So one of the most useful things to know how to value is something called a growing perpetuity. So for a growing perpetuity, we have a payment that grows at a fixed rate forever. So how does this work? Now, when anything gets a little bit complicated, it's very useful to write down the timeline. So for our growing perpetuity, we won't have anything at time 0. At time 1, we will have our cash flow, C. At time 2, we'll have our cash flow, but now we're going to grow it up by g, g stands for growth. At time 3, we're going to take our g and grow it up again. Then dot, dot, dot, we're going to continue this forever. So when we calculate the present value, we have our first payment, our second payment, which we discount back, our third payment which we discount back, and on, and on. So we have a very beautiful formula for this. Let's just write out what we had to begin with. C discounted back one year, C times 1 plus g discounted back two years, and on, and on. A typical term in the future is going to have C times 1 plus g_t minus 1 discounted back to t years, and on, and on. Now, as you could tell from our previous discussion, what's really going to matter here is whether these terms are decreasing exponentially or are they growing exponentially? For them to be decreasing exponentially, we need the condition r bigger than g. So if r is greater than g, then that formula reduces to C divided by r minus g. The growing perpetuity formula. If r is less than g or if r is equal to g, then the formula does not exist. So notice a couple things. When g equals 0, the present value is C over r. That's good. That's the formula we got from before. Notice also that the larger is r, the lower is the present value. That makes sense. Because the higher is r, the greater is the time value of money, the less you need to put aside today. Notice also that the greater is g, the greater the growth rate, the greater is the present value. Because g appears in the denominator with a negative sign. That makes sense too. Because the bigger are these g's, the bigger are these terms, and the bigger is the thing that you're discounting back. So that's a growing perpetuity. Another important type of cash flow to value is a delayed perpetuity. This can be surprisingly tricky. So a delayed perpetuity says we have cash flow C every year, starting three years from now. A cash flow C every year, starting three years from now. Let's write down that cash flow diagram. So here we are at time 0, we don't get anything. At time 1, we don't get anything. At time 2, we don't get anything. At time 3, we get our cash flow. At time 4, we get our cash flow, and on, and on. What's the value of this asset? Well, let's just remember for a moment, what is the value of our usual perpetuity? What do the cash flows look like? So if we don't delay the perpetuity, the first cash flow starts at time 1. So this guy has a present value of C over r. Now let's go to our delayed perpetuity, 0, 0, three years cash flow starts and then it continues in perpetuity forever. So what you want to do is put yourself at time 2. Because remember for the perpetuity, the first cash flow starts one year from now. That's our convention to get that nice formula C over r. You could have a different convention, then you'd have a different formula. That's our convention. So at time 2, the present value of this perpetuity, is C over r. At time 2, it's just an ordinary run of the mill perpetuity. So now to get it back to time 0, what do we do? We discount it. The present value is, this present value discounted back to time 0. How do we do that? We divide it by 1 plus r squared. So it's very counter-intuitive. A perpetuity whose cash flows begin in three years is the perpetuity formula discounted back two years. In general, if we have a perpetuity, cash flows begin in t years. The present value is 1 over 1 plus r_t minus 1 times C over r. A little strange. All we have to do is remember, when we have an ordinary perpetuity, the cash flows are one year from now.
