# Pricing new-issue and seasoned preferred stocks: a comparison of valuation models.

Over $1 billion in preferred stock is traded on the New York Stock
Exchange each month -- a volume sufficient to make an investigation of
alternative valuation techniques meaningful to both current and
prospective investors evaluating opportune trading times. An
understanding of alternative valuation models for preferred stocks will
also be useful in more general assessments of shareholder wealth, since
common stock value equals firm value less the value of its senior
securities.

The trading value of nonconvertible preferred securities can be derived by using discounted cash flow methods (perpetuity model) or the option pricing theory (OPT) models presented in the literature. A number of studies ([1], [6], [9], [11], [12] and [17]) have considered these models and the pricing of preferred stocks, but no previous study has compared the merits of alternative models. It is worth investigating whether pricing techniques more complicated than the widely used perpetuity model might provide significantly better estimates of preferred stock prices.

This paper uses three models to estimate the prices of fixed-rate-dividend nonconvertible preferred stocks, looking separately at entirely new issues and at outstanding (seasoned) issues of preferred stock. Model estimates were generated with the security perpetuity pricing model, Merton's OPT-derived pricing model, and a numerically solved OPT-derived pricing model. The parameters necessary to use these models were first approximated using estimation samples of new-issue and seasoned preferred stocks, and subsequently used to price validation samples of preferred stocks.(1) We then compared each model's predicted prices with actual market prices.

Our results suggest that the simple perpetuity model more accurately prices new issues of preferred stocks. The evidence regarding seasoned issues is less conclusive; however, the option models appear more accurate. In general, results of the perpetuity model appear to be less sensitive to violations of key assumptions than is true for the OPT-derived models.

The remainder of this paper is organized as follows: Section I presents the models that are used to estimate the prices of preferred stocks. Section II discusses the data used in the estimation procedure. Section III develops the methodological procedures used to price the securities. Section IV presents the empirical results of the models and compares their ability to price issues of preferred stocks. The final section concludes the paper.

I. Valuation Models

Fixed-dividend preferred stocks traditionally have been valued using the perpetuity model, which assumes an infinite and constant stream of dividend payments. Under these conditions

[P.sub.0] = D/k,(1)

where [P.sub.0] is the preferred stock's value, D is the constant annual dividend, and k is the required rate of return.

Alternatively, Merton [12] has suggested the use of Equation (2) to value preferred stocks, when they are viewed as consol bonds. [Mathematical Expression Omissions] where

d = (C/r)/V,

a = 2r/[[sigma].sup.2], [Mathematical Expression Omitted]

Equation (2) is Merton's original equation which was later modified by Ingersoll [9], where P(a, ad) and P(a+1, ad) are incomplete gamma functions, [Gamma](a) and [Gamma](a+1) are gamma distributions, C is the preferred stock's annual dividend, r is the risk-free rate of interest, V is the firm's value, and [[sigma].sup.2] is the instantaneous variance of the change in V. Equation (2) holds only if several major conditions are met:(2) (i) no dividend payments are made to common stockholders, (ii) the market value of the firm follows an Ito diffusion process with variance ([[sigma].sup.2]) per unit of time proportional to the square of the market value of the firm (V), and (iii) for simplicity, the preferred issue is the only senior obligation of the firm, so that the firm's value (V) equals the sum of the market values of the common and preferred stocks. (Ingersoll [9] later states that the firm's value (V) in Equation (2) may be represented by the sum of the market values of the common and preferred stocks even when the firm's capital structure includes additional senior securities.)

An OPT-generated model, such as Equation (2), requires the use of capital-market parameters related to characteristics of the issuing firm and a risk-free interest rate that are presumably simply and objectively obtained. In many cases, it is desirable to expand the valuation process to incorporate such considerations as common stock dividend payments and call provisions, and their effect on the security's value.(3) A second OPT-derived model incorporating these additional conditions is represented by Equation (3) and the set of conditions indicated below: [Mathematical Expression Omitted] where [[sigma].sup.2] is the instantaneous variance of the change in the firm's value per unit of time, V is the firm's value, r is the risk-free rate of interest, P is the preferred stock's value, and t represents time.(4) In addition, Black-Scholes [2] and Merton [12] have shown that V follows a stochastic process, such as dV/V = u dt + [sigma] dz, where dz is a Gauss-Wiener process.

As Merton [12] shows, the stochastic differential Equation (3) may represent the value of any security. As a practical matter, if it is used to value preferred stocks, it is necessary to define boundary conditions that affect the preferred stock's value. These boundary conditions can be summarized as follows:(5)

(i) If the preferred stock is callable, its highest expected price is assumed to be the security's call price. For non-callable securities, an upper limit on value is determined by dividing the preferred stock's annual dividend by the risk-free interest rate. In practice, this upper boundary value is never attained since the security's yield is always greater than that risk-free rate.

(ii) At any time, the aggregate value of the outstanding preferred stock cannot exceed the value of the entire firm. As Brennan and Schwartz [3] show, this condition guarantees that when the firm's value is zero, the preferred stock's value is also zero.

(iii) The preferred stock's value immediately after a preferred dividend payment date is equal to its value immediately before that date minus the amount of dividends paid.

(iv) At any time, if the firm's value is not enough to allow for the payment of the preferred stock dividends, common stock dividends will not be paid. This condition becomes relevant in cases of very low firm values.

(v) The preferred stock's value immediately following a common stock dividend payment is a function of the post-distribution firm value.

II. The Data

Two samples of nonconvertible fixed-rate dividend preferred stocks were collected to begin our study. One comprises 142 new issues sold during the years 1976 through 1980; the other consists of 322 seasoned issues. The seasoned issues are on companies in the new-issue sample having at least two seasoned preferred stocks outstanding on the issue date of the new issue prices with OPT-derived models, at least two seasoned preferred stocks must be outstanding for each issuer. This is because the implicit value of the firm (V) and variance ([[sigma].sup.2]) that are required for the solution of the OPT-derived models are found by a trial-and-error process that requires at least a pair of the company's seasoned preferred stocks.

The data were collected from the following sources: Drexel Burnham Lambert's Public Offerings of Corporate Securities; Moody's Bond Survey and Public Utilities Manual; and Standard & Poor's Fixed Income Investor, Security Owner's Stock Guide, and Industrial COMPUSTAT Files. Actual seasoned preferred stock prices were collected from the Standard & Poor's Daily Stock Prices.

III. Methodology

To assess the performance of each valuation model, we divided the new issues into an estimation sample and a validation sample. The estimation sample is used to estimate the parameter k used in pricing the preferred stocks in the validation sample. Of the 142 new issues in our sample, 53 are from companies that had at least two seasoned preferred stocks outstanding on the date of the new issue. Thus, as discussed above, our OPT-derived models can be used to estimate prices for these 53 new issues; they become our validation sample, and the remaining 89 new issues are used as the estimation sample.

