# Share Buyback Documentation

## Introduction

A share buyback is when a company repurchases some of its outstanding shares. In USA the majority of corporate profits are now being used for share buybacks instead of dividend payouts. The common belief is that a share buyback is a perfect substitute for a dividend payout, possibly even with a tax advantage depending on the tax-rates for dividends and capital gains. This belief originated in academia where it is known as the "substitution hypothesis". We will now show that it is quite foolish, as share buybacks may have very large positive or negative impacts on long-term shareholder value, depending on the share-price relative to the intrinsic value.

## Models

- SB models simulate Share Buybacks.

## Advice

- If in doubt then do nothing! Don't make a share buyback if you don't understand its effect on long-term shareholder value. You cannot destroy shareholder value if you just make a dividend payout instead.
- Avoid making share buybacks when the stock seems to be over-valued because it destroys long-term shareholder value. It would be much better to make a dividend payout, or retain the cash to improve the financial strength of the company, so you can take advantage of the low share-prices the next time the stock-market crashes, and buyback a large number of shares for the benefit of long-term shareholders.
- Try different inputs for your simulations to better understand how share buybacks would affect the long-term shareholder value in different scenarios. If you need to make wildly optimistic assumptions about the future, in order for the share buyback to result in a gain to long-term shareholders, then you should probably refrain from making the share buyback. Conversely, if your assumptions about the future are very conservative and the simulations still show a significant gain in shareholder value, then you may want to make a large share buyback.
- Once you understand share buybacks, you can use them to greatly increase long-term shareholder value, without having to grow the company's sales and earnings, cut expenses, overwork your employees, etc. All you have to do is repurchase shares when their prices are significantly below their intrinsic value.
- Inform your shareholders about your share buyback plans for full transparency: How do you value share buybacks to determine if they are good or bad for long-term shareholders? Do you get the money from excess cash, profits or borrowings? And regularly inform the shareholders about recent share buybacks and your future plans. The shareholders are your business partners, not your enemies.

## Valuation

To properly understand the effect of a share buyback to long-term shareholder value, you should always keep in mind that a share buyback has two direct consequences:

- A share buyback reduces the amount of cash that the company has.
- A share buyback reduces the number of shares outstanding.

So the company is using some of its cash to reduce the number of shares outstanding. The question is whether this benefits or harms the remaining shareholders. The answer is that it depends on the share-price relative to the intrinsic value to long-term shareholders.

To model this mathematically, let \( v \) be the intrinsic value to long-term shareholders. This is the total amount and not per-share. It can be defined in different ways. The simplest is to consider the intrinsic value to be the company's excess cash that could either be used for dividends or share buybacks now, plus the present value of all future earnings for eternity that could also be paid out as dividends: \[ \label{eq_v} v = Excess\ Cash + \sum_{t=1}^{\infty} \frac{Earnings_t}{(1+d)^t} \] Let \( TaxRateDividend \) be the tax-rate for dividends, and let \( Shares \) be the number of shares outstanding prior to the buyback. Then the per-share, after-tax intrinsic value is denoted \( V \) and is: \[ \label{eq_V} V = v \cdot \frac{1 - TaxRateDividend}{Shares} \] The share buyback reduces the intrinsic value by the buyback amount so the value becomes \( v - Buyback \) instead. And the number of shares is reduced to \( Shares - Buyback / SharePrice \). So after the share buyback, the per-share, after-tax intrinsic value is denoted \( W \) and becomes: \[ \label{eq_W} W = (v - Buyback) \cdot \frac{1 - TaxRateDividend}{Shares - \frac{Buyback}{SharePrice}} \] The relative value of the share buyback versus a dividend payout for the same amount, is the ratio between the value with the share buyback \( W \), and the value without the share buyback \( V \), which can be reduced to the following formula when using the definition \( MarketCap = Shares \cdot SharePrice \): \[ \label{eq_relative_value} \frac{W}{V} = \frac{1 - \frac{Buyback}{v}}{1 - \frac{Buyback}{MarketCap}} \]

