# Share Issuance Documentation

## Introduction

When a company issues more shares, it dilutes the ownership of the current shareholders. This may be done for different reasons, e.g. to raise cash for the company's general use, or to repay some of its debt, or for the company to make a specific investment, or to acquire another company in whole or in part.

The question is whether this adds or destroys value for the shareholders of the company issuing the shares, and for the other party that buys or receives the newly issued shares. The short answer is that it depends on the intrinsic value of the newly issued shares, compared to the cash payment or other value that the company receives in exchange for the new shares.

The following describes the mathematical formulas used in the simulation models for share issuances. It is recommended that you read this entire document, even though you may only be interested in the latter parts, because some of the formulas are reused throughout the document.

Share issuance is related to the valuation of share buybacks and was also described in [Pedersen 2012] , although many of the formulas below are new and were first presented here.

## Models

- SI models simulate Share Issuance.

## Advice

- Avoid issuing new shares when the company's shares are under-valued, because it incurs a loss to the current shareholders. This is especially true when the company is in financial distress and the share-price is extremely low. Companies should avoid this by not taking on excessive debt in the first place!
- When a company's shares are over-valued, it is beneficial for the current shareholders that the company issues more shares, especially if the proceeds are used to pay down excessive debt.
- In most valuation scenarios, a share issuance is only a zero-sum game if the fees are zero. The exception is when the company issues shares to fund an investment, where both the company and the buyer of the new shares can gain or lose together, depending on the return on the investment.

## Intrinsic Value

To measure the valuation impact of a new share issuance, we first need to define the so-called intrinsic value of a company to its shareholders. The notation here is similar to that used for share buybacks. We use subscripts to denote two companies A and B. Formulas are identical for the two companies unless noted otherwise.

Let \( v_A \) be the intrinsic value of company A to its long-term shareholders, and similarly \( v_B \) for company B. 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 be used for dividends, plus the present value of all future earnings for eternity that could also be paid out as dividends: \[ \label{eq_v} v_A = Excess\ Cash_A + \sum_{t=1}^{\infty} \frac{Earnings_{A,t}}{(1+d)^t} \] Instead of only taking the future earnings into account, you can also use simulation models that take the future share-prices into account. But then you should set the tax-rates for both dividends and capital gains to zero in those models, because otherwise it might distort the valuation of a share issuance. It may even be argued that this is perhaps the most fair base-line valuation of a share issuance.

Let \( TaxRateDividend \) be the tax-rate for dividends. We could use different dividend tax-rates for the two companies, but the formulas would become unwieldy, and it is a reasonable simplification to use the same tax-rates, which also means they cancel out in the formulas below.

Let \( Shares_A \) be the number of shares outstanding prior to the new share issuance. Then the per-share, after-tax intrinsic value is denoted \( V_A \) and is: \[ \label{eq_V} V_A = v_A \cdot \frac{1 - TaxRateDividend}{Shares_A} \] The market capitalization (or market-cap for short) of the company, is the total market-value of all the currently outstanding shares for the company: \[ \label{eq_mcap} MarketCap_A = Shares_A \cdot SharePrice_A \] Let \( New\ Shares_A \) be the number of new shares issued in company A, and let \( Issuance_A \) be the market-value of those new shares, so we have the following relations: \[ \label{eq_issuance} Issuance_A = New\ Shares_A \cdot SharePrice_A ~~~ \Leftrightarrow ~~~ New\ Shares_A = \frac{Issuance_A}{SharePrice_A} \] The total number of shares after the new issuance is then: \[ Shares_A + New\ Shares_A = Shares_A + \frac{Issuance_A}{SharePrice_A} = Shares_A \cdot \left( 1 + \frac{Issuance_A}{MarketCap_A} \right) \]

Let \( W_A \) denote the intrinsic value of company A after the share issuance, which is per diluted share and adjusted for the dividend tax. Further below we will consider different scenarios for valuing a share issuance, and the exact definition of \( W_A \) depends on the scenario for the share issuance. But the following formulas are common in most scenarios.

