Estimation of the Liquidity Premium of TIIS
Hoi Fung
Christian–Albrechts–Universität zu Kiel
1.
Motivation
5.
• Since 1997, the USA starts to issue the Treasury Inflation Indexed Securities (TIIS).
Algorithm of the K ALMAN–Filter
1. Initiation:
ξˆ1|0 = E (ξ1)
• Although the reputation and outstanding of this new class of securities has grown
P1|0 = E [ξ1 − E (ξ1)] [ξ1 − E (ξ1)]′
n
rapidly since 1997, the transaction volume is still significant lower than its nominal
counterpart.
o
2. Prediction:
• Hence, the liquidity of TIIS is relative small compare to US nominal bonds.
ŷt|t−1 = A′xt + H ′ξˆt|t−1
• The market price of its illiquidity (the so–called liquidity premium) is signifikant for
pt|t−1 = H ′Pt|t−1 H + R
TIIS [Sack (2000) and Shen (2006)].
• This is important due to its influence on the Break–Even Inflation Rate (BEIR).
3. Updating:
• This paper investigates whether the liquidity premium has been declined continous-
ξˆt|t = ξˆt|t−1 + Pt|t−1 H H ′Pt|t−1 H + R
ly [Shen and Corning (2001)].
−1 yt − A′xt − H ′ξˆt|t−1
Pt|t = Pt|t−1 − Pt|t−1 H H ′Pt|t−1 H + R
−1
H ′Pt|t−1
4. Forecast:
ξˆt+1|t = F ξˆt|t
2.
Pt+1|t = F Pt|t F ′ + Q
Economics
• The BEIR is a synonym of the yield spread between nominal and indexed bonds.
5. Smoothing:
ξˆt|T = ξˆt|t + Jt ξˆt+1|T − ξˆt+1|t
• The yield of nominal bonds contain the following economic information [Shen
(1998)]:
Y
• The Variables
ψ
rt ,
e,ψ
πt
and
ieldnom,ψ
t
ψ
IPt
=
rtψ
+
πte,ψ
+
Pt|T = Pt|t + Jt Pt+1|T − Pt+1|t Jt′
IPtψ
are:
ψ
– (rt ) the annualized expected real rate of return at t until the end of Maturity ψ ,
e,ψ
– (πt ) the annualized expected inflation rate at t until the end of Maturity ψ and
ψ
– (IPt ) the inflation risk premium of Treasury Bond at t for the Period ψ − t.
6.
Results
Yield Spread and Estimated Liquidity Premium
• The yield of indexed bonds contain the expected real rate of return and the liquidity
ψ
premium LPt [Emmons (2000)]:
Yield Spread (5Y)
Exp. Inflation (5Y Median)
3.2
ψ
ψ
Y ieldind,ψ
=
r
+
LP
t
t
t
3.3
3.2
2.8
3.1
• The above mentioned liquidity premium
ψ
LPt
2.4
is due to lack of liquidity in TIIS–
market. It represents the compensation for the risk of illiquidity during the Period t
to ψ in terms of premium.
2.8
1.6
2.7
1.2
of the nominal bonds and liquidity premium of indexed bonds:
=
e,ψ
πt
3.
+
ψ
IPt
−
2.9
2.0
• Hence, the yield spread contains the expected inflation rate, inflation risk premium
ψ
Spreadt
3.0
2.6
2003
ψ
LPt
2004
2005
2006
2007
2003
Est. Inflation Risk Premium
Data
• In this paper, the following data are applied to estimate the liquidity premium of
TIIS:
c
, daily (1320
– Yield of treasury security with a constant maturity of 5 year (FRED
Obs.))
– Yield of treasury inflation indexed security with a constant maturity of 5 year
c
, daily (1320 Obs.))
(FRED
– CPI-All Urban Sample-Annual Inflation (Datastream, monthly (60 Obs.))
– Median of 5 years–ahead annualized expected inflation of University of Michigan
(Datastream, monthly (60 Obs.))
1.6
.08
1.2
.06
0.8
.04
0.4
.02
0.0
.00
-0.4
2004
2005
2006
2005
2006
2007
Est. Liquidity Premium
.10
2003
2004
2007
2003
2004
2005
2006
2007
• The result of this paper shows a significant negative impact of liquidity premium on
yield spread. This is reflected by Ĥ = −0.05. This finding is consistent with the
economic theory.
• The estimated liquidity premium of 5–year TIIS has not been decreasing continously as former expected.
• It shows a high volatility of perceived liquidity risk by the market.
• The result above rejects the finding of Shen and Corning (2001).
4.
• The reason of this inconsistency might be a term–specific liquidity premium due to
Model
• The following state–space model is chosen to describe the stochastic feature of
a segment–specific market volume of TIIS.
the sum of the inflation risk and liquidity premia:
yt = A′xt + H ′ξt + wt
7.
ξt = cstate + F ′ξt−1 + vt
• The ML–technique is applied in order to estimate the system matrices above.
• The corresponding log–likelihood function is:
f (yt |xt, υt−1 ) =
1
r
(2π)m/2 pt|t−1 · exp −
1
2
yt − ŷt|t−1
′
p−1
t|t−1
yt − ŷt|t−1
the yield spread into account simutaneously to estimate the liquidity premium of
TIIS.
• The result shows a significant contribution of liquidity premium within the 5–year
yield spread.
sample period.
Â
Ĥ
F̂
R̂
ML–Estimates 0,16815 0,05495 -0,05133 0,98729 0,056234
t–Statistics
2,6742 2,0791 -29,94 215,48 37,504
• The K ALMAN–Filter [Kalman (1960)] is implemented in order to estimate the liquidity premium of TIIS.
• This paper uses a state–space approach in order to take all economic factors within
• Hence, the liquidity of the 5–year TIIS has not been improved steadily during the
• The result of the estimated system matrices are as follow:
ĉstate
Conclusions
• Despite that, a declining liquidity premium is expected in future due to new issue
and hence increasing outstanding of 5–year TIIS [Shen (2006)].