We then subtract 1 from the relative value, to make it easier to see when the share buyback results in a gain or loss relative to a dividend payout. We call this for the Return on Intrinsic Value (ROIV): \[ \label{eq_roiv} ROIV = \frac{W}{V} - 1 \] When the share-price is equal to its intrinsic value so that \( MarketCap = v \), then ROIV is zero which means the share buyback does not change the intrinsic value. This is the so-called equilibrium where it does not matter whether the company makes a share buyback or dividend payout. But this is only true when the share-price is exactly equal to the intrinsic value.

Note that both the number of shares and the dividend tax-rate cancels out in the formula for the relative value of a share buyback, so if we are considering the value to "eternal" shareholders who will never sell their shares, then the dividend tax-rate is irrelevant when valuing the share buybacks. Further below we rewrite the formulas to also consider capital gains, so both the capital gains and dividend tax-rates become relevant.

Also note that the ROIV ratio is ill-defined when the intrinsic value is zero or negative, so those simulations are marked as invalid. And because the formula is non-linear there are a lot of profound implications that are explained in [Pedersen 2012] .

## Return on Buyback (ROB)

The ROIV ratio above tells us how a share buyback affects the intrinsic value of a company to its long-term shareholders. But it is also useful to consider the effect on shareholder value relative to the amount that was spent on the share buyback. This ratio is named Return on Buyback (ROB) and is defined as follows: \[ \label{eq_rob} ROB = \frac{W - V}{\left( Buyback / Shares \right) \cdot (1 - TaxRateDividend)} \] Unlike the ROIV ratio, the ROB ratio can also be calculated for intrinsic values that are zero or negative.

The numerator \( W - V \) is the net effect of the share buyback, as it is the difference between the intrinsic value of the company with and without the share buyback. The first part of the denominator \( (Buyback / Shares) \) is the amount used for the share buyback on a per-share basis, because \( W \) and \( V \) are also on a per-share basis.

The second part of the denominator \( (1 - TaxRateDividend) \) adjusts for dividend-tax, so the ROB ratio compares the net effect of the share buyback to the buyback amount, as if the buyback amount had been paid out as a dividend instead. This is done to make a fair comparison. Using the simple definitions of \( W \) and \( V \) from Eqs.\eqref{eq_W} and \eqref{eq_V} above, the dividend tax-rate cancels out so the ROB formula can be reduced to the following: \[ \label{eq_rob_reduced} ROB = \frac{v}{Buyback} \cdot ROIV \] This means the ROB ratio is the inverse fraction of the intrinsic value that is used for a share buyback, multiplied with the gain/loss ratio for the intrinsic value to shareholders. Further below we also consider capital gains in the calculation of \( W \) and \( V \), so the dividend tax-rate no longer cancels out in the ROB formula, and the original ROB formula in Eq.\eqref{eq_rob} must be used instead.

Note that the ROB ratio was not used in [Pedersen 2012] and was first presented here.

## Alternative Investment

We may compare a share buyback to other investments or acquisitions that the company could make, to assess which would be the most profitable for the company's shareholders. There are two ways of doing this:

- The ROB ratio can be compared to NPV ratios for the alternative investments, where the purchase price should be the same as the buyback amount.
- The IRR of the company's stock can be compared directly to the IRR of the alternative investment.

It is very important that you do not mix these two methods! You must not compare IRR and NPV ratios, because it would grossly distort the comparison. This would be like using a metric ruler for the company's stock, but an imperial ruler for the alternative investment. You must use the same valuation method for both.

Which method to use depends on your personal preference. The IRR method is somewhat easier, because it does not require you to select discount rates for the company's stock and the alternative investment. You don't even need to use special simulation models for share buybacks, as the normal IRR models are sufficient.

But perhaps you prefer to work with present values instead of IRR, in which case you must use a specialized share buyback model, to simulate the ROB ratio and compare it to NPV ratios for the alternative investments.