The relative value of the share issuance is calculated as the ratio \( W_A / V_A \) between the intrinsic value after and before the share issuance. It can sometimes be reduced to a simple expression, depending on the definition of \( W_A \). Subtracting 1 gives us the gain/loss ratio for the intrinsic value to the current shareholders of the company. We call this for the Return on Intrinsic Value (ROIV): \[ \label{eq_roiv} ROIV_A = \frac{W_A}{V_A} - 1 \]

## Return on Issuance (ROIS)

It is often useful to consider the change in shareholder value relative to the issuance amount. We call this the Return on Issuance (ROIS) ratio which is defined as follows: \[ \label{eq_rois} ROIS_A = \frac{W_A - V_A}{\frac{Issuance_A}{Shares_A} \cdot (1 - TaxRateDividend)} \] The numerator \( W_A - V_A \) is the net effect of the share issuance, as it is the difference between the intrinsic value of the company with and without the share issuance. Using Eq.\eqref{eq_roiv} the numerator can also be written as: \[ ROIS_A = \frac{ROIV_A \cdot V_A}{\frac{Issuance_A}{Shares_A} \cdot (1 - TaxRateDividend)} \] The denominator is the issuance amount adjusted for the number of shares prior to the new share issuance, as well as the dividend tax. This is done to make a fair comparison to \( V_A \) in the numerator. Then using the definition of \( V_A \) from Eq.\eqref{eq_V}, the ROIS formula can be reduced to the following: \[ \label{eq_rois_reduced} ROIS_A = \frac{ROIV_A \cdot v_A}{Issuance_A} \]

A share issuance can be considered a kind of investment for the company's current shareholders, because they are giving up value by diluting their ownership of the company, but they are also getting something in return. We can compare the value of the share issuance to other investments the company can make, to assess which would be the most profitable for the current shareholders. This is done by comparing the ROIS to NPV ratios for the alternative investments, where the purchase price should be the same as the share issuance amount.

It is very important that you do not compare ROIS to the IRR of the alternative investment, as this would grossly distort the comparison. It 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. We use present values for the share issuance, so we must also use present values for the alternative investment.

## Cash Payment

The first scenario that we consider, is when a company issues shares in exchange for a cash payment. This adds the issuance amount minus fees to the intrinsic value of the company, and it also increases the number of shares outstanding.

In this scenario, the intrinsic value \( W_A \) of company A after the share issuance, per diluted share and adjusted for dividend tax, is defined as follows: \[ \label{eq_W_A_cash} W_A = \frac{v_A + Issuance_A - Fees_A}{Shares_A + New\ Shares_A} \cdot (1 - TaxRateDividend) \] The fees are specified because they may be very large when issuing new shares, as opposed to making share buybacks where the fees are minimal in open-market trades. For example, the fees typically range between 5-10% of the issuance amount in Initial Public Offerings (IPO).

The ROIV ratio measures the gain/loss of intrinsic value to the current shareholders, which reduces to the following: \[ ROIV_A = \frac{W_A}{V_A} - 1 = \frac{1 + \frac{Issuance_A - Fees_A}{v_A}}{1 + \frac{New\ Shares_A}{Shares_A}} - 1 \] Using Eqs.\eqref{eq_mcap} and \eqref{eq_issuance} the fraction between the number of new and old shares can also be rewritten as follows: \[ \label{eq_roiv_cash} ROIV_A = \frac{1 + \frac{Issuance_A - Fees_A}{v_A}}{1 + \frac{Issuance_A}{MarketCap_A}} - 1 \] Using the original definition of ROIS in Eq.\eqref{eq_rois} with the definition of \( W_A \) from Eq.\eqref{eq_W_A_cash} we get: \[ ROIS_A = \frac{\frac{v_A + Issuance_A - Fees_A}{1 + New\ Shares_A / Shares_A} - v_A}{Issuance_A} \] The fraction between the number of new and old shares can again be rewritten as follows: \[ ROIS_A = \frac{\frac{v_A + Issuance_A - Fees_A}{1 + Issuance_A / MarketCap_A} - v_A}{Issuance_A} \] Now consider the return on the issuance amount from the perspective of the buyer of the newly issued shares, which is denoted \( ROIS_{Buyer} \) and defined as follows: \[ \label{eq_rois_buyer} ROIS_{Buyer} = \frac{W_A}{\frac{Issuance_A}{New\ Shares_A} \cdot (1 - TaxRateDividend)} - 1 \] The numerator is the per-share intrinsic value after the share issuance. The denominator is the amount that the buyer has paid for each of the newly issued shares, and adjusted for dividend tax like the numerator.

