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## NAR convertible every X years

February 20th, 2009

When the NAR is convertible less frequently than a year it seems to give lots of students trouble.  Here is an email I sent recently to a student struggling with this idea.  I hope it helps.

A nominal annual rate is an ANNUAL rate and a rate in NAME ONLY (that’s where the nominal comes).  So when you have a nominal annual rate the first thing to do is convert it to an effective rate per the convertible period.

For example, if we have a nominal rate of 12% convertible semi-annually, then the first thing we do is divide by the number of compounding periods per year (since it is ANNUAL).  Thus 12% / 2 = 6% effective per 6 months (the convertible period).  Now we can actually use that 6% rate to move money around.  To accumulate money for 6 months we simply multiply by 1.06.  To move it for a year (which is 2 six-month periods) we multiply by (1.06)^2.  To discount money for 10 years we would divide by (1.06)^20 since there are 20 six-month periods in 10 years.

Now assume that we are given a nominal rate of 12% convertible once every 4 years.  Remember the 12% is an NOMINAL ANNUAL rate so first thing is to get the effective rate that we can use.  12% x 4 = 48% effective rate per 4 years (the convertible period).  So if we want to accumulate money for 1 year then we multiply by (1.48)^(1/4) since 1 year is 1/4 of 4 years.  If we want to accumulate money for 4 years then we multiply by 1.48.  If we want to discount money for 40 years then we divide by (1.48)^10 since there are 10 four-year periods in 40 years.