Tag Investing

Gordon model (Constant-Growth Valuation Model)

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The Gordon model is one of the models used in dividend valuation. It is very simple, as long as one knows the formula, which is:

P0=D1/(rs-g)

Also sometimes presented as: P = D/(k-g)

Where:

P0 or P – price of the stock

D1 or D – per share dividend expected at the end of year 1 (at the end of the next financial period)

rs or k – required return for equity investor

g – constant growth rate (expected annual growth of dividends)

Gordon model is usually used for mature companies only since it is assumed that annual growth of dividends remains constant.

It is very important to note that if you are only given the current per share dividend (D0, per share dividend received in this financial period), then you will need to adjust it for the next financial period before you can use it in the Gordon model. To do this you will need to take the current dividend and multiply it by (1 + g). The calculation is as follows:

D1=D0*(1+g)

The original equation of the Gordon model (P0=D1/(rs-g)) calculates the price of the share. However, you are looking for the cost of common stock. Therefore, you need to rearrange equation of the Gordon model as follows:

rs = (D1/Po) + g

Now you just plug in the numbers into the adjusted Gordon equation and you will be able to obtain the cost of common stock. Because common stock is paid out of the after-tax earnings, the tax adjustment is irrelevant.

Sometimes it is necessary to find the growth rate (g) first, before you can calculate the cost of the common stock (rs) with the help of the Gordon model. To do so, you need to find out what was the per share dividends applicable to common stock over the last few years (this information will be given). After obtaining this information, you can calculate the growth rate.

It is best to explain this with an example.

EXAMPLE 1:

Calculating the growth rate, which is necessary for usage of the Gordon model:

The information given below is on per share dividends applicable to common stock over the last few years. You need to find the growth rate of dividends over the given period.

Per share dividends from 2005-2010:

2010 – $4

2009 – $3.96

2008 – $3.76

2007 – $3.27

2006 – $3.25

2005 – $3

Now, by using a financial calculator, you can calculate the growth rate as follows:

PV = -3 (per share dividend in 2005, the first year from which per share dividend information is available)

FV = 4 (per share dividend in 2010, the per share dividend in the current period)

N = 5 (number of periods over which growth occurred)

Find I = it will be 5.92% (this number represents growth of dividends over the given period)

EXAMPLE 2:

Using the growth rate (found above) in the Gordon model:

Now, if we know that the growth of the dividends is expected to be the same into the future and the price of the stock is $55, we can compute the cost of common stock (rs) as follows:

rs=(4*(1+0.0592)/55)+0.0592

rs=0.0770+0.0592

rs=0.1362=0.14%

The cost of common stock also represents the return that investors expect to earn from their shares. If the actual return is less – investors will sell their stock.

Test yourself

The ordinary share is currently sold for $40 each. The growth of shares was 10% over the last 5 years and is expected to be the same in the future. A dividend of $3.5 dollars was paid to shareholders in the current period.

REQUIRED: What is the cost of an ordinary share?

SOLUTION:

We need to use the adjusted Gordon model. In other words, we need to use the formula: rs = (D1/Po) + g

Rs=(3.5*(1+.1)/40) +.1

Rs=(3.85/40) +.1

Rs=19.63%

Note that the dividend is adjusted for growth in the next period by multiplying the current dividend by (1+g).

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Finding the specific after-tax cost of common stock (rp)

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Our next concern is to find the after-tax cost of common stock, after attending to finding the after-tax cost of long-term debt and after-tax cost of preferred stock.

Common stock

Common stock, which is also called common shares or ordinary shares, refers to the category of ownership of the enterprise. Common shares generally have voting rights and better potential for appreciation of shares compared to preferred stock.

However, holders of common stock generally do not have fixed dividends and cannot receive dividends until dividends are paid out to preferred stock holders. Moreover, in case of liquidation, holders of common stock only have claim on company’s assets if claims of all creditors as well as holders of preferred stock are satisfied. Therefore, common stock is more risky than preferred stock.

Cost of common stock (rp)

To determine the specific after-tax costs of common stock (rp), you can use two techniques: Gordon model or the CAPM (Capital Asset Pricing Model)

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Finding the after-tax cost of preferred stock (rp)

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After discussing the cost of long-term debt, we must now find the cost of preferred stock (after-tax). Preferred stock, which is also called preferred shares or preference shares, refers to the category of ownership that has preferential claim on earnings and assets of the firm, compared to common stock ownership.

The preferential claim is generally manifested in the fact that dividends cannot be distributed to common stockholders until it is distributed to holders of preferred stock first. Further, in case of liquidation, holders of the preferred stock also have preferential claim on assets of the firm, compared to the holders of common stock.

Preferred stock is a hybrid instrument as it has characteristics of both debt and equity. The drawback of preferred shares, compared to the common stock, is lower potential for appreciation of shares as well as absence of voting rights.

