The bond equivalent yield and effective annual return are both annualized returns, but there are few similarities beyond that. They are calculated differently and are used for different purposes. The BEY is used in Treasury bill quotes and the EAR is a transformation of the annual percentage rate quoted in loans such as mortgages, credit cards and car loans.

## Bond Equivalent Yield

If a Treasury Bill (a discount bond with par value of $10,000) can be bought for $9,950.00, and has 30 days left to maturity, the BEY is calculated by first dividing the par value by the price and subtracting 1 -- $10,000/$9,950.00 - 1 -- to arrive at a 0.005025, or 0.5025 percent, growth in value over 30 days. Multiplying this growth by the number of 30-day periods in a year (365 days per year divided by 30 days left to maturity) -- 0.005025 x (365/30) --- results in a BEY for this example of 6.11 percent.

The formula is BEY = ($10,000/$9,950.00 - 1)(365/30), or more generally, BEY = ($Par Value/$Price- 1)(365/days left to maturity).

## Effective Annual Return

The EAR converts a stated annual percentage rate to a rate that indicates the actual amount of interest paid when the frequency of compounding is accounted for. If a stated APR is 6.0302 percent, compounded monthly, then the EAR is found by dividing the APR by 12 months -- 0.060302/12 = 0.00502517; adding 1 -- 1.00502517 -- and finding the 12 power (^12) of the sum -- 1.00502517^12 = 1.062; and subtracting 1 -- 1.062 - 1. The EAR for this example is 6.20 percent.

The formula is EAR = (1+0.060302/12)^12-1, or more generally, EAR = (1+APR/compounding frequency)^(compounding frequency)-1.

## Comparing the BEY and EAR

The 6.0302 percent divided by 12 months in the EAR example is the same monthly rate as 0.5025 percent used for the 30-day rate in the BEY example, but the annualized rates for the BEY and EAR are not the same. The difference is due to the fact that the BEY does not assume the yield will be compounded, whereas the EAR explicitly accounts for the compounding of interest.

## The Benefits of the BEY and EAR

The benefit of calculating the EAR is that it provides an accurate rate of interest on a loan in contrast to the stated APR. The benefit of the BEY may be less apparent. The Treasury bill bid and asked yields indicate the price that a dealer is willing to buy and sell the T-bills for as a fraction of the par value. For example, an asked yield of 6 percent for a $10,000 par bill with 30 days left to maturity is calculated as $10,000(1 - [0.06(30/360)]) = $9,950.00.

The asked yield determines the price, which can then be used to calculate the more correct BEY. The asked yield is faulty because it assumes 360 days instead of 365 days and is based on a percentage of the par value instead of the actual price. It remains in use due to the tradition of using simpler calculations established before computers were available.

#### References

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