1. VU University
Amsterdam
Master Thesis
The effectiveness of interest rate swaps
Evaluated by yield curve modeling
Author:
Roy Meekel (2131897)
Supervisor:
Dr. Norman Seeger
June 30, 2014
Abstract
This master thesis analyses the interest rate swap and uses yield curve simulation to evaluate
the effectiveness of an interest rate swap portfolio. A fictive pension fund is created with
certain liabilities. Because of a difference between the duration of the assets and liabilities,
interest rate risk arises. A swap portfolio is set up to hedge this risk. The effectiveness
of this swap portfolio will be evaluated by simulating 10,000 yield curves. In the model for
describing the yield curve, λ is assumted to be fixed. However, this study shows that the λ to
use depends on the maturities that are involved. By using λ=0.01, simulation results provide
no evidence that the interest rate swap is not working. By modifying the data with the UFR
method, volatility of zero rates with high maturities is reduced.
Keywords: Pension fund; Interest rate swaps; Interest rate sensitivity; Interest rate simulation;
Hedging.

2. Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Dutch pension system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Funding ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Type of pension agreements . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Financial Assessment Framework . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Ultimate Forward Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Asset Liability Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Data and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.1 Liability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.2 Interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2.1 Swap rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.2 Yield curve forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.4 Ultimate Forward Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Interest rate simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Influence of lambda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 Modified data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.1 Basis model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 Adjusted-lambda model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.3 Modified-data model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1

3. 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
A Liabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B Description of weights determination . . . . . . . . . . . . . . . . . . . . . . 66
C Notional values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
D Descriptive statistics modified data . . . . . . . . . . . . . . . . . . . . . . . 69
2

4. 1. Introduction
May 28, 2014; Reuters reports that Dutch banks may have mis-sold interest rate derivatives
to small and medium sized enterprises1
. Complex interest rate derivatives were sold as
hedging product, as protection against rising interest rates. Rabobank, ABN AMRO and
SNS Reaal provided about 90% of the derivatives. These contracts have a negative value,
because interest rates decreased over the last few years2
. This should not be a problem if the
interest rate swap is used for hedging, but can be, if the derivative only is used as protection
for rising interest rates.
Since 2007, pension funds have to value their assets and liabilities according the principles
of market valuation. This means that all investments and liabilities of pension funds will be
valued according the same principles. The interpretation of this concept is that all expected
future cash flows will be discounted against the current term structure of the risk free interest
rate (De Nederlandsche Bank, 2007). For investments (such as equities and bonds), prices
are directly observable in the market. The Dutch regulator (the Dutch Central Bank, DNB)
state that the discounted liabilities must be fully covered by the investments. Due to the
low interest rates on the market, liablities were overvalued. Therefore, pension funds had
too low funding ratios. To reduce the effect of interest rate changes, the ultimate forward
rate was introduced (De Nederlandsche Bank, 2012).
1
http://www.reuters.com/article/2014/05/28/netherlands-derivatives-idUSL6N0OE3J420140528
2
http://www.statistics.dnb.nl/index.cgi?lang=uk&todo=Rentes; Table T1.3.
3

5. Market valuation determines that the value of the discounted liabilities depends on the
current value of the interest rate. Over time, the term structure of the interest rate is not
constant, so this principle will cause interest rate risk for pension funds. To hedge this risk,
interest rate swaps were invented in the early 1980’s. Bicksler and Chen (1986) were one of
the first researchers who analyzed the interest rate swap. This study will analyze the effect
of adding interest rate swaps to the portfolio of a pension fund. The main research question
for this study will be:
What is the effectiveness of the use of interest rate swaps for pension funds to
manage the risk of interest rate changes, evaluated by yield curve simulation?
The changes of the interest rates can be simulated by a model that forecasts the yield curve.
For Asset-Liability Management studies, the Nelson-Siegel model is often applied to describe
a the yield curve. This model includes three factors, namely the level, slope and curvature
of the yield curve. This model will be used as a tool to estimate the parameters level, slope
and curvature of the yield curve on data ranging from 2003:12 until 2014:04. An vector
autoregressive model will be estimated on the estimated parameters to simulate yield curve
changes.
This study is different from previous research because it analyses the interest rate swap
as a stand alone financial instrument by use of yield curve simulation. The remainder of
this report is structured as follows. The second section addresses all relevant information
about the Dutch pension fund system, which types of systems are used in the Netherlands
4