For seasoned issues, the estimation sample of 106 preferred stocks comprises two securities for each of the 53 stocks in the new-issue validation sample. Again, these seasoned preferred stocks are necessary to estimate the implicit values of the firm (V) and variance ([[sigma].sup.2]). Another 216 seasoned issues constitute the validation group.

A. The Perpetuity Model Estimation

The perpetuity model (Equation (1)) is used to estimate prices for both new and seasoned issues of preferred stocks. First, ordinary least squares (OLS) is used with the estimation sample of new issues to estimate k (required rate of return). We base our estimation of k on past studies ([11], [17]) using OLS to estimate preferred stock yields. The independent variables in the regression are the credit rating assigned by Moody's Investor Service (with zero-one dummy variables representing the ratings Aa, A, Baa or lower; issues rated Baa and lower are the excluded set in the regression), a zero-one variable indicating that the issues does (1) or does not (0) have a sinking fund, the interest rate on long-maturity U.S. Treasury issues on the date of sale (the Treasury issues are neither due nor callable in less than ten years and are taken from the Moody's Bond Survey), a measure of the recent volatility of interest rates (it is determined as the mean absolute deviation in long-maturity Treasury rates over the 20 days preceding the sale date of the issue as reported in Moody's Bond Survey), the natural logarithm of the size of the issue measured in thousands of real 1976 dollars as deflated by the implicit GNP deflator, and a zero-one variable indicating that the issue was competitively bid (1) or negotiated (0). Essentially, the independent variables reflect daily market conditions and specific characteristics of the new issues.

The estimated coefficients from the above estimation of k using the new-issue estimation sample are then used to approximate the value of k for each of the 53 new issues in the validation sample. The perpetuity model (Equation (1)) is then used to price each new issue in the validation sample.

The same procedure is followed to estimate k using the seasoned-issue estimation sample. The issue size and type of bid variables are not included in the seasoned-issue regression since Hopewell and Kaufman [8, p. 1656] suggest that the issue size and the type of bid are important only for new issues. The coefficient estimates obtained from the seasoned issues are then used to estimate k for the 216 preferred stocks in the seasoned-issue validation sample. The required rate of return (k) is then used in Equation (1) to estimate each seasoned stock's price in the validation sample.

B. OPT-Derived Models Estimation

The estimation process for the OPT-derived models involves the following procedure. The prices of the 106 securities in the seasoned-issue estimation sample are used as inputs in both Equations (2) and (3) to estimate the implicit firm and variance values. The value of the preferred stock's annual dividend and the risk-free rate of interest are also used as inputs to both equations;(6) in addition, Equation (3) uses the security's call price, the deferred call period, and the common stock's annual dividend.(7) A numerical solution algorithm is used to solve Equation (3) subject to the boundary conditions previously indicated.

A trial-and-error procedure is then used with both Equations (2) and (3) to

solve simultaneously for each pair of preferred stock prices.(8) When the calculated prices of both stocks for each firm equal the actual prices, then the firm and the variance values computed at that point are the implicit values for firm (V) and variance ([[sigma].sup.2]). Because numerous pairs of implicit values are found to exist for a given firm at a given point in time, true values were estimated by taking an equally weighted average of their implicit values. Similar procedures have been previously used in the literature (e.g., Schmalensee and Trippi [15] and Patell and Wolfson [13]). Finally, once the true firm and variance values have been estimated, the predicted price is calculated with Equations (2) and (3) for each of the 53 securities in the new-issue validation sample and the 216 securities in the seasoned-issue validation sample.

IV. Empirical Results

A. Comparing Predictive Accuracy

Since the actual prices of the preferred stock issues in our sample range from $25.00 to more than $100.00, the prediction errors of the models are expressed as percentages of the actual price. This percentage error, (((predicted price - actual price)/actual price) x 100), hereafter will be referred to as the prediction error for each model.

The new-issue prediction errors for each model are graphed from the largest to the smallest error in Exhibit 1. The model PN errors (from the perpetuity model, using new-issue data to estimate rates of return) are computed using the prices of the 53 stocks predicted by Equation (1). The errors that result from using the prices predicted from Equation (2) are referred to hereafter as the model M (Merton's) errors; the errors from Equation (3) are denoted as the model N (numerically solved OPT) errors. Exhibit 2 shows the errors for each model for the seasoned-issue validation sample, again graphed from the largest error to the smallest. The model PS (perpetuity, seasoned) errors use the prices predicted from Equation (1).(9) The model M errors are computed from prices obtained using estimators of the true implicit firm and variance values in Equation (2). The model N errors are computed from prices predicted using the same estimators of the true implicit firm and variance values with Equation (3).

Exhibits 1 and 2 provide preliminary evidence on the predictive ability of the models. The prediction errors for the OPT models in both the new-issue sample and the seasoned-issue sample are biased upward, suggesting that these models overprice the preferred stocks. Only the perpetuity models estimated with parameters obtained from their respective samples (model PN for the new-issue sample and model PS for the seasoned-issue sample) do not appear to exhibit significant bias.

Since we are concerned with the predictive accuracy of our models, we compare the models in terms of the absolute size of their prediction errors, without reference to the sign (+ or -) of the errors. Exhibit 3 presents the descriptive statistics of the mean absolute prediction errors ([Aberror.sub.ij]). The mean absolute error for the perpetuity model ([[mu].sub.pn] = 6.51%) is lower than that of either OPT-derived model ([[mu].sub.m] = 11.43% and [[mu].sub.n] = 9.41%). In the seasoned-issue sample, however, the mean absolute prediction error is lower for both Merton's OPT-derived model ([[mu].sub.m] = 8.70%) and the numerically solved OPT-derived model ([[mu].sub.n] = 10.72%) than for the perpetuity model ([[mu].sub.ps] = 13.56%).