Using the definition of \( W_A \) from Eq.\eqref{eq_W_A_cash} and reducing the formula, we get the following: \[ \label{eq_rois_buyer_reduced} ROIS_{Buyer} = \frac{v_A - MarketCap_A - Fees_A}{MarketCap_A + Issuance_A} \] It can be proven that the share issuance is only a so-called "zero-sum game" when the fees are zero: \[ ROIS_A = -ROIS_{Buyer} ~~~ \Leftrightarrow ~~~ Fees_A = 0 \]

The \( ROIS_{Buyer} \) ratio can be compared to NPV ratios for alternative investments, to assess if those would be more profitable than the newly issued shares in company A. As noted further above, it is very important that you do not compare ROIS ratios to the IRR of the alternative investment, as this would grossly distort the comparison.

### Example of Cash Payment

Let us consider an example where the stock is somewhat over-valued. Say the market-value of all the company's shares is \( MarketCap_A = $10b \), but the intrinsic value of the company is only \( v_A = $8b \). The company wants to issue new shares for \( Issuance_A = $1b \) and they have to pay \( Fees_A = $50m \). The gain/loss of intrinsic value to the company's current shareholders is then calculated using Eq.\eqref{eq_roiv_cash}: \[ ROIV_A = \frac{1 + \frac{Issuance_A - Fees_A}{v_A}}{1 + \frac{Issuance_A}{MarketCap_A}} - 1 = \frac{1 + \frac{$1b - $50m}{$8b}}{1 + \frac{$1b}{$10b}} - 1 \simeq 1.7 \% \] This may seem like a minor gain of only 1.7%, but remember that this is relative to the intrinsic value of the entire company. To better understand the magnitude of the gain, we may calculate the ROIS ratio, which measures the net effect of the share issuance, relative to the amount that was raised from the issuance of new shares. We use Eq.\eqref{eq_rois_reduced} as follows: \[ ROIS_A = \frac{ROIV_A \cdot v_A}{Issuance_A} = \frac{1.7\% \cdot $8b}{$1b} = 13.6\% \] So although the number of shares is diluted, the original shareholders actually benefit from the share issuance, because the shares are over-valued compared to their intrinsic value. Conversely, the buyer of the new shares will incur a significant loss on their invested amount, because they are paying more than the shares are intrinsically worth. This is measured by the \( ROIS_{Buyer} \) ratio in Eq.\eqref{eq_rois_buyer_reduced}: \[ ROIS_{Buyer} = \frac{v_A - MarketCap_A - Fees_A}{MarketCap_A + Issuance_A} = \frac{$8b - $10b - $50m}{$10b + $1b} \simeq -18.6\% \] So the company's original shareholders gain \( ROIS_A = 13.6\% \) while the buyers lose \( ROIS_{Buyer} = -18.6\% \) on the amount paid for the newly issued shares. This is not a "zero-sum game" because of the fees involved. If you repeat the calculations above with \( Fees_A = 0 \) you should get \( ROIS_A = -ROIS_{Buyer} \).

## Investment

Now consider a scenario where the share issuance for company A is used to fund an investment that the company wants to make, such as the construction of a new factory, and whose return is uncertain so it is a random value that can be simulated. Let \( Invest \) denote the investment amount, and let \( Return \) denote the present value of the return on the investment.

