Discounted Cash Flow (DCF) Method Calculation

DCF methods are based either on the equilibrium model of Modigliani and Miller (1958), or on the capital asset pricing model (Sharpe, 1964; Lintner, 1965; Mossin, 1966).

The Equilibrium Model of Modigliani-Miller

According to the equilibrium model of Modigliani and Miller (1958), investors can invest in shares and bonds of companies on a capital market. The investors’ expectations about the expected return, the risk of the companies’ profit distributions and a risk-free interest rate are identical.

The value of a company results from the discounted profit distributions and bonds. Profit distributions are discounted using the cost of equity and debt using the cost of debt.

• The total cost of capital always remains the same and thus also the company value (value additivity).
• The cost of debt capital in the Modigliani and Miller equilibrium model is determined by the risk-free interest rate.
• The cost of equity capital in the equilibrium model of Modigliani and Miller is a linear function of the debt ratio and takes risk into account.

Basically, this also assumes tax neutrality of equity and debt capital as well as a lack of bankruptcy risk.

By the way, there is a competing model with the classical thesis. In this model, the debt-equity ratio has an impact on the total cost of capital, which is not linear.

The assumptions are, among others:

• homogeneous expectations
• an infinite time horizon
• perfect capital market
• tax neutrality
• no insolvency risk
• risk neutrality

It can be seen that these assumptions are not fulfilled in reality. It is a simplified explanatory model.

Capital Asset Pricing Model

The capital asset pricing model is based on the portfolio theory of Markowitz (1952) and the equilibrium model of Modigliani and Miller (1958). It can be attributed to three authors (Sharpe, 1964; Lintner, 1965; Mossin, 1966).

In Markowitz’s (1952) model, investors have various risky investment opportunities and skillfully combine them so that the expected return is realised with minimal risk. In Markowitz’s model, one speaks of an efficient frontier, which specifies the optimal combination possibilities of expected returns and risks.

The model can be extended to include a risk-free interest rate for investing or borrowing money. One then has a so-called tangential portfolio. In summary, an attempt is made to achieve an expected return with the lowest possible risk (variance) by cleverly combining investments.

The CAPM model now assumes that all investors have the same expectations about the returns on the capital market. Investors can invest money in assets and invest or borrow money on the financial market accordingly. Through an “invisible hand”, the market values and thus the company values are determined for the market equilibrium. The result is a so-called market portfolio with a capital market line.

The expected return is determined by a price of time (risk-free interest rate) and price of risk (risk premium). For the mathematical derivation, please refer to the sources mentioned. They will probably be familiar to some readers. The systematic risk can be expressed as market risk premium * beta factor.

The formula for the capital market line in the CAPM model is:

expected return = risk-free rate + (expected return on market portfolio – risk-free rate)/ standard deviation of market portfolio * standard deviation

For the individual investor, the expected value depends upon the portfolio risk taken (standard deviation). More information on this can be found in the cited literature.

The formula for the security market line in the CAPM model is:

expected return of the asset = risk-free interest rate + (expected return of market portfolio - risk-free interest rate)/variance of market portfolio * covariance of the asset in relation to the market portfolio

However, the formula of the security market line in the CAPM model is often presented as follows:

expected return of the asset = risk-free interest rate + (market risk - risk-free interest rate) * beta factor

The beta factor represents the covariance of the asset in relation to the market portfolio divided by the variance of the market portfolio.

The assumptions of the model are, among others:

• homogeneous expectations
• a single period
• perfect capital market
• no taxes
• no transaction costs
• no time delay
• risk aversion

When considering the assumptions, it also becomes clear that this is a simplified explanatory model and does not reflect reality. While some assumptions are removed by model extensions, such as in Brennan (1970) for tax integration, the basic critique remains.

The risk measurement based on variance also causes positive desired deviations to lead to greater discounting, which does not seem reasonable from the theoretical side.

It should be noted that the equilibrium model of Modigliani-Miller and CAPM are actually not combinable from the risk propensity and time horizon (Hering, 2021).

