# Level 1 CFA® Exam:

Equity Valuation - Introduction

In this lesson, we're going to discuss valuation of financial instruments. First, we briefly present the main categories of equity valuation models and then we move on to talk about the first group, namely dividend discount models.

The most important aspect in financial instrument valuation is the ratio of the intrinsic value estimated by analysts to the market value. If the former exceeds the latter, we say that a security is undervalued. If the intrinsic value is lower than the market value, we deal with the situation in which the security is overvalued.

You probably know that the valuation of a security is complicated and affected by many factors that can distort the final result. With high uncertainty, the difference between a market value and an intrinsic value must be big enough to allow us to recognize that a financial instrument is overvalued or undervalued.

An analyst valued the shares of PIT Company whose current price per share is USD 21 and she estimated their value at USD 20.6 per share. Can we conclude that the company is undervalued or overvalued based on such data? What steps should an investor take?

Based on the ratio of the intrinsic value to the market value, we can say that the company is slightly overvalued. It is so because the market price of USD 21 is higher than the intrinsic value. However, in this case, the difference is so small that entering into an appropriate transaction by long-term investors based on such an analysis would be very risky. What's more, valuation is burdened with many assumptions, which in turn affect the results. In such a case, it's good to look at the conclusions of other analysts and refrain from entering into transactions immediately.

We distinguish 3 categories of such models:

- multiplier models (aka. market multiple models),
- asset-based valuation models, and
- present value models (aka. discounted cash flow models).

Multiplier models use either:

- multiples, such as share price to earnings or share price to sales per share, which depend on the share price, or
- multipliers like EV/EBITDA and EV/total revenue, which depend on the enterprise value.

To value the company using a multiplier model, we simply multiply the value of a given multiplier by a given value of the company’s financial statements. For example, if we assume that the price-to-earnings ratio for a given company is 10, and the company’s net income from the financial statement is equal to USD 20 million, we will estimate the company value as 10 times USD 20 million, that is USD 200 million.

In asset-based valuation models the intrinsic value of a company is the difference between its assets and liabilities minus the value of preferred shares issued by the company.

Present value models are based on an assumption that the value of an instrument is equal to the present value of the instrument’s future benefits or free cash flows that are to be distributed to shareholders.

The most important present value models are:

- the dividend discount models, and
- the free-cash-flow-to-equity valuation model.

The dividend discount models are based on a basic assumption stating that financial market participants expect future benefits when they defer consumption. And so, we can say that the value of an instrument is equal to the present value of future benefits expected by the investor. In the case of the dividend discount models, the future benefits are dividends.

Therefore, the dividend discount model has the following formula that you can use in your level 1 CFA exam:

\(V_{0} = \Sigma^{\infty}_{t=1} \frac{D_{t}}{(1+r)^{t}}\)

- \(V_{0}\) - value of a share of stock today, at t=0
- \(D_{t}\) - expected dividend in year "t", assumed to be paid at the end of the year
- \(r\) - required rate of return on the stock

This formula assumes that we will never sell a share.

Now, imagine that we want to compute the value of the share if we assume that we get the dividends for N years and we sell the stock after year N. How can we compute this value?

Notice that the income from shares includes both the proceeds from their sale and the dividend. If an investor buys a share and after a year decides to sell it, the formula looks like this:

\(V_{0} = \frac{D_{1} + P_{1}}{(1+r)^{1}} = \frac{D_{1}}{(1+r)^{1}} + \frac{P_{1}}{(1+r)^{1}}\)

- \(V_{0}\) - value of a share of stock today, at t=0
- \(D_{1}\) - expected dividend in year 1, assumed to be paid at the end of the year
- \(r\) - required rate of return on the stock
- \(P_{1}\) - expected price per share at t=1

In this formula, \(P_1\) is the expected price at the end of Year 1 and is equal to:

\(P_{1}=\frac{D_{2}+P_{2}}{1+r}\)

Thus, the value of the share today equals:

\(V_{0}=\frac{D_{1}}{(1+r)^{1}}+\frac{D_{2}+P_{2}}{(1+r)^{2}}\)

We can repeat the procedure and then we will get the formula for the present value of a share for 'n' holding periods (years):

