2. A Basic Model of Bond Markets
In chapter 1 we revisited the concept of time value of money and its relation to opportunity cost, but over a one period model. We did touch upon multi period models but only under the assumption that the interest rate from period to period is the same: a phenomena referred to as a flat term structure of interest rate.
In this chapter we expand our model to a more realistic situation of a multi period time and relax our assumption about a flat term structure. Rather, here we will look into the prices of bonds in the market and infer the term structure of interest rate for these prices. As before, our guideline for the inferences is the stratification of the no arbitrage condition. However, in this multi period model the formulation of the no arbitrage condition and the consequences of its satisfaction are somewhat different and more structured than our presentation of the one period model. We start with some basic notation and definitions so that we can be in a position to formulate the no arbitrage condition.
2.1 Setting the Framework
The debt market, as its name implies, is a market in which debts are bought and sold. This market is also referred to as the bond market. A bond is a security issued by a particular entity which promises to pay the holder of the bond a fixed amount of money at fixed times in the future, ![t[1]](images/Chp2-1_1.gif){kind=small}
, ![t[2]](images/Chp2-1_2.gif){kind=small}
,...,![t[k]](images/Chp2-1_3.gif){kind=small}
. At each of the payment times except the last one, ![t[k]](images/Chp2-1_4.gif){kind=small}
, the bond pays the same amount of money which we will denote by 
. On the last payment date, referred to as the maturity of the bond, the bond pays an amount equal to 
. The amount 
is referred to as the principal or the face value of the bond and the payments of the amount 
are called coupon payments.
The value of 
is a certain percentage of the face value, 
, and is called the coupon rate. The coupon rate is specified when the bond is issued and remains fixed for the life of the bond. Thus for example, a bond may pay $
every year for the next three years and at the end of the three years pay 
. Such a bond has a maturity of three years and a face value of $100 and the annual coupon payments are $5 each. One immediately recognizes that the cash flow from a bond is like a repayment of a loan that was taken for three years at an interest rate of 5 percent, paid annually. Indeed, buying a bond is giving a loan to the issuer. Such a bond, as in our example, is called a 5 percent bond since it is like a loan taken at 5 percent. In most countries though, the payments from a bond are made semiannually: a 5 percent bond will pay 2.5 percent of the face value every six months.
The coupon rate at which the bond is issued depends, of course, on the interest rate that prevails at the time of issue in the market. In order to induce investors to buy the bond (lend their money), the bond must offer a competitive interest rate. Similarly, after the bond has been issued, it can also be bought and sold in the market (called the secondary market). The price in the secondary market will reflect current market conditions with respect to the interest rate prevailing at that time.
Consider the bond in our example that was issued with a coupon rate of 5 percent. An investor holding the bond for the first six months and then selling it in the secondary market may get more or less than $100 when it is sold. Buying a bond six months after the bond was issued is like giving a loan of $100 to the issuer for 2.5 years. If at that time the interest rate prevailing in the market for loans over 2.5 years is, for example, 4 percent, the bond will not be sold for $100. If the bond did sell for $100, it would constitute a lending at an interest rate higher than the one prevailing in the market. The holder of this bond will not like to pass such good deal to others. Furthermore, the owner realizes that the bond will attract buyers if it will offer a rate competitive with the current market rate. Hence the bond will be sold at a price, say 
, such that
| > | P=5*(1+0.04)^(-0.5)+5*(1+0.04)^(-1.5)+105*(1+0.04)^(-2.5); |
 |
| > |
In such an environment, the bond will sell for more than its face value. Such a bond is called a premium bond.
Suppose instead that interest rates rise to 8 percent. The competitive forces in the market will alter the price of the bond in such a way that
| > | P=5*(1+0.08)^(-0.5)+5*(1+0.08)^(-1.5)+105*(1+0.08)^(-2.5); |
 |
| > |
Thus, the bond will be sold at less then its face value. Such a bond is called a discount bond. We thus see that there is an inverse relationship between the price of a bond and the level of interest rates. Therefore, implicit in the prices of bonds in the market is some information about the interest rates in the market. This information can be uncovered using a technique that we will soon introduce.
Moreover, recovering information about interest rates implicit in bond prices is intimately related to the no-arbitrage condition. A condition with which we have already familiarized ourselves in the simplistic model and of the former chapter.
The fuzzy term we just used ("competitive forces in the market") will soon be seen to be the force of investors seeking arbitrage opportunities. Consequently, these investors affect the market so that prevailing prices eliminate such opportunities. Stating it differently, prices in the market satisfy the no-arbitrage condition. Furthermore, an explanation of the way the condition was formulated in the former Chapter will make it adaptable to a realistic model of the bond market. As well, the discount factors of the former Chapter will be replaced with a function of discount factor - the term structure of interest rates in the market.
We limit our focus, almost throughout this Chapter, to national government bonds. These securities are regarded as risk-free securities, since governments do not usually default on their obligations. Footnote 1. Bonds, as we see, represent fixed payment amounts which are paid at fixed, deterministic times. For this reason, bonds are also referred to as fixed income securities. Thus, if an investor holds a bond to its maturity, the amount of the payments and their timing are certain, provided that the issuer does not default on some payments. Hence, national government bonds are considered non-risky securities.
A bond which is issued by a less creditworthy issuer must offer a higher interest rate, in comparison to a government bond, in order to compensate the investor for taking the risk of the issuer defaulting on the bond. The lower the creditworthiness of the issuer, the higher the interest rate the bond must offer. Indeed, we observe this in the market for corporate bonds (issued by corporations) which offer higher interest rates than government-issued bonds. Agencies exist in the market which engage in rating the creditworthiness of different issuers. The lower the rating is, the higher the interest rate they must offer on their bonds. There are other factors that may affect the interest rate at which the bond is issued.
Certain bonds have features that affect the interest rate. For example, some bonds, called callable bonds, allow the issuer to call the bond back prior to its maturity. The issuer can pay the holder the principal plus a certain amount and so buy back the bond, at certain times subject to certain conditions. If interest rates decrease it might be advantageous to the issuer to "call" the bond. If it is advantageous to the issuer, it is disadvantageous to the holder of the bond. The investor holding such a bond would require compensation for this callable feature. The compensation is in terms of a higher interest rate offered on the bond. Bonds with no extra features are sometimes referred to as straight bonds.
Studying the interest rate structure is conducted in the market for national government bonds and includes only straight bonds. In this market, the interest rate implicit in the prices of these bonds does not include compensation for risk. It reflects only the economic competitive conditions in the market. For this reason we limit our attention to the government bond market and, for the time being, to straight bonds. As we will proceed we will discover that there are a few rates in the market and that what we referred to as the "interest rate" in the market is a more complicated structure of interest rates.