In this scenario, the intrinsic value \( W_A \) of company A after the share issuance, per diluted share and adjusted for dividend tax, is defined as follows: \[ W_A = \frac{v_A + Issuance_A - Fees_A - Invest + Return}{Shares_A + New\ Shares_A} \cdot (1 - TaxRateDividend) \] Assume that \( Invest = Issuance_A - Fees_A \) so we have: \[ \label{eq_W_A_invest} W_A = \frac{v_A + Return}{Shares_A + New\ Shares_A} \cdot (1 - TaxRateDividend) \] The ROIV ratio measures the gain/loss of intrinsic value to the current shareholders, which reduces to the following: \[ ROIV_A = \frac{W_A}{V_A} - 1 = \frac{1 + \frac{Return}{v_A}}{1 + \frac{New\ Shares_A}{Shares_A}} - 1 \] The fraction between the number of new and old shares can again be rewritten as follows: \[ \label{eq_roiv_invest} ROIV_A = \frac{1 + \frac{Return}{v_A}}{1 + \frac{Issuance_A}{MarketCap_A}} - 1 \] Using the original definition of ROIS in Eq.\eqref{eq_rois} with the definition of \( W_A \) from Eq.\eqref{eq_W_A_invest} we get: \[ ROIS_A = \frac{\frac{v_A + Return}{1 + New\ Shares_A / Shares_A} - v_A}{Issuance_A} \] The fraction between the number of new and old shares can again be rewritten as follows: \[ \label{eq_rois_invest} ROIS_A = \frac{\frac{v_A + Return}{1 + Issuance_A / MarketCap_A} - v_A}{Issuance_A} \] Using the definition of \( ROIS_{Buyer} \) in Eq.\eqref{eq_rois_buyer} with the definition of \( W_A \) from Eq.\eqref{eq_W_A_invest} we get: \[ \label{eq_rois_buyer_invest} ROIS_{Buyer} = \frac{v_A + Return}{MarketCap_A + Issuance_A} - 1 \] It can be proven that the share issuance is only a "zero-sum game" when the issuance amount exactly equals the return on the investment: \[ ROIS_A = -ROIS_{Buyer} ~~~ \Leftrightarrow ~~~ Issuance_A = Return \]

### Example of Investment

Let us consider an example where the stock is somewhat under-valued. Say the market-value of all the company's shares is \( MarketCap_A = $10b \), but the intrinsic value of the company is \( v_A = $12b \). The company wants to issue new shares for \( Issuance_A = $1b \), and although the fees may be very high, they are actually not used in these valuation formulas. The company makes an investment for the money it raises, and the present value of the return on the investment is \( Return = $1.3b \). The gain/loss of intrinsic value to the company's current shareholders is then calculated using Eq.\eqref{eq_roiv_invest}: \[ ROIV_A = \frac{1 + \frac{Return}{v_A}}{1 + \frac{Issuance_A}{MarketCap_A}} - 1 = \frac{1 + \frac{$1.3b}{$12b}}{1 + \frac{$1b}{$10b}} - 1 \simeq 0.76 \% \] This may seem like a negligible gain, but remember that this is relative to the intrinsic value of the entire company. To better understand the magnitude of the gain, we may calculate the ROIS ratio, which measures the net effect of the share issuance, relative to the amount that was raised from the issuance of new shares. We can use the general and simple Eq.\eqref{eq_rois_reduced}, or the more lengthy Eq.\eqref{eq_rois_invest} for this particular scenario: \[ ROIS_A = \frac{\frac{v_A + Return}{1 + Issuance_A / MarketCap_A} - v_A}{Issuance_A} = \frac{\frac{$12b + $1.3b}{1 + $1b / $10b} - $12b}{$1b} \simeq 9.1\% \] So although the number of shares is diluted through a new share issuance, at a share-price that is lower than the intrinsic value, the original shareholders of company A still benefit from the share issuance, because the return on the investment is greater than the loss from issuing new shares at prices that are below their intrinsic value.

The buyer of the new shares gains both from the shares being under-valued, and from the return on the company's investment. This is measured by the \( ROIS_{Buyer} \) ratio in Eq.\eqref{eq_rois_buyer_invest}: \[ ROIS_{Buyer} = \frac{v_A + Return}{MarketCap_A + Issuance_A} - 1 = \frac{$12b + $1.3b}{$10b + $1b} - 1 \simeq 20.9\% \] So the company's original shareholders gain \( ROIS_A = 9.1\% \) while the buyer of the new shares gain \( ROIS_{Buyer} = 20.9\% \) measured relative to the amount paid in the share issuance. Both parties gain from the share issuance in this example, even though the company's shares are under-valued, because the company's original shareholders gain even more from the return on the investment.

This is only a "zero-sum game" when \( Issuance_A = Return \). If you repeat the calculations above with such values then you should get \( ROIS_A = -ROIS_{Buyer} \).