6. and which regulation is relevant for Dutch pension funds. The third section will introduce
the interest rate swap. Relevant formulas and hedging principles are explained. The fourth
section divided into two subsections. The first subsection discusses the data that is used for
the analyses and the second subsection provides extensive methodology descriptions. Section
5 provides the estimated models. Section 6 comments on the results of the simulations with
the different models and section 7 concludes.
5

7. 2. Dutch pension system
The structure will be outlined, to explore the Dutch pension system (2.1). The health of
an pension fund depends on the funding ratio (2.2). There are different kind of pension
agreements in the Dutch pension sector(2.3). In the last decade, the Dutch Central Bank
introduced several major regulation changes (2.4).
2.1. Structure
In the Netherlands, the pension system is based upon three pillars, just like in many other
countries. The pension system’s three pillars are:
• The state pillar
• The supplementary pension pillar
• The private savings pillar
These three pillars combined is the total pension for a pensionable personKakes and Broeders
(2006, page 29 & 30). For each citizen, the benefits received from the three different pillars
depend on their personal situation. The state pillar is called the Algemene Ouderdomswet
(AOW) or the general old-age law, which is a pay as you go scheme. This means that the
benefits are mainly financed by the contributions that are paid by all employees in that same
period. This scheme was introduced in 1957.
6

8. Figure 2.1: This figure shows the value of the assets of pension funds as a percentage of the Gross
Domestic Product (GDP) for several European Countries. (Source: OECD)
The second pillar, the supplementary pension pillar is a mandatory collective agreement.
Usually, the employer pays the contribution, where after the employee receives pension rights.
Because of this compulsory participation, the participation rate is close to 100% (Kakes and
Broeders, 2006). A pension fund invests all contributions in, the most profitable way at the
lowest acceptable risk.
The third pillar is voluntarily and consists of private savings for retirement, such as
life annuities. This pillar is designed to fill the gap that arises when people switch jobs
between different companies. Their pension will be ’broken’, thus extra savings are required
to cover for the losses that have emerged. The Dutch pension system is compared with other
European countries substantial (see figure 2.1). This figure shows that pension funds in the
Netherlands manange assets worth 1.5 times GDP).
7

9. 2.2. Funding ratio
The health of a pension fund can be expressed as the funding ratio. The funding ratio can
be calculated with equation 1
FRt =
At
Lt
, (1)
where the At is equal to the value of the assets at time t and Lt is equal to the value of the
liabilities at time t (equation 2)
Lt =
T
i=t
CFi
(1 + yt:i)i−t)
. (2)
In this equation, CFi is the expected cash flow at time i, and yi:t is the zero rate with
maturity t. The funding ratio is the ratio between the value of the assets and the value of
the nominal liabilities. The funding ratio at time t+1 can be determined with equation 3
FR(t + 1) =
At × (1 + rA
t:t+1) − Bt+1 + Ct+1
Lt × (1 + it) × (1 + rL
t:t+1) − Bt+1 + PRt+1 + t+1
, (3)
where At is the current value of the assets, rA
t:t+1 is the return on the asset portfolio,Bt+1
are the paid benefits to pensioners, Ct+1 are the new contributions paid by the contributor
(usually the employer), Lt is the current value of the liabilities, it is the provided indexation,
rL
t:t+1 is the return on the liability portfolio, PRt+1 is the present value of the new pension
rights and t+1 is an error term, for example longevity risk.
8

10. Figure 2.2: This figure shows the evolution of the funding ratio in the period 2007 - 2012. On
the right hand scale is the value of the assets and liabilities presented, on the left hand scale the
funding ratio in percentage points. (Source: DNB)
The Dutch pension system is very sensitive for market movements, because of the enormous
second pillar, relative to GDP. In 2007, new regulations were introduced (De Nederlandsche
Bank, 2007), where liabilities have to be discounted against market rates. In those days, the
zero rate yield curve was declining. Lower interest rates caused pension funds to have a lower
funding ratio, compared to when the fixed discount rate was 4%. The decline of funding
ratio’s of the Dutch pension sector can be seen in figure 2.2. The figure shows that before
the start of the financial crisis, pension funds were quite healthy with an average funding
ratio of ±140%. In 2008, the value of the pension funds’ assets declined while the value of
the liabilities increased, due to the decreasing interest rates. This caused that the funding
ratio reached an all-time low of ±95%. During the next years, the funding ratio could not
recover because of low interest rates.
9

