1 Introduction
You finally got to the DCF stage of company valuation.
In front of you there is a big Excel table, with neatly separated sheets for each stage of the analysis and/or DCF component, just like Koller taught you.
And now, you got to the point of using WACC, which asks you question on ponders of equity and debt.
Given that the company is operating in such and such industry, you head over to prof. Damodaran’s web page, and go to the section of Discount Rate Estimation.
You get your ponders, and you call it a day.
Time for celebration? Not quite.
See, Damodaran & Koller expected that you will be dealing with public, not private equity. Yet, in your professional life, you’re almost always getting private equity for valuation.
Let’s see why you have to change your methodology in determining optimal WACC.
2 Weighted Average Cost of Capital (WACC)
Let’s remind ourselves what WACC is:
The weighted average cost of capital (
WACC) is the rate that a company is expected to pay on average to all its security holders to finance its assets. TheWACCis commonly referred to as the firm’s cost of capital. Importantly, it is dictated by the external market and not by management. TheWACCrepresents the minimum return that a company must earn on an existing asset base to satisfy its creditors, owners, and other providers of capital, or they will invest elsewhere.
WACC can be calculated from the following formula:
\[ WACC = \frac{D}{D + E} \times K_{d} \times (1 - t) + \frac{E}{D + E} \times K_{e}\]
The meaning of symbols are as follows:
- \(E\): market value of equity.
- \(D\): market value of debt.
- \(E + D\): market value of equity and debt.
- \(K_{d}\): cost of debt.
- \(t\): tax rate.
- \(K_{e}\): cost of equity.
I’ve bolded especially relevant parts. WACC asks for the following input: market value of equity and debt, and this is troubling given that we want to get to calculate market value of equity. First of all, how can I supply value of equity if I am trying to calculate it?
Let’s add even more complexity.
3 CAPM
How will you calculate \(K_{e}\)? Plenty of methods exist, but most popular one is CAPM:
\[ K_{e} = R_{f} + \beta{} \times (E(R_{m}) - R_{f}) \]
Where:
- \(R_{f}\): risk-free rate.
- \(E(R_{m})\): expected return of the market.
- \(\beta{}\): sensitivity of the expected excess asset returns to the expected excess market returns.
Let’s dig deeper into \(\beta{}\).
4 \(\beta{}\) (Hamada’s Equation)
There are plenty of \(\beta{}\) to choose from, but most popular one is based on Hamada’s equation:
\[ \beta{}_{L} = \beta{}_{U} \times (1 + (1 - T) \times \frac{D}{E}) \]
Where:
- \(\beta{}_{L}\): levered beta, which goes into the CAPM.
- \(\beta{}_{U}\): unlevered beta that we get from relevant set of Guideline Public Companies (GPC).
- \(T\): tax rate.
- \(D\): market value of debt.
- \(E\): market value of equity.
I like this formula, because it shows that stakeholders will demand more return if company goes into more debt.
But, see the formula one more time: it asks of us market value of debt and equity. We don’t know market value of equity, we’re trying to calculate it.
5 Challenge
How can we solve this circularity? It’s simple, we embrace it.
- Step 01: we randomly choose market value of equity. It could be any positive number, it doesn’t matter.
- Step 02: calculate \(\beta{}_{L}\).
- Step 03: calculate \(K_{e}\) by using
CAPM. - Step 04: calculate
WACC, by using market value of equity from Step 01. - Step 05: get the FMV of the company from
DCF. - Step 06: go back to Step 01, with FMV from Step 05.
Do this until:
\[ FMV_{Step_{1}} = FMV_{Step_{5}}\]
For reference purposes, formula of DCF is:
\[ PV = \frac{CF_{1}}{(1 + r)^{1}} + \frac{CF_{2}}{(1 + r)^{2}} + \ldots{} + \frac{CF_{n}}{(1 + r)^{n}}\]
Of course, I will not be going into \(CF_{n}\), but I will call it a day with Gordon’s growth model.
6 Example
After importing relevant modules, let’s define our free cash flows:
Let’s define our starting parameters:
FMV_EQUITY : 1,000,000 EUR
FMV_DEBT : 1,000,000 EUR
RETURN_DEBT : 5.00%
BETA_UNLEVERED : 1.20000
TAX_RATE : 20.00%
RISK_FREE_RATE : 5.00%
EQUITY_RISK_PREMIUM : 15.00%
SIZE_PREMIUM : 3.50%
GROWTH_RATE : 2.00%
CASH_BALANCE_SHEET : 100,000 EURNow, we go into the iterating procedure. First, we define our starting \(\beta{}_{L}\):
Acme_Beta = BetaHamada(
e=_const_data["FMV_EQUITY"],
d=_const_data["FMV_DEBT"],
t=_const_data["TAX_RATE"],
bu=_const_data["BETA_UNLEVERED"]
)This gives us \(\beta{}_{L}\) of 2.16. Next step is to calculate \(K_{e}\) with CAPM methodology. In version of CAPM below, I will incorporate size premium (smaller companies are more risky, and by the size of these cash flows, we can safely conclude that):
Acme_Capm = CapmModified(
rf=_const_data["RISK_FREE_RATE"],
bl=Acme_Beta.levered(),
erp=_const_data["EQUITY_RISK_PREMIUM"],
sp=_const_data["SIZE_PREMIUM"]
)\(K_{e} = 40.90%\)% Yeah, a risky company, and these kind of rates are not unusual for small companies. On to WACC:
Acme_WACC = Wacc(
e=_const_data["FMV_EQUITY"],
coe=Acme_Capm.coe,
d=_const_data["FMV_DEBT"],
cod=_const_data["RETURN_DEBT"],
t=_const_data["TAX_RATE"]
)
print_dictionary(Acme_WACC.get_report())FMV Equity : 1,000,000 EUR
FMV Debt : 1,000,000 EUR
Enterprise Value : 2,000,000 EUR
Cost of Equity : 40.90%
Weight - Equity : 50.00%
Cost of Equity (Weighted): 20.45%
Cost of Debt : 5.00%
Tax Rate : 20.00%
Cost of Debt (Net) : 4.00%
Weight - Debt : 50.00%
Cost of Debt (Weighted) : 2.00%
WACC : 22.45%Finally, we enter the DCF:
Acme_DCF = DCF(
cf=cf.values,
r=Acme_WACC.get_wacc(),
g=_const_data["GROWTH_RATE"],
cash=_const_data["CASH_BALANCE_SHEET"],
mid=True,
debt=_const_data["FMV_DEBT"]
)
# for reference purposes:
ITERATION_ZERO_FMV = Acme_DCF.get_fmv()
ITERATION_ZERO_WACC = Acme_WACC.get_wacc()Currently, our company is worth 356,550.14 EUR. But, let’s do basic sensitivity analysis:
Unfortunately, we are not done with valuation. If I use calculated FMV for calculating new \(\beta{}_{L}\), I will get something completely different:
OldBeta: 2.16000
NewBeta: 3.89247This means that I will have to repeat this methodology until I get no more differences. Here is the history:
| Iteration | FMV | BL | CAPM_COE | WACC | FMV_DEBT | EV | PERC_EQUITY |