## Full Acquisition

Now consider a scenario with two companies A and B, where company A acquires company B in full, and pays for the acquisition with newly issued shares in company A. This is also called a Stock Swap because the shareholders of company B exchange all their shares for newly issued shares in company A. The question is whether the shareholders in either company will gain or lose from this exchange.

The Swap Ratio is how many new shares in company A are issued and exchanged for each share in company B: \[ Swap\ Ratio = \frac{New\ Shares_A}{Shares_B} \] The issuance amount for company A is defined in Eq.\eqref{eq_issuance} and is the market-value for the new shares in company A. If the stock swap is done according to current market-values for the shares in both companies, then the issuance amount must also equal the entire market-cap of company B as defined in Eq.\eqref{eq_mcap} (with subscript B instead). Combining these two formulas, the swap ratio can also be written as the ratio between the two share-prices: \[ \label{eq_swap_ratio_shareprice} Swap\ Ratio = \frac{SharePrice_B}{SharePrice_A} \] This is useful if you want to use a swap ratio with current market-values for the shares of the two companies. But this does NOT mean the shareholders of the two companies have equal chance of gain or loss. Because that also depends on the intrinsic values of the two companies relative to their market-values, as well as the earnings synergies that arise from the merger. That is why the simulation model allows you to set another swap ratio, than the one implied by the share-prices.

Let \( Synergy \) denote the present value of the earnings synergies that arise from the merger of the two companies. This could be the present value of all future savings from removing redundant factories or staff.

In this scenario, the intrinsic value \( W_A \) of the merged company A after the stock swap, per diluted share and adjusted for dividend tax, is defined as follows: \[ \label{eq_W_A_full_acq} W_A = \frac{v_A + v_B + Synergy - Fees_A}{Shares_A + New\ Shares_A} \cdot (1 - TaxRateDividend) \] For the original shareholders in company A, the ROIV ratio measures their gain/loss of intrinsic value, which reduces to the following: \[ ROIV_A = \frac{W_A}{V_A} - 1 = \frac{1 + \frac{v_B + Synergy - Fees_A}{v_A}}{1 + \frac{New\ Shares_A}{Shares_A}} - 1 \] The fraction between the number of new and old shares in the merged company A can be rewritten as follows: \[ \label{eq_roiv_full_acq_A} ROIV_A = \frac{1 + \frac{v_B + Synergy - Fees_A}{v_A}}{1 + \frac{Issuance_A}{MarketCap_A}} - 1 \] But to calculate the relative value of the stock swap for the original shareholders of company B, we need a slightly different definition of its intrinsic value. So let the per-share, after-tax intrinsic value of company B prior to the stock swap be denoted \( \hat{V}_B \) with a so-called "hat" marker, because it is calculated differently by using the number of newly issued shares in company A, rather than the original number of shares in company B: \[ \hat{V}_B = v_B \cdot \frac{1 - TaxRateDividend}{New\ Shares_A} \] For the original shareholders in company B, the ROIV ratio is calculated using \( \hat{V}_B \), so we make a proper comparison of the number of shares in the merged company. The formula reduces to: \[ ROIV_B = \frac{W_A}{\hat{V}_B} - 1 = \frac{1 + \frac{v_A + Synergy - Fees_A}{v_B}}{1 + \frac{Shares_A}{New\ Shares_A}} - 1 \] The fraction between the number of old and new shares in company A can again be rewritten as follows: \[ \label{eq_roiv_full_acq_B} ROIV_B = \frac{1 + \frac{v_A + Synergy - Fees_A}{v_B}}{1 + \frac{MarketCap_A}{Issuance_A}} - 1 \] The ROIS ratio for company A is the same as Eq.\eqref{eq_rois}, but for the original shareholders of company B it is defined using \( \hat{V}_B \) and \( New\ Shares_A \) to make a proper comparison to the per-share numbers in the merged company: \[ ROIS_B = \frac{W_A - \hat{V}_B}{\frac{Issuance_A}{New\ Shares_A} \cdot (1 - TaxRateDividend)} \] which can be reduced to the following: \[ \label{eq_rois_B_full_acq} ROIS_B = \frac{ROIV_B \cdot v_B}{Issuance_A} \] It can be proven that this is only a "zero-sum game" when the fees equal the earnings synergies: \[ ROIS_A = -ROIS_B ~~~ \Leftrightarrow ~~~ Fees_A = Synergy \]

