Note on Valuing Equity Cash Flows
9 -2 9 5 -0 8 5 REV: SEPTEMBER 20, 2012
TIMOTHY LUEHRMAN
Note on Valuing Equity Cash Flows This note introduces a discounted cash flow (DCF) methodology for valuing highly levered equity claims on operating assets. The equity cash flow methodology is an alternative to option-pricing, which is theoretically preferable, but frequently impractical. It also is an alternative, or a complement, to other DCF approaches, such as valuing assets and then subtracting the value of the debt. This latter method is practical, but has other limitations and biases the net effect of which is unclear. Equity cash flows have the advantage that, though they also give a biased result, the sign of the bias is known. Properly applied, methodology gives a lower bound for the value of highly levered equity, which is often a useful result. This note introduces the methodology; shows how to use it; discusses the sources and signs of its built-in biases; and provides some guidance about when to use it.
The Methodology It is well known that holders of highly levered equity essentially own a call on the assets of the firm. They can exercise the call by paying off the outstanding debt. They will do so if, when the debt comes due, the assets are worth more than the face amount of the obligation. If, when the debt is due, the assets are worth less this amount, the equity holders will not exercise—that is, they will not pay off the debt in cash, but rather let the lenders keep the assets. This simple description is clear enough. Unfortunately, in reality the structure of a typical long-term debt contract creates multiple call options with complex characteristics. Every time an interest or principal payment comes due, the equity holders have an exercise decision to make. In effect, they hold a sequence of nested options— that is, options on options—rather than a simple call option. Other common features of corporate debt contracts (for example, callability, convertibility, covenants, etc.) introduce still more options, and the valuation problem becomes even harder. When it is impossible to apply option-pricing techniques to highly levered equity, it still will be desirable to value it somehow—a flawed analysis generally will be better than no analysis. It is natural in this situation to fall back on DCF techniques. To determine whether one or another DCF approach is preferable, consider first a firm with low leverage, illustrated in Figure 1. Because there is little debt, the chance that the firm will default is remote. Consequently, the equity holders' option to walk away from the debt and the assets is not very valuable and ignoring it will not seriously distort the analysis. The firm's assets, including tax ________________________________________________________________________________________________________________ Professor Timothy A. Luehrman prepared this note as the basis for class discussion. Copyright © 1994, 2005, 2009, 2012 President and Fellows of Harvard College. To order copies or request permission to reproduce materials, call 1-800-545-7685, write Harvard Business School Publishing, Boston, MA 02163, or go to http://www.hbsp.harvard.edu. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of Harvard Business School.
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shields from its modest leverage, can be valued with standard DCF techniques.1 Similarly, the value of the debt is the present value of promised interest and principal payments, discounted at the cost of debt. Subtracting the value of the debt from the value of the assets gives the value of the equity. Figure 1: Applying DCF techniques to a firm with low leverage A firm with low leverage Assets (including interest tax shields) Value assets using unlevered cash flows and an unlevered discount rate.
Value of the assets
Debt
Value of debt cash flows, discounted at the cost of debt.
Equity Value of the claims
Value of equity cash flows, discounted at the cost of (levered) equity.
With low leverage, it does not matter whether we value equity directly or as assets-minus-debt.
Alternatively, the equity may be evaluated directly, by discounting expected equity cash flows at the cost of equity. Equity cash flows are simply the cash flows associated with the assets and tax shields, less those associated with the debt. This alternative is also shown in Figure 1. When leverage is low, it generally does not matter whether we value the assets and then subtract the debt, or discount the equity cash flows themselves. Now suppose that leverage is high and that the amount of debt changes over time. Figure 2 shows the same assets, now including both more tax shields and the expected costs of financial distress. It also shows three layers of debt on top of a small amount of equity. Even if we stipulate that this firm is solvent, default is clearly much more likely and the option held by equity owners is accordingly a more important feature of the problem. In this setting, it is not a matter of indifference whether we value the assets and subtract the debt, or value the equity cash flows directly. First note that either way is more difficult than before, with low leverage. Valuing the assets is a bit trickier because there are more and riskier tax shields and costs of financial distress to worry about. Valuing the debt to be subtracted is much harder now. It is unlikely to be traded, so market values cannot be observed. All but the most senior debt is risky, and there are three different layers, which in the event of default have interdependent values. In the end, after executing a difficult analysis, it will be hard to tell whether we have systematically over- or undervalued the equity by using the assetsminus-debt approach.
1 This presumes there are no real options on the asset side of the balance sheet. A treatment of real options is beyond the scope of this note, but see, for example, Capital Projects as Real Options: An Introduction, HBS case no. 295-074.
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Figure 2: Applying DCF techniques to highly levered equity The same firm, with high leverage
Value assets using unlevered cash flows and an unlevered discount rate.
Same assets (including more interest tax shields and costs of financial distress) New value of the assets
Debt #1 Debt #2 Debt #3 Equity
Value by layer, using debt cash flows and the costs of debt.
Use equity cash flows and the cost of (levered) equity.
New value of the claims
With high leverage, valuing equity directly is probably simpler than assets-minus-debt and, under certain assumptions, the sign of the bias in the equity-cash-flow approach is known.
Valuing the equity directly—i.e., discounting equity cash flows at the cost of equity—requires one complex analysis rather than four. But more importantly, by making certain assumptions, we can be fairly confident that this approach will underestimate the value of the equity. Collectively, these assumptions imply that equity holders will not exercise their option to walk away, even when it is in their best interests to do so. In general, equity is more valuable than this unrealistic assumption makes it appear, so we know the sign of the bias is negative. The magnitude of the bias is unclear; it depends on all the usual determinants of option value.
