So I already mentioned a consol is one that pays a quantity. What I was saying c or whatever x forever. And the consol present this kind of value is x over or coupon over r. So it's kind of obvious in the case of a, why do we call them consols by the way? Because in Britain, in the 1700s, the government of the United Kingdom issued bonds with no maturity date. They promised to pay this coupon forever, and the only way they could get out of it was by buying them back. And those bonds, well, there was some adjustments made in the 1880s. They're still paying the coupon. The British government hasn't defaulted in all that time. And so it's not forever, but it's close to forever. I mean, several hundred years, a long time to be paying a coupon. Now it's simple to understand the consol present that says the price of the consol p is equal to c over r. You can just return that, turn it around. It says the yield to maturity on a consol is c over p, and that's kind of obvious, right? If the consol is paying a 3 pound coupon, and it's selling for 200 pounds, I say what is the yield to maturity on it? Well it's 3 pounds divided by 200 pounds. In that case it would be, if it was 3% yield to maturity, I'm sorry if it was a 3% coupon issued, it's now paying one and a half percent yield to maturity because the price has gone up. This is an important point with bonds. The coupon is fixed at the time the bond is issued but the market price of the bond changes through time so if the British government issued a 3 pound consol for 100 pounds and that consol is selling for Â£200 today, the yield to maturity is down to 1.5% instead of 3%. But this is no fault of the British government, they are true to their word. They are paying the consol as promised, it's the market that does it so bonds are risky, they have market risk even if there is no default risk. If you buy a consol, you know you won't live forever. The British government may live forever but you won't, so you're going to want to sell the coupon at some point. The British government does not guarantee the rate that you will get, the price you will get for selling your consol. So, there's market risk for our consol and for any debt instrument, long term debt instrument. By the way, if a bank or a company or a government were to issue a usually companies don't issue consols because nobody believes they'll last forever. Patriots in Britain might believe that the British government will last forever. Although I can tell you it won't [LAUGH], not forever. Nothing is forever. But they imagine it's forever, so they're willing to buy British consols. But what if the British government did something even more dramatic? They say, we're going to have the coupon on the Consol growing at a constant rate. So, it's going to start out at 3 pounds, and then it's going to grow at a rate g per year. G is a percent of growth per year forever. Well what is the present value of an amount x if the rate is the yield to maturity or interest rate is r and the x is going to grow at rate g? Well, it turns out the growing consol present discounted value is x over r minus g. So, by the way, what happens if g is greater than r [LAUGH] or g equals r? Any idea on that? Well, I think you probably can figure this out. It's, if g equals r it's x divided by 0, it's infinite. That's because it's the series that you're summing is not convergent. If you're getting an amount that's growing at the rate of interest, the present value is infinite and if it's growing at greater than the rate of interest, well the formula breaks down. It's still infinite. So that's why we get puzzled about the present situation. If the interest rate is 0 then consols don't even converge. How can that be? Doesn't the interest rate always have to be above the growth rate. How can anything, nothing can have an infinite price. Well, I guess the answer is short term interest rates have 0 but longer term interest rates are still positive so and the US doesn't have consols, but we have something analogous to consols, like land, for example. That lasts forever, as far as we know. You can rent it out or you can plant crops on it, as far as we know, forever. Maybe those crop, what about land? Aren't crop values rising through time? So that g is a positive number? And if interest rates are, they can't be 0, they have to be, bond rates have to be above 0 otherwise assets that have growing payments would be worth an infinite amount. Now this is an annuity present discounted value. Now an annuity is like a consol except that it stops after a certain number of years. In a typical annuity, a typical annuity is a home mortgage. When you buy a house, the typical financing you'll get, is that you will pay now the compounding integral is monthly. They think it's not realistic to ask ordinary people to pay their mortgage every six months because it's too hard for them. They'd have to, they wouldn't remember to save and they wouldn't have enough money to pay after six months. So we gotta move to a monthly schedule for individuals. A typical and that's not, this is not compounding monthly. This is the present value of X dollars every year, starting in one year and then again in two years, then again in three years and the last payment is T years. This is different from the present value for a corporate bond, or for a coupon carrying bond because there's no principle repayment at the end. I'm talking about, I think I have another slide on mortgages. I'll come back to that. But I tell you realistically, these financial instruments are designed around human imperfections. People find it difficult to pay back a mortgage. And what they'll find especially difficult, is to pay a balloon payment at the, they call it a balloon payment. There used to be mortgages like this. You would borrow to buy your house, you would say, I'm going back to the 1920s. The house cost $10,000, typical house in the 1920s. You borrowed $9,000. You've got your own money to put up. The $9,000, then you pay back the money at a rate interest. And then, but at the end, you don't owe anything except the last monthly payment. So that's what an annuity is. This is a little history of thought, actually the Growing Consol Formula has been called the Gordon Rule according to Myron Gordon who was a professor of economics about a half century ago. But actually it goes back to Jacob Bernoulli, I learned that from Will Goetzmann and his co author Geert Rouwenhorst here. So it's an old formula