# Yield Curve Building in Excel with Central Bank Meeting Dates and STIR OIS Trading Arbitrage Opportunities: The GBP Sonia case

One of the central banks' duties is to ensure the cost of money (aka interest rates) is optimal for the economy.

If a) the economy is overheated and needs to slow down to avoid a future burst or b) inflation rates are too high due to increased consumer demand, higher interest rates will a₁) have a cooling effect on the economy by raising the cost of capital of new projects and b₁) mitigate inflationary pressures by tempting the consumers to save their disposable income rather than spend it into buying goods and services.

Reversely, a sluggish economy or too low inflation rates because of low demand will benefit from lower interest rates.

In any free-market economy exist hundreds of different types of interest rates that constantly fluctuate because of varying supply and demand for the respective money borrowing-lending contracts. Central banks are not able to set the level of these interest rates directly, but can affect them indirectly by several means, such as changing the rate by which they charge the banks that borrow funds from them or increasing (decreasing) the money supply in circulation as a result of buying (selling) sovereign and corporate bonds in the secondary market.

For each major currency, there is a published schedule of dates when the respective central bank announces the undertaken market intervention measures. At any given time, the level of short term interest rates – typically with tenor of less than a year – is determined by the actions of the interest rate traders and reflects – among else - the latter's views on what the central bankers will announce on the forthcoming scheduled dates. In other words, the market prices of the various interest rate instruments incorporate the average sentiment held by the market participants regarding the central bank's impending monetary policy.

In what follows, I will show you how to use the market rates of short-term GBP OIS referencing the overnight Sonia index as of 21 Nov 2022 to build a yield curve that implies specific and quantifiable jumps of the forward Sonia index set to occur on the scheduled announcement dates.

This can be seen as some sort of "reverse engineering", whereby the future overnight rate spikes expected to be caused by the central bank announcements are extracted out of today's prices of certain instruments. It is "reverse engineering" because the extracted overnight rate spike estimates have a causal effect on today's market prices, but not vice versa.

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### Central Bank Meetings in the UK

The active central bank in the UK is the Bank of England (BoE).

Its Monetary Policy Committee (MPC) consists of nine members who convene eight times a year. Each meeting lasts three days that culminates to a vote on a set of measures that include the setting of the Bank of England Base Rate (BOEBR), which is the rate commercial banks pay when they borrow funds from the BoE overnight.

The thus decided base rate is published on 7 am UK time of the following day, which is always a Thursday.

All that is needed below for the construction of the yield curve is the list of publication dates.

For example, in the current study when 21 Nov 2022 is the trade date, the forthcoming MPC publication dates are:

- 15 Dec 2022
- 2 Feb 2023
- 23 Mar 2023
- 11 May 2023
- 22 Jun 2023
- 3 Aug 2023
- 21 Sep 2023
- 2 Nov 2023

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The STIR Sonia OIS Market as of 21 Nov 2022

If you are not familiar with *Overnight Index Swaps* (OIS), you may want to visit my posts titled Overnight Index Swap (OIS): Pricing and Understanding using Excel and Overnight Index Swap (OIS): Observation Lags, Lookbacks, Rate Cutoffs and step-by-step Pricing in Excel.

When it comes to Short Term Interest Rate (STIR) trading that involves *swaps* maturing in less than a year, alongside the regular *spot swaps* exist also *forward starting swaps* of which the start and end dates coincide with the *base rate* publication dates.

Regarding the *spot swaps*, their common start date, also known as *effective date*, equals the trade date, i.e the 21 Nov 2022, because of the Sonia OIS convention of a settlement of 0 business days. Note that *swaps* in other currencies often have a settlement of 2 business days.

Their end dates correspond to a standard set of tenors.

Below is a table with the *spot swaps* and their market rates observed in ICAP on 21 Nov 2022:

Regarding the *forward swaps*, these are defined by direct reference to the MPC publication dates. Below is a table with the *forward swaps* and their market rates observed in ICAP on 21 Nov 2022:

The maturities of these two sets of instruments span the same time period with max horizon the 2 Nov 2023, a bit less than 12 months after the trade date of 21 Nov 2022.

Their market rates are plotted below:

The blue points correspond to the *forward swap rates* and appear much higher than the orange points that correspond to the *spot swap rates*. This is due to the *forward* nature of the first set of rates and the upward slope of the implied *forward overnight rates*.

In fact, I will shortly show that both sets of points imply yield curves that are very close – within one bp - to each other.

