# Share Buyback Tutorial

## Introduction

There are a lot of myths and disagreements about share buybacks. Academia does not have a proper framework for valuing share buybacks, because finance professors have believed for half a century, that stock-markets are "efficient" so share-prices always equal the "risk-adjusted value" to long-term investors. From this belief they concluded that there was no difference between share buybacks and dividends, possibly with the exception of taxes. But their arguments are pseudo-science.

The reality is that a share buyback has two immediate effects: It reduces the company's cash holdings, and it reduces the number of shares outstanding. So a share buyback is a kind of investment the company is making in itself, on behalf of the remaining shareholders. It obviously matters a great deal whether the shares are priced high or low compared to the future earnings of the company.

We can easily see what happens to the shareholder value when simulating share buybacks. In this tutorial we first forecast the company's future earnings, and then we also forecast the future share-prices. You can view the full simulation reports here and here, and clone the simulations to run them yourself with different input values.

We use the company Foot Locker as an example. We do not have any special insights into this company, so we merely use historical data to estimate the probability distributions for the future earnings and share-prices. We are using the same data from the other tutorial on present values. We will not update these tutorials with new data in the future.

## Valuation Method

We can use two different valuation methods to assess whether a share buyback is good or bad for the remaining shareholders:

- The stock's future rate of return, which can be simulated using the standard IRR models and does not require any specialized models for share buybacks. If the IRR is higher than some threshold, such as the expected return on a broad stock-market index plus a risk-premium, then a share buyback would likely be beneficial to the remaining shareholders. But this cannot tell us by how much the shareholder value will change as a result of the share buyback.
- The stock's intrinsic value, which can be simulated using present values in specialized share buyback models. The disadvantage is that we have to select a discount rate, but the advantage is that it allows us to measure how much the shareholder value changes as a result of the share buyback. We will use this method here.

## Simulate Earnings

It seems obvious that a share buyback should only be made when it benefits the remaining shareholders. From this statement it can be argued, that there is no need to consider a share buyback's impact on future share-prices, because we are always valuing the share buyback from the perspective of an "eternal" shareholder who will never sell their shares, so the valuation should only consider the future earnings that could be paid out as dividends. We will do that here, but we will also simulate the share-prices further below.

The future earnings are simulated from the company's historical Net Profit Margin and recent sales. We assume zero real earnings growth to make a conservative valuation, that is, we assume the earnings merely grow the same as the inflation, which means that we should use a real or inflation-adjusted discount rate in the present value simulations. We use a fixed discount rate of 9% here, which is the historical average 5-7 year real return on the S&P 500 stock-index, plus a 5% risk-premium for this particular stock. We don't use the full probability distribution of historical returns on the S&P 500 as the discount rate, because it causes problems with simulation models that use terminal growth-rates, as explained in another tutorial.

The model outputs several plots with the simulation results. The plot below shows a histogram with the so-called Return on Intrinsic Value (ROIV), which is the percentage gain/loss to the intrinsic value per-share, when making a share buyback instead of a dividend payout now. Most of the values are between 1-3%, but significant losses are also possible if the company’s future earnings are negative. This may seem like a small change, but the share buyback is only $100m for a company with a $2.5b market-cap, and an intrinsic value that is likely somewhere between negative $2b and positive $9b, with a mean around $5b. If the buyback amount is $500m instead, then the ROIV is mostly between 10-20%, which would be a very significant gain in shareholder value, achieved simply by repurchasing the shares at a significantly lower price than their intrinsic value.

The plot below is for the same simulation, but shows the so-called Return on Buyback (ROB) instead, which measures the percentage gain/loss relative to the buyback amount. With these input assumptions, the range is around -200% to +400%, so the share buyback could result in a big loss, but would more likely result in a very large gain, when compared to the amount of money used for the share buyback.

The plot below varies the current share-price on the x-axis and shows the ROB ratios on the y-axis. This lets us easily see how the ROB ratios change with the current share-price. The histogram above is actually the slice shown as the vertical dashed blue line. The shade of blue indicates the density of simulation results in a region of the plot, where darker blues indicate a higher density.

You can view the full report and clone the simulation to run it yourself with different input values.

## Simulate Share-Prices

A common argument in favor of companies making share buybacks instead of dividend payouts, is when capital gains taxes are lower than dividend taxes. In this tutorial we have set all tax-rates to zero, but you can clone the simulation to experiment with different tax-rates. You will find that different tax-rates do affect the value of share buybacks, but the share-price relative to the intrinsic value is far more important.

In addition to the future earnings, we now also simulate the company's future share-prices by splitting the price into two components: A valuation metric such as P/E or P/S ratio, and the earnings or sales. We simulate these independently of each other and multiply the values together to get the share-price. We personally prefer using historical P/S ratios because they are usually much more stable than P/E ratios, which are also ill-defined when the earnings are negative or zero.

Below is a so-called violin-plot which shows the distributions vertically with simulation results for each future year (these sometimes resemble violins). This plot shows the ROIV for each of the 10 future years, that is, if the company makes a share buyback at the current price, and you sell your shares in one of the future 10 years, then how much has the intrinsic value of the shares gained or lost from making the share buyback today, compared to the company making a dividend payout instead.

The plot below is for the same simulation, but shows the ROB instead, which is the percentage gain/loss relative to the buyback amount.

The plot below varies the current share-price on the x-axis and shows the ROIV ratios on the y-axis. This shows that if the current share-price is lower, then the ROIV ratios are higher, and vice versa. So a share buyback becomes increasingly beneficial to the remaining shareholders, as the share-price becomes lower relative to the stock's intrinsic value, and vice versa. That is also common sense.

You can view the full report and clone the simulation to run it yourself with different input values.

## Other Investment

We can also compare the value of a share buyback to another alternative investment the company could make for the money. The simplest way to do this, is to simulate the IRR for both the company's stock and the alternative investment, and then compare the two resulting probability distributions to see which is generally higher, so that would be the more profitable use of the money.

We can also compare the ROB ratio that we simulated above to the NPV ratio for the alternative investment. It is important that the buyback amount is the same as the price paid for the alternative investment, because the ROB ratio changes with the buyback amount.

The plot below shows an example where the ROB ratio is copied from the plot further above, and the NPV ratio is just a normal distribution for demonstration purposes. In this example the share buyback has a wider distribution with the potential for both greater gains and losses compared to the alternative investment.

It is very important that you do not compare IRR and ROB ratios, because they measure investment returns in very different ways, so it would be like using both imperial and metric rulers to measure and compare different things.