Cost of Capital Estimation Models

 

1. Latent-Variable CAPM with Expected Risk Premium (ERP)

Wayne E. Ferson and Dennis H. Locke (1997)

 

2. Miller's Indifferent-Capital-Structure CAPM with Tax Premium

Gordon Sick (1997)

 

3. Multi-Beta CAPM with Size Premium

Roger G. Ibbotson, Paul D. Kaplan, and James D. Peterson (1997)

 

4. Limited-Dependent-Variable CAPM with Transactions-Cost Premium

David A. Lesmond (1995)

 

5. Multi-Beta CAPM with Turnover Premium

Shing-yang Hu (1997)

 

6. Multi-Beta CAPM with Distress Premium and Dividend Yield

S.P. Kothari and Jay Shaken (1997)

 

7. Multi-Beta CAPM with Country-Risk Premium

Roger G. Ibbotson (1997)

 

8. CAPM with Bondholder-Stockholder-Conflict Premium

Robert Parrino and Michael S. Weisbach (1997)

 

 

Project/Division Cost of Capital Estimation Models

 

1. Pure-Play Beta and Idiosyncratic Variances Minimization

Marlena Akhbari, David Geltner, and Brian Kluger (1997)

 

2. Full-Information Industry Beta and Conglomerate Size Effect

Paul D. Kaplan and James D. Peterson (1997)

 

3. Accounting-Based Beta and Cluster Analysis

Marcus A. Ingram (1997)

 

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Latent-Variable CAPM with Expected Risk Premium (ERP) Back to Top

 

Practitioners needing estimates of a firm's cost of equity have long relied on the CAPM. Recent evidence casts doubt on the validity of the CAPM and Beta. However, there is not much evidence to gauge the importance of the rejections of the CAPM in a practical decision-making context. This paper presents evidence on the sources of error in the estimating cost of capital over time. We use a number of proxies for the true MVE portfolio, allowing that the CAPM is the wrong model. The analyst is assumed to rely on a standard market index. We find that the great majority of the error in estimating the cost of capital is found in the risk premium estimate or the ERP, and relatively small errors in the estimate of risk measure or the Beta. This suggests that analysts should improve estimation procedures for market risk premiums, which are commonly based on historical averages. This can be done by using regression models, such as appeared in the finance literature, or by purchasing forecasts from firms which specialize in producing them.

 

To calculate a cost of equity, analysts often use historical average returns as an estimate of the risk premium on a market index, combined with estimates of beta according to the CAPM. Our simulations suggest that the accuracy of cost of capital figures are likely to benefit from improving the estimates of the expected risk premium on a standard market benchmark, even if the CAPM is the wrong model. Estimates that reflect the current state of the economy may be purchased from specialists or generated in house using straightforward regression analysis. When errors in the cost of capital over time are the issue, improving the market risk premium estimate is much more important than are concerns about using the wrong beta. Recent evidence that expected risk premiums vary with the state of the economy raises a host of issues about the practice of capital budgeting, and this article has merely scratched the surface. For example, our analysis followed the common practice of using monthly data to develop the estimates, even though the results may be used to evaluate cash flows many years into the future. However, studies find that market indicators like the ones we use have higher explanatory power for long-horizon returns than for short-horizon returns. Therefore, state dependence in required returns is likely to be very important when formally incorporated into long-term capital budgeting problems. Future research is needed to deepen understanding of the issues.

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Miller's Indifferent-Capital-Structure CAPM with Tax Premium Back to Top

 

This paper examines the valuation implications of a Miller-style model of capital structure. The popular weighted average cost of capital (WACC) and adjusted present value (APV) formulas are generally cast in a model consistent with the MM model in which there is value associated with interest tax shields. This paper addresses the question of how to calculate value as those interest tax shields converge to zero because of differential taxation of debt and equity income at the personal level. The paper shows that the zero-beta rate of return in the CAPM for equity assets should be an after-tax debt rate, not a risk-free rate. This suggests that the equity risk premium for a unit of beta risk is larger than thought before and this paper explores the impact of this on the cost of equity. With this modification to the cost of equity, the WACC formula still holds. Modifications to the APV formula are shown that properly recognize leverage and tax issues. It is derived on a certainty-equivalent basis and discounts interest tax shields at a cost of riskless equity (zero-beta CAPM) rather than riskless debt.

