Comparing bond yields can be daunting, mainly because their coupon payment frequency may be different. Moreover, because fixed income investments use multiple yield conventions, when comparing different bonds, you must convert the yield to a common basis.

Taken individually, these conversions are simple. But when a problem includes both the compound period and the number of days conversion, it is difficult to find the correct solution.

## Factors to consider when comparing bond yields

U.S. T-bills (T-bills) and corporate commercial paper investments are quoted and traded in the market in a discounted manner. Investors will not receive any coupon interest. Profit is the difference between its current purchase price and its maturity face value.That is *Implicit* Interest payment.

The discount amount is expressed as a percentage of the face value and then annualized in a 360-day year.

Key points

- Investors in Treasury bills have no interest payments. The return is the difference between the purchase price and the face value at maturity.
- To complicate matters, the ratio is based on an assumed 360 days in a year.
- In CD, the annual percentage rate (APR) underestimates the return. A better number is the annual rate of return (APY), which takes into account compound interest.

There are inherent problems with the rates quoted on a discount basis. On the one hand, the discount rate underestimates the true rate of return within the maturity period. This is because the discount is expressed as a percentage of the face value.

It is more reasonable to think of the rate of return as the interest earned divided by the current price rather than the face value. Since the purchase price of Treasury bills is lower than their face value, the denominator is too high and the discount rate is underestimated.

The second problem is that the ratio is based on a hypothetical year with only 360 days.

### Bank CD yield

Historically, the return on the bank certificate of deposit was quoted in 360 days as a year, and some are still the same to this day. However, due to the slightly higher price for the year with 365 days, most retail CDs are now quoted with the year for 365 days.

The return is published along with its annual percentage return (APY). This should not be confused with the annual interest rate (APR), which is the interest rate quoted by most banks in mortgage loans.

In the APR calculation, the interest rate received during the period is simply multiplied by the number of periods in the year. But the effect of compounding is not included in the APR calculation-unlike APY, it takes into account the effect of compounding.

The APR for a six-month CD that pays 3% interest is 6%. However, APY is 6.09%, which is calculated as follows:

APY = (1 + 0. 0 3) 2 − 1 = 6. 0 9% APY = (1 + 0.03)^2-1 = 6.09%

OnephosphorusYes=(1+.3)2–1=6.9%

The yields of Treasury bills and bonds, corporate bonds and municipal bonds are quoted on a semiannual bond basis (SABB) because their coupon payments are paid semiannually. Compound interest occurs twice a year and is used 365 days a year.

## Bond yield conversion

### 365 days and 360 days

In order to correctly compare the rates of return of different fixed-income investments, the same rate of return must be used for calculation. The first and simplest conversion changes the 360-day yield to a 365-day yield. To change the interest rate, simply “add up” the 360-day yield to a factor of 365/360. A 360-day rate of return of 8% is equal to a 365-day rate of return of 8.11%. That is:

8% × 3 6 5 3 6 0 = 8. 1 1% 8% times frac{365}{360} = 8.11%

8%X36365=8.11%

### Discount Rate

The discount rate commonly used on Treasury bills is usually converted into bond equivalent yield (BEY), sometimes called coupon equivalent yield or investment yield. The conversion formula for “short-term” notes with a maturity date of 182 days or less is as follows:

BEY = 3 6 5 × DR 3 6 0 − (N × DR) where: BEY = bond equivalent yield DR = discount rate (expressed as a decimal) N = number of days between settlement and maturitybegin{aligned} &BEY = frac{365 times DR}{360-(N times DR)}\ &textbf{where:}\ &BEY=text{bond equivalent yield}\ &DR=text{the discount Rate (expressed as a decimal)}\ &N=text{Days between settlement and maturity}\ end{aligned}

SecondSecondYes=36–(NXDresistance)365XDresistanceWhere:SecondSecondYes=Bond equivalent yieldDresistance=Discount rate (expressed as a decimal)N=Number of days between settlement and maturity

### Long date

The so-called “long-term” Treasury bills have an expiry date of more than 182 days. In this case, due to compounding, the usual conversion formula will be slightly more complicated. The formula is:

BEY = − 2 N 3 6 5 + 2

[

(

N

3

6

5

)

2

+

(

2

N

3

6

5

−

1

)

(

N

×

D

R

3

6

−

(

N

×

D

R

)

)

]

1/2 ÷ 2 N − 1 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

SecondSecondYes=365–2N+2[(365N)2+(3652N−1)(36−(N×DR)N×DR)]1/2÷2N–1

### Short date

For short-term government bonds, the implied compounding period of BEY is the number of days between settlement and maturity. However, the long-term Treasury bill BEY does not have any clearly defined compound hypothesis, which makes its interpretation difficult.

BEY is systematically lower than the half-yearly compounded annualized rate of return. Generally speaking, for the same current and future cash flows, compounding more frequently at a lower interest rate corresponds to compounding less frequently at a higher interest rate.

The rate of return that is compounded more frequently than half-yearly (such as the implicit assumption of short-term and long-term BEY conversion) must be lower than the corresponding rate of return of actual half-year compounding.

### BEY and the Ministry of Finance

The BEY reported by the Federal Reserve and financial market institutions should not be used for comparison with long-term bond yields. The problem is not that the widely used BEY is inaccurate. They serve different purposes—that is, to facilitate the comparison of the yields of Treasury bonds, Treasury bonds, and Treasury bonds that mature on the same date.

In order to make accurate comparisons, the discount rate should be converted to a semi-annual bond basis (SABB), as this is a commonly used basis for bonds with longer maturities.

To calculate SABB, use the same formula to calculate APY. The only difference is that compound interest occurs twice a year. Therefore, APY using 365 days a year can be directly compared with the SABB-based rate of return.

The discount rate (DR) of N-day Treasury bills can be directly converted to SABB by the following formula:

SABB = 3 6 0 3 6 0 − (N × DR) × 1 8 2. 5 N − 1 × 2 SABB = frac{360}{360-left (N times DR right )} times frac{182.5}{N-1} times 2

secondOneSecondSecond=36–(NXDresistance)36XN–1182.5X2

.