Slide #1: Lecture 18 M&M Proposition I with Taxes
Welcome to Lecture 18: M&M Proposition I with taxes but no bankruptcy costs.
Slide #2: Topics covered
In this lecture, we will cover six topics.
First, we will talk about the assumptions of the model, and then we will move on to the valuation formulae which allow us to value a firm based on the formulae that come out of the M&M I with taxes model.
Then we will interpret the model with a graph, which makes things a bit clearer and easier to see. We then move on discuss the implications of the model.
Next, we will do a numerical example which will allow us to use the formulae that come out of the model. Last but not least, a practice problem will be provided with the appropriate check answers.
Slide #3: Assumptions
So, what are the assumptions?
The obvious one you may first notice is the fact that we now allow for the presence of corporate taxation.
The remaining assumptions in the model are:
- A perfect market, which assumes that anybody and any firm can go into the market, and borrow and lend at the same rate;
- No bankruptcy costs, which means that there are no disadvantages for one to borrow as much as possible; and,
- A perpetual going concern, which allows us to use the present value of perpetuity formula (PV = PMT/r) to value a firm’s future cash flows. So, when you see a similar formula being used in the next few slides, you will realize that it is because of this perpetual going concern assumption that the formula can be derived from the model.
Slide #4: Valuation Formulae
What are the valuation formulae derived from the M&M Proposition I with taxes? There are two sets of formulae. The first set has to do with the unlevered firm (i.e., the firm that has zero debt), and the second set has to do with the levered firm (i.e., the firm that is in debt).
There is only one formula in the first set. It says that the value of an unlevered firm can be calculated as its earnings before interest and taxes (EBIT), multiplied by 1 minus the corporate tax rate, all divided by the cost of capital of the unlevered firm:
VU = EBIT(1 - TC) / RU
So, to find the firm value of an unlevered firm (VU), we will need three things: EBIT, corporate tax rate (TC), and the cost of capital of that unlevered firm (RU).
There are two formulae in the second set of formulae derived from the M&M Proposition I with taxes. The first formula says that the value of a levered firm can be calculated as the value of an identical unlevered firm, plus the corporate tax rate multiplied by the level of debt:
VL = VU + TC D
TC D represents the tax shields obtained from paying interest on debt. So, to find the value of a levered firm (VL), we will need three things: the value of the unlevered firm (VU), the corporate tax rate (TC), and the level of debt (D).
Next, in this same set of formulae for the levered firm, we have a formula that says that we can find the equity value of the levered firm by deducting the debt-level (D) from the levered firm’s value (VL):
EL = VL - D
Below is a summary of the variables used in the formulae above:
VU= value of unlevered (0 Debt) firm
EBIT = Earnings before interest and taxes
TC= corporate tax rate
RU= cost of capital for unlevered firm
VL= value of levered (In Debt) firm
D = dollar amount of debt in the levered firm
EL= dollar amount of equity in the levered firm
Slide #5: Graph
The graph that depicts M&M Proposition I with taxes is shown here:
Firm Value (V)
VL = VU + TCD
TC x D
VUVU
VU
0 Debt (D)
The graphical interpretation of the M&M I with taxes model is quite simple. We draw the graph, with the x-axis (horizontal axis) representing the debt-level (D), and the y-axis (vertical axis) representing the firm value (V). As you go towards the right on the graph, the debt-level increases. As you go higher on the graph, firm value increases.
What this graph says is that the value of the unlevered firm will remain constant at any level of debt, as long as the EBIT, corporate tax rate, and cost of unlevered capital remain constant. This makes sense, as by definition, the unlevered firm has no debt, and therefore its value would not be affected by debt-level.
The difference between the value of the levered firm and that of the unlevered firm is made up simply of the debt tax shields (TC x D). So the value of the levered firm gets higher and higher, towards infinity, as the debt-level goes up. Conceivably, this means that one would want to borrow as much as possible in order to increase the value of the levered firm as much as possible. Conversely, this also implies that the cost of capital for the levered firm must be decreasing, as its value goes up and up when debt is increased.
Slide #6: Implications
The conclusions from the last slide are exactly the implications that we obtain from the M&M I with taxes model. The first implication is that debt-financing is good, and therefore, in extreme cases, one would want to have infinite debt-financing. The second implication is that as debt-level increases, the cost of capital (WACC, or weighted average cost of capital) of the levered firm decreases.
Slide #7: Numerical example
Now, without further ado, let’s do a numerical example using the model and formulae we have just learned.
ABC Inc. and XYZ Co. are identical except for their financing policy. ABC is financed only with equity, whereas XYZ has $1,000,000 in debt-financing. Both firms have earnings before interest and taxes (EBIT) of $500,000, and both are subject to 35% corporate tax rate. ABC’s cost of capital is estimated to be 18%.