 

This paper has explored the valuation implications of the Miller (1977) equilibrium for capital structure indifference with personal and corporate taxation. It has extended the model to allow cross-sectional variation in corporate tax rates and explored the necessary revisions to the CAPM and to the APV formula. The model requires a distinction between the return on riskless debt and on zero-beta equity. The zero-beta equity return is an after-tax return on riskless debt where the tax rate is the tax rate of a marginal firm that is indifferent to the taxation of debt and equity income in a Miller debt and taxes world. This marginal firm's tax rate is not directly observable, but a lower bound for the zero-beta equity return is provided by the return on riskless tax-exempt municipal bonds and an upper bound is provided by the expected return on the global minimum variance portfolio of equity assets. Given the model, the market price of risk is larger than typically measured by the average excess return on an equity market portfolio relative to riskless bonds. This means that the capital market line in the CAPM is steeper than originally thought. This theory suggests that there is merit in empirically estimating the zero-beta return on riskless equity and reassessing historical market risk premia. Given the revised measure of the cost of equity, the model is consistent with traditional WACC. However, APV formulas should be modified to recognize that the certainty equivalent of the rate of return on riskless debt is the return on riskless debt and that resulting interest tax shield should be discounted at the cost of riskless equity rather than at a cost of debt.

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Multi-Beta CAPM with Size Premium Back to Top

 

The authors adjust estimates of systematic risk (the betas) for cross-autocorrelations in security returns. They show that substantial positive adjustments to beta are necessary for small firms. Traditional estimates of beta are unrelated to future returns over the 1931 through 1994 time period, whereas adjusted estimates are positively correlated with future returns. In addition, adjusted beta estimates partially account for the size effect in common stock returns.

 

No commercial beta services provide estimates of systematic risk that account for the lagged price response of small firms to market wide information. Our results indicate that beta estimates for small firms are severely biased downwards. Traditional beta estimates are unrelated to future returns. However, adjusted estimates of beta display the positive risk-return tradeoff implied by the CAPM. In addition, adjusted beta estimates are capable of accounting for part of the size effect in stock returns. Based on these results, we recommend commercial beta services incorporate the information contained in prior market returns.

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Limited-Dependent-Variable CAPM with Transactions-Cost Premium Back to Top

 

Downward biased systematic risk estimates are caused by transaction-costs- induced zero returns which alter the relation between daily security returns and the market index. These zero returns dominate the return structure for both small and large firms. On average, small firms exhibit over 40% zero returns while the largest size firm deciles experience over 12% zero returns. The CAPM based market model, which does not specifically reflect the preponderance of zero returns, produces beta estimates that are negatively related to average portfolio returns. Unfortunately, firm size better explains portfolio returns than does beta. However, specifically incorporating these zero returns in the return generating process produces beta estimates that are positively and significantly related to average portfolio returns. Firm size is largely reduced to insignificance.

 

Zero returns dominate the trading patterns of small firms and are even significantly present in the trading behavior of large firms. The number of zero returns experienced by small firms are as high as 85% of the daily return observations for a given year. Larger firms experience as many as 40% zero returns for a given year. The large number of zero returns is caused by transaction costs. A regression test of this premise reveals a strong association, 35% R2 for the aggregate regression tests, between transaction costs and the induced number of zero returns. These zero returns reveal the effects that transaction costs exert on daily security returns. The influence of these zero returns necessitates a regression technique that incorporates the effects of transaction costs in systematic risk estimation.

 

The technique used to better estimate market risk is the Limited Dependent Variable (LDV) regression model. This model uses the preponderance of zero returns as an endogenously specified return variable that proxies for the influence of transaction costs. These transaction costs are modeled by the LDV model's intercept terms and are shown to be vary in inversely with firm size. The OLS market model does not specifically reflect the influence of the transaction-costs induced zero returns. The market model specification for the return generating process assumes that security returns continuously reflect the market return index because systematic risk is the only priced element. The zero returns signal that this assumption is not valid. Failure to specifically reflect the transaction-costs induced zero returns results in an omitted variable problem because systematic risk is not universally applicable across all market returns. In essence, the risk-return relation is no longer linear. The non-linearity causes a downward bias in the OLS market model's systematic risk estimator because of omitted variables. The omitted variables are transaction costs, LDV systematic risk, and the asset's own variance. The more zero returns a security experiences, the more downward biased the OLS systematic risk estimates become.