11. 2.3. Type of pension agreements
There are two main financing methods for the second pillar. Both are funded, but there is
a difference in what is guaranteed:
• Defined Benefit (DB)
• Defined Contribution (DC)
The defined benefit scheme is where contributions are variable and the pension rights are
’guaranteed’. The defined contribution scheme is where the contribution is fixed, but
the pension rights can vary, where as it depend on the performance of the pension fund.
The better the pension fund performs, the higher the return, the higher the benefits for
pensioners.
The DB scheme can be subdivided into an average salary scheme and an final salary
scheme. If for example an employee was for 40 years an active member of the scheme, and
she receives for each active year 1/60th
of her average salary, she receives 2/3rd
of her average
salary as pension. In the final salary scheme this will be the final salary instead of her average
salary.
In the Netherlands, the majority of the pension plans are DB schemes. According
numbers of the Pensioen Federatie (2010) 87% of all active members had an average salary
scheme.
In general, the average salary scheme has an indexation policy. In equation 3 is the it
parameter the granted indexation. Indexation means that the pension rights of all members
10

12. will be revised for inflation or wage increases. However, this indexation is not guaranteed,
because when it is not feasible for the pension funds’ financial situation to provide indexation,
it will not be applied. The real funding ratio (equation 4) corrects the funding ratio for
expected inflation, where the pension fund will always grant full indexation
FRr
=
At
Lr
t
. (4)
where the real liabilities are defined as equation 5
Lr
t =
T
i=t
CFi × (1 + πt:i)i−t
(1 + yt:i)i−t)
, (5)
where CFi is the expected cash flow at time i, yi:t is the discount rate observed at time t and
πt:i is the expected inflation. Therefore, the liabilities will be higher, and thus the funding
ratio lower.
The DC scheme is where the amount of the pension a person receives depends on
the returns that are made on the paid contributions. Also, the more a person pays as
contribution, the more pension is received. Therefore, the investment risk and interest rate
risk rest with the employee rather than the pension fund.
A third scheme is called the hybrid pension scheme or Collective Defined Contribution.
In those schemes the DB features are combined with the DC features. The pension rights
are based upon salary and how long a person participates in the scheme. Contributions are
fixed, but if it transpires that these contributions were insufficient, the pensions will be cut.
11

13. Due to new regulations, there is a movement from DB schemes where contributions are
variable and payouts are fixed towards DC schemes where contributions are fixed and payouts
are variable. (Ponds and van Riel, 2007) This movement can be explained by demographic
changes. People get older, so the pension funds have to pay more than expected (longevity
risk). The younger people have to pay for this. Those young people will invest more
individually rather than collectively. Therefore, the shift will continue towards an CDC
scheme (Kocken, 2007).
2.4. Regulation
In the Netherlands, The Dutch Central Bank (DNB) is the governor for pension funds.
DNB examines the financial position of pension funds and assesses whether a pension fund
is financially healty or not. DNB is also responsible for monitoring pension funds that
they comply with regulation. In the Netherlands, the ’pensioenswet’ is the law where all
regulation is stated.
2.4.1. Financial Assessment Framework
The Financial Assessment Framework (FTK) sets out the requirements for a pension fund,
regarding the financial situation. As stated in paragraph 2.2, the financial position is largely
reflected by the funding ratio. The FTK requires a minimum funding ratio of 105%. In
addition to this requirement, the pension fund has to hold sufficient liquidity buffers to
be able to cope with financial setbacks. This is on top of the 105%, so the final required
funding ratio has to be approximately 125%. The required buffer depends among others on
12