### Example of Full Acquisition

Let us consider an example where the stock of company A is somewhat under-valued, and the stock of company B is somewhat over-valued. Say the market-cap and intrinsic value of company A is: \[ MarketCap_A = $10b, ~~~ v_A = $12b \] For company B the numbers are: \[ MarketCap_B = $1b, ~~~ v_A = $800m \] The present value of the earnings synergies from the merger is \( Synergy = $200m \). The fees involved in the merger are \( Fees_A = $50m \). We will make the stock swap at the current market-value for the two companies, so the newly issued shares in company A must have the same market-value as all the shares in company B: \[ Issuance_A = MarketCap_B = $1b \]

The gain/loss of intrinsic value to the original shareholders of company A is calculated using Eq.\eqref{eq_roiv_full_acq_A}: \[ ROIV_A = \frac{1 + \frac{v_B + Synergy - Fees_A}{v_A}}{1 + \frac{Issuance_A}{MarketCap_A}} - 1 = \frac{1 + \frac{$800m + $200m - $50m}{$12b}}{1 + \frac{$1b}{$10b}} - 1 \simeq -1.9\% \] The gain/loss of intrinsic value to the original shareholders of company B is calculated using Eq.\eqref{eq_roiv_full_acq_B}: \[ ROIV_B = \frac{1 + \frac{v_A + Synergy - Fees_A}{v_B}}{1 + \frac{MarketCap_A}{Issuance_A}} - 1 = \frac{1 + \frac{$12b + $200m - $50m}{$800m}}{1 + \frac{$10b}{$1b}} - 1 \simeq 47.2\% \] There is a very large difference in ROIV ratios for the two companies, because these ratios measure the change in intrinsic values for the two companies, and company A has 15x higher intrinsic value than company B. If we instead consider the ROIS ratios from Eqs.\eqref{eq_rois_reduced} and \eqref{eq_rois_B_full_acq}, then we compare the change in shareholder value relative to the issuance amount, so we get more similar numbers for the two companies, but with opposite signs: \[ ROIS_A = \frac{ROIV_A \cdot v_A}{Issuance_A} = \frac{-1.9\% \cdot $12b}{$1b} \simeq -22.7\% \] \[ ROIS_B = \frac{ROIV_B \cdot v_B}{Issuance_A} = \frac{47.2\% \cdot $800m}{$1b} \simeq 37.7\% \] This means the shareholders of company A have lost -22.7% on the issuance amount, while the shareholders of company B have gained 37.7%. This is because the shares of company A were somewhat under-valued, while the shares of company B were somewhat over-valued. The earnings synergies were not enough to counter-balance the fees and mis-valuations that caused losses for the original shareholders of company A.

This is only a "zero-sum game" when the fees equal the earnings synergies. If you repeat the calculations above with \( Fees_A = Synergy \) then you should get \( ROIS_A = -ROIS_B \).

## Partial Acquisition

Now consider a scenario with two companies A and B that both issue new shares and swap them, so each company owns a part of the other. The question is whether the shareholders in either company will gain or lose from this exchange.

The Swap Ratio is how many new shares in company A are exchanged for each new share in company B: \[ Swap\ Ratio = \frac{New\ Shares_A}{New\ Shares_B} \] If the stock swap is done according to current market-values for the shares in the two companies, then we must have \( Issuance_A = Issuance_B \), and using the definition from Eq.\eqref{eq_issuance}, it means the swap ratio can again be written as the ratio between the two share-prices as in Eq.\eqref{eq_swap_ratio_shareprice}. This is useful if you want to use a swap ratio with current market-values for the shares of the two companies. But this does NOT mean the shareholders of the two companies have equal chance of gain or loss. Because that also depends on the intrinsic values of the two companies relative to their market-values. That is why the simulation model allows you to set another swap ratio, than the one implied by the current share-prices.