## Examples

Let us first consider an example where the stock is slightly over-valued. Say the market-cap of all the company's shares is $10b, and the company has $1b in excess cash that we want to either use for a share buyback or dividend payout. If the present value of all future earnings is $7b then the total intrinsic value is \( v = $8b \) because it also includes the excess cash. The gain/loss of intrinsic value from making the share buyback versus dividend payout is then calculated using Eq.\eqref{eq_roiv}: \[ ROIV = \frac{W}{V} - 1 = \frac{1 - \frac{Buyback}{v}}{1 - \frac{Buyback}{MarketCap}} - 1 = \frac{1 - \frac{$1b}{$8b}}{1 - \frac{$1b}{$10b}} - 1 \simeq - 2.8\% \] So the share buyback would incur a loss to long-term shareholders of about -2.8% relative to a dividend payout. This may seem like a minor loss, but remember that this is relative to the intrinsic value of the entire company, so it may actually be a substantial loss. To better understand the magnitude of the loss, we may calculate the Return on Buyback (ROB) ratio, which measures the net effect of the share buyback, relative to the amount that was used on the buyback. Because this example uses the simple definitions of \( W \) and \( V \), we can use the reduced formula for the ROB ratio from Eq.\eqref{eq_rob_reduced} as follows: \[ ROB = \frac{v}{Buyback} \cdot ROIV = \frac{$8b}{$1b} \cdot (-2.8\%) = -22.4\% \] The share buyback can be considered a type of investment that the company is making on behalf of its remaining shareholders, and the ROB ratio measures the return on that investment. In this example it would result in a loss of about -22% on the invested amount. No financial manager would make such a stupid investment, if they knew the math!

If financial managers consistently buyback shares when they are overvalued, then the value destruction is compounded through the years and will become massive over time. This example assumed the shares were only 25% over-valued. You should try using the above formulas with larger over-valuations of the stock.

Let us consider another example where the stock is under-valued by 50%, so the market-cap is still $10b but now the intrinsic value is actually \( v = $20b \), which again contains excess cash of $1b that we will either use on a share buyback or a dividend. The gain/loss of intrinsic value is then calculated using Eq.\eqref{eq_roiv}: \[ ROIV = \frac{W}{V} - 1 = \frac{1 - \frac{Buyback}{v}}{1 - \frac{Buyback}{MarketCap}} - 1 = \frac{1 - \frac{$1b}{$20b}}{1 - \frac{$1b}{$10b}} - 1 \simeq 5.6 \% \] So the share buyback would result in a gain to long-term shareholders of about 5.6% relative to a dividend payout. This may seem like a small gain, but remember again that it is relative to the intrinsic value of the entire company. So let us calculate the ROB ratio using Eq.\eqref{eq_rob_reduced} to see the valuation impact relative to the amount used for the share buyback: \[ ROB = \frac{v}{Buyback} \cdot ROIV = \frac{$20b}{$1b} \cdot 5.6\% = 111.2\% \] So the effect of the share buyback would be similar to the company making an investment for $1b whose present value was about 111% more than that amount. The return is more than twice the invested amount, simply because the repurchased shares were under-valued by 50%. Every financial manager would make such an investment, if they knew the math!

You should try and experiment with different inputs in the formulas above, to see how a share buyback would impact the shareholder value. This is essentially what the simulation models are doing, by using many thousands of random values from probability distributions that you provide as input.

These examples show that share buybacks may have a significant impact on long-term shareholder value when the share-price does not exactly equal the intrinsic value, and the impact can be very large in case of extreme over- and under-valuation of the stock. We can usually only estimate the intrinsic value without knowing its exact value, so making a share buyback is like walking a razor's edge between gains and losses. That is why it is essential to have a large margin of safety between the share-price and the estimated intrinsic value, to avoid destroying long-term shareholder value when making a share buyback.

