Polanitzer Function for Estimating a Firm’s Normative Cost of Debt without a Synthetic Credit Rating

This paper proposes a technique that provides an indication of the normative cost of debt of a company, either privately-held or publicly-traded, called “Polanitzer Function”.

Photo Credit: Intrinsic Value

Polanitzer fucnction is a fundamental analysis method of estimating a firm’s normative cost of debt as it uses additional financial leverage, and how that relates to the default risk of the firm. The measure is used to summarize the effects this type of leverage has on a firm’s cost of debt.

Polanitzer Function Assumptions

• The Polanitzer function is a method of estimating a firm’s normative cost of debt as it uses additional financial leverage.
• Once you get the firm’s normative cost of debt using the Polanitzer function — it is easy also to measure the credit spread as well as the default probability implied by that cost of debt.
• The Polanitzer function draws upon the Modigliani-Miller theorem on capital structure.
• The higher the Polanitzer function’s cost of debt, the higher the default risk of the firm.

Note #1: In this paper, where ever it says “debt” or “debt capital” it means interest bearing liabilities only (both short term and long term).

Cost of Debt

Cost of debt is the market rate the business would incur on invested capital debt borrowing. The cost of debt is obtained from the market and is sensitive to economic conditions and the company’s ability to service debt, indicated by its capital structure and debt coverage ratios.

The company’s normative cost of debt as of the valuation date may or may not resemble the interest rates on existing debt. Before assigning a cost of debt, the valuator should first investigate and analyze:

1. The current capital markets
2. Availability and cost of debt
3. Subject company’s current financial condition and collateral availability

Firms with public debt and/or equity capital incur additional operating costs required for a public firm. These costs relate to SEC filings and other “public company” costs. When spread across a large amount of debt and/or equity capital, these costs may be moderate.

For a smaller firm, these costs may suggest that publicly sourced capital is not an economically viable financing option. While the cost of debt is often viewed as a simple cost of interest relative to the amount borrowed, inclusion of compliance costs could lead to a higher cost of debt.

As a certain share of offering costs is fixed relative to the amount of funds raised, a smaller financing would also be expected to provide reduced proceeds (as a percent of the gross borrowings) relative to a larger financing. Standard & Poor’s (S&P) publishes its Corporate Ratings Criteria annually. Assessment of these ratings criteria provides important insights into understanding and estimating the cost of debt.

S&P’s ratings process includes: (1) Examination of financial measures; (2) Review of business fundamentals; (3) Prospects for growth; and (4) Vulnerability to technological change, labor unrest, regulatory actions. S&P prepares short term and long term ratings: (a) Short term rating is an assessment of an issuer’s credit quality with respect to an instrument considered short term in the relevant market; (b) Long-term rating is more frequently seen in practice.

Debt ratings do not comment on the liquidity of a rated instrument or any other element affecting the suitability of an investment for a particular investor. Market liquidity issues in the 2008 / 2009 timeframe resulted in significant volatility in debt yields due to price fluctuations.

This issue complicates estimation of an appropriate cost of debt. The following chart summarizes the investment grades as reported by Moody’s, S&P, and Fitch:

Figure 1

The following chart reflects the financial risk profile of companies followed by S&P from 1981–2006 by S&P rating:

Figure 2

When estimating the cost of debt for a subject company, the valuator may make comparisons of the “financial risk indicative ratios” above with those of the subject company. This paper proposes a technique that provides an indication of the normative cost of debt of a company, either privately-held or publicly-traded, called “Polanitzer Function”.

What Is the Polanitzer Function?

Polanitzer function is a fundamental analysis method of estimating a firm’s normative cost of debt as it uses additional financial leverage, and how that relates to the default risk of the firm. The measure is used to summarize the effects this type of leverage has on a firm’s cost of debt.

Polnitzer function has 2 limitations:

1. The company’s normative debt-to-equity ratio, which is a parameter in the function, ranges between 0.10% and 99.90% (it cannot be 0% nor 100%)
2. The company’s normative cost of debt cannot exceed its long-term normative cost of equity.

How the Polanitzer Function Works?

Roi Polanitzer is a well-known authority in Israel in the fields of business valuation, actuarial science, financial risk management and financial engineering and has published numerous papers that articulate many of the concepts used in those fields. Polanitzer developed models, functions and approximation equations in those fields. His equation first appeared in this paper.