 

The OLS systematic risk estimates demonstrate the trend that smaller firms have lower systematic risk estimates than do large firms. Evidently, when using the OLS systematic risk estimates, small firms are less risky than large firms yet earn higher returns. Comparisons between the OLS and the LDV model demonstrate the downward bias in the OLS systematic risk estimates. The LDV model's systematic risk estimates, on the other hand, more properly show that small firms are three times as risky as large firms and that the returns associated with small firms are commensurate with risk. Fama-MacBeth's tests of the relation between monthly portfolio returns and OLS systematic risk estimates along with firm size show that as average returns increase, the OLS systematic estimates decrease. This result is opposite to the economic intuition of the CAPM and illustrates the degree of the downward bias in OLS systematic risk estimates due to the zero returns. Market capitalization is invariably negative and significant, thus reinforcing the firm size effect when using an OLS methodology. On the other hand, the LDV risk estimates demonstrate a positive and significant relation with average portfolio returns. In addition, the market capitalization variable (size premium) is insignificant in almost every sub-period tested. The LDV model provides a more economically rational relation between average portfolio returns and market risk as well as reducing the explanatory power of firm size. Indeed, the effects of firm size are all but removed from explaining average portfolio returns once the zero returns are reflected in the market risk estimation process. The viability of beta in explaining portfolio returns is still apparent.

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Multi-Beta CAPM with Turnover Premium Back to Top

 

This paper tries to find a widely accessible measure of liquidity and studies its impact on asset pricing. Using trading turnover as a measure of liquidity and the 1976-1993 Tokyo Stock Exchange data, cross-sectionally, stocks with higher turnover have lower expected returns. This evidence is consistent with predictions derived from an Amihud-Mendelson type of transaction cost model in which the turnover measures marginal investors' trading frequency. The trading frequency hypothesis also predicts that the cross-sectional expected return is a concave function of the turnover and time-series expected return is an increasing function of the turnover. The Japanese data supports both predictions.

 

Turnover can forecast expected stock returns and the observed relation is consistent with the hypothesis that the turnover measures marginal investors' trading frequency. The evidence provided here, however, can only suggest that the trading frequency effect exists and is stronger than the information-based trading and the order-processing cost effects. To further disentangle these effects, future research need to measure the transaction costs and examine the relation between expected returns, turnover, and transaction costs at the same time.

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Multi-Beta CAPM with Distress Premium and Dividend Yield Back to Top

 

We find reliable evidence that both book-to-market and dividend yield track time-series variation in expected real stock returns over the period 1926-91 (in which B/M is stronger) and the subperiod 1941-91 (in which dividend yield is stronger). A Bayesian bootstrap procedure implies that an investor with prior belief 0.5 that expected returns on the equal-weighted index are never negative comes away from the full-period B/M evidence with posterior probability 0.08 for the hypothesis (0.14 with the impact of the 1933 outlier tempered). Although this raises doubts about market efficiency, the post-1940 evidence is consistent with expected returns always being positive.

 

The B/M results suggest that expected return variation over the 1926-91 period was not driven entirely by equilibrium changes in compensation for risk. Rather, it appears that the market may have been inefficient, particularly in the late 1920s and early 1930s, a period of great economic volatility. The stronger results for the equal-weighted index, which is influenced more by small-firm returns, is consistent with this conclusion, although greater variation in risk-related predictability is also plausible for smaller firms.

 

The market inefficiency conclusion should be tempered by several observations. First, there is the familiar element of data mining which implies that the significance of extreme results is overstated, although by how much it is difficult to say. While the financial ratios considered here can be motivated theoretically, dividend yield is examined in part because of its prominence as compared to earnings yield in past time-series study. B/M is considered primarily because of its recently acquired celebrity status in explaining cross-sectional stock return variation, although the extent to which this biases time-series results at the market level is less clear. These concerns and the fact that the forecasting power of B/M and dividend yield varies considerably over different subperiods give further credence to the idea that data mining consideration should temper one's assessment of the value of these ratios in making future investment decisions.

