# Converting US Treasury CMT to Discount Yields

I'd like to convert the US Treasury Constant Maturity series (par, semi-annual coupon, Actual/365 daycount convention) into Discount Factors (for appropriate comparison for certain money-market series, calculation of forward rates, conversion to alternative daycounts, etc.).

The "hypothetical security" reflecting the CMT quote is straightforward for 6 months to 30 years: $$PV=FaceValue$$ that pays $$FaceValue * CMT Rate/2$$ every 6 months before maturity, $$FaceValue*(1+CMTRate/2)$$ at maturity (even on weekends/holidays, since it's an interpolated curve).

What is the hypothetical security for the 1, 2, and 3 month CMT quotes? Is it a discount bond (and, if so, what's the discount formula to arrive at $$PV$$)? Or is it a coupon bond with $$PV=FaceValue$$ that pays $$FaceValue*(1+CMTRate * YearFrac365)$$ at maturity?

I think it is a coupon bond with semiannual coupons of $$CMTRate$$, thus a payment of $$FaceValue*(1+CMTRate/2)$$ at maturity and no other payments due. The $$PV$$ of this bond is $$FaceValue*CashflowatMaturity/(1+CMTRate/2)^{2*YearFrac365}$$