The formula for the Polanitzer function is:

where:

where:

where:

Rf = Risk-free rate

Rp = Equity risk premium

Rs = Size premium

βa = Asset beta

T = Tax rate

D/E = Debt-to-equity ratio, at market value

d = Weight of debt capital in the capital structure, at market value

* Asset beta is the market risk of a company without the impact of debt.

* Debt-to-equity ratio is a measure of a company’s financial leverage.

How to Calculate the Polanitzer Function

The Polanitzer function is calculated in three steps:

1. Estimating the company’s long-term normative cost of equity (Ke).
2. Estimating the company’s a prior cost of debt (Kd*).
3. Estimating the company’s posterior cost of debt (Kd) as the minimum of the result from no. 1 and no. 2.

What Does the Polanitzer Function Tell You?

The function draws upon the Modigliani-Miller theorem on capital structure and extends an analysis to quantify the effect of financial leverage on a firm. Financial leverage which is represented by the debt-to-equity ratio is a measure of the company’s default risk. The Polanitzer function, then, shows how the firm’s cost of debt changes with leverage. The higher the cost of debt, the higher the default risk of the firm.

Example of using the Polanitzer Function in order to Estimate a Firm’s Long-Term Normative Cost of Debt

Assume we were asked to estimate the long-term normative cost of debt of a private firm which operates in the retail (building supply) industry in Israel, as of March 29, 2024 (hereinafter: “the Valuation Date”). In the absence of a quote for the firm’s cost of debt and in the absence of data regarding debt instruments for companies similar in characteristics to the subject firm, to the extent required for the purpose of estimating a reliable representative sample, I decided to estimate the firm’s long-term normative cost of debt based on the Polanitzer function.

Note #2: The figures displayed below may contain slight insignificant deviations caused by rounding off numbers

Step 1— Estimating the firm’s long-term normative cost of equity

First, lets’ estimate the firm’s long-term cost of equity since which it constitutes a CAP (i.e., the maximum possible cost) for its long-term normative cost of debt, based on the following formula:

where:

• Rf — is the risk-free real interest rate. I estimated this parameter at 1.95%, based on the yield to maturity derived from a real risk-free yield curve in Israel with a duration of 25 years, as of the valuation date (Source: Intrinsic Value, March 29, 2024).
• βa— is the asset beta. I estimated this parameter at 1.69, based on a study on Unlevered Betas by Industry of the scholar Aswath Damodaran for the retail (building supply) industry in United States, as of the valuation date. (Source: Damodaran Online, January 5, 2024).
• T — is the tax rate. I estimated this parameter at 23.0%, based on the long-term normative statutory tax rate applies to companies in Israel, as of the valuation date.
• D/E — is the firm’s long-term normative debt-to-equity ratio. I estimated this parameter at 19.92%, based on a study on D/E Ratios by Industry of the scholar Aswath Damodaran for the retail (building supply) industry in United States, as of the valuation date. (Source: Damodaran Online, January 5, 2024).
• Rp — is the equity risk premium. I estimated this parameter at 5.63%, based on a study on Risk Premiums for Other Markets of the scholar Aswath Damodaran for Israel, as of the valuation date. (Source: Damodaran Online, January 5, 2024).
• Rs — is an additional rate of return, attributed to the firm’s stock and reflecting specific risks, including an additional premium required in case of lack of liquidity (DLOL — Discount for Lack of Liquidity). Empirical data and various studies show that investors in the capital market often demand to receive an additional risk premium for their investment in companies, which expresses the aforementioned various parameters of their investment. The appropriate discount rate for the firm’s equity was composed from a size premium (10z) which was estimated at 10.73% (reflecting an additional return required for publicly-traded companies according to the firm’s size).

Based on the above 6 empirical estimates for the 6 parameters of the above formula, I estimated the firm’s long-term normative cost of equity (Ke) at 23.63%, as of the valuation date. As mentioned, this rate constitutes a CAP for the firm’s long-term normative cost of debt.

Step 2 — Estimating the firm’s a prior long-term normative cost of debt

Now, lets’ estimate the firm’s a prior long-term normative cost of debt based on the following formula:

where,

where:

• Rf — I estimated this parameter at 1.95% (for explanation see above).
• Rs — I estimated this parameter at 10.73% (for explanation see above).
• d — is the firm’s long-term normative weight of debt capital in the capital structure. I estimated this parameter at 16.61%, based on a study on D/(D+E) Ratios by Industry of the scholar Aswath Damodaran for the retail (building supply) industry in United States, as of the valuation date. (Source: Damodaran Online, January 5, 2024)
• Rp —I estimated this parameter at 5.63% (for explanation see above).
• βa — I estimated this parameter at 1.69 (for explanation see above).
• T — I estimated this parameter at 23.0% (for explanation see above).
• D/E —I estimated this parameter at 19.92% (for explanation see above).