A Simple Example A simple leveraged buyout provides an illustration of how to apply the methodology. Suppose an LBO firm has tentatively agreed to pay $1.1 billion to acquire a certain business. The managers of the business anticipate EBIT of $125 million for the coming year and this is expected to grow in perpetuity at 3% per year. The LBO sponsors are confident that they can raise $900 million in debt financing, provided they contribute $200 million of equity and that all excess cash generated by the business is used to pay down debt. The sponsors expect to divest their holdings after five years. Should they invest the $200 million and acquire the business? In other words, is the equity worth more than $200 million?
Cash flows and terminal value xValuing the equity in this LBO will require some additional information, but we already know a lot about how to frame the analysis. The equity cash flows over the first five years are very simple: they equal zero, by construction, because all available cash is going to debt holders. In fact, the only equity cash flow to be discounted is the terminal value of the equity at the five-year horizon. This simple set-up is shown in Figure 3.
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Figure 3:
Equ ity cash flow s for a sim p le LBO
Yearly cash flow s
Closing
Equ ity cash flow Equ ity term inal valu e
EOY 1
2
0.0
3 0.0
4 0.0
5 0.0
0.0 ?
One way to estimate the terminal value of the equity is to multiply projected net income at the terminal horizon by a price-earnings ratio. However, p-e ratios are affected by leverage, and selecting the right one is tricky. A safer alternative is to estimate the terminal value of the assets (for example, as a growing perpetuity, or via a market multiple) and subtract the terminal value of the debt. The amount of debt at the end of year 5 depends on how much of the original $900 million can be retired using internally generated funds during years 1-5. Figure 4 presents projections for interest and principal payments, under the additional assumptions that the tax rate is 36%; the interest rate on the debt is 7%; capital expenditures equal depreciation (both grow at 3% per year); and additions to net working capital start at $10 million in year 1 and grow at 3%. Figure 4 shows that cash flow available to retire debt equals $29.7 million in year 1 and grows to $44.7 million by year 5. Applying these funds to debt reduction lowers the outstanding debt to $715.1 million at the end of the fifth year.
Figure 4:
Financial Projections for a simp le LBO
Yearly cash flow s
Closing
EBIT Interest p aym ents Pre-tax p rofits Taxes @36% N et income Dep reciation Cap ital exp end itu res N WC ad d itions Cash flow available Princip al p aym ents Equ ity cash flow Yearly debt schedule Beginning d ebt Interest p aym ents (7%) Princip al p aym ents End ing d ebt
EOY 1 125.0 -63.0 62.0 -22.3 39.7 20.0 -20.0 -10.0 29.7 -29.7 0.0
Closing
900.0
EOY 1 900.0 63.0 29.7 870.3
2 128.8 -60.9 67.8 -24.4 43.4 20.6 -20.6 -10.3 33.1 -33.1 0.0 2 870.3 60.9 33.1 837.2
3 132.6 -58.6 74.0 -26.6 47.4 21.2 -21.2 -10.6 36.8 -36.8 0.0 3 837.2 58.6 36.8 800.5
4
5
136.6 -56.0 80.6 -29.0 51.6 21.9 -21.9 -10.9 40.6 -40.6 0.0 4
140.7 -53.2 87.5 -31.5 56.0 22.5 -22.5 -11.3 44.7 -44.7 0.0 5
800.5 56.0 40.6 759.8
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759.8 53.2 44.7 715.1
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Assuming a risk-free rate of 4%, a market risk premium of 6%, an asset beta of 0.85, and a perpetuity growth rate of 3%, the terminal value of the assets at the end of year 5 is about $1,450 million.2 Subtracting the terminal value of the debt gives a terminal value for the equity of $734 million. This value goes in Figure 3, to be discounted back to the present at the cost of equity.
Discount rates The cost of equity is the expected return that investors require as compensation for holding the stock. It depends on the amount of systematic risk borne by stockholders. This in turn depends on the systematic risk inherent in the business and on the firm's capital structure. As the capital structure changes, so does the cost of equity. In this example, leverage falls as debt is paid off. Systematic risk borne by stockholders falls accordingly, assuming the asset beta remains unchanged. As risk goes down, the cost of equity goes down, too. Figure 5 shows how to use the estimated terminal value of equity to generate an estimated cost of equity at the terminal horizon: 1.
The terminal values of debt and equity together are sufficient to determine an equity ratio at the end of year 5.
2.
Dividing the asset beta of 0.85 by the equity ratio of 0.51 gives a levered equity beta of 1.68.
3.
This levered beta can be used in the CAPM, along with a risk-free rate of 4% and a market risk premium of 6%, to get a cost of equity at the end of year 5 of 14%.
4.
Applying this discount rate to the $734 million terminal value of equity gives $643.5 million for the year-4 value of equity.
5.
Now we can repeat the process to obtain a year-4 cost of equity and year-3 value of equity, and so forth.
6.
The analysis terminates with an "answer," the value of equity at the closing of the transaction.
Figure 5:
Discou nting equ ity cash flow s by w orking backw ard s
Yearly discount ing End ing MV of equ ity End ing d ebt + equ ity Debt/ cap ital Equ ity/ cap ital Asset beta Levered (equ ity) beta Cost of equ ity
Closing answ er
EOY 1
2
3
4 643.5
5 734.0 1449.1 0.49 0.51 0.85 1.68 14.07%
2 The terminal value of the assets should include the terminal value of interest tax shields. This in turn depends on the expected capital structure. The estimate given in the text assumes a constant debt ratio following year 5, so the weighted average cost of capital, given by assumption in the Appendix as 8.6%, can be used in the perpetuity formula. An alternative assumption is that the dollar amount of debt (and hence the dollar amount of the annual tax shield) stays constant in perpetuity. A third possibility is that the existing debt is paid off on schedule and not refinanced. Note that each assumption will give a different terminal value for interest tax shields.