- What is XYZ’s firm value and equity value?
- What happens to the firm value and equity value if XYZ increases its debt financing to $2,000,000?
Part a:
The first thing we do is write down all the information given:
D = $1,000,000
EBIT = $500,000
TC = 35% = 0.35
RU = 18% = 0.18
That is all the information we need to calculate the value of the levered firm.
Slide #8: Numerical example (cont.)
First, we must calculate the value of the unlevered firm, using the formula:
VU = EBIT(1 - TC) / RU
We plug in the numbers for EBIT, TC, and RU, and we get
VU = $500,000(1 - 0.35) / 0.18 = $1,805,555.56
This is the value of the unlevered firm, ABC.
The value of the levered firm can then be calculated using the formula:
VL = VU + TC D
We plug in the numbers for VU, TC, and D, and we get
VL = $1,805,555.56 + (0.35)($1,000,000)
= $1,805,555.56 + $350,000
= $2,155,555.56
This is the value of the levered firm, XYZ.
Slide #9: Numerical example (cont.)
To calculate the equity value of the levered firm, we use the formula:
EL = VL - D
We plug in the values for EL, VL, and D, and we get
EL = $2,155,555.56 - $1,000,000 = $1,155,555.56
Based on this equity number, we can now calculate the debt-equity ratio:
D/EL = $1,000,000 / $1,155,555.56 = 0.865384615
We can put all these numbers into the graph we drew before:
Firm Value (V)
VL = VU + TCD
VL =
$2,155,555.56
TC x D = $350,000
VU$1,805,555.56
0$1,000,000 Debt (D)
We can see that the value of the unlevered firm stays constant at the all-debt-level, at $1,805,555.56.
The tax shields on debt is TC x D = 0.35 x $1,000,000 = $350,000.
The value of the levered firm, at the debt-level of $1,000,000, is simply equal to the value of the unlevered firm, at $1,805,555.56, plus the tax shields on debt of $350,000.
Part b:
To answer part b, which asks us to calculate the firm value and equity value if the debt-level is $2,000,000, we simply redo all the calculations using the appropriate formulae.
We know that the value of the unlevered firm will not change with debt-level, and therefore, we have
VU = $1,805,555.56
The value of the levered firm is calculated as:
VL = VU + TC D = $1,805,555.56 + (0.35)($2,000,000) = $2,505,555.56
The equity value of the levered firm is then calculated as:
EL = VL - D = $2,505,555.56 - $2,000,000 = $505,555.56
The debt-equity ratio is therefore
D/EL = $2,000,000 / $505,555.56 = 3.95604956
Slide #10: Practice problem
Here is a practice problem for you.
We have two firms, Firm U and Firm L, that are identical in every respect except for their financing policy. Firm U has zero debt, and its assets have an estimated value of $10 million. Firm L is partly financed with debt, and its assets have an estimated value of $12.5 million. Given that both firms are subject to a corporate tax rate of 40%, and both have an EBIT of $2 million, calculate the cost of capital for Firm U and the debt-equity ratio for Firm L.
Slide #11: Check answer - Cost of equity of Firm U
Information given:
VU = $10,000,000
VL = $12,500,000
TC = 0.4
EBIT = $2,000,000
To find RU:
VU = EBIT(1 – TC) / RU
Multiplying both sides by RU, we get
RU x VU = EBIT(1 – TC)
Dividing both sides by VU, we get
RU = EBIT(1 – TC) / VU
Plugging in EBIT = $2 million, TC = 0.4, and VU = $12.5 million, we get
RU = $2,000,000 (1 – 0.4) / $12,500,000 = 0.12 = 12%
Slide #12: Check answer - D/E ratio of Firm L
Information given:
VU = $10,000,000
VL = $12,500,000
TC = 0.4
EBIT = $2,000,000
First, we calculate the debt-level in Firm L by using the levered firm valuation formula:
VL = VU + TC D
Subtracting VU from both sides, we get
VL – VU = TC D
Dividing both sides by TC, we get
(VL – VU) / TC = D
Plugging in VL = $12.5 million, VU = $10 million, and TC = 0.4, we get
D = ($12,500,000 - $10,000,000) / 0.4= $2,500,000 / 0.4 = $6,250,000
We can now calculate the equity of the levered firm:
EL = VL – D = $12,500,000 - $6,250,000 = $6,250,000
The debt-equity ratio of Firm L is therefore:
D/EL = $6,250,000 / $6,250,000 = 1
Here ends Lecture 18 on M&M Proposition I with taxes but no bankruptcy costs.