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TABLE OF CONTENT
TABLE OF CONTENT
LIST OF FIGURE
LIST OF TABLE
INTRODUCTION .......................................................................................................... 1
CHAPTER 1: INTRODUCTION TO YIELD CURVE RISK MANAGEMENT ... 3
1.1. Yield curve ............................................................................................................ 3
1.1.1. Shape of yield curve ....................................................................................... 3
1.1.2. Type of yield curve ........................................................................................ 5
1.2. Yield curve risk management................................................................................ 5
1.2.1. Interest rate sensitivities measurement .......................................................... 5
1.2.2. Yield curve factor model ................................................................................ 9
1.2.3. Interest rate risk immunization .................................................................... 10
CHAPTER 2: REDUCED-FORM YIED CURVE MODELLING ......................... 16
2.1. Nonparametric approach the Principal analysis .................................................. 16
2.1.1. Statistical basis of Principal Component Analysis ...................................... 16
2.2. Parametric Approach The Nelson Sigel model ................................................... 27
2.2.1. Original model ............................................................................................. 27
2.2.2. The dynamic Nelson-Sigel model ................................................................ 31
2.2.3. State space model with Kalman Filter ......................................................... 32
2.3. Application to England Government Bond yield ................................................ 33
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2.3.1. Data .............................................................................................................. 34
2.3.2. Estimating Parameter ................................................................................... 36
2.3.3. Dynamic modeling ....................................................................................... 43
2.3.4. Forecasting Interest rate ............................................................................... 46
CHAPTER 3: RISK SENSITIVITY AND IMMUNIZATION ............................... 58
3.1. Risk sensitivity .................................................................................................... 58
3.1.1. Reitano Partial Duration model .................................................................... 58
3.1.2. Example ........................................................................................................ 61
3.2. Factor duration models ........................................................................................ 63
3.2.1. PCA duration ............................................................................................... 63
3.2.2. NS duration based approach ........................................................................ 64
3.3. Immunization approaches ................................................................................... 65
3.3.1. Cash flow matching...................................................................................... 65
3.3.2. Traditional Duration Matching .................................................................... 65
3.3.3. Duration Vector Model ................................................................................ 66
3.3.4. Key rate immunization model ...................................................................... 68
3.3.5. Factor duration based immunization ............................................................ 71
3.4. Portfolio Optimization in immunization ............................................................. 71
3.5. An application to England Government Bond Data ........................................... 73
3.5.1. Portfolio Design ........................................................................................... 73
3.5.2. Result of portfolio optimization ................................................................... 77
CHAPTER 4: APPLICATION TO VIETNAMESE BOND MARKET ................ 83
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4.1. Data ..................................................................................................................... 83
4.2. Yield curve modeling .......................................................................................... 83
4.3. Forecasting Yield curve ...................................................................................... 88
CHAPTER 5: CONCLUSIONS ................................................................................. 91
APPENDIX ................................................................................................................... 93
A. Mathematic Basis ............................................................................................. 93
B. Summary of Immunization Models Features .................................................. 97
C. Net Worth Immunization - Gap Analysis ......................................................... 99
REFERENCE ............................................................................................................. 104
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LIST OF FIGURE
Figure 1.1: The US dollar yield curve as of February 9, 2005. The curve has a typical
upward sloping shape .......................................................................................................... 4
Figure 1.2: History of the term structure, USA, January 1990 February 2013................ 4
Figure 1.3: Illustration of target-date immunization ......................................................... 13
Figure 1.4: The Contingent Immunization ........................................................................ 14
Figure 2.1: Sensitivities of term structure with PCs ........................................................ 22
Figure 2.2: Variation of Eigenvalues with respective years (for all moving windows) ... 25
Figure 2.3: PC1 for all windows ....................................................................................... 26
Figure 2.4: PC2 for all windows (Standardized data) ....................................................... 27
Figure 2.5a,b,c: level, slope and curvature factor loading. ............................................... 29
Figure 2.6: Factor loading with fixed 0 0609. ............................................................. 30
Figure 2.7: Nominal Government Bond Yield Bank Of England ................................. 34
Figure 2.8: Actual and fitted average yield curve. ............................................................ 38
Figure 2.9: Selected tted (model-based) yield curves. .................................................... 38
Figure 2.10: Residual surface ............................................................................................ 40
Figure 2.11 a, b, c: The empirical level, slope and curvatures (red) and estimated
component factor 2 31 , ,t tt (blue) ............................................................................. 42
Figure 2.12a,b,c: Autocorrelation 1
,2
,3
................................................................... 43
Figure 2.13a,b,c: Autocorrelation of residual 1 2 3 , , .................................................... 45
Figure 2.14: RMSE comparison of seven methods........................................................... 56
Figure 2.15: Residual Autocorrelation of different methods ............................................ 57
Figure 3.1: Bond Price Change by Interest Rate Movements ........................................... 66
Figure 3.2: Yield curve shift at a key rate ......................................................................... 69
Figure 3.3: Yield curve shift in key rates .......................................................................... 70
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Figure 4.1: PCA analysis of VN bond yield from 29/08/2008 to 8/7/2013 ...................... 84
Figure 4.2: Surface of VN bond yield ............................................................................... 85
Figure 4.3: Fitted yield curve at 29/08/2008 using mean data and mean estimated
parameters. ........................................................................................................................ 86
Figure 4.4: Fitted yield curve at selected date .................................................................. 87
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LIST OF TABLE
Table 1.1: Benchmark yield curve of theoretical bond ........................................................8
Table 2.1: Principal Components for total period of all moving windows ....................20
Table 2.2: Principal Component factor weight statistical description of 1 month
moving windows ............................................................................................................21
Table 2.3: Eigenvalues for 1 and 3 month windows ......................................................... 23
Table 2.4: Eigenvalues for 6 and 12 month window ........................................................23
Table 2.5: Explanatory factor (%) of the first three principal components (6 month) .....24
Table 2.6: Descriptive statistics for monthly yields at different maturities and for the
yield curve level, slope and curvature. ..............................................................................35
Table 2.7: Statistical description of maturities minimizes curvature (third component).
PCA base on daily data of yields from 1990 2011 ........................................................36
Table 2.8: Descriptive statistic of 1 2 3, , t t t using monthly yield data from
Jan1990Dec2011, with t xed at 42.2925.....................................................................37
Table 2.9: The Descriptive Statistic of model Residual ....................................................39
Table 2.10: The correlation between empirical slope, level curvature and 1 2 3 , , ..40
Table 2.11: The forecasted result comparison of seven methods. ....................................53
Table 2.12 P-value of test on mean error of RW forecast. ................................................54
Table 2.13: P-value of test on mean error of PCA forecast ..............................................55
Table 2.14 P-value of test on error mean of DNS AR forecast .......................................55
Table 2.15: P-value of test on error mean of DNS ARI forecast ......................................55
Table 2.16: P-value of test on error mean of Direct AR forecast .....................................55
Table 2.17 Dikey-Fuller test on estimated beta coefficient. .............................................58
Table 3.1: Yield curve of England Government Bond in 31/12/2006 ..............................74
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Table 3.2: Portfolio designs ..............................................................................................75
Table 3.3: Correlation Matrix of Spot Rates Shifts ..........................................................76
Table 3.4: Portfolio allocation for Bullet Strategy ............................................................77
Table 3.5: Portfolio allocation for Laddering Strategy ....................................................77
Table 3.6: Portfolio allocation for Barbell Strategy ..........................................................78
Table 3.7: Change in benchmark yield curve ...................................................................79
Table 3.8: Surplus of each strategy in small change case .................................................80
Table 3.9: Surplus of each strategy in medium change case ............................................80
Table 3.10: Surplus of each strategy in extreme change case ...........................................80
Table 3.11: Surplus of each strategy in small change case (610 ) ..................................81
Table 3.12: Surplus of each strategy in medium change case (410 ) .............................81
Table 3.13: Surplus of each strategy in extreme change case (%) ....................................81
Table 4.1: Statistical Description of VN Bond Yield .......................................................85
Table 4.2: Statistical description of residual ....................................................................87
Table 4.3: Result of forecasted yield curve from 7 methods. Starred method means its
RMSE is lowest one in that case. ......................................................................................90
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INTRODUCTION
Asset and liability management (often abbreviated ALM) is the practice of managing
risks that arise due to mismatches between the assets and liabilities in financial
intermediaries. Among many targets, ALM still focuses on interest rate risk and
liquidity risk because they represent majority loss in profit from financial operation.
Financial institutions managing fixed income portfolio of any types always have their
ultimate goal that is to keep a positive interest margin. This makes interest rate risk
management become a vital task in daily activity. Interest rate risk is defined as the risk
originated from change in interest rate that affects the value of interest bearing security.
In calculating this value; one usually uses a benchmark yield curve which is
interpolated from a universe of risk free bonds traded in the market. Therefore,
managing interest rate risk is to take into account the change in the total term structure
of interest rates (that is, yield curve).
Yield curve risk management is always attached to measure the sensitivity of
asset/liability value to change in the yield curve then hedge risk by matching it with
sensitivity of a benchmark portfolio or risk free asset. For example, in asset/liability
management, one controls the sensitivity of assets relative to fixed-income liabilities.
In total return fixed income management, one typically controls the sensitivity of the
asset portfolio relative to a fixed-income benchmark index which defines the
performance objective of the portfolio, finally, in fixed-income assets relative to a short
portfolio of fixed-income derivatives, such as interest rate future contracts.
Therefore, independent of the objective of the yield curve risk management program,
the first fundamental problem is to quantify the interest rate sensitivities of a given
fixed-income portfolio. The next fundamental problem is either to develop defensive
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risk management strategies from a longer term model of yield curve movement. The
last fundamental problem relates to the development of yield curve models.
Traditionally, one usually uses duration model which bases on the parallel shift
assumption of yield curve, which is impractical. Many researches have been working
on relaxation of this assumption. However, on broadening the concept of duration to
meet that requirement, models usually use intuitive choice of parameter and become
highly dimensional, which increase the computational complexity. Reduced form
factor were another group of models with partially overcome these problem.