## Capital Gains

In [Pedersen 2012] it is argued that share buybacks should always be made for the sake of remaining shareholders, which ultimately means that the valuation should only consider the "eternal" shareholders who will never sell their shares, so there is no need to consider the effect of a share buyback on the future share-prices. This is done in the simulation model that uses terminal values in the calculation of present values.

Some people like to have at least an estimate of how the share buyback affects the future share-prices, so that is done in the simulation models that use valuation ratios such as P/E and P/S, using the following formulas that are derived from the present value of stocks.

The intrinsic value of the company to its long-term shareholders is now calculated from the excess cash, the present value of the earnings until the future year \( N \) which are assumed to be paid out as dividends, and the present value of \( MarketCap_N \) for the future year \( N \): \[ v = Excess\ Cash + \sum_{t=1}^{N} \frac{Earnings_t}{(1+d)^t} + \frac{MarketCap_N}{(1+d)^N} \]

This allows us to consider the effect of different tax-rates for dividends and capital gains. The capital gains tax depends on each investor's purchase price for the shares, so it may be different for all investors and we therefore have to make a reasonable compromise and calculate the capital gains between the current \( MarketCap \) and \( MarketCap_N \) for future year \( N \): \[ Capital\ Gains_N = MarketCap_N - MarketCap \] If the capital gains are negative then there is no tax, which is calculated as follows: \[ TaxCapGains_N = (1 - TaxRateCapGains) \cdot \max(Capital\ Gains_N, 0) \] To shorten the following formulas, let us denote the after-tax, present value of the market-cap as follows: \[ PV\ MCap_N = \frac{MarketCap_N - TaxCapGains_N}{(1+d)^N} \] We can now calculate the after-tax, per-share intrinsic value WITHOUT the share buyback:

The ratio for the relative value of the share buyback \( W/V \) can only be reduced slightly, so as to remove \( Shares \) which is the share-count before the buyback. So it is still a quite complicated formula, and it actually gets even more complicated as explained below.

## Simulation

In the formulas above the unknown numbers are the future \( Earnings_t \) for each year \( t \) and the future \( MarketCap_N \) for the final year \( N \). These numbers are simulated from various formulas and user-defined probability distributions, so the valuations are calculated with many thousands of different inputs.

To make a fair comparison between the two scenarios for dividend payouts and share buybacks, all the simulated numbers are the exact same for the two scenarios, except for the parts of the valuation formulas \( W \) and \( V \) that are different. This accurately shows the difference that share buybacks make to long-term shareholder value.

## Valuation Scale

The user can provide a probability distribution that is used to scale the simulated valuation ratios after the share buyback. By default the scale is set to 1 so it has no effect. But it is useful if you think investors may want to pay either a higher or lower valuation ratio after the share buyback.

For example, if you think investors may be unhappy with the share buybacks then you could set the scale to a value lower than 1. And conversely you can set the scale above 1 if you think the investors are willing to pay a higher valuation ratio after the share buyback.

## Buyback Tax

Some countries may impose a tax on share buybacks because they no longer get to tax the dividend payouts. If you set a non-zero tax-rate for share buybacks, then the reduction in the number of shares from the buyback is calculated as follows: \[ Shares \cdot \left(1 - \frac{Buyback \cdot (1 - BuybackTaxRate)}{MarketCap} \right) \] This replaces \( Shares \cdot (1 - Buyback / MarketCap ) \) in the denominator in the formulas for \( W \) above.

The buyback tax-rate can also be used to deduct other fees relating to the buyback, such as trading fees.

## Borrowing

The share buyback can be partially or entirely funded from borrowed money. To make a fair comparison between the dividend and share buyback scenarios, the money must be borrowed in both scenarios, otherwise the gain/loss comparison would get distorted. This means the borrowed money is assumed to be either used for a share buyback, or paid out immediately as a dividend, so the company would change its capitalization structure equally in both scenarios.