Here, the formula for the discount factor is presented. ρ stands for the discount factor, t for the respective year, τ is a running variable for the time, i for the interest rate and r for the risk. If the risk in the cash flow is taken into account by a discount, no risk premium may be added to the interest rate.

The discounting of the cash flow of a company with a perpetual life is represented by the following formula. The first part shows the discounting in the planning period, and the second part represents the annuity at the planning horizon. C stands for the net present value, t for the respective year, T for the planning period, e for the cash flow, ω for the growth rate.

If a finite lifespan is assumed instead of a perpetual lifespan, a present value is applied to the planning horizon. n stands for the years.

The weighted average cost of capital (WACC) approach is a variant of the DCF method and is based on the CAPM model (Sharpe, 1964; Lintner, 1965; Mossin, 1966).

WACC Approach (Steady State)

In addition to the initial data, the cost of equity is taken from the APV approach of 11%. Equivalence to the APV approach can be achieved without any difficulties. The formula for WACC in the steady state is:

WACC = cost of equity * value of equity / value of total capital +

cost of debt * value of debt / value of total capital +

cost of debt * value of the tax shield / value of total capital.

WACC = 0.11*70000/90000 + 0.05*20000/90000 + 0.05*6000/90000 = 9.33%

Often, the last two terms are combined. However, this is only possible under very simplified assumptions regarding tax law.

WACC = 0.11*70000/90000 + (1-0.3)*0.05*20000/90000 = 9.33%

If the equity-financed net cash flow of 8400 MU is discounted with WACC, the result is 8400/0.0933 = 90000 MU.

From the total company value of 90000 MU, the debts of 20000 MU must again be subtracted, so that the equity is worth 70000 MU.

In practice, WACC is not seen as a reformulation of the APV approach, but the cost of equity is derived from the CAPM model.

WACC Approach with Detailed Planning Period

If an attempt is made to establish equivalence with the APV approach in the case of a detailed planning period and a perpetual annuity with a growth rate, the results with the above formula may be unsatisfactory. One way of achieving equivalence between the APV approach and the WACC is to distinguish between the present value of the tax shield and the value of the tax shield. The present value is determined with the interest rate on borrowed capital. The value is first discounted using the borrowing rate (existing debt) and then using the cost of equity (new debt). For a derivation and reconciliation, please refer to Massari et al. (2008).

However, the fact that the tax shield for existing debt is discounted using the cost of debt and that of new debt using the cost of equity is not very convincing to me, especially in view of the theoretical basis of the equilibrium model of Modigliani and Miller (1958).

For equivalence between the APV approach and the WACC approach, the growth rate of the tax shield must be taken into account. WACC stands for the WACC interest rate, i for the cost of equity (debt), i_f for the cost of debt, ω_S for the growth rate of the tax shield and t is a running variable for time.

In practice, the cost of equity for WACC is not calculated as a reformulation of the APV approach, but is derived from the CAPM model.

WACC Approach Calculator

Here you have the option of using a calculator for the WACC approach. To do this, enter the cost of equity, the cost of debt, the equity-financed net cash flow, the present value of debt, the value of equity, the company value and the present value of the tax shield. You can enter the values as a series of figures (vectors). You can also enter a growth rate for the planning horizon. The last input field is optional. If the field remains empty, a perpetual annuity is calculated on the planning horizon. If you enter a number, a present value with the given years is taken into account.

In the following, the subjective earnings value method will be compared to the DCF method (APV), and an example will be given for both.

The data are from the initial scenario. An inflation and thus a growth of the net cash flow of 2% is assumed, resulting in a perpetual growth rate at the planning horizon. The interest rate is 5% but must be taxed at 25% in private assets, so the net interest rate is 3.75%. No taxes are due on the profit from the sale of the company.

The seller perceives the expectations about the company's development as subjectively certain. Furthermore, he is risk-averse and wants to invest the proceeds from the sale of the company in the bank.

Excursus: Risk can generally be taken into account either via the discount factor (risk premium method) or via an adjustment in the cash flow (certainty equivalent method). Both can be mathematically converted into each other (Terstege, 2023). It seems sensible to me to capture it as an adjustment in the cash flow using various scenarios. This simplifies a correct tax calculation, as it is based on earnings figures, is often non-linear, and also includes allowances. In addition, one must explicitly think about the risk instead of capturing it indirectly. In the above example, however, the seller sees his forecasts as subjectively certain.

