Calculating and comparing bond yields isn't easy. Bonds can have varying frequencies of coupon payments; the number of days in the year may also differ. Because fixed-income investments use a variety of yield conventions, it is important to convert the yield to a common basis when comparing different bonds. Taken separately, these conversions are straightforward; however, when a problem contains both compounding period and day count conversions, the correct solution is not so obvious. In this article, we'll take a look at a couple of the common problems investors run into when calculating these yields and show you how to work them out. (To learn all about bonds, see our *Bond Basics* and *Advanced Bond Concepts* tutorials.)

## Current Conventions on Calculating and Comparing Bond Yields

U.S. Treasury bills (T-bills) and corporate commercial paper are quoted and traded in the market on a discount basis. This means that there is no explicit coupon interest payment â€” the difference between the face value at maturity and the current price is the implicit interest payment. The amount of the discount is stated as a percentage of the face value, which is then annualized over a 360-day year. (Keep reading about commercial paper in *Money Market: Commercial Paper* and *Asset-Backed Commercial Paper Carries High Risk*.)

The problems with rates quoted on a discount basis are well-known. First, a discount rate is a downwardly biased representation of the investor's rate of return (or the borrower's cost of funds) over the term to maturity. Second, the rate is based on a hypothetical year that has only 360 days. The downward bias comes from stating the discount as a percentage of the face value. In investment analysis, one naturally thinks of a rate of return as the interest earned divided by the current price, not the face value. Since the price of a T-bill is less than its face value, the denominator is too high, so the discount rate understates the true yield.

Bank Certificates of Deposit have historically been quoted on a 360-day year also, and institutionally, many still are. However, because the rate is a little higher using a 365-day year, most retail CDs are now quoted using a 365-day year. Returns are marketed using annual percentage yield (APY). This rate is not to be confused with APR or annual percentage rate, the rate at which most banks quote mortgages. In an APR calculation, the interest rate received during the period is simply multiplied by the number of periods in a year. The effect of compounding is not included. APY, however, takes effects of compounding into account. (To learn more, read *APR Vs. APY: Why Your Bank Hopes You Can'** t Tell The Difference*)

A six-month CD that pays 3% interest has an APR of 6%. However, the APY is 6.09%, calculated as follows:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â APY = (1 + 0.03)^2 â€“ 1 = 6.09% |

Yields on Treasury notes and bonds, corporate bonds and municipal bonds are quoted on a semiannual bond basis (SABB) because their coupon payments are made semiannually. Compounding is twice a year, and a 365-day year is used.

## Bond Yield Conversions

*365 Days Vs. 360 Days*

In order to properly compare the yields on different fixed-income investments, it is important to use the same yield calculation. The first and easiest conversion is changing a 360-day yield to a 365-day yield. To change the rate, simply "gross up" the 360 day yield by the factor 365/360. A 360-day yield of 8% would equate to a 8.11% yield based on a 365-day year.

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 8% * (365 / 360) = 8.11% |

*Discount Rates - 182 Days*

Discount rates, commonly used on T-bills, are generally converted to a bond-equivalent yield (BEY), sometimes called a coupon-equivalent or an investment yield. The conversion formula for "short-dated" bills with a maturity of 182 or fewer days is the following:

Â

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

Where:**BEY** = the bond-equivalent yield**DR **= the discount rate (expressed as a decimal)**N** = the number of days between settlement and maturity

*Long Dates*

For "long-dated" T-bills that have a maturity of more than 182 days, the usual conversion formula is a little more complicated because of compounding. The formula is:

**Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â BEY = - 2N / 365 + 2 [ (N / 365) ^{2} + (2N / 365 - 1) (N * DR / 360 - (N * DR) ]^{1/2} / 2N - 1**

*Short Dates*

For short-dated T-bills, the implicit compounding period for the BEY is the number of days between settlement and maturity. However, the BEY for a long-dated T-bill does not have any well-defined compounding assumption which makes its interpretation rather difficult.

BEYs are systematically less than the annualized yields for semi-annual compounding. In general, for the same current and future cash flows, more frequent compounding at a lower rate corresponds to less frequent compounding at a higher rate. A yield for more frequent than semiannual compounding - such as is implicitly assumed with both short-dated and long-dated BEY conversions - must be lower than the corresponding yield for actual semiannual compounding.

*BEYs and the Treasury*

BEYs reported by the Federal Reserve and other financial market institutions should not be used as a comparison to the yields on longer maturity bonds. The problem is not that the widely used BEYs are inaccurate, they just serve a different purpose. That purpose is to facilitate comparison of yields on T-bills, T-notes and T-bonds maturing on the same date. To make an accurate comparison, discount rates should be converted to a semiannual bond basis (SABB), because that is the basis commonly used for longer maturity bonds.

To calculate SABB, the same formula to calculate APY is used. The only difference is that compounding happens twice a year. Therefore, APYs using a 365-day year can be directly compared to yields based on SABB.

A discount rate (DR) on an N-day T-bill can be converted directly to a SABB with the following formula:

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â SABB = [ (360 / 360 - (N * DR) )^{182.5/N} - 1] * 2 |

A convenient feature in this equation is that it is stated as a function only of N and DR, which are directly observable for any traded T-bill. It is not necessary to calculate the price of the bill, making the equation a little easier to program into a spreadsheet and avoiding unnecessary rounding errors. Another key feature is that this conversion formula applies to both short-dated and long-dated T-bills. Unlike BEYs, the SABB presents the yields in a form fully comparable to the yields on Treasury notes and bonds. The formula converts the T-bill discount rate, quoted for a 360-day year and 360/N compounding periods per year, to a more reasonable investment yield, quoted for a 365-day year and two compounding periods.

## The Bottom Line

In summary, comparison of alternative fixed-income investments always requires conversion of yields to a common basis. The general rule is that the effects of compounding should be included and conversions should always be done on a 365-day bond basis. Comparing bond yields may not be easy, but it shouldn't be too difficult for the average investor either.