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Cost of equity (Ke): CAPM and the capital asset pricing model

CAPM: CALCULATION OF THE COST OF EQUITY (“Ke”) OR THE MINIMUM YEARLY RETURN IN PERCENTAGE REQUIRED BY AN INVESTOR IN A PROJECT, USING THE CAPITAL ASSET PRICING MODEL

Let’s assume that we are a group of entrepreneurs that are founding a start-up project in Spain.

The entrepreneurs want to enter as shareholders of the company, entrepreneurship or business project. The total funding required by this project in t=0 is called TOTAL FUNDING NEEDS OF THE PROJECT,  and the funding will be secured by banking debt (“D%”) with an annual cost of interests Kd = 5% and by the contribution made by the founding partners = “equity owners” = “shareholders”. As the banking debt, the shareholders will also demand a minimum yearly profit for their investment, that is called “Ke” or cost of equity, being the CAPM model used to calculate its value.

1. How an investor who enters a business project earns money:

Just as the bank requires a 5% annual return (Kd) + the return of the principal amount borrowed; we, as shareholders, will demand a minimum annual return for the money we invest in this project, that is called “Ke: Cost of equity”. This profitability must be paid by the project with the cash flows it generates in two ways:

1) Payment of dividends from the profits generated by the company / project each year. The metric that defines the percentage of net profit that goes to pay dividends is called “Pay-Out”.

2) Increase in the value of the company (share price increase). Depending on the valuation method, the share price can be calculated as: a) net equity of the company on the balance sheet, b) intrinsic value of the company, c) market value (share price in the market). The value of the company will depend on the valuation method. If we invest in a company €10,000 for a 10% of the shares, and now these shares have a price in the market of €100,000, our profitability should be calculated taking into account the dividends received during the time we held the shares + value of the 10% shares, calculating the profitability received as investors (cash-flow invested at t0 = € 10,000 versus cash-flows received over the investment time horizon consisting of dividends and a final value of € 100,000). The calculation of the profit should be undertaken using investment appraisal techniques such as Net Present Value (“NPV”), Internal Rate of Return (“IRR”) and Payback period (“PB”).

To calculate the minimum annual return that we will demand as shareholders, and which we will call “Ke”,  the CAPM model will be used (“Capital Asset Pricing Model”). It seems obvious that if the project does not provide at least a return equal to or greater than Ke, it will not be profitable for the shareholders.

In general terms, and depending on the type of entrepreneurship project (=risk), the Ke internationally moves in the range between 15% and 40%, with an average value of  25% circa, although this metric must to be supported with a relevant report/research, and will depend on the risk of the investment project.

2. CAPM model (Capital Asset Pricing Model)

Before commenting on this model, it must be said that the CAPM model has many detractors for its limitations, although it is widely taught in Corporate Finance subjects.

The CAPM model conceptually establishes that an investor will demand a higher profitability if the risk the project bears is greater. Therefore a project with more risk should provide a higher return to an investor. The investor will demand for each element of perceived risk, a return or profit (risk premium = profitability). It is important to note that although we speak of “risk premium” in the following formula, what we are adding is profitability in %, and not risks (the risks are measured as standard deviations!). Therefore, we will identify risks, then we will quote that risk in terms of profitability (“risk premium”), and the formula will be as follows:

Ke = Risk premium 1 + Risk premium 2 + … + Risk premium “n”.

But before adding risk premiums, we must ask ourselves: is there an investment that does not bear any risk at all? The answer is simple, NO, but in finance the 10-year bond of a government is considered as the risk free investment.

Let’s now imagine that we have the option to invest our money in a risk-free project (option A) or investing the money in this start-up project that we’ve been offered and bears more risk (option B). It seems logical to think that we will demand as shareholders more profitability to our money in that project that bears more risk and less to what we consider as “investment without risk” (in the case of existing such investment). We will see below, that we will assume that there is a risk-free investment, and therefore, the Ke formula will remain as:

Ke = Profitability risk-free investment + Risk premium 1 + Risk premium 2 + … + Risk premium “n”.