To test for differences in the models' predictive accuracy, we estimate the following analysis of variance statistical model which tests for the equality of mean absolute prediction errors:

[Aberror.sub.ij] = [mu] + [[tau].sub.i] + [[epsilon].sub.ij], (4) where [Aberror.sub.ij] is the jth absolute prediction error taken from model i, [mu] is the overall mean absolute prediction error across prediction models, [[tau].sub.i] is a parameter measuring deviations of the ith model's absolute prediction error from the overall mean ([mu]), and [[epsilon].sub.ij] is a random error component that is NID(0, [[sigma].sup.2]).(10)

The first hypothesis test concerns whether the [[tau].sub.i] = 0. This jointly tests whether there is a significant difference in the absolute prediction errors of the models. Paired orthogonal contrasts of mean absolute prediction errors (e.g., [[mu].sub.pn] = [[mu].sub.m]) are also tested.(11) These results, presented in Exhibit 4, indicate that the perpetuity model (model PN) performs best in pricing the new issues. The mean absolute error for the perpetuity model is lower (at the five percent significance level) than that of either OPT-derived model (models M and N). In the seasoned-issue sample, the mean absolute prediction error is significantly lower for Merton's OPT-derived model ([[mu].sub.m] = 8.70%) than for the perpetuity model ([[mu].sub.ps] = 13.56%). The mean absolute error of the numerically solved OPT-derived model ([[mu].sub.n] = 10.72%) is not significantly different from the mean absolute error for the perpetuity model. The mean absolute errors for the OPT-derived models are also not significantly different from each other.

These results suggest that the simple perpetuity model does a better job of pricing new issues of preferred stocks. While the option models appear to more accurately price the seasoned issues, the evidence is less conclusive. Although the absolute pricing errors are significantly larger for the perpetuity model than for Merton's OPT-derived model, they are not significantly larger than the numerically solved OPT model.(12)

B. Sensitivity to Violations of Key Assumptions

The findings described above suggest that, on average, the simple perpetuity model more accurately prices new issues than the OPT-derived models, but may be worse in pricing the seasoned issues, over the sample period 1976-1980. In this section of the paper, we investigate potential explanations for these findings. The fact that the perpetuity model does better in pricing new issues than seasoned securities may reflect a more accurate estimation of k used in Equation (1). An [R.sup.2] of 0.809 for the new-issue estimation sample and 0.548 for the seasoned-issue estimation sample support this hypothesis.

A further explanation may be provided by examining key assumptions of the pricing models. The perpetuity model assumes that the term structure of interest rates is flat and known with certainty and that the dividend payments are infinite (noncallable and without a sinking fund). Merton's OPT-derived model makes the same assumptions and also assumes that the value of the firm is invariant to its capital structure. The numerically solved OPT-derived model maintains the same assumptions as the Merton model, but it allows the preferred stock to be callable.

Virtually all preferred stock is callable, while most recently issued fixed-rate preferreds contain a sinking fund. The sample period, 1976 to 1980, was chosen partly to allow inclusion of issues with and without sinking funds, permitting us to examine the differential effects on each model's pricing abilities. This sample period also witnessed major swings in market interest rates. The term structure of interest rates was positively sloped for most of the 1976 to 1978 time period, but negatively sloped for much of 1979 to 1980. Volatility was also much greater in the latter period, in part because of the Federal Reserve Board's change in monetary policy in October 1979. Thus, we can examine the effects of the slope of the term structure and the volatility of market interest rates.

Our results may also have been influenced by the assumption of the OPT-derived models that the value of the firm is invariant to its capital structure. As discussed earlier, these models assume that the preferred issue is the only senior obligation of the firm, so that the firm's value (V) is equal to the sum of the market values of the common and preferred stocks. Although Ingersoll [9] states that this equality holds even when the capital structure includes additional senior securities, violation of this assumption may influence the relative pricing accuracy of the OPT-derived models.

Violating any one of these assumptions could affect the pricing accuracy of our models. Exhibit 5 shows the correlations of the absolute pricing errors with variables proxying each assumption. SINK is a dummy variable equaling 1 if the issue contains a sinking fund and 0 otherwise. TERM measures the slope of the term structure of interest rates on the date the preferred stock is being priced. (The slope is determined by subtracting the three-year U.S. Treasury note yields from the interest rate on long-maturity U.S. Treasury issues proxied by an index of Treasury issues that are neither due nor callable in less than ten years. Both rates are taken from the Moody's Bond Survey.) VOL proxies the recent volatility of interest rates. It is computed as the mean absolute deviation in the long-term U.S. Treasury index yields over the 20 days preceding the date the preferred stock is being priced, as reported in the Moody's Bond Survey. DEBTEQ measures the debt-to-equity ratio of the issuing firm during the year the issue is being priced.

Only Merton's OPT-derived model appears to be sensitive to the presence of a sinking fund. Model M misprices sinking fund issues more severely than nonsinking-fund issues, in both the new- and seasoned-issue samples. Since model M assumes that the dividend payments are infinite (noncallable and without a sinking fund), it is not surprising that it misprices sinking-fund issues more than does model N, which accounts for the potentially finite life of the preferred stock. It is less clear why model M misprices the issues more than the perpetuity model does. Perhaps the more accurate pricing of model PN is due in part to the fact that the estimation of k in Equation (1) incorporates the presence of a sinking fund.

There is a significant negative correlation between the absolute pricing error and the term structure of interest rates for each model in both samples, except for model PN in the new-issue sample. We also observe, though not specifically reported, that the models' pricing estimates are biased downward when the term structure is more positively sloped, and biased upward when the term structure is more negatively sloped. Perhaps the use of the yield on an index of long-term Treasury issues overestimates the risk-free rate when the term structure is positive (resulting in downwardly biased estimates), and underestimates the risk-free rate when the term structure is negative (resulting in upwardly biased estimates).

Both of the OPT-derived models appear to misprice more severly at higher levels of interest rate volatility, while the performance of the perpetuity model seems to be uncorrelated with the level of interest rate volatility. It may be that the estimated implicit firm value (V) and variance ([sigma.sup.2]) values exaggerate the true firm value and variance during periods of high market uncertainty, so that mispricing in the OPT-derived models is more pronounced when interest rates are highly volatile. Finally, the capital structure (debt-to-equity ratio) of the firm does not appear to influence the performance of any of the models.(13)

C. Checking the Models' Pricing Accuracy Over Another Sample Period

As mentioned in the previous section, the sample period, 1976-1980, was chosen because it contained significant variation in the factors used in the previous section. A potential concern is whether the results obtained over this period are robust to different time periods. In this section of the paper, we check if the performance of the pricing models is similar for a more recent five-year time period. We repeat the procedure discussed in Section III using samples of new issue and seasoned nonconvertible fixed-rate dividend preferred stocks selected over the years 1985 to 1989. The prediction errors for each model from a validation sample of 23 new issues are graphed from the largest to the smallest error in Exhibit 6. The model PN errors are computed using the prices of the stocks predicted by Equation (1). Again, the errors obtained from Equation (2) are the model M (Merton's) errors and the errors from Equation (3) are the model N (numerically solved OPT) errors. Exhibit 7 presents the errors for each model for a seasoned-issue validation sample of 35 stocks, again graphed from the largest error to the smallest. The model PS (perpetuity, seasoned) errors use the prices predicted from Equation (1). Model M errors again represent Merton's model and model N represents the numerically solved OPT model.