This thesis will introduce a general framework in addressing above fundamental
problems with relaxed assumption. The first chapter of this thesis will be a general
introduction on yield curve risk management. The second chapter will discuss about
yield curve model and forecast. The third chapter will focus on portfolio immunization.
Finally, we will apply these forecasting models to Vietnamese Bond Yield data.
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CHAPTER 1: INTRODUCTION TO YIELD CURVE RISK MANAGEMENT
1.1. Yield curve
Yield curve present graphical interpretation of term structure of interest rates, as a
functional relationship between maturity of debt instruments of different length of time
to maturity and its yield to maturity.
1.1.1. Shape of the yield curve
As we can observe by analyzing yield curves in different markets at any time, a yield
curve can be one of four basic shapes, which are:
- Normal: in which yields are at average levels and the curve slopes gently
upwards as maturity increases;
- Upward sloping: in which yields are at historically low levels, with long rates
substantially greater than short rates;
- Downward sloping (or inverted) : in which yield levels are very high by
historical standards, but long-term yields are significantly lower than short rates;
- Humped: where yields are high with the curve rising to a peak in the medium-
term maturity area, and then sloping downwards at longer maturities.
On the three-dimensional plot (Fig. 1.2), we can see changes of yield curve shape for a
long period. The lower-right portion of the graph presents term structure for January
1990. The upper-left portion of the diagram presents term structure for February 2013.
The first term structure is upward sloping, with long-term yields above short-term
yields. The second is downward sloping, and is maintained by inverting the
relationship between short-term yields and long-term yields. The upward-sloping term
structures are more common than downward-sloping term structures, as it is obvious
from the figure.
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Figure 1.1: The US dollar yield curve as of February 9, 2005. The curve has a typical
upward sloping shape (Source: Wikipedia)
Figure 1.2: History of the term structure, USA, January 1990 February 2013
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1.1.2. Type of yield curve
1.1.2.1. Yield to maturity yield curve
The curve is constructed by plotting the yield to maturity against the term to maturity
for a group of bonds of the same class. The bonds used in constructing the curve will
only rarely have an exact number of whole years to redemption; however it is often
common to see yields plotted against whole years on the x-axis.
1.1.2.2. The par yield curve
The par yield curve plots yield to maturity against term to maturity for current bonds
trading at par. The par yield is, therefore, equal to the coupon rate for bonds priced at
par or near to par, as the yield to maturity for bonds priced exactly at par is equal to the
coupon rate.
1.1.2.3. The zero-coupon (or spot) yield curve
The zero-coupon (or spot) yield curve plots zero-coupon yields (or spot yields) against
term to maturity. The zero-coupon yield curve is also known as the term structure of
interest rates.
1.2. Yield curve risk management
As mentioned in the introduction, yield curve risk management has three fundamental
problems
1.2.1. Interest rate sensitivities measurement
Duration and convexity are main tools in measuring the interest rate sensitivity of fixed
income portfolio. Duration measure the change of bond price in response to interest
rate changes. Usually, if we calculate this change only proportionate to duration
measurement, the approximation will not bear good result as there is a certain
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convexity in the value curve. The more curved the price function of the bond is, the
more inaccurate the valuation is. Convexity is a measure of the curvature of how the
price of a bond changes as the interest rate changes, i.e. how the duration of a bond
changes as the interest rate changes.
If we assume a smooth price function, duration can be formulated as the first derivative
of the bond price with respect to the interest rate, and then the convexity would be the
second derivative of the price function with respect to the interest rate. In this section,
the duration and convex analysis are based on a flat yield curve and parallel shift, this
assumption is far from the reality and will be relaxed later. However, both cases start
from the Taylors theorem below
Taylors theorem
( 1)
2 1
( ) ( 1)
If and its first ( 1) derivatives , , ... , are continuous on , .
Then it exists , such :
( ) ( ) ' " ...2! ! 1 !
n
n n
n n
f n f f f a b
c a b
b a b a b af b f a b a f a f a f a f c
n n
The second order Taylor approximation gives the change in value derived from a
change in interest rates as
V
V yy
(0.1)
2
2
2
1
2( )
V VV y y
y y
(0.2)
1.2.1.1. Duration
There are 2 different forms of Duration [30], [44]. We will start from the variation of
the price for an infinitesimal variation of interest
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7
11
1 1
n
tt
tt CFV
y y y
(0.3)
Then
1
1
11
1
1
nt
tt
nt
tt
t CFV
yyS
CFV y
y
(0.4)
Set
1 1
1
1 11
1
( )
n nt tt t
t t
ntt
t
t CF t CF
y yD S y
CF V
y
We call D as Macaulay Duration and S the modified duration. The greater the duration
of a security, the greater the percentage change in the market value of the security for a
given change in interest rates. Therefore, the greater the duration of a security is, the
greater its interest-rate risks are. In the condition of continuous compounding of
interest rate, Macaulay Duration will equal modified duration.
1.2.1.2. The Convexity
The convexity measure of a security can be used to approximate the change in price
that is not explained by duration [30], [44]
22
2 2 21 1
1 1
1 1 1 1
n nt tt t
t t
t CF t CFV
y y y y y
.
Then
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22
2 21
1
1 1
n
t tt
t tVCF
y y y
And
2
2
212
1
1 1
n
t tt
t tCFV
y yyConv
V V
(0.5)
Conv is the percentage change of market value due to the convexity. The percentage
price change due to convexity is 21
2Conv dy .
1.2.1.3. An application
There is a 18Y bond with coupon rate 10% and will be paid annually. We assume a
parallel shift of yield curve by 5bp. The benchmark yield curve for this bond is
1Y-6Y (%) 6Y-12Y (%) 12Y-18Y (%)
5.1791 5.0972 5.0354
4.9749 4.9117 4.8471
4.7842 4.7257 4.6731
4.6261 4.584 4.5452
4.5085 4.4729 4.4377
4.4023 4.3667 4.3307
Table 1.1: Benchmark yield curve of theoretical bond
The present value of this bond is 167.3162805. As above formula uses univariate
model, therefore, we must find the yield maturity of this bond and use it as the discount
yield. We can use IRR function of Excel to do it.
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(cash flow) = 4.475%y IRR
Using (0.3) and (0.5), we can estimate the duration and convexity of this bond:
D=10.96087954 and C=154.4658629
Assume there is a parallel shift 5bp to the yield curve, the new present value is
166.4338992. We can approximate this value using (0.1) and (0.2), we achieve
166.3993137 and 166.4025443. Clearly, by combining with convexity, we get more
accurate result.
1.2.2. Yield curve factor model
Modeling the yield curve risk is an important part of a good risk management practice.
It is also a basic step to hedge a fixed income portfolio. As opposed to the class of no
arbitrage and equilibrium models, there are reduced-form models based on a statistical
approach, where interest rates are often modeled with a univariate time series or a
multivariate time series. The univariate class includes the random walk model, the
slope regression model, and the Fama-Bliss forward rate regression model (Fama and
Bliss, 1987), where interest rates are modeled for each term of maturity individually.
This class of models cannot, however, efficiently explore the cross-sectional
dependence of interest rates of different maturities for estimation and forecasting
purposes.
The multivariate class includes the vector autoregressive (VAR) [19] models and the
error correction models (ECMs), where interest rates of several maturities are
considered simultaneously to utilize the dependence structure and cointegration.
However, we cannot include all maturities to the model as this will increase the
dimension, this can be solved by selecting some key maturities in model. In that case,
this will reduce the comprehension of analyze process.
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Therefore, the first problem faced in term structure modeling is how to summarize the
term structure information at any point in time. Yield curve models should employ a
structure that consists of a small set of factors and the associated factor loadings that
link yields of different maturities to those factors. Besides providing a useful
compression of information, a factor structure is also consistent with the celebrated
parsimony principle, which conducts the model following a broad sight, not to dig
deep in fitting model but increase the out-sample fitting. Parsimony is also a
consideration for determining the number of factors needed; along with the demands of
the precise application, for example in principal analysis, we can omit the remaining
component which only represents a small portion in variance.
This thesis will introduce two most common approaches. The first is principal
component analysis, a general approach places structure only on the estimated factors.
For example, the factors could be the first few principal components, which are
restricted to be mutually orthogonal, while the loadings are relatively unrestricted.
Indeed, the first three principal components typically closely match simple empirical
proxies for level (e.g., the long rate), slope (e.g., a long minus short rate), and curvature
(e.g., a mid-maturity rate minus a short and long rate average). The second approach is
a fitted Nelson-Siegel curve (introduced in Charles Nelson and Andrew Siegel, 1987)
[24]. In this thesis, we will apply this method under the dynamic form following the
work of Diebold and Li (2005) [9], this representation is effectively a dynamic three-
factor model of level, slope, and curvature.
1.2.3. Interest rate risk immunization
Traditional immunization focused on the concept of duration firstly introduced by
Macaulay (1938) for implementing immunization techniques, which work well on
parallel shift environment. These models relying on duration were targeting interest
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rate risk and not really yield curve risk, since different points of the yield curve were
not allowed to move independently from each other as in the non-parallel world.
Klotz (1985) [18] made first step to move from generic interest rate risk to yield curve
risk with the introduction of the concept of convexity. Convexity is related to the
second derivative of the priceyield relationship of a bond. Later researches worked on
duration and convexity matching: socalled M-square and M-vector models were
introduced by Fong and Fabozzi (1985) [47-Appendix E], Chambers et al. (1988) [5],
and Nawalka and Chambers (1997) [23]. Another multifactor model which captures the
higher order of duration into account is duration vector (DV) models. An accurate
review of them is given in Nawalka et al (2003) [22], who also introduces a
generalization of the DV approach identified as generalized Duration vector (GDV).
A parallel development of immunization models relied on a statistical description of
the factors underlying yield curve shifts. This description was based on principal
component analysis (PCA). Barber and Cooper (1996) [1] adapted the duration model
of Fisher-Weil to estimate PCA duration then applied to fixed income portfolio
immunization. Jayathilaka, S. S. [16] also based on work of Cooper to develop a
duration immunization strategy on certain kind of fixed income portfolio.
