Methodological Note on the Construction of the Yield Curve for Public Securities of CEMAC States

Banque des États de l'Afrique Centrale (BEAC)

Construction of the Yield-to-Maturity Curve for Public Securities of the Economic and Monetary Community of Central Africa (CEMAC) States Methodological Note November 2015

Table of Contents
A. Context and Justification ..... 3
B. Instruments Used and Data Sources ..... 3
1) Instruments used in the construction of the yield curve ..... 3
2) Data sources ..... 3
C. Description of the Methodology ..... 4
1) Methodology for constructing the BTA segment ..... 5
2) Methodology for constructing the OTA and OT segments ..... 5
3) Construction of the extrapolated segment ..... 6
D. Publication of Sovereign Yield Curves at BEAC ..... 7

A. Context and Justification
The term structure of interest rates, or yield curve, is the function that, for each maturity, indicates the level of the yield rate associated with a given date.
The construction of a sovereign yield curve falls within the framework of improving public securities market reporting and could contribute to the development of this market, particularly its secondary segment. The regular updating and publication of the sovereign yield curve by BEAC establishes a reference for the prices of Treasury bonds issued by issuers on the market, and allows estimation of risks associated with other non-sovereign issuers.

B. Instruments Used and Data Sources
1) Instruments used in the construction of the yield curve:
Three instruments are selected for constructing sovereign yield curves at BEAC: Treasury Bills and Assimilable Bonds (BTA and OTA) issued on the public securities market through auctions organized by BEAC, as well as Treasury Bonds (OT) issued via syndication on the sub-region's financial markets. Given the differences observed in the characteristics of these instruments (see table in annex), data harmonization is performed through: (i) the conversion of BTA rates to an exact actuarial basis 365, equivalent to OTA rates; and (ii) the calculation, for OTs, of the weighted average time to maturity, taking into account their amortization. Depending on the availability and reliability of secondary market data, three approaches are generally used by central banks to construct yield curves:

• The ideal approach relies on secondary market data, thereby obtaining a yield curve reflecting the prices at which financial counterparties have actually exchanged securities of an issuer;
• However, in the absence of a secondary market, the yield curve can also be constructed solely from primary market data;
• A third approach resulting from the synthesis of the first two is adopted by some central banks in developing countries, where secondary markets are very underdeveloped. Under this approach, the yield curve is obtained by combining data from primary market issuances and secondary market transactions.
  Given the weakness of transactions in the secondary segment of the public securities market in the CEMAC zone, BEAC has adopted the third approach. Thus, curves are primarily constructed from primary market data and, secondarily, from secondary market transactions when these are significant and reliable. Primary market data are retained if the following conditions are met respectively:
• issuance amount exceeding 1 billion FCFA and serviced by a minimum of two Treasury Value Specialists (SVT), for OTAs and BTAs;
• issuance amount exceeding 1 billion FCFA, for OTs.
  Regarding the secondary market, only data from firm transactions for amounts equal to or greater than 250 million are taken into account.

C. Description of the Methodology
The approach used generally consists of retaining three segments in the curve for determining yields for benchmark maturities whose associated rates are not directly observed.
The three considered curve segments are:

• A first segment, constructed based on primary market data from auctions, as well as transactions in the secondary market for BTAs. The benchmark maturities associated with this segment are 3 months, 6 months, and 1 year;
• A second segment, elaborated from primary market data (auctions and issuances) and transactions in the secondary market for OTAs and OTs, with benchmark maturities of 1.5 years, 2 years, 3 years, and 3.5 years;
• A third segment, constructed by extrapolation for benchmark maturities of 4 and 5 years, to inform the market on long-term yields and create a certain appetite for using these maturities.

1) Methodology for constructing the BTA segment:
The benchmark points of this curve segment are determined by following these sequential steps:

• Step 1: * Based on the results of auctions conducted during the considered month, a weighted average interest rate is calculated for each maturity. These rates are converted to an actuarial basis 365 and plotted on the curve as monthly yield rates.

• Secondary market BTA operations conducted during the considered month for an amount exceeding 250 million FCFA per operation are grouped by average remaining maturity and mapped to the nearest benchmark maturity. A weighted average yield rate is calculated for each benchmark maturity and plotted on the curve.