Exhibits 6 and 7 show that the prediction errors in both the new-issue sample and the seasoned-issue sample are again biased upward in the numerically solved OPT model, but are not for Merton's model. The perpetuity model still does not appear to exhibit significant bias in either sample. Exhibit 8 presents the descriptive statistics of the mean absolute prediction errors (Aberrorij). The mean absolute error for the perpetuity model ([mu]pn = 7.53%) is again lower than that of either OPT-derived model [mu]m = 11.54% and [mu]n = 10.62%) in the new-issue sample. As before, the mean absolute prediction error in the seasoned-issue sample is less for both Merton's OPT-derived model ([mu]m = 11.58%) and the numerically solved OPT-derived model [mu]n = 10.87%) than for the perpetuity model ([mu]ps = 15.37%).

The results in Exhibit 9 support that the perpetuity model (model PN) more accurately prices the new issues. The mean absolute error for the perpetuity model is lower (at the five percent significance level) than that of Merton's model and also lower than that of the numerically solved model (at the ten percent level). The mean absolute prediction error in the seasoned-issue sample is again significantly higher for the perpetuity model than for Merton's OPT-derived model (at the ten percent level) and, unlike before, is significantly higher than for the numerically solved OPT-derived model. The mean absolute errors for the OPT-derived models are again not significantly different from each other. These findings support that the simple perpetuity model does a better job of pricing new issues of preferred stocks and that the option models more accurately price the seasoned issues.

V. Conclusion

Three models are used to estimate the price of fixed-rate-dividend preferred stocks (both new and seasoned issues): the perpetuity pricing model, Merton's OPT-derived pricing model, and a numerically solved OPT-derived pricing model. For each model, the necessary parameters are first estimated using an estimation sample of preferred stocks, and then used to price a separate validation sample of stocks. The ability of the models to price the stocks is then compared. We also examine each model's sensitivity to violations of key assumptions.

The results provide evidence that the perpetuity model more accurately prices new issues of preferred securities. The results for the seasoned-issue securities are not as robust, but suggest that the OPT model is superior to the perpetuity model. Overall, the perpetuity model appears to be less sensitive to violations of key assumptions than the OPT-derived models.

(1)The necessary parameters are the preferred stock's required in the perpetuity model and the firm's value and its variability in the OPT- derived models.

(2)For additional assumptions, see Merton [12,p. 450].

(3)Emanuels's [6] models also incorporates more constraints into the analysis than Merton's.

(4)Since some preferred stocks are callable and have no specified time to maturity, the upper limit in the range from which the values of the variable time were drawn was set equal to 30, 40, 50, 60, and 80 years. One security price was determined for each upper limit; the average among these upper limits represents the callable preferred stock price. If the security was not callable, the variable time was set equal to 100 years.

(5)For the sake of brevity and readability, the solution algorithm and the finite difference equations that represent these conditions are not inclueded; they are available on request from the authors. Information on the numerical solution of partial differential equations by finite difference methods can be found, for example, in Smith [16]. For an application of this valuation method for convertible bonds, see Brennan and Schwarts [3].

(6)The literature (see, for example, Cox and Rubinstein [4, p. 379]) typically recommends using the yield to maturity on long-term Treasury bonds as the risk-free rate of interest when OPT models are applied to long-term securities.

(7)The common stock's annual dividend used was the most recent one paid.

(8)The pair of preferred stocks for each firm is necessary to solve for V and [[sigma].sub.2].

(9)Model PN pertains only to the new-issue sample.

(10)Since the analysis of variance procedure assumes that the experimental errors are normally and Independently distributed, the equality of means was also tested using the nonparametric test are consistent with the analysis of variance results; thus, the more powerful parametric analysis of variance testing procedures are employed.

(11)[[mu].sub.i] = [mu] + [[tau].sub.i].

(12)We posit that our absolute pricing errors are within tolerable limits compared to past studies involving the pricing of securities. Most past studies have simply compared root mean squared errors (RMSE) of different pricing models and have argued that the lowest RMSE is best. They do not actually test if the pricing accuracy is significantly different across models. Lauterbach and Schultz [10] do use mean absolute pricing errors as measures of predictive accuracy. The mean absolute pricing errors for the two models in their study were 13.5% and 11.3%. In general, our mean absolute pricing errors are comparable.

(13)As an additional assessment of our models, we examined the performance of several preferred stocks over an eight-week period following the actual pricing by each of our models. We wished to see if stocks that were highly underpriced by the models (models PN, PS, and N) showed positive market-adjusted returns, and if stocks that were highly overpriced by the models (all models) showed significantly negative market-adjusted abnormal returns. We also wished to see if any model was significantly better selecting under(over)priced stocks.

While our findings showed that, in general, the underpriced stocks did demonstrate positive cumulative abnormal returns and that the overpriced stocks demonstrated cumulative negative abnormal returns and that the overpriced stocks demonstrated cumulative negative abnormal returns over the two-month period, the cumulative returns were not significantly different from zero. The cumulative returns also were not significantly different across the models. This evidence suggests that the deviations between model prices and market prices are due to model pricing inaccuracies (as examined in Section IV) and not due to market inefficiency.

References

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The trading value of nonconvertible preferred securities can be derived by using discounted cash flow methods (perpetuity model) or the option pricing theory (OPT) models presented in the literature. A number of studies ([1], [6], [9], [11], [12] and [17]) have considered these models and the pricing of preferred stocks, but no previous study has compared the merits of alternative models. It is worth investigating whether pricing techniques more complicated than the widely used perpetuity model might provide significantly better estimates of preferred stock prices.

This paper uses three models to estimate the prices of fixed-rate-dividend nonconvertible preferred stocks, looking separately at entirely new issues and at outstanding (seasoned) issues of preferred stock. Model estimates were generated with the security perpetuity pricing model, Merton's OPT-derived pricing model, and a numerically solved OPT-derived pricing model. The parameters necessary to use these models were first approximated using estimation samples of new-issue and seasoned preferred stocks, and subsequently used to price validation samples of preferred stocks.(1) We then compared each model's predicted prices with actual market prices.

Our results suggest that the simple perpetuity model more accurately prices new issues of preferred stocks. The evidence regarding seasoned issues is less conclusive; however, the option models appear more accurate. In general, results of the perpetuity model appear to be less sensitive to violations of key assumptions than is true for the OPT-derived models.

The remainder of this paper is organized as follows: Section I presents the models that are used to estimate the prices of preferred stocks. Section II discusses the data used in the estimation procedure. Section III develops the methodological procedures used to price the securities. Section IV presents the empirical results of the models and compares their ability to price issues of preferred stocks. The final section concludes the paper.