A third class of widely used immunization models relies on the concept of key rate
duration (KRD) introduced by Ho (1992) [35]. These models explain yield curve shifts
based on a certain number of points along the curve the key rates and on linear
approximations based on time to maturity for the remaining rates.
Yield curve hedging techniques base on these duration models named immunization
approach. If we classify by the purpose of organization managing fixed income
portfolio we have net worth immunization and target date immunization, by managing
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style there are passive and contingent immunization. Here we will go through these
mentioned types.
1.2.3.1. Net Worth Immunization
Net worth immunization approach is adopted by institutions that would like to reduce
variation in their net market worth, e.g. banks. There may be regulatory enforcement
regarding to keep a certain level of net worth in some type of financial institution.
Generally, this objective can be broadly achieved by ensuring that the duration of the
asset portfolio equal the duration of the liability portfolio if the present value of the
assets equals the present value of the liabilities. Gap analysis is used in net worth
immunization. This approach is adopted in institutions like banks. (see APPENDIX C)
1.2.3.2. Target Date Immunization
Not just that the net worth remains relatively constant, but bank and other institutions
like pension fund, insurance company also concern that each promised payment
(liability) be made on the appropriate date. Obviously, if each liability is immunized,
all liabilities together are also immunized. However, if net worth is immunized, each
liability by itself might not be immunized, and it would be necessary to depend upon
higher asset revaluations overall to make up shortfalls in the funding for particular
liabilities, this leads to the target date immunization.
The return of an asset calculated over a horizon equal to its duration is immune to any
interest rate variation. With fixed rate assets, obtaining the yield to maturity requires
holding the asset until maturity. When selling the asset before maturity, the return is
uncertain because the price at the date of sale is unknown. If rates increase, prices will
decrease, and vice versa.
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Figure 1.3: Illustration of target-date immunization
The holding period return combines the current yield (the interest paid) and the capital
gain or loss at the end of the period. The total return from holding an asset depends on
the usage of intermediate interest ows. Investors might reinvest intermediate interest
payments at the prevailing market rate. The return, for a given horizon, results from the
capital gain or loss, plus the interest payments, and plus the proceeds of the
reinvestments of the intermediate flows up to this horizon. If the interest rate increases
during the holding period, there is a capital loss due to the decline in price,
simultaneously, intermediate flows benefit from a higher reinvestment rate up to the
horizon at a higher rate. If the interest rate decreases, there is a capital gain at the
horizon. At the same time, all intermediate reinvestment rates get lower. These two
effects tend to offset each other. There is some horizon such that the net effect on the
future value of the proceeds from holding the asset, the reinvestment of the
intermediate ows, plus the capital gain or loss cancels out. When this happens, the
future value at the horizon is immune to interest rate changes. This horizon is the
duration of the asset
time
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1.2.3.3. Passive and contingent immunization
A portfolio manager may have a minimum portfolio value in mind at a pre-specified
horizon point. He has a switching regime: fully immunizes the portfolio and attains
the required minimum portfolio value at the horizon point, this is a passive
immunization. Alternatively, if he currently has a greater sum of money that is
required in order to attain that minimum value, he may choose a more active
approach: contingent immunization.
Figure 1.4: The Contingent Immunization
a) Triggering case b) Non Triggering case
Contingent immunization started from the work of Leibowitz and Weinberger (1981,
1982 and 1983) [36] [37]. It can be considered as a midpoint between passive
immunization and active bond management strategies. Contingent immunization is a
stop loss strategy that allows portfolio managers to take advantage of their ability to
forecast interest rate movements as long as their forecasts are successful, but switches
to a passive immunization strategy should the stop loss limit be encountered.
Specifically, contingent immunization consists of forming a bond portfolio with
duration larger or smaller than the investors planning period depending on interest
V
Horizon
V
Horizon
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rate expectations. If the investor thinks that interest rates are going to rise more than
the market expects she would buy a bond portfolio with a duration D' smaller that her
planning period H and vice versa. However, if interest rates move opposite to the
investors expectations and the portfolio value falls to a given stop loss limit then she
would immunize and guarantee this lower limit for the eventual portfolio return. This
strategy gives contingent immunization an option like feature: limiting losses but
preserving upside potential if interest rates movements are favorable.
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CHAPTER 2: REDUCED-FORM YIELD CURVE MODELLING
2.1. Nonparametric approach: the Principal analysis
2.1.1. Statistical basis of Principal Component Analysis
Principal Component Analysis (PCA) techniques can be employed to explore the
variability and correlations of various yields. PCA input is a set of N correlated series;
the input will be decomposed into N uncorrelated components, which are called
factors. In order to do so, the covariance structure is computed and its eigen-structure is
produced. The eigenvectors with the largest eigenvalues point towards the most
important factors, and can be utilized to investigate which proportion of the variability
of the original series is explained by individual factors. Typically, we expect a set of
factors that will explain 90-95% of the total variability.
Also, yields are strongly auto-correlated through time. It makes then perfect sense to
work with the time-differenced series: in essence the factors will then explain changes
in the yield curve behavior through time, rather than the yield curve level. We will
denote with ( ; )y t the yield of bond with maturity , recorded at time 1 2, ,....t
Therefore, each yield change 1( ; ) ; )(j j jy t yy t is written as the weighted sum
of n factors [46]
11, , ,... ...
i nj j j j i j ny c f f f
The coefficients ,j i
are called factor loadings, and essentially determine the sensitivity
of yield jy to factor
if , and
jc is a constant
j jc yE . If we assume that the factors
are uncorrelated, and they are normalized with zero mean and unit variance, then we
can write the covariance of different yields as
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1
, ,( , )
b
j k j m k mm
yCov y
Therefore, if we denote with L the matrix that collects all factor loadings, then the
covariance matrix of the yield changes will be equal to
LL
Given that the covariance matrix is not singular, an eigen-decomposition will produce a
matrix V with the linearly independent eigenvectors, together with a diagonal matrix of
eigenvalues M, such that
1VMV
But as is symmetric, the eigenvectors form an orthonormal matrix 1V V , which
implies essentially that the factor loadings matrix can be expressed as
L V M
Using this representation we can write the yield changes in the terms of the elements of
the eigenvector matrix V and the eigenvalues in M
1 1 1, , ,... ...
j j j j i i i j n n nc v m f v m f v m fy
It is therefore intuitive that factors that are associated with higher eigenvalues will
contribute more to the total variability of the series. In particular, if we consider the
overall variance of all yields, then we can write as the sum of all eigenvalues.
2
1 1 1 1,
( )n n n n
j j i i jj j i j
Var y v m m
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18
Where the last equality follows from the fact that the eigenvectors are normalized to
unit length. In factor analysis we use only the largest n eigenvalues, and implicitly the
contributions of the rest as being independent across all maturities
1 1 1, , ,... ...
j j j j i i i j n n n jc v m f v m f v my f
The number of factors that we retain should be used as to make the variance of the
remainder component j
small. As a rule of thumb, n should ensure that at least 95%
of the total variance is explained by the corresponding factors. If one finds that such a
value of n is large compared to the total number of variables, it is evidence that factor
analysis might not appropriate for this case
1.1. Application to the Bank of England Data
To apply the model, I employ database of the England government bond nominal spot
rates, where it covers a time period starting from year 2000 to year 2013. There will be
50 evenly spaced maturities ranging from 0.5 to 25 year.
Data which comes as inputs to PCA should be stationary but data we are using here
will be like term structure of interest rates, prices or yields. Generally these factors are
non-stationary but the PCA is based on the stationary data, therefore these historical
data or original data must be transformed generally into returns. These returns should
be normalized before analyzing, if not the first principal component will be led by the
inputs with a greater instability.
We can normalize by estimate the z-score:
j j
ij
y yX
-
19
To test the stability of PCA result, PCs are determined using the daily changes of
interest rate under different moving windows such as one month (M1), three months
(M3), semi-annually (M6) and yearly (M12). The correlation matrix has been found
using these data with daily changes of interest rate in relevant moving windows.
1 Month 3 months 6 months 12 months
0.065 0.280 0.511 0.064 0.287 0.537 0.063 0.290 0.543 0.062 0.289 0.543
0.096 0.303 0.260 0.096 0.315 0.289 0.095 0.324 0.309 0.095 0.325 0.319
0.110 0.277 0.123 0.110 0.293 0.146 0.110 0.304 0.162 0.110 0.306 0.168
0.118 0.248 0.051 0.118 0.268 0.067 0.118 0.279 0.079 0.118 0.281 0.081
0.124 0.226 0.012 0.124 0.245 0.020 0.124 0.255 0.028 0.123 0.257 0.027
0.128 0.204 -0.019 0.128 0.224 -0.016 0.128 0.234 -0.009 0.128 0.235 -0.011
0.131 0.186 -0.036 0.132 0.205 -0.036 0.132 0.215 -0.033 0.132 0.216 -0.036
0.135 0.167 -0.054 0.135 0.185 -0.058 0.135 0.194 -0.057 0.135 0.197 -0.058
0.137 0.149 -0.071 0.138 0.166 -0.080 0.138 0.175 -0.081 0.138 0.177 -0.082
0.140 0.133 -0.076 0.140 0.149 -0.087 0.140 0.157 -0.090 0.140 0.159 -0.092
0.141 0.114 -0.085 0.142 0.130 -0.100 0.142 0.138 -0.105 0.142 0.139 -0.107
0.143 0.098 -0.090 0.144 0.112 -0.106 0.144 0.119 -0.113 0.144 0.120 -0.116
0.144 0.082 -0.096 0.145 0.096 -0.115 0.145 0.103 -0.124 0.145 0.104 -0.126
0.145 0.069 -0.101 0.145 0.081 -0.120 0.146 0.087 -0.129 0.146 0.088 -0.131
0.146 0.057 -0.104 0.146 0.068 -0.123 0.146 0.074 -0.133 0.146 0.075 -0.136
0.146 0.045 -0.101 0.147 0.056 -0.123 0.147 0.060 -0.133 0.147 0.061 -0.135
0.147 0.034 -0.101 0.147 0.043 -0.125 0.147 0.047 -0.136 0.147 0.047 -0.139
0.147 0.023 -0.101 0.147 0.031 -0.123 0.147 0.035 -0.134 0.148 0.036 -0.136
0.147 0.014 -0.100 0.148 0.022 -0.121 0.148 0.025 -0.133 0.148 0.025 -0.135
0.148 0.006 -0.096 0.148 0.012 -0.116 0.148 0.015 -0.128 0.148 0.015 -0.130
0.148 -0.003 -0.090 0.148 0.003 -0.110 0.148 0.005 -0.121 0.148 0.005 -0.123
0.148 -0.009 -0.080 0.148 -0.005 -0.099 0.148 -0.003 -0.110 0.148 -0.003 -0.112
-
20
0.149 -0.016 -0.073 0.149 -0.012 -0.091 0.149 -0.011 -0.101 0.149 -0.012 -0.103
0.149 -0.023 -0.068 0.149 -0.020 -0.084 0.149 -0.019 -0.093 0.149 -0.020 -0.095
0.149 -0.028 -0.061 0.149 -0.026 -0.076 0.149 -0.026 -0.084 0.149 -0.027 -0.086
0.149 -0.035 -0.050 0.149 -0.035 -0.063 0.149 -0.035 -0.071 0.149 -0.036 -0.072
0.149 -0.040 -0.043 0.149 -0.041 -0.054 0.149 -0.042 -0.060 0.149 -0.043 -0.062
0.149 -0.046 -0.034 0.149 -0.048 -0.043 0.149 -0.049 -0.049 0.149 -0.050 -0.050
0.149 -0.052 -0.023 0.149 -0.056 -0.029 0.149 -0.057 -0.033 0.149 -0.058 -0.034
0.149 -0.056 -0.015 0.149 -0.060 -0.019 0.149 -0.063 -0.022 0.149 -0.063 -0.023
0.149 -0.061 -0.007 0.149 -0.067 -0.009 0.149 -0.070 -0.011 0.149 -0.070 -0.011
0.148 -0.066 0.002 0.148 -0.073 0.003 0.148 -0.076 0.003 0.149 -0.076 0.003
0.148 -0.070 0.012 0.148 -0.078 0.014 0.148 -0.082 0.016 0.148 -0.083 0.016
0.148 -0.074 0.020 0.148 -0.083 0.022 0.148 -0.088 0.024 0.148 -0.088 0.024
0.147 -0.079 0.030 0.147 -0.090 0.035 0.147 -0.094 0.038 0.147 -0.095 0.039
0.147 -0.083 0.036 0.147 -0.094 0.044 0.147 -0.099 0.049 0.147 -0.100 0.049
0.146 -0.087 0.047 0.146 -0.099 0.056 0.147 -0.104 0.062 0.147 -0.105 0.063
0.146 -0.090 0.054 0.146 -0.103 0.065 0.146 -0.109 0.071 0.146 -0.110 0.072
0.146 -0.094 0.059 0.146 -0.107 0.072 0.146 -0.114 0.080 0.146 -0.114 0.081
0.145 -0.097 0.067 0.145 -0.111 0.083 0.145 -0.118 0.090 0.145 -0.119 0.092
0.145 -0.100 0.072 0.145 -0.115 0.090 0.145 -0.122 0.098 0.145 -0.122 0.100
0.144 -0.103 0.082 0.144 -0.119 0.100 0.144 -0.126 0.109 0.144 -0.126 0.110
0.144 -0.105 0.085 0.144 -0.122 0.105 0.144 -0.129 0.114 0.144 -0.130 0.116
0.143 -0.108 0.091 0.143 -0.125 0.112 0.143 -0.133 0.122 0.143 -0.133 0.124
0.143 -0.110 0.092 0.143 -0.128 0.116 0.143 -0.136 0.126 0.143 -0.136 0.128
0.142 -0.113 0.098 0.142 -0.131 0.122 0.142 -0.140 0.132 0.142 -0.140 0.135
0.141 -0.115 0.101 0.142 -0.134 0.125 0.142 -0.142 0.136 0.142 -0.142 0.139
0.141 -0.117 0.106 0.141 -0.136 0.131 0.141 -0.145 0.142 0.141 -0.145 0.145
0.141 -0.118 0.108 0.141 -0.138 0.134 0.141 -0.147 0.146 0.141 -0.147 0.149
0.140 -0.120 0.111 0.140 -0.140 0.139 0.140 -0.149 0.151 0.140 -0.149 0.154
Table 2.1: Principal Components for total period of all moving windows
Table 2.1 shows the principal components for total period of all moving window, The
first component eigenvector is shown in the first, forth, seventh and tenth columns.
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21
Exclude several first rows, the remaining values varies in a narrow band ranging from
0.13 to 0.153. Although this is relative flat in comparison with last two components.
Component Shape Max Min Standard Deviation
1 Flat 0.149 0.065 0.015
2 Steepness 0.303 -0.120 0.122
3 Curvature 0.511 -0.104 0.109
Table 2.2: Principal Component factor weight statistical description of 1 month moving
windows
From table 2.2, we can confirm this flat characteristic of first component through the
standard deviation of factor weights of each component as the first component has the
smallest one. And this result is similar to other moving window. The second
component factor weights almost monotonically decreases from 0.303 to -0.12.
Third principal component, factor weights are positive in the short maturity and when
the maturity dates goes up it decreases and the middle part of the term structure gives
negative values where it becomes positive in the longer maturity years. This third
principal component corresponds to the curvature. The above description is based on
the one month moving window. With three months or six months moving windows, the
value may slightly change but the shape of the PCs will not change. Three main
components of one month window are plot in figure 2.1, from this we can clearly see
the shape of these components.
The first component can be interpreted as the parallel shift of yield curve, hedging
against Factor 1 is therefore close to duration hedging, which is denoted as the level
factor. PC1 is the most important component to explain the term structure movements,
table 2.3 gives the average eigenvalue for each moving window in each year and table
2.4 is the correspondent percentage of variation explanation of each component. The
-
22
one month window eigenvalue of first component is 41.01 and it explains 88.01% of
the total variation.
Figure 2.1: Sensitivities of term structure with PCs
Maturity
1 month 3 months
1
2
3
1
2
3
2000 41.02 6.26 1.63 41.28 5.80 1.77
2001 40.71 7.03 1.26 40.90 6.76 1.30
2002 45.19 3.39 0.65 45.31 3.16 0.67
2003 43.98 4.45 0.91 43.79 4.36 1.17
2004 45.29 3.17 0.60 45.40 3.00 0.56
2005 45.78 3.04 0.59 46.09 2.77 0.55
2006 45.06 3.71 0.62 45.08 3.69 0.62
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23
2007 44.80 3.83 0.70 44.84 3.72 0.65
2008 44.16 4.60 0.83 44.19 4.56 0.76
2009 42.58 5.17 1.60 42.04 5.46 1.74
2010 44.68 3.61 1.00 44.41 3.75 1.00
2011 44.53 3.70 1.12 44.70 3.42 1.22
2012 44.30 3.26 1.32 44.44 2.94 1.27
Table 2.3: Eigenvalues for 1 and 3 month windows
Maturity
6 months 12 months
1
2
3
1
2
3
2000 41.17 5.80 1.90 41.61 5.47 1.62
2001 41.38 6.26 1.31 41.79 6.05 1.13
2002 45.40 2.99 0.72 44.94 3.53 0.63
2003 43.62 4.37 1.31 43.71 4.14 1.40
2004 45.46 2.96 0.52 45.19 3.19 0.55
2005 46.11 2.76 0.58 45.75 3.02 0.62
2006 45.26 3.52 0.65 44.91 3.76 0.67
2007 45.01 3.62 0.61 45.31 3.33 0.65
2008 43.93 4.77 0.78 43.47 4.81 0.98
2009 41.67 5.72 1.79 41.70 5.98 1.47
2010 44.31 3.76 1.05 44.79 3.31 1.03
2011 44.39 3.59 1.37 43.44 4.51 1.39
2012 44.11 3.11 1.33 43.93 3.19 1.34
Table 2.4: Eigenvalues for 6 and 12 month window
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24
M1 M2 M3 M4
2000 82.04 12.51 3.27 82.57 11.60 3.55 82.33 11.59 3.80 83.21 10.95 3.24
2001 81.41 14.07 2.51 81.79 13.52 2.60 82.77 12.52 2.63 83.58 12.10 2.27
2002 90.37 6.78 1.29 90.63 6.31 1.33 90.81 5.98 1.43 89.89 7.05 1.25
2003 87.96 8.90 1.81 87.57 8.72 2.34 87.25 8.74 2.62 87.41 8.28 2.81
2004 90.58 6.34 1.20 90.79 6.00 1.12 90.92 5.93 1.03 90.38 6.37 1.11
2005 91.57 6.08 1.19 92.17 5.55 1.10 92.23 5.51 1.16 91.49 6.04 1.23
2006 90.12 7.42 1.24 90.17 7.39 1.23 90.52 7.03 1.30 89.83 7.53 1.35
2007 89.60 7.67 1.40 89.68 7.44 1.31 90.03 7.25 1.23 90.63 6.65 1.31
2008 88.31 9.21 1.67 88.38 9.12 1.52 87.86 9.54 1.55 86.94 9.62 1.97
2009 85.17 10.33 3.19 84.08 10.93 3.47 83.34 11.44 3.58 83.40 11.95 2.93
2010 89.37 7.22 2.00 88.82 7.49 2.00 88.63 7.53 2.10 89.58 6.63 2.06
2011 89.07 7.41 2.24 89.40 6.85 2.43 88.78 7.18 2.73 86.88 9.03 2.78
2012 88.61 6.51 2.64 88.87 5.88 2.53 88.22 6.23 2.66 87.86 6.37 2.67
Table 2.5: Explanatory factor (%) of the first three principal components (6 month)
For the second principal component factor, steepness has opposite effects to the short
term maturity and long term maturity. As the value of the PC2 increases, the term
structure of short maturity increases and long term maturity decreases.
The third eigenvector corresponds to Curvature factor and the reason is that it has
reduced the middle term structure while increasing the short maturity and long maturity
term structures. Because of that, the curvature of the term structure depends on this
third principal component. The third principal component is relatively not very
important and the total variation explained by the PC3 is 3.27 percent for the total time
period concerned.
