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CAPITAL STRUCTURE: PROBLEMS & DETAILED SOLUTIONS (copyright © 2016 Joseph W. Trefzger)

Work These for Sure

1. Alberta Appliances is considering a new investment project: producing the Portable Perfect Chicken Chum, a small rotisserie grill that plugs into a car’s cigarette lighter. The Chums are expected to be sold for $135 each. The variable cost (materials, production wages, power to run the machines, etc.) of producing each unit is expected to be $94. Cash outlays for building rent, management salaries, and other fixed annual operating costs are expected to be $2,600,000 each year. Alberta also would have to buy manufacturing machinery at a cost of $13,250,000. This equipment is expected to have a 10-year life (equal to the life of the project), and then to have no salvage value. What is the grill project’s annual operating (sometimes called accounting) break-even point? If Alberta’s weighted average cost of capital is 12.5% per year, what is the project’s annual financial break-even point?

Type: Break-even. A break-even point for an investment project (which is computed for a given time period; here we think, as typically is done in break-even analysis, in terms of a year) can be found, most generally, as

Annual Break-Even Point = Year ' s Total¿Costs ¿contribution margin per unit sold .

Let’s start with the easy part of the formula, the denominator. The contribution margin (CM) is the expected selling price per unit, minus the variable cost per unit. If this difference is positive, then each unit sold “pulls its own weight” (covers its own variable, or differential, cost) and contributes something to the company’s “greater good” (meeting fixed costs and, once fixed costs are met, contributing to the company’s net income). Thus a higher contribution margin is better, because we want each item made and sold to contribute more to the company’s well-being. (A negative CM is bad; production should be stopped if the CM is negative.) Here the contribution margin is $135 – $94 = $41.00. (Note that we treat the selling price per unit as being the same no matter how many are sold – a point on which this type of analysis might be criticized, since we know that to sell more a company normally has to reduce the price charged for each unit. A defense would be that, for a moderate increase in quantity sold, the price would not likely have to decline by much.)

Using contribution margin as the denominator lets us see how many units made and sold (i.e., how many $41 contributions) it takes for the project to break even (to cover fixed costs, which are the costs not accounted for in the contribution margin). So in this approach we find break-even as the output level at which the producer covers fixed costs, after having accounted for the total variable costs by subtracting variable cost per unit from selling price per unit in finding the CM.

The numerator is a more difficult issue, because we might define “fixed” costs in a less-inclusive or a more-inclusive fashion. Fixed costs are costs that do not vary with the level of output. Fixed costs that are easy to measure are those paid for in cash during the year. Examples are rent on buildings, salaries to managers whose pay does not change with the company’s output (or with hours worked; sorry, future managers), and heating/electricity costs at the minimum level to keep the facilities safe and functional. But another component of fixed cost is not easily measured: each year’s share of the up-front outlays that are expected to provide benefits over multiple years. (It is this hard-to-measure component of fixed cost that we can compute in a less-inclusive or more-inclusive manner.) With this information in mind, we can restate our break-even formula more precisely as:

Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 1

Annual BEP = Year ' s Cash ˗ Based ¿Operating Costs+Year' s Share of Cost of Long ˗ Lived Equipment ¿

contribution margin per unit sold .

The main issue here is the decline in value of long-lived equipment used in the project. A machine’s value arguably drops more in a given year if it is used more intensively, but equipment becomes obsolete with the passage of time, and thus to some degree its loss in value over time is unaffected by the intensity of use. But perhaps more importantly, we are trying to determine an equal break-even point for each year of the project’s life, so we implicitly assume that use of long-lived assets is at a constant level from year to year. How do we annualize the cost of something that will provide service for multiple years? The less-inclusive way is simply to divide the item’s cost by the number of years n in its expected useful life (or the life of the associated project): straight line depreciation. (There would be nothing wrong with computing a different BEP for each year, perhaps by using accelerated depreciation, but it is convenient to have one number we can think of as applying equally to each year.) Break-even based on the less-inclusive view of fixed costs is called the operating (or accounting) break-even point, which is how “break-even” is traditionally computed. We find this value with the formula:

Operating BEP = Year ' s Cash ˗ Based ¿Operating Costs+(Cost of Long ˗ Lived Equipment

n ) ¿contributionmargin per unit sold

.Based on the values given for this problem, we compute

Operating BEP = $ 2,600,000+ $ 13,250,00010

$ 135−$ 94 = $2,600,000+$ 1,325,000

$ 41 = 3,925,000

41 =

95,731.71

or 95,732 grills (if the units were tons of gravel it would be possible to sell a fractional unit, but we round the answer to a whole number if partial units could not be sold, as with grills, and in break-even analysis we always round upward because missing the break-even point even by a fractional unit would leave some costs unmet). Making and selling 95,732 grills annually would seem to allow Alberta to cover all variable and fixed costs, and thus to break even, each year. But this answer is based on a less-inclusive view of fixed costs. What fixed costs have we ignored?

At the operating break-even point, the firm breaks even only with regard to operating costs (such that EBIT = $0), but not with regard to financing costs. Consider that the lenders and owners who provided Alberta with the $13,250,000 to buy the equipment expect a 12.5% average annual return on their invested money. Dividing the $13,250,000 by 10 tells how much the company must collect each year, through sales revenue minus variable costs, to get the money paid for the equipment back over the life of the project, but what about a fair rate of return to the investors who paid for that equipment? Think of it this way: if you invested $13,250,000 for ten years, and simply got back $13,250,000/10 = $1,325,000 each year, you would feel cheated. You would be getting back your investment, but no rate of return on that money.

However, there is some amount – an amount greater than $1,325,000 – that you could be paid each year and feel you were being treated fairly, getting back your investment over the project’s life and also getting a fair rate of return each year. To compute an annualized figure that includes this rate of return, we divide the equipment cost not by n, but rather by Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 2

the n-year PV of an annuity factor. Using the more-inclusive fixed cost measure that combines the equipment’s purchase cost with its financing cost (to provide a fair annual rate of return to the investors), we compute the financial break-even point:

Financial BEP = Year ' s Cash ˗ Based ¿Operating Costs+( Cost of Long ˗ Lived Equipment

n ˗ Period PV of Annuity Factor ) ¿contributionmargin per unit sold

.If the risk of the investment indicates that a 12.5% annual return would be in order, we find the applicable 10-year PV of an annuity factor:

( 1−( 11.125 )

10

.125 ) = 5.536431, and we then compute

Fin. BEP = $ 2,600,000+ $13,250,0005.536431

$ 135−$ 94 = $ 2,600,000+$ 2,393,238.61

$ 41 = 4,993,238.61

41 =

121,786.31

or 121,787 grills. The only computational difference between the operating and financial break-even points is whether we divide the equipment cost by n (here, 10) or by the n-year PV of an annuity factor (here the 10-year, 12.5% factor 5.536431). The substantive difference between them is that the operating BEP indicates the level of output that leads to a $0 EBIT, while the financial BEP indicates the level of output that leads to a $0 economic value added, or EVA. The financial break-even point is always higher than the operating BEP; if we measure fixed cost to be higher because we think of it in a more-inclusive way that includes financing costs, then we have to think in terms of making and selling more units to cover all relevant costs (i.e., to break even).

2. In its most recent operating year, British Columbia Lumber Products sold 450,000 folding tables. Its total fixed operating costs (in an accounting sense) were $3,500,000, and variable cost per unit produced was $27.75. The tables were sold to houseware stores for $46 each. British Columbia paid $765,000 in interest during the year. Compute the company’s degree of operating leverage (DOL), degree of financial leverage (DFL), and degree of total leverage (DTL, also called degree of combined leverage DCL), and interpret these values.

Type: Operating, financial, total leverage. We compute the degree of operating leverage as

DOL = Q( p−vc)Q ( p−vc )−FC

= Q( p−vc)EBIT

= 450,000($ 46−$ 27.75)450,000 ($ 46−$27.75 )−$3,500,000

= $ 8,212,500$ 4,712,500

=

1.743

and the degree of financial leverage as

DFL = EBIT

EBIT−∫¿¿ = $ 4,712,500

$ 4,712,500−$ 765,000= $ 4,712,500

$ 3,947,500 = 1.194

The degree of total (or combined) leverage is the product of the DOL x DFL:

DTL = DOL x DFL = 1.743 x 1.194 = 2.081

We interpret a DOL of 1.743 as: If financial relationships that have held in the past continue


to hold in the future, a 1% change in sales can be expected to bring about a 1.743% change in operating income, or earnings before interest and taxes (EBIT). So a 1% increase in sales causes EBIT to increase by 1.743%, while a 1% decrease in sales causes EBIT to decline by 1.743%. (If none of the operating costs were fixed, but rather all operating costs varied proportionally with output, then DOL would be 1, and a 1% change in sales would lead to a 1% change in EBIT.)