I. Valuation Models

Fixed-dividend preferred stocks traditionally have been valued using the perpetuity model, which assumes an infinite and constant stream of dividend payments. Under these conditions

[P.sub.0] = D/k,(1)

where [P.sub.0] is the preferred stock's value, D is the constant annual dividend, and k is the required rate of return.

Alternatively, Merton [12] has suggested the use of Equation (2) to value preferred stocks, when they are viewed as consol bonds. [Mathematical Expression Omissions] where

d = (C/r)/V,

a = 2r/[[sigma].sup.2], [Mathematical Expression Omitted]

Equation (2) is Merton's original equation which was later modified by Ingersoll [9], where P(a, ad) and P(a+1, ad) are incomplete gamma functions, [Gamma](a) and [Gamma](a+1) are gamma distributions, C is the preferred stock's annual dividend, r is the risk-free rate of interest, V is the firm's value, and [[sigma].sup.2] is the instantaneous variance of the change in V. Equation (2) holds only if several major conditions are met:(2) (i) no dividend payments are made to common stockholders, (ii) the market value of the firm follows an Ito diffusion process with variance ([[sigma].sup.2]) per unit of time proportional to the square of the market value of the firm (V), and (iii) for simplicity, the preferred issue is the only senior obligation of the firm, so that the firm's value (V) equals the sum of the market values of the common and preferred stocks. (Ingersoll [9] later states that the firm's value (V) in Equation (2) may be represented by the sum of the market values of the common and preferred stocks even when the firm's capital structure includes additional senior securities.)

An OPT-generated model, such as Equation (2), requires the use of capital-market parameters related to characteristics of the issuing firm and a risk-free interest rate that are presumably simply and objectively obtained. In many cases, it is desirable to expand the valuation process to incorporate such considerations as common stock dividend payments and call provisions, and their effect on the security's value.(3) A second OPT-derived model incorporating these additional conditions is represented by Equation (3) and the set of conditions indicated below: [Mathematical Expression Omitted] where [[sigma].sup.2] is the instantaneous variance of the change in the firm's value per unit of time, V is the firm's value, r is the risk-free rate of interest, P is the preferred stock's value, and t represents time.(4) In addition, Black-Scholes [2] and Merton [12] have shown that V follows a stochastic process, such as dV/V = u dt + [sigma] dz, where dz is a Gauss-Wiener process.

As Merton [12] shows, the stochastic differential Equation (3) may represent the value of any security. As a practical matter, if it is used to value preferred stocks, it is necessary to define boundary conditions that affect the preferred stock's value. These boundary conditions can be summarized as follows:(5)

(i) If the preferred stock is callable, its highest expected price is assumed to be the security's call price. For non-callable securities, an upper limit on value is determined by dividing the preferred stock's annual dividend by the risk-free interest rate. In practice, this upper boundary value is never attained since the security's yield is always greater than that risk-free rate.

(ii) At any time, the aggregate value of the outstanding preferred stock cannot exceed the value of the entire firm. As Brennan and Schwartz [3] show, this condition guarantees that when the firm's value is zero, the preferred stock's value is also zero.

(iii) The preferred stock's value immediately after a preferred dividend payment date is equal to its value immediately before that date minus the amount of dividends paid.

(iv) At any time, if the firm's value is not enough to allow for the payment of the preferred stock dividends, common stock dividends will not be paid. This condition becomes relevant in cases of very low firm values.

(v) The preferred stock's value immediately following a common stock dividend payment is a function of the post-distribution firm value.

II. The Data

Two samples of nonconvertible fixed-rate dividend preferred stocks were collected to begin our study. One comprises 142 new issues sold during the years 1976 through 1980; the other consists of 322 seasoned issues. The seasoned issues are on companies in the new-issue sample having at least two seasoned preferred stocks outstanding on the issue date of the new issue prices with OPT-derived models, at least two seasoned preferred stocks must be outstanding for each issuer. This is because the implicit value of the firm (V) and variance ([[sigma].sup.2]) that are required for the solution of the OPT-derived models are found by a trial-and-error process that requires at least a pair of the company's seasoned preferred stocks.

The data were collected from the following sources: Drexel Burnham Lambert's Public Offerings of Corporate Securities; Moody's Bond Survey and Public Utilities Manual; and Standard & Poor's Fixed Income Investor, Security Owner's Stock Guide, and Industrial COMPUSTAT Files. Actual seasoned preferred stock prices were collected from the Standard & Poor's Daily Stock Prices.

III. Methodology

To assess the performance of each valuation model, we divided the new issues into an estimation sample and a validation sample. The estimation sample is used to estimate the parameter k used in pricing the preferred stocks in the validation sample. Of the 142 new issues in our sample, 53 are from companies that had at least two seasoned preferred stocks outstanding on the date of the new issue. Thus, as discussed above, our OPT-derived models can be used to estimate prices for these 53 new issues; they become our validation sample, and the remaining 89 new issues are used as the estimation sample.

For seasoned issues, the estimation sample of 106 preferred stocks comprises two securities for each of the 53 stocks in the new-issue validation sample. Again, these seasoned preferred stocks are necessary to estimate the implicit values of the firm (V) and variance ([[sigma].sup.2]). Another 216 seasoned issues constitute the validation group.

A. The Perpetuity Model Estimation

The perpetuity model (Equation (1)) is used to estimate prices for both new and seasoned issues of preferred stocks. First, ordinary least squares (OLS) is used with the estimation sample of new issues to estimate k (required rate of return). We base our estimation of k on past studies ([11], [17]) using OLS to estimate preferred stock yields. The independent variables in the regression are the credit rating assigned by Moody's Investor Service (with zero-one dummy variables representing the ratings Aa, A, Baa or lower; issues rated Baa and lower are the excluded set in the regression), a zero-one variable indicating that the issues does (1) or does not (0) have a sinking fund, the interest rate on long-maturity U.S. Treasury issues on the date of sale (the Treasury issues are neither due nor callable in less than ten years and are taken from the Moody's Bond Survey), a measure of the recent volatility of interest rates (it is determined as the mean absolute deviation in long-maturity Treasury rates over the 20 days preceding the sale date of the issue as reported in Moody's Bond Survey), the natural logarithm of the size of the issue measured in thousands of real 1976 dollars as deflated by the implicit GNP deflator, and a zero-one variable indicating that the issue was competitively bid (1) or negotiated (0). Essentially, the independent variables reflect daily market conditions and specific characteristics of the new issues.

The estimated coefficients from the above estimation of k using the new-issue estimation sample are then used to approximate the value of k for each of the 53 new issues in the validation sample. The perpetuity model (Equation (1)) is then used to price each new issue in the validation sample.