3. Risk-free return or risk free investment (“Rf”)

In general terms, there is not an asset without 100% no risk, but in finance, the risk free investment or “Rf” is considered to be the profitability offered by the 10-year bond of a country (certain countries only). The reason for taking the 10-year bond is because this instrument has high liquidity (we can buy-sell at any time that financial asset in the financial market) and it is considered a risk-free investment  due to the payment capacity of selected countries (although we know that countries also go bankrupt).

Thus, if we are Americans, we will consider the risk-free investment as the 10-year Americans bond (“Rf10a_U”), if we are Spaniards, the Spanish 10-year bond (“Rf10_ESP”) and if we are Germans, the 10-year German bond (“Rf10_GE”). Analyzing this data on May 28, 2016, the annual coupon (= annual interest that the US, Spanish or German state would pay to investors for a 10 year-bond would be as follows):

• Rf10a_US: 1.818%
• Rf10_ESP: 1.476%
• Rf10_GE: 0.143%

From this data, it is observed that Spain as a state, gets funding from the investors in a 10-year horizon with an annual cost of 1.476% while Germany does it at 0.143%. As it can be seen, Spain bears a 1.33% (1.476% -0.143%) higher funding costs than Germany, and the reason is investors perceive more risk in lending money to Spain as a country than to Germany (mainly related to stability of public finances, country global competitiveness, labour market, unemployment rates, and probability to payback the amount borrowed).

The 1.33% spread (=133 basis points) is known as “Country Risk Premium”, defined as the additional/excess of profitability over the 10-year German Bond demanded by investors from the Spanish government in order to get funding and due to Spain’s country’s intrinsic risks. Here we see again that at higher risk, the investor demands greater profitability. In our specific case, when investing in Spain, our risk-free return will be the 10-year Spanish bond, Rf = 1.476%.

The value of the risk premium is very important, because if Spain has €1 trillion of public debt at an average annual rate of 3% and has annual debt maturities of €250,000 millions this fiscal year, an increase in the country risk premium from 1.33% to 8%, it would imply that Spain in order to raise public debt funding in the stock markets, it should pay an annual return or coupon to its bonds of 0.143% + 8% = 8.143%, and should refinance the total debt maturities of €250,000 millions at a yield/cost of 8.143% rather than 3%; having to assume the Spanish finances an annual interest cost of 8.143% * 250,000 million = €20,358 millions (only because of the increase in risk perceived by investors), and if we assume that the other €750,000 millions remain at 3% level (3% * €750,000 millions = €22,500 millions). So the total annual cost of the Spanish debt would be €42,858 millions, which makes it a situation very difficult to maintain from a long-term perspective and solvency of the Spanish finances that could lead to the country’s bankruptcy.

4. Rate of returns for perceived risk.

Now, we will have to identify the risks that affect the business. The risks impacting  a project can be of different nature, among others:

(1) Risks due to the nature of the company / industry in which it is developed (IT, food and beverage, banking, construction, etc.).

(2) Financial (prices, interest rates, credit default, exchange rates, etc).

(3) Liquidity of the investment (ability to convert into cash the financial assets)-

(4) Environment (country, legal, fiscal, etc).

(5) Others.

The basic CAPM model only quotes the risk-free investment and the specific risks for the type of project/company/sector undertaken. This risk is defined by a numerical constant called BETA (equal or greater than zero), that conceptually means if our company (or project) has the same risk as the comparable equity stockmarket  (beta = 1), higher risk than the equity market (if Beta = 2, we would say that the project has twice the risk of the stockmarket) or lower risk (Beta = 0.5, our company/project has half the risk of the comparable equity market). If we were investing in Spain, the comparable equity market would be IBEX-35, in the United States S&P 500 and in Europe, Eurostoxx 50.