\(V_{0} = \Sigma^{n}_{t=1} \frac{D_{t}}{(1+r)^{t}} + \frac{P_{n}}{(1+r)^{n}}\)

- \(V_{0}\) - value of a share of stock today, at t=0
- \(D_{t}\) - expected dividend in year "t", assumed to be paid at the end of the year
- \(r\) - required rate of return on the stock
- \(P_{n}\) - value of the expected price at the end of year "n"

An investor purchased a share of RDG Company, which he'll sell after three years at an expected price of USD 16.75. During the holding period, he'll receive annual dividends three times. Their expected values are USD 1.10, USD 0.60, and USD 1.70, respectively. If the expected rate of return is 7%, what is the present value of the share?

We will solve the problem in 3 different ways.

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Please remember that if there is an investor with a certain investment horizon, the present value of a financial instrument is directly affected by the dividends received before the sale of the shares and it is indirectly affected by the expected dividends to be paid after the sale because these determine the selling price of the instrument.

In the dividend discount models, we deal with the problem of an infinite series of expected dividends, which in reality is difficult to predict. One of the solutions is the Gordon growth model which assumes that dividends grow indefinitely at a constant rate:

\(V_{0} = \frac{D_{0}\times (1+g)}{r-g} = \frac{D_{1}}{r-g}\)

- \(V_{0}\) - value of a share of stock today, at t=0
- \(D_{0}\) - current dividend
- \(r\) - required rate of return on the stock
- \(g\) - dividend expected growth rate
- \(D_{1}\) - expected dividend in year 1, assumed to be paid at the end of the year

The share value is equal to the dividend at the end of the first year divided by the required rate of return less the constant growth rate of the dividend.

How can we interpret the constant growth rate of the dividend?

If for example, a company paid a dividend today that amounts to USD 100 per share (\(D_0=100\)) and the constant growth rate of the dividend (\(g\)) is 5%, it means that the expected dividend at the end of the first year (\(D_1\)) will be equal to:

\(D_1=D_0\times(1+g)=100\times1.05=105\)

It also means that the value of the expected dividend at the end of the second year (\(D_2\)) will be equal to:

\(D_2=D_0\times(1+g)^2=100\times1.05^2=110.25\)

And so on.

Andrea Bassini, an Italian investor, expects that a dividend at the end of Year 4 will amount to USD 4.5. What is the expected value of a dividend at the end of Year 1 and Year 9? Assume that the constant growth rate of the dividend amounts to 6%.

(...)

The constant growth rate of the dividend may be estimated using return on equity and earnings retention rate:

\(g = b \times ROE=\frac{E-D}{E}\times{ROE}=(1-\frac{D}{E})\times{ROE}\)

- \(g\) - dividend growth rate
- \(b\) - earnings retention rate
- \(ROE\) - return on equity
- \(E\) - earnings
- \(D\) - dividend
- \(\frac{D}{E}\) - dividend payout ratio

The Gordon growth model assumes that dividends grow at a constant rate, which means that it can only be applied in the case of companies that pay dividends regularly.

The dividend on a share paid at the end of the last year is USD 4 and the expected rate of return is 8%. If ROE is 12% and the earnings retention rate is 60%, what is the value of the share according to the Gordon growth model?

(...)

If the rate of return on a company's equity is 12% and the growth rate of the dividend is 3%, then what is the dividend payout ratio?

(...)

After some practical problems, it’s time we had a look at the theoretical aspect of the Gordon growth model in more detail:

The assumptions of the Gordon growth model are as follows:

- Dividends are useful in the valuation process.
- Dividends are characterized by a constant growth rate.
- The required rate of return does not change.
- The dividend growth rate is lower than the required rate of return.

It may happen that a company does not pay dividends or that the Gordon growth model may not reflect all factors that influence the company's characteristics. An analyst may then use one of the alternatives, namely:

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\(V_{0} = \Sigma^{n}_{t=1} \frac{D_{0}\times (1+g_{s})^{t}}{(1+r)^{t}} + \frac{V_{n}}{(1+r)^{n}}\)

- \(V_{0}\) - value of a share of stock today, at t=0
- \(r\) - required rate of return on the stock
- \(D_{0}\) - current dividend
- \(g_{S}\) - short-term dividend growth rate
- \(V_{n}\) - intrinsic value per share in year "n"

\(V_{n} = \frac{D_{n+1}}{r-g_{L}}\)

- \(V_{n}\) - intrinsic value per share in year "n"
- \(D_{n+1}\) - dividend in year "n+1"
- \(r\) - required rate of return on the stock
- \(g_{L}\) - long-term dividend growth rate

\(D_{n+1} = D_{0}\times (1+g_{S})^{n}\times (1+g_{L})\)

- \(D_{n+1}\) - dividend in year "n+1"
- \(D_{0}\) - a level dividend
- \(g_{S}\) - short-term dividend growth rate
- \(g_{L}\) - long-term dividend growth rate
- \(n\) - moment of time

We can expand dividend discount models so they could reflect the characteristics of companies even more accurately.

In the three-stage DDM model, we distinguish among three stages: growth, transition, and maturity. The growth phase is characterized by a high growth rate which is followed by a lower one. In the third period, there is a sustainable growth rate into perpetuity. This model is more appropriate for companies in their early stages, unlike the two-stage model, which is suitable for valuing more developed companies.

As mentioned in one of the previous lessons, preferred shareholders enjoy certain rights before common shareholders. The value of preference shares is given by the following formula to be used in your level 1 CFA exam:

\(V_{0} = \frac{D}{r}\)

- \(V_{0}\) - present value of a preferred stock
- \(D\) - constant dividend paid at the end of the consecutive periods
- \(r\) - constant required rate of return over period

If you have already watched our Quantitative Methods videos, you will probably recognize the formula. It is the formula for the present value of the perpetuity. This formula is also the same as the formula for the Gordon growth model if we assume that \(g=0\).

We're going to consider a non-callable perpetual preferred stock. Its par value is USD 100. The share pays an annual dividend of USD 4 and the required rate of return is 6%. What is the value of the share?

\(V_{0}=\frac{D}{r}=\frac{4}{6\%}=66.67\)

For a holding period of 'n' years, the value of the preference shares is calculated according to the formula:

\(V_{0} = \Sigma^{n}_{t=1} \frac{D_{t}}{(1+r)^{t}} + \frac{F}{(1+r)^{n}}\)

- \(V_{0}\) - estimated intrinsic value for a preferred stock
- \(D_{t}\) - expected dividend in year "t", assumed to be paid at the end of the year
- \(r\) - required rate of return on the stock
- \(F\) - preferred stock's par value

Another valuation model used especially in the case of companies that don't pay dividends regularly is the free-cash-flow-to-equity valuation model (FCFE), which accounts for cash flows available to shareholders after all the company's liabilities are paid. Such cash flows are given by the following formula:

\(FCFE = CFO - FCInv + B_{net}\)

- \(FCFE\) - free-cash-flow-to-equity
- \(CFO\) - net income plus non-cash expenses minus investment in working capital
- \(FCInv\) - fixed capital investment
- \(B_{net}\) - net borrowing

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- We distinguish among different groups of valuation models: discounted cash flow models, multiplier models, and asset-based valuation models.
- Multiplier models use either multiples which depend on the share price or multiples which depend on the enterprise value.
- To value the company using a multiplier model, we simply multiply the value of a given multiplier by a given value of the company’s financial statements.
- In asset-based valuation models the intrinsic value of a company is the difference between its assets and liabilities minus the value of preferred shares issued by the company.
- Present value models are based on an assumption that the value of an instrument is equal to the present value of the instrument’s future benefits or free cash flows that are to be distributed to shareholders.
- The most important present value models are the dividend discount models (DDMs) and the free-cash-flow-to-equity valuation model.
- One of the DDM models is the Gordon growth model.
- The Gordon growth model assumes that dividends grow indefinitely at a constant rate.
- In the case of fast-growing companies, we use multistage dividend discount models, classified into two-stage and three-stage models.
- In the three-stage DDM model we distinguish among three stages: growth, transition, and maturity. The growth phase is characterized by a high growth rate which is followed by a lower one. In the third period, there is a sustainable growth rate into perpetuity.
- The three-stage DDM model is more appropriate for companies in their early stages, unlike the two-stage model, which is suitable for valuing more developed companies.
- In the case of the FCFE model, the appropriate discount rate is a required rate of return on equity.