Let \( \Delta Shares \) (the Greek letter "Delta") denote the number of new shares in a company relative to its total number of shares after the issuance: \[ \label{eq_delta_shares_A} \Delta Shares_A = \frac{New\ Shares_A}{Shares_A + New\ Shares_A} \] \[ \label{eq_delta_shares_B} \Delta Shares_B = \frac{New\ Shares_B}{Shares_B + New\ Shares_B} \] Let \( w_A \) denote the intrinsic value of company A after the stock swap, but not per-share, and before dividend tax. This consists of the company's intrinsic value before the share issuance \( v_A \), plus the part of company B that is owned by company A after the stock swap and written as \( w_B \cdot \Delta Shares_B \), minus the fees for company A: \[ w_A = v_A + w_B \cdot \Delta Shares_B - Fees_A \] Similarly for company B, the intrinsic value after the stock swap, but not per-share and before dividend tax, is: \[ w_B = v_B + w_A \cdot \Delta Shares_A - Fees_B \] The two formulas above are defined in terms of each other. That is, \( w_A \) is defined in terms of \( w_B \), which is then defined in terms of \( w_A \) again. This seems like an impossible circular definition, but it can actually be solved by inserting one formula into the other and reducing. The results are: \[ \label{eq_w_A_part_acq} w_A = \frac{v_A + (v_B - Fees_B) \cdot \Delta Shares_B - Fees_A}{1 - \Delta Shares_A \cdot \Delta Shares_B} \] \[ \label{eq_w_B_part_acq} w_B = \frac{v_B + (v_A - Fees_A) \cdot \Delta Shares_A - Fees_B}{1 - \Delta Shares_A \cdot \Delta Shares_B} \] We can now calculate the per-share, after-tax intrinsic values of the two companies after the stock swap: \[ W_A = w_A \cdot \frac{1 - TaxRateDividend}{Shares_A + New\ Shares_A} \] \[ W_B = w_B \cdot \frac{1 - TaxRateDividend}{Shares_B + New\ Shares_B} \] The ROIV ratios measure the gain/loss of intrinsic value to the current shareholders of each company. We do not expand these formulas further as they would become very long: \[ ROIV_A = \frac{W_A}{V_A} - 1 = \frac{w_A / v_A}{1 + \frac{New\ Shares_A}{Shares_A}} - 1 \] \[ ROIV_B = \frac{W_B}{V_B} - 1 = \frac{w_B / v_B}{1 + \frac{New\ Shares_B}{Shares_B}} - 1 \] The fraction between the number of new and old shares can again be rewritten as follows: \[ \label{eq_roiv_part_acq_A} ROIV_A = \frac{w_A / v_A}{1 + \frac{Issuance_A}{MarketCap_A}} - 1 \] \[ \label{eq_roiv_part_acq_B} ROIV_B = \frac{w_B / v_B}{1 + \frac{Issuance_B}{MarketCap_B}} - 1 \] For the original shareholders in each company, the ROIS ratios are the same as Eq.\eqref{eq_rois}, with the respective subscripts for company A and B.

In the other scenarios above, we determined whether they were "zero-sum games" by comparing their ROIS ratios. That was possible because the two ROIS ratios were calculated with the same issuance amount. But in this scenario we use Eq.\eqref{eq_rois} to calculate the ROIS ratios with different issuance amounts for the two companies — they may be equal but they don't have to be. So to determine if this kind of stock swap is also a "zero-sum game", we will instead compare the change of intrinsic value for the two companies. It can then be proven that this stock swap is also a "zero-sum game" when the fees are zero: \[ ROIV_A \cdot v_A = - ROIV_B \cdot v_B ~~~ \Leftrightarrow ~~~ Fees_A = Fees_B = 0 \]

### Example of Partial Acquisition

Let us consider an example where the stock of company A is somewhat under-valued, and the stock of company B is somewhat over-valued. Say the intrinsic value of company A is \( v_A = $12b \) and it is \( v_B = $8b \) for company B. Both companies have the same number of shares and share-price e.g. \( Shares = 1b \) and \( SharePrice = $10 \) so that \( MarketCap = $10b \) for both companies. The fees involved are \( Fees_A = Fees_B = $25m \).