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Figure 6 shows the completed analysis, and a present value for the LBO equity of about $340 million, well above the sponsors' acquisition cost of $200 million. [The Appendix consolidates Figures 3-6 to summarize the analysis on one page.] Note that, as expected, the cost of equity declines over time, from 20% to 14%. This is due solely to the drop in the debt-to-value ratio, from 68% at closing to 49% at the terminal horizon. Figure 6:
Discou nting equ ity cash flow s: the com p leted analysis
Yearly discount ing End ing MV of equ ity End ing d ebt + equ ity Debt/ cap ital Equ ity/ cap ital Asset beta Levered (equ ity) beta Cost of equ ity
Closing 339.1
EOY 1 407.0 1277.3 0.68 0.32 0.85 2.67 20.01%
2 480.2 1317.4 0.64 0.36 0.85 2.33 17.99%
3 559.0 1359.4 0.59 0.41 0.85 2.07 16.40%
4 643.5 1403.3 0.54 0.46 0.85 1.85 15.12%
5 734.0 1449.1 0.49 0.51 0.85 1.68 14.07%
Biases The equity cash flow analysis presented above is biased in the sense that, given unbiased inputs, the methodology produces a biased-low output. Obviously, an analyst choosing to use very aggressive forecasts of EBIT, for example, still can obtain a biased-high result—biased inputs do indeed produce biased outputs. But the important point is that there is a bias in the methodology independent of the forecasts. It comes from two main sources.
Promised vs. expected cash flows When leverage is high, debt is risky. It quite possibly will not be repaid as promised—the lenders might receive only a fraction of what they are owed or they might receive it later than promised. For risky debt, then, expected cash flows must be less than promised cash flows. This is why the lenders in our LBO example demand a promise of 7% interest rather than the risk-free rate of 4%. Yet in constructing the equity cash flows in Figure 4, we assumed that expected payments to lenders equaled the promised payments. This overestimate of the expected payments to debt holders causes a corresponding underestimate of expected cash flows to equity holders. The systematic underestimation of equity cash flows causes a systematic underestimation of equity value. If equity cash flows are zero by agreement in the deal and by construction in the analysis, how can they be underestimated? After all, zero cannot be a biased estimate of zero. True, but the terminal values matter. Another way to see the bias is to understand that we have surely overestimated the expected terminal value of the debt. In effect, our analysis assumes that debt holders always are paid in full, which implies that equity holders make up shortfalls out of their own pockets when the amount owed exceeds the value of the business. In reality, equity holders will not do this, which means their true expected cash flow must be higher than this presumption allows.
The beta of debt
The same presumption of no default is implied by our treatment of the levered equity betas that determine the costs of equity. In Figures 5 and 6 we used the following formula to compute the levered betas: unlevered = (E/V)levered, where unlevered is the asset beta (0.85 in our example) and levered is the beta of the stock. This formula assumes that debt is riskless—that the beta of debt is zero. The formula is derived from the following relationship: 6 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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Vassets = Eequity + Ddebt, where V, E, and D are the values of the firm, its equity and its debt, respectively. Now divide both sides of the equation by V and let debt be zero. This gives: assets = (E/V)equity. This is exactly our formula for computing levered betas, because assets and equity correspond to unlevered and levered, respectively. The assumption that the beta of debt is zero causes us to overestimate the equity beta. We are assuming, in effect, that all the systematic risk inherent in the assets is borne by equity holders and none by debt holders. In fact, debt holders do bear some of this risk, which means equity holders have to bear less—the true equity beta is accordingly lower than our formula suggests. To see this, note that the cost of debt is greater than 4%. There is a risk premium in the cost of debt, which implies a positive beta.3 Fill in some positive beta for the debt and solve for the beta of the equity using the relationship above. The resulting equity betas will be less than the figures shown in Figure 6. By overestimating the levered beta, we overestimate the cost of equity. This causes us to overdiscount (i.e., undervalue) the equity cash flows.
Using the Methodology Fortunately, all of the biases described above run in the same direction. They combine to produce a low estimate of the value of equity. Rather than settling for a biased analysis, it is tempting to try to fix it with adjustments here and there to the cash flows and betas. Unfortunately, though the impulse is understandable, the attempt is futile and probably misguided. Both risky debt and highly levered equity have option-like characteristics. Although the beta of an option is a well-defined idea, option betas are not stable and cannot be made to fit our simple DCF framework. The only way to eliminate the biases is to perform an option-pricing analysis rather than a DCF analysis. Recall, though, that we resorted to DCF, knowing it was imperfect, because option pricing was impractical. Indeed, depending on the situation, a careful DCF analysis with known biases may be more useful than no analysis or one with biases of unknown signs. And sensitivity analyses that examine different assumptions may yield more insight than anti-bias tweaking of the methodology.
Non-zero equity cash flows One of the simplifying features of the LBO example was the stipulation that all excess cash was used to reduce debt. In some cases, debt contracts specify a fixed schedule of principal payments, which means that equity cash flows can be non-zero. When they are positive, the algorithm for working backwards through the analysis must be modified somewhat. Over a given year, the return expected by the equity holder can come in two forms: cash flows received (e.g., a dividend) and changes in the value of the equity (a capital gain or loss). In our simple example, dividends were zero, so we could solve for year-4 equity value by discounting year5 value at the cost of equity. If there is a dividend during year 5, then this dividend, plus the change in value from year 4 to 5, all divided by the beginning (year-4) equity value, must equal the cost of equity.
3 It is tempting to suppose that the beta of the debt in the LBO example must be about 0.5 because the CAPM relationship suggests that 7% = 4% + debt(6%). But this is misleading because the cost of debt is not really 7%. The promised return is 7%. For the reasons stated in the text, 7% cannot also be the expected return. So 0.5 has to be an overestimate of the debt beta, just as zero is an underestimate.
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Note on Valuing Equity Cash Flows
When equity cash flows are negative, they must be carefully interpreted and perhaps modified. Does a negative equity cash flow in a given year signify an expected default (in which case specific scenarios should be examined, or a different financing program created)? Or does it represent an expected new equity issue (in which case the expected proceeds of the issue must be added to the analysis)? Either is possible, but the two have quite different implications for both managers and investors.