We interpret a DFL of 1.194 as: If financial relationships that have held in the past continue to hold, a 1% change (increase or decrease) in EBIT can be expected to bring about a 1.194% change (increase or decrease) in net income. (If there were no fixed financing costs, but rather all financing costs varied proportionally with output, then DFL would be 1, and a 1% change in EBIT would be expected to lead to a 1% change in net income.)

Finally, we interpret a DTL of 2.081 as: If financial relationships that have held in the past continue to hold, a 1% change in sales can be expected to bring about a 2.081% change in net income. (DOL tells us how sales affects operating income, DFL tells us how operating income affects net income, and DTL cuts out the middle term, telling us how sales affects net income.) Here the existence of fixed costs, in British Columbia’s operations and financing, causes the expected change in net income to be more than double any expected change in sales. We can use DTL as a forecasting tool in both percentage and dollar terms. Note that total sales were $450,000 x $46 = $20,700,000, and let’s say that net income was $2,500,000. If sales were expected to increase (decrease) by $1,000,000 (a $1 million ÷ $20.7 million = 4.8309% change), net income would be expected to increase (decrease) by 2.081 x 4.8309% = 10.0531% (a .100531 x $2,500,000 = $251,329 change).

DOL and DFL have capital structure implications because a high DTL could be seen as dangerous; a slight downturn in sales could cause net income to fall dramatically (as revenue falls but cost, much of which is fixed, does not). Therefore a firm with a high DOL (typically a function of technology, dictated by the type of product or service it produces) might want to be less aggressive in its use of borrowed money (strive for a lower DFL, so that DTL does not become too high).

3. The managers of Canada Consolidated Copper are analyzing the company’s capital structure. They feel that the costs of various components of the financing mix (debt, preferred stock, and common stock) would be as follows, under the indicated capital structure possibilities:

Proportion Cost of Proportion Cost of Proportion Pre-Tax from Preferred Preferred from Common Commonfrom Lenders Cost of Debt Stockholders Stock Stockholders Stock (D/V or wd) (kd) (P/V or wp) (kp) (E/V or wc) (ke) 0 N/A .10 6.0% .90 14.5% .20 8% .10 7.5% .70 16.5% .40 10% .10 9.5% .50 18.0% .60 14% .10 15.0% .30 28.0% .80 26% .10 30.0% .10 52.0%

Compute the weighted-average cost of capital (WACC) under each of the representative capital structures shown(do not forget to adjust the cost of debt to reflect the expected income tax savings under Canada’s 34% marginal income tax rate). Which financing mix results in the lowest WACC and is, therefore, our estimate of the optimal capital structure? Why does that outcome seem to occur?

Type: Optimal capital structure. The weighted average cost of capital (WACC, which we compute as WACC = wd � kd (1 – t) + wp � kp + we � ke) for each respective possibility shown is:

WACC = 0 (1 – .34) + .10 � .06 + .90 � .145Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 4

= 0 + .006 + .1305 = .1365 or 13.65%

WACC = .20 � .08 (1 – .34) + .10 � .075 + .70 � .165= .01056 + .0075 + .1155 = .13356 or 13.356%WACC = .40 � .10 (1 – .34) + .10 � .095 + .50 � .18

= .0264 + .0095 + .09 = .1259 or 12.59% � Lowest

WACC = .60 � .14 (1 – .34) + .10 � .15 + .30 � .28= .05544 + .015 + .084 = .15444 or 15.444%

WACC = .80 � .26 (1 – .34) + .10 � .30 + .10 � .52= .13728 + .03 + .052 = .21928 or 21.928%

Here we wish to find the mix of lender and owner money that keeps the firm’s overall cost of financing (“weighted average cost of capital,” or WACC, also abbreviated as kA) as low as possible. Managers make the company’s owners wealthier by taking on investment projects whose expected returns exceed the cost of the invested money (the WACC), so finding the lowest possible WACC opens the door to more money-making investments, and thus is consistent with maximizing the owners’ wealth.

The lowest computed WACC, 12.59%, accompanies the 40% debt/10% preferred stock/50% common stock capital structure; thus that combination would seem to be, from among the representative list of “round number” financing mixes considered, the optimal capital structure. (There is no reason why the breakdown could not include combinations such as 37%/15%/48%, but in examining specific capital structure figures we typically stick with fairly broad generalities and look at directional changes, because it would be impossible to precisely pinpoint the truly optimal capital structure.) Finding we have an optimal capital structure that contains not-too-much debt, yet not-too-little debt, is in keeping with the theoretical “smile-shaped” WACC function.

Note that the firm’s pre-tax cost of debt financing rises as debt becomes a higher proportion of the capital structure. The reason is that if there is a large total claim at the “front of the line,” a slight downturn in the firm’s sales could cause even the lenders to face substantial losses (while the cost of equity financing is yet higher, because the stockholders receive no returns unless and until the lenders’ claims have been satisfied in full). Note also that the highest combined proportion of debt and preferred stock we consider is 90% (we assume, for simplicity, that preferred stock would provide 10% of the money to pay for Canada’s assets under any of the capital structure possibilities considered). It would be essentially impossible for the firm to get 100% of the funds it buys assets with from parties other than the true owners, the common stockholders (in fact, it is sometimes unrealistic to assume that the proportion could get much higher than 50%).

Finally, note that the expected cost of preferred stock financing is typically less than the pre-tax cost of debt, because of the income tax breaks that corporations enjoy when they receive dividends as preferred stockholders (they can be happy with lower returns because they pay lower tax on dividend income). However, as the debt proportion of the capital structure becomes increasingly high, we might wonder whether the preferred stockholders would seek comparatively higher returns in light of the high risk of standing in the “middle of the line” behind such a high debt position, whose claim is settled in full before preferred stockholders receive anything.

4. The treasurer of Hudson Bay Handbrakes, Inc. is trying to decide what the company’s optimal capital structure would be. Hudson Bay has $100,000 in total assets, and expects operating income, or EBIT (a figure largely unaffected by the chosen capital structure) to be $8,000 in each of the next several years. The company’s common stock sells for a market price of $10 per share. The treasurer believes that Hudson Bay would pay an 8% annual Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 5

interest rate on borrowed money under any capital structure choice. A simple “EBIT/EPS” analysis the treasurer performs results in the projections shown below. Debt Ratio 0% 25% 50% 75% 99%Amount of Debt $0 $25,000 $50,000 $75,000 $99,000Amount of Equity $100,000 $75,000 $50,000 $25,000 $ 1,000Shares of $10 Common Stock 10,000 7,500 5,000 2,500 100

Operating Income (EBIT) $8,000 $8,000 $8,000 $8,000 $8,000Minus Interest (8% x debt) $ 000 $2,000 $4,000 $6,000 $7,920Net Income (ignoring tax) $8,000 $6,000 $4,000 $2,000 $ 800

Net Income/Equity = ROE 8% 8% 8% 8% 8%Net Income/Shares = EPS $.80 $.80 $.80 $.80 $.80

(Income taxes are ignored in this analysis.) Based on these projections, it seems that any selected capital structure would lead to the same level of return on equity (ROE) and earnings per share (EPS), two important measures of financial return to the owners whose wealth the firm’s managers are supposed to maximize. Is this finding realistic; is capital structure irrelevant? Why does financial leverage here have a neutral (rather than a positive or negative) effect? In other words, what has caused projected ROE and EPS to be the same, in this situation, for any debt ratio?

Type: EBIT/EPS analysis. What we are trying to determine is whether debt financing a good or bad thing. Here we address the owners’ wealth maximization question in a more direct way than we do when looking for the minimum WACC (albeit with some unrealistic assumptions, as discussed below). It should be intuitively clear that borrowing is good if the cost (the interest rate the borrower would pay) is low, thereby contributing to a low WACC, and bad if the cost is high. Of course, “low” and “high” are relative terms; specifically, a company would wish to borrow to a greater extent if debt’s annual percentage cost were lower than the annual percentage return that could be earned on the assets bought with borrowed money, and borrow to a lesser extent if the annual percentage cost were higher than the annual percentage return that could be earned on those assets.

It should follow that if the annual percentage cost of borrowed money (the interest rate that would be paid) were equal to the expected annual percentage return earned on the purchased assets, there would be no incentive to borrow or to refrain from borrowing. And that situation is what we see in the Hudson Bay treasurer’s projections. Recall that financial leverage has a positive impact on owners’ financial returns if the interest rate paid is less than the company’s basic earning power ratio, which we compute here as

Basic Earning Power = EBITTotal Assets

= $8,000$ 100,000

= .08 or 8%

Because the interest rate the firm would pay on borrowed money is also expected to be 8%, the impact of financial leverage is neutral; if Hudson Bay borrows money to pay for assets, the assets produce exactly enough in returns to pay for the borrowed money, so the return to the owners is the same regardless of whether the company pays for the $100,000 in assets entirely with owners’ (equity) money or almost entirely with borrowed money. (We use 99% as the highest conceivable debt ratio, both because it is impractical to think of lenders providing 100% of the money to pay for a company’s assets, and because with zero equity the ROE and EPS figures are undefined.)