We will make a stock swap at the current market-value for the two companies, so the newly issued shares of the two companies must have the same market-value. Because the share-prices are equal we know from Eq.\eqref{eq_swap_ratio_shareprice} that the swap ratio is one. Let us say that we want to issue and swap \( New\ Shares = 100m \) in each company, corresponding to the amount: \[ Issuance_A = Issuance_B = New\ Shares \cdot SharePrice = 100m \cdot $10 = $1b \]

We first calculate the delta-values for the number of new shares in each company using Eqs.\eqref{eq_delta_shares_A} and \eqref{eq_delta_shares_B}. In this simple example both companies have the same number of old and new shares, so we get: \[ \Delta Shares_A = \Delta Shares_B = \frac{New\ Shares}{Shares + New\ Shares} = \frac{100m}{1b + 100m} \simeq 0.0909 \] We then use Eq.\eqref{eq_w_A_part_acq} to calculate the intrinsic value of company A after the share-issuance, not per-share, and before dividend tax: \[ \displaylines{ w_A & = & \frac{v_A + (v_B - Fees_B) \cdot \Delta Shares_B - Fees_A}{1 - \Delta Shares_A \cdot \Delta Shares_B} \\ & = & \frac{$12b + ($8b - $25m) \cdot 0.0909 - $25m}{1 - 0.0909 \cdot 0.0909} & \simeq & $12.81b } \] Similarly we use Eq.\eqref{eq_w_B_part_acq} for company B: \[ \displaylines{ w_B & = & \frac{v_B + (v_A - Fees_A) \cdot \Delta Shares_A - Fees_B}{1 - \Delta Shares_A \cdot \Delta Shares_B} \\ & = & \frac{$8b + ($12b - $25m) \cdot 0.0909 - $25m}{1 - 0.0909 \cdot 0.0909} & \simeq & $9.14b } \] The ROIV ratios for the original shareholders of each company are calculated using Eqs.\eqref{eq_roiv_part_acq_A} and \eqref{eq_roiv_part_acq_B}: \[ ROIV_A = \frac{w_A / v_A}{1 + \frac{Issuance_A}{MarketCap_A}} - 1 = \frac{$12.81b / $12b}{1 + \frac{$1b}{$10b}} - 1 \simeq -2.99\% \] \[ ROIV_B = \frac{w_B / v_B}{1 + \frac{Issuance_B}{MarketCap_B}} - 1 = \frac{$9.14b / $8b}{1 + \frac{$1b}{$10b}} - 1 \simeq 3.85\% \] So the original shareholders of company A have lost -2.99% on the intrinsic value of their shares, while the original shareholders of company B have gained 3.85% on the intrinsic value of their shares. Although the companies swapped the same number of shares, to obtain the same ownership size in each other, the intrinsic values of the two companies were different, so their gain/loss from the stock swap are also different.

In this example, the two ROIV ratios are of similar scale, but with opposite signs. This is because the intrinsic values of the two companies are fairly close and the market-caps are identical. Otherwise the ROIV ratios could be very different for the two companies. This is because the ROIV ratios measure the change in intrinsic value as a result of the stock swap. It is often useful to consider the ROIS ratios as well, which measure the change in intrinsic value relative to the issuance amount. The ROIS ratios are calculated using Eq.\eqref{eq_rois_reduced}: \[ ROIS_A = \frac{ ROIV_A \cdot v_A }{Issuance_A} = \frac{ -2.99\% \cdot $12b }{$1b} \simeq -35.9\% \] \[ ROIS_B = \frac{ ROIV_B \cdot v_B }{Issuance_B} = \frac{ 3.85\% \cdot $8b }{$1b} \simeq 30.8\% \] For company A it means the stock swap had a similar effect as a $1b investment with a loss of -35.9% in present value. Conversely, for company B the stock swap had a similar effect as a $1b investment with a gain of 30.8% in present value. This is because the shares in company A were under-valued, and the shares in company B were over-valued. So the original shareholders of company A were issuing shares at a too low price, in exchange for new shares in company B that were issued at a too high price, compared to their intrinsic values.

This is only a "zero-sum game" when the fees are zero. If you repeat the calculations above with zero fees, then you should get that \( ROIV_A \cdot v_A = - ROIV_B \cdot v_B \). In this example we could also have compared the ROIS ratios, because the issuance amount is the same for both companies, but that may not always be the case.