14. the investment risk and the average age of the members of the pension fund.
When a pension funds’ funding ratio is below the regulatory minimum ratio (the 125%
described above), a recovery play has to be drawn up. This recovery plan has to describe
how it will eliminate the reserve deficit. Only in extreme cases may the pension reduce
entitlements and rights.
2.4.2. Ultimate Forward Rate
In 2012, the Dutch government introduced the ’September pension package’ (Ministirie
van Sociale Zaken en Werkgelegenheid, 2013). This package was introduced to reduce the
sensitivity of liabilities on fluctuations in the interest rate. One of the measures that was
taken, was the introduction of the Ultimate Forward Rate (UFR). The UFR is a method to
move the interest rate for very long maturities towards a preagreed level. This preagreed
level is fixed at 4.2% in the euro area and is in line with the Solvency II directive for insurers
(De Nederlandsche Bank, 2012).
The zero rates that are used for discounting the liabilities can be derived from market
information when the market is liquid enough. Therefore, it is questionable whether the
interest rate generated out of the market information for long maturities is reliable, because
this interest rate is based on a few transactions in the market. The zero rate yield curve
that should be used for discounting the liabilities is published monthly by DNB.
13

15. 3. Asset Liability Management
Clark et al. (2006) state that the financial position of a pension fund depends on several
exogenous and three endogenous variables. Those exogenous variables are economic variables
such as interest rates and inflation. Endogenous variables are the policy decisions such as
contribution policy, indexation policy and investment policy. The study which policy is
used and what the effects are of those policies on the balance sheet is called Asset Liability
Management (ALM).
Since the early 1980’s is the use of interest rate swaps significantly increased (Bicksler
and Chen, 1986). Bicksler and Chen (1986) provide an economic analysis of the interest rate
swap and state that it is a useful tool for active liability management and that it is a helpful
hedging instrument against interest rate risk. They also state that interest rate risk arises
for firms in which the duration of assets does not match the duration of the liablities.
Duration is the measure of interest rate sensitivity of a fixed income investment and is
expressed as a number of years. The higher the duration, the more sensitive the fixed income
investment is for interest rate changes. The duration formula can be derived from the bond
pricing formula. For a zero coupon bond the price can be determined by equation 6
b(T, t) = e−r(T−t)
. (6)
14

16. Via Taylor’s rule, it can be obtained that the change of the bond price can be expressed as
equation 7
dBt
Bt
= rdt − (T − t)dr +
1
2
(T − t)2
(dr)2
. (7)
In equation 7 is (T −t) the duration and 1
2
(T −t)2
equals convexity. Convexity is the measure
of sensitivity of the duration of a bond. So if the interest rate changes, how much is will the
duration change. This will be smaller for more short term bonds and for small interest rate
changes.
Sun et al. (1993) declares an interest rate swap as an agreement between two institutions
in which each commits to make periodic payments. Payments are based on a predetermined
notional for a predetermined period, which is called the maturity of the interest rate swap.
A pension fund can enter into a pay fixed/receive floating swap or a pay floating/receive
fixed swap. The value of a fixed lag is much more sensitive to interest rate changes. With a
pay fixed/receive floating swap, duration is bought, while with a pay floating/receive fixed
swap, duration is sold. Figure 3.1 shows a diagrammed interest rate swap transaction.
Financial institutions, such as banks, often behave as an intermediary party. It arranges
contracts with both sides and asks a premium for this. They act as market makers, because
two companies are unlikely to contact the financial institution at the same time to arrange
a swap for a certain notional with the same maturity (Hull, 2011).
As stated above, an interest rate swap consists of two separate parts. A fixed part and
a floating part. The floating part can be seen as a floating rate bond. This is a bond where
15

17. Figure 3.1: This figure shows an interest rate swap agreement between a pension fund and an
investment bank. This is a pay fixed receive floating swap from the pension funds’ point of view.
the coupons depend on future interest rates. It can be shown that a floating bond is traded
at par value and the value of the swap after a coupon payment is again traded at par value.
The fixed part is a coupon bond. The coupon rate of the fixed part is called the swap rate.
This swap rate can be derived from the zero term structure (equation 8)
K =
P(T0, T0) − P(T0, Tn)
n
i=1 P(T0, Ti)
, (8)
where P(T0, Ti) stands for the discount factor of time i. The discount factor can be calculated
with equation 9
P(T0, Ti) =
1
(1 + Ri)i
, (9)
where Ri is equal to the zero rate with maturity i, and i is equal to maturity (in years).
16