A couple of practical points to note: the situation depicted in this problem is not good for the company’s owners; with a basic earning power ratio equal to the interest rate on debt, the assets do not produce any extra value to magnify the financial returns to the owners. Indeed, the owners here end up with the same 8% return as the lenders on a pre-tax basis (and would earn a lower return than the lenders if income taxes were included; for example, if Hudson Bay paid a 30% average income tax rate the owners would end up with just 8% [1 – .30] = Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 6

5.6% under any capital structure). Because they face more risk (stand farther back in the line) than lenders, the owners would be unhappy with a percentage return no greater than the return the lenders receive. In this case, if the basic earning power ratio (EBIT/Total Assets) were higher than 8%, a simple EBIT/EPS analysis would predict higher levels of ROE and EPS for successively higher proportions of borrowed money; with basic earning power less than 8% an EBIT/EPS analysis would predict lower levels of ROE and EPS for successively higher proportions of borrowed money.

This type of analysis can help us to think more clearly about general capital structure issues, but it suffers from some unrealistic underlying assumptions. The most unrealistic is that the interest rate on borrowed money would be the same for any capital structure (debt/equity split). In fact, a very real practical problem in “real world” capital structure analysis is that as the debt ratio increases and lenders perceive more risk, the interest rate they charge is likely to increase, thereby making it more and more unlikely that basic earning power will exceed the interest rate. It is also strange to assume that income taxes would play no role, and perhaps somewhat unrealistic to assume that the value per share of common stock would be the same no matter how many shares were created.

Work These for Extra Practice

5. Labrador Electric, Ltd. plans to invest in new equipment so it can produce heavy-duty electrical extension cords. The equipment costs $2,850,000 and has an expected life of 8 years, which is also the expected life of the project. There is no expected salvage value for the equipment at the end of year 8. Fixed operating costs paid in cash each year (rent and management salaries, for example) are expected to be $465,000 annually. The variable cost of producing the cords is expected to be $2.67 per unit, and Labrador expects to sell each cord to a retailer for $3.89. What is the project’s annual operating (sometimes called accounting) break-even point? If the company’s annual weighted average cost of capital is 9.75%, what is the project’s annual financial break-even point?

Type: Break-even. Recall that the general formula for break-even is

Annual Break-Even Point = Year ' s Total¿Costs ¿contribution margin per unit sold ,

which we can state more specifically as

Annual BEP = Year ' s Cash ˗ Based ¿Operating Costs+Year' s Share of Cost of Long ˗ Lived Equipment ¿

contribution margin per unit sold .

If we use the less-inclusive measure of the annualized cost of the equipment (the purchase cost but not a fair rate of return on the investors’ money), then we compute the operating break-even point (for a $0 EBIT) as:



.Based on the values given for this problem:

Operating BEP = $ 465,000+ $2,850,0008

$3.89−$2.67 = $ 465,000+$ 356,250

$ 1.22 = 821,250

1.22 = 673,155.74


or 673,156 cords (we must round the answer up to the next whole number, because it would not be possible to sell a portion of a cord, and rounding downward would leave the company a tiny bit short of covering all its costs). Making and selling 673,156 cords annually would appear to allow Labrador to cover all its variable and fixed costs, and thus to break even, each year. But recall that this answer is based on a less-inclusive view of fixed costs, which does not include a rate of return for the providers of the money that paid for the equipment (EBIT of $0 but a negative EVA). If it costs Labrador 9.75% annually to deliver appropriate financial returns to the lenders and owners who provided the $2,850,000 to buy the equipment, then under a more inclusive view of the annualized fixed cost of equipment we compute the financial break-even point (for a $0 EVA) as

Financial BEP = Year ' s Cash ˗ Based ¿Operating Costs+( Cost of Long ˗ Lived Equipment

n ˗ Period PV of Annuity Factor ) ¿contributionmargin per unit sold

.With the specific values given in this problem, the applicable 8-year PV of an annuity factor is

( 1−( 11.0975 )

8

.0975 ) = 5.383828, and we compute

Fin. BEP = $ 465,000+ $2,850,0005.383828

$ 3.89−$2.67 = $ 465,000+$ 529,363.09

$ 1.22 = 994,363.09

1.22 = 815,051.72 ,

or 815,052 cords. If we ignore financing cost as a fixed cost, then we can think in terms of breaking even when 673,156 cords are made and sold, but if we include financing costs as a fixed cost then we must think in terms of making and selling a greater 815,052 units to break even.

Again, the only computational difference between the operating and financial break-even points is dividing the equipment cost by 8 vs. the 8-year PV of Annuity factor. The annualized cost of having the equipment in place has to be greater in the financial break-even case ($529,363.09) than in the operating break-even case ($356,250); the more-inclusive measure that includes a financial return must exceed the less-inclusive measure that includes only the equipment’s purchase cost.

6. One of Manitoba Confections’ many fine candy products is chocolate-covered Sophomore Mints. Five million boxes are sold each year, at a price of $.43 per box, to retailers (who then sell them to customers for a higher price). The variable cost of producing each box is $.17. Total fixed operating costs (in an accounting sense) incurred in producing the mints are $480,000 per year, and mint production accounts for $310,000 each year in interest payments. Compute Manitoba’s degree of operating leverage (DOL), degree of financial leverage (DFL), and degree of total leverage (DTL, also called degree of combined leverage DCL), and interpret these values. If sales increase by 30% in the coming year, what would we expect the change in net income to be?

Type: Operating, financial, total leverage. We compute the degree of operating leverage as


= Q( p−vc)EBIT

= 5,000,000($ .43−$ .17)5,000,000 ( $ .43−$ .17 )−$ 480,000

= $ 1,300,000$ 820,000

=

1.585

and the degree of financial leverage asTrefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 8

DFL = EBIT

EBIT−∫¿¿ = $ 820,000

$ 820,000−$310,000= $ 820,000

$ 510,000 = 1.608

The degree of total (or combined) leverage is the product of the DOL x DFL: 1.585 x 1.608 = 2.549.

We interpret a DOL of 1.585 as: If financial relationships that have held in the past continue to hold in the future, a 1% increase (decrease) in sales can be expected to bring about a 1.585% increase (decrease) in operating income, or earnings before interest and taxes (EBIT). (If there were no fixed operating costs, but rather all operating costs varied proportionally with output, then DOL would be 1, and a 1% change in sales would be associated with a 1% change in EBIT.)

We interpret a DFL of 1.608 as: If financial relationships that have held in the past continue to hold, a 1% change (increase or decrease) in EBIT can be expected to bring about a 1.608% change (increase or decrease) in net income. (If there were no fixed financing costs in the form of interest payments, but rather all financing came from common stockholders whose returns vary proportionally with the company’s output, then DFL would be 1, and a 1% change in EBIT would be associated with a 1% change in net income.)

Finally, we interpret a DTL of 2.549 as: If financial relationships that have held in the past continue to hold, a 1% change in sales can be expected to bring about a 2.549% change in net income. (DOL tells us how sales affects operating income, DFL tells us how operating income affects net income, and DTL cuts out the middle term, telling us how sales affects net income.) So if sales were to rise by 30%, net income would be expected to increase by 2.549 x 30% = 76.47% over its prior level.

Here the existence of fixed costs, in Manitoba’s operations and financing, causes the expected change in net income to be about two-and-a-half times any expected change in sales. If this degree of total leverage seems too high, Manitoba might wish to replace some debt financing with equity in order to reduce the DFL and, in turn, the DTL.

7. Maple Leaf Navigation Equipment, Inc.’s operating results have been very stable over the past few years. In each recent year the balance sheet has shown $65 million in assets, the income statement has shown operating income (EBIT) of $12 million, and the average federal-plus-state income tax rate has been 38%. The only major change has been a recent increase in the company’s debt/assets ratio (a financing issue that does not directly affect operating performance). The proportion of assets paid for with debt financing was recently increased from 40% (at which Maple Leaf paid a 7.5% average annual interest rate) to 55% (with an accompanying increase to 9.5% in the average annual interest rate, reflecting lenders’ perception that they face greater risk). Because this increase in debt financing increased Maple Leaf’s return on equity (ROE), some managers have proposed further increasing the debt ratio, to 70%. If the increase to this high level of debt would cause the average annual interest rate to rise to 16%, should the proposed change be made?

