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

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

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

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

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

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

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

1

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

2

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.

3

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.

4

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

5

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

6

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

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

8

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.

9

(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.

10

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

11

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

12

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.

13

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

14

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

15

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.

16

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

17

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

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.

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

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

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.

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

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

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

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

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

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.

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:

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)

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:

61

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:

62

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