 

The dividend yield slope estimate for the 1926-91 period and all of the estimates for the 1941-91 subperiod are more reasonable than the B/M results, but still substantial, implying expected return movements from three to eight percentage points for one-standard-deviation changes in the financial ratios. These estimates all result in varying degrees of increased confidence in the proposition that expected returns were never negative. While the considerable estimated variation in expected returns might be difficult to reconcile with market efficiency, the hypothesis that the slope is greater than three is never rejected at the 0.05 level. Definitive statements concerning the extent to which estimated variation in expected returns is related to risk or systematic mispricing (or both) require further analysis.

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Multi-Beta CAPM with Country-Risk Premium Back to Top

 

The law of one price implies that any differences in the rates of inflation will be offset by a change in the exchange rate. It suggests that in order to estimate changes in the spot rate of exchange, we need to estimate differences in inflation rates. It also implies that inflation-adjusted exchange rate (or real exchange rate) should have been constant. The law of one interest rate (or capital market equilibrium) assumes that there is a single real rate of interest in the world capital markets if they are fully integrated. Since government cannot directly control interest rates in the international markets, we might expect that in these markets differences between the expected real rates of interest would be small. But the government has more control in domestic short-term interest rates. It is possible for a country to have a real interest rate that is below the real rate in other countries. Individuals and companies are capable of transferring their capital from countries with low real interest rates to those with high real rates. The international CAPM assumes that all securities are priced in perfectly integrated markets and investors measure returns in different currencies. The riskfree rate can be in any currency if international fisher parity holds. The risk premium and the beta can be measured relative to the integrated market. In the segmented markets, the costs of capital vary for different countries but are in equilibrium assuming fair pricing. However, in the emerging markets where inflation rates are high and asset prices are unstable and volatile, the cost of capital must take into account the country risk in addition to the market risk premium. In this event, controlling cost of capital is more important than measuring it. The designated currency for receipts of cash flow has large effect.

 

In summary, cost of capital is an equilibrium rate assuming fair pricing. International parity relates interest rate differences to differences in expected inflation rates, forward currency rates, and expected currency movements. The international CAPM can be based on any currency and estimates of risk premium and beta that are mutually consistent. Emerging markets costs of capital are very high primarily because of country risk (political and domestic economic risks). some of which can be mitigated.

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CAPM with Bondholder-Stockholder-Conflict Premium Back to Top

 

This paper examines the importance of stockholder-bondholder conflicts in capital structure choice. We compute the expected magnitude of the wealth transfer between stockholders and bondholders when a firm accepts a new equity-financed project and then characterize the set of positive NPV projects that stockholders would prefer to ignore (the underinvestment problem) and the set of negative NPV projects that stockholders would like to accept (the overinvestment problem). The results quantify the distortions from both underinvestment and overinvestment and verify that both increase with the quantity of debt in the capital structure. However, these distortions appear too small to offset the value of the tax shields associated with debt, and thus are unable to explain the vast majority of cross-sectional variation in capital structures.

 

Despite over 40 years of research on the topic, we still know little about the determinants of capital structure. There is general agreement that debt has a tax advantage over equity, but we know little about the relative importance of the costs of debt that potentially offset this tax advantage. Among the most popular explanations for the relatively low debt levels that we observe are those proposed by Myers (1977) and Jensen and Meckling (1976) that agency problems inherent in the differing objectives of stockholders and bondholders offset the tax advantage of debt. It has been argued that stockholder-bondholder conflicts are an important determinant of capital structure (Smith and Watts, 1992), have major consequences for the way firms reorganize when in financial distress (Gertner and Scharfstein, 1991), and even have macroeconomic implication (Lamont, 1995).

 

We document that the distortion exists and that it increases with debt levels. The stockholder of the median firm on Compustat would choose to accept a low risk project only if the return is at least 8,12%, assuming that the project pays annual cash flows equal to 10% of expected annual cash flows and the projects has a zero NPV with a discount rate of 7.96%. This hurdle rate rises with the amount of debt in the capital structure. If we increase the debt ratio to 40%, the hurdle rate increases to 8.35%. This calculation quantifies the underinvestment effect first proposed by Myers (1977). As the project's risk increases, the relevant hurdle rate drops closer to the all-equity benchmark rate. Eventually, the project enters the overinvestment range, in which the hurdle rate for new projects is lower than that required for the project to have a zero NPV. In this range, stockholders have incentive to overinvest in risky negative NPV projects.