Therefore the intrinsic risk of a project is measured by a number, called Beta, the higher the Beta, the higher the risk, and the higher should be the demanded profit. In general terms, Beta is calculated based on the correlation between the profitability of our company compared to the equity market (in terms of stock prices profit), however, if our project is new, we can not calculate it by this procedure since we do not have quoted prices of our company nor we can calculate price returns (unlike publicly traded companies where on a daily basis we can calculate the price of the shares and the equity index, calculate daily returns to both and therefore calculate a Beta (share, market).

When you do not have price quotes for the start-up, you should look for a betas report by country and sector. It is important to understand very well what Beta is provided by the report , because a Betas report of American automotive companies could not be used for a European automotive companies, nor an automotive beta report made for multinational companies when our company is a SME automotive (the SME-Small, Medium Enterprises have more risk than a multinational because of its size). A source for obtaining the Beta of a company could be the Beta report from professor Aswath Damoradan, professor of finance at Stern University in New York:

• Betas report by sector for US companies.
• Betas report by sector for European companies.

Additionally, Dr. Damoradan publishes reports of all kinds on its website about market risk premiums, interest rates, etc.

The next consideration to make is that if the Beta provided is calculated on a sample of multinational companies or large capitalization companies; we can not use the same Beta report and we will have to increase the Beta to incorporate the risk we have due to the size of our project or company. This is what is known as the “Size Risk Premium” and that in an entrepreneurial project would increase the value of  Damoradan’s Beta between x1 and x4 times (A reasonable value, although it must be justified, could be x2-x3 times , although as I mentioned this will depend on the risk of the project). Therefore, the formula of Beta = Beta Damoradan * Beta size of the company. This calculation therefore would tale into account that our project does not have the same scale than the companies sample analyzed by Damoradan in his Beta report (designed for large size companies).

In our example, if the Beta for the “Advertising” sector is 0.87, and we increase it by the risk premium due to size by 400% (Beta size: x4), the Beta of our project would be: 0.87 * (400%) = x3.48. Note: In our calculations we are not considering leveraged and unlevered Betas (i.e, there are betas per sector depending on whether the company has debt or does not have debt). In this example, we do not enter into these considerations or calculations. But it seems logical that the beta of an advertising company that does not have banking debt is lower than the risk (beta) of an advertising company that has debt (a company with leverage). Betas reports can give us unlevered betas (that is, without a debt structure), where the beta would have to be leveraged based on the specific debt structure of our company. If this consideration and calculation is not made, we will take leveraged betas instead of unlevered betas for having more risk.

Once we know that the risk of our project measured as Beta is x3,48 (=348%)) and  the comparable equity market (Ibex 35 in this case) has a Beta of 1, it seems logical that if we know the profitability that investors demand from the Ibex 35 for the risk of the companies that make up the index (known as “MRP = Market Risk Premium”), we will demand as risk premium for the sector-industry of the project: Beta * MRP = x3.48 * PRM. For the calculation of the MRP of the Ibex 35 we have 3 options:

a. To obtain the expected MRP that as of today for the Ibex 35 (difficult to obtain, but it is the right way). MRP = Expected profitability of the Ibex 35 – Profitability of the Spanish 10-year bond.

b. Calculate the historical MRP of the Ibex 35 in a period of time (easy to obtain, but conceptually incorrect, although widely used in finance). MRP=Historical profitability of the Ibex 35 – Profitability of the Spanish 10-year bond. According to various studies (see Pablo Fernández) and the period considered, the Ibex-35 MRP can range between 3.7% -5.5%. We could take a value of 5%.

c. Take the average value used by companies in the financial sector for MRP. This value according to a survey conducted by Pablo Fernández in 2011 and for the case of Spain has an average of 5.9% and a median of 5.5%, with a minimum value of 1.5% and a maximum value of 15.5% (on a sample of 900 surveys). If instead of considering Spain, we consider the entire sample of 1,500 surveys made to managers of financial companies, analysts and finance professors of universities at the international level, the most relevant results of the survey are: Large dispersion of responses (teachers use market risk premiums MRP between 3% and 8%, analysts between 2% and 11.9%, and companies between 1.5% and 15%). The average premium used by companies (6.1%) is higher than that used by teachers (5.5%) and that used by analysts (5.6%).

