Overview
In order to put comparisons between rates on an apples to apples basis, a set of quotation conventions have developed that quotes rates on an annualized basis. However, this complicates the present value formulas somewhat.
Quotation Conventions
When quoting interest rates or yields, market conventions often come into play. In order to put comparisons between rates on an apples to apples basis, the market likes to quote things on an annual basis. However, the present value calculations assume that interest earned is reinvested, and this requires the use of a rate that is stated for the length of time between payments. That is, if interest is earned monthly, the formulas above will state the rate, r, as a monthly rate.
In order to comply with market conventions, the market commonly uses the following formula for calculating present values. (see Smith chpt 2)
{% PV = FV / [(1 + \frac{APR}{periodicity}) ^{years * periodicity}] %}
- PV = present value
- FV = future value
- periodicity = number of periods in the year
- APR = the yield to maturity stated as an annual percentage rate corresponding to the given periodicity.
This way of quoting rates has some downsides. For instance, if you have two instruments, both quoted at an annual rate of 12%, but one with a periodicity of 12 (monthly payments) and one with a semi-annual periodicity. (2 payments in the year). Then the future value of the first instrument is higher, given the earlier reinvestments. This means that the quoted rate cannot be be used to strictly determine value.
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