# Term Structure of Interest Rate

Section 2

In this section, I will introduce some essential components about term structure, explain the IS/LM model to reveal the relation between term structure and GDP growth and lastly bring in some empirical evidence to support this relation.

2.1 Some basic terminologies and equations

Bond, being one of the most popular financial products, is one example of firm’s and nation’s lending and borrowing. There are two ways a bond delivers its return. (Please note that when comparing the yield of different bonds, only the terms to maturity vary. All other characteristics are identical.)

The first way is to offer a coupon every period and the principle along with a coupon when the bond matures. Face value is denoted by D. coupon payment by C, maturity by N, price by P, yield by Y. The log of each variable is expressed in lower case. Now, we can calculate the price of bond with a yearly coupon payment by [1]:

And if we assume the payment is in continuous stream, the time difference is dt and the coupon payment is there for Cdt. Then the price equation is [1]:

The second way is to only offer its face value on a specified date, no coupon payment before it matures. This type is called zero-coupon bond and the price of it is just [2]:

Again, if we rearrange and show it in a continuous form [2]: ; (Please note that in the above three equations, Professor John H. Cochrane considers Y(N) 1 / 19

Term Structure of Interest Rate. Candidate number 25909

and y(N) as one plus the yield to maturity (YTM), namely Y(N)=1+YTM and log(Y(N))=log(1+YTM)=y(N).)

In a more widely expressed form, the yield of a zero-coupon bond, for purchasing a bond at its current price and holding it till maturity at time N to receive £1, is the following:

Since

Another vital rate for the term structure is the forward rate, maturity and N’ is the years of holding. For instance,

, where N is the year of means the forward rate for a

two-year bond after holding it for one year at time t. Theoretically, this rate should be the same as the expected yield of a one-year bond at time t+1.

Also, the product of one plus the one-year yield and one plus the forward rate of next year should equal to the square of one plus the two-year yield:

This is how the forward rate is calculated.

Additionally, when we assume that the interest rates on bond are small, and in reality they usually are, we can simplify the above equation through a linear approximation [3]:

Since Hence:

and

are small, the product of either two is very close to zero.

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Term Structure of Interest Rate. Candidate number 25909

2.2 Introducing the spread and the yield curve

When comparing yields at different maturities, one common term is used as the spread or, some refer it as, the slope. Spread indicates the difference between the yields of longer and shorter terms by deducting the latter from the former:

This relationship can also be expressed graphically; it is called the yield curve.

A yield curve plots the yields of bonds with different maturities against their associated maturities. It is the most straight-forward way to examine the term structure of a typical type of bond, with the same characteristics except its maturity. The slope of the yield curve is the spread we mentioned above.

Fig. 1 shows the most common type of a yield curve – upward sloping yield curve. This is regarded as the normal yield curve. Not difficult to notice that a bond with longer terms delivers higher returns, the spread is therefore positive. Due to the preference towards shorter term financial instruments (i.e. bonds) over longer term, the demand of shorter term ones is higher and the price is, therefore, higher. A higher price indicates a lower rate of return as the principal remains unchanged. So the yield of longer term bonds excesses shorter term ones. It can also be explained due to a...

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