The same procedure is followed to estimate k using the seasoned-issue estimation sample. The issue size and type of bid variables are not included in the seasoned-issue regression since Hopewell and Kaufman [8, p. 1656] suggest that the issue size and the type of bid are important only for new issues. The coefficient estimates obtained from the seasoned issues are then used to estimate k for the 216 preferred stocks in the seasoned-issue validation sample. The required rate of return (k) is then used in Equation (1) to estimate each seasoned stock's price in the validation sample.

B. OPT-Derived Models Estimation

The estimation process for the OPT-derived models involves the following procedure. The prices of the 106 securities in the seasoned-issue estimation sample are used as inputs in both Equations (2) and (3) to estimate the implicit firm and variance values. The value of the preferred stock's annual dividend and the risk-free rate of interest are also used as inputs to both equations;(6) in addition, Equation (3) uses the security's call price, the deferred call period, and the common stock's annual dividend.(7) A numerical solution algorithm is used to solve Equation (3) subject to the boundary conditions previously indicated.

A trial-and-error procedure is then used with both Equations (2) and (3) to

solve simultaneously for each pair of preferred stock prices.(8) When the calculated prices of both stocks for each firm equal the actual prices, then the firm and the variance values computed at that point are the implicit values for firm (V) and variance ([[sigma].sup.2]). Because numerous pairs of implicit values are found to exist for a given firm at a given point in time, true values were estimated by taking an equally weighted average of their implicit values. Similar procedures have been previously used in the literature (e.g., Schmalensee and Trippi [15] and Patell and Wolfson [13]). Finally, once the true firm and variance values have been estimated, the predicted price is calculated with Equations (2) and (3) for each of the 53 securities in the new-issue validation sample and the 216 securities in the seasoned-issue validation sample.

IV. Empirical Results

A. Comparing Predictive Accuracy

Since the actual prices of the preferred stock issues in our sample range from $25.00 to more than $100.00, the prediction errors of the models are expressed as percentages of the actual price. This percentage error, (((predicted price - actual price)/actual price) x 100), hereafter will be referred to as the prediction error for each model.

The new-issue prediction errors for each model are graphed from the largest to the smallest error in Exhibit 1. The model PN errors (from the perpetuity model, using new-issue data to estimate rates of return) are computed using the prices of the 53 stocks predicted by Equation (1). The errors that result from using the prices predicted from Equation (2) are referred to hereafter as the model M (Merton's) errors; the errors from Equation (3) are denoted as the model N (numerically solved OPT) errors. Exhibit 2 shows the errors for each model for the seasoned-issue validation sample, again graphed from the largest error to the smallest. The model PS (perpetuity, seasoned) errors use the prices predicted from Equation (1).(9) The model M errors are computed from prices obtained using estimators of the true implicit firm and variance values in Equation (2). The model N errors are computed from prices predicted using the same estimators of the true implicit firm and variance values with Equation (3).

Exhibits 1 and 2 provide preliminary evidence on the predictive ability of the models. The prediction errors for the OPT models in both the new-issue sample and the seasoned-issue sample are biased upward, suggesting that these models overprice the preferred stocks. Only the perpetuity models estimated with parameters obtained from their respective samples (model PN for the new-issue sample and model PS for the seasoned-issue sample) do not appear to exhibit significant bias.

Since we are concerned with the predictive accuracy of our models, we compare the models in terms of the absolute size of their prediction errors, without reference to the sign (+ or -) of the errors. Exhibit 3 presents the descriptive statistics of the mean absolute prediction errors ([Aberror.sub.ij]). The mean absolute error for the perpetuity model ([[mu].sub.pn] = 6.51%) is lower than that of either OPT-derived model ([[mu].sub.m] = 11.43% and [[mu].sub.n] = 9.41%). In the seasoned-issue sample, however, the mean absolute prediction error is lower for both Merton's OPT-derived model ([[mu].sub.m] = 8.70%) and the numerically solved OPT-derived model ([[mu].sub.n] = 10.72%) than for the perpetuity model ([[mu].sub.ps] = 13.56%).

To test for differences in the models' predictive accuracy, we estimate the following analysis of variance statistical model which tests for the equality of mean absolute prediction errors:

[Aberror.sub.ij] = [mu] + [[tau].sub.i] + [[epsilon].sub.ij], (4) where [Aberror.sub.ij] is the jth absolute prediction error taken from model i, [mu] is the overall mean absolute prediction error across prediction models, [[tau].sub.i] is a parameter measuring deviations of the ith model's absolute prediction error from the overall mean ([mu]), and [[epsilon].sub.ij] is a random error component that is NID(0, [[sigma].sup.2]).(10)

The first hypothesis test concerns whether the [[tau].sub.i] = 0. This jointly tests whether there is a significant difference in the absolute prediction errors of the models. Paired orthogonal contrasts of mean absolute prediction errors (e.g., [[mu].sub.pn] = [[mu].sub.m]) are also tested.(11) These results, presented in Exhibit 4, indicate that the perpetuity model (model PN) performs best in pricing the new issues. The mean absolute error for the perpetuity model is lower (at the five percent significance level) than that of either OPT-derived model (models M and N). In the seasoned-issue sample, the mean absolute prediction error is significantly lower for Merton's OPT-derived model ([[mu].sub.m] = 8.70%) than for the perpetuity model ([[mu].sub.ps] = 13.56%). The mean absolute error of the numerically solved OPT-derived model ([[mu].sub.n] = 10.72%) is not significantly different from the mean absolute error for the perpetuity model. The mean absolute errors for the OPT-derived models are also not significantly different from each other.

These results suggest that the simple perpetuity model does a better job of pricing new issues of preferred stocks. While the option models appear to more accurately price the seasoned issues, the evidence is less conclusive. Although the absolute pricing errors are significantly larger for the perpetuity model than for Merton's OPT-derived model, they are not significantly larger than the numerically solved OPT model.(12)

B. Sensitivity to Violations of Key Assumptions

The findings described above suggest that, on average, the simple perpetuity model more accurately prices new issues than the OPT-derived models, but may be worse in pricing the seasoned issues, over the sample period 1976-1980. In this section of the paper, we investigate potential explanations for these findings. The fact that the perpetuity model does better in pricing new issues than seasoned securities may reflect a more accurate estimation of k used in Equation (1). An [R.sup.2] of 0.809 for the new-issue estimation sample and 0.548 for the seasoned-issue estimation sample support this hypothesis.