We will take a value of 5.5% as MRP for the Ibex 35. Therefore, and for the case of our project, we will demand a return for the intrinsic risk of the company due to the industry/sector:

(Beta = x3.48): x3.48 * 5, 5% = 19.14%.

5. Minimum annual return required by the shareholder to enter into this investment project.

Once we have calculated the risk free investment yield (10-year Spanish Bond) and the risk premium demanded due to the industry/sector risk of our start-up project, we apply the CAPM formula:

Ke = Rf + PRM = Spanish bond 10 years + risk premium of the company = 1.476% + 19.14% = 20.61%. As shareholders, we will demand a at least an annual return of 20.61%. If the project does not provide us with such minimum profitability, it will not be profitable, and the investment should not be undertaken.

WACC: Weighted average cost of capital, definition

WHAT IS THE FUNDING MIX OF A PROJECT AND THE WACC (WEIGHTED AVERAGE COST OF CAPITAL)

Once the total funding needs of a start-up project has been calculated, the following step would be to consider the potential funding sources where the funding can be obtained. The two major funding sources of any company/project are mainly:

  1. Equity funding (“E”) or amount of money borrowed from investors/shareholders/equity owners of a project/company.
  2. Banking debt (“D”) or amount of money borrowed from financial institutions or banks.

The funding mix ratio depends on the project risk, credit scoring/rating, country framework and start-up ecosystem for that specific country. For the case of Spain, according to Webcapital (2016), the funding mix ratio for start-up projects in Spain was 86% from shareholders (“E”) and 14% from banking debt (“D”).

It is very important to understand that this funding is not for free, and it has a yearly cost in percentage defined as “Ke” (cost of equity funding) and “Kd” (cost of banking debt funding). The “Ke” depends on the project characteristics and risk profile, and the “Kd” depends on the interest rates yield curve and risk profile of the project too (credit spread). As a general rule we could assume that Ke will be in the range of 15%-40% with an average value of 25%, and the Kd could range from 3%-8% with an average value of 5%. These metrics should be supported by any study/research depending of the project characteristics. The Ke metric usually is calculated using the “CAPM”, Capital Asset Pricing Model.

The Ke is defined as the minimum yearly profit expectations in percentage required by shareholders that invest in a project. If this Ke or minimum profit expectation is not achieved by the project, the shareholder or equity owner will not be satisfied with the project’s profitability according to the project’s risk profile. In the same way, the project should be able to pay back the amount of money borrowed from the bank (principal of the banking loan) and the yearly interest determined by Kd.

Once the funding mix ratio has been defined and the Ke and Kd have been calculated, the cost of the funding mix can be calculated, being defined as the“WACC” or Weighted Average Cost of Capital. This KPI or metric would be defined as the yearly weighted cost in percentage of the funding mix taking into account the Ke and Kd and the ratio %E and %D. The WACC formula is as follows:

WACC=%E*Ke+%D*Kd*(1-corporate taxes)

Let’s imagine a project with the following financing scheme:

  • Equity funding (E: 86%), Baking debt funding (D: 14%).
  • Ke (25%) and Kd (5%).
  • Corporate tax level: 25%.
  • The WACC=86%*25%+14%*5%*(1-25%)=22.03%. For every euro borrowed with the funding mix 86% (E) and 14% (D), and with a cost of funding Ke (25%) and Kd (5%), the weighted cost is €0.05, and the project should generate at least a profit/return of €0.05 to be able to pay back the funding sources and profit expectations, otherwise the project will be unprofitable.

EXAMPLE

In the following link it is explained how to determine the total funding needs of a start-up project. Using the methodology explained there, the entrepreneurs require for their start-up €55,213,500. If the funding mix is 86% equity (E) and 14% financial debt (D), the amount borrowed from the investors will be “E”=86%€55,213,500= €47,483,610 and from banks “D”=14%€55,213,500= €7,729,890.

The E=86%=€47,483,610 should be invested in the project through shares.

