8.2 Day Counts
If a value-at-risk measure is to reasonably reflect such factors as the accrual of interest, theta effects, ex-dividend dates, or “riding down” the yield curve, it must accurately account for the passage of time during the value-at-risk horizon. It is inexact to speak of an asset’s value without specifying a point in time. Consider an at-the-money call option that has 8 days to maturity at time 0. If we employ a 1-week value-at-risk horizon, the option’s price behavior will be quite different at time 1 than at time 0.
To avoid confusion, we label quantities with time superscripts. Both 0p and 1P denote a portfolio’s value, but 0p denotes it at the present time and 1P denotes it at the end of the value-at-risk horizon. In this section, we address some technical issues relating to the measurement of time. The material is more important for some portfolios than others. For a repo portfolio, settlement dates are precisely defined and should be recognized by a value-at-risk measure. For a coffee portfolio, ex-dock settlement occurs at the ship’s arrival in port, whenever that might be.
8.2.1 Actual Days, Basis Days, and Trading Days
We distinguish between:
- actual days, each comprising a 24-hour period;
- basis days, counted on an actual, 30-day or other basis, consistent with interest rate conventions, such as actual/actual, 30/360, or 30/actual; and
- trading days, which are any actual day during which a specified market conducts business.
If basis days are calculated on an actual basis, then basis days equal actual days. Trading days may be defined with regard to the days of operation of a specific market or more generally with regard to the business days of some collection of markets or geographic region. Consequently, trading days and nontrading days are not universally defined.
It is common to measure value-at-risk over short horizons of a day or a week. We might define such horizons in terms of actual days or trading days. To avoid having the end of a horizon fall on a weekend or holiday, we adopt the latter convention. A “1-day horizon” comprises 1 trading day. A “5-day horizon” comprises 5 trading days.
This convention is endorsed by empirical studies1 indicating that market fluctuations from one trading day to the next exhibit little effect from intervening nontrading days. For this reason, value-at-risk measurements provide a more consistent indication of risk if they are measured over a trading day than over an actual day.
We might feel inclined to measure time with trading days and not bother at all with basis days. Unfortunately, things aren’t so simple. In finance, some computations depend upon the passage of basis days. Most obvious of these is the accrual of interest. As a practical matter, many value-at-risk measures must account for time in both trading days and basis days.
To facilitate writing formulas, we define the basis days function τ. For integers t > 0, τ(t) indicates the number of basis days from time 0 until trading day t. There is no simple formula for τ(t). It depends upon the present date, the calendar, and our convention (actual, 30-day, etc.) for counting basis days.
For example, count basis days as actual days. If today is October 15, 2002, what is τ(5)? Using the calendar in Exhibit 8.2, we count 5 trading days from October 15. Skipping weekends and other nontrading days, we conclude that trading day t = 5 coincides with October 22. There are 7 basis days between October 15 and October 22, so τ(5) = 7.
Suppose today is November 8, 2002, count basis days as actual days, and assume a 1-day value-at-risk horizon. How many basis days are in our horizon? Because November 9, 10 and 11, are nontrading days, our value-at-risk horizon ends on November 12. There are τ(1) = 4 basis days in the value-at-risk horizon.
8.2.2 Time Versus Maturity
An important convention is that we always measure time from the current time 0, but the maturity of an instrument may be measured from any point in time. Measure basis days as actual days and suppose today is October 11, 2002. Our value-at-risk horizon comprises 1 trading day, which is 4 basis days. The horizon ends on October 15. At time 0 a loan has 12 basis days until it matures. As of time 1, that same loan has 8 basis days until it matures. Maturity is relative, based upon the point in time at which it is measured. Time, on the other hand, is always measured from time 0.
If today is November 14, 2002, what is τ(10)? Assume basis days are actual days.
Suppose today is October 23, 2002, and we are employing a 1-day value-at-risk horizon. How many actual days are in our horizon?
Suppose today is November 27, 2002, and we are employing a 1-day value-at-risk horizon. How many actual days are in our horizon?
Suppose today is October 11, 2002, and we are employing a 1-day value-at-risk horizon. How many actual days are in our horizon?
The month of February 2003 has 28 days. How many actual days are there between February 25 and March 5, 2003? How many 30-day basis days are there?