Treasury notes (or T-notes) are debt instruments that are issued by the U.S. Treasury. With maturities of more than a year but no more than 10 years, these semi-annual coupon bonds are usually issued with maturities of 2,3,5, 7, and 10 years and the next business-day settlement. While a 10-year T-note is auctioned only in the months of February, May, August, and November, the T-notes of other maturities are auctioned monthly.^{[1]} The formula that is used to compute the accrued interest between coupon dates is of the form

(2.2)

where AI represents the accrued interest owed to the owner of the note since the last coupon date, d represents the number of calendar days from the last coupon date to the settlement date, N represents the number of calendar days between the last and next coupon dates, and c represents the annual coupon rate associated with the bond.^{[2]} Since bonds are traded interchangeably using both prices and yields-to-maturity (or yields), it is important for any practitioner to understand the following interplay between these two forms of representations:

■ The prices on which bonds are traded (quoted) are called clean prices, because they do not include the accrued interest that must be paid to the seller of the bond in order to compensate the seller for holding the bond from the last coupon date to the settlement date.

■ The yields on which bonds are traded are calculated using dirty prices that are obtained by adding the bond clean prices to their accrued interests [which are computed using equation (2.2)].

■ The relationship that connects a bond yield to a bond price takes the form

(2.3)

where QPBond represents the quoted price, d/N/C are as given in equation (2.2), n denotes the number of periods taken to receive the last coupon, у denotes the yield-to-maturity of the bond, and x denotes the stub period between the settlement date and the next coupon date which is given by the formula

The Microsoft Excel function “=Price()” gives the clean price of a bond and can be used to value the type of bonds discussed above by using a basis of 1.^{[3]} Microsoft Excel also has a function, “=Accrint()”, that provides the

TABLE 2.2A Calculating Bond Yield from Bond Price

accrued interest a bond buyer has to pay in addition to the bond's clean

price.^{[4]}Table 2.2a shows how Microsoft Excel's “=Price()” function can be used to obtain the yield-to-maturity of a bond from its quoted price.

As can be seen from Table 2.2a, a guess on the yield-to-maturity is initially done and then iterated in a manner to make the result in cell B6 zero. One can alternatively obtain the solution in cell B5 using another Microsoft Excel function – “=Yield()” – with the appropriate arguments. In practice, it is not uncommon for a portfolio to contain millions of bonds, bonds with uneven coupon sizes, or infrequent coupon dates. As a consequence, due to the lack of efficiency and standardization between the bonds, it is impractical to use Microsoft Excel functions to compute bond prices and yields. In such instances, practitioners resort to the use of numerical methods (e.g., Newton's Method).

To illustrate the use of Newton's Method to compute bond yields, I will redo the example in Table 2.2a. In order to do this, one needs to first rewrite equation (2.3) as follows:

(2.4a)

In doing so, the required yield of the bond is simply the solution of the equation f (y) = 0.

By Newton's method, the solution to the equation f (y) = 0 can be obtained by recursively solving

(2.4b)

for n = 1, 2, 3, ..., where y0 is the initial guess.in equation (2.4b) can be obtained by differentiating equation (2.4a) with respect to у and shown to give rise to equation (2.4c)

(2.4c)

Table 2.2b shows the implementation of Newton's method using equations (2.4a), (2.4b), and (2.4c) to obtain the required bond yield.

As can be seen from Table 2.2b, one is able to converge to the bond yield obtained in Table 2.2a using Newton's method in 3 iterations.

[1] The 10-year T-note is also auctioned as a reopening in the other months of the year, where the definition of reopening means that the already issued (i.e., existing) 10-year T-notes are reissued with the same maturity dates, coupon sizes, and coupon dates but with different issue dates. The reader is referred to the link treasurydirect.gov/instit/marketables/tnotes/tnotes.htm for details.

[2] This is sometimes also called the A/A or Actual/Actual day-count convention, where the Actual in the numerator refers to the actual number of days between the last coupon date and settlement date, while the Actual in the denominator refers to the number of calendar days between the last and next coupon dates. Other conventions that are sometimes used are A/360 (or Actual/360), A/365 (or Actual/365), 30/360, and so on.

[3] The last argument of the “=Price()” function is called the basis, which can either be 0, 1, 2, 3, or 4, where each value refers to a different type of day-count convention. The interested reader is referred to Microsoft Excel for further details.

[4] It is important for the reader to realize that as long as the price is known (whether it is a clean or dirty price), the yield-to-maturity of a bond can be manually calculated by varying the yield input until the price is matched.

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