Fixed Income security question

Question

Hello, This question if for Level 1 in fixed income securities. I need help please.  :( I am having trouble with the conversion from annual to semiannual and converting to quarterly. ( It is section 3.3 Yield measures for fixed-rates bonds. It is the formula for conversion based on periodicity i believe is what i don't understand.I am assuming that (1.03) to the power of 2 is semiannual. Not sure about (1.03) to the power of 0.5 ( i have no idea why it is 0.5)here is the question: A firm has issued a bond with YTM of 6% on a semiannual basis. What yield should be used to compare it with an annual pay bond and a quarterly pay bond?AFor annual pay bond – 6.09%, for quarterly pay bond – 5.96%.BFor annual pay bond – 6.15%, for quarterly pay bond – 5.90%.CFor annual pay bond – 615%, for quarterly pay bond – 6.20%.Explanation:A is correct. A general formula to convert an annual percentage rate for m periods per year, denoted APRm, to an annual percentage rate for n periods per year, APRn, isFor annual pay bond: (1.03)2 - 1 = 6.09%For quarterly pay bond: (1.03)0.5 - 1 =1.49%.and for quarterly basis 1.49 * 4 = 5.96%  Thank you,

fmccray · Answer

This explanation is so helpful. English is not my first language either :). Thank you so so much.

janpeter · Answer

The 6% number is the annual percentage rate (APR), which is just a convention that banks used. It is simply the interest rates received multiplied by the number of periods in a year. So 6% on a semiannual basis means you get paid 3% every 6 months. The question is asking "If I have a bond that pays 6% on a semiannual basis, what is the equivalent APR I should shop for if I want an annual or quarterly pay bond?"To convert, you compound it accordingly to calculate annual and quarterly interest payments, then recalculate the APR.So if you're receiving 3% every 6 months,to make it annual you'd compound it by 2 periods (6 months * 2): 1.03^2 -1 = 6.09%to make it quarterly you'd have to compound it by 0.5 periods (6 months * 0.5): 1.03^0.5 -1 = 1.49%, which is 1.49% * 4 = 5.96%Bonus info: Because of the nature of compounding, frequent interest payments will always result in a lower APR. So without calculating you can already tell that C is wrong, because the quarterly pay bond APR is higher than the semiannual (6%).Hope that is clear, English is not my first language.