When one is considering an interest rate that holds to a single date in the future, it is straightforward to convert an interest rate from a source compounding frequency and day count to a target compounding frequency and day count. The key insight is that, however constructed using compounding frequencies and day counts, there is only one discount factor (the present value of $1) associated with a given future date for a given credit quality.
The algorithm is as follows. One starts by computing the discount factor for the future date using the source compounding frequency and day count. Then one uses the discount factor to solve for the rate that recovers the same discount factor using the target compounding frequency and day count.
R_S = source interest rate in decimal form
R_T = target interest rate in decimal form (what we are solving for)
F_S = source rate comp freq (1=ann. ,2=semi-ann,…12=monthly etc.)
F_T = target rate compounding frequency
AF_S = accrual factor (time in years) using the source rate day count
AF_T = accrual factor (time in years) using the target rate day count
DF = the discount factor for the future date (the PV of $1)
Assuming discrete compound rates, we have from the source rate information (slightly different equations are used for continuous compounding but the idea id the same):
DF = 1/(1 + R_S/ F_S)^( AF_S*F_S)
Suppose R_S = 0.02, F_S = 2, and AF_S = 2, we then have
DF = 1/(1 + .02/2)^4 = 0. 0.9609803445
This is the discount factor we must recover. Now assume that the target rate has the following: F_T = 1 and AF_T = 1.98 (different due, say, to an assumed different day count)
- 0.9609803445 = 1/(1 + R_S/ 1)^( 1.98*1)
Solving, R_S = 0.02030507783 or 2.0305… percent
Finally, we verify that the solved target interest rate generates the same discount factor
DF = 1/(1 + R_T/ F_T)^( AF_TF_T) = 1/(1+0.02030507783/1)^(1.981)
DF = 0. 0.9609803445
If, on the other hand, one is dealing with a bond or a swap that has multiple cash flow dates, the work gets more complicated and involves a cumulative or summed discount factor.