Type: Optimal capital structure. The computation of ROE for each of the three scenarios is as follows:

Previous: 40% Debt Current: 55% Proposed: 70%Earnings Before Interest & Taxes (EBIT) $12,000,000 $12,000,000 $12,000,000Minus Interest 1,950,0001 3,396,2503 7,280,0005

Earnings Before Taxes (EBT) 10,050,000 8,603,750 4,720,000Minus Taxes (38%) 3,819,000 3,269,425 1,793,600Net Income $6,231,000 $5,334,325 $2,926,400Stockholders’ Equity $39,000,0002 $29,250,0004 $19,500,0006


ROE = Net IncomeEquity

$6,231,000$ 39,000,000

$5,334,325$29,250,000

$ 2,926,400$ 19,500,000

= 15.9769% = 18.2370% = 15.0072%

1) .075 x (40% of $65,000,000) = .075 x $26,000,000 = $1,950,000. 2) Debt is 40% of $65 M = $26,000,000. Equity is the other (60% of $65 M) = $39,000,000. 3) .095 x (55% of $65,000,000) = .095 x $35,750,000 = $3,396,250. 4) Debt is 55% of $65 M = $35,750,000. Equity is the other (45% of $65 M) = $29,250,000.5) .16 x (70% of $65,000,000) = .16 x $45,500,000 = $7,280,000. 6) Debt is 70% of $65 M = $45,500,000. Equity is the other (30% of $65 M) = $19,500,000.

So consider what has happened. Initially, Maple Leaf made too little use of debt financing; its return on equity increased from 15.9769% to 18.2370% when the proportion of assets paid for with borrowed money (the debt/assets, or debt, ratio) was increased from 40% to 55%. (We are using ROE as a proxy for maximizing the value of the owners’ investment, an approach that is sensible if we can think of this year’s ROE as an indicator of long-term prospects.) The higher interest rate paid to the greater number of lender dollars was (in light of the income tax savings) more than offset by the proportional gain the owners realized by sharing the net income over fewer fellow owners’ dollars. So having more debt financing seems to work well for the owners. Or does it? Note that a further increase from 55% to 70% debt would cause ROE to fall not just below the current 18.2370%, but actually below the original 15.9769%.

Thus the proposed change should not be made. While 55% debt seems to be better for the owners than 40% debt, 70% debt does not seem to be better for the owners than 55% debt. These values would seem to illustrate the “smile-shaped” weighted average cost of capital function, with a lower cost of capital at an intermediate level of debt than at a too-low or too-high level.

8. The managers of New Brunswick Nutrients, a company with $10 million in assets, are investigating what their firm’s capital structure should be. The current financing mix is somewhat heavily “leveraged,” with a 60% debt ratio ($6 million in debt financing, $4 million in equity from 80,000 shares of common stock valued at $50 per share). Under one alternative structure being considered, the company would move to a 40% debt ratio (create 40,000 new $50 shares and use the proceeds to pay off $2 million worth of debt, leaving $4 million in debt and $6 million in equity). Under a second alternative, New Brunswick would move to a low 20% debt ratio (create 80,000 new $50 shares and use the proceeds to pay off $4 million in debt, leaving $2 million in debt and $8 million in equity). Operating income, or EBIT (which is largely unaffected by capital structure), is expected to be $900,000 per year under any chosen financing mix, and New Brunswick’s average federal-plus-state income tax rate is 32%. If the average annual interest rate paid on debt under any capital structure alternative would be 10%, what would return on equity (ROE) and earnings per share (EPS) be under each of the three capital structure alternatives? What if the average annual interest rate were instead 8%, or 9%?

Type: EBIT/EPS analysis. Here we want to compute ROE and earnings per share EPS based on three different interest rates and three possible capital structures. For a 10% interest rate, we find:

$6 million = $4 million = $2 million = 60% Debt 40% Debt 20% Debt

Operating Income (EBIT) $900,000 $900,000 $900,000Minus Interest (10% x debt) 600,000 400,000 200,000 Earnings Before Taxes (EBT) $300,000 $500,000 $700,000Minus Income Tax (32% of EBT) 96,000 160,000 224,000Net Income $204,000 $340,000 $476,000

Stockholders’ Equity1 $4,000,000 $6,000,000 $8,000,000Net Income ÷ Equity = ROE 5.10% 5.67% 5.95%Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 10

Number of Shares2 80,000 120,000 160,000Net Income ÷ Shares = EPS $2.55 $2.83 $2.98

However, for an 8% interest rate we find:$6 million = $4 million = $2 million = 60% Debt 40% Debt 20% Debt


Stockholders’ Equity1 $4,000,000 $6,000,000 $8,000,000Net Income ÷ Equity = ROE 7.14% 6.57% 6.29%Number of Shares2 80,000 120,000 160,000Net Income ÷ Shares = EPS $3.57 $3.29 $3.15

Yet for a 9% interest rate we find:$6 million = $4 million = $2 million = 60% Debt 40% Debt 20% Debt



1) If debt financing is $6 million, equity is $10 million – $6 million = $4 million; if debt financing is $4 million, equity is $10 million – $4 million = $6 million; and if debt financing is $2 million, equity is $10 million – $2 million = $8 million. 2) With stock worth $50 per share, $4 million in equity means $4 million ÷ $50 = 80,000 shares; $6 million in equity means $6 million ÷ $50 = 120,000 shares; and $8 million in equity means $8 million ÷ $50 = 160,000 shares.

Perhaps the first thing to notice is that, in absolute terms, a company is always better off if its inputs (materials, labor, money) cost less, holding all else equal. Here we see the highest range of ROE & EPS values in the 8% average annual interest rate case (ROE of 6.29% – 7.14%; EPS of $3.15 – $3.57); the next highest in the 9% average annual interest rate case (ROE of 6.12%, EPS of $3.06); and the lowest with the costly 10% average annual interest (ROE of 5.10% – 5.95%; EPS of $2.55 – $2.98). But we are not asked to make an absolute comparison; we must make a relative comparison based on a given interest rate assumption. After all, the interest rate may reflect market conditions outside the company’s control, whereas capital structure is the company’s choice.

So is New Brunswick too heavily financed with debt? Recall that financial managers’ goal should be to maximize the value of the owners’ investment in the firm, an outcome the managers would achieve by consistently delivering a high return on equity and high accompanying earnings per share. Note that if the average interest rate paid on borrowed money is 10% yearly, the estimated ROE and EPS steadily rise as the presumed debt ratio is reduced from 60% to 40% to 20%. So we might initially be inclined to conclude that replacing debt financing with equity, thereby achieving a lower debt ratio, is generally a good idea.


But then look at the results for the 8% average annual interest rate case: ROE and EPS would be expected to decline if the financing mix were to contain progressively less debt financing. Based on this observation, we might be inclined to conclude that maintaining a higher debt ratio is generally a good idea. So which is better, more debt financing or less? It should not be surprising to finance students that the answer depends on how much the debt financing costs. If debt is cheap, then we should incorporate more of it into the capital structure; if debt is expensive, then we should try to use less of it. But how do we define “cheap” or “expensive”?

“Financial leverage” (using relatively more debt financing) works out to a company’s advantage if the assets purchased with the borrowed money generate an annual rate of return greater than the annual interest rate paid on the borrowed money, because the excess provides an enhancement to the financial returns for the company’s owners. For example, if a company can borrow at an 8% annual interest rate, and use the borrowed money to buy assets that generate a 9% annual rate of return, then the 1% difference in the rates times the amount borrowed becomes an added dollar return to the owners (who get to keep what is left after all other parties, including the lenders, have been compensated), an outcome known as “positive leverage.” But if the assets purchased with the borrowed money provided a 9% annual return while the company paid a 10% annual interest rate, the – 1% difference in the rates times the amount borrowed would reduce the dollars available for the owners (“negative leverage”).

This result is exactly what we see in the 8% and 10% interest cases above. If the basic earning power ratio, computed as EBIT ÷ Total Assets, is greater than the interest rate, then financial leverage (borrowed money) has a positive, or beneficial, impact on the owners’ returns. In this problem, basic earning power is computed as


= $ 900,000$ 10,000,000

= .09 or 9%,

so it stands to reason that if it could borrow at less than 9% the firm would benefit from a higher debt ratio (60% debt provides the highest ROE and EPS in the 8% interest case). Conversely, if its average annual borrowing cost were greater than 9%, New Brunswick would benefit from a lower debt ratio (20% debt provides the highest ROE and EPS in the 10% interest case). If the annual borrowing cost were equal to the annual return (basic earning power) earned on the assets bought with the borrowed money, then the company’s managers would be indifferent regarding capital structure choice; the purchased assets would just exactly pay for the borrowed money that financed them, so no advantage would be gained by borrowing more or borrowing less. Note that for a 9% annual interest rate, ROE and EPS would be expected to be the same for any capital structure. (Income tax causes ROE and EPS to be lower than they would be in the absence of taxes, but the average income tax rate does not affect whether leverage has a positive or negative impact on ROE and EPS, as long as the rate is assumed to be the same for all capital structure choices – unless that rate is 100%, in which case net income, ROE, and EPS would all be zero.)