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25
Figure 2.2: Variation of the Eigenvalues with respective years (for all moving windows)
From figure 2.2, the eigenvalues of each component are almost stable through thirteen
years, with a slightly increase of the first component and decrease of the second one.
The third component is the most stable. This again confirms the importance of first
component. Any 3 factor model bases on these components should bear a good
analysis result. Immunization that bases on parallel assumption therefore is still
meaningful.
Figure 2.3 and 2.4 plot the first and second component average factor weight for 3
moving window. From this we can see that, the factor weight remain unchanged in any
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26
specific time frame. For the second component, the larger the moving window is, the
more the short-term rate increases, the long-term rate decreases. But in general, the
shapes of steepness factor are the same.
Figure 2.3: PC1 for all windows
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27
Figure 2.4: PC2 for all windows (Standardized data)
2.2. Parametric Approach: The Nelson Sigel model
2.2.1. Original model
We look into the two main variants of the model, namely the original formulation of
Nelson and Siegel (1987) [24], and the Dynamic one developed by Diebold Li (2006)
[9].
Nelson and Siegel (1987) suggested modeling the yield curve at a point in time as
follows: let ( )y be the zero rate for maturity , then
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28
1
21 3
1 1
st
nd rd2 component co
component
mponent3
ex / / )) / )
p( ) exp(( exp(
/ /y
Thus, for given a given cross-section of yields, we need to estimate four parameters:
1 2 3, , and . For m observed yields with different maturities 1 ,, m , we have m
equations. There is a simple strategy to obtain parameters for this model: x , and
then estimate the -values with Least Squares.
In this model, the yield y for a particular maturity is hence the sum of several
components.
In first component, factor loading on 1
is one, it is independent of time to maturity
and interpreted as the long-run yield level.
In second component, factor loading on 2
is 1 /e
/
t
t
, a function of time to maturity.
It starts at 1 but decays monotonically and quickly to 0, hence the influence of 2
is
only felt at the short-end of the yield curve, it can be consider as a short-term factor.
Factor loading on 3
is 1
//e
/
tte
t
and also a function of , but this function is
zero for 0 , increases, and then decreases back to zero as grows. It thus adds a
hump to the curve.
The parameter affects the weight functions for 2
and3
; in particular does it
determine the position of the hump. An example is shown figure 2.1.a to 2.1.c. The
parameters of the model thus have, to some extent, a direct (observable) interpretation,
which brings about the constraints1
0 , 1 2
0 . We also need to have 0 .
-
29
Figure 2.5a: Level. The left graph shows 1
3( )y . The right graph shows the
corresponding yield curve, in this case also 1
3( )y . The influence of 1
is constant
for all
Figure 2.2b: Short-end shift. The left graph shows 2
1 exp()
/ )(
/y
for
22 . The right graph shows the yield curve resulting from the effects of
1 and
2 , i.e.,
1 2
exp( /)
/
)(y
for
13 ,
22 . The short-end is shifted down by
2%, but then curve grows back to the long-run level of 3%.
2
0
4
0 5 10
Yie
ld in %
Component
2
0
4
0 5 10
Resulting yield curve
0
-2
4
5 10
Yie
ld in %
Component
2
0
4
0 5 10
Resulting yield curve
-
30
Figure 2.5c: Hump. The left graph shows 3
1 exp( / )exp( / )
/
for
36 .
The right graph shows the yield curve resulting from all three components. In all graphs, value
of is 2.
We plot three factors on figure 2.6 below:
Figure 2.6: factor loading with fixed 0 0609.
2
0
4
0 5 10
Yie
ld in %
Component
2
0
4
0 5 10
Resulting yield curve
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31
2.2.2. The dynamic Nelson-Sigel model
Recall the original model above:
1 2 3
1 1
/ //( )
/ /
e ey e
A dynamic version is required to understand the evolution of the bond market over
time. Hence Diebold and Li (2006) suggest allowing the coecients to vary over
time, resulting in:
1 2 3
1 1/ / /
/ /( )
t t t
e ey e
,
in which case they show that, given their Nelson-Sigel loadings, the coefficients may
be interpreted as time-varying level, slope and curvature factors.
We can use AR(1) process or VAR(1) model to specify the dynamic of these
coefficients. In matrix notation, both are formulated:
1( )
tt tX AX I A u N
Where
C is a constant and 0,tN Q
1
2
3
t
t
t
tX
,
1
2
2
( )
( )
( )
t
t t
t
N
,
3
1
2
u
For AR(1) model,
-
32
2
33
11
2
0 0
0 0
0 0
a
A a
a
,
2
2
3
11
2
3
2
2
0 0
0 0
0 0
q
Q q
q
VAR(1) model
11 12 13
21 22 23
31 32 33
a a a
A a a a
a a a
'Q qq where 22
31 32
1
3
21
3
10 0
0
q
q q q
q q q
2.2.3. State space model with Kalman Filter
The Kalman Filter is an iterative estimation algorithm designed to solve the problem of
estimating state variables of a linear dynamical system with unobservable data. This is
typically done by writing the model in terms of a linear state-space representation (or a
linear dynamical system). The state-space representation means formulating two
equations, a transition and measurement equation.[4]
The measurement is
0, ( , )t t ttR BX E E H
And the transition equation is
10( ( , )) ,
t t t tX AX I A u N N Q
Having specified the state-space representation, the focus turns to the algorithm. The
optimal estimator in a Kalman Filter is the conditional mean of tX independent on
information known up to time t-1 or t, denoted 1|t t
X
and |t t
X respectively.
-
33
Using the transition equation, the recursive prediction step can be calculated as, where
information up until time t-1 is known
1 1 1|
[ ] ( )t t t t tX E X AX I A u
Encumbered with mean square error matrix (again the predictive step)
| 1 1 | 1 | 1 1
t t t t t t t t t t
X X X X A A Q
Using the measurement equation these estimates can be improved by observing tR and
using its addition information (the update step)
1
1 1| | |[ ]
t t t t t t t t t tX X X X B BF v
11 1 1| | | | |t t t t t t t t t t tt t t tX X X BFX B
With the forecasting errors being
1|t t t tv R BX
With variance
1|( )t t t t
Var v F B HB
The Kalman Filter iterative process begins with 0X and
0 being set at the
unconditional mean and co-variance. We can estimate parameters by using Kalman
Filter Maximum-Likelihood. More about this method can be found in Appendix A.
2.3. Application to England Government Bond yield
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34
2.3.1. Data
Continue with the data from end of month England government bond yield but we will
sample a different maturity set: 1Y, 2Y, 3Y, 4Y 5Y, 6Y, 7Y, 8Y, 9Y, 10Y, 11Y, 12Y,
13Y, 14Y, 15Y, 16Y, 17Y, 18Y from 01/1990 to 12/2011, leading to 258 observations
of monthly yield.
Figure 2.7: Nominal Government Bond Yield Bank Of England
Figure 2.7 plot the yield surface of data, there is a general decrease in yield level from
1990 to 2011. The yield curve in general has shown a persistent flat shape through
years before the upward sloping trend emerges from 2006 2011. We can predict a
low variation of curvature factor.
-
35
Maturity
(Months) Mean
Standard
Deviation Min Max 1( ) 6( ) 12( ) 24( )
12 5.348 2.838 0.376 14.311 0.972 0.797 0.616 0.328
24 5.481 2.662 0.328 13.725 0.972 0.803 0.644 0.395
36 5.616 2.548 0.487 13.315 0.973 0.812 0.666 0.440
48 5.723 2.467 0.733 13.075 0.974 0.821 0.684 0.472
60 5.806 2.404 1.001 12.932 0.975 0.828 0.698 0.496
72 5.870 2.352 1.264 12.828 0.976 0.834 0.710 0.516
84 5.919 2.306 1.510 12.733 0.976 0.839 0.720 0.533
96 5.955 2.264 1.733 12.629 0.977 0.845 0.729 0.547
108 5.981 2.225 1.931 12.508 0.977 0.850 0.737 0.560
120 5.999 2.189 2.105 12.368 0.978 0.855 0.745 0.571
132 6.010 2.156 2.257 12.211 0.978 0.860 0.753 0.582
144 6.016 2.125 2.390 12.039 0.979 0.865 0.761 0.592
156 6.017 2.096 2.507 11.855 0.980 0.870 0.769 0.601
168 6.013 2.069 2.611 11.663 0.980 0.874 0.776 0.609
180 6.005 2.045 2.703 11.465 0.981 0.879 0.783 0.617
192 5.994 2.023 2.786 11.263 0.982 0.883 0.790 0.624
204 5.980 2.003 2.861 11.059 0.982 0.887 0.796 0.631
216 5.963 1.984 2.929 10.854 0.983 0.891 0.802 0.638
Level 5.963 1.984 2.929 10.854 0.983 0.891 0.802 0.638
Slope 0.615 1.777 -4.182 4.121 0.966 0.750 0.513 0.081
Curvature 0.301 0.695 -1.303 2.988 0.923 0.485 0.141 0.022
Table 2.6: Descriptive statistics for monthly yields at different maturities and for the
yield curve level, slope and curvature. The last three columns contain the sample
autocorrelations at lag 1, 12, 30 months.
From the statistical description, the major trend is upward sloping by the mean of
yields through years, and that the long rates are less volatile and more persistent than
short rates, and it is even more stable if we consider that long-term means are smaller
than short term ones, that the slope is less persistent than any individual yield but quite
-
36
highly variable relative to its mean, and the curvature is the least persistent of all
factors and the most highly variable relative to its mean.
2.3.2. Estimating Parameter
Initially, Nelson and Siegel use a grid of value for . Which each value of , we can
calculate the values of two loading. The estimation of beta coefficient will then become
linear regression.
In Diebold-Li work, they instead work with a unique value of that maximizes the
loading on curvature component. This not only makes the study ahead simple and
convenient, but also numerical trustworthiness, by enabling us to replace hundreds of
potentially challenging numerical optimizations with trivial least-squares regressions.
Of course, we must choose the most appropriate value of . Recall that determines
the maturity at which the loading on the medium-term or curvature, factor achieves it
maximum.
