In this video, we'll examine two important concepts in finance, perpetuities and annuities. Let's take a look at each of these. A perpetuity is a stream of equal payments that goes on forever. As an example, you install solar PV on your rooftop, it will save you $1,000 a year in power bills. Assume that goes on forever, at least for the moment. You can borrow money at five percent annual interests. Suppose you take a loan out for this project and you're paying five percent annual interest on the loan or you could invest your money if you're paying for it out of savings, you can invest and earn five percent somewhere else. What's the present value of this long stream of payments? Here's the calculation is pretty simple. Again, r is the interest rate, A is the annual perpetuity payment, in this case, savings and PV is the present value of that perpetuity. The equation or the formula for calculating the present value of the perpetuity is simply the perpetuity payment A divided by the interest rate A over r couldn't be simpler. In this case, the present value is equal to 1,000 divided by 0.05, the interest rate to get $20,000. The present value of that infinite series of payments is $1,000 a year, is $20,000 today. Somebody says, I'll buy this stream of payments from you for $5,000. You'd say, no, I'm not interested. Now, it's worth $20,000 to me today. But if they said I'll buy it for $30,000, You'd probably be interested. An annuity is simply a perpetuity with the back-end cut off. So an annuity is a stream of equal payments for a set period of time. Looking at our example, we install solar PV in our roof, saving $1,000 a year in power bills. But now we admit that the panels will only last 20 years instead of forever. Then we'll have to be replaced or perhaps we sell the house or any case, 20 years is the right horizon. We can still borrow money at five percent annual interest. We ask again, what is the present value of this annuity? Notation is the same. r is the interest rate, A is the annual annuity payment, PV is the present value, and now n is equal to the length of the annuity, in this case 20 years. The formula of the present value for the annuity is a bit more complicated. It's equal to the value of the perpetuity but we have to subtract off the back end of the perpetuity in order to get the annuity. In this case, it's the perpetuity value times 1 divided by 1 plus the interest rate to the nth power. It's not really hard to plug in the numbers and do the calculations. When we do that, we find that the present value of the annuity is equal to the perpetuity value, $20,000 times 0.62, 62 percent, giving us $12,422 significantly less than the perpetuity. But since we only have 20 years, this is the present value. In this brief summary, we've looked at perpetuities and annuities. They're equal annual payments of different lengths. A perpetuity is equal annual payments that will continue forever. An annuity is equal annual payments that continue for a set number of years. These are both important concepts that we'll return to you as we further discuss renewable energy project finance. Let's keep going.