The preceding discussion is an example of what is known as “EBIT/EPS” analysis. The same level of EBIT (which relates to a company’s operating activities, not its financing choices) can lead to varying levels of EPS, depending on the amount, and the cost, of debt financing. This type of analysis can help us to think more clearly about general capital structure issues, but it suffers from some unrealistic underlying assumptions. The most unrealistic is that the interest rate on borrowed money would be 10% (or 9% or 8%) regardless of whether the company borrows a smaller or larger amount of money. In fact, a very real practical problem in “real world” capital structure analysis is that as the debt ratio increases and


lenders perceive more risk, the interest rate they charge is likely to increase, thereby making it increasingly unlikely that basic earning power will exceed the interest rate. It is also at least somewhat unrealistic to assume that the average income tax rate would be the same (here, 32%) no matter what level of measured taxable income the company realized, and under some circumstances it would be unrealistic to assume that the value per share of common stock would be the same no matter how many new shares were created.

Work These for Extra Insights

9. Each professional grade premium fishing rod manufactured by Newfoundland Premium Reel (NPR) is sold to sporting goods retailers at a $226 price. The variable cost per unit is $159.50, and the operation incurs $5,970,000 in yearly fixed costs ($3,970,000 in annual fixed payments and $2 million per year in straight-line depreciation on equipment that had a $14 million purchase cost and a seven-year expected life), as long as the year’s production does not exceed 125,000 units. How much annual net income does NPR earn, in an accrual accounting sense, if it produces and sells 100,000 units? How much does it lose if it makes and sells only 75,000 units? What is the operating (also called operating) break-even point? What change would we see in the operating break-even point if the price at which NPR sells each unit were to rise to $240, while all costs remained unchanged? What change would we see if the selling price were to rise to $240, but at the same time variable cost per unit rose to $176?

Type: Break-even. This problem is designed to help you put the break-even idea into a broader profitability context. Accrual accounting-based net income is measured as total revenue, minus fixed costs, minus total variable costs. At 100,000 units produced NPR’s accounting profit is

($226 x 100,000 units) – $5,970,000 – ($159.50 x 100,000 units) = $22,600,000 – $5,970,000 – $15,950,000 = $680,000

But at a much lower 75,000 unit output level the accounting profit is negative, at

($226 x 75,000 units) – $5,970,000 – ($159.50 x 75,000 units) = $16,950,000 – $5,970,000 – $11,962,500 = –$982,500

We can therefore see that the operating break-even point must be more than 75,000 units and less than 100,000 units, specifically:



= $ 3,970,000+ $ 14,000,0007

$ 226−$ 159.50 = $ 5,970,000

$226−$ 159.50 = $ 5,970,000$ 66.50 = 89,774.44, or 89,775

units

(we round to a whole unit because the company could not sell a fraction of a fishing rod, and we always round upward in break-even analysis because the lower value would leave the company slightly short of breaking even). The operating break-even point is the annual sales level that would provide a $0 operating income (which is also $0 EBIT if there is no non-operating income) for the year; let’s double check to make sure that the number we computed is correct:

($226 x 89,774.44 units) – $5,970,000 – ($159.50 x 89,774.44 units) = $20,289,023.44 – $5,970,000 – $14,319,023.18 = $0

It should be intuitively clear that if the price goes up and costs do not change, the firm can


break even without having to sell as many units. Here, with price rising to $240 and variable cost per unit remaining at $159.50, the contribution margin rises from $226 – $159.50 = $66.50 to $240 – $159.50 = $80.50. Each time NPR produces and sells another rod it covers the $159.50 differential cost and provides an extra $80.50 contribution toward meeting fixed costs (or generating an operating profit). So now we ask: how many $80.50 contributions does the company need to cover $5,970,000 in fixed costs?

Operating BEP = $ 5,970,000$ 240−$ 159.50 = $5,970,000

$ 80.50 = 74,161.49, or 74,162 units

Let’s again double-check:

($240 x 74,161.49 units) – $5,970,000 – ($159.50 x 74,161.49 units) = $17,798,757.60 – $5,970,000 – $11,828,757.66 = $0.

Now the break-even point is lower. But if NPR falls short of the break-even point, it now loses $80.50 rather than $66.50 for each unit by which it falls short.

Finally, it should be clear that a $14 increase in the selling price per unit (from $226 to $240), accompanied by a $16.50 increase in variable cost per unit (from $159.50 to $176), does not benefit the company. Here the contribution margin declines to $240 – $176 = $64 per unit. How many $64 contributions does NPR need to realize if it is to cover $5,970,000 in fixed costs?

Operating BEP = $ 5,970,000$ 240−$176

= $5,970,000$64

= 93,281.25, or 93,282 units,

a figure higher than NPR experienced before the price increase. Let’s double-check once again:

($240 x 93,281.25 units) – $5,970,000 – ($176 x 93,281.25 units) = $22,387,500 – $5,970,000 – $16,417,500 = $0.

With this increase in variable costs the break-even point becomes quite high, and each added unit sold after the break-even point contributes only $64 to profits. The good news (if there is any) is that each unit by which NPR falls short of the high break-even point represents only $64 of loss.

10. Northwest Territories Fabricating is buying new equipment so it can manufacture tiny high-carbon steel pins for industrial use. The equipment costs $4,000,000. The market for the pins is expected to remain strong for six years, and at the end of year six the equipment is expected to have a $400,000 salvage value. Fixed operating costs paid in cash are expected to be $1,100,000 each year. The variable cost of producing a pound of pins, which the company expects to sell for $12.99, is $8.00. Northwest’s managers feel that the weighted average cost of capital for a project like this one is 14.5%. Compute the annual operating (also called accounting) and financial break-even points.

Type: Break-even. A major difference between this example and earlier break-even questions is that here we have an expected salvage value. In computing the operating BEP we ignore the time value of money, so we adjust for the equipment’s expected salvage value simply by subtracting it from the purchase price: $4,000,000 – $400,000 = $3,600,000, and can compute:


Op. BEP = $ 1,100,000+ $ 3,600,0006

$ 12.99−$ 8.00 = $1,100,000+$ 600,000

$ 12.99−$8.00 = 1,700,000

4.99 = 340,681.36

pounds

of pins (we do not have to round, since it would be possible to produce a fraction of a pound of the small pins). But to compute Northwest’s financial break-even point, in which we consider the time value of money, we must first subtract the present value of the expected salvage value from the equipment’s purchase price to find the PV of the equipment cost:

$ 4,000,000(1.145 )0

– $ 400,000(1.145 )6

= $4,000,000 – $177,512 = $3,822,488

(when time value is taken into account, the right to collect $400,000 in six years is worth less than $400,000 today). So we can compute the appropriate PV of an annuity factor as

( 1−( 11.145 )

6

.145 ) = 3.836005 , and then can compute

Fin. BEP = $ 1,100,000+ $ 3,822,4883.836005

$ 12.99−$8.00 = $ 1,100,000+$996,476.40

$ 4.99 = 2,096,476.40

4.99 =

420,135.55

pounds of the pins. As always, the financial break-even point (based on the view that the financing cost is part of the equipment’s fixed cost, such that we think in terms of breaking even at a $0 EVA) must be greater than the operating break-even point (in which we ignore the financing cost, such that we think in terms of breaking even at $0 EBIT). While operatting BEP is the traditional break-even measure, we can argue that it understates the true break-even level because it is based on an incomplete measure of the project’s fixed costs.

11. Nova Scotia Paper Company’s degree of operating leverage (DOL) is 1.29, and its degree of financial leverage is 1.46. What is its degree of total leverage (DTL, also called degree of combined leverage DCL)? How do we interpret these values? What would the DOL be if Nova Scotia were producing at its operating break-even point?

Type: Operating, financial, total leverage. Operating leverage reflects a company’s reliance on fixed cost activities in its operations. Financial leverage reflects its reliance on fixed cost financing (debt, since interest payments to lenders are not related to the firm’s output). Total leverage is a combination of the two, which we compute as

DOL x DFL = DTL1.29 x 1.46 = 1.8834

We interpret a DOL of 1.29 as: If financial relationships that have held in the past continue to hold in the future, a 1% change (increase or decrease) in sales can be expected to bring about a 1.29% change (increase or decrease) in operating income, or earnings before interest and taxes (EBIT). We interpret a DFL of 1.46 as: If financial relationships that have held in the past continue to hold, a 1% change (increase or decrease) in EBIT can be expected to bring about a 1.46% change (increase or decrease) in net income.

We interpret a DTL of 1.8834 as: If financial relationships that have held in the past continue to hold, a 1% change (increase or decrease) in sales can be expected to bring about a Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 15

1.8834% change (increase or decrease) in net income. (DOL tells us how sales affects operating income, DFL tells us how operating income affects net income, and DTL cuts out the middle term, telling us how sales affects net income.)

Finally, notice that at the operating BEP, the DOL denominator Q(p – vc) – FC = EBIT is, by definition, equal to 0. So when the company is producing at its operating break-even point, the DOL equation reduces to


= Q( p−vc)EBIT

= Q( p−vc)$ 0

=

So when it is producing at its operating BEP, a company’s DOL is infinite. Recall that DOL is a proportional measure: the percentage change in EBIT that follows a given change in sales. If EBIT is $0, then any change (positive or negative) to a nonzero level is, proportionally, an infinite change.

