This document discusses parsimonious yield curve modeling in less liquid markets. It begins with the main goals of the study, which are to offer insights on quantitative solutions to challenges in estimating yield curves in less liquid government bond markets. It then defines less liquid markets and discusses innovative elements and stylized facts of the project using Polish government bond market data. The document reviews relevant literature on yield curve modeling in developed versus less liquid markets. It also discusses the data sources and liquidity measures, filtering rules and weight system framework, Nelson-Siegel-Svensson yield curve form, and key findings regarding improving yield curve fit and testing the Pure Expectations Hypothesis in Poland.

1. Parsimonious yield curve modelling
in less liquid markets
Marcin Dec, SGH and GRAPE
Warsaw, 24th June, 2021
The financial support of National Science Centre (grant UMO-2020/37/N/HS4/02202) is
gratefully acknowledged.
SGH and GRAPE
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2. Main goals of this study
1 to offer novel insights on possible quantitative solutions to typical of less liquid government
bond markets (such as Polish one) challenges when estimating yield curve, namely:
1 generally shorter recorded history compared to advanced economies,
2 extreme sensitivity of risk premium inference due to the issues with estimation of the short
end of the yield curve,
3 insufficient diversity of maturities of available bonds,
4 insufficiently precise price quotes for many of the off-the-run securities.
2 to develop a class of weighting schemes which improves fit relative to conventionally used
methods.
3 to verify that Pure Expectations Hypothesis (PEH) does not hold universally in LLMs
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3. Definition of less liquid markets
1 at least 80 bln USD equivalent outstanding amounts of general government debt
securities in local currency
2 excluding sovereign issuers from the United States, the United Kingdom, Japan, the
Euro Area, Switzerland, Canada and Australia.
Using Bank for International Statistics data as of the end of December 2019 the following
countries would fall into this group: Brazil, Chile, Czechia, Hungary, India, Indonesia,
Israel, Malaysia, Mexico, Korea, Poland, Singapore, South Africa, Thailand and Turkey.
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4. Innovative elements of the project (Preludium: YIELDS)
1 a joint study of yield curve fit (estimation) and term premia structure estimation is a
unique proposal not found in the literature covering less liquid government bond markets.
Our approach exploits to the maximum the available micro-structural data of bond
market: turnover (monthly and daily, where available), outstanding amounts, bid-ask
spreads, number of transactions traded daily, bid and ask yields.
2 a delivery of more meaningful, realistic and interpretable time-series of yield curves than
the ones produced by methods overly concentrated on their smoothness and perfect fit to
all observed data without market practise view on weights of different bond series.
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5. Stylised facts on Polish government bonds’ market
part 1
1 Monthly secondary TS market’s turnover is of the same magnitude as monthly sum of
NBP bills auctioned at reference rate.
2 The volume traded (both on BS platform and on the market as a whole), as well as daily
number of transactions follow common pattern during the lifetime of a certain bond type.
3 The bid-ask spread (on BS) rises approximately two- or threefold above bond’s lifetime
mean in the last year.
4 Zero trading days patterns resemble mirror reflection of the ones observed for volume and
number of trades.
5 Historically the lowest bid-ask spreads were observed in the segment of (6, 12] years to
maturity and the highest in the short-end of the curve [0, 1.5] years.
6 Ultra long end of Polish yield curve (12, 30] is very erratically inhabited with only one or
two series quoted on fixing, and no representation since 2018 till now.
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6. Stylised facts on Polish government bonds’ market
part 2
7 Switch auctions influence prices by increasing BAS due to very limited motivation on both
sides: potential buyers’ side who have alternative strategy of rolling NBP bills and potential
sellers’ who maybe better off using these bonds to buy longer and more liquid ones.
8 Historically the least liquid segments were [0, 1.5] and (12,30] years to maturity.
9 Switch operations make short bonds (eligible to switch from) richer than the interpolated
interest rate between NBP bills and [1.0, 1.5] segment of bonds.
10 All segment-wise average yield time series are trend stationary when corrected for long
term variance a mode de Newley-West for lags of at least 18-months
11 Share of BondSpot in total secondary market turnover is erratic and the list of bonds
traded is periodically shallow.