The above formulas for the intrinsic values \( v \) have the borrowed amount added and the interest payments subtracted, similar to this basic formula: \[ v = Excess\ Cash + Borrow + \sum_{t=1}^{\infty} \frac{Earnings_t - Borrow \cdot Interest\ Rate}{(1+d)^t} \]

The interest payments will also affect the simulated market-cap of the company if it is calculated from e.g. the future earnings and P/E ratios, because the earnings have the interest payments subtracted.

The interest-rate should be given as the after-tax rate. For example, if the interest-rate is 5% and the corporate tax-rate is 20%, then the after-tax interest rate is only 4% which is calculated as \( 5\% \cdot (1 - 20\%) = 4\% \).

## Drift

You may sometimes notice a peculiar phenomenon in the simulation results, where the valuation seems to drift over time. The plot below shows an example where the buyback amount is $1b and equals the excess cash, and there is no tax, constant earnings of $1b per year, a constant discount-rate of 20%, and the only source of variation is the P/E ratio which is normal-distributed with mean 15 and std.dev. 3. Notice how the simulated share buybacks mostly result in a gain for the first year, but then the valuations drift lower and get progressively worse until the last year, where all of the simulated share buybacks result in a loss.

This seems bizarre at first and you may wonder if there are errors in the simulations. But it is a phenomenon that we also saw for present values of stocks, and it is just a natural result of comparing present values for successive years. If you try running this example yourself, you will see that the valuation drift gets worse for larger discount rates. It is also possible for the valuations to drift upwards instead of downwards. It all depends on the choice of discount rate, earnings, and market-cap.

One way of making sense of this, is to imagine that you have 10 different shareholders: Person 1 will own the shares for 1 year, person 2 will own the shares for 2 years, and so on. In the example above, the first few investors will most likely experience a gain if you make a share buyback today instead of a dividend payout. But the investors who will hold on to their shares for up to 10 years will almost certainly experience a loss from the share buyback, and they would be much better off with a dividend payout now.

This may seem paradoxical but it is because we assumed there is no future growth in the earnings of the company, and the future valuation ratio will also not grow (although it will fluctuate randomly around its mean), and because we have set the discount rate so high at 20%. So the present value will decrease as the number of years \( N \) increases.

What this means in practice, is that you should always consider the gain/loss of a share buyback with regard to different holding periods and choices of discount rate. You should also view the second simulation plot that makes it easier to see the overall gain/loss, as it combines all years into a single plot instead of showing the years individually. In this example the plot shows there is roughly 50/50 overall chance of gain versus loss from the share buyback at the current share-price. The plot also shows that we would need to buyback shares at a significantly lower share-price to improve the overall chance of gain. And conversely, if the current share-price becomes higher then there is a greater risk of the share buyback causing a loss to shareholders, regardless of their holding period.

## Internal Rate of Return

The Internal Rate of Return (IRR) can be interpreted as the rate of return an investor would get from buying shares at their current price. In the simple case where the valuation is done for "eternal" shareholders who will never sell their shares, so they only receive value from future dividends, and if the share-price does not change after the share buyback, then the IRR also remains the same after the buyback. This is shown in Section 4.3.8 of [Pedersen 2012] where the IRR is called "value yield" instead.

When modelling the changes in share-prices and having different tax-rates for capital gains and dividends, the IRR can change very slightly as a result of a share buyback. But for all practical purposes, the IRR can be considered to remain nearly constant following a share buyback.

So if you are able to make a reasonable estimate of the IRR for a stock, then you can simply compare the IRR to a threshold that you deem to be an acceptable rate of return, to determine if you should make a share buyback or not. If the IRR is very low or even negative then you should not make the share buyback, because the money would likely be better invested elsewhere for a higher return.

But only using the IRR to decide whether or not to make a share buyback, does not tell us by how much the shareholder value is expected to change as a result of the share buyback. For this we need to compare the present values before and after the share buyback, as we have done in the formulas above. The only case where the shareholder value remains unchanged following a share buyback, is when the present value is exactly equal to the current share-price, which only happens when the discount rate is equal to the IRR.