13.8.1 Timescale Conversion
We introduced two timescales, the normal timescale, and the equal-month timescale, to satisfy three requirements:
- Monthly buckets have to be an integer number of days in length because the cash flow engine works on a daily timescale.
- Performance should generate rates with an equal-month timescale.
- We cannot set the bucket length to be 30 days, because buckets will start 5 days earlier each year, and this conflicts with reporting requirements.
The equal-month timescale is used only internally in the Rate Generator. It assumes that each month is constant and is equal to 1/12 of a year. The normal timescale counts the actual number of days, that is, each monthly bucket has a different length. In other words, the normal timescale assumes an Actual/Actual day count basis, whereas the equal-month timescale assumes a 30/360 count basis. The convention is the regular Oracle ALM convention for a month: if bucket zero starts on day n then all next buckets start on day n except when this day does not exist (February 30 for instance), in which case it reverts to the last existing day of the month (for example, February 28).
There is a one-to-one relationship between the timescales.
Let us suppose that the As-of-Date is January 15, 2011. By definition, every bucket will then start on the 15th of that corresponding month.
Table 13-1 Timescale for As-of-Date
| Calendar Time | Time on the normal timescale | Bucket Number | Length of the Bucket | Time in equal month timescale | Discrete rate on the normal timescale | onverted rate on equal month timescale |
|---|---|---|---|---|---|---|
|
1/15/2011 |
0 |
0 |
31/365 |
0 |
- |
- |
|
1/20/2011 |
5/365 |
0 |
31/365 |
5/(12*31) |
0.05 |
0.049726 |
|
2/14/2011 |
30/365 |
0 |
31/365 |
30/(12*31) |
0.05 |
0.049726 |
|
2/15/2011 |
31/365 |
1 |
28/365 |
1/12 |
0.05 |
0.049726 |
|
3/14/2011 |
58/365 |
1 |
28/365 |
1/12+27/(12*28) |
0.05 |
0.047364 |
In this example, the discrete yield (quoted Actual/Actual) is constant. However, the 2-month smoothed yield is lower than the 1 month smoothed yield because the timescale transformation overestimates the length of the second month.