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# The hidden math of corporate share buybacks

Updated: Feb 3

#buybacks #corporatefinace #M&A #valuation #investmentManagement

Share buyback, corporations repurchasing their shares, is an argument that might appear extremely easy in principle to seasoned investors and managers. However, if asked about the fundamental reasons for buybacks, many start listing things probably just collateral, often marginal, and sometimes even wrong. When we rely on lists to answer questions, we probably do not know the answer. Yes, the corporation might be trying to “send the message that its share is undervalued”, they might be trying to “sustain the stock price”, they might be “giving cash back to shareholders”, and yes, “buybacks are ways to increase earnings per share (EPS)”… but again, all the above do not seem to get to the core of repurchases.

Truth is, share buybacks are by no means an easy argument, the reason being, buybacks are investments, and no investment is easy. Considering the true nature of share repurchases, getting a better understanding of their dynamics allows for better managerial and decision-making abilities.

*It is worth demystifying the argument on EPS briefly mentioned above and often discussed: while it is true that EPS (earnings of the business divided by outstanding shares) increase being the same net profit of the company divided by fewer shares outstanding after repurchasing, very often the net-worth of shareholders and the value of their shares decrease – shareholders are getting poorer while appearing richer; we will discuss this below.*

Before going through the financial math and dynamics related to buybacks, let us present a quick list of reasons often used to explain share repurchases. They are to some extent correct and effective in specific situations, but still do not give a complete [financial] explanation:

__Strategic reasons__

Discourage takeovers from other corporations (mostly hostile ones) by not allowing the share to trade at low levels and appear excessively attractive to competitors and other players.

Discourage “sharks” investors and speculators from amassing big chunks of ownership at cheap prices (discourage in general those players usually buying big ownership to then take control of the board and ultimately of the corporation).

…

__Good-practice reasons__

Create artificial demand for the share therefore sustaining its price in the open market

Spread the message that the corporation believes its share is undervalued by the market and it should be bought

Reduce the pile of cash sitting in the bank which might send a negative message to investors that the corporation and its management do not see profitable re-investment opportunities

…

__Operational____reasons__

In-house shares to fulfill employees’ stock option plans without the need to reissue additional shares which would dilute further current shareholders (beyond the dilution from the exercise of those options)

Sustain the value of stock-option plans through an optimal level of the stock price (related to the previous point)

In-house shares to fund M&A activities paid for with shares without the need to reissue additional shares which would dilute further current shareholders

…

As the reader may have perceived, all the above are optimal fixes for accidental and short-term situations rather than producers of long-term value for shareholders - arguably the ultimate duty of a corporation. Finally, having already briefly discussed Earnings Per Share in the *note section* above, let us now try to get to the core of the argument.

__Math and dynamics of buybacks__

Shares repurchase is often seen as one of the two main ways corporations can use to give money back to shareholders, the other being dividends. While the latter is an obvious process, the former is a subtle one. In some ways, both dividends and buybacks imply that the corporation either has already enough capital to undertake all expansion projects or worse, it does not see good investment opportunities. In those situations, management correctly prefers to give the cash in excess back to the owners, investors, who can go then invest it somewhere else at higher returns. Repurchases of shares, however, have a longer reach than dividends, and historically successful managers have increased shareholders’ wealth through good buyback programs. Investing is all about the return investors get on their capital, which in corporations can be measured in terms of return on equity (ROE = Earnings / Equity) – and the multiples paid for it since investors usually pay more than the book value on the balance sheet. When a good ROE cannot be maintained because either the corporation cannot produce enough earnings or good runs in the past have increased equity too much through retained earnings, managers [correctly] prefer to *deflate* the corporation by buying part of shareholders out and letting them go - reducing assets through cash and at the same time reducing equity through lower equity outstanding. Consequently, the same earnings in time – growing or not – can produce a higher return to owners simply because they are fewer. Note, as we will see below and have already mentioned above, while this is related in general to growing earnings per share (EPS), the underneath real advantage for the shareholders in terms of intrinsic value is not guaranteed.

Say a corporation has the following characteristics:

Only two shareholders

No debt (all equity funded)

Constant annual net earnings (after-tax) of $100 – earnings per share (EPS) to each share and shareholder of $50 each year

Say the cost of capital of the corporation (Weighted Average Cost of Capital), therefore the required return on that stock of a diversified investor, is 10%. The annuity in perpetuity of that constant $50 EPS has a present value (PV from Earnings) of $50 / 10% =** $500**; that is the investment cost that would allow shareholders to realize 10% each year on their investment through the $50 a single share would receive each year. Also, say that the corporation has $600 in cash in excess of operating and reinvestment needs. Since the value of a company is the result of discounting its future cash-flows (earnings) generated by its assets, decreased by the liabilities (say debt and unfunded liabilities), cash in excess can, in general, be considered non-operating assets — neglecting the interest it can earn by sitting at the bank. Therefore, cash above operating and reinvesting needs can be added at face value to the value calculated through the cash flows.

*We can think about it in this way: after calculating the value of a corporation through discounting of its forecasted cash-flows or earnings in time, any asset not contributing to those cash-flows is a net addition to the valuation being something that could be used in some other way to produce extra earnings without changing anything in the cash-flows of the company. Similarly, any debt or unfunded liability (e.g. unfunded pension fund) is a net subtraction from the cash-flow valuation, having to be repaid or funded in case of acquisition (e.g. M&A).*

So, the total present value of a single share (of the two outstanding) in our corporation is equal to the present value (PV) from earnings calculated above and equal to $500, plus half the cash in excess, that is $600 / 2 = $300 — half of the $600 in excess cash because it is divided by the two shareholders. The total fair price of each one of the two stocks is **PV total = $800**.

Now the interesting part. In case the corporation decides to repurchase one share of the two outstanding because they are selling at the *correct* PV total — intrinsic value reflecting anything the corporation can return in time to that share — we would have the following scenarios:

Enterprise value

*before*the repurchase: ( $100 total earnings / 10% ) Earnings + $600 cash in excess =**$1,600**— note, this is correctly twice the PV total of $800 of a single shareEnterprise value

*after*the repurchase: ( $100 total earnings / 10% ) Earnings + ($600 cash in excess — $800 acquisition price) =**$800**— for sake of simplicity, we can think that the remaining $200 = $600 — $800 in cash needed are momentarily taken from a pile of operating cash used to run the operations and sitting underneath the $600 cash in excess.

*Dividing earnings by the required return calculates the present value of that constant annuity in perpetuity*

So, first, we can note that the repurchase decreases the value and size of the corporation. This is coherent with what we said at the beginning: management usually deflates the business through repurchases to allow earnings to catch up with equity and re-establish a proper Return on Equity (ROE). However, we should answer the following: “Is this good for shareholders?”. Let us try to re-frame the same calculation in terms of shares (i.e. shareholders).

Value per share

*before*the repurchase of one of the two shareholders: [ ($100 / 2) EPS / 10% ] + $600 / 2 cash in excess =**$800**— equal to the PV total calculated above and correctly equal to half the enterprise value above $1,600Share value of the only shareholder left

*after*the repurchase: [ ($100 / 1) EPS / 10% ] — $200 / 1 cash in excess =**$800**— correctly equal to the enterprise value above, $800

So, nothing changed for the single shareholder?! It seems that the investor who was bought-out has received in cash the fair value s/he already had by staying inside the corporation ($800 cash received), and the remaining one still has the same intrinsic value ($800). “Yes”, however, isn’t it always the case in finance? From Black-Scholes used to price financial derivatives to the mortgage calculation on our houses, we could think about everything in finance being structured — we should probably say priced — on the hypothesis of null arbitrage, or zero free gain/loss. Then, those who manage to spot opportunities when those arbitraging conditions are not met, end up being winners. To a similar extent, with buybacks there is always somebody gaining and somebody losing, and as we are about to show, it usually depends on the acquisition price of the shares. Before going further, please note that after the repurchase EPS would increase from $100 / 2 = $50 to $100 / 1 = $100, however, the remaining and only shareholder would still have the same net worth as shown above (i.e. intrinsic value, wealth, etc.).

**How profit and loss can be made through buybacks**

Say in the example above, the share of the business sold on the market for slightly more than the $800 fair value calculated above, say $825. The calculation for the remaining and only shareholder would change according to the following:

Share value

*after*the repurchase of the only shareholder left: [ ($100 / 1) EPS / 10% ] + ($600 — $825) cash in excess =**$775**— compared to the previous $800 obtained (net loss of $25) or the $800 before the repurchase

So, the remaining shareholder, while seeing his/her Earnings per Share double from $50 to $100 (apparent gain), s/he would experience an underneath net loss of -$25 (real loss). Conversely, had the corporation bought the share at slightly less than the fair price, say at $775, the remaining shareholder would experience a net gain of +$25:

Share value

*after*the repurchase of the only shareholder left: [ ($100 / 1) EPS / 10% ] + ($600 — $775) cash in excess =**$825**— compared to the previous $800 obtained (net gain of +$25) or the $800 before the repurchase

Please note that in the first case (repurchase at $825) the remaining shareholder loses -$25 and the leaving one gains the same amount — receiving $25 more than its intrinsic value as a shareholder. The opposite is true in the second case where the share is repurchased at $775. So, the acquisition price plays a critical and opposite role in the outcome of the buyback for the leaving and remaining shareholders.

**Buybacks financed through debt (if readers want to skip this, they can go directly to the Takeaway section)**

A final comment on buybacks financed through new debt rather than cash. Say our corporations above does not have the $800 in cash needed to buy out one of the two investors at the fair price. It decides to raise $800 in new debt for a cost of 2.5% per year (annual interest of $20 per year) and a final bullet repayment of the principal of $800 in 5 years. The Weighted Average Cost of Capital would probably decrease to about 7.5% considering almost an equal average between debt and equity in the capitalization (original WACC or discounting rate 10%). The remaining investor would be in the following situation after the buyout of the other shareholder:

Intrinsic value after the purchase: [ ($100 / 1) EPS / 7.5% ] — $20 interest x (1 - 0.3 Tax Rate) x { [(1+ 7.5%)⁵ -1 ] / 7.5% } — $800 principal / (1 + 7.5%)⁵ = $1,429 — $82 — $570 =

**$715**< $800 before the repurchase (see above)

*The second and third terms are the actualization of the annuity for 5 yrs of $20 interest (2.5% of $800), and the actualization of the final repayment. Interests are considered after-tax (therefore multiplied by 1–0.3, assuming a 30% tax rate) because they would have the beneficial effect of reducing taxes. We could use a discounting of 2.5% instead of 7.5% in those last two terms, obtaining $688, still <$800; this last choice would imply the computation of the market value of the debt, meaning, how much that debt would be valued by somebody willing to invest in debt of a company with the same risk profile of ours — that could be more appropriate for equity valuation and M&A, let us here stick to the valuation of the internal subtraction of cash-flow at 7.5% for sake of this argument.*

Real situations are complex, but the calculation above shows that while debt reduces the cost of capital (debt cheaper than equity), the remaining investor will bear the entire burden of the cost of the capital raised to purchase the other share. This is critical to understand: when the corporation used cash to repurchase one share, half of that cash was of the leaving shareholder — the remaining one benefited from that. When using debt, not only the remaining shareholders bear the whole cost of the acquisition, but it also pays interest on that.

We said that share repurchases are investments, and like any investment raising debt to fund it is not necessarily bad — the calculation above could be easily flipped positive through a slightly lower discounting rate. However, the math has to be run accurately because if debt is used to repurchase shares, chances of reducing the remaining shareholders’ value are high. In general, we could say that if the stock is selling for a considerable discount compared to the fair value, we could make a profit for the remaining shareholders even by raising debt. However, that positive outcome does not seem easy to accomplish. Moreover, raising debt to repurchase shares seems to defy the whole argument of using buybacks to give money back to shareholders … what cash? We are raising as debt that cash needed to buy shareholders out, and we are also paying interest on that … In facts, reinvesting in a company that cannot produce cash in excess does not seem great; maybe the company does produce that cash, which is already returning as dividends and it does not want to scale back on that because it would be a bad signal to investors. Anyway, none of those situations seems ideal.

**Takeaway**

What we just discussed are simplified examples, but the dynamics are the same within big public companies, small Start-Ups, and many other *partnerships*. A lot revolves around the acquisition price, therefore, much of the outcome depends on the capacity of the buyer (i.e. corporation) to assess the real intrinsic value (i.e. correct purchasing price) of the stock — that is, the correct cash-flow valuation from future earnings of the business. The selling shareholder might have a different idea for the future of the business with respect to the purchasing one [and the corporation] who might be forecasting higher and even growing future earnings of the business. Consequently, the latter would be attaching each share a higher intrinsic value and be willing to pay more for each share. Only time can tell who is right …

A possible summary:

Buybacks (i.e. repurchases) could be a good thing to guarantee investors a good ROE.

Buybacks should not change anything in theory (i.e. $0 net gain/loss to shareholders); where practice produces either gains or losses is extremely dependent on repurchasing price.

Buybacks financed through debt have an extra cost that could eat into the possible gain obtained at point 2 (repurchases executed below fair value per share). Therefore, the result of such implementation is related to the comparison of the discount we can get on the market for the shares and the cost of the debt raised to purchase them.

It could be added that in practice buybacks are often a strategic game between investors and corporations. Shares are often priced initially as a multiple common to similar businesses (e.g. Price to Earnings in the financial sector, say 10). Therefore, the simple practice of pumping up EPS by reducing the number of shares outstanding can make the stock look better and trade higher. That is all good till the true fundamental of the business come up in time and the stock readjusts. For example, if the increase in EPS was realized through a repurchase financed through debt, in time the cost of that liability would eat into the earnings of the company, and finally, reduce the valuation of the stock for the same PE multiple. Anyway, as seen above, in any case of bad implementations there is an immediate loss in true intrinsic value — arguably the only thing that matters to fundamental and long-term investors. In the short term, a speculator might benefit from brief EPS growth managing to sell shares at inflated prices.

To conclude, if directly asked “are buybacks good or bad for shareholders?”, I would answer that they are by any means an investment, with the only additional comment that financing a repurchase with debt has additional [quantitative] drawbacks compared to other types of investments — repurchases should be done with cash in excess, see the section above discussing debt. Like any investment, the outcome of a buyback depends on how good the company is at forecasting the return from that investment (earnings of the business in time), and how it fits that forecast within the current possible acquisition price and the possible financing cost — in real situations there is always a financing cost, even when neglecting debt and considering only cash in excess.

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