# Internal Rate of Return Tutorial

## Introduction

The Internal Rate of Return (IRR) is the discount-rate where the current price equals the present value of future amounts. It is a way of estimating the annualized rate of return that an investor would get from making an investment at the current price.

In this tutorial, we will show how to use the IRR to value a stock for long-term investing. We first do this by forecasting the future earnings that could be paid out as dividends to the shareholders, 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 Sally Beauty Holdings 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 will not update this tutorial with new data in the future.

## Simulate Earnings

The future earnings are simulated from the company's historical Net Profit Margin and recent sales. For the first 4 years we also deduct $100m each year to repay debt.

When we only want to simulate the future earnings and not the future share-prices, we use a model that also simulates terminal growth-rates for the earnings of the final year, which are then assumed to grow at that rate for eternity. However, for this particular company, a reasonable assumption is zero growth, which is also a good starting point for making conservative valuations in general.

The model outputs two plots with the simulation results. The first plot shows a histogram with the IRR values on the x-axis, where higher vertical bars means there were more simulation results for those IRR values. The full range of outcomes is about 10-22%, but most of the simulated IRR values fall between 16-21%. If the input distributions are reasonable estimates for the future earnings, then the stock would seem to be an attractive investment at the current share-price.

The second plot below varies the current share-price on the x-axis and shows the IRR values on the y-axis. The histogram above is actually the slice shown as the vertical dashed blue line. The light-blue colors show low occurrences of simulated IRR values, while darker blues show increasingly higher occurrences of simulated IRR values. This plot lets us easily see how the IRR changes with the current share-price. For example, if we only want to invest when most of the IRR values are above 20%, then we can see from this plot that we probably need to pay less than $8 per share.

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

## Simulate Share-Prices

The problem with only simulating future earnings, is that it doesn't show us anything about the impact of price volatility on the investment return, so we might be unpleasantly surprised by large losses in the share-price.

When simulating the future share-prices, we split 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 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.

In this example we use two periods of historical P/S ratios. For the first 4 years where we assume the company will use a large part of its earnings to repay its debt, we use the most recent years which had lower P/S ratios. For year 5 onwards we use the full historical range of P/S ratios.

We also have two plots for this model. The first 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 IRR for each of the 10 future years, that is, if we buy shares at the current price and sell the shares in one of the future 10 years. This shows that in the first year, it is actually possible to have a loss of about -25% because the stock may drop in price. But as the years progress, the future earnings become more important to the overall return, so the IRR values become positive.

The second plot below varies the current share-price on the x-axis. The shade of blue again indicates how many simulated IRR values are in a region of the plot. Because the IRR varies greatly for the future years, it tends to smear this plot, but you can still get an idea of how the current share-price changes the IRR values.

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