No sailor, plotting their position, will risk their life on one sighting. Navigating the choppy and dark waters of finance is just the same. In this article I show how multiple measures of the cost of equity capital can be brought together to increase confidence in the ultimate choice of rate.

Most practitioners use the Capital Asset Pricing Model to establish the cost of equity. Many academic studies have declared the death of the CAPM and other models such as the Fama and French 3F model and the Arbitrage Pricing Theory model have vied for its crown. However none of these methods offer significant advantages over the simplicity of the CAPM but, it is well recognised that the CAPM can be unreliable, especially for smaller firms.

My chosen company is Disney (DIS) which is a multinational corporation publishing its accounts under USGAAP. Getting the principles right for a large firm is the first, albeit small step, towards estimating the equity cost of capital for smaller less actively traded companies. To do the estimation for Disney I have home-grown all the data inputs. My concern with using public data sources is that they can be quite opaque in explaining their measurement bases and when they do the results are not acceptable.

The Disney Equity Cost of Capital – 1st bearing

When estimating the Disney beta I undertook a five year regression of the monthly returns on the stock against the returns on a broadly based index. The theory of CAPM suggests that the most representative index of the whole market should be used. With Disney I opted for the S&P500 (for a UK company I would invariably use the FTSE All Share Index).

There are a number of problems using betas derived from historical data. Over a five year time series the implied point of estimation has a 2.5 year lag. My first correction is to ungear and regear the beta to the company’s current gearing using market caps. The next step is to correct for mean reversion. Disney’s raw beta was close to one and thus any such correction will be tiny but in principle it should be done. These two adjustments gave a corrected beta of 1.09.

To check the accuracy of my result I re-computed beta using implied volatility estimates and the ratios of the skew of the returns on the company versus the skew on the returns on the index. This more forward looking measure was very close to my corrected beta and so I was able to carry a figure of 1.09 into the CAPM estimate without too many concerns. The R2 on the beta estimate revealed that 65.75% of the risk of Disney was driven by the market and the remainder from non-market sources.

There is much debate whether the yield on short term Treasury Bills or long term Treasury Bonds should be used for the risk free rate. With a beta close to one it doesn’t matter providing the rate is used consistently in the estimation of the equity risk premium. The principal differences between the two rates arise because of various types of market risk and the long term rate reflects the reward for that risk. However, the beta is the means by which market risk is captured, and using the long rate double counts that element of risk. So, being a purist and following the methods of the pioneers of the subject I opt for the short rate (currently 0.15%).

The most difficult input into the CAPM is the equity risk premium. With the current turmoil in the markets historical estimates are likely to be unreliable. To get around this used a forward looking approach implying the market return from the dividend yield on the index (1.93%) and an estimate (from the Congressional Budget Office) of US GDP levelling out to 4.1% in the years 2015-2020. Putting it all together my estimate of the market required rate of return was 6.11% and my cost of equity capital was 6.65%.

The Disney Equity Cost of Capital – 2nd bearing

To corroborate this number we should look for a measurement basis that is different to the CAPM. One less well known method is based upon the Market Derived Capital Pricing Model (MCPM) . This model relies upon the concept of risk management through hedging. The MCPM includes a risk adjustment reflecting the premium an investor would expect to pay to ensure that their returns do not fall below that offered on Disney’s bonds. Presumably no rational investor would be willing to accept a rate of return less than the yield on the company’s debt.

In order to implement the MCPM we need to know the following: the yield on Disney’s corporate debt at its average term to maturity. Currently the average term to maturity is approximately 10 years with a yield of 3.11%. Disney’s dividend yield is 1% with a share price of $34.64. Finally we need the volatility of the company’s equity. The historical volatility over 100 days, terminal weighted, is 22.5%. This agrees closely with the implied volatility on near the money calls for the company.

Here are the steps to establish the MCPM rate:

Step 1:

Estimate the forward stock price which would just give investors the same rate of return as that required by the debt investors. With yield on the debt of 3.11% and dividend yield of 1% the potential capital gain required over the life of the debt is 2.11%. Compounding the current price over 10 years at 2.11% gives a price at maturity of $42.68. This is the exercise price required in 10 years to lock in a minimum return of 3.11% overall.

