Now we have all the tools to calculate the equity value, to apply DCF methodologies, and to use comparable companies. Everything is ready and we have to apply them to PE deals. In PE deals, as you know, there is an issue of double valuation, because we have a problem to calculate the equity value at time 0. Thatâ€™s when the PEI decides to invest, but we have, as well, a problem to calculate the equity value at the end. That means in the exit, and the difference between the two equity values accordingly with the time of investment are crucial because all together they affect IRR. To face this problem of double valuation, we can use two different approaches. The first one, and weâ€™ll start with that, is based on the fact that the issue to calculate the equity value at time 0 is not a problem. Itâ€™s not a problem in that the business plan is solid and the assumptions are robust. That means the amount of the equity value calculated with DCF is really reasonable. It will be a completely different story with startups or in cases in which the business plan is not solid as well. The first case is very common, for example, in expansion, in replacement, in which companies are mature, and it's honestly relatively easy to calculate a business plan. In this case, we have to decide how to approach the issue of valuation. The first step is to calculate the equity value at time 0, and we know everything about it. The second problem is not exactly to calculate the equity value at time n, which is a bit more complicated, because if we want to calculate the equity value at time n using the DCF again, we need another business plan for the years after the end of the business plan we have, and honestly that's very difficult. So, the problem we have to deal with is to see if considering the numbers we have in our hands, that means in the business plan, there is enough room for the PEI to get to an IRR that is really reasonable. Let's apply all these concepts using an example, an exercise. You have, in your hands, a business plan of the company: six years of business plan starting from 2015 till 2020. The issue is to apply the concept of double valuation of the equity value. That means that to calculate the equity value at time 0, using the concept of DCF, where you know everything. And to calculate the equity value at the exit using the concept of multiples, because it's impossible to calculate again, as I said before, the equity value using the DCF. This approach is very relevant because we have to see if there is enough room within the business plan to get to a certain IRR that makes sense for the private equity investor accordingly with the expectation of return overall on the entire portfolio, but now we have to use some numbers. Let's imagine that the investment made by the PEI is 4 and a half million, and using DCF, 4 and a half million represents 30% of the companyâ€™s equity. Now the problem is to calculate the equity value at the exit, and to do that we have to run another assumption and the other assumption is related to the holding period. Let's imagine the holding period is three years. That means we have to consider three years of our business plan, 2015, 2016, and 2017. We have to focus on 2017 which is the year of the exit and in this year we simply need to identify two numbers: the EBITDA and the net financial position. Why are they important? The EBITDA is important because, using a multiple it's possible to calculate the enterprise value. And so, for example, if you use a multiple of four, we multiply the EBITDA, which is 6 million, and we get what is the value of the enterprise value that in this case is 24 million. But if we have 24 million, and we also know what is the net financial position, that using again a business plan. The net financial position is represented by 4 million, itâ€™s possible to calculate what is the amount of the equity value, which is 24 million minus 4 million. That represents 20 million euros. 20 million euros using multiples represent the equity value of the company at the exit. Now in a certain sense the job is done, because if we have 30% of shares in our hands as a PEI, in the moment of the exit, 30% of 20 million represents six million Euros. So we invested, at time 0, 4 and a 1/2 million, and the expectation, after 3 years, is to have, in our hands as a PEI, 6 million. If we calculate the IRR considering three years of holding period, the expected IRR is 10.06%. Our problem is not to say if 10% is good or bad. Itâ€™s not possible to say, but every PEI considering his or her expectation of return can say if it is a good return or not. However, as you can imagine, itâ€™s possible to run a sensitivity analogy, and in your hands, you also have the table and matrix in which you can combine two different parameters of the matrix itself. On one hand, you can use different multiples you can expect. On the other hand, you can consider differing holding periods. For example, if we keep our assumption of a three year holding period, and you jump into the matrix, you see that, considering a multiple of four, you see 10% of IRR. If we increase the multiple, that means we bet on the growth of the market. In this case, the expected IRR is going to increase. On the contrary, if we decide to run another simulation, that means our expectation is to stay in the company, not for three years, but to stay for four years or five years, in this case, you can see in the matrix, that the values of the IRR are completely modified. A concept which is quite intuitive but simple. If the stay, the holding period, is longer to have a higher IRR we have to bet on higher multiples. And as you can imagine the game is quite risky but is a huge big responsibility of any kind of PEI to make this kind of decision.