Moving window Mean (Year) Median (Year) Mode (Year)
1 month 6.3522 5 6
3 month 6.0791 7 6
6 month 6.2202 7 6
12 month 6.3202 8 7
Table 2.7: Statistical description of maturities minimizes curvature (third component).
PCA base on daily data of yields from 1990 2011
From PCA result, we can find the maturities that maximize loading on curvature on
each sample. Table 2.6 gives the statistical description of these values. We choose the
mean of these maturities of 1 month case. The value of lambda that maximizes loading
at 6 month is 42.2925.
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37
Using the empirical formula to estimate these feature of yield curve:
18( )tl y
18 1( ) ( )ts y y
18 12 ( ) ( ) ( )tc y x y y
Where tl , ts , tc are empirical level, slope, curvature at time t. In the curvature
empirical, x is a medium-term yield. We choose 5x . Then:
2 3
0.6762 + 0.0703tt t
s (2.1)
2 3
0 0026. 0.2786t t tc (2.2)
We use these empirical values as a benchmark to check whether 1 2 3, , t t t are
corresponding to the level, slope and curvature of model.
Mean Standard
Deviation
Min Max 1( ) 8( ) 20( )
6.0559 1.9393 2.9751 10.2630 0.9834 0.8995 0.7931
-1.0395 2.4980 -6.4450 5.1675 0.9653 0.7551 0.5313
0.7718 2.4491 -5.5818 8.9322 0.9306 0.5161 0.1934
Table 2.8: Descriptive statistic of 1 2 3, , t t t using monthly yield data from Jan1990
Dec2011, with t xed at 42.2925
-
38
Figure 2.8: Actual and fitted average yield curve.
Figure 2.9: Fitting selected day of yield curves.
-
39
From figure 2.8, 2.9, the model can replicate different shape of yield curve including
upward sloping, downward sloping, humped , except for period with many local
extremes.
Maturity
(Months) Mean Std.Dev Min Max MAE RMSE 1( ) 8( ) 20( )
12 0.042 0.091 -0.174 0.463 0.065 0.042 0.899 0.461 0.072
24 -0.028 0.062 -0.281 0.106 0.047 0.028 0.886 0.453 0.161
36 -0.034 0.085 -0.428 0.198 0.058 0.034 0.903 0.449 0.020
48 -0.027 0.066 -0.335 0.170 0.042 0.027 0.894 0.400 -0.057
60 -0.015 0.034 -0.176 0.075 0.022 0.015 0.861 0.290 -0.144
72 -0.002 0.015 -0.046 0.040 0.012 0.002 0.810 0.141 -0.058
84 0.010 0.034 -0.102 0.158 0.023 0.010 0.893 0.408 0.020
96 0.019 0.052 -0.142 0.259 0.034 0.019 0.893 0.415 -0.009
108 0.026 0.061 -0.150 0.312 0.040 0.026 0.890 0.412 -0.020
120 0.029 0.062 -0.133 0.323 0.041 0.029 0.888 0.410 -0.024
132 0.029 0.056 -0.099 0.296 0.038 0.029 0.887 0.409 -0.024
144 0.026 0.045 -0.058 0.238 0.031 0.026 0.886 0.408 -0.024
156 0.020 0.029 -0.028 0.158 0.022 0.020 0.883 0.399 -0.026
168 0.010 0.011 -0.011 0.069 0.011 0.010 0.833 0.305 -0.067
180 -0.002 0.016 -0.063 0.057 0.011 0.002 0.857 0.375 -0.034
192 -0.017 0.038 -0.192 0.084 0.025 0.017 0.888 0.422 -0.013
204 -0.034 0.062 -0.331 0.102 0.042 0.034 0.894 0.431 -0.009
216 -0.053 0.086 -0.476 0.113 0.061 0.053 0.898 0.437 -0.007
Table 2.9: The Descriptive Statistic of model Residual
The residual surface shows a turbulent in initial period, however, residual becomes
stable later without extreme value. We move on to consider the correlation between
empirical slope, curvature and level and estimated ones. Figure 2.11 a, b, c plot these
-
40
values in pairs and for the ease of comparison, we modified 2 31
, , according to
(2.1) (2.2) but omit the small term.
Figure 2.10: Residual surface
1t
2t
3t
tl 0.9560 0.0968 0.1262
ts 0.1028 -0.9956 0.0609
tc 0.3510 -0.3599 0.9467
Table 2.10: The correlation between empirical slope, level curvature and
1 2 3 , ,
-
41
The correlation between estimated component factors and empirical slope, level and
curvature, benchmark values we initially defined, are very high. Therefore the three
factors in our model correspond to level, slope and curvature.
Figure 2.11a: The empirical level (red) and estimated component factor 1 t (blue)
-
42
Figure 2.12b, c: The empirical slope and curvatures (red) and estimated component
factor 2 3 ,t t (blue)
-
43
2.3.3. Dynamic modeling
We use AR(1) to model the dynamic of component factors. To check the fitness of the
model, many aspects should be checked.
Recall table 3.4, from the autocorrelations of the three factors, we can see that the rst
factor is the most persistent, and that the second factor is more persistent than the third.
From figure 2.12 a, b, c, the AR(1) model succeeds to preserve this persistency
characteristic of the 3 component factors.
Figure 2.13a: Autocorrelation 1
-
44
Figure 2.12b: Autocorrelation 2
Figure 2.12c: Autocorrelation 3
-
45
Figure 2.14a: Autocorrelation of residual 1
Figure 2.13b: Autocorrelation of residual2
-
46
Figure 2.13c: Autocorrelation of residual2
Figures 2.13 a, b, c have shown residual autocorrelation of estimated level, slope and
curvature factors. The autocorrelations are very small, indicating that the models
accurately describe the conditional means of level, slope and curvature.
2.3.4. Forecast Interest rate
In this section, we will forecast yields using three factor NS model with underlying
dynamic component factors stylized by the AR(1), VAR(1) ARI(1,1). Besides, several
different methods will be employed to compare with results using NS model. These
methods are:
Random walk without drift
)( ( )t h ty y
-
47
The forecast is always no change
Direct regression on three AR(1) principal component.
The PCA will be carried on the yield data instead of yield change. We define iit t
x f y
i=1, 2, 3. Then we use a univariate AR(1) model to produce a h-step-ahead forecast of
the principal components:
, / i t h t i i it
c xx
, i=1, 2, 3
Then the forecasted yield can be produced as:
1 2 2 3 31 , // /, ,/ ( ) ( ) ( ) ( )t h t tt h t t h tt h
f x f xy x f
,
Where 1( )f is the element in the eigenvector
iq that corresponds to maturity .
AR(1), ARI(1) and VAR(1) on yields
These methods forecast directly yields.
Below is the result of forecast
Case Method Mean Std
dev AME RMSE ( )a ( )b
R-
square
1 y
ear
to m
aturi
ty -
1 m
on
th
fore
cast
(a=
1, b
=1
2)
RW -0.0692 0.2640 0.1477 0.2711 0.6748 0.1236 0.9830
PCA* -0.0270 0.2712 0.1519 0.2706 0.6760 0.0716 0.9831
DNSAR(1) -0.0437 0.2883 0.1662 0.2895 0.7021 0.0462 0.9806
DNSARI(1) 0.0597 0.2729 0.2004 0.2774 0.5944 0.0851 0.9822
DNSVAR(1) -0.1425 0.3184 0.2144 0.3468 0.7536 0.1408 0.9722
-
48
AR(1) -0.0855 0.2656 0.1536 0.2772 0.6776 0.1131 0.9820
VAR(1) -0.2565 0.3071 0.2700 0.3984 0.7563 0.2959 0.9616
1 y
ear
to m
aturi
ty -
3 m
on
th f
ore
cast
(a=
1, b
=1
2)
RW -0.2210 0.6823 0.3651 0.7124 -0.3055 -0.1869 0.8762
PCA -0.2027 0.6806 0.3840 0.7054 -0.2873 -0.2289 0.8786
DNSAR(1)* -0.2884 0.6878 0.4436 0.7411 -0.2405 -0.3000 0.8660
DNSARI(1) 0.0035 0.6846 0.4208 0.6796 -0.2859 -0.1807 0.8873
DNSVAR(1) -0.5787 0.7496 0.6390 0.9426 -0.0807 -0.1470 0.7832
AR(1) -0.2691 0.6770 0.3930 0.7239 -0.3037 -0.2139 0.8702
VAR(1) -0.7456 0.7574 0.7580 1.0588 -0.0710 -0.1370 0.7137
1 y
ear
to m
aturi
ty -
6 m
onth
fore
cast
(a=
1, b=
12)
RW -0.4788 1.0866 0.5924 1.1797 -0.1664 -0.1906 0.6162
PCA -0.5011 1.0494 0.6585 1.1556 -0.1813 -0.2324 0.6317