Calculating the cost of preferred stock

To calculate the specific after-tax cost-of-preferred-stock all we need to do is to take the preferred stock dividend and divide it by the net proceeds from the sale of the preferred stock (funds received minus flotation cost).

Cost-of-preferred-stock (rp) = Preferred stock dividend/(Funds received – Flotation costs)

Because preferred stock is paid out of the after-tax earnings, the cost-of-preferred-stock is already after-tax.

EXAMPLE:

If Company A issued 9% preferred stock at $100 and the flotation cost is $8, then the calculation will be as follows:

rp = 100*9%/100-8

rp =9/92

rp =9.8%

Test yourself

A corporation is issuing 10% preferred stock that should be sold for $15 each. The business will incur flotation costs of $2 per share.

REQUIRED: What is the cost-of-preferred-stock?

SOLUTION:

10%*15/15-2

1.5/13

The answer is 11.54%

Note: If you struggle with a calculation, read using a financial calculator article for some simple tips on using a financial calculator.

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Capital Rationing

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Many firms operate under capital rationing. Firms ration capital because more often than not firms do not have unlimited funds to invest. Therefore, not all acceptable projects can be actually accepted. This is, of course, contradictory with goal of maximizing shareholders value.

We can formally define the rationing of capital as follows: It is a situation when firms do not accept all acceptable projects due to a limited amount of funds or due to limits imposed on investments. The goal is to select portfolio of projects with the highest net present value.

Under situations involving scarce capital, businesses will select a portfolio of projects with the highest NPV and which does not exceed the allocated budget. There are two commonly used techniques to select projects in these situations, the net present value NPV approach and the internal rate of return (IRR) approach.

The IRR approach graphs return against the total investment on the investment opportunities schedule (IOS) and by drawing the budget constraint shows the group of projects that are acceptable to be invested in. The NPV approach ranks projects by IRR and than generates a portfolio of projects with the highest overall present value.

When selecting projects, the net present value (NPV) approach is preferred because it maximizes shareholders’ returns whereas an internal rate of return (IRR) approach just generates a portfolio of acceptable projects.

 

EBIT-EPS Capital Structure Approach

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The EBIT-EPS capital structure approach focuses on finding a capital structure with the highest EPS (earnings per share) over the expected range of EBIT (earnings before interest and taxes).

The reason why we are interested in finding a capital structure which will permit maximization of the EPS over the expected range of EBIT is because it partially helps us to achieve the ultimate objective of the enterprise. The ultimate objective of the enterprise is to maximize shareholders’ wealth by maximizing its stock price. Two key variables that affect stock price are return (earnings attributed to owners of the enterprise) and risk (which can be measured by required return (rs)). This approach explicitly considers maximization of returns (EPS). However, it is important to note that this approach ignores risk (does not explicitly consider risk).

Major shortcoming of the EBIT-EPS approach

The fact that this approach fails to explicitly consider risk is the major shortcoming of this method. As firm obtains more debt (its financial leverage increases), the risk also increases and shareholders will require higher returns to compensate for the increased financial risk. Therefore, this approach is not completely appropriate because it does not consider one of the key variables (risk), which is necessary for maximization of shareholders’ wealth.

Considering financial risk

As per above, the approach does not explicitly consider financial risk. However, when utilizing the approach, financial risk can be considered in two ways:

1) The approach measures financial risk by the financial breakeven point. The higher the breakeven point the greater the financial risk.

2) The approach also measures the financial risk by the slope of the capital structure line. The steeper the capital structure line the greater the financial risk.

EBIT-EPS graph

It is a graphical approach. EPS is plotted on the vertical axis (x-axis) and EBIT on the horizontal axis (y-axis). By connecting the coordinates for different capital structures (different variations of equity versus debt), capital structure lines for each capital structure are graphed.

We will need to represent EBIT-EPS coordinates (capital structure lines) for different capital structures to ascertain at which levels of EBIT which capital structure is preferred. This will allow us to find a capital structure with the highest EPS over the expected range of EBIT.

For the purposes of this article it is sufficient to mention that to find EBIT-EPS coordinates we can assume particular EBIT values (and associated earnings available for common stockholders values) and calculate EPS in line with such values for different capital structures.

The formula to calculate EPS is as follows:

EPS = Earnings Available for Common Stockholders/ Number of Shares of Common Stock Outstanding Another easy way to find one of the EBIT-EPS coordinates is to use the financial breakeven point calculation. Financial break-even point occurs at the level of EBIT (earnings before interest and taxes) at which EPS (earnings per share) equals zero. At this level of EBIT all fixed financial costs are covered. The formula for calculation of the financial break-even point is as follows:

Financial break-even point = I + PSD/1-T

Where:

I – interest charges

PSD – preferred stock dividends

T – tax rate

***

This capital structure approach does NOT allow us to determine the point where weighted average cost of capital is at a minimum and where stock price is at a maximum (where wealth of the owners of the firm is maximized). The approach focuses on maximizing earnings rather than on maximizing wealth. Therefore, although it is helpful to use when analyzing alternative capital structures, the major shortcoming of this approach should be taken into account.