12. Responding to some directors’ concerns that they are making too little use of debt financing, the managers of Nunavut Natural Flavorings are reviewing the company’s capital structure. The $96,000,000 in assets are currently financed with 15% debt and 85% common equity. Two other capital structure possibilities being considered would involve 35% debt or 70% debt (with the remainder from common equity). Nunavut common stock sells in the market for $135 per share. The company’s operating income, or EBIT (which is largely unaffected by capital structure), is expected to be $10,560,000 per year, over the next few years, no matter what financing mix is employed; and income tax is paid at a 38% combined federal-plus-state average rate. If the average annual interest rate Nunavut pays for debt financing, regardless of its capital structure choice, would be 7%, what would return on equity (ROE) and earnings per share (EPS) be under each of the three capital structure possibilities being considered? What if the company instead paid a 14%, or 11%, average annual interest rate?

Type: EBIT/EPS analysis. Here we want to compute ROE and EPS based on three different possible capital structures. Note that the amount of borrowed money would be either 15% of $96,000,000 = $14,400,000; 35% of $96,000,000 = $33,600,000, or 70% of $96,000,000 = $67,200,000. If the interest rate is 7%, we find:

$14,400,000 = $33,600,000 = $67,200,000 =

15% Debt 35% Debt 70% DebtOperating Income (EBIT) $10,560,000 $10,560,000 $10,560,000Minus Interest (7% x debt) 1,008,000 2,352,000 4,704,000 Earnings Before Taxes (EBT) $ 9,552,000 $ 8,208,000 $ 5,856,000Minus Income Tax (38% of EBT) 3,629,760 3,119,040 2,225,280Net Income $ 5,922,240 $ 5,088,960 $ 3,630,720


However, if the int. rate is 14%, we find: $14,400,000 = $33,600,000 = $67,200,000 =

15% Debt 35% Debt 70% DebtOperating Income (EBIT) $10,560,000 $10,560,000 $10,560,000Minus Interest (14% x debt) 2,016,000 4,704,000 9,408,000 Earnings Before Taxes (EBT) $ 8,544,000 $ 5,856,000 $ 1,152,000Minus Income Tax (38% of EBT) 3,246,720 2,225,280 437,760Net Income $ 5,297,280 $ 3,630,720 $ 714,240



Yet if the interest rate is 11%, we find: $14,400,000 = $33,600,000 = $67,200,000 =

15% Debt 35% Debt 70% DebtOperating Income (EBIT) $10,560,000 $10,560,000 $10,560,000Minus Interest (11% x debt) 1,584,000 3,696,000 7,392,000 Earnings Before Taxes (EBT) $ 8,976,000 $ 6,864,000 $ 3,168,000Minus Income Tax (38% of EBT) 3,410,880 2,608,320 1,203,840Net Income $ 5,565,120 $ 4,255,680 $ 1,964,160


1) If debt is $14.4 million, equity is $96 million – $14.4 million = $81.6 million; if debt is $33.6 million, equity is $96 million – $33.6 million = $62.4 million; and if debt is $67.2 million, equity is $96 million – $67.2 million = $28.8 million. 2) With stock worth $135 per share, $81.6 million in equity leaves $81.6 million ÷ $135 = 604,444 shares; $62.4 million in equity leaves $62.4 million ÷ $135 = 462,222 shares; and $28.8 million equity leaves $28.8 million ÷ $135 = 213,333 shares.

The highest range of ROE and EPS possibilities comes with the lowest interest rate (7%); the lowest range accompanies the highest interest rate (14%). A company is always better off if its inputs, including money, cost less, holding all else equal. But our task here is to make a relative comparison based on a given interest rate assumption. After all, the interest rate may reflect market conditions outside the company’s control, whereas capital structure is the company’s choice.

So are the directors right; is Nunavut not making enough use of debt financing? We can decide by looking at whether the managers are meeting the goal of maximizing the value of the owners’ investment in the firm, an outcome they would achieve by consistently delivering a high return on equity and high accompanying earnings per share. Note that if the average interest rate paid on borrowed money is just 7% yearly, the estimated ROE and EPS steadily rise as the presumed debt ratio is increased from 15% to 35% to 70%. So we might initially be inclined to conclude that replacing equity financing with debt, thereby achieving a higher debt ratio, is generally a good idea.

But then look at the results for the 14% average annual interest rate case: ROE and EPS would be expected to decline if the financing mix were to contain progressively more debt financing. Based on this observation, we might be inclined to conclude that maintaining a lower debt ratio is generally a good idea. So which is better, more debt financing or less? It should not be surprising to finance students that the answer depends on how much the debt financing costs. If debt is cheap, then we should incorporate more of it into the capital structure; if debt is expensive, then we should try to use less of it. We define debt financing as “cheap” or “expensive” depending on whether it costs less or more than the company’s basic earning power ratio. Here we find:


= $10,560,000$ 96,000,000

= .11 or 11%


“Financial leverage” (using relatively more debt financing) works out to a company’s advantage if the assets purchased with the borrowed money generate an annual rate of return (basic earning power) greater than the annual interest rate paid on the borrowed money, because the excess enhances the financial returns for the company’s owners. Here, if Nunavut borrows at a 7% annual interest rate and uses the borrowed money to buy assets that generate an 11% annual rate of return, then there is “positive leverage;” the 4% difference in the rates times the amount borrowed becomes an added dollar return to the owners (who get to keep whatever is left after other parties, including the lenders, have been paid).

But if the assets bought with the borrowed money provided an 11% annual return while the company paid a 14% annual interest rate, there would be “negative leverage” (the total dollar return to the owners would be reduced by the – 3% difference in the rates times the amount borrowed). And if basic earning power equals the interest rate paid, as in the 11% interest case above, ROE and EPS are unaffected by the firm’s capital structure choice; the reason is that the assets purchased with borrowed money would just exactly pay for the borrowed money.

The preceding discussion is an example of what is known as “EBIT/EPS” analysis. The same level of EBIT (which relates to a company’s operating activities, not its financing choices) can lead to varying levels of EPS, depending on the amount, and the cost, of debt financing. This type of analysis can help us to think more clearly about general capital structure issues, but it suffers from some unrealistic underlying assumptions. The most unrealistic is that the interest rate on borrowed money would be 7% (or 14% or 11%) regardless of whether a smaller or larger amount of money is borrowed. In fact, a serious practical problem in “real world” capital structure analysis is that as the debt ratio rises and lenders perceive more risk, the interest rate they charge is likely to go up, thereby making it increasingly unlikely that basic earning power will exceed the interest rate.

It also may be at least somewhat unrealistic to assume that the value per share of common stock would be the same no matter how many shares were created, and that the average income tax rate would be the same (here, 38%) no matter what level of measured taxable income the company realized. (Income tax causes ROE and EPS to be lower than they would be in the absence of taxes, but the average income tax rate does not affect whether leverage has a positive or negative impact on ROE and EPS, as long as the rate is assumed to be the same for all capital structure choices – unless that rate is 100%, in which case net income, ROE, and EPS would all be zero.)

13. Mr. Ontario owns 6,720 of the 3.36 million outstanding shares of common stock in Prince Edward Island Corporation (PEIC). Each share is worth $25 (we will treat market and book values as being the same). The company has $120 million in assets, financed 30% with debt and 70% with common equity. PEIC’s managers, who feel the firm could benefit from making greater proportional use of borrowed money, have considered increasing the debt ratio to 44%. The average annual interest rate PEIC pays on debt has been 6.85%, a figure that would not be expected to change even if debt financing increased. EBIT (which is essentially unaffected by the capital structure) is expected to be $9,500,000 annually in each of the next several years, and for purposes of this analysis we ignore income taxes. How much financial return does PEIC generate for Mr. Ontario each year, under the current capital structure? How much financial return would it generate for him each year if the proposed change were made, and Mr. Ontario kept all of his shares? How might he return to a position financially equivalent (in terms of risk and total financial return) to the one he was in before the change? How does this situation relate to Modigliani & Miller’s capital structure irrelevance ideas?

Type: Homemade leverage. The common stockholders’ total investment in PEIC is 3,360,000 shares worth $25 per share = $84,000,000. Mr. Ontario owns 6,720 shares x $25 = $168,000 worth of that equity, so he has 6,720/3,360,000 (or think $168,000/$84,000,000) = .002 or .2% of the outstanding total. With .30 x $120 million = $36 million in debt financing (so .0685 x $36,000,000 = $2,466,000 of the EBIT is paid as interest to lenders), and with no income tax (so none of EBIT Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 18

is paid to the government; it all goes to the lenders and owners such that EBIT = NOPAT), the net income should be $9,500,000 EBIT – $2,466,000 interest = $7,034,000 (for $7,034,000 total earnings ÷ 3,360,000 total shares = $2.093452 earnings per share). And with a .2% claim, Mr. Ontario gets .2% of the $7,034,000 net income, or .002 x $7,034,000 = $14,068 as his total financial return each year under the current capital structure.