 

The rate at which the hurdle rate drops depends on the correlation between the firm's and project's cash flows. If the correlation is low, the project provide diversification, reducing the overall risk and transferring wealth to bondholders. If the correlation is high, then this diversification effect is reduced and a lower risk level is required to get overinvestment. Our model tells us which types of projects the firm should accept and ignore when investment decisions are made in the interest of stockholders. Its limitation is that it is impossible to know which projects firms are actually passing up, since firms' investment opportunity sets are inherently unobservable (i.e. strategic real options). However, our analysis does tell us the rates of return for any projects that the firm does pass up because of these conflicts. We construct a test of whether Myers's arguments can explain the capital structure decisions of typical firms. Given the rates of return implicit in the projects that are passed up, we estimate how much investment the firm would have to make to justify ignoring the tax shields. Our results imply that the firm would have to invest approximately 38.5 times annual after-tax cash flow, or 4.5 times total firm value, in order to create enough value to justify giving up the tax shields that the firm could realize if it increased its debt ratio to 40%. This calculation suggests that the underinvestment problem, while one of a number of costs of debt, is unable to explain capital structure choices for typical firms.

 

Given the amount of debt used by a firm, the increase in stockholder-bondholder conflicts associated with additional debt are too small to explain why the firm does not use more debt. Thus, conflicts alone cannot explain the well-documented cross-sectional patterns between firms' asset and capital structures. One characteristic of the equilibria in a number of recent theoretical papers is that the maturities of a firm's assets and liabilities are matched. Since the growth options of many equity-financed firms have extremely long horizons while the assets-in-place associated with debt-financed firms typically have shorter durations, this characteristic is consistent with the empirical evidence. Nonetheless, it seems clear that progress in explaining capital structure will require distinguishing between the theoretical arguments in the literature. Numerical analysis in this study is one way in which this can be accomplished. We should emphasize that there are a number of limitations on this analysis that should be addressed in future work. We have argued that our basic conclusions are likely to be robust to the choice of discount factors and to the fact that we ignore intertemporal project interactions. In addition, we assume throughout the paper that cash flows follow a random walk. A more sophisticated model of cash flows would yield more accurate measures of investment distortions. Finally, we assume that strict priority always holds and that there are no direct or indirect bankruptcy costs. Clearly, it would be an improvement to replace these assumptions with more realistic ones.

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Pure-Play Beta and Cost of Capital Back to Top

 

The pure-play method for estimating the cost of capital for a project requires the analyst to identify one or more publicly traded firms that operate in a similar line of business. These firms must be pure-play, i.e., they must not have significant operations in other lines of business. If pure-play firms can be identified, market information on these firms can be used to estimate the cost of capital for the project. A major difficulty is that many firms operate in multiple lines of business.

 

Prior researchers have addressed this difficulty using mathematical programming (Boquist and Moore, 1983) or regression (Erhardt and Bhagwat, 1991) to estimate industry (sector) betas using estimates of security (company) betas along with the industry compositions of the securities. Specifying betas for each industry implies a specific company beta for a company with given proportions in each industry. The mathematical program or regression selects the sector betas to minimize the deviations between the implied company betas and observed company betas.

 

This paper takes a different approach by framing the problem as an investment question. We demonstrate how to construct a portfolio which approximates a pure-play in a target sector, using a combination of publicly traded stocks each of which is not purely involved in the target sector. The idea is to build a portfolio to replicate the investment returns to a specified target sector without any direct exposure to other sectors. This is accomplished using long positions in some publicly traded stocks and short positions in others in order to eliminate exposure in unwanted sectors. Geltner and Kluger (1997) illustrates how to construct such a portfolio and consider its properties and suitability for real estate investment (both speculative and hedging) as well as for constructing real estate indices. This paper illustrates how the methodology can also be used for estimating a project's cost of capital.

 

The pure-play portfolio obtains maximum diversification in the pure-play sector, and simultaneously eliminates exposure in non-target sectors by using short positions. This methodology has several potential applications. It can be used as an investment vehicle to either speculate or hedge in a specific industry, or geographic sector of the economy. Similarly, sectoral or geographic return indices are a by-product of the returns to the pure play.