### Yield Curve Building using the Forward Sonia OIS Market Rates on the MPC Publication Dates

The image below shows all the input parameters to the Deriscope Function in **cell J15** that constructs a corresponding Deriscope Object of Deriscope Type **Yield Curve OIS**, uniquely identified by the returned handle name **&OisRates.1**.

The red-colored cell contains the Deriscope formula. Details on the meaning of the shown colors and the syntax of the Deriscope spreadsheet formulas can be found in this introductory post about using Deriscope in Excel.

The object **&OisRates.1** is essentially a container for the ICAP market rates of the *forward swaps*, except for the first rate of **2.9275%** that is entered as a *stub rate* for the initial interval up to the first MPC publication date of **15 Dec 2022**.

The bootstrapping of these rates and the creation of the implied *yield curve* is accomplished by the formula =ds(J5:K13) shown in **cell J4** below:

The object created in **cell J4** is named **&SoniaCurve.1** and is of Deriscope type *Yield Curve*.

The entries shown here in red circle associated with the keys **Modelled Qty**= and **Interp Method**= are critical and must equal **Fwd Rate** and **Backwd Flat** respectively, in order to create a stepwise implied *forward overnight rate* as expected by the market.

### Generation of Implied Forward Overnight Rates

The object **&SoniaCurve.1** has a *local function* named *Fwd Rate* that returns the implied *forward rates* for any input set of dates.

In the screenshot below, this function is evoked by the formula =ds("Object=",J$4,P5:Q8,P10,P11#,Q10,Q11#) entered as a dynamic array in **cell R11**.

It references the dates shown in the two **columns P** and **Q** and returns the corresponding implied *forward rates*.

Since the dates are consecutive calendar dates, the returned rates represent *forward overnight rates*.

Also in red circle is shown the key-value pair **DayCount= %ACT/365**, which defines the input daycount convention as ACT/365 to match that of the Sonia index.

Below is the plot of all implied *forward overnight rates*:

It must be noted that all jumps occur exactly on the MPC publication dates,

Their sizes have been determined by the bootstrapping algorithm in such a way that the implied *forward OIS rates* match the supplied market rates.

For example, as is shown in the red rectangle section of the above table, on the first MPC publication date of **15 Dec 2022** (Thursday) the "forecasted" *forward* Sonia rate jumps to **3.5029%** from the previous level of **2.9248%**, which indicates a jump size of **0.5781%** or **57.81** basis points.

### Comparing the Two Sets of Market Rates

We have seen that the *forward* OIS market rates imply a *yield curve* represented by the object **&SoniaCurve.1**, which in turn implies the *forward* Sonia rates shown above.

In fact, the curve **&SoniaCurve.1** also implies specific rates for the *spot* OIS traded in the market. It is interesting to know to what extent these implied *spot* OIS rates match the corresponding market rates shown at the beginning of this article.

Normally, to calculate the implied OIS rate for any *spot* OIS, one would need to first construct the *spot* OIS instrument and then solve for its *fair rate* in the presence of the given curve **&SoniaCurve.1**.

But if the *spot* OIS matures in less than a year, one can prove that its *fair rate* equals the *forward rate* from the swap's *effective date* until the swap's *termination date*.

Since all our swaps mature in less than a year, I will use the much simpler *forward rate* formula below that evokes the Deriscope function **Fwd Rate**.

The dates shown in **column V** are input to the dynamic array formula =ds("Object=",J4,V21:W25,"EndDate=",V5#) in **cell W5** and have been set equal to the maturities of the market *spot* swaps.

My next step is to place the known market rates to the right of the returned implied rates and calculate the differences in **column Y**:

The **column Y** contains the differences (Implied Spot – Market Spot) expressed in basis points.

My first impression is that these are generally very small. Apart from the **-0.74** bp for the swap maturing on **21 Jan 2023**, all other differences are mostly less than half a basis point.

It is also interesting to note that apart from the single maturity of **21 Jan 2023**, the implied spot rates are all higher than the market spot rates. It seems that for some reason the market trades forward swaps at a premium relative to spot swaps.

Below is the corresponding chart:

Nevertheless, one should not reach any conclusions by examining these rates alone, because they reflect a single snapshot of the market taken at a particular time during the day of 21 Nov 2022. A range of disturbances (insufficient liquidity, stale live feeds etc) may be responsible for the observed differences. One should repeat this study over several times and days before making any statements regarding pricing discrepancies between the forward and spot OIS markets.

Click on **StirOisSonia.xlsx** to download the spreadsheet produced with the above steps.

At https://www.deriscope.com you can find more information about Deriscope and download the Excel Add-In required to run some of the formulas in the spreadsheet.