New borrowing In some situations, debt is expected to increase rather than decrease over time. This can occur with or without a rising debt ratio. Equity cash flows naturally reflect increases in the firm's obligations by incorporating rising interest and principal payments into the projections. It is important to also include the proceeds of new debt issues as positive cash flows—the equity holders should not be expected to repay debts for which they never received the cash proceeds. In effect, proceeds from new borrowings amount to negative principal payments. End-of-year discount rates In the LBO example, we used discount rates computed from endof-year capital structures and applied them over the preceding year. Because leverage was constantly falling, our discount rates were a little lower than they should have been. This bias is sometimes worth worrying about because it may go in the opposite direction of those discussed above and diminish our confidence in the sign of the overall bias. [Note, though, that when leverage is rising, the bias goes in the opposite direction.] This effect is unlikely to be large unless leverage changes very quickly and substantially. If necessary, it can be addressed in two ways. One is to use average rather than end-of-year discount rates. The other is to increase the frequency of the cash flow projections. That is, to use semi-annual or quarterly projections and discount rates rather than annual ones. Insolvent firms Finally, the expedient of assuming that debt is riskless can sometimes cause this methodology to break down entirely. This clearly is the case when all equity cash flows are negative. When the value of the assets is less than the face amount of obligations due and payable, the firm is insolvent. The equity cash flow methodology as presented here will give a negative value for the equity. And yet, the equity in an insolvent firm clearly has a value greater than zero. The equity holders' call option is out of the money, but it will have a positive value as long as some time and variance remain. This is exactly the situation when option pricing is most needed. Unfortunately, tinkering with a DCF analysis is an unpromising approach to evaluating an out-ofthe-money call option. Instead, option pricing, decision tree analysis, simulation, scenario analysis, or experienced judgment should be examined as possible alternatives to DCF analyses. All have one or more advantages over a naive application of DCF techniques.
When To Use Equity Cash Flows Recommendations for when to use equity cash flows follow naturally from the preceding discussion of how the methodology works and where its biases come from. It will be most useful when leverage is high but not too high—that is, when debt is clearly risky, but the firm is clearly solvent. For firms within this range, the equity cash flow methodology is a good way to obtain a lower bound for the value of the equity. To complement this analysis, it may also be useful to locate an upper bound for the equity value. This may be possible using other (presumably aggressive) assumptions within a different DCF analysis (e.g., adjusted present value and assets-minus-debt). When leverage is extremely high—the firm is either insolvent or quite close to it—equity cash flows can still be counted on to underestimate equity value. But in this range, the negative bias is so 8 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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severe that the estimate cannot possibly be very useful—one could as easily guess that equity value is a bit above zero and be no farther from the truth. For the methodology to be useful, it must constitute an improvement over mere educated guessing. At the other extreme, where leverage is low and debt is not very risky, equity cash flows still "work." But here also the methodology has no obvious, compelling advantages over the assetsminus-debt approach. It might turn out to be computationally simpler, or easier to communicate, or have some other practical advantage, but then again it might not. This depends more on the case at hand and the skill of the analyst than on the methodology. Finally, the ECF methodology is frequently used in settings where the capital structure is complex, and yet the main point of the analysis is the valuation of an equity claim. Corporations entering alliances or joint ventures, for example, in which they own an equity interest in a large and/or risky venture that may or may not be managed by themselves will frequently focus on equity cash flows: what did we invest, and what are our expected dividends? Project financing presents a similar opportunity to apply the methodology.
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Appendix:
Equity cash flow calculations for a simple LBO
Assumptions Growth rate Risk-free rate
3.0% 4.0%
Asset beta EMRP
0.85 6.0%
Yearly cash flow s
Closing
EBIT Interest payments Pre-tax profits Taxes @36% Net income Depreciation Capital expend itures NWC ad d itions Cash flow available Principal payments Equity cash flow Yearly debt schedule Beginning debt Interest payments (7%) Principal payments Ending debt
EOY 1 125.0 -63.0 62.0 -22.3 39.7 20.0 -20.0 -10.0 29.7 -29.7 0.0
Closing
900.0
EOY 1 900.0 63.0 29.7 870.3
2
3
4
5
128.8 -60.9 67.8 -24.4 43.4 20.6 -20.6 -10.3 33.1 -33.1 0.0
132.6 -58.6 74.0 -26.6 47.4 21.2 -21.2 -10.6 36.8 -36.8 0.0
136.6 -56.0 80.6 -29.0 51.6 21.9 -21.9 -10.9 40.6 -40.6 0.0
140.7 -53.2 87.5 -31.5 56.0 22.5 -22.5 -11.3 44.7 -44.7 0.0
2
3
4
5
870.3 60.9 33.1 837.2
837.2 58.6 36.8 800.5
800.5 56.0 40.6 759.8
759.8 53.2 44.7 715.1
Terminal value estimates Terminal value, assets Terminal value, d ebt
1449.1 -715.1
Terminal value, equity Yearly discounting Ending MV of equity Ending debt + equity Debt/ capital Equity/ capital Asset beta Levered (equity) beta Cost of equity
734.0 Closing 339.1
EOY 1
2
3
4
TV assets calculation Year 6 EBIT
144.9
EBIT(1-t) less NWC FCF assets
92.7 -11.6 81.1
WACC TV assets
8.6% 1449.1
5
407.0
480.2
559.0
643.5
734.0
1277.3 0.68 0.32 0.85 2.67 20.01%
1317.4 0.64 0.36 0.85 2.33 17.99%
1359.4 0.59 0.41 0.85 2.07 16.40%
1403.3 0.54 0.46 0.85 1.85 15.12%
1449.1 0.49 0.51 0.85 1.68 14.07%
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TIMOTHY LUEHRMAN
Note on Valuing Equity Cash Flows This note introduces a discounted cash flow (DCF) methodology for valuing highly levered equity claims on operating assets. The equity cash flow methodology is an alternative to option-pricing, which is theoretically preferable, but frequently impractical. It also is an alternative, or a complement, to other DCF approaches, such as valuing assets and then subtracting the value of the debt. This latter method is practical, but has other limitations and biases the net effect of which is unclear. Equity cash flows have the advantage that, though they also give a biased result, the sign of the bias is known. Properly applied, methodology gives a lower bound for the value of highly levered equity, which is often a useful result. This note introduces the methodology; shows how to use it; discusses the sources and signs of its built-in biases; and provides some guidance about when to use it.