|---|---|---|---|---|---|---|---|
| 1 | 1,000,000 | 2.16000 | 40.9000% | 22.4500% | 1,000,000 | 2,000,000 | 50.00% |
| 2 | 356,550 | 3.89247 | 66.8870% | 20.5290% | 1,000,000 | 1,356,550 | 26.28% |
| 3 | 564,469 | 2.90071 | 52.0107% | 21.3225% | 1,000,000 | 1,564,469 | 36.08% |
| 4 | 472,491 | 3.23178 | 56.9767% | 20.9991% | 1,000,000 | 1,472,491 | 32.09% |
| 5 | 508,856 | 3.08659 | 54.7988% | 21.1317% | 1,000,000 | 1,508,856 | 33.72% |
| 6 | 493,769 | 3.14423 | 55.6634% | 21.0775% | 1,000,000 | 1,493,769 | 33.06% |
| 7 | 499,908 | 3.12035 | 55.3053% | 21.0997% | 1,000,000 | 1,499,908 | 33.33% |
| 8 | 497,390 | 3.13008 | 55.4511% | 21.0906% | 1,000,000 | 1,497,390 | 33.22% |
| 9 | 498,420 | 3.12609 | 55.3913% | 21.0943% | 1,000,000 | 1,498,420 | 33.26% |
| 10 | 497,998 | 3.12772 | 55.4158% | 21.0928% | 1,000,000 | 1,497,998 | 33.24% |
| 11 | 498,171 | 3.12705 | 55.4058% | 21.0934% | 1,000,000 | 1,498,171 | 33.25% |
| 12 | 498,100 | 3.12732 | 55.4099% | 21.0932% | 1,000,000 | 1,498,100 | 33.25% |
| 13 | 498,129 | 3.12721 | 55.4082% | 21.0933% | 1,000,000 | 1,498,129 | 33.25% |
| 14 | 498,117 | 3.12726 | 55.4089% | 21.0932% | 1,000,000 | 1,498,117 | 33.25% |
| 15 | 498,122 | 3.12724 | 55.4086% | 21.0932% | 1,000,000 | 1,498,122 | 33.25% |
| 16 | 498,120 | 3.12725 | 55.4087% | 21.0932% | 1,000,000 | 1,498,120 | 33.25% |
| 17 | 498,121 | 3.12724 | 55.4087% | 21.0932% | 1,000,000 | 1,498,121 | 33.25% |
| 18 | 498,120 | 3.12725 | 55.4087% | 21.0932% | 1,000,000 | 1,498,120 | 33.25% |
| 19 | 498,120 | 3.12724 | 55.4087% | 21.0932% | 1,000,000 | 1,498,120 | 33.25% |
| 20 | 498,120 | 3.12724 | 55.4087% | 21.0932% | 1,000,000 | 1,498,120 | 33.25% |
So, I started with the same FMV value of debt and equity. These could have been my so-called target capital structure. While this might make sense for public equity,1 for private equity target capital structure does not make any sense precisely because it’s share prices are not public/visible.
This iteration process does not depend on starting value of FMV: I could easily put 100,000,000 EUR, and the final value would have been the same.
But, this process does depend on:
- Cash Flows: larger cash flows \(\rightarrow{}\) less risk (all else constant).
- Level od debt: bigger debt \(\rightarrow{}\) more risk (all else constant).
