In a previous post I mentioned that the economic return, or internal rate or return, or yield to maturity (YTM) of a money market instrument (one with a year or less of a term to run) can be calculated. But this can't be all there is to the bond equivalent yield. After all, why the name bond equivalent when yield to maturity or internal rate of return would be more on the money? The answer is that the bond equivalent yield of that money market instrument is a yield translated to facilitate a direct comparison with a (US) Treasury bond with less than a year until it matures. It is assumed the bond will pay semi-annual coupon, so the bond equivalent yield ought to take into consideration the compounding effect at the 6-month horizon. If the last but one coupon on a US treasury has been paid, then the treasury bond's YTM calculation is the bond equivalent yield as described in the earlier post. If, however, there are two puffs left on the cigar, so to speak, then the YTM calculation is slightly more awkward.
Why would you want to prefer to work out the equivalent (final year US Treasury) bond yield, and not just the economic internal rate of return? Well, you'd do both, depending on your purposes, but there's be many reasons why you'd want to compare the money market instrument with the equivalent maturity US Treasury.
So, if you really want to know what a money market instrument's economic yield is (YTM), you are best to just ignore the bond equivalent yield (which would bear a lower yield number, since there is an additional internal compounding for those instruments expiring in more than 6 months and the more frequently an instrument compounds the lower the rate can be to result in the same present value), but instead apply the $\frac{F-P}{P} \times \frac{365}{d}$ if you happen to have the money market instrument's price already to hand.
US Treasury (notes and) bonds have a semi-annual coupon and they have an ACT/ACT day count, which amounts to 365/365 three years out of four and 366/366 on the leap year. The indicator 'ACT' implies that the convention doesn't bother about months and instead counts actual days, making sure that a full year of days makes up precisely 1.0 years.
All Yield to maturity calculations share this in common - namely that they relate the market price of a series of cash flows with its valuation. It is that single yield which equates the market price and theoretical price. So you can see that to translate a discount basis into a YTM money market basis you equate their prices. By the way you can think of the terminology bond equivalent yield as a shorthand for: 'money market rate re-based into coupon bearing equivalent-riskiness bond yield'
Recall that discount yield is $y_d = \frac{F-P}{F} \times \frac{360}{d}$ and YTM (no intervening compounding) is $y_m = \frac{F-P}{P}\times \frac{365}{d}$. Express these both in terms of $P$ and equate them, then solve for $y_m$ and you get $\frac{365}{360} \times \frac{y_d}{1-\frac{d}{360}y_d}$. The best way to remember this formulation is as a pair of boosters applied to the smaller $y_d$ number. The first booster corrects upwards for the 360 day basis of $y_d$ and the second corrects up for the economically irrelevant 'divide by face value' convenience of the definition of $y_d$.
There's a cleaner formula for this, which is $y_m = \frac{365 y_d}{360-y_d}$, assuming that you'd like your annualised real yield to be based on a 365 day year. You'll sometimes see $y_m$ referred to as the money market investment yield, and perhaps it is quoted on a 360 day basis, which would be simply $y_m = \frac{360 y_d}{360-y_d}$
All Yield to maturity calculations share this in common - namely that they relate the market price of a series of cash flows with its valuation. It is that single yield which equates the market price and theoretical price. So you can see that to translate a discount basis into a YTM money market basis you equate their prices. By the way you can think of the terminology bond equivalent yield as a shorthand for: 'money market rate re-based into coupon bearing equivalent-riskiness bond yield'
Recall that discount yield is $y_d = \frac{F-P}{F} \times \frac{360}{d}$ and YTM (no intervening compounding) is $y_m = \frac{F-P}{P}\times \frac{365}{d}$. Express these both in terms of $P$ and equate them, then solve for $y_m$ and you get $\frac{365}{360} \times \frac{y_d}{1-\frac{d}{360}y_d}$. The best way to remember this formulation is as a pair of boosters applied to the smaller $y_d$ number. The first booster corrects upwards for the 360 day basis of $y_d$ and the second corrects up for the economically irrelevant 'divide by face value' convenience of the definition of $y_d$.
There's a cleaner formula for this, which is $y_m = \frac{365 y_d}{360-y_d}$, assuming that you'd like your annualised real yield to be based on a 365 day year. You'll sometimes see $y_m$ referred to as the money market investment yield, and perhaps it is quoted on a 360 day basis, which would be simply $y_m = \frac{360 y_d}{360-y_d}$
