In early July 2008, the U.S. Treasury auctioned off a series of T-bills. The official reported results for the auctions are shown in Table 1.1. Each T-bill was issued on July 3, 2008, and uses an actual/360 day-count. The Investment Rate also is called the bond equivalent yield (BEY), the term that I use in this section. The intent is to report to investors an interest rate for the security that is more meaningful than the discount rate and that allows a comparison to Treasury note and bond yields.

The given prices are straightforward applications of pricing on a discount rate basis using equation 1.6:

TABLE 1.1 T-Bill Auction Results

Term

Maturity Date

Discount Rate

Investment

Rate

Price (per $100 in par value)

4 week

07-31-2008

1.850%

1.878%

99.856111

13 week

10-02-2008

1.900%

1.936%

99.519722

26 week

01-02-2009

2.135%

2.188%

98.914708

52 week

07-03-2009

2.295%

2.368%

97.679500

The 4-week, 13-week, 26-week, and 52-week T-bills almost always have 28, 91, 182, and 364 days to maturity, respectively. They typically are issued and settled on a Thursday and mature on a Thursday. The 26-week T-bill this time had 183 days in its time period because New Year's Day got in the way.

The bond-equivalent yield for each T-bill can be calculated by working with the cash flows or with a conversion formula. First, use equation 1.4 for add-on rates, letting Year = 365.

The first three BEY results confirm the reported Investment Rates; the fourth is wrong. The “official” APR – the one reported by the Treasury – on the 52-week T-bill is 2.368% while our calculation here is 2.382%. Quips like “close enough for government work” are not acceptable in bond math.

Before resolving this discrepancy, we can attempt to confirm the reported Investment Rates using a conversion formula similar to equation 1.8.

(1.11)

This directly converts a 360-day discount rate to a 365-day add-on rate.

Notice that identical results are obtained using either the cash flows or the conversion formula and that again we have the wrong Investment Rate for the 52-week T-bill.

The source of the discrepancy is that the U.S. Treasury uses a different method to calculate its official Investment Rate (i.e., the bond equivalent yield) when the time to maturity exceeds six months. The BEY for the 52-week T-bill is based on this impressive formula.

(1.12)

Enter Days = 364 and BV = 97.679500 to obtain the “correct” result that BEY = 2.368% for the long-dated T-bill.

Where does equation 1.12 come from? Mathematically, it is the solution to this expression found using the quadratic rule.

(1.13)

The equation is derived in the Technical Appendix. The Treasury's intent is to provide an interest rate for the T-bill that is comparable to a Treasury note or bond that would mature on the same date and that still has one more coupon payment to be made.

A problem is that the annual interest rate in equation 1.13 does not have a well-defined periodicity – and knowing the periodicity of an interest rate is critical in my opinion. The first term in parenthesis in 1.13 looks like semiannual compounding for a periodicity of 2 (the annual rate of BEY is divided by two periods in the year). The second term suggests compounding more frequently than semiannually. For example, if Days = 270, it looks like close to quarterly compounding (BEY is divided by about four periods in the year). Frankly, the official Investment Rates reported in financial markets on long-dated T-bills are not particularly transparent: Knowing the rate and one cash flow does not allow one to calculate easily the other cash flow. Even discount rates, despite their inadequacy as rates of return, are transparent in that sense.

Suppose that we need to construct a Treasury yield curve. The idea of any yield curve in principle is to display visually the relationship between interest rates on securities that are alike on all dimensions except maturity. Ideally, all the observations would be for securities that have the same credit risk, same liquidity, and same tax status. That is why Treasury yield curves in the financial press typically are based on the most recently auctioned instruments (these are called the “on-the-run” securities). They not only are the most liquid, they also are priced close to par value. That mitigates tax effects due to prices at a premium or a discount to par value. That said, it is common in practice to see the short end of the Treasury yield curve – that is, money market rates – display interest rates having varying periodicities.

Which T-bill rates should one include in a Treasury yield curve? Surely not the discount rates (1.850%, 1.900%, 2.135%, and 2.295%). Those understate the investor's rate of return. In my opinion, the best visual display of market conditions would report annual percentage rates having the same periodicity. A natural candidate is semiannual compounding because that is how yields to maturity on Treasury notes and bonds are calculated and presented.

Therefore, I suggest that T-bill discount rates first be converted to a 365-day add-on basis and then be converted to a semiannual bond basis (SABB). Note that SABB = APR2 in equation 1.9 – it is the APR for a periodicity of 2.

Each APR on the left side of each equation is the BEY calculated above, including the “wrong” rate for the 52-week T-bill. The conversions of the 4-week and 13-week T-bills entail more frequent to less frequent compounding, so their SABB rates are higher than the BEY. The 26-week SABB is the same as the BEY because 365/183 is so close to 2. Notice that the 52-week SABB is the same as the “correct” BEY obtained with equation 1.12. That is because when Days = 364, equation 1.13 effectively implies semiannual compounding.

Market practice, in any case, is to use the reported Investment Rates (1.878%, 1.936%, 2.188%, and 2.368%) at the short end of Treasury yield curves. This imparts a systematic bias for an upwardly sloping term structure because the shortest maturity rates have higher periodicities than the others. Best practice, I contend, would be to use the rates that have been converted to the SABB (1.886%, 1.941%, 2.188%, and 2.368%).

The differences between the SABB and the BEY results in the example are quite small because the interest rates are low. Suppose instead that money market rates in the U.S. someday are much higher than they were in 2008. If the discount rates for each of these four T-bills are 12%, the “official” Investment Rates would be 12.281%, 12.547%, 12.957%, and 13.399%. Converted as above, the corresponding SABB rates would be 12.605%, 12.745%, 12.956%, and 13.400%. The difference at the short end of the yield curve then would be quite significant – 32.4 basis points (12.605% minus 12.281%) for the 4-week bills and 19.8 basis points (12.745% minus 12.547%) for the 13-week bills.

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