We can repeat iteration process for various level of debt, and see what would final values look like:
| FMV | BL | CAPM_COE | WACC | FMV_DEBT | EV | PERC_EQUITY |
|---|---|---|---|---|---|---|
| 1,019,023 | 1.24710 | 27.2066% | 26.1211% | 50,000 | 1,069,023 | 95.32% |
| 992,538 | 1.29672 | 27.9508% | 25.7586% | 100,000 | 1,092,538 | 90.85% |
| 965,933 | 1.34908 | 28.7362% | 25.4112% | 150,000 | 1,115,933 | 86.56% |
| 939,211 | 1.40443 | 29.5664% | 25.0780% | 200,000 | 1,139,211 | 82.44% |
| 912,374 | 1.46305 | 30.4457% | 24.7579% | 250,000 | 1,162,374 | 78.49% |
| 885,427 | 1.52527 | 31.3790% | 24.4501% | 300,000 | 1,185,427 | 74.69% |
| 858,373 | 1.59144 | 32.3716% | 24.1539% | 350,000 | 1,208,373 | 71.04% |
| 831,213 | 1.66198 | 33.4296% | 23.8684% | 400,000 | 1,231,213 | 67.51% |
| 803,952 | 1.73735 | 34.5602% | 23.5932% | 450,000 | 1,253,952 | 64.11% |
| 776,592 | 1.81809 | 35.7713% | 23.3275% | 500,000 | 1,276,592 | 60.83% |
| 749,135 | 1.90481 | 37.0722% | 23.0708% | 550,000 | 1,299,135 | 57.66% |
| 721,586 | 1.99824 | 38.4736% | 22.8226% | 600,000 | 1,321,586 | 54.60% |
| 693,945 | 2.09921 | 39.9881% | 22.5824% | 650,000 | 1,343,945 | 51.63% |
| 666,216 | 2.20868 | 41.6302% | 22.3499% | 700,000 | 1,366,216 | 48.76% |
| 638,402 | 2.32782 | 43.4173% | 22.1245% | 750,000 | 1,388,402 | 45.98% |
| 610,503 | 2.45798 | 45.3697% | 21.9059% | 800,000 | 1,410,503 | 43.28% |
| 582,524 | 2.60080 | 47.5120% | 21.6938% | 850,000 | 1,432,524 | 40.66% |
| 554,466 | 2.75826 | 49.8739% | 21.4879% | 900,000 | 1,454,466 | 38.12% |
| 526,331 | 2.93275 | 52.4913% | 21.2878% | 950,000 | 1,476,331 | 35.65% |
| 498,120 | 3.12724 | 55.4087% | 21.0932% | 1,000,000 | 1,498,120 | 33.25% |
| 469,837 | 3.34542 | 58.6813% | 20.9040% | 1,050,000 | 1,519,837 | 30.91% |
| 441,483 | 3.59194 | 62.3790% | 20.7199% | 1,100,000 | 1,541,483 | 28.64% |
| 413,060 | 3.87273 | 66.5910% | 20.5405% | 1,150,000 | 1,563,060 | 26.43% |
| 384,569 | 4.19556 | 71.4334% | 20.3658% | 1,200,000 | 1,584,569 | 24.27% |
| 356,013 | 4.57067 | 77.0600% | 20.1956% | 1,250,000 | 1,606,013 | 22.17% |
| 327,392 | 5.01195 | 83.6792% | 20.0295% | 1,300,000 | 1,627,392 | 20.12% |
| 298,707 | 5.53870 | 91.5805% | 19.8675% | 1,350,000 | 1,648,707 | 18.12% |
| 269,960 | 6.17851 | 101.1777% | 19.7094% | 1,400,000 | 1,669,960 | 16.17% |
| 241,151 | 6.97231 | 113.0846% | 19.5550% | 1,450,000 | 1,691,151 | 14.26% |
So, as percentage of equity drops down, shareholders are asking for bigger and bigger return, since the company is getting more risky. Debt obligations are fixed, and defaulting those leads to forced liquidation of the company.
Picture is worth a thousand words, I guess.2
Do note that I am not a fan of CAPM (far better methods exist) due to it’s methodological weaknesses and it’s abuse in practice, but if you want to use CAPM method in valuation, you have to incorporate this process. I’ve seen valuation reports where final FMV value would have been 50% less than stated (implying overvaluation by 100%) if this process had been implemented.
7 References
- Damodaran, A. (2012b, April 17). Investment Valuation: Tools and Techniques for Determining the Value of any Asset, University Edition (3rd ed.). Wiley.
- McKinsey & Company Inc., Koller, T., Goedhart, M., & Wessels, D. (2020b, June 30). Valuation: Measuring and Managing the Value of Companies (Wiley Finance) (7th ed.). Wiley.
- Hitchner, J. R. (2017b, May 1). Financial Valuation: Applications and Models (Wiley Finance) (4th ed.). Wiley.
- Abrams, J. B. (2010b, March 29). Quantitative Business Valuation: A Mathematical Approach for Today’s Professionals (2nd ed.). Wiley.