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7. Literature review
Developed (liquid) vs less liquid markets
Developed (liquid) markets
McCulloch (1975), Vasicek (1977), Nelson i Siegel (1987), Fama
i Bliss (1987), McCulloch (1990), Campbell i Shiller (1991),
McCulloch i Kwon (1993), Svensson (1994), Campbell (1995),
Fisher, Nychka, i Zervos, (1995), Duffie i Kan (1996), Waggoner
(1997), Anderson i Sleath (1999), Bolder i Gusba (2002), Duffee
(2002), Garbade (2004), Sack i Elsasser (2004), Kim i Wright
(2005), Gurkaynak, Sack i Wright (2006), Diebold i Li (2006),
Cochrane i Piazzesi (2008), Duffie (2010), Joslin, Singleton i Zhu
(2011), Bauer, Rudebusch i Wu (2012), Gurkaynak, Refet, i
Wright (2012), Hamilton i Wu (2012), Kim i Orphanides (2012),
Adrian, Crump i Moench (2013), Malik i Meldrum (2016),
Bauer (2016), Crump, Eusepi i Moench (2016), Bauer i
Hamilton (2017), Bauer i Rudebusch (2017), McCoy (2019).
Less liquid markets
Świętoń (2002), Cięciwa
(2003), Marciniak (2006),
Liberadzki, Wójcik (2006),
Kliber (2009), Dziwok
(2012, 2013), Jabłecki,
Raczko i Wesołowski
(2016), Rubaszek (2016),
Kucera, Dvorak, Komacrek i
Komarkova (2017), Kolasa i
Wesołowski (2020).
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8. Data
Consolidation of wide range of sources: BondSpot, MinFin and NBP for the period spanning over more than 15 years
In the course of preparatory work, a unique database of Polish government bonds was
collected, described and processed:
1 more than 8,000 tables from Bond Spot S.A. were consolidated for the period 2005:01 -
2020:06 containing for each series of bonds fixing prices and yields, buy and sell prices and
yields, data on turnover (number, volume and transaction values).
2 data from the National Bank of Poland on official interest rates and the rate of one-day
deposits on the POLONIA interbank market were used, as well as information on NBP
bills tenders.
3 Statistical information from the Ministry of Finance was consolidated and used: (1)
coupon tables and interest periods for particular series of bonds, (2) databases of
operations on securities (ordinary auctions, swaps, early redemptions, special operations),
(3) monthly trading on the market secondary in particular series, (4) issuance letters and
regulations concerning the Primary Dealers system
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9. Liquidity measures of Polish fixed coupon bonds by segments
2005:01 - 2020:06
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10. Filtering rules and weight system framework
Principles
1 Principle 1. Minimise the share of arbitrary decisions.
2 Principle 2. Do not exclude bonds from the sample entirely, but diminish their weight
accordingly, unless the pricing is systemically distorted.
3 Principle 3. Include as much reliable and useful information (static and dynamic data) as
possible to reflect importance of a certain bond in the yield curve formation.
4 Principle 4. Include only risk free rates
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11. Filtering rules and weight system framework
Rules
1 Rule 1. From the broad set of Polish government bonds we exclude CPI-linkers, floaters,
foreign denominated and retail bonds. This is a usual choice in yield curve estimation
literature
2 Rule 2. We take all pricing information of fixed and zero coupon bonds which are subject
to fixing on BondSpot, with an exception in the next bullet.
3 Rule 3. we exclude bonds with less than 0.85 or 1.20 years to maturity (till mid 2017 and
after that period) because their prices and therefore ytm are distorted significantly by the
switch operations of Polish MinFin (as it was clearly shown).
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12. Parsimonious yield curve form
Nelson Siegel Svensson
The most popular parsimonious yield curve forms are based on the works of Nelson and
Siegel with extension proposed by Svensson
Instantaneous forward rates in NSS models
f j
(x) = β0 + β1e
x
τ1
+ β2
x
τ1
e
x
τ1
+ β3
x
τ2
e
x
τ2
| {z }
second hump
Integrating the above on (0, x) and dividing by x we receive:
Yield curve form in NSS models
y(x) = β0 + (β1 + β2)
1 − e
− x
τ1
x/τ1
− β2e
− x
τ1
+ β3

1 − e
− x
τ2
τ/τ2
− e
− x
τ2


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13. Parsimonious yield curve form
Nelson Siegel Svensson
Possible interpretation of parameters
β0 - long term interest rates level
β0 + β1 - level of ultra-short interest rate
τ1, τ2 - position of humps
β2, β3 - humps’ intensity
Constraints (Θ):
1 β0 0
2 β0 + β1 −0.02
3 τ1 0
4 τ2 τ1
Minimisation problem
min
Θ
O = min
Θ
( N
X
i=1
Wi (Pi (Θ) − pi )
2
+ W nbp
R(Θ) − rnbp
2
)
s.t. C(Θ)
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14. Constrained minimisation problem
Starting seed
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15. Parsimonious yield curve
Example
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16. Key findings
part 1
We have found a class of weighting systems for Polish government bonds yield curve to be used
in NSS estimation which significantly improves the fit and smoothness as compared to the
traditional approach of all equal weight.