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Using a financial calculator

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Using a financial calculator is a skill, similar to typing. You just need to know which steps to take and then you need to practice to the point when you feel comfortable with using a calculator.

In all explanations with a financial calculator we will be using an HP 10bll. Other financial calculators are similar, yet we find it easier to work with the HP. Most text books use HP calculators when providing guidance on using a financial calculator, so if you have a different calculator you may need to spend more time learning slightly different calculation steps. Before investing further time, it may be wise to get a universally used calculator.

Before using a financial calculator to make specific calculations such as calculating NPV or IRR, it is important to make sure that you:

1 – Clear the calculator – by pressing second function followed by “C All”

2 – Set calculator for the “END” by pressing second function followed by “BEG/END” and ensuring that the word “BEGIN” is not displayed. Exceptions to this rule occur when it is specifically stated in the problem that cash flows occur at the beginning of the period (for example, at the beginning of the year).

Again, if no sign appears on the display then you do not need to reset it as it is set for “END” by default. If it says “BEGIN” on the display, you need to press second function followed by “beg/end.”

When you set the calculator for the “END” of the period you do that because in the problem you are working with, cash inflow or outflow occurs at the end of the period. If the problem does not state when cash flows occur, you need to assume that it occurs at the “END” of the period.

The majority of calculations will require the “END” setting. If it is by mistake set for “BEGIN” but cash flows occur at the end of the period, then incorrect answers will be generated.

Therefore, it is advisable to keep it set for the “END” at all times as a default and only reset it for “BEGIN” when a calculation requires that to be done. Right after a calculation is completed that requires the “BEGIN” setting, it is important to develop a habit to reset it to the “END”.

In the explanations using a financial calculator, for convenience and clarity purposes, we will generally display explanations of calculations as presented in the example below:

PV: -900 I: 7 N: 5 FV: 1,262.3

When using a HP 10bll financial calculator, or using any financial calculator, you need to first insert the number (number, e.g. -900) and then insert the purpose of the number (e.g. PV).

For example, as per above, you need to press:

900 followed by the minus sign followed by PV

7 followed by I

5 followed by N

Than press FV, and the calculator will display the correct answer

Financial calculators sometimes give false answers. It is advisable to check each calculation 3-4 times to make sure that the same answer is given consistently.

Throughout the site, if you ever struggle with a calculation, always come back to this page for some simple tips on using a financial calculator.

Test yourself

ABC Corporation plans to invest in project C which has an initial investmentof $500,000. ABC’s cost of capital is 8%. The operating cash flows to be generated from the project will be as follows:

End of 1st year: $100,000 End of 2nd year: $300,000 End of 3rd year $250,000

1 – What is the Profitability Index (PI) for project C?

2 – What is the NPV for project C?

3 – Taking the NPV found in the previous step into account, is the project acceptable according to the NPV technique?

4 – Based on the Profitability Index (PI), is project C acceptable?

SOLUTION:

1 – First we need to find present values of the mixed stream of operating cash inflows. Using a financial calculator, we need to take the following steps:

End of 1st year:

FV: $100,000

N: 1

I: 10

Calculate PV: $90,909.09

End of 2nd year:

FV: $300,000 N: 2

I: 10

Calculate PV: $247,933.88

End of 3rd year:

FV: $250,000

N: 3

I: 10

Calculate PV: $187,828.7

Next we need to add up all present values from operating cash inflows to obtain the total PV of operating cash inflows:

= $90,909.09 + $247,933.88 + $187,828.7

Total PV of operating cash inflows = $526,671.67

Next we will follow the equation for Profitability Index (PI):

PI = Total present value of cash inflows/Initial investment

PI=$526,671.67/$500,000

PI=1.05

Therefore, the profitability index (PI) for project C is 1.05.

2 – To find NPV, we follow the formula for NPV:

NPV=Present value of cash inflows – Initial investment

Therefore, NPV for project C = $526,671.67 – $500,000

NPV for project C = $26,671.67

3 – Since NPV is more than zero ($26,671.67), project C is acceptable according to NPV technique.

4 – Since Profitability Index (PI) is greater than 1 (1.05), the project may be considered to be acceptable.

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Internal Rate of Return method (IRR)

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Sophisticated capital budgeting techniques include Net present value method (NPV), Internal Rate of Return method (IRR), Profitability index (PI) and Equivalent Annual Annuity (EAA). Internal Rate of Return method (IRR) is discussed below.

Internal Rate of Return (IRR) is a widely used technique.