DNSAR(1) -0.6672 1.0451 0.8120 1.2331 -0.1388 -0.3366 0.5807
DNSARI(1)* -0.1142 1.1085 0.6766 1.1059 -0.1797 -0.1833 0.6627
DNSVAR(1) -1.2310 1.1345 1.2310 1.6681 0.0562 -0.1263 0.2326
AR(1) -0.5722 1.0562 0.6669 1.1941 -0.1674 -0.2238 0.6006
VAR(1) -1.4337 1.1349 1.4337 1.8231 0.1542 -0.0984 0.0383
1 y
ear
to m
aturi
ty
- 1
2 m
onth
fore
cast
(a=
1, b
=1
2) RW -1.0421 1.4510 1.0657 1.7765 -0.0961 -0.2169 -0.3063
PCA -1.1520 1.2495 1.1520 1.6917 -0.1941 -0.2004 -0.1847
DNSAR(1) -1.4090 1.2525 1.4175 1.8781 -0.3118 -0.1799 -0.4601
-
49
DNSARI(1)* -0.4008 1.4802 1.0516 1.5214 -0.0264 -0.2472 0.0419
DNSVAR(1) -2.4718 1.3514 2.4718 2.8116 0.0818 -0.3414 -2.2722
AR(1) -1.2165 1.3532 1.2165 1.8111 -0.1773 -0.1954 -0.3816
VAR(1) -2.6628 1.5035 2.6628 3.0516 0.0565 -0.2526 -3.0650
3 y
ear
to m
aturi
ty -
1 m
onth
fore
cast
(a=
6, b=
18)
RW* -0.0655 0.2509 0.1915 0.2576 0.3769 0.2578 0.9779
PCA -0.1369 0.2613 0.2330 0.2933 0.4347 0.2514 0.9714
DNSAR(1) -0.1935 0.2617 0.2715 0.3240 0.4699 0.2052 0.9651
DNSARI(1) -0.0853 0.2503 0.2050 0.2627 0.3910 0.1978 0.9771
DNSVAR(1) -0.2808 0.2622 0.3129 0.3829 0.4601 0.2577 0.9512
AR(1) -0.0753 0.2518 0.1931 0.2611 0.3817 0.2576 0.9770
VAR(1) -0.2626 0.2824 0.3036 0.3842 0.5249 0.3255 0.9487
3 y
ear
to m
aturi
ty -
3 m
onth
fore
cast
(a=
6, b
=1
8)
RW -0.2086 0.5619 0.4552 0.5955 -0.5291 -0.0295 0.8758
PCA -0.3061 0.5605 0.4999 0.6350 -0.5384 -0.0460 0.8588
DNSAR(1) -0.4487 0.5549 0.5814 0.7104 -0.4889 -0.1307 0.8233
DNSARI(1)* -0.1472 0.5609 0.4463 0.5759 -0.5287 -0.0433 0.8839
DNSVAR(1) -0.7058 0.5744 0.7548 0.9074 -0.4270 0.0155 0.7117
AR(1) -0.2382 0.5614 0.4666 0.6060 -0.5207 -0.0357 0.8694
VAR(1) -0.7576 0.6163 0.7928 0.9737 -0.3141 -0.0049 0.6524
3
yea
r
to
mat
uri
t
y -
6
mon
th
fore
cast
(a=
6,
b=
1
8)
RW -0.4571 0.8219 0.6714 0.9349 -0.5654 -0.0187 0.6547
-
50
PCA -0.5914 0.7965 0.7304 0.9871 -0.5696 -0.0370 0.6150
DNSAR(1) -0.8347 0.7819 0.9182 1.1396 -0.4587 -0.1769 0.4869
DNSARI(1)* -0.2761 0.8343 0.6346 0.8727 -0.5636 -0.0193 0.6991
DNSVAR(1) -1.3354 0.8049 1.3354 1.5560 -0.4007 0.0586 0.0434
AR(1) -0.5170 0.8115 0.6892 0.9569 -0.5557 -0.0232 0.6324
VAR(1) -1.4467 0.8298 1.4467 1.6646 -0.2315 0.0666 -0.1487
3 y
ear
to m
aturi
ty -
12 m
onth
fore
cast
(a=
6, b=
18)
RW -0.9758 0.7696 0.9786 1.2387 -0.0964 -0.3327 0.1456
PCA -1.1709 0.6983 1.1709 1.3602 -0.3213 -0.2391 -0.0303
DNSAR(1) -1.5294 0.7441 1.5294 1.6981 -0.4795 -0.0993 -0.6056
DNSARI(1)* -0.5570 0.8209 0.7139 0.9862 -0.1247 -0.3143 0.4584
DNSVAR(1) -2.4804 0.7339 2.4804 2.5849 -0.0070 -0.3967 -2.7207
AR(1) -1.0962 0.7478 1.0962 1.3234 -0.1590 -0.3174 0.0077
VAR(1) -2.6193 0.8505 2.6193 2.7517 -0.0042 -0.2603 -3.4463
5 y
ear
to m
aturi
ty -
1 m
on
th f
ore
cast
(a=
6, b
=1
8)
RW -0.0575 0.2416 0.1945 0.2467 0.2241 0.2222 0.9724
PCA -0.0885 0.2473 0.2061 0.2610 0.2332 0.2093 0.9691
DNSAR(1) -0.1610 0.2514 0.2448 0.2970 0.2805 0.1944 0.9600
DNSARI(1)* -0.0568 0.2405 0.1937 0.2454 0.2009 0.1869 0.9727
DNSVAR(1) -0.2400 0.2480 0.2755 0.3439 0.2580 0.1955 0.9464
AR(1) -0.0627 0.2421 0.1956 0.2484 0.2268 0.2220 0.9716
-
51
VAR(1) -0.2400 0.2639 0.2818 0.3553 0.3436 0.2458 0.9402 5 y
ear
to m
aturi
ty -
3 m
on
th f
ore
cast
(a=
6, b
=1
8)
RW -0.1837 0.4964 0.4414 0.5259 -0.5248 0.0266 0.8697
PCA -0.2327 0.4974 0.4572 0.5458 -0.5212 0.0173 0.8597
DNSAR(1) -0.4008 0.5005 0.5344 0.6383 -0.4245 -0.0542 0.8081
DNSARI(1)* -0.1121 0.5010 0.4273 0.5097 -0.5063 0.0304 0.8776
DNSVAR(1) -0.6338 0.5003 0.6876 0.8052 -0.4521 0.0885 0.6946
AR(1) -0.1996 0.4968 0.4467 0.5320 -0.5207 0.0256 0.8647
VAR(1) -0.6918 0.5327 0.7317 0.8707 -0.3864 0.0544 0.6261
5 y
ear
to m
aturi
ty -
6 m
onth
fore
cast
(a=
6,
b=
18)
RW -0.4046 0.7087 0.6278 0.8113 -0.6358 0.0486 0.6602
PCA -0.4776 0.7023 0.6527 0.8448 -0.6190 0.0292 0.6316
DNSAR(1) -0.7589 0.7028 0.8465 1.0307 -0.4115 -0.0964 0.4517
DNSARI(1)* -0.2297 0.7262 0.6074 0.7563 -0.6398 0.0597 0.7048
DNSVAR(1) -1.2136 0.6828 1.2231 1.3899 -0.5537 0.1336 0.0029
AR(1) -0.4381 0.7045 0.6334 0.8249 -0.6318 0.0519 0.6432
VAR(1) -1.3192 0.7070 1.3263 1.4942 -0.4152 0.1172 -0.2090
5 y
ear
to m
aturi
ty -
12
mo
nth
fore
cast
(a=
6, b
=1
8)
RW -0.8661 0.5967 0.8899 1.0489 -0.2491 -0.3482 0.2575
PCA -0.9729 0.5822 0.9824 1.1313 -0.3638 -0.2540 0.1363
DNSAR(1) -1.3861 0.6772 1.3861 1.5401 -0.3722 -0.0495 -0.6007
DNSARI(1)* -0.4858 0.6304 0.6254 0.7916 -0.2391 -0.3429 0.5771
-
52
DNSVAR(1) -2.2550 0.5274 2.2550 2.3148 -0.1716 -0.4062 -2.6161
AR(1) -0.9367 0.5880 0.9471 1.1033 -0.2914 -0.3241 0.1641
VAR(1) -2.3796 0.6363 2.3796 2.4618 -0.1625 -0.2332 -3.3130
18 y
ear
to m
aturi
ty -
1 m
onth
fore
cast
(a=
12, b=
24)
RW -0.0211 0.2350 0.1819 0.2343 -0.1321 0.1170 0.8867
PCA -0.0776 0.2501 0.2045 0.2601 -0.0877 0.0689 0.8604
DNSAR(1) -0.1320 0.2509 0.2167 0.2819 -0.0417 0.0947 0.8360
DNSARI(1) -0.0727 0.2440 0.1934 0.2529 -0.0340 0.0749 0.8680
DNSVAR(1) -0.1892 0.2346 0.2360 0.3001 -0.1369 0.0464 0.8142
AR(1)* -0.0178 0.2352 0.1821 0.2342 -0.1291 0.1184 0.8852
VAR(1) -0.1464 0.2316 0.2110 0.2726 -0.1271 0.0642 0.8397
18
yea
r to
mat
uri
ty -
3 m
onth
fore
cast
(a
=12,
b=
24)
RW -0.0693 0.3567 0.2961 0.3608 -0.2743 -0.2189 0.7377
PCA -0.1065 0.3688 0.3122 0.3813 -0.2807 -0.1739 0.7071
DNSAR(1) -0.2393 0.3898 0.3419 0.4550 -0.1117 -0.2072 0.5829
DNSARI(1) -0.0759 0.3713 0.3061 0.3763 -0.2197 -0.2016 0.7146
DNSVAR(1) -0.4092 0.3276 0.4294 0.5227 -0.3424 -0.1812 0.4494
AR(1)* -0.0594 0.3581 0.2958 0.3604 -0.2638 -0.2164 0.7343
VAR(1) -0.4180 0.3297 0.4407 0.5309 -0.3565 -0.1259 0.4054
18 y
ear
to
mat
uri
ty -
6 m
onth
fore
cast
(a=
12
,
b=
24
) RW -0.1643 0.4762 0.3648 0.5003 -0.3655 -0.1921 0.5052
PCA -0.1779 0.4962 0.3851 0.5235 -0.4031 -0.1452 0.4582
-
53
DNSAR(1) -0.4019 0.5346 0.5114 0.6655 -0.0366 -0.1977 0.1245
DNSARI(1) -0.1050 0.5002 0.3841 0.5073 -0.3041 -0.1750 0.4912
DNSVAR(1) -0.7370 0.4118 0.7454 0.8427 -0.5294 -0.1373 -0.4038
AR(1)* -0.1444 0.4794 0.3655 0.4971 -0.3404 -0.1872 0.5037
VAR(1) -0.7916 0.4229 0.7985 0.8960 -0.5533 -0.1312 -0.6650
18 y
ear
to m
aturi
ty -
12 m
onth
fore
cast
(a=
12, b=
24)
RW -0.3519 0.5376 0.4379 0.6387 -0.0876 -0.0601 0.2090
PCA -0.3289 0.5384 0.4355 0.6270 -0.0826 -0.1612 0.2376
DNSAR(1) -0.6564 0.6815 0.7093 0.9420 -0.0111 -0.0142 -0.7208
DNSARI(1)* -0.1586 0.5803 0.4511 0.5968 -0.0691 -0.0752 0.3094
DNSVAR(1) -1.3089 0.3573 1.3089 1.3560 -0.2333 -0.1665 -2.5658
AR(1) -0.3104 0.5510 0.4368 0.6284 -0.0619 -0.0594 0.2209
VAR(1) -1.3917 0.3860 1.3917 1.4434 -0.2387 -0.1438 -3.2604
Table 2.11: The forecasted result comparison of seven methods. 3 methods base on DNS
model with dynamic of factor component modeled by AR(1), VAR(1), ARI(1,1). Principal
Analysis, Random Walk and two direct method AR(1), VAR(1) forecast directly the yield
data.(Starred methods as ones with minimum RMSE in its group)
We will estimate and forecast recursively, using data from Jan-1990 to the time that the
forecast is made, beginning in Feb-2004 and extending through the end of database.