A further explanation may be provided by examining key assumptions of the pricing models. The perpetuity model assumes that the term structure of interest rates is flat and known with certainty and that the dividend payments are infinite (noncallable and without a sinking fund). Merton's OPT-derived model makes the same assumptions and also assumes that the value of the firm is invariant to its capital structure. The numerically solved OPT-derived model maintains the same assumptions as the Merton model, but it allows the preferred stock to be callable.

Virtually all preferred stock is callable, while most recently issued fixed-rate preferreds contain a sinking fund. The sample period, 1976 to 1980, was chosen partly to allow inclusion of issues with and without sinking funds, permitting us to examine the differential effects on each model's pricing abilities. This sample period also witnessed major swings in market interest rates. The term structure of interest rates was positively sloped for most of the 1976 to 1978 time period, but negatively sloped for much of 1979 to 1980. Volatility was also much greater in the latter period, in part because of the Federal Reserve Board's change in monetary policy in October 1979. Thus, we can examine the effects of the slope of the term structure and the volatility of market interest rates.

Our results may also have been influenced by the assumption of the OPT-derived models that the value of the firm is invariant to its capital structure. As discussed earlier, these models assume that the preferred issue is the only senior obligation of the firm, so that the firm's value (V) is equal to the sum of the market values of the common and preferred stocks. Although Ingersoll [9] states that this equality holds even when the capital structure includes additional senior securities, violation of this assumption may influence the relative pricing accuracy of the OPT-derived models.

Violating any one of these assumptions could affect the pricing accuracy of our models. Exhibit 5 shows the correlations of the absolute pricing errors with variables proxying each assumption. SINK is a dummy variable equaling 1 if the issue contains a sinking fund and 0 otherwise. TERM measures the slope of the term structure of interest rates on the date the preferred stock is being priced. (The slope is determined by subtracting the three-year U.S. Treasury note yields from the interest rate on long-maturity U.S. Treasury issues proxied by an index of Treasury issues that are neither due nor callable in less than ten years. Both rates are taken from the Moody's Bond Survey.) VOL proxies the recent volatility of interest rates. It is computed as the mean absolute deviation in the long-term U.S. Treasury index yields over the 20 days preceding the date the preferred stock is being priced, as reported in the Moody's Bond Survey. DEBTEQ measures the debt-to-equity ratio of the issuing firm during the year the issue is being priced.

Only Merton's OPT-derived model appears to be sensitive to the presence of a sinking fund. Model M misprices sinking fund issues more severely than nonsinking-fund issues, in both the new- and seasoned-issue samples. Since model M assumes that the dividend payments are infinite (noncallable and without a sinking fund), it is not surprising that it misprices sinking-fund issues more than does model N, which accounts for the potentially finite life of the preferred stock. It is less clear why model M misprices the issues more than the perpetuity model does. Perhaps the more accurate pricing of model PN is due in part to the fact that the estimation of k in Equation (1) incorporates the presence of a sinking fund.

There is a significant negative correlation between the absolute pricing error and the term structure of interest rates for each model in both samples, except for model PN in the new-issue sample. We also observe, though not specifically reported, that the models' pricing estimates are biased downward when the term structure is more positively sloped, and biased upward when the term structure is more negatively sloped. Perhaps the use of the yield on an index of long-term Treasury issues overestimates the risk-free rate when the term structure is positive (resulting in downwardly biased estimates), and underestimates the risk-free rate when the term structure is negative (resulting in upwardly biased estimates).

Both of the OPT-derived models appear to misprice more severly at higher levels of interest rate volatility, while the performance of the perpetuity model seems to be uncorrelated with the level of interest rate volatility. It may be that the estimated implicit firm value (V) and variance ([sigma.sup.2]) values exaggerate the true firm value and variance during periods of high market uncertainty, so that mispricing in the OPT-derived models is more pronounced when interest rates are highly volatile. Finally, the capital structure (debt-to-equity ratio) of the firm does not appear to influence the performance of any of the models.(13)

C. Checking the Models' Pricing Accuracy Over Another Sample Period

As mentioned in the previous section, the sample period, 1976-1980, was chosen because it contained significant variation in the factors used in the previous section. A potential concern is whether the results obtained over this period are robust to different time periods. In this section of the paper, we check if the performance of the pricing models is similar for a more recent five-year time period. We repeat the procedure discussed in Section III using samples of new issue and seasoned nonconvertible fixed-rate dividend preferred stocks selected over the years 1985 to 1989. The prediction errors for each model from a validation sample of 23 new issues are graphed from the largest to the smallest error in Exhibit 6. The model PN errors are computed using the prices of the stocks predicted by Equation (1). Again, the errors obtained from Equation (2) are the model M (Merton's) errors and the errors from Equation (3) are the model N (numerically solved OPT) errors. Exhibit 7 presents the errors for each model for a seasoned-issue validation sample of 35 stocks, again graphed from the largest error to the smallest. The model PS (perpetuity, seasoned) errors use the prices predicted from Equation (1). Model M errors again represent Merton's model and model N represents the numerically solved OPT model.

Exhibits 6 and 7 show that the prediction errors in both the new-issue sample and the seasoned-issue sample are again biased upward in the numerically solved OPT model, but are not for Merton's model. The perpetuity model still does not appear to exhibit significant bias in either sample. Exhibit 8 presents the descriptive statistics of the mean absolute prediction errors (Aberrorij). The mean absolute error for the perpetuity model ([mu]pn = 7.53%) is again lower than that of either OPT-derived model [mu]m = 11.54% and [mu]n = 10.62%) in the new-issue sample. As before, the mean absolute prediction error in the seasoned-issue sample is less for both Merton's OPT-derived model ([mu]m = 11.58%) and the numerically solved OPT-derived model [mu]n = 10.87%) than for the perpetuity model ([mu]ps = 15.37%).

The results in Exhibit 9 support that the perpetuity model (model PN) more accurately prices the new issues. The mean absolute error for the perpetuity model is lower (at the five percent significance level) than that of Merton's model and also lower than that of the numerically solved model (at the ten percent level). The mean absolute prediction error in the seasoned-issue sample is again significantly higher for the perpetuity model than for Merton's OPT-derived model (at the ten percent level) and, unlike before, is significantly higher than for the numerically solved OPT-derived model. The mean absolute errors for the OPT-derived models are again not significantly different from each other. These findings support that the simple perpetuity model does a better job of pricing new issues of preferred stocks and that the option models more accurately price the seasoned issues.

V. Conclusion

Three models are used to estimate the price of fixed-rate-dividend preferred stocks (both new and seasoned issues): the perpetuity pricing model, Merton's OPT-derived pricing model, and a numerically solved OPT-derived pricing model. For each model, the necessary parameters are first estimated using an estimation sample of preferred stocks, and then used to price a separate validation sample of stocks. The ability of the models to price the stocks is then compared. We also examine each model's sensitivity to violations of key assumptions.

