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Given the Daily US Treasury Yield Curve Rates for a specific date I will fit the curve with the cubic spline method, but first I need to know how to use the data points given by the Treasury.

For example, if today is June 1st, 2015, and the 3 months rate is 0.8%, does this mean that I need to simply add 3 to today's month (resulting in September 1st, 2015) or I need to add 90 days to today's date (resulting in August 30th, 2015) ?

Similarly with the 30 years rate, should I add 365 x 30 days, should I consider leap years, or simply add 30 to the year (following the example above, it will result in June 1st, 2045) ?

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Your question is really about how to map between term-to-maturity and calendar dates, which can be a tricky problem since the number of days in a month varies by month. In short, you should increment using months if your incremental term is in months, and increment using years if your incremental term is in years.

In your examples, three months (or 0.25 years) forward from 2015-06-01 is 2015-09-01, and 30 years forward from 2015-06-01 is 2045-06-01. In the case of annual increments, this is consistent with Treasury note and bond issuance conventions.

In the case of monthly increments, this is consistent with common sense. As far as Treasury securities are concerned, the closest "3-month" securities are 13-week T-bills, so you might consider counting by weeks if you're looking for a T-bill maturity date. These pages from the TreasuryDirect website might help:

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