Step 2

Estimate the volatility of the equity either from the Black Scholes model or using historical estimates. In this case I opted for the historical estimate of 22.5%.

Step 3

Estimate the value of the limited liability protection offered by the implied put option using either the Black and Scholes model or a numeric modelling technique over the term to maturity. Because, there is nothing to stop the equity investor closing their equity account early I have modelled the American style option price using a 100 time step binomial model. So, the inputs are: spot price $34.64, required exercise price $42.68, term to maturity 10 years, risk free rate (the corporate debt rate) 3.11%, volatility 22.5% and dividend yield 1%.

The valuation model values a 10 year put option at $11.18.

Step 4

Divide the put value by the current share price and annualise this figure by further dividing by the 10 year annuity at 3.11% (8.48). The result (3.8%) is the additional return over the corporate debt rate required to compensate for the equity risk involved.

Thus the cost of equity capital equals the rate of return on the company’s debt (3.11%) plus the additional return of 3.80%. Overall 6.91% which is just 26 points different from the CAPM method.

The corroboration – 3rd bearing

On the basis of 44 study groups each working to provide an estimate of the next 5 years EPS and DPS for Disney we arrived at a consensus result of:

Year 1 2 3 4 5

Earnings per share 2.06 2.26 2.53 2.76 3.07

Dividend per share 0.44 0.46 0.49 0.52 0.57

Using a simple stepped valuation model we came to the following results: CAPM valuation = $37.27, MCPM valuation = $35.59. Split the difference = $36.41.

This result is clearly very close to the current share price of Disney. Given the EPS/DPS were conducted at different points over the preceding 6 months I also collected the share price recorded at the date of each analysis and took the average. The answer $36.65!

Conclusion

We have in effect found three compass points to locate Disney’s cost of equity. First, the CAPM, second the option based MCPM and finally the market implied rate. Or, to put it another way: using two methods based upon the CAPM and the MCPM gives a nearly perfect estimate of the equity price. Following the procedure described above on a large number of publicly quoted companies over the years has convinced me of two things: one estimate of the cost of equity is not enough, and the problem is less often the models, more often the data.

Ps: Many thanks to the splendid efforts of the financial analysis teams at Manchester Business School who provided the forecasts.

## 6 comments:

Dear Mr. Bob

Reading the original article from Mc. Nulty et. al., we observe that their view of risk differs significantly from the Sharpe model (CAPM). They claim that investors are not diversified and hence idiosyncratic risk has to be priced or equivalently, that the management of a company has to take into account all the risks in ongoing investments if not accounted otherwise. Due to the portfolio selection approach in the CAPM, only systematic risk has to be priced there. Hence, it is purely accidentally that both methods (MCPM or CAPM) yield similar results. I would appreciate if you could make it clearer in your post.

Thanks

An interesting comment. I do not think McNulty et al provided a full and thorough justification for their method. I also think there are flaws with their technique which I and other researchers are looking at. A good starting point in resolving this issue is to consider the relationship between the OPM and the CAPM (Fischer Black showed how to derive the latter from the former and Cessari and D-Adda show in A Simple Approach to CAPM, Option Pricing and Asset Valuation how both models are reducible to a commen intertemporal general equilibrium valuation model).

At an intuitive level, and leaving theory aside, I think the difference is this: CAPM is an expectations model of course. What the McNulty model does is to reveal the return implied by real prices in competitive markets: the actual bond yield, the current share price and the current value of a put option with the same TTM is the outstanding bond. The actual share price is being formed in a market where non-market risk is not priced so that is being captured by the McNulty model in determining the premium over the bond yield that an investor is willing to pay. Have a think on that and see if it makes sense.

Prof. Bob,

Thank you for your post on cost of equity calculation. I was wondering how what time frame you use to calculate skew ratios? Also, I assume that the skew ratio can be greater than 1 but some papers call approximate that as the correlation between the index and the stock. If you have an example, that would be awesome. If you can also provide an example of calculating option implied beta, that would be very helpful also. Thank you again for the post as it cleared a few things up for me as far as MCPM is concerned.

Sincerely,

Anish

Hi Prof Bob

I attended a short seminar of yours through PMI in September 2015.

I working on a weekly essay for my MSc in Project Management and would like to reference you triagulation of CoC. Do you have any paper or books I can access through my institutions Eselvier etc account?

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