Based on the above 7 empirical estimates for the 7 parameters of above the formula, I estimated the firm’s a prior long-term normative cost of debt (Kd*) at 4.63%, as of the valuation date.

Step 3 — Estimating the firm’s posterior long-term normative cost of debt

Now, lets’ estimate the firm’s posterior long-term normative cost of debt based on the following formula:

where:

• Ke— I estimated this parameter at 23.63% (for explanation see above).
• Kd* — I estimated this parameter at 4.63% (for explanation see above).

Based on the above 2 empirical estimates for the 2 parameters of above the formula, I estimated the firm’s posterior long-term normative cost of debt (Kd) at 4.63%, as of the valuation date. Therefore, the firm’s long-term normative cost of debt as of the valuation date is 4.63%.

Optional Step — Sanity check

For the purpose of an indicative reasonableness check, I examined the reasonableness of my result for the firm’s long-term normative cost of debt, by using a synthetic credit rating model. From the result of the synthetic credit rating model, it appears that the firm’s synthetic (indicative but not public) credit rating is ilAA.

The expected return to debt (e.g., bond) investors derived from the real yield curve for a credit rating of ilAA, as of the valuation date and with a duration of 25 years, is estimated at 3.64%. My model missed the cost of debt derived from the real yield curve for the credit rating of ilAA by only 0.99%. Not bad at all!!!

Example of Measuring the Firm’s Credit Spread Implied by the Polanitzer Function’s Cost of Debt

Now, after we estimated the long-term normative cost of debt of the above private firm as of the valuation date, let’s measure the firm’s credit spread implied by the Polanitzer function’s cost of debt, as of the valuation date. The firm’s implied credit spread is the spread between the Polanitzer function’s cost of debt and a default free bond is driven to a considerable extent by credit risk, but also the liquidity premium has a great impact on that spread.

Let’s, measure the firm’s credit spread implied by the Polanitzer function’s cost of debt, as of the valuation date, based on the following formula:

where:

• Kd — I estimated this parameter at 4.63% (for explanation see above).
• Rf — I estimated this parameter at 1.95% (for explanation see above).

Based on the above 2 empirical estimates for the 2 parameters of above the formula, I estimated the firm’s credit spread implied by the Polanitzer function’s cost of debt, as of the valuation date. Therefore, the firm’s implied long-term normative credit spread as of the valuation date is 2.68%.

Example of Estimating the Firm’s Default Probability Implied by the Polanitzer Function’s Implied Credit Spread

Now, after we measured the implied long-term normative credit spread of the above private firm as of the valuation date, let’s estimate the firm’s probability to default implied by the Polanitzer function’s credit spread, as of the valuation date. The firm’s implied default probability is the likelihood that over a specified period, usually one year, the firm will not be able to make their scheduled repayments on a particular debt.

Let’s, estimate the firm’s default probability implied by the Polanitzer function’s credit spread, as of the valuation date, based on the following formula:

where:

• Sp — I estimated this parameter at 2.68% (for explanation see above).
• Kd — I estimated this parameter at 4.63% (for explanation see above).

Based on the above 2 empirical estimates for the 2 parameters of above the formula, I estimated the firm’s default probability implied by the Polanitzer function’s credit spread, as of the valuation date. Therefore, the firm’s implied long-term normative default probability as of the valuation date is 2.56%.

The Difference Between Polanitzer Function and Synthetic Credit Rating Models

The Polanitzer function is a technique that provides an indication of the long-term normative cost of debt of a company (both privately-held and publicly-traded) without examining the credit worthiness of the company, but relies on financial parameters to determine its long-term normative cost of debt in the calculation of the weighted average cost of capital (WACC). The Polanitzer function effectively eliminates the need to examine the company’s financial statements and the characteristics of its activities and therefore to estimate its long-term normative cost of debt based on synthetic credit rating models.

Synthetic credit rating models are methods for estimating the company’s long-term normative cost of debt, which are based on the results of statistical models for estimating credit rating and/or default probability, such as Altman, Merton, Chesser and CART models. In the absence of a public credit rating from rating agencies for a subject company, valuators in Israel usually use linear discriminant models that classify the subject company’s debt into groups with different default risks, based on past business characteristics.