The Methodology It is well known that holders of highly levered equity essentially own a call on the assets of the firm. They can exercise the call by paying off the outstanding debt. They will do so if, when the debt comes due, the assets are worth more than the face amount of the obligation. If, when the debt is due, the assets are worth less this amount, the equity holders will not exercise—that is, they will not pay off the debt in cash, but rather let the lenders keep the assets. This simple description is clear enough. Unfortunately, in reality the structure of a typical long-term debt contract creates multiple call options with complex characteristics. Every time an interest or principal payment comes due, the equity holders have an exercise decision to make. In effect, they hold a sequence of nested options— that is, options on options—rather than a simple call option. Other common features of corporate debt contracts (for example, callability, convertibility, covenants, etc.) introduce still more options, and the valuation problem becomes even harder. When it is impossible to apply option-pricing techniques to highly levered equity, it still will be desirable to value it somehow—a flawed analysis generally will be better than no analysis. It is natural in this situation to fall back on DCF techniques. To determine whether one or another DCF approach is preferable, consider first a firm with low leverage, illustrated in Figure 1. Because there is little debt, the chance that the firm will default is remote. Consequently, the equity holders' option to walk away from the debt and the assets is not very valuable and ignoring it will not seriously distort the analysis. The firm's assets, including tax ________________________________________________________________________________________________________________ Professor Timothy A. Luehrman prepared this note as the basis for class discussion. Copyright © 1994, 2005, 2009, 2012 President and Fellows of Harvard College. To order copies or request permission to reproduce materials, call 1-800-545-7685, write Harvard Business School Publishing, Boston, MA 02163, or go to http://www.hbsp.harvard.edu. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of Harvard Business School.
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shields from its modest leverage, can be valued with standard DCF techniques.1 Similarly, the value of the debt is the present value of promised interest and principal payments, discounted at the cost of debt. Subtracting the value of the debt from the value of the assets gives the value of the equity. Figure 1: Applying DCF techniques to a firm with low leverage A firm with low leverage Assets (including interest tax shields) Value assets using unlevered cash flows and an unlevered discount rate.
Value of the assets
Debt
Value of debt cash flows, discounted at the cost of debt.
Equity Value of the claims
Value of equity cash flows, discounted at the cost of (levered) equity.
With low leverage, it does not matter whether we value equity directly or as assets-minus-debt.
Alternatively, the equity may be evaluated directly, by discounting expected equity cash flows at the cost of equity. Equity cash flows are simply the cash flows associated with the assets and tax shields, less those associated with the debt. This alternative is also shown in Figure 1. When leverage is low, it generally does not matter whether we value the assets and then subtract the debt, or discount the equity cash flows themselves. Now suppose that leverage is high and that the amount of debt changes over time. Figure 2 shows the same assets, now including both more tax shields and the expected costs of financial distress. It also shows three layers of debt on top of a small amount of equity. Even if we stipulate that this firm is solvent, default is clearly much more likely and the option held by equity owners is accordingly a more important feature of the problem. In this setting, it is not a matter of indifference whether we value the assets and subtract the debt, or value the equity cash flows directly. First note that either way is more difficult than before, with low leverage. Valuing the assets is a bit trickier because there are more and riskier tax shields and costs of financial distress to worry about. Valuing the debt to be subtracted is much harder now. It is unlikely to be traded, so market values cannot be observed. All but the most senior debt is risky, and there are three different layers, which in the event of default have interdependent values. In the end, after executing a difficult analysis, it will be hard to tell whether we have systematically over- or undervalued the equity by using the assetsminus-debt approach.
1 This presumes there are no real options on the asset side of the balance sheet. A treatment of real options is beyond the scope of this note, but see, for example, Capital Projects as Real Options: An Introduction, HBS case no. 295-074.
2 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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Figure 2: Applying DCF techniques to highly levered equity The same firm, with high leverage
Value assets using unlevered cash flows and an unlevered discount rate.
Same assets (including more interest tax shields and costs of financial distress) New value of the assets
Debt #1 Debt #2 Debt #3 Equity
Value by layer, using debt cash flows and the costs of debt.
Use equity cash flows and the cost of (levered) equity.
New value of the claims
With high leverage, valuing equity directly is probably simpler than assets-minus-debt and, under certain assumptions, the sign of the bias in the equity-cash-flow approach is known.
Valuing the equity directly—i.e., discounting equity cash flows at the cost of equity—requires one complex analysis rather than four. But more importantly, by making certain assumptions, we can be fairly confident that this approach will underestimate the value of the equity. Collectively, these assumptions imply that equity holders will not exercise their option to walk away, even when it is in their best interests to do so. In general, equity is more valuable than this unrealistic assumption makes it appear, so we know the sign of the bias is negative. The magnitude of the bias is unclear; it depends on all the usual determinants of option value.
A Simple Example A simple leveraged buyout provides an illustration of how to apply the methodology. Suppose an LBO firm has tentatively agreed to pay $1.1 billion to acquire a certain business. The managers of the business anticipate EBIT of $125 million for the coming year and this is expected to grow in perpetuity at 3% per year. The LBO sponsors are confident that they can raise $900 million in debt financing, provided they contribute $200 million of equity and that all excess cash generated by the business is used to pay down debt. The sponsors expect to divest their holdings after five years. Should they invest the $200 million and acquire the business? In other words, is the equity worth more than $200 million?
Cash flows and terminal value xValuing the equity in this LBO will require some additional information, but we already know a lot about how to frame the analysis. The equity cash flows over the first five years are very simple: they equal zero, by construction, because all available cash is going to debt holders. In fact, the only equity cash flow to be discounted is the terminal value of the equity at the five-year horizon. This simple set-up is shown in Figure 3.
3 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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Note on Valuing Equity Cash Flows
Figure 3:
Equ ity cash flow s for a sim p le LBO
Yearly cash flow s
Closing
Equ ity cash flow Equ ity term inal valu e
EOY 1
2
0.0
3 0.0
4 0.0
5 0.0
0.0 ?
One way to estimate the terminal value of the equity is to multiply projected net income at the terminal horizon by a price-earnings ratio. However, p-e ratios are affected by leverage, and selecting the right one is tricky. A safer alternative is to estimate the terminal value of the assets (for example, as a growing perpetuity, or via a market multiple) and subtract the terminal value of the debt. The amount of debt at the end of year 5 depends on how much of the original $900 million can be retired using internally generated funds during years 1-5. Figure 4 presents projections for interest and principal payments, under the additional assumptions that the tax rate is 36%; the interest rate on the debt is 7%; capital expenditures equal depreciation (both grow at 3% per year); and additions to net working capital start at $10 million in year 1 and grow at 3%. Figure 4 shows that cash flow available to retire debt equals $29.7 million in year 1 and grows to $44.7 million by year 5. Applying these funds to debt reduction lowers the outstanding debt to $715.1 million at the end of the fifth year.
Figure 4:
Financial Projections for a simp le LBO
Yearly cash flow s
Closing
EBIT Interest p aym ents Pre-tax p rofits Taxes @36% N et income Dep reciation Cap ital exp end itu res N WC ad d itions Cash flow available Princip al p aym ents Equ ity cash flow Yearly debt schedule Beginning d ebt Interest p aym ents (7%) Princip al p aym ents End ing d ebt
EOY 1 125.0 -63.0 62.0 -22.3 39.7 20.0 -20.0 -10.0 29.7 -29.7 0.0
Closing
900.0
EOY 1 900.0 63.0 29.7 870.3
2 128.8 -60.9 67.8 -24.4 43.4 20.6 -20.6 -10.3 33.1 -33.1 0.0 2 870.3 60.9 33.1 837.2
3 132.6 -58.6 74.0 -26.6 47.4 21.2 -21.2 -10.6 36.8 -36.8 0.0 3 837.2 58.6 36.8 800.5
4
5
136.6 -56.0 80.6 -29.0 51.6 21.9 -21.9 -10.9 40.6 -40.6 0.0 4
140.7 -53.2 87.5 -31.5 56.0 22.5 -22.5 -11.3 44.7 -44.7 0.0 5
800.5 56.0 40.6 759.8
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759.8 53.2 44.7 715.1
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Assuming a risk-free rate of 4%, a market risk premium of 6%, an asset beta of 0.85, and a perpetuity growth rate of 3%, the terminal value of the assets at the end of year 5 is about $1,450 million.2 Subtracting the terminal value of the debt gives a terminal value for the equity of $734 million. This value goes in Figure 3, to be discounted back to the present at the cost of equity.
Discount rates The cost of equity is the expected return that investors require as compensation for holding the stock. It depends on the amount of systematic risk borne by stockholders. This in turn depends on the systematic risk inherent in the business and on the firm's capital structure. As the capital structure changes, so does the cost of equity. In this example, leverage falls as debt is paid off. Systematic risk borne by stockholders falls accordingly, assuming the asset beta remains unchanged. As risk goes down, the cost of equity goes down, too. Figure 5 shows how to use the estimated terminal value of equity to generate an estimated cost of equity at the terminal horizon: 1.
The terminal values of debt and equity together are sufficient to determine an equity ratio at the end of year 5.
2.
Dividing the asset beta of 0.85 by the equity ratio of 0.51 gives a levered equity beta of 1.68.
3.
This levered beta can be used in the CAPM, along with a risk-free rate of 4% and a market risk premium of 6%, to get a cost of equity at the end of year 5 of 14%.
4.
Applying this discount rate to the $734 million terminal value of equity gives $643.5 million for the year-4 value of equity.
5.
Now we can repeat the process to obtain a year-4 cost of equity and year-3 value of equity, and so forth.
6.
The analysis terminates with an "answer," the value of equity at the closing of the transaction.
Figure 5:
Discou nting equ ity cash flow s by w orking backw ard s
Yearly discount ing End ing MV of equ ity End ing d ebt + equ ity Debt/ cap ital Equ ity/ cap ital Asset beta Levered (equ ity) beta Cost of equ ity
Closing answ er
EOY 1
2
3
4 643.5
5 734.0 1449.1 0.49 0.51 0.85 1.68 14.07%
2 The terminal value of the assets should include the terminal value of interest tax shields. This in turn depends on the expected capital structure. The estimate given in the text assumes a constant debt ratio following year 5, so the weighted average cost of capital, given by assumption in the Appendix as 8.6%, can be used in the perpetuity formula. An alternative assumption is that the dollar amount of debt (and hence the dollar amount of the annual tax shield) stays constant in perpetuity. A third possibility is that the existing debt is paid off on schedule and not refinanced. Note that each assumption will give a different terminal value for interest tax shields.
