Fundamentals of Bond

A coupon bond is a bond that pays a interest payment (coupon) from the date of issuance until the date of maturity.

• face/par value (): the fixed price that the issuer pays at the time of maturity.
• maturity (): the time when the bond issuer must repay the original bond value.
• coupon (): a fixed dividend that a bond pays until the time of maturity.

A zero-coupon bond is a bond that pays no coupons, i.e. . Note that any coupon bond that pays a coupon can be seen as a composite of various zero-coupon bonds with the face value of . For the rest of the section, we only consider a zero-coupon bond, where and . Now, by denoting period bond for a zero-coupon bond with periods left to maturity,

• price of the period bond at : .
• yield of the period bond at : .
• yield at the maturity is zero: , as .
• short rate (risk-free rate of return) is essentially the yield of the zero-coupon bond with one-period left to maturity:

Since the price of the bond must equals to the average rate of the bond until maturity, we have by taking logs on the both sides, thus, the yield of the period zero-coupon bond can be defined as Note that where is the rate of return from investing the bond at and holding it to maturity. In other words, represents the average rate of return of the bond until maturity.

Also, we can conversely recover bond price from the yields:

Now, consider for a case where an investor buys a period bond at time and holding it until time . For this case, we call such return as period ahead.

• period ahead rate of return for period bond: .
• period ahead excess return for period bond: .
• period ahead expected excess return (risk premium) of period bond: .

Here, the is derived by

If , we call it one-period ahead holding period. Then

• one-period ahead rate of return for period bond: .
• one-period ahead excess return for period bond: .
• one-period ahead risk-premium: .

Suppose a case where we are interested in the future risk-free (one period ahead) rate , at time .

1. Issue a zero-coupon period bond at . then we have units of numeraire.
2. Buy worth of period bond at . then we have units of bonds maturing at time .

Then,

3. At time , the issued bond from step 1 will mature, thus cost of incurs.
4. At time , the purchased bond from step 2 will mature, thus we receive the payoff of .

From the given setting, now we can define the forward rate.

Using Taylor approximation, we have then by summing up from up to , we have Thus the yield of period bonds at time can be described as Similarly, which equals to This implies that the difference between the forward rate from time to and the risk-free rate in the same time intervals, equals to the difference between the returns from period holdings for bond and the returns from period holdings for bond.

Note that in Affine Term Structure Models > ^15ea9cAffine Term Structure Models > Proposition 11 (equivalence of expectation hypothesis), we show that the three forms of the ^4f1b61Proposition 2 (expectation hypothesis) are all equivalent.