 

The pure-play technique, as described in many Corporate Finance textbooks (e.g., Rao) is a well accepted method for approximating a project's cost of capital. The primary difficulty in applying the technique occurs if there are no or few publicly traded firms in the industry of interest. In such cases, the methodology presented here can be used to build a pure-play portfolio using firms that operate in multiple lines of business.

 

Building a pure-play portfolio requires two main inputs: the percentage of each firms' value broken down by industry (sector), and the relative idiosyncratic variance terms. There can be up to nxt distinct idiosyncratic variance terms, one for each industry (j=1,...,t) for every firm (i=1,...,n), x being the fraction of ith firm invested in jth industry. For our illustration, we used very simple assumptions about these inputs. We used the firm's revenue shares in each industry as a proxy for the corresponding percent of market value. In addition, we assumed that the idiosyncratic variance term were all equal (homoskedasticity). Clearly more complex and realistic alternative assumptions are possible. Further research needs to be carried out to enumerate and evaluate such assumptions, and to determine when and if the idiosyncratic variances might be estimated using past market data.

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Full-Information Industry Beta Back to Top

 

Since estimates of beta for individual firms contain a great deal of statistical noise, analysts often estimate beta by averaging the betas of firms who specialized in a similar line of business. Conglomerates are typically excluded from the set of potential pure play since their operations span more than one line of business. Yet conglomerates can be large firms that account for a significant market share in a particular line of business. Large firms tend to have lower betas than small firms. Therefore, pure-play beta estimates will likely be upwardly biased. In this study, betas are estimated for 66 industries using the pure-play approach and a full-information approach. The full-information approach incorporates industry-specific information contained in the betas of conglomerates into the beta estimation process. Full-information betas and their corresponding standard errors are substantially smaller than their pure-play counterparts.

 

Industry cost of capital calculations are complicated by the fact that many firms operate in multiple industries. The observable betas for those firms who operate in more than one line of business are weighted averages of the unobservable betas of the individual operations. Therefore, these conglomerates are typically excluded from a pure-play industry analysis. However, conglomerates can often large players in a particular industry. The analysis of this study reveals that traditional calculations pure-play industry betas are biased up. This follows from the fact that conglomerates tend to be large firms and large firms tend to have low betas. Conglomerates can be incorporated into the industry beta estimate via regression equation. An important advantage associated with the cross-sectional regression methodology is that the resultant full-information industry betas are more precise.

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Accounting-Based Beta Back to Top

 

This study proposes and tests a practical method for selecting publicly traded firms which can serve as proxies for private firms, division, or projects in divisional cost of capital estimation problems. The cluster analysis method introduced herein is an approach to divisional cost of capital estimation which utilizes accounting variables to select a portfolio of publicly traded proxy firms. This method builds on the intuition of Fuller and Kerr's (1981) pure-play proxy technique and utilizes cluster analysis, which Elton and Gruber (1970) and others have shown to be useful in grouping firms by their systematic risk. Empirical tests of this method are performed on a random sample of 30 firms. These tests utilize CAPM estimated expected returns as the benchmark by which to compare firms' suitability to serve as divisional cost of capital proxies. These tests show that the estimates given by the cluster analysis method approach the CAPM benchmarks, and the hypothesis that they are equivalent to these benchmarks cannot be rejected. As a result, we conclude that cluster analysis is useful for identifying divisional cost of capital proxies.

 

The divisional cost of capital problem continues to be important for practitioners and researchers alike. It is frequently desirable to be able to estimate the market-priced risk of assets (e.g., divisions or projects) which do not have publicly-traded dividend-paying equity securities. CAPM is a widely accepted method of estimating the market-priced risk for firms with publicly-traded securities, so CAPM estimates of expected returns were taken as benchmarks. Cluster analysis was used to select proxies for target firms (pseudo-divisions) based on their accounting data. The cluster analysis method was shown to be useful in predicting the CAPM expected return of a target using that target's accounting data by selecting proxies for the target from publicly traded firms. Further, it was established that some of the measured discrepancies between targets' returns and proxies' returns are likely attributable to shortcomings in using the CAPM on ex post data for benchmark returns rather than to deficiencies in the cluster analysis approach. These results establish the usefulness of the cluster analysis approach to divisional cost of capital estimation and the potential benefits of further research in this area.

 

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