The results provide evidence that the perpetuity model more accurately prices new issues of preferred securities. The results for the seasoned-issue securities are not as robust, but suggest that the OPT model is superior to the perpetuity model. Overall, the perpetuity model appears to be less sensitive to violations of key assumptions than the OPT-derived models.

(1)The necessary parameters are the preferred stock's required in the perpetuity model and the firm's value and its variability in the OPT- derived models.

(2)For additional assumptions, see Merton [12,p. 450].

(3)Emanuels's [6] models also incorporates more constraints into the analysis than Merton's.

(4)Since some preferred stocks are callable and have no specified time to maturity, the upper limit in the range from which the values of the variable time were drawn was set equal to 30, 40, 50, 60, and 80 years. One security price was determined for each upper limit; the average among these upper limits represents the callable preferred stock price. If the security was not callable, the variable time was set equal to 100 years.

(5)For the sake of brevity and readability, the solution algorithm and the finite difference equations that represent these conditions are not inclueded; they are available on request from the authors. Information on the numerical solution of partial differential equations by finite difference methods can be found, for example, in Smith [16]. For an application of this valuation method for convertible bonds, see Brennan and Schwarts [3].

(6)The literature (see, for example, Cox and Rubinstein [4, p. 379]) typically recommends using the yield to maturity on long-term Treasury bonds as the risk-free rate of interest when OPT models are applied to long-term securities.

(7)The common stock's annual dividend used was the most recent one paid.

(8)The pair of preferred stocks for each firm is necessary to solve for V and [[sigma].sub.2].

(9)Model PN pertains only to the new-issue sample.

(10)Since the analysis of variance procedure assumes that the experimental errors are normally and Independently distributed, the equality of means was also tested using the nonparametric test are consistent with the analysis of variance results; thus, the more powerful parametric analysis of variance testing procedures are employed.

(11)[[mu].sub.i] = [mu] + [[tau].sub.i].

(12)We posit that our absolute pricing errors are within tolerable limits compared to past studies involving the pricing of securities. Most past studies have simply compared root mean squared errors (RMSE) of different pricing models and have argued that the lowest RMSE is best. They do not actually test if the pricing accuracy is significantly different across models. Lauterbach and Schultz [10] do use mean absolute pricing errors as measures of predictive accuracy. The mean absolute pricing errors for the two models in their study were 13.5% and 11.3%. In general, our mean absolute pricing errors are comparable.

(13)As an additional assessment of our models, we examined the performance of several preferred stocks over an eight-week period following the actual pricing by each of our models. We wished to see if stocks that were highly underpriced by the models (models PN, PS, and N) showed positive market-adjusted returns, and if stocks that were highly overpriced by the models (all models) showed significantly negative market-adjusted abnormal returns. We also wished to see if any model was significantly better selecting under(over)priced stocks.

While our findings showed that, in general, the underpriced stocks did demonstrate positive cumulative abnormal returns and that the overpriced stocks demonstrated cumulative negative abnormal returns and that the overpriced stocks demonstrated cumulative negative abnormal returns over the two-month period, the cumulative returns were not significantly different from zero. The cumulative returns also were not significantly different across the models. This evidence suggests that the deviations between model prices and market prices are due to model pricing inaccuracies (as examined in Section IV) and not due to market inefficiency.

References

[1.] J.S. Bildersee, "Some Aspects of the Performance of Non-Convertible Preferred Stocks," Journal of Finance (December 1973), pp. 1187-1201.

[2.] F. Black and M. Scholes, "The Pricing of Options and Corporate Liabilities," Journal of Political Economy (May-June 1973), pp. 637-654.

[3.] M.J. Brennan, and E.S. Schwartz, "Convertible Bonds: Valuation and Optimal Strategies for Call and Conversions," Journal of Finance (December 1977), pp. 1699-1715.

[4.] J.C. Cox and M. Rubinstein, "Generalizations and Applications," in Options Markets, Englewood Cliffs, NJ, Prentice Hall, Inc., 1985, pp. 359-426.

[5.] P.J. Davis, "Gamma Function and Related Functions," in Handbook of Mathematical Functions, tenth printing, Washington, D.C., U.S. Government Printing Office (December 1972), pp. 644-682.

[6.] D. Emanuel, "A Theoretical Model for Valuing Preferred Stock," Journal of Finance (September 1983), pp. 1133-1155.

[7.] M.J. Gordon and L.I. Gould, "Comparison of the DCF and HPR Measures of the Yield on Common Shares," Financial Management (Winter 1984), pp. 40-47.

[8.] M.H. Hopewell and G.G. Kaufman, "Commercial Bank Bidding on Municipal Revenue Bonds: New Evidence," Journal of Finance (December 1977), pp. 1647-1656.

[9.] J.E. Ingersoll, "A Contingent-Claims Valuation of Convertible Securities," Journal of Financial Economics (May 1977), pp. 289-321.

[10.] B. Lauterbach and P. Schultz, "Pricing Warrants: An Empirical Study of the Black-Scholes Model and Its Alternatives," Journal of Finance (September 1990), pp. 1181-1209.

[11.] M.W. Marr and M.F. Spivey, "The Cost Relationship Between Competitive and Negotiated Preferred Stock Sales Under Different Credit Market Conditions," Quarterly Journal of Business and Economics (Summer 1988), pp. 23-40.

[12.] R.C. Merton, "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance (May 1974), pp. 449-470.

[13.] J.M. Patell and M.A. Wolfson, "Anticipated Information Releases Reflected in Call Option Prices," Journal of Accounting and Economics (April 1979), pp. 117-140.

[14.] M. Rubinstein, "Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes From August 23, 1976 Through August 31, 1978," Journal of Finance (June 1985), pp. 455-480.

[15.] R. Schmalensee, and R.R. Trippi, "Common Stock Volatility Expectations Implied by Option Premia," Journal of Finance (March 1978), pp. 129-143.

[16.] G.D. Smith, Numerical Solution of Partial Differential Equations, second edition, Oxford University Press, 1978.

[17.] E.H. Sorensen and C.A. Hawkins, "On the Pricing of Preferred Stock," Journal of Financial and Quantitative Analysis (November 1981), pp. 515-528.

[18.] R.E. Whaley, "Valuation of American Futures Options: Theory and Empirical Tests," Journal of Finance (March 1986), pp. 127-150.

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Title Annotation: | Practical Issues in Valuations |
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Author: | Ferreira, Eurico J.; Spivey, Michael F.; Edwards, Charles E. |

Publication: | Financial Management |

Date: | Jun 22, 1992 |

Words: | 5400 |

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