According to the earnings value method, a company value of 7700/(0.0375-0.02) = 440,000 monetary units results. To get the same net cash flow as through the company shares, the seller must invest an amount of 440,000 monetary units in the bank.

If the interest rate or cash flow fluctuates within the planning horizon, the more complicated formula with period-specific discounting must be applied. As an example, it is applied here to the case of the perpetual annuity with a growth rate:

Year 1: 7700 * (1+0.0375)^-1 = 7421.69

Year 2: 7854 * (1+0.0375)^-2 = 7296.50

Year 3: [8011.08/(0.0375-0.02)] * (1+0.0375)^-3 = 425,281.81

The sum also equals 440,000 monetary units.

Earnings Value Method as Base

The calculations show that he receives the same cash flow each year as through his company. Since the cash flow grows perpetually, the invested assets also increase in each period.

Now, the company value is calculated using the Equity approach, which is equivalent to the APV approach from above. The data are taken from the above example of the APV approach and the Equity approach with a perpetual growth rate of 2% in the steady state.

(8700-1000)/(0.1142857142857143-0.02) = 81,666.67 monetary units.

If the interest rate or the cash flow fluctuates within the planning horizon, the more complicated formula with period-specific discounting must also be applied here. As an example, it is applied here to the case of the perpetual annuity with a growth rate:

Year 1: (8700-1000) * (1+0.1142857142857143)^-1 = 6,910.26

Year 2: (8874-1020) * (1+0.1142857142857143)^-2 = 6,325.54

Year 3: [(9051.48-1040.4)/(0.1142857142857143-0.02)] * (1+0.1142857142857143)^-3 = 68,430.87

The sum also equals 81,666.67 monetary units.

The seller sold his company for this amount because he relied on an external consultant. However, he does not invest his money in the capital market but in his local bank at 5% gross interest because he is risk-averse. He tries to receive the same cash flow as through the company. However, this is not possible for him, and his assets decrease year after year. The problem is that the calculation was not made using the endogenous marginal interest rates of the party involved. A value was determined for a fictitious investor in the capital markets. The external consultant simply relied on the textbook formulas known to him without taking into account the individual situation of his client.

DCF Method

The DCF method is based on capital market theoretical explanatory models that do not correspond to real conditions. In the above example, the alternative investment of the bank was the marginal interest rate (endogenous marginal interest rate) and not an interest rate derived from the CAPM model. The extreme example shows that the decision value of the party involved only coincidentally matches the company value determined using a DCF method.

The deviations of the company value according to the DCF method can occur both above and below compared to the earnings value method or functional business valuation.

Brennan, M. J. (1970). Taxes, Market Valuation And Corporate Financial Policy. National Tax Journal, 23(4), 417-427.

Hering, T. (2017). Investment Theory (5th ed.). De Gruyter Oldenbourg.

Hering, T. (2021). Business Valuation (5th ed.). De Gruyter Oldenbourg.

Lintner, J. (1965). Security Prices, Risk, and Maximal Gains from Diversification. The Journal of Finance, 20(4), 587-615.

Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77-91.

Massari, M., Roncaglio, F., & Zanetti, L. (2008). On the Equivalence between the APV and the WACC Approach in a Growing Leveraged Firm. European Financial Management, 14(1), 152–162.

Matschke, M. J., & Brösel, G. (2013). Business Valuation: Functions—Methods—Principles (4th ed.). Springer.

Modigliani, F., & Miller, M. H. (1958). The Cost of Capital, Corporation Finance and the Theory of Investment. The American Economic Review, 48(3), 261-297.

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Myers, S. C. (1974). Interactions of Corporate Financing and Investment Decisions—Implications for Capital Budgeting. The Journal of Finance, 29(1), 1-25.

Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3), 425-442.

Terstege, U., Bitz, M., & Ewert, J. (2023). Investment Calculation Plain & Simple. Springer Gabler.