It is also very easy to utilize Internal Rate of Return with the help of a financial calculator. It is much more challenging to calculate it by hand. Again, as in utilizing the NPV method, it is important to first understand the logic behind the calculation.

In simple terms, the IRR is the rate of return that would equate NPV with zero. If IRR higher than cost of capital than project should be accepted and vice versa. If IRR at least equals cost of capital than we know that business will earn at least rate equal to its cost of capital on this particular project.

Below is shown how to calculate IRR using the financial calculator.

IRR for annuity is calculated as follows:

Initial investment, minus sign – CFi

Annual cash inflow – CFi1

Number of periods – second function Ni

Second function IRR

IRR for a mixed stream is calculated as follows:

Initial investment, minus sign – CFi

Put in each cash inflow separately following with CFi1, CFi2 etc

Second function IRR

Both NPV and IRR will show whether the project is acceptable. However, the ranking of specific acceptable projects may differ between two techniques.

Test yourself

ABC have an option to invest in project B. The initial investment for project B is $35,000. Operating cash inflows from project B expected to be $5,000 per year for 8 years. The cost of capital of ABC is 5%.

What is the Internal Rate of Return (IRR) for project B?

Find out if project B is acceptable based on IRR calculation.

Solution:

With the help of financial calculator, we can determine IRR of project B as follows:

CFio: -35,000

CFi1: 5,000 (annual operating cash inflow)

Second function Nj: 8 (8 years)

Second function IRR: calculate – 3.07

The IRR of project B is 3.07%. The cost of capital of ABC is 5%. Since IRR (3.07%) is below cost of capital (5%), the project is not acceptable.

Test yourself

ABC have an option to invest in project D. The initial investment is $300,000. The operation cash inflows are expected to be $100,000 at the end of year 1, $110,000 at the end of year 2 and $130,000 at the end of year 3. The cost of capital of ABC is 10%.

  1. Calculate IRR
  2. Recommend if based on IRR technique project D is acceptable.

Solution:

1. With the help of financial calculator, the calculation is as follows:

Clear calculator: second function followed by C ALL

CFo: -300,000

CF1: 100,000

CF2: 110,000

CF3: 130,000

Second function IRR: calculate – 6.24%

2. Since cost of capital of ABC is 10% and IRR is only 6.24% (less than cost of capital), based on IRR technique, project D is not acceptable.

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Unsophisticated capital budgeting techniques

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Simple (unsophisticated) capital budgeting techniques include average rate of return (ARR) and the payback method (also called PB or payback period).

Average Rate of Return

Average Rate of Return (ARR) is an unsophisticated budgeting technique and generally considered to be ineffective. Average Rate of Return (ARR) evaluates relative profitability of the investment. In other words, it evaluates how project affects accounting profits. Average Rate of Return (ARR) is calculated as follows:

ARR = Average income / Average investment

Average income refers to annual average net profits after tax (refer to income statement to see how net profits after tax are determined). Annual average net profits after tax is found by taking total net profits after tax over the useful life of the project and dividing it by number of years over useful life of the project. Average investment refers to average investment over the economic life of the project. The ARR capital budgeting technique does not consider the time value of money. It also considers net profits rather than cash inflows. Consequently, the technique overlooks the possibility of reinvestment of returns.

The positive side of this technique, as compared to payback period discussed below, is that it considers returns on investment over entire useful life of the project. However, this technique is generally not recommended.

Payback method

Payback period (PB), also called a payback method, is another unsophisticated budgeting technique. It determines how long it takes to recover the initial investment by taking into account cash inflows from the investment. If we deal with an annuity (an equal periodic cash flow over a specific period) than all we need to do is to divide initial investment by an annuity.

However, if we deal with a mixed stream of cash inflows (unequal cash flows during specific period with no precise pattern) than we need to add up cash flows until the initial investment is recovered.

Management needs to subjectively determine the maximum payback period and then projects are evaluated according to this. If the project’s payback period is below maximum than the project is acceptable and vice versa.

The payback period budgeting technique measures business’s risk exposure because the project’s risk level depends on how long it takes to recover the initial investment. However, it does not explicitly consider the time value of money.

Moreover, this budgeting technique is weak because it is subjective in nature since the minimum payback period is subjectively determined. Furthermore, it does not take into account the cash flows that occur after the payback period.

A variation of payback period capital budgeting technique allows to account for time value of money and risk (due to usage of discount rate which incorporates risk). Such variation is called discounted payback period technique. This technique determines how long it takes for discounted cash flows to recover the investment. However, this variation still does not consider cash flows after the payback period.

Test Yourself

ABC Corp has a proposed project A, which has expected cash inflows of $4,000 over 10 years period. The initial investment is $30,000. Find the payback period.

SOLUTION:

Payback period = 30,000/4,000 = 7.5 years

This means that it will take 7.5 years for ABC to recover its investment in project A.

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