk Structure of Interest Rates," Journal of Finance29 (No. 2. May). 449-470.

Mikkelson. Wayne H,, 1985. "Capital Structure Changes andDecrease.s in Stockholders' Wealth: A Cross-Sectional Studyof Convertible Security Calls." in B. M. Friedman. Ed.,Corporate Capita! Structure in the United Slates, Chicago,IL. University of Chicago Press for National Bureau ofEconomic Research.

Ramanlal, Pradipkumar. Sleven V. Mann, and William T.Moore, 1997, "Convertible Preferred Stock Valuation: Testsof Alternative Models," Journal of Quantitative Financeand Accounting (forthcoming).