Comparing bond yields can be daunting, mainly because they can have varying frequencies of coupon payments. And, because fixed-income investments use a variety of y♍ield conventions, you have to convert the yield to a common basis when comparing different bonds.

Taken separately, these conversions are straightforward. But when a problem contains both 澳洲幸运5官方开奖结果体彩网:compounding period and 澳洲幸运5官方开奖结果体彩网:day-count conversions, t🍸he correct solution is harder to reach.

Factors to Consider when Comparing Bond Yields

澳洲幸运5官方开奖结果体彩网:U.S. Treasury bills (T-bills) and corporate 澳洲幸运5官方开奖结果体彩网:commercial paper investments are quoted and traded in the market on a discount basis. The investor does not receive any coupon interest payments. The profit is in the difference between its current purchase price and its face value at maturity. That 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.

There are baked-in problems with rates quoted on a discount basis. For one thing, discount rates understate the true 澳洲幸运5官方开奖结果体彩网:rate of return over the 澳洲幸运5官方开奖结果体彩网:term to maturity. This is because the dis෴count is stated as a percentage of fac🐻e value.

It is more reasonable to think of a rate of return as the interest earned divided by the current price, not the face💛 value. Since the T-bill is purchased at less than its face value, th൲e denominator is overly high and the discount rate is understated.

The second problem is that the rate is based on a hypotheti𓄧cal year that has o✱nly 360 days.

The Yields on Bank CDs

The returns of bank 澳洲幸运5官方开奖结果体彩网:certificates of deposit historically were quoted on a 360-day year also, and some are to this ♕day. However, since the rate is mode⛎stly higher using a 365-day year, most retail CDs are now quoted using a 365-day year.

The returns are posted with their 澳洲幸运5官方开奖结果体彩网:annual percentage yield (APY). This is not to be confused with the annual percentage rate (APR♉), which is the rate which most banks quote with their mortgages.

In 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 APY, which takes the effects of compounding into account.

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

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

Yields on Treasury notes and bonds, corporate bonds, and municipal bonds are quoted on a 澳洲幸运5官方开奖结果体彩网:semi-annual bond basiS (SABB) because tಌheir coupon payments are made semi-annually. Compounding occurs twice per year, using a 365-day year.

Bond Yield Conversions

365 Days versus 360 Days

In order to properly compare the yields on different 澳洲幸运5官方开奖结果体彩网:fixed-income investments, it’s essential to use the same yield calculation. The first and easiest 澳洲幸运5官方开奖结果体彩网:conversion changes a 360-day yield to a 365-day yield. To c🐈hange the rate, simply "gross up" the 360-𒊎day yield by the factor 365/360. A 360-day yield of 8% is equal to a 365-day yield of 8.11%. That is:

 $8\% \times \frac{365}{360} = 8.11\%$8%×360365​=8.11%          &nbꦐsp;        &nbs⛄p;                     

Discount Rates

Discount rates, commonly used on T-bills, are generally converted to a 澳洲幸运5官方开奖结果体彩网:bond-equivalent yield (BEY), sometimes called a 澳洲幸运5官方开奖结果体彩网:coupon-equivalent or an investment yield. The conversion formula for "short-dated" bills with a maturity of 182 or fewer days is the following:

$\begin{aligned} &BEY = \frac{365 \times DR}{360 - (N \times DR)}\\ &\textbf{where:}\\ &BEY=\text{the bond-equivalent yield}\\ &DR=\text{the discount rate (expressed as a decimal)}\\ &N=\text{\# of days between settlement and maturity}\\ \end{aligned}$​BEY=360−(N×DR)365×DR​where:BEY=the bond-equivalent yieldDR=t💝he discount rate (expressed as a decimal)N=# o👍f days between settle𒀰ment and maturity​

Long Dates

So-called "long-dated" T-bills have a maturity of more than 182 days. In thജis case, the usual converꦛsion formula is a little more complicated because of compounding. The formula is:

 $BEY = \frac{-2N}{365} + 2[(\frac{N}{365})^2 + (\frac{2N}{365} - 1)(\frac{N \times DR}{360 - (N \times DR)})]^{1/2} \div 2N - 1$BEY=365−2N​+2[(365N​)2+(3652N​−1)(360−(N×DR)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. But the BEY for a long-dated T-bill does not have any well-defined compounding assumption, which mak💎es its interpretation difficult.

BEYs are systematically less than the annualized yields for semi-annual compounding. In general, for the same current and future cash flo🍌ws, more frequent compounding at a lower rate corresponds to less frequent compounding at a higher rate.

A yield for more frequent than 澳洲幸运5官方开奖结果体彩网: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 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. 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 an๊ accurate comparison, discount rates s🐽hould 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 differen🐽ce is that compounding happens twice a year. Therefore, APYs using a 365-day year can be directly compared to y🍌ields based on SABB.

A 澳洲幸运5官方开奖结果体彩网:discount rate (DR) on an N-day T-bill can be converted directly to a SABB with the following formulওa:

$SABB = \frac{360}{360-\left ( N \times DR \right )} \times \frac{182.5}{N-1} \times 2$SABB=360−(N×DR)360​×N−1182.5​×2