5 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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Note on Valuing Equity Cash Flows
Figure 6 shows the completed analysis, and a present value for the LBO equity of about $340 million, well above the sponsors' acquisition cost of $200 million. [The Appendix consolidates Figures 3-6 to summarize the analysis on one page.] Note that, as expected, the cost of equity declines over time, from 20% to 14%. This is due solely to the drop in the debt-to-value ratio, from 68% at closing to 49% at the terminal horizon. Figure 6:
Discou nting equ ity cash flow s: the com p leted analysis
Yearly discount ing End ing MV of equ ity End ing d ebt + equ ity Debt/ cap ital Equ ity/ cap ital Asset beta Levered (equ ity) beta Cost of equ ity
Closing 339.1
EOY 1 407.0 1277.3 0.68 0.32 0.85 2.67 20.01%
2 480.2 1317.4 0.64 0.36 0.85 2.33 17.99%
3 559.0 1359.4 0.59 0.41 0.85 2.07 16.40%
4 643.5 1403.3 0.54 0.46 0.85 1.85 15.12%
5 734.0 1449.1 0.49 0.51 0.85 1.68 14.07%
Biases The equity cash flow analysis presented above is biased in the sense that, given unbiased inputs, the methodology produces a biased-low output. Obviously, an analyst choosing to use very aggressive forecasts of EBIT, for example, still can obtain a biased-high result—biased inputs do indeed produce biased outputs. But the important point is that there is a bias in the methodology independent of the forecasts. It comes from two main sources.
Promised vs. expected cash flows When leverage is high, debt is risky. It quite possibly will not be repaid as promised—the lenders might receive only a fraction of what they are owed or they might receive it later than promised. For risky debt, then, expected cash flows must be less than promised cash flows. This is why the lenders in our LBO example demand a promise of 7% interest rather than the risk-free rate of 4%. Yet in constructing the equity cash flows in Figure 4, we assumed that expected payments to lenders equaled the promised payments. This overestimate of the expected payments to debt holders causes a corresponding underestimate of expected cash flows to equity holders. The systematic underestimation of equity cash flows causes a systematic underestimation of equity value. If equity cash flows are zero by agreement in the deal and by construction in the analysis, how can they be underestimated? After all, zero cannot be a biased estimate of zero. True, but the terminal values matter. Another way to see the bias is to understand that we have surely overestimated the expected terminal value of the debt. In effect, our analysis assumes that debt holders always are paid in full, which implies that equity holders make up shortfalls out of their own pockets when the amount owed exceeds the value of the business. In reality, equity holders will not do this, which means their true expected cash flow must be higher than this presumption allows.
The beta of debt
The same presumption of no default is implied by our treatment of the levered equity betas that determine the costs of equity. In Figures 5 and 6 we used the following formula to compute the levered betas: unlevered = (E/V)levered, where unlevered is the asset beta (0.85 in our example) and levered is the beta of the stock. This formula assumes that debt is riskless—that the beta of debt is zero. The formula is derived from the following relationship: 6 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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Vassets = Eequity + Ddebt, where V, E, and D are the values of the firm, its equity and its debt, respectively. Now divide both sides of the equation by V and let debt be zero. This gives: assets = (E/V)equity. This is exactly our formula for computing levered betas, because assets and equity correspond to unlevered and levered, respectively. The assumption that the beta of debt is zero causes us to overestimate the equity beta. We are assuming, in effect, that all the systematic risk inherent in the assets is borne by equity holders and none by debt holders. In fact, debt holders do bear some of this risk, which means equity holders have to bear less—the true equity beta is accordingly lower than our formula suggests. To see this, note that the cost of debt is greater than 4%. There is a risk premium in the cost of debt, which implies a positive beta.3 Fill in some positive beta for the debt and solve for the beta of the equity using the relationship above. The resulting equity betas will be less than the figures shown in Figure 6. By overestimating the levered beta, we overestimate the cost of equity. This causes us to overdiscount (i.e., undervalue) the equity cash flows.
Using the Methodology Fortunately, all of the biases described above run in the same direction. They combine to produce a low estimate of the value of equity. Rather than settling for a biased analysis, it is tempting to try to fix it with adjustments here and there to the cash flows and betas. Unfortunately, though the impulse is understandable, the attempt is futile and probably misguided. Both risky debt and highly levered equity have option-like characteristics. Although the beta of an option is a well-defined idea, option betas are not stable and cannot be made to fit our simple DCF framework. The only way to eliminate the biases is to perform an option-pricing analysis rather than a DCF analysis. Recall, though, that we resorted to DCF, knowing it was imperfect, because option pricing was impractical. Indeed, depending on the situation, a careful DCF analysis with known biases may be more useful than no analysis or one with biases of unknown signs. And sensitivity analyses that examine different assumptions may yield more insight than anti-bias tweaking of the methodology.
Non-zero equity cash flows One of the simplifying features of the LBO example was the stipulation that all excess cash was used to reduce debt. In some cases, debt contracts specify a fixed schedule of principal payments, which means that equity cash flows can be non-zero. When they are positive, the algorithm for working backwards through the analysis must be modified somewhat. Over a given year, the return expected by the equity holder can come in two forms: cash flows received (e.g., a dividend) and changes in the value of the equity (a capital gain or loss). In our simple example, dividends were zero, so we could solve for year-4 equity value by discounting year5 value at the cost of equity. If there is a dividend during year 5, then this dividend, plus the change in value from year 4 to 5, all divided by the beginning (year-4) equity value, must equal the cost of equity.
3 It is tempting to suppose that the beta of the debt in the LBO example must be about 0.5 because the CAPM relationship suggests that 7% = 4% + debt(6%). But this is misleading because the cost of debt is not really 7%. The promised return is 7%. For the reasons stated in the text, 7% cannot also be the expected return. So 0.5 has to be an overestimate of the debt beta, just as zero is an underestimate.
7 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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Note on Valuing Equity Cash Flows
When equity cash flows are negative, they must be carefully interpreted and perhaps modified. Does a negative equity cash flow in a given year signify an expected default (in which case specific scenarios should be examined, or a different financing program created)? Or does it represent an expected new equity issue (in which case the expected proceeds of the issue must be added to the analysis)? Either is possible, but the two have quite different implications for both managers and investors.
