# Difference between a 3-months UK nominal spot rate and a 3-months UK treasury bill discount rate?

I am trying to collect data I could use for calibration of a short-rate modeling process, so I need data which represents the historical short-rates.

On the Bank of England webpage I came across the historical Government liability curve data, and also the 3-months UK treasury bills discount rates.

It turns out they are not the same. For example, on the 31 May 2017 the UK nominal 3-month spot rate was 0.04, and the 3-months UK treasury bill discount rate was 0.0575. Also, in the 3-months UK nominal spot rate table, values of 3-months spots are actually mostly not given.

I don't understand what is the difference between these two values, and which ones should I use for my historical data?

Many thanks for any insights on this.

They are based on completely different assets.

• Gilts:

As this document states:

the government liability nominal yield curves are derived from UK gilt prices and General Collateral (GC) repo rates.

Later on, the same document defines:

A conventional gilt is a guarantee by the Government to pay the holder of the gilt a fixed cash payment (coupon) every six months until the maturity date, at which point the holder receives the final coupon payment and the principal.

These assets are usually long term. As this page states:

In recent years the Government has concentrated issuance of conventional gilts around the 5-, 10- and 30-year maturity areas, but in May 2005 the DMO issued a new 50-year maturity conventional gilt. In June 2013, following market consultation the DMO issued a new 55 year maturity conventional gilt.

Meanwhile, the document referred earlier defines "General Collateral (GC) repo rates" as:

General collateral (GC) repo rates refer to the rates for repurchase a greements in which any gilt may be used as collateral. Hence, GC repo rates should in principle be close to true risk- free rates. Repo contracts are actively traded for maturities out to one year; the rates prevailing on these contracts are very similar to the yields on comparable maturity conventional gilts.

Notice the phrase in bold. The rates are similar to comparable (e.g. 3-month T-Bills), but not identical. Unfortunately, they do not state why.

• 3-months T-Bills:

According to Bank of England definitions, the "Treasury bill tender 3 month (91 days) bills" is:

Treasury Bills are bearer Government Securities representing a charge on the Consolidated Fund of the UK issued in minimum denominations of £5,000 at a discount to their face value for any period not exceeding one year. Although they are usually issued for 3 month (91 days), on occasion they have been issued for 28 days, 63 days and 182 days. They are issued:

• by allotment to the highest bidder at a weekly (Friday) tender to a range of counterparties;
• in response to an invitation from the Debt Management Office to a range of counterparties;
• at any time to Government departments (non-marketable bills only).

Regarding why they have less data available, the same article points out that:

The secondary market in Treasury bills has in recent years become illiquid and representative rates are no longer obtainable other than those for the most recently issued 91 day bills. The rates shown are the average rates of discount at the weekly tender for 91 day bills.

Finally, regarding to which one to use, I would be definitively inclined to use the data related to the yield curve (gilts). This, as indicated above, are traded daily in the secondary market. Thus, data is more extensive, and perhaps more "reliable".

• As a result of arbitrage, Treasury bills and general colateral repo are both risk-free rates that trade close to each other. In practice, general collateral repo has less institutional distortions than Treasury bills. Also, if I read the question correctly, the TBill data has more holes than the fitted curve data. Jul 7 '17 at 13:31
• I've made a mistake in my original question, apologies - the nominal yield curve actually lacks data for 3-month spot rates. I just corrected it in my question. Jul 7 '17 at 13:42
• Except historical short rates, for the calibration I also need historical spot rates for various maturities. The problem is that the nominal yield curve lacks in short-rate data, and if I understood right - I shouldn't be using both at the same time, that is - I shouldn't be using historical T-bills rates as short-rate proxies, and historical nominal yield curve spot rates at the same time - it wouldn't be consistent. Is maybe than a solution to completely drop the nominal yield curve and just use historical discount rates for government bonds of various maturities? Jul 7 '17 at 14:19
• Milan - The answer depends upon how much data are missing. However, the quote convention consistency issue I highlighted in my answer is of limited importance, unless you are attempting to something like fit an arbitrage-free curve across all maturities (or relative value trading). For an econometric exercise, the quote convention differences would likely be smaller than other sources of error, and could be reduced with some detective work. Jul 7 '17 at 14:49
• @BrianRomanchuk Thanks. Corrected some errors, and changed my opinion on which one to use hehe. Jul 7 '17 at 15:54

(The question was clarified as specifying that the yield curve data had more missing observations. That may outweigh my theoretical preference for the yield curve in this answer.)

are a few potential sources for the discrepancy.

• The samples used to infer the 3-month rate could be different; for example, they could be done at a different time of day (or different instruments).
• The liability curve is a fitted curve; the fitted data will diverge from the observed market data. This is probably the largest potential defect of the fitted curve as a source.
• The quote conventions are different. The rest of this answer will discuss quote conventions.

The liability curve data uses a simple non-market convention that is consistent across maturities. If I am reading the document correctly, it's based on semi-annual compounding, and so the discount factor associated with a rate $r$ is $\frac{1}{1+\frac{r}{2}}$. For other maturities, the discount factor is calculated by taking to the power of the time period divided by six months.

• You would take that term to the power 2 to get the annual discount factor.
• The 3-month discount factor would be the square root of that term.

As a disclaimer, I read the description documents quickly, and I am unsure that this is the convention used. However, the key is that it is consistent across maturities, and so if you want to compare to longer maturity interest rates, the discount curve data has the advantage of internal consistency.

The discount rate data presumably reflects market conventions for the rate quotation. You would need to find a source for the convention used in the U.K. market, I do not have a reference. Treasury bill discount rates were developed in the pre-digital computer era to offer an easily-calculated interest rate that was roughly comparable across maturities. (The invoice price for Treasury bills depends upon the maturity, even if the discount rate is the same.) We need comparability so that we can answer questions like the following: would I get a better return from holding a 3-month bill, or rolling the 1-month bill 3 times (assuming the interest rate is unchanged). The mathematically clean continuously-compounded interest rate answers that question, but it is not easy to work with if you do not have digital computers/calculators.

In pretty well all analytical applications, you want to convert from the market convention to a clean discount rate; you might only convert back to the market convention if you want your final results to be comparable to market quotations. In other words, the liability curve data is probably a better starting point, as otherwise, you need to track down the market convention formula to do the conversion to a clean convention in order to do your calculations.

• Thank you for your detailed answer, it definitely makes a couple of things clearer to me. Though, I am not sure what does "consistent across maturities" exactly mean, i.e. if I were to use UK treasury bills/bonds/notes rates, in what way would they not be consistent across maturities? Jul 7 '17 at 12:59
• Bond yield conventions are different than Treasury bill conventions, and they differ across markets. (For example, a semi-annual bond yield of 4% is different than an annual yield of 4%.) My point about consistency across maturities refers to the following: how do we compare the rate on 1-month bill versus a 3-month? Try doing that without digital computing, and you see why market participants came up with easily-computed approximations of mathematical discount factors (market conventions). Jul 7 '17 at 13:24