Next, we want to expand the ideas going from the cash flows to fixed income instruments. Fixed income instruments are securities that guarantee a fixed cash flow. They guarantee you a dollar amount. But are these instruments risk-free? Not at all. There is default risk associated with them, which means that at some point this entity that is backing this fixed-income security might go bankrupt, and therefore the cash flow that you were really promised, is not going to come through. About the only entity that is there that is risk-free is the US government or equally stable governments that are going to guarantee you that the cash flow is going to come. Corporate fixed income securities are always open to default risk. Another risk associated with them is the inflation risk. Even if the entity that gave you the fixed income security does not default, the value of the currency might go down over the periods, and as a result, the fixed cash flow is now of lower value. This happens because the buying power of the currency comes down over time. In order to hedge against this, people have started thinking about tips which are inflation protected securities. Even if there wasn't very heavy inflation risk in the market, there's also a market risk, meaning these securities might become less or more valuable as time goes by, and if you wanted to sell the securities in the market, the price that you might end up getting is going to fluctuate over time. As a result, you are opening yourself up to market risk. Having said all of that, let's consider some typical fixed income securities and try to price them using a no-arbitrage condition. One of them is the perpetuity, which says that it's going to give you a fixed amount A for all times in the future. What is the no-arbitrage price for this security? If you assume that you can borrow or lend at the rate r, an unlimited amount then the no-arbitrage price for this security is going to be the sum of k, going from one to infinity of A divided by one plus r to the power k. If you sum this infinite series, you end up getting that the price is nothing but A divided by r, which is the per-period interest rate. Annuity it's a fixed income instrument that pays an amount A for periods k equals 1 through n. You can write this as a difference between two perpetuaties. One perpetuity that starts right now. It pays from, so here's t equal to zero. You have a perpetuity that pays at an amount A at all these times in the future, and another perpetuity, that starting at time n plus 1. Meaning here's time n, here's time n plus 1. This is the first time you are going to get a negative amount from this perpetuity. After that, everything will cancel. All of these will cancel and you'll end up getting that the cash-flows associated with that is A for t equals 1, 2 up through n. The difference between these two is exactly the price of the annuity. Therefore, the price of the annuity is A over r, this is the price of a perpetuity that starts right now. The price of the perpetuity that starts at time n is going to be A over r, but this is going to be at time t equal to n. You have a discount this back end period. This is the discount factor. We discount it back, you get the price for the annuity. This is going to be A over r, 1 minus 1 plus r to the power n. The next fixed income instrument that we are going to be interested in is a bond. A bond is characterized by five different parameters or five different qualities, is a face value. Face value is typically a hundred dollar thousand. There's a coupon rate, Alpha, which is paid every six months. Every six months a bond pays you Alpha times F, which is the face value divided by 2. There's the maturity, there's the date of the payment or the face value in the last coupon, and there is a price, which is the price at which these bonds are going to be selling. In addition to this, there is also a quality rating. SNP arrange them as triple A, double A, triple B, double B, triple C, double C, and so on. The quality essentially is trying to get at the default risk. A high-quality bond has very low default risk. The chance that it would stop paying its coupons are going to be very low. A lower quality bond, there's a higher default risk, meaning that there's a higher chance that it would not pay the payments. Here's the story. Every six months, so half, one year, one-half, two, and so on, it pays Alpha, which is the coupon rate times F, which is the face value divided by 2. At maturity capital T, you get the face value back plus you get Alpha F divided by 2, which is the last coupon payment. Now, bonds are going to be characterized by at least four numerical quantities. The face value, the coupon rate, the maturity, and the price. In addition, there is this other quantity called quality. Since bond differ in so many different dimensions, it's hard to compare bonds. One quantity that has been introduced in order to be able to compare bonds of different maturities, different coupon rates, and face values, is this idea known as the yield to maturity. The yield to maturity is the annual interest rate at which the current price for the bond P is exactly equal to the present value of the coupon payments plus the face value. You take all the coupons and you discount them at the rate Lambda over 2, because that's, the coupons come every six months. Lambda is the annual rate of interest and therefore you have to discount it by Lambda over 2. Similarly, you discount the face value that appears at time capital T, which is to say after 2T, 6 month periods at the rate Lambda over 2. What does yield to maturity does, is that it gives you a single number to start thinking about bonds. It summarizes the face value, coupon maturity, quality, and so on. It's a number that has understandable movements with respect to quality. Lower quality means lower price, which means higher yield to maturity, which means essentially the intuitively to think about it is that cash payments in the future are going to be discounted with a higher interest rate. Why with a higher interest rate? Because I'm not certain that those cash payments are going to come and therefore, I want to discount them very strongly. With lower quality, I'm even more uncertain, which means that I should have a higher yield to maturity. If the interest rate in the market were to change and the yield to maturity would change in a similar manner. Yield to maturity, therefore, gives us a way to think about different bonds and compare them, but remember, yield to maturity is a single number. It's a very crude measure. It's trying to summarize four different numbers by a single number. It's not going to be able to capture everything.