This class has three core characteristics:
1 at least the same weight for the short end of the curve as a sum for all other tenors of
bonds
2 exclusion of eligible-for-switch bonds from the estimation
3 bonds’ weights based on at least outstanding amounts
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17. PEH testing regressions
part 1
1 Fama Bliss (1): Excess return on term premium regressions
rx
(n)
t+h = α + β
f t+h,t+n
t − y
(h)
t
+
(n)
t+h
2 Fama Bliss (2): Realized change in the spot rate on term premium regressions
y
(n)
t+h − y
(h)
t = α + β
f t+h,t+n
t − y
(h)
t
+
(n)
t+h
3 Cochrane Piazzesi: One year excess return on average one year forward rates
regressions
rx
(n)
t+1 = β
(n)
0 + β
(n)
1 y1
t + β
(n)
2 f t+1,t+2
t + β
(n)
3 f t+2,t+3
t + β
(n)
4 f t+3,t+4
t + β
(n)
5 f t+4,t+5
t +
(n)
t+1
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18. PEH testing regressions
part 2
4 Thornton (1): Rolling realised returns on term premia regressions
1
k
k−1
X
i=0
y
(h)
t+i×h − y
(h)
t = α + β
y
(n)
t − y
(h)
t
+ t
5 Thornton (2): Realised spread on pro rata temporis current spread regressions
y
(n−h)
t+h − y
(n)
t = α + β
h
n − h
y
(n)
t − y
(h)
t
+ t
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19. Pure expectations hypothesis testing
Fama and Bliss style
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20. Key findings
part 2
1 Contrary to the research conducted in the US markets, where PEH is almost always
rejected, we found that in Poland there is a limited scope (in a space spanning various
investment horizons and selected bond maturities) where PEH cannot be rejected
2 The scope where pure expectations hypothesis probably holds in Poland is bounded by
(1) the investment horizon of approximately 12 months and (joined condition) and (2) by
maturity of the bond of circa 36 months.
3 It is still unclear (as in the bewildering variety of research) what causes the rejection of
PEH for all other combinations of horizon length and maturity: existence of some
kind of risk premia or unexpected excess yield (we have only a mixture of these two
contained in β coefficient estimators).
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21. Thank you for your time!
SGH and GRAPE
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22. Typowe zależności empiryczne - ilustracja
Miesięczne średnie obroty, udział dni bez handlu w miesiącu, wartość niewykupiona - obligacje 5-letnie
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23. Typowe zależności empiryczne - przykładowa ilustracja
Miesięczne średnie dzienne liczby transakcji, spread bid-ask - obligacje 5-letnie
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24. Typowe zależności empiryczne - przykładowa ilustracja
Spread aukcji zamiany. Istotne zaburzenie struktury krzywej stóp do 1 roku
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25. Premie terminowe
Szczegóły podejścia ACM - część 1
Adrian, Crump i Moench
1 zmienne stanowe (czynniki z PCA) Xt ewoluują zgodnie z modelem VAR:
Xt+1 = µ + ΦXt + vt+1
gdzie: vt+1| {Xs}
t
s=0 ∼ N(0, Σ)
2 istnieje jadro wyceny {pricing kernel} P
(n)
t = Et
h
Mt+1P
(n−1)
t+1
i
3 zakładają, że to jądro M jest wykładniczo affiniczne:
Mt+1 = exp −rt − 1
2 λ0
tλt − λ0
tΣ−1/2
vt+1
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26. Premie terminowe
Szczegóły podejścia ACM - część 2
Adrian, Crump i Moench
4 zakładają, że rynkowe ceny za ryzyko mają nastepującą afiniczną formę:
λt = Σ−1/2
(λ0 + λ1Xt)
5 modelują logarytmy nadwyżkowych zwrotów miesięcznych {log excess holding returns}
obligacji zapadajacych w n-miesiącach:
rx
(n−1)
t+1 = ln P
(n−1)
t+1 − ln P
(n)
t − rt
6 dekomponują nieoczekiwany nadwyżkowy zwrot na dwa komponenty: skorelowany z vt+1 i
drugi - warunkowo ortogonalny do vt+1:
rx
(n−1)
t+1 − Et
h
rx
(n−1)
t+1
i
= γ
(n−1)0
t vt+1 + e
(n−1)
t+1
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27. Premie terminowe
Długoterminowe średnie
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