The forecasted error can be defined as:
/
t h t h tyE y
-
54
The forecast will carry on 16 cases of 4 maturities (1Y, 3Y, 5Y, 18Y) and 4 forecast
period (1, 3, 6 and 12 months)
Table 2.11 shows the statistical description of residual from seven methods. If we take
RMSE as a benchmark, DNS-ARI(1) is the best model. Among the DNS group
method, DNSVAR is the lowest performance one. This can be seen from the student
test of error mean (zero mean hypotheses). P-values of the test on every case in this
method are almost zero. However, even P-value of DNARI model is high; we should
Nelson Siegel as a bias model. This can be an inherent disadvantage of model itself:
the accuracy trade-off between two ends of yield curve. Small values of correspond
rapid decay in the regressors and therefore will be able to fit curvature at low maturities
well, while being unable to fit the excessive curvature over longer maturity ranges and
vice versa. However, as mentioned previously, we would not expect the high fitness
from a parsimonious model.
RW 1Y 3Y 5Y 18Y
1 month 0.0317 0.0324 0.0503 0.4554
3 months 0.0095 0.0032 0.0033 0.1137
6 months 0.0007 0.0000 0.0000 0.0071
12 months 0.0000 0.0000 0.0000 0.0000
Table 2.12 P-value of test on mean error of RW forecast.
0
1
0
0
E
E
H :
H :
PCA 1Y 3Y 5Y 18Y
1 month 0.4078 0.0000 0.0038 0.0115
3 months 0.0167 0.0000 0.0003 0.0201
6 months 0.0003 0.0000 0.0000 0.0052
-
55
12 months 0.0000 0.0000 0.0000 0.0000
Table 2.13: P-value of test on mean error of PCA forecast
DNAR 1Y 3Y 5Y 18Y
1 month 0.2086 0.0010 0.0000 0.0000
3 months 0.0000 0.0000 0.0000 0.0000
6 months 0.0000 0.0000 0.0000 0.0000
12 months 0.0000 0.0000 0.0000 0.0000
Table 2.14 P-value of test on error mean of DNS AR forecast
DNARI 1Y 3Y 5Y 18Y
1 month 0.0716 0.9667 0.4093 0.0420
3 months 0.0057 0.0340 0.0097 0.0000
6 months 0.0522 0.0694 0.0132 0.0000
12 months 0.0150 0.0965 0.0954 0.0402
Table 2.15: P-value of test on error mean of DNS ARI forecast
DirectAR 1Y 3Y 5Y 18Y
1 month 0.0088 0.0147 0.0336 0.5283
3 months 0.0017 0.0008 0.0015 0.1756
6 months 0.0000 0.0000 0.0000 0.0180
12 months 0.0000 0.0000 0.0000 0.0001
Table 2.16: P-value of test on error mean of Direct AR forecast
We check the stationary characteristic of beta coefficient estimated from data. The unit
root test shows that, the first two beta coefficient test results fail to reject the null
hypothesis. They may contain unit root and their first order of difference series may be
stationary. Therefore, the DNARI(1) model performs better than any other method.
-
56
Result p-value Test Statistic
1 0 0.1466 -1.4142
2 0 0.0718 -1.7780
3 1 0.0245 -2.2405
Table 2.17 Dikey-Fuller test on estimated beta coefficient. Critical value is 1.9420
at 5%
Figure 2.15: RMSE comparison of seven methods
:Direct VAR(1); :RW ; :Direct AR(1); :DNS-ARI(1); :DNS-AR(1);
:DNS-VAR(1); :PCA
-
57
If RMSE has given us a clear choice of optimal method, correlated residual tells a
different story. While DNS-VAR(1) shows a good forecast result, it does not come
with a small autocorrelation of residual. We cannot say firmly any optimal method
with low RMSE and correlated residual.
Figure 2.16: Residual Autocorrelation of different methods
:Direct VAR(1); :RW ; :Direct AR(1); :DNS-ARI(1); :DNS-AR(1);
:DNS-VAR(1); :PCA
-
58
CHAPTER 3: RISK SENSITIVITY AND IMMUNIZATION
In previous chapter, we have modeled and forecasted the yield curve with various
methods. A comprehensive management of yield curve risk must include solving the
last two fundamental problems mentioned in chapter 1: quantifying the interest rate
sensitivities of a given fixed-income portfolio and developing defensive risk
management strategies from a longer term model of yield curve movement with the
relaxed assumption: non-parallel of yield curve shift. I will introduce the most classic
model from Robert Reitano. This model is a natural way to deal with the problem of
non-parallel shift. In later section of this chapter, more methods will be introduced.
3.1. Risk sensitivity
To measure the sensitivity of fixed income portfolio to change in interest rate, shift in
the benchmark yield curve or any latent factors that belong to a factor model like PCA
or DNS in the non-parallel shift assumption, we use duration analysis are the main
tools.
3.1.1. Reitano Partial Duration model
3.1.1.1. Directional Duration and Convexity:
Let 0 01 02 0
, ,..., )(m
i i i i represents an m-point benchmark yield curve.
Let 1
( ,..., )m
N n n be a direction vector, 0N .
Price of bond after a shift t units in the direction N: ( ) ( )o
P t P i tN [20][21][22][23]
Again, we want to measure the change of portfolio value to the yield curve movement.
Assumed P(t) to be twice continuously differentiable and recall Taylor expansion in
(0.2), P(t) can be approximated to first and second order in t as follows:
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59
(t) (0) (0)tP P P (3.1)
21 2(0)(t) (0 (0)) t / tP P PP (3.2)
Let jP(i) denote the j-th partial derivative of P(i) and
jkP (i) denote the corresponding
mixed second-order partial derivative. We then obtain:
0(t) +tN(i )j jP n P (3.3)
0+tN(i )j jkkP Pnn (3.4)
Therefore:
0 0
0j j
PP
( n) i )iN
P(
(3.5)
2
0 020
j k jk( ) i )
N
PP n n P (i
(3.6)
Let
N
PD (i) / P(i)
N
(3.7)
2
2NC (i) / P(i)
N
(3.8)
Combine (3.5), (3.6) and (3.1), (3.2) and replace with (3.7), (3.8)
0 0 0
1N
iN) / P(i ) DP(i )(i (3.9)
20 0 0 0
1 21N N
iN) / P(i ) D (i )(P(i ) / C i(i ) . (3.10)
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60
We call N ND ,C the directional duration and directional convexity function in the
direction of N. If =(1,...,1)N , the parallel shift direction vector, DN(i0) and 0)(iNC can
be calculated like parallel shift approach.
Formulas (3.9) and (3.10) are consistent even though there are infinitely many ways to
specify the direction vector N, for example, given N, let N' = 1/2N. The corresponding
shift magnitudes satisfy: 2i i then 1/ 2N ND D and 1/ 4 N NC C . One can
normalize the model by requiring the direction vector N to satisfy |N|=1 . The
magnitude variable, i , is then uniquely defined as the length of the shift vector Ni .
3.1.1.2. Partial Duration and Convexity
Consider first and second order of Taylor expansion in m-dimensional versions:
0 0 0 jj ii) P(i )P(i ( iP ) (3.11)
0 0 0 0
1 2j j jk j k
i) P(i ) P(i ) iP(i / P i) i(i (3.12)
These approximations naturally motivate the following definitions:
The j-th partial duration function, denoted jD (i) , is defined for 0P( i) as follows:
1j j(i) P(i) / P(i), j,k , .,mD ..
The jk partial convexity function, denoted
1jk jk
(i) / P(i), j,C ( k ,.i) P ..,m
Given the above definition, the total duration vector, denoted D(i), and the total
convexity matrix, denoted C(i), are defined as follows:
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1 m
D(i) (D(i),...,D (i)) (3.13)
11 1
1
m
m mm
C (i) C ( i)
C( i)
C ( i) C ( i)
(3.14)
Then
0 0 0
1i) / P(i )P(i D(i ) i (3.15)
0 0 0 0
1 1 2i) / P(i ) D(iP(i ) i / i ii )C( (3.16)
We use as dot product or inner product.
3.1.2. Example
Assume a simple portfolio of three fixed cash flows:
Year Cash flow Spot rate
0 -75
1 20 0.105
2 15 0.1
3 25 0.1
4 30 0.09
5 15 0.085
Then we can calculate the price of portfolio at the spot rate vector:. At any spot vector
1 5( )i i ,...,i
(1+y )iii
i
CFP( i)
The partial derivatives:
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2
11
3
22
4
3 3
5
4 4
6
5 5
20 1
30 1
75 1
120 1
75 1
/ (y )P
/ (y )P
P / (y )
P / (y )
P / (y )
and
11 22 333 4 5
1 2 3
44 556 7
4 5
40 90 300
1 1 1
600 450
1 1
(y ) (y ) (y )
(y ) (y )
P ,P ,P ,
P ,P
At i0
1
2
3
4
5
2 974
4 0925
9 3011
14 1609
8 3469
D .
D .
D .
D .
D .
11 22 33
44 55
5 3829 11 1613 33 822
64 9582 46 1579
C . ,C . ,C . ,
C . ,C .
0 (i j)ijC .
Then, from (3.15):
0 01) ( )(( )i P i DP i i
Now for a shifted vector 0 0005 0 001 0 0002 0 001