But if equity financing were to be replaced with debt up to a point at which lenders provided 44% of the $120 million invested = $52,800,000, then equity investors would provide only the other 56%, or $67,200,000. To get from $84,000,000 down to $67,200,000 in equity, the company would buy $84,000,000 – $67,200,000 = $16,800,000 worth of common stock (from stockholders other than Mr. Ontario, who keeps all his shares). At $25 per share, it would therefore retire $16,800,000 ÷ $25 = 672,000 shares from the market, so the remaining shares would number only 3,360,000 – 672,000 = 2,688,000.

With $168,000 coming from Mr. Ontario, he would have $168,000/$67,200,000 (or 6,720 ÷ 2,688,000) = .25% of the outstanding total. With a 6.85% interest rate the firm would pay .0685 x $52,800,000 = $3,616,800 each year to the lenders, thus leaving only $9,500,000 – $3,616,800 = $5,883,200 as net income for the equity investors (for $5,883,200 total earnings ÷ 2,688,000 total shares = $2.188960 earnings per share). Since Mr. Ontario would, after the change, account for .25% of the equity base, his total financial return (his share of the net income) would be a higher .0025 x $5,883,200 (or 6,720 shares x $2.188960 EPS) = $14,708 under the 44% borrowing arrangement.

So shouldn’t Mr. Ontario be happy? Not necessarily; the change in capital structure would also change his risk/return profile. A higher debt ratio reduces the proportional number of equity dollars with which the net income must be shared, leaving Mr. Ontario with a chance for a higher return on equity (more net income on his unchanged equity investment). But it also increases the proportional number of lender dollars that must be paid interest before we can view what is left as net income for the owners, so each remaining owner is left in a riskier position. How can Mr. Ontario replicate the results of the earlier plan that he was happier with ($14,068 annual financial return and a 30% debt ratio), in which he did not get as much money but also faced less risk because there were fewer lenders standing in front of him in the line of money providers? First, since the company has moved to using 44% – 30% = 14% more borrowed money, he can undo that unwanted capital structure change’s impact on him by becoming a lender (you un-borrow by lending). He sells shares equal in value to the percentage change in the debt ratio, times his original ownership percentage, times the company’s asset total, here .14 x .002 x $120,000,000 = $33,600 (which involves $33,600 ÷ $25 = 1,344 shares). Then he lends that money out (lending to undo the impact of the company’s borrowing) at a 6.85% rate, realizing .0685 x $33,600 = $2,301.60 per year in interest. But he then has only 6,720 – 1,344 = 5,376 shares on which the $2.188690 annual EPS is earned: 5,376 x $2.188690 = $11,766.40. So he should end up with $2,301.60 from lending + $11,766.40 in net income on his remaining shares of stock = $14,068 in total financial return.

In the M&M world of no taxes, perfect capital markets (no differences in interest rates among borrowers/lenders, big/small transactors), and no transaction costs, the individual investor can take steps (using “homemade leverage”) to undo any unwanted risk/return impact of the company’s capital structure maneuverings. In such a world, it would make no sense for corporate managers to devote time and resources to finding the optimal capital structure, because any mistakes they made could be quickly, easily, and costlessly corrected by individual investors. But recall that M&M’s genius is not that they proved capital structure to be irrelevant; in the world we actually inhabit we are almost sure that capital structure choices are important to maximizing stockholders’ value or wealth. Their great insight was that they assumed away a few simple real-world issues to get an artificial world where capital structure does not matter. Those things they assumed away (taxes, transaction costs, imperfect capital markets, lenders’ lack of certainty on how their money may be used) thus are the issues that are likely to cause capital structure to matter. Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 19

14. Ms. Quebec owns 625 of the 12,500 outstanding shares of common stock in Saskatchewan School Services. Each share is worth $12 (we will treat market and book values as being the same). One feature that attracted Ms. Quebec to the stock was Saskatchewan’s aggressive use of debt financing; the company’s $375,000 in assets have been financed 60% with debt and 40% with common equity. Saskatchewan’s managers, on the other hand, fear that the company is overburdened with debt, and they have decided to reduce the debt ratio to 36%. The average annual interest rate Saskatchewan pays on debt has been 8.75%, a figure that would not be expected to change even if debt financing were reduced. EBIT (which is essentially unaffected by the capital structure) is expected to be $46,500 annually in each of the next several years, and for purposes of this analysis we ignore income taxes. How much total financial return (her proportional share of net income) does Saskatchewan generate for Ms. Quebec each year, under the current capital structure? How much total financial return will it generate for her each year after the change is made, if she holds the same number of shares after the change? How might she return to a position financially equivalent (in terms of risk and total financial return) to the one she was in before the change? How does this situation relate to Modigliani & Miller’s capital structure irrelevance ideas?

Type: Homemade leverage. The common stockholders’ total investment in Saskatchewan is 12,500 shares worth $12 per share = $150,000. Ms. Quebec owns 625 shares x $12 = $7,500 worth of it, so she has 625/12,500 (or look at it as $7,500/$150,000) = .05 or 5% of the outstanding total. With .60 x $375,000 = $225,000 in debt financing (so .0875 x $225,000 = $19,687.50 of the EBIT is paid each year as interest to lenders), and with no income tax (so none of EBIT is paid to the government; it all goes to the lenders and owners such that EBIT = NOPAT), net income should be $46,500 EBIT – $19,687.50 interest = $26,812.50. With 12,500 shares outstanding, earnings per share (EPS) is $26,812.50 ÷ 12,500 = $2.145. And with a 5% claim, Ms. Quebec gets 5% of the $26,812.50 net income, or .05 x $26,812.50 (or think of it as 625 shares x $2.145 EPS) = $1,340.625 in total financial return each year under the current capital structure arrangement.

But if debt financing were to be replaced with equity up to a point at which lenders provided just 36% of the $375,000 invested = $135,000, then equity investors would provide the other 64%, or $240,000 (for $240,000 ÷ $12 = 20,000 shares). With an 8.75% interest rate the firm would pay .0875 x $135,000 = $11,812.50 each year to the lenders, thus leaving $46,500 – $11,812.50 = $34,687.50 as net income for the equity investors ($34,687.50 ÷ 20,000 shares = $1.734375 EPS). With $7,500 coming from Ms. Quebec (she is not among those who buy new shares that give the company money to repay some debt), she would have $7,500/$240,000 = 3.125% of the new outstanding total. Since Ms. Quebec would, after the change, account for only 3.125% of the equity base, her claim on the net income would be a lower .03125 x $34,687.50 (or 625 shares x $1.734375 EPS) = $1,083.984375 under the 36% borrowing arrangement.

The lower debt ratio reduces the proportional number of lender dollars that must be paid interest before we can view what is left as net income for the owners, so each remaining owner is left in a less risky position. But it also increases the proportional number of equity dollars with which net income must be shared, leaving Ms. Quebec with a lower expected return on equity (less net income on her unchanged equity investment). How can the risk-tolerant Ms. Quebec replicate the results of the original high-debt plan, with which she was happy? Note that to get from $150,000 up to $240,000 in equity, the company would sell $240,000 – $150,000 = $90,000 worth of new stock (to people other than Ms. Quebec, who buys none of the new shares). At $12 per share, it would therefore create $90,000 ÷ $12 = 7,500 new shares. So the resulting total number of shares would be 12,500 + 7,500 = 20,000 (as shown above), of which Ms. Quebec would own 625 ÷ 20,000 = .03125, or 3.125% of the new outstanding total. Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 20

But Ms. Quebec was happier with the old system (she faced more risk, but also got more net income, because there were fewer fellow owners standing alongside her in the line of money providers). What can she do to restore her earlier situation ($1,340.625 annual financial return and a 60% debt ratio)? First, since the company has moved to using 60% – 36% = 24% less borrowed money, she can undo that unwanted capital structure change’s impact by borrowing money on her own, to replace what she thinks the company should borrow. She borrows an amount equal to the percentage change in the debt ratio, times her original ownership percentage, times the company’s asset total, here .24 x .05 x $375,000 = $4,500. She pays an 8.75% annual interest rate for the use of this money, so it costs her .0875 x $4,500 = $393.75 per year. Then she uses that money to buy $4,500 ÷ $12 = 375 shares from other investors in the market, bringing her total number of shares up to 625 + 375 = 1,000. Her share of the firm’s annual net income, with this larger number of shares, will be 1,000 x $1.734375 = $1,734.375. So she should end up with $1,734.375 in net income – $393.75 interest paid on what she borrowed = $1,340.625 in total annual financial returns.