The models are mainly based on various econometric/statistical methods for estimating a company’s credit rating and, as a result, its default probability and its ability to repay its debt, by combining financial ratios from its financial statements, in various linear regression coefficients. Using the synthetic credit ratings obtained from the aforementioned models, the valuators in Israel estimate the company’s long-term normative cost of debt, based on the yield curves quoted by a recognized quoting entity in Israel (e.g., Intrinsic Value), as much as possible in accordance with the specific debt conditions, which include, among other things, the maturity dates and various linkage indices.

Example of using the Polanitzer Function in order to Estimate 30 Companies’ Long-Term Normative Costs of Debt

Assume we were asked to estimate the long-term normative cost of debt of 30 private companies which operate in 30 different industries in Israel, as of the valuation date using the Polanitzer function.

Step 1 — Estimating the companies’ cost of equity

First, lets’ estimate the companies’ costs of equity, based on the following formula:

Step 2 — Estimating the companies’ a prior costs of debt

Now, lets’ estimate the companies’ a prior costs of debt, based on the following formula:

Step 3 — Estimating the companies’ posterior costs of debt

Now, lets’ estimate the companies’ posterior costs of debt, based on the following formula:

Optional Step — Sanity check

Conclusion

This paper suggests a way to estimate a firm’s normative cost of debt without a synthetic credit rating called “Polanitzer function”. Polanitzer’s function relates the required return on the firm’s asset to its WACC counterpart, when the required return on the firm’s equity is estimated based on the modified CAPM using Hamada’s equation for measuring the firm’s levered beta based on the firm’s asset beta. It can be useful in several areas of finance, including valuation, actuarial science, portfolio management and risk management, to name just a few.

This formula can ease up the work of valuators while they are estimating the firm’s cost of debt, in order to get the firm’s WACC, as part of performing a valuation to that firm.

About the Author

Roi Polanitzer, Actuary and Valuator

Mr. Roi Polanitzer is an economist (with a undergraduate degree and graduate degree) and a valuator (of business, intangible assets and complex financial instruments) and an actuary (specializing in financial actuarial science, general insurance actuarial science, and life & pension insurance actuarial science) certified by an actuarial association in Israel recognized by the state authorities. In addition, he is a well-known authority in Israel in the fields of finance, actuarial, valuation, risk management, options, financial engineering, algorithmics and data science.

Mr. Polanitzer founded the Israel Association of Valuators and Financial Actuaries (IAVFA) and currently serves as the chairman of the IAVFA. In addition, he founded the Actuaries’ Israel Association (AIA, formerly the “Professional Data Scientists’ Israel Association” — PDSIA) and currently serves as the president of the AIA.

Mr. Polanitzer is the owner of the valuation firm “Intrinsic value — independent business appraisers” and serves as the firm’s chief actuary and the responsible valuator on its behalf. In addition, since 2006, he has been conducting actuarial opinions on the matter of balancing resources between spouses due to divorce proceedings, both as part of mediation proceedings as a consulting actuary and as part of legal proceedings as an expert actuary on behalf of the courts, appointed by family courts and rabbinic courts in Israel.

Mr. Polanitzer studied actuarial science at Ben-Gurion University of the Negev (BGU), University of Haifa, Ariel University and John Bryce Training College. In addition, he previously taught courses in the fields of actuarial science and valuation in academic and other institutions and currently teaches in the IAVFA’s courses and professional training in these fields.

Mr. Polanitzer was certified as an actuary by the Global Association of Actuaries, the Israeli Association of Actuaries, the IAVFA and the AIA. In addition, he is a member of the International Association of Actuaries.

Mr. Polanitzer led and took a very significant part in writing the IAVFA’s statements of opinion in the fields of actuarial science and valuation. In addition, he published over 600 papers in Hebrew and over 400 papers in English in the fields of actuarial science and valuation.

Mr. Polanitzer developed the IAVFA’s certification programs in the fields of actuarial science and valuation and has been certifying actuaries and valuers in Israel since 2015. In addition, he founded the program for certificate studies in actuarial science and risk management at the at Ben-Gurion University of the Negev (BGU).

Mr. Polanitzer developed models, functions and approximation equations in the fields of actuarial science and valuation. In addition, he served as the chairman of the committee for determining a valuation policy the Israel Tax Authority’s employees (“Polanitzer Committee”).

Mr. Polanitzer previously ran for office as the chief actuary of the Israel Capital Market, Insurance and Savings Authority and successfully passed all the stages of the tender. In addition, he ran for office as the supervisor of banks at the Bank of Israel and for the position of director of valuations at the Israel Tax Authority and successfully passed all stages of the tender.