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
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
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
bond yields are calculated on an actual/actual basis, while discount
yields are usually calculated on an actual/360 basis.
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.
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
The first formula for bond-equivalent yield is
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  because this makes it intuitively
clear how the formula addresses the first two of the above three issues.
By multiplying by the factor:
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
approximately addresses the second issue.
If a discount instrument has more than 182 days remaining
to maturity, the formula for bond-equivalent yield is
where the instrument's face value is assumed to be 100,
and "price" is its quoted price.
Formula  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
and there will be
years remaining until the discount instrument's maturity
date. Reinvesting the proceeds from  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
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 .