# Valuing private companies Part 1: Quantifying systematic and unsystematic risk

Compared to a hypothetical diversified investor in a public company, a hypothetical investor in a private business can potentially suffer from two key disadvantages:

- reduced opportunity to diversify away unsystematic risk; the business may be their main investment
- a lack of liquidity; an investor in a public company can sell their shares instantaneously, the investment in a private company can take time to sell

Consequently, a hypothetical buyer of a private business may only be prepared to pay a price that reflects both systematic and unsystematic risk.

This article (part one of two) considers diversification and the impact on systematic and unsystematic risk.

## What is risk?

The word risk derives from the Latin resecum. It means “the one that cuts”; a nautical expression relating to the cliffs which endanger ships[1].

Risk can be categorised in many ways and in valuation that risk relates to both the danger of future lower cash flows but also the opportunity for higher cash flows. In finance, risk is translated into a rate of return that investors require for bearing that risk[2].

In modern portfolio theory the variance of actual returns around an expected return is used as a measure of risk[3]. Can this be used as a guide to measure risk in a private closely held company?

## Modern portfolio theory and diversification

In 1952 Harry Markowitz, a professor of finance at the University of California, introduced modern portfolio theory (MPT)[4]. In 1990 the professor was recognised with the Noble Memorial Prize in Economic Sciences.

According to the professor’s theory, risk and return of an investment should not be viewed of itself, but in the context of how that investment impacts the risk and return of an investment portfolio.

The theory assumes investors are risk adverse and that they can reduce portfolio risk through diversification.

Unsystematic risk is the specific risk associated with investments. According to MPT, investors can diversify away unsystematic risk within their portfolio.

Systematic risk is the risk common to all investments. This is the exposure to events that effect the total economy and market returns. Systematic risk cannot be diversified.

Critics state that MPT is flawed; the model does not match the real world and assumptions are unrealistic, for example, returns do not follow a normal distribution. According to Nassam Taleb if you remove the normal distribution assumption you are left with nothing but “hot air”[5].

## Capital Asset Pricing Model (CAPM) and beta

The Capital Asset Pricing Model, built on MPT, determines an expected rate of return for an investment by considering the expected return of a risk-free investment, the expected return of the market and the sensitivity of an investment to systematic risk[6].

The sensitivity of an investment to systematic risk is reported as beta. Beta measures how an investment’s returns move, on average, when the overall stock market returns increase or decrease. Beta is calculated as the joint variability of the returns of the market and the investment, against the variance of the returns of the market[7].

CAPM assumes that the variance of returns reflects risk, there is an efficient trade-off between risk and return, investors can hold portfolios that maximise returns for the level of risk and that all investors have access to the same information. These assumptions are criticised for not reflecting reality.

Despite his caution against using the model, Rojo-Ramirez’s examination reveals that most valuation experts however still apply the CAPM model for company valuation[8].

## CAPM, beta, and private businesses

At the heart of CAPM is the assumption that an investor can and will hold a diversified portfolio. Unsystematic risk can be diversified.

A hypothetical investor, however, in a private company may be limited in their ability to create a diversified portfolio; a significant portion of their wealth being invested in the company.

If a cost of equity is estimated using the CAPM model for a private company and the hypothetical buyer is likely to be unable to diversify risk, then unsystematic risk may need to be considered. The problem then becomes how to quantify that unsystematic risk?

## Unsystematic risk and total beta

In 2001 Professor Aswath Damodaran published The Dark Side of Valuation and introduced the controversial topic of total beta[9]. Known as the Dean of Valuation, Damodaran is a professor of finance at the Stern School of Business, New York University.

Unlike beta which measures only an investments alignment with systematic risk, total beta is a measure of both systematic and unsystematic risk. Total beta is simply calculated as the standard deviation of the returns of an investment against the standard deviation of the returns of the market[10].

Valuation practitioners Peter Butler and Keith Pinkerton subsequently expanded the total beta discussion and developed the Butler Pinkerton model[11].

## Impact of diversification on the cost of equity

Professor Damodaran publishes annual global total betas on the Damodaran Online website[12]. The 2022 data set provides average betas and total betas for ninety-five industries taken from 47,606 companies.

The figure below is a sample extract of unlevered industry betas and total betas from the 2022 data set; with those betas applied to a nominal equity risk premium of 7.0%[13]. The figure illustrates how lack of diversification on the part of the investor can significantly increase the cost of equity.

## Criticisms of total beta

According to valuation practitioner Sarah von Helenstein, valuers search for silver bullets that can replace the uncertainties and difficulties of informed professional judgement with nice, simple quick quantitative formulas. Unfortunately, the world of investing and markets is complex, nonlinear, and dynamic, and thus there is no quick fix and total beta is not a viable silver bullet[14].

Elsewhere Pratt and Grabowski state the central problem with using total beta is that it is based on a myriad of unknown factors with unknown effects on the cost of equity[15].

## Summary

Despite its critics, the Capital Asset Price Model is still widely used in the valuation of companies. CAPM assumes investors can diversify away unsystematic risk. A hypothetical willing buyer in a private company may not be able to diversify away unsystematic risk. Consequently, unsystematic risk may need to be considered.

One method proposed to reflect both systematic and unsystematic risk is to use the concept of total beta. Total beta simply uses the variability of the returns of an investment compared to the variability of the returns of the market. Like MPT and CAPM though, total beta has its critics and flaws.

Estimating the risks and thus the cost of capital for a private business is not an exact science or mathematical model. Further, value may be influenced by the context of the valuation and the pool of likely hypothetical buyers.

Part two of this paper considers how the lack of liquidity in a private company, compared to the investment liquidity in a listed company, impacts value.

To find out more on this topic please register for the upcoming CA ANZ event: Quantifying risk and lack of liquidity in a privately held companies.

## Simon Cook

Simon specialises in valuing private businesses and quantifying damages. He is a Chartered Accountant Business Valuation Specialist and Forensic Accounting Specialist with Chartered Accountants Australia and New Zealand (CA ANZ). He chairs the CA ANZ Business Valuation group for Queensland and is a member of the CA ANZ Trans-Tasman Business Valuation Committee.

[1] Gaetano Liuzzo, Stefano Bentley, Federica Giacometti, Elena Bonfante, and Andrea Serraino (2014) “The Term Risk: Etymology, Legal Definition and Various Traits”

[2] Damodaran, A “what is Risk”

[3] Damodaran, A “what is Risk”

[4] Markowitz, H.M., March 1952, “Portfolio Selection”. The Journal of Finance.

[5] Taleb, Nassim Nicholas (2007), “The Black Swan: The Impact of the Highly Improbable”

[6] ERi = Rf + Bi (Erm — Rf); where ERi is the expected return of the investment, Rf is the risk-free rate, Bi is the beta of the investment and (Erm — Rf) = market risk premium

[7] Beta = Covariance (Ri, Rm)/ Variance (Rm); where Rm = average expected rate of return on the market and Ri = expected return on an asset I; expressed another way B = r (Std dev i)/(St dev m); where r is the correlation coefficient and std dev is the standard deviation of the investment and the market

[8] Rojo-Ramirez, Alfonso “Privately Held Company Valuation and Cost of Capital”, Journal of Business Valuation and Economic Loss Analysis, 2014

[9] A. Damodaran (2001). “The Dark Side of Valuation”, New York: Prentice-Hall

[10] Total Bi = Standard deviation of the returns of the investment / standard deviation of the returns of the market; an alternative way to calculate total beta is the beta of the investment divided by r, where r is the correlation coefficient

[11] P. Butler and K. Pinkerton (2006). “Company-Unsystematic Risk — A Different Paradigm: A New Benchmark,” Business Valuation Review; “The Butler Pinkerton Model: Empirical Support for Company-unsystematic Risk” (2008), The Value Examiner

[12] https://pages.stern.nyu.edu/~adamodar/New_Home_Page/datacurrent.html

[13] In the figure the risk-free rate is 3.0% and the equity risk premium of 7.0%. Unlevered betas are used, which assumes no debt in the business. The rates are simply for comparison purposes and should not be relied upon in anyway.

[14] Helfenstein, Sarah von (2003). “Revisiting Total Beta”, Business Valuation Review, vol 28, American Society of Appraisers; also see response

[15] Pratt, Shannon and Grabowski, Roger (2010) “Cost of capital: Applications and example”, Wiley, fourth edition