The cash-flow generated by the project should be able to payback to the investors (via dividends and share price increase) at least the minimum profit expectations defined by Ke=25%. In addition, the project cash-flow generation should be able to payback the principal of the banking loan (“D”=€7,729,890) and the interests (“Kd”=5%). In the following link there is an explanation of how a banking loan is solved following a French amortization method.

 

Cómo descontar flujos de caja: Discounted Cash Flow (DCF)

DCF: CÓMO CALCULAR LA RENTABILIDAD ECONÓMICO-FINANCIERA DE UN PROYECTO (TIR, VAN, PAYBACK) Y DESCONTAR FLUJOS DE CAJA

Cómo determinar la rentabilidad de un proyecto, TIR, VAN, Payback

Este post es la continuación de otro donde se explican los procedimientos existentes en finanzas para calcular la rentabilidad económico-financiera de un proyecto de inversión. 

De forma conceptual el valor de un proyecto se define como el potencial de generar caja (cash flow) del proyecto a lo largo de su vida. Revise por favor el siguiente link, para entender los tipos de flujos de caja existentes y las fórmulas que hay que aplicar para su cálculo, ya que en este post el dato de partida que usamos para los cálculos es el flujo de caja del accionista.

Supongamos un proyecto que tiene una duración finita de 5 años, genera los siguientes flujos de caja para el accionista, es decir, la caja que le queda al accionista del proyecto de inversión. Este flujo de caja es el que se conoce como “Equity Free Cash Flow” o flujo de caja del accionista (“EFCF”).

Una vez calculados los flujos de caja del accionista, necesitamos saber la rentabilidad mínima anual en % que le exige el accionista al proyecto de inversión, o dicho de otra forma, el coste que tiene el proyecto por financiarse vía accionistas. Este ratio se conoce como “Ke”, coste del equity o rentabilidad mínima anual que el accionista exige a los flujos de caja por realizar la inversión en ese proyecto. El cálculo del Ke se puede realizar usando el modelo CAPM, o Capital Asset Pricing Model. Para este ejemplo Ke=25%. Valores comunes suelen estar en el rango del 20-35% para proyectos de emprendimiento, aunque esto depende de la naturaleza del proyecto de inversión, por lo que un 25% podría ser un buen proxy de Ke.

Conocidos los flujos de caja del accionista y el coste de la financiación Ke=25%; el siguiente paso es “mover” (=descontar) los flujos de caja a t=0, debido al valor temporal del dinero. Es decir flujos monetarios en distinto instantes de tiempo, no son equivalentes por el valor temporal del dinero o la capacidad que tiene un flujo monetario hoy de generar rentabilidad a futuro, motivo por el cual 100€ hoy no son equivalentes a 100€ dentro de 1 año. Para mover flujos monetarios usaremos la fórmula del interés compuesto: Vf=Vo*(1+int)^t. Si definimos “Fd” (Factor de Descuento) como 1/(1+int)^t, el valor actual (“Vf”) de un flujo futuro en t=0 (“Vo”) será igual a: Vf=Fd*Vo.

Para el ejercicio en cuestión, el valor de los flujos del accionista sin tener en cuenta el valor temporal del dinero es:

  • (T=0, -200.000€). Sería la aportación en patrimonio neto que realiza el accionista de la empresa/proyecto para tener “derecho” a percibir los EFCF de los años siguientes.
  • (T=1, -100.000€)
  • (T=2, +50.000€)
  • (T=3, +150.000€)
  • (T=4, +250.000€)
  • (T=5, +400.000€)

El factor de descuento, “Fd” aplicando la fórmula: Fd=1/(1+25%)^t

  • (T=0, Fd=1)
  • (T=1, Fd=0,8)
  • (T=2, Fd=0,64)
  • (T=3, Fd=0,51)
  • (T=4, Fd=0,41)
  • (T=5, Fd=0,33)

Y el valor de cada flujo futuro descontado a t=0 (valora actual neto de cada flujo en t=0, y que tiene en cuenta el valor temporal del dinero), se obtiene al multiplicar Vf*Fd:

  • (T=0, -200.000€)
  • (T=1, -80.000€)
  • (T=2, +32.000€)
  • (T=3, +76.800€)
  • (T=4, +102.400€)
  • (T=5, +131.072€)

Y por tanto el “Valor Actual Neto (VAN)”, o valor en t=0 de todos los flujos caja del accionista o “EFCF” es VAN=-200.000-80.000+32.000-76.800+102.400+131.072= 62.272€. Es decir, como el VAN>0€, el proyecto es rentable y debe realizarse porque este proyecto genera flujos de caja a nivel del accionistas que cubren el coste de la financiación “Ke”, y además le proporcionan al accionista un exceso de caja sobre sus expectativas mínimas de rentabilidad de 62.272€. Téngase en cuenta que VAN no es la rentabilidad del proyecto, es el exceso de caja que se lleva el accionista sobre su expectativa mínima de rentabilidad. Por tanto, a mayor VAN, accionista más feliz porque obtiene un mayor exceso de caja que no exigía pero que el proyecto es capaz de pagar.

Una vez calculado el valor del proyecto en Euros (VAN), calcularemos la rentabilidad anual en % del proyecto, que ya sabemos debe ser igual o superior al coste de la financiación Ke=25%. Se define como “TIR”, o Tasa Interna de Rentabilidad, a aquel tipo de interés compuesto que hace el VAN=0. Es decir, la TIR es una tasa de interés que al descontar flujos EFCF nos da un valor de VAN=0. Y la ecuación queda de la siguiente forma:

VAN=0=-200.000-100.000/(1+TIR)^1+50.000/(1+TIR)^2+150.000/(1+TIR)^3+250.000/(1+TIR)^4+400.000/(1+TIR)^5

Resolviendo esta ecuación, obtenemos una TIR=32,04%, es decir, la rentabilidad anual en % del proyecto es 32,04%>25% (como ya adelantaba un VAN>0€). Es importante entender que la TIR es un tipo de interés compuesto anualizado o TAE, y que no es la rentabilidad real que año a año siguen los flujos de caja, ni la rentabilidad acumulada del proyecto a 4 años pero que en el periodo analizado de 5 años, sí da una rentabilidad igual a la del proyecto.

Calculado el VAN y TIR, el siguiente paso es calcular cuánto tiempo nos lleva recuperar la inversión del accionista, es decir, cuándo los flujos de caja del accionista acumulados positivos, igualan a los acumulados negativos, es decir, el inversor este proyecto desembolsa en (t=0, -200.000€) y (t=1, -100.000€). Al proyecto, le “cuesta” 3,4 años en recuperar los -300.000€. Para el cálculo del “Payback” o “Plazo de recuperación”, es necesario calcular los flujos de caja acumulados y ver en qué momento los flujos acumulados pasan de negativo a positivo (en ese año al cambiar el signo, se produce que los flujos acumulados + igualan a los -). Esto ocurre en t=3, donde el acumulado negativo es -100.000€ y en t=4 es 150.000€. Como queremos que entre caja por +100.000 para que el flujo acumulado sea “0€”, el cálculo será: 3 años + 100.000/250.000=3,4 años. Es decir, a este proyecto le lleva 3,4 años que los flujos de caja acumulados negativos invertidos por el accionista (-300.000€), igualen a los acumulados positivos (+300.000€). El plazo de recuperación depende del sector e industria, pero como regla general podríamos aceptar proyectos con Paybacks inferiores a 4 años.

El último índice de rentabilidad calculado es el “Profitability Index: PI”. Que se define como VAN/Flujos de caja negativos, y nos da una idea del exceso de rentabilidad que consigue el accionista por cada Euro invertido, en este caso PI=1,22= (76.800+102.400+131.072)/(100.000+200.000)

Por tanto este proyecto presenta los siguientes ratios a nivel de estudio de rentabilidad económico-financiera, y se llevaría a cabo.

  • VAN: 62.272€.
  • TIR: 32,04%
  • PAYBACK: 3,4 años.
  • PROFITABILITY INDEX: 1,22.

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