Comparing bond yields can be daunting, mainly due to the fact that they can have varying frequencies of coupon payments. And because fixed-income investments use a variety of yield conventions, it is vital to convert the yield to a common basis when comparing different bonds. When taken separately, these conversions are straightforward. But when a problem contains both compounding period and day-count conversions, the correct solution is harder to come by.

(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 there’s no explicit coupon interest payment. Rather, there is an *implicit* interest payment, which is the difference between the face value at maturity and the current price. 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*.)

There are baked-in problems with rates quoted on a discount basis. For one thing, discount rates are downwardly biased representations of investors’ rates of return, over the term to maturity. Secondly, 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 overly high, consequently, the discount rate understates the true yield.

Bank Certificates of Deposit have historically been quoted on a 360-day year also. Institutionally, many still are. However, since the rate is modestly 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 is not to be confused with the APR (annual percentage rate), which is the rate at which most banks quote for mortgages. With APR calculations, the interest rates received during the period are simply multiplied by the number of periods in a year. But the effect of compounding is not included with APR calculations—unlike the APY, which takes the 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:

$APY = (1 + 0.03)^2 - 1 = 6.09\%$

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

### Bond Yield Conversions

#### 365 Days versus 360 Days

In order to properly compare the yields on different fixed-income investments, it’s essential to use the same yield calculation. The first and easiest conversion entails 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 an 8.11% yield based on a 365-day year.

$8\% \times \frac{365}{360} = 8.11\%$

#### Discount Rates

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:$\begin{aligned} &BEY = \frac{365 \times DR}{360 - (N \times DR)}\\ &\textbf{where:}\\ &BEY=\text{the bond-equivalent yield}\\ &DR=\text{the discount rate (expressed as a decimal)}\\ &N=\text{\# of days between settlement and maturity}\\ \end{aligned}$

#### 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:

$BEY = \frac{-2N}{365} + 2[(\frac{N}{365})^2 + (\frac{2N}{365} - 1)(\frac{N \times DR}{360 - (N \times DR)})]^{1/2} \div 2N - 1$

#### Short Dates

For short-dated T-bills, the implicit compounding period for the BEY is the number of days between settlement and maturity. But the BEY for a long-dated T-bill does not have any well-defined compounding assumption, which makes its interpretation 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 isn’t that the widely used BEYs are inaccurate, however, they serve a different purpose—namely, 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:

$SABB = \frac{360}{360-\left ( N \times DR \right )} \times \frac{182.5}{N-1} \times 2$

### The Bottom Line

Comparing alternative fixed-income investments requires a conversion of yields to a common basis, where the effects of compounding should be included, and conversions should always be done on a 365-day bond basis.