• Step 2: In the absence of new auctions for a given maturity during subsequent months, the last observed yield rate is retained on the curve for one month in the case of the 13-week maturity, and for 3 months in the case of 26- and 52-week maturities.
• Step 3: If the yield rate of one of the three benchmark maturities is not yet determined, this rate will be completed by linear interpolation.
• Step 4: If yield rates of two benchmark maturities remain undetermined, these rates will be completed by adding or subtracting a margin to the known benchmark yield rate. Margins will be calculated based on rate spreads across different maturities observed over the previous 6 months, as calculated in Step 1.

2) Methodology for constructing the OTA and OT segments:
The benchmark points of this curve segment are determined by following these sequential steps:

• Step 1: * Plot on the curve the yield rate resulting from the monthly OTA auction for the used maturity.

• If an OT issuance is conducted during the considered month, the yield rate of this issuance is plotted on the curve for a maturity equal to the average maturity of that issuance.
• OTA and OT secondary market operations conducted during the considered month for an amount exceeding 250 million FCFA per operation are grouped by average remaining maturity and mapped to the nearest benchmark maturity. A weighted average yield rate is calculated for each benchmark maturity and plotted on the curve.

• Step 2: In the absence of new OTA/OT issuances and secondary market transactions for a given maturity during subsequent months, the last observed yield rate is retained on the curve for 6 months.
• Step 3: Complete missing benchmark maturity yield rates using linear interpolation.

3) Construction of the extrapolated segment:
The determination of yields for 4- and 5-year maturities, not previously used by the market, is done by extrapolation using Brandt's method based on complete information from the two preceding segments. The Brandt method presents itself as a parametric model specifying a functional form of the yield as follows:
Rt = α + β1t + γ2t(t+4) + δ3(t-4) where:
Rt is the yield-to-maturity; ti is the theoretical/effective maturity; n = 1 to N, is the number of maturities.
This equation is estimated based on data from the other two curve segments using the ordinary least squares method. The obtained parameters are used to estimate the two points of the long-term segment, namely the yields for 4- and 5-year maturities. ^ For more discussion on the Brandt method, see Bruce Tuckman 2011, "Fixed Income Securities", Wiley.

D. Publication of Sovereign Yield Curves at BEAC The yield curve developed by BEAC is updated and published on a monthly basis.

Annex: Characteristics of Instruments Used for Constructing the Yield-to-Maturity Curve at BEAC (Short-Term Treasury Bills/Assimilable Bonds (BTA); Assimilable Treasury Bonds (OTA) and Treasury Bonds (OT) issued via syndication, medium- and long-term)

Characteristics	Treasury Bill/Assimilable Bond (BTA) issued via BEAC-organized auction	Assimilable Treasury Bond (OTA)	Treasury Bond (OT) issued via syndication
Emission	Auction via weekly calls for bids	Auction via monthly calls for bids	Issuance via syndication on the CEMAC financial market
Nominal Value (N)	1,000,000 FCFA	10,000 FCFA	10,000 FCFA
Maturity	13 Weeks (91 days); 26 Weeks (182 days); and 52 weeks (364 days)	Equal to or greater than 2 years	Equal to or greater than 2 years
Interest Payment	Interests are prepaid and deducted in advance, determined based on the interest rate proposed by the bidder at auction (T) and the number of actual days relative to a 360-day year, according to: I = (N × T × i) / 360. Amount of prepaid interest. T: Nominal interest rate; N: Nominal value of the Assimilable Treasury Bill; i: days from settlement to maturity.	Interests are paid annually at term, calculated based on actual days relative to a 365-day year: C = N × T × (j/365). C: Annual interest/coupon; N: Nominal value of the Assimilable Treasury Bond; T: Nominal interest rate; j: days from settlement to maturity.	Interests are paid annually at term, calculated based on actual days relative to a 365-day year: C = N × T × (j/365). C: Annual interest/coupon; N: Nominal value of the unamortated Treasury Bond; T: Nominal interest rate; j: days from settlement to maturity.
Principal Repayment	In a single lump sum at maturity	In a single lump sum at maturity	In equal annual fractions after a grace period
Pricing Formula	P = Price (n: number of years); R: yield rate. P = (N - C) / (1 + R)^n ... [standard bond pricing]