Bond Equivalent Yield Formula

People, who are interested in purchasing fixed income securities that are backed by the full faith and credit of the US government, have the option of investing in Treasury bills, notes and bonds; I Savings Bonds, EE/E Savings Bonds and Treasury Inflation-Protected Securities (TIPS). These securities are low-risk investments that have a great deal of liquidity in the secondary market.

A Treasury bill (T-bill) is a money market instrument or a short term debt instrument that is issued by the US govt. T-bills have a maturity of 12 months or less and are quoted and traded in the market at discount to par. Hence, there is no explicit coupon payment on Treasury bills. Treasury notes (T-notes), are issued with a maturity of 2, 3, 5, 7, or 10 years and pay a fixed rate of interest every six months until maturity. Treasury bonds (T-bonds) have a maturity period of 30 years and like Treasury notes, they pay a fixed rate of interest every six months until maturity.

In general, Treasury bonds (notes) are traded in the capital market and are issued with a maturity of 10 years or more and carry a coupon, whereas Treasury bills do not carry a coupon since they are quoted and traded in the market on a discount basis. Hence, the need for the formula that helps to restate the yield on a T-bill in terms of semiannual investment yield, in order to facilitate comparison with T-bonds and T-notes assumes importance.

The Need for Bond Equivalent Yield Formula
The investor's return on investment (ROI) or return if the bond is held till maturity is called the yield to maturity or simply the yield. The yield needs to be determined so that one can compare alternative fixed income investments. The yield on a T-bill is calculated using either the discount yield formula or the investment yield formula.

The discount yield formula is useful for determining the discount yield for T-bills that have a 3 -month (91 days) or a 6-month (182 days) maturity. The formula is as follows:

Discount Yield Formula or Discount Rate = [(FV - PP)/FV] * [360/M]

Where
FV = Face Value
PP = Purchase Price
M = Maturity of T-bill (in days)

Ideally, the rate of return is calculated on the purchase price and not on the face value. However, the discount yield formula calculates the ROI on face value. Since, the purchase price is typically less than face value, the formula tends to understate the yield. Moreover, the rate is based on a hypothetical year that has only 360 days.

The investment yield formula is used to overcome the aforementioned drawbacks of the discount yield formula.

Investment Yield Formula = [(FV - PP)/PP] * [365 or 366/M]

The Treasury uses the discount yield formula and the investment yield formula for calculating yields on all T-bills, except the one-year T-bill.

The investment yield formula can be used to compare the return on investment on T-bills to other short-term investments in order to choose the best alternative. However, one cannot use the investment yield formula to compare the yields on T-bills with those of T-notes and T-bonds maturing on the same date as the former.

Formula for Bond Equivalent Yield
The discount rate for the T-bill is converted to a coupon equivalent using the bond equivalent yield formula. The formula helps to restate the yield on a T-bill or any other debt instrument in terms of semiannual investment yield, in order to facilitate direct comparison with an interest bearing coupon security, viz. T-bond, T-note.

For T-bills with a maturity of 182 days or less, discount rate/discount yield on the T-bill can be converted to bond equivalent yield (to maturity) using the following formula:

BEY= (365*DR)/(360-(N*DR))

Where
DR = Discount Rate or Discount Yield (in decimal)
N = Time between settlement date and maturity date (in days)

The settlement date is the date when the ownership of the security changes hands. For government securities, the settlement date is typically one day from the date of executing the trade.

For Treasury bills with maturity periods greater than 182 days, the calculation becomes more complicated. Thankfully, BEY is reported by the Federal Reserve and other financial market institutions and is quoted in newspapers.

Hopefully, the above article would has answered your questions on bond equivalent yield and emphasized the necessity of the bond equivalent yield formula for facilitating comparison between fixed income securities, that do not have an explicit coupon payment, with other coupon bearing securities that mature on the same date.