New borrowing In some situations, debt is expected to increase rather than decrease over time. This can occur with or without a rising debt ratio. Equity cash flows naturally reflect increases in the firm's obligations by incorporating rising interest and principal payments into the projections. It is important to also include the proceeds of new debt issues as positive cash flows—the equity holders should not be expected to repay debts for which they never received the cash proceeds. In effect, proceeds from new borrowings amount to negative principal payments. End-of-year discount rates In the LBO example, we used discount rates computed from endof-year capital structures and applied them over the preceding year. Because leverage was constantly falling, our discount rates were a little lower than they should have been. This bias is sometimes worth worrying about because it may go in the opposite direction of those discussed above and diminish our confidence in the sign of the overall bias. [Note, though, that when leverage is rising, the bias goes in the opposite direction.] This effect is unlikely to be large unless leverage changes very quickly and substantially. If necessary, it can be addressed in two ways. One is to use average rather than end-of-year discount rates. The other is to increase the frequency of the cash flow projections. That is, to use semi-annual or quarterly projections and discount rates rather than annual ones. Insolvent firms Finally, the expedient of assuming that debt is riskless can sometimes cause this methodology to break down entirely. This clearly is the case when all equity cash flows are negative. When the value of the assets is less than the face amount of obligations due and payable, the firm is insolvent. The equity cash flow methodology as presented here will give a negative value for the equity. And yet, the equity in an insolvent firm clearly has a value greater than zero. The equity holders' call option is out of the money, but it will have a positive value as long as some time and variance remain. This is exactly the situation when option pricing is most needed. Unfortunately, tinkering with a DCF analysis is an unpromising approach to evaluating an out-ofthe-money call option. Instead, option pricing, decision tree analysis, simulation, scenario analysis, or experienced judgment should be examined as possible alternatives to DCF analyses. All have one or more advantages over a naive application of DCF techniques.
When To Use Equity Cash Flows Recommendations for when to use equity cash flows follow naturally from the preceding discussion of how the methodology works and where its biases come from. It will be most useful when leverage is high but not too high—that is, when debt is clearly risky, but the firm is clearly solvent. For firms within this range, the equity cash flow methodology is a good way to obtain a lower bound for the value of the equity. To complement this analysis, it may also be useful to locate an upper bound for the equity value. This may be possible using other (presumably aggressive) assumptions within a different DCF analysis (e.g., adjusted present value and assets-minus-debt). When leverage is extremely high—the firm is either insolvent or quite close to it—equity cash flows can still be counted on to underestimate equity value. But in this range, the negative bias is so 8 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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severe that the estimate cannot possibly be very useful—one could as easily guess that equity value is a bit above zero and be no farther from the truth. For the methodology to be useful, it must constitute an improvement over mere educated guessing. At the other extreme, where leverage is low and debt is not very risky, equity cash flows still "work." But here also the methodology has no obvious, compelling advantages over the assetsminus-debt approach. It might turn out to be computationally simpler, or easier to communicate, or have some other practical advantage, but then again it might not. This depends more on the case at hand and the skill of the analyst than on the methodology. Finally, the ECF methodology is frequently used in settings where the capital structure is complex, and yet the main point of the analysis is the valuation of an equity claim. Corporations entering alliances or joint ventures, for example, in which they own an equity interest in a large and/or risky venture that may or may not be managed by themselves will frequently focus on equity cash flows: what did we invest, and what are our expected dividends? Project financing presents a similar opportunity to apply the methodology.
9 This document is authorized for use only in Investor by Prof. Raju Majumdar at IILM-GSM Institute for Integrated Learning in Management from January 2014 to March 2014.
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Appendix:
Equity cash flow calculations for a simple LBO
Assumptions Growth rate Risk-free rate
3.0% 4.0%
Asset beta EMRP
0.85 6.0%
Yearly cash flow s
Closing
EBIT Interest payments Pre-tax profits Taxes @36% Net income Depreciation Capital expend itures NWC ad d itions Cash flow available Principal payments Equity cash flow Yearly debt schedule Beginning debt Interest payments (7%) Principal payments Ending debt
EOY 1 125.0 -63.0 62.0 -22.3 39.7 20.0 -20.0 -10.0 29.7 -29.7 0.0
Closing
900.0
EOY 1 900.0 63.0 29.7 870.3
2
3
4
5
128.8 -60.9 67.8 -24.4 43.4 20.6 -20.6 -10.3 33.1 -33.1 0.0
132.6 -58.6 74.0 -26.6 47.4 21.2 -21.2 -10.6 36.8 -36.8 0.0
136.6 -56.0 80.6 -29.0 51.6 21.9 -21.9 -10.9 40.6 -40.6 0.0
140.7 -53.2 87.5 -31.5 56.0 22.5 -22.5 -11.3 44.7 -44.7 0.0
2
3
4
5
870.3 60.9 33.1 837.2
837.2 58.6 36.8 800.5
800.5 56.0 40.6 759.8
759.8 53.2 44.7 715.1
Terminal value estimates Terminal value, assets Terminal value, d ebt
1449.1 -715.1
Terminal value, equity Yearly discounting Ending MV of equity Ending debt + equity Debt/ capital Equity/ capital Asset beta Levered (equity) beta Cost of equity
734.0 Closing 339.1
EOY 1
2
3
4
TV assets calculation Year 6 EBIT
144.9
EBIT(1-t) less NWC FCF assets
92.7 -11.6 81.1
WACC TV assets
8.6% 1449.1
5
407.0
480.2
559.0
643.5
734.0
1277.3 0.68 0.32 0.85 2.67 20.01%
1317.4 0.64 0.36 0.85 2.33 17.99%
1359.4 0.59 0.41 0.85 2.07 16.40%
1403.3 0.54 0.46 0.85 1.85 15.12%
1449.1 0.49 0.51 0.85 1.68 14.07%
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