## Determination of market-wide implied cost of capital

### 1. Implied cost of capital – The very basics

The value of a company is equal to the discounted value of the dividend payments ("Dividend Discount Model"). Using three years of explicit dividend forecasts and a constant-growth assumption from year 4 on, the market value MV_{0} can be written as:

(1)

where k is the implied cost of capital and

- MV
_{0}: Current market value - D
_{1}, D_{2}, D_{3}: 1-/2-/3-year ahead dividend forecasts - g: Long-term growth rate

All we need to estimate implied cost of capital are estimates for these three input parameters: The current market value, dividend forecasts and a long-term growth rate.

### 2. Long-term growth rate – The very basics

A lot of discussions on implied cost of capital centers around the long-term growth rate. Naively applied, it can have a huge impact on implied cost of capital estimates. For example, if the current market value is MV_{0}=100 and dividend forecasts are D_{1}=4, D_{2}=4, D_{3}=4 then a growth rate of 0% results in an implied cost of capital of 4%, if the growth rate assumption is 5%, the implied cost of capital is 8.6%.

However, growth cannot come from nothing, in particular not in the long-run. Let us assume the earnings forecast for year 3 is E_{3}=4. In this case, the dividend forecast assumes a payout ratio of 100%. It seems unreasonable that the company can grow by 4% and, at the same time, pay out 100% of its earnings. Such a company would very quickly end up having an extremely high profitability. If the company started with a book value of BV_{0}=40 so that the return on equity is 10%. After 10 years, return on equity would grow to 16%, after 50 years to 115% and after 100 years return on equity would be above 1,000%. This is unlikely to happen as competition would certainly erode these high returns. We will make a very simple assumption: Payout ratios and growth rates from year 3 on must be consistent:

(2)

The left-hand side of equation (2) is the retention in percent of year-2 book value of equity. Our assumption means that earnings and dividends cannot grow faster than book values over the long-run. This assumes that return on equity will stay at the level it reached reached directly before the terminal value period started. Equation (1) then simplifies to

(3)

We describe and discuss equation (2) in more detail in the separate document "Long-run growth rates".

### 3. Implied cost of capital on the market level

How can we apply equation (1) to estimate implied cost of capital for whole markets? There are two possibilities, one that is frequently applied in the academic literature, and one that we prefer

1. Determine implied cost of capital for each company using (1), and then take a weighted average of these estimates.

2. Aggregate the input parameters across all companies, i.e. determine the total $-amount of dividends and the total market capitalization of all companies. Apply equation (1) to these aggregate values.

We use the latter approach for two reasons: First, estimates are better. It can be shown that the first approach inhibits a bias in the estimation of the market wide implied cost of capital. Second, results using the latter approach are much easier to interpret. The following table provides an overview of aggregate statistics for the German market as of March, 31st, 2013:^{1}

MV_{0} |
BV_{0} |
BV_{2} |
D_{1} |
D_{2} |
E_{3} |
---|---|---|---|---|---|

€ 776 bn | € 484 bn | € 568 bn | € 25 bn | € 27 bn | € 79 bn |

Applying equation (3) using g=0% results in implied cost of capital of 9.14%. The 10-year German government bond yield was 1.28% as of end-of-March 2013, resulting in an implied equity risk premium of 7.86%. Investors who are more skeptical might also want to apply the most pessimistic dividend and earnings forecast across all analysts. If we do that, dividends forecast reduce to D_{1} = € 21bn and D_{2} = € 22bn, the three-year ahead earnings forecast reduces to € 63bn, the two-year ahead book value forecast to € 549bn. This results in an implied cost of capital estimate of 7.37% and an equity premium of 6.09%.

^{1} These numbers are based on free-float adjusted and are based on all companies for which sufficient analyst forecast data is available. The total market capitalization (€ 776bn) is therefore somehow lower than the total market capitalization of all stocks traded in the German market.

### 4. Documents

This document has provided an overview on our philosophy of how to estimate implied cost of capital. The following documents describe each of the important topics in more detail- Methodology: Dividend discount model versus residual income model
- Methodology: Long-run growth rates
- Sample selection (Criteria for inclusion/exclusion of companies, Free float adjustment, Treatment of different stock classes)
- Market value and dividend, earnings and book value forecasts