In the money market, discount instruments are generally quoted with discount yields, which are not directly comparable with yields calculated for other types of instruments. To facilitate comparisons, discount yields may be converted to **bond-equivalent yields**.

Conceptually, here is the idea. Bonds have long terms whereas money market instruments have short terms. However, during their last year prior to maturity, bonds have short terms just like money market instruments. When we calculate the bond-equivalent yield of a discount instrument, we calculate a number comparable to the yield to maturity for a US Treasury bond that is in its last year prior to maturity. There are three issues that must be addressed by the formula for bond-equivalent yield:

- Treasury bond yields are calculated on an actual/actual basis, while discount yields are usually calculated on an actual/360 basis.
- The formula for discount yield differs from the formula for calculating a Treasury bond’s yield to maturity because it divides by the instrument’s face value as opposed to its price.
- Bonds with more than a half year to maturity have one more coupon to pay whereas discount instruments pay no coupons.

To address the third problem, there are two formulas for bond-equivalent yield. One is for discount instruments with up to 182 days to maturity. The other is for discount instruments with more than 182 days to maturity.

The first formula for bond-equivalent yield is

[1]

Where “actual days in year” is either 365 or 366, depending on whether this is a leap year. “Days to maturity” is the actual number of days to maturity. You may see this formula expressed more simply elsewhere. I like to write it as in [1] because this makes it intuitively clear how the formula addresses the first two of the above three issues. By multiplying by the factor:

[2]

the formula converts the discount yield from an actual/360 basis to an actual/actual basis. This addresses the first issue. Dividing the discount yield by the factor

[3]

approximately addresses the second issue.

If a discount instrument has more than 182 days remaining to maturity, the formula for bond-equivalent yield is

[4]

where the instrument’s face value is assumed to be 100, and “price” is its quoted price.

Formula [4] is derived by assuming that the discount instrument’s price is invested for six months to earn simple interest at a rate equal to the unknown bond-equivalent yield. At the end of the six months, the investment will be worth

[5]

and there will be

[6]

years remaining until the discount instrument’s maturity date. Reinvesting the proceeds from [5] for this period, again at a rate equal to the unknown bond-equivalent yield, our investment will have a value on the discount instrument’s maturity date of

[7]

Setting this equal to the discount instrument’s maturity value, which is its face value of 100, we obtain a quadratic equation in the unknown bond-equivalent yield. Applying the quadratic formula, we then solve for the bond-equivalent yield. The result is formula [4].

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