In the M&M world of no taxes, perfect capital markets (no differences in interest rates among borrowers/lenders, big/small transactors), and no transaction costs, the individual investor can take steps (using “homemade leverage”) to easily undo any unwanted risk/return impact of the company’s capital structure maneuverings. In such a world, it would make no sense for corporate managers to devote time and resources to finding the optimal capital structure, because any mistakes they made could be quickly, easily, and costlessly corrected by individual investors. But recall that M&M’s genius is not that they proved that capital structure is irrelevant; in the world we actually inhabit it is almost certain that capital structure choices are important to maximizing stockholders’ value. Their great insight was that they assumed away a few real-world issues and got an artificial world in which capital structure does not matter. Those few simple things they assumed away (taxes, transaction costs, imperfect capital markets, lenders’ lack of information on how their money may be used) thus are the issues that are most likely to cause capital structure to matter.

15. Yukon Yogurt, Inc. sells 15,000 cases of yogurt to food stores each year for $39 per case. The variable cost (labor, electricity, materials) of producing each case is $34, and accrual-based annual fixed operating costs are $50,000 ($40,000 in cash-based fixed costs such as rent, and $10,000 in annual straight-line depreciation on equipment with a $70,000 cost and a 7-year expected life). Consultants feel that Yukon could reduce the variable cost of producing each case (primarily overtime labor and wasted materials) by switching to a more capital-intensive production process, in which accrual-based annual fixed operating costs would rise to $60,000 ($40,000 cash-based, $20,000 annual straight-line depreciation on costlier equipment with a $140,000 cost and a 7-year life). Because they believe that the more efficient equipment would allow variable cost to fall to $33.00 or even $32.50 per case, the consultants conclude that the switch should unquestionably be made. Are they correct? If you, as Yukon’s chief financial officer, feel that using the more expensive equipment would bring variable costs down only to $33.50 per case, should you approve the change?

Type: Break-even. Under the existing production setup, Yukon’s annual accounting profit at 15,000 cases sold is

Total Revenues – Fixed Costs – Total Variable Costs= ($39.00 x 15,000 units) – $50,000 – ($34.00 x 15,000 units)

= $585,000 – $50,000 – $510,000 = $25,000

Under the more capital-intensive arrangement, with $60,000 in annual fixed costs, the annual accounting profit would be a much higher

($39.00 x 15,000 units) – $60,000 – ($32.50 x 15,000 units) = $585,000 – $60,000 – $487,500 = $37,500

if variable cost per case were $32.50; or a still attractive


($39.00 x 15,000 units) – $60,000 – ($33.00 x 15,000 units) = $585,000 – $60,000 – $495,000 = $30,000

if variable cost per case were $33.00. So the consultants would seem to be right, based on the variable cost figures they have assumed. But note that accounting profit would be only

($39.00 x 15,000 units) – $60,000 – ($33.50 x 15,000 units) = $585,000 – $60,000 – $667,500 = $22,500

if variable cost per case were $33.50. So switching to a process with higher fixed costs is good if variable cost per case is reduced substantially, but bad if it is reduced only slightly. How can we examine this issue systematically? First, compute the operating break-even point under the existing production arrangement:



$ 40,000+ $70,0007

$39.00−$34.00 = $ 40,000+$ 10,000

$39.00−$ 34.00 = $50,000

$ 5.00 = 50,000

5 = 10,000 cases

If the switch were made, and variable cost per case were reduced to $32.50, then the contribution margin (CM) would rise from the existing $39.00 – $34.00 = $5.00 per case to a higher $39.00 – $32.50 = $6.50 per case, and the operating break-even point would be a considerably lower

$ 40,000+ $ 140,0007

$39.00−$ 32.50 = $ 40,000+$ 20,000

$39.00−$ 32.50 = $60,000

$6.50 = 60,000

6.5 = 9,231 cases

When the contribution margin rises, the amount contributed to meeting fixed costs (and ultimately generating a profit) by each added case rises. But because we expect a higher CM to be possible only if we incur higher fixed costs, the key is to have CM rise proportionally by enough to offset the increased fixed cost. Here CM rises to $6.50/$5.00 = 130% of (meaning a 30% increase over) its original level, whereas fixed costs rise to only $60,000/$50,000 = 120% of (a 20% increase over) their original level. (If the contribution margin rises by a higher percentage than fixed costs, the BEP falls.) This result is the “best of both worlds;” Yukon would break even at a lower level of output, and each added unit produced and sold would contribute more to accrual accounting-based net income. (The only downside would be that the new plan would lead to greater losses at very low output levels, but here the clear expectation is for 15,000 cases to be made and sold each year.) If the switch were made and variable cost per case fell only to $33.00, the CM would rise, but only to $39.00 – $33.00 = $6.00 per case, for an operating break-even point unchanged at

$ 40,000+ $ 140,0007

$39.00−$33.00 = $ 40,000+$ 20,000

$39.00−$33.00 = $ 60,000

$6.00 = 60,000

6 = 10,000 cases

Here the CM rises proportionally to the same $6.00/$5.00 = 120% of its original level as we see with fixed costs: $60,000/$50,000 = 120%. (If fixed costs rise by the same percentage as the contribution margin, the BEP stays the same.) This result is still quite good; Yukon would break even at the same annual 10,000 cases made and sold that prevails under the existing arrangement, and then each unit sold beyond that break-even point would contribute more ($6.00 rather than $5.00) to accrual accounting-based net income. So the Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 22

consultants would seem to be correct, based on the variable cost possibilities they considered and an annual 15,000 case sales level.

However, if the switch were made and variable cost per case fell just a small amount, to $33.50, the CM would rise to only to $39.00 – $33.50 = $5.50 per case, for an operating BEP of

$ 40,000+ $ 140,0007

$39.00−$33.50 = $ 40,000+$ 20,000

$39.00−$33.50 = $ 60,000

$5.50 = 60,000

5.5 = 10,910 cases

Under your assumptions, the CM would rise proportionally to only $5.50/$5.00 = 110% of the original level, less than the 120% we see for fixed costs. (If fixed costs increase by a greater percentage than the CM, the BEP rises.) As shown above, the resulting accounting profit would be only $22,500 (less than the existing $25,000 annual figure), so you should be reluctant to approve the change. It is true that if the change were made each case sold beyond 10,910 would contribute $5.50 (not just $5.00) to income. But under the existing system, each unit sold beyond 10,000 already contributes $5.00 to income. So to compare the proposed change to the existing system, we must recognize that at the new plan’s break-even level (10,910 cases, if variable cost is $33.50 per case), the original plan (with a 10,000 case break-even) would already be generating 910 x $5 = $4,550 in accounting-based net income.

So incurring higher fixed costs that reduce variable costs to $33.50 per case would be good for Yukon only if a sufficient increase in production and sales were expected. Thus the important question to ask is not simply whether Yukon can hit a higher break-even point. It is whether the firm can reach a still-higher sales level, beyond the BEP, at which it is indifferent between existing and proposed production arrangements. We find the output level at which a company is indifferent between an existing production plan and a proposed change by solving for quantity Q in

p•Q – FC1 – vc1• Q = p•Q – FC2 – vc2•Q ,

an equation that equalizes revenue minus total cost = profit under both plans (p is expected selling price per unit, FC1 is existing level of total annual fixed costs, FC2 is expected annual fixed cost total if the switch is made, vc1 is variable cost per unit under existing arrangement, vc2 is expected variable cost per unit under proposed plan). The price would not change just because the fixed/ variable cost mix changes; customers are not likely to know or care about the production process (although we could criticize this type of analysis for treating the price as being the same no matter how many cases are sold). This equation simplifies to

Q (vc1 – vc2) = FC2 – FC1

Q = FC 2−FC1

vc2−vc1 ,

which for the $33.50 per case variable cost possibility gives us

Q = FC 2−FC1

vc2−vc1 = $ 60,000−$ 50,000

$ 34.00−$ 33.50 = $ 10,000

$ .50 = 10,000

.50 = 20,000 cases

Under the more capital-intensive process, each case sold beyond 10,910 contributes $5.50 (not just $5.00) to profit. But for output up to 20,000 cases Yukon would be better off with the existing arrangement. Then at 20,000 cases sold the accounting income is the same under either plan: Trefzger/FIL 240 & 404 Topic 8 Problems & Solutions: Capital Structure 23

($39.00 x 20,000 units) – $50,000 – ($34.00 x 20,000 units) = $780,000 – $50,000 – $680,000 = $50,000

($39.00 x 20,000 units) – $60,000 – ($33.50 x 20,000 units) = $780,000 – $60,000 – $670,000 = $50,000

And then for each case made and sold beyond 20,000 the new plan would contribute an extra $5.50 to accounting profit, while the existing plan would contribute only an extra $5.00. For example, at 21,000 cases sold the respective accounting income levels would be

($39.00 x 21,000 units) – $50,000 – ($34.00 x 21,000 units) = $819,000 – $50,000 – $714,000 = $55,000 and

($39.00 x 21,000 units) – $60,000 – ($33.50 x 21,000 units) = $819,000 – $60,000 – $703,500 = $55,500;

the higher fixed cost/higher BEP arrangement would generate the greater profit for output levels beyond the 20,000 annual case indifference point.


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