18. 4. Data and methodology
This chapter provides the relevant data descriptions (4.1) and brief explanations on the
different applied methodologies (4.2).
4.1. Data
Various types of data is required for this study. First, a fictive pension fund has to be created,
so a liability distribution is generated (4.1.1). To discount the liabilities and for simulation
purposes, interest rates are required. The DNB provides the required interest rates (4.1.2).
4.1.1. Liability distribution
As stated by Bicksler and Chen (1986), the duration of assets and liabilities have to be
the same. Because pension funds have liabilities with long maturities, the duration of
the liabilities portfolio is higher than the duration of the asset portfolio. Data about the
liabilities is subtracted from a theoretical course called Institutional Investments and ALM.
A representative distribution of liabilities of a pension fund is shown in figure 4.1.
Appendix A provides all values of the liabilities in a table. The peak is around the
20-25 years to maturity. In this study, each cash flow is completely paid at the beginning
of the year. This is a big assumption but is still a reasonable approximation. A trade-off
between simplification of the research and influence on the conclusion is made. The asset of
the pension fund is only cash. The duration of cash is assumed to be zero, and creates the
biggest possible gap between duration of assets and duration of liabilities.
17

19. Figure 4.1: This figure presents the liability distribution of a typical pension fund. All values are
determined by the sum of expected future cash flows to each individual per year. Both build up
pension rights as well as mortality tables are used to determine the expected cash flows.
4.1.2. Interest rates
Interest rates that are used by Dutch pension funds to calculate the current value of their
liabilities, are subtracted from The Dutch Central Bank database. The DNB provides a
monthly updated zero yield curve with maturities from 1 to 60 years. Descriptive statistics
of the interest rates are provided in table 4.1. This zero curve is based upon European swap
rates for 1-10 year maturities (yearly intervals) and 12, 15, 20, 25, 30, 35, 40 and 50-year
maturities, listed by Bloomberg.
The intervening maturities are considered as less liquid and therefore not used. The swap
rate is exchanged against the 6-month EURIBOR and the curve is based on the lower bid
rate of Bloomberg bid/offer spread. The sample that is used contains data ranging from
December 2003 to April 2014.
18

20. Figure 4.2: This figure shows yield curves over time from a sample 2003:12-2014:04 at maturities
of 1-60 years.
Figure 4.2 provides a three-dimensional plot of the data available. From these zero curves,
theoretical swap rates are calculated. As stated above, swaps with maturities of 1, 2, 3, 4,
5, 6, 7, 8, 9 ,10, 12, 15, 20, 25, 30, 40 and 50 years are considered as liquid in the market.
Therefore, swap rates that will be calculated are for swaps with these maturities.
19

21. Maturity Mean St. dev. Min Max Obs Maturity Mean St. dev. Min Max Obs
1 2.21% 0.0143 0.34% 5.36% 125 31 3.81% 0.0086 2.21% 5.35% 125
2 2.34% 0.0136 0.41% 5.36% 125 32 3.81% 0.0085 2.22% 5.35% 125
3 2.51% 0.013 0.52% 5.28% 125 33 3.80% 0.0084 2.22% 5.35% 125
4 2.68% 0.0123 0.68% 5.19% 125 34 3.80% 0.0084 2.23% 5.35% 125
5 2.84% 0.0116 0.85% 5.11% 125 35 3.79% 0.0083 2.23% 5.36% 125
6 2.99% 0.011 1.04% 5.06% 125 36 3.79% 0.0083 2.24% 5.36% 125
7 3.12% 0.0105 1.20% 5.02% 125 37 3.78% 0.0083 2.25% 5.36% 125
8 3.24% 0.0101 1.37% 5.02% 125 38 3.78% 0.0082 2.25% 5.36% 125
9 3.34% 0.0098 1.52% 5.02% 125 39 3.77% 0.0082 2.25% 5.36% 125
10 3.44% 0.0095 1.66% 5.03% 125 40 3.77% 0.0082 2.26% 5.36% 125
11 3.52% 0.0093 1.79% 5.05% 125 41 3.77% 0.0081 2.27% 5.36% 125
12 3.60% 0.0091 1.90% 5.06% 125 42 3.76% 0.0081 2.28% 5.35% 125
13 3.66% 0.009 2.00% 5.08% 125 43 3.76% 0.008 2.28% 5.35% 125
14 3.72% 0.0088 2.09% 5.09% 125 44 3.76% 0.0079 2.29% 5.34% 125
15 3.77% 0.0087 2.16% 5.10% 125 45 3.76% 0.0079 2.30% 5.34% 125
16 3.79% 0.0088 2.20% 5.09% 125 46 3.76% 0.0079 2.30% 5.33% 125
17 3.81% 0.0088 2.23% 5.11% 125 47 3.75% 0.0078 2.31% 5.33% 125
18 3.83% 0.0088 2.23% 5.16% 125 48 3.75% 0.0078 2.31% 5.32% 125
19 3.85% 0.0088 2.24% 5.20% 125 49 3.75% 0.0077 2.32% 5.32% 125
20 3.87% 0.0088 2.25% 5.24% 125 50 3.75% 0.0077 2.32% 5.31% 125
21 3.86% 0.0088 2.25% 5.26% 125 51 3.75% 0.0077 2.32% 5.31% 125
22 3.86% 0.0088 2.25% 5.28% 125 52 3.74% 0.0077 2.32% 5.31% 125
23 3.86% 0.0088 2.24% 5.30% 125 53 3.74% 0.0076 2.32% 5.30% 125
24 3.86% 0.0088 2.24% 5.31% 125 54 3.74% 0.0076 2.31% 5.30% 125
25 3.86% 0.0088 2.24% 5.33% 125 55 3.74% 0.0076 2.31% 5.30% 125
26 3.85% 0.0088 2.23% 5.33% 125 56 3.74% 0.0076 2.31% 5.29% 125
27 3.84% 0.0087 2.22% 5.34% 125 57 3.74% 0.0075 2.30% 5.29% 125
28 3.83% 0.0087 2.21% 5.34% 125 58 3.74% 0.0075 2.30% 5.29% 125
29 3.83% 0.0087 2.21% 5.34% 125 59 3.74% 0.0075 2.30% 5.28% 125
30 3.82% 0.0086 2.20% 5.35% 125 60 3.73% 0.0075 2.30% 5.28% 125
Table 4.1: This table provides descriptive statistics of the zero rates for different maturities which
are provided by DNB. The sample period is 2003:12-2014:04.
4.2. Methodology
This section describes which methods will be used to answer the research question. Now the
pension fund is set up, all swap rates can be determined. Notional values of the swaps are
determined by use of duration matching (4.2.1). To model the interest rate or yield curve, a
model designed by Nelson and Siegel (1987) is used. Diebold and Li (2006) present in their
paper how to use the Nelson-Siegel model to forecast government bond yields. That paper
will be the basis for forecasting the zero curve of The Dutch Central Bank (4.2.2).
20

22. Possible unit root in the parameters forces the vector autoregressive model to be estimated
on differences instead of absolute values. Presence of cointegration between the estimated
parameters requires to add error correction terms into the model (4.2.3). The ultimate
forward rate is used by the DNB since 2012:09 to reduce volatility in the yield curve of
bonds with high maturities (4.2.4).
4.2.1. Swap rates
The latest zero-curve of the DNB is used to determine the swap rates. As noticed before, the
swap rate for time t can be calculated by equation 8. Because the zero rates are provided
by DNB, all swap rates can be determined (table 4.2).
Maturity Zero rate Discount factor Swap rate
1 0.40% 0.99604 0.40%
2 0.46% 0.99086 0.46%
3 0.59% 0.98245 0.59%
4 0.78% 0.96928 0.78%
5 0.99% 0.95203 0.98%
6 1.19% 0.93143 1.18%
7 1.38% 0.90846 1.36%
8 1.56% 0.88352 1.53%
9 1.72% 0.85741 1.68%
10 1.87% 0.83071 1.82%
12 2.12% 0.77735 2.05%
15 2.37% 0.70384 2.27%
20 2.55% 0.60447 2.43%
25 2.67% 0.51813 2.53%
30 2.81% 0.43609 2.64%
40 3.08% 0.29741 2.82%
50 3.28% 0.19886 2.93%
Table 4.2: This table provides swap rates for swaps with maturities of 1-10 years, 12, 15, 20, 25,
30, 40 and 50 years, based on the zero rates provided by DNB. Swap rates are determined on the
zero curve of 2014:04.
21

23. With table 4.2, the fixed lag payments of a swap position are determined. For the floating
part, forward rates are required. The price for a 2-year floating bond can be priced with
equation 10
FRB0 = EQ
0
R0,1
1 + R0,1
+
1 + R1,1
(1 + R0,2)2
, (10)
where R1,1 is the interest rate observed at time t = 1. In the risk neutral world, the
expectation of the future interest rate is the forward rate. The forward rate can be calculated
with equation 11
Ft−1,t =
(1 + Rt)t
(1 + Rt−1)t−1
− 1, (11)
where Ft−1,t is the forward rate between time t and time t − 1, Rt is the zero rate at time
t4, Rt−1 is the zero rate at time t − 1 for t = 1, 2, .., 60 and R0 ≡ 0. The forward rate can be
substituted into equation 10, so the equation of a 2 year floating rate bond can be rewritten
as equation 12
FRB0 =
R0,1
1 + R0,1
+
1 + F1,2
(1 + R0,2)2
. (12)
The value of a floating rate bond can be derived from equation 11 and equation 12. A floating
rate bond trades always at par value. As stated in chapter 3, the bond will always trade at
par value after a payment is made. Therefore, it only creates 1-year duration exposure. The
value of the swap is at initiation equal to zero, because the fixed part also trades at par value.
The notional value of the various swaps can be determined now. As stated by Bicksler and
Chen (1986), an interest rate swap can be used to absorb the difference between the duration
22

24. of the assets and liabilities. To measure interest rate sensitivity of a zero-coupon bond, the
basis point value (BPV) is used. A coupon bond can be seen as a set of zero-coupon bonds.
Therefore, a coupon bond will create duration exposure for all maturities. BPV measures
the change of the value of the bond for any yield change. Equation 13 determines the BPV
of liability i with maturity t.
BPVi =
M
(1+Rt)
∗ Bt
10.000
, (13)
where M is the maturity, Rt is the zero rate of time t and Bt is the value of the liability
at time t. BPV exposure for year t of swap i with swap rate K and maturity T can be
calculated with equation 14 where the notional is equal to 1.
BPVexposure =
KM
(1 + Rt)t
·
t
1+Rt
10000
, (14)
where KM is the swap rate of maturity M, Rt is the zero rate of time t. The BPV value of
the liability with maturity i has to be equal to the BPV value of the swap with maturity i.
Relevant for the 50 year swap are the BPV of all liabilities with maturities of 41-60 years,
and some of the BPV of liabilities with maturities 1-40. Relevant for the 40 year swap are the
BPV of the liabilities with maturities of 31-40 years, and some of the BPV of liabilities with
maturities 1-30, for the 30 year swap the BPV of the liabilities with maturities of 26-30 years,
and some of the BPV of liabilities with maturities 1-25, for the 25 year swap the BPV of the
liabilities with maturities of 21-25 years, and some of the BPV of liabilities with maturities
23

25. 1-20, for the 20 year swap the BPV of the liabilities with maturities of 16-20 years, and some
of the BPV of liabilities with maturities 1-15, for the 15 year swap the BPV of the liabilities
with maturities of 13-15 years, and some of the BPV of liabilities with maturities 1-12, for
the 12 year swap the BPV of the liabilities with maturities of 11-12 years, and some of the
BPV of liabilities with maturities 1-10. The 10-1 year swaps cover the BPV of the liability
with the same maturity as the swap and some of the BPV of the liabilities with maturities
of i − 1 where i is the swap maturity. All notional values are determined with equation 15
Notional =
BPVcoverage
BPVexposure
, (15)
Where BPVcoverage is the sum of relevant BPVi minus the BPV that is already covered by
swaps with higher maturities and BPVrelevant is the sum of the BPV exposure of the relevant
maturities. The notional values for the seventeen swaps are listed in table 4.3.
Value of the swap
It can be derived that the value of the floating part of the swap is equal to equation 16
VFloating = Ni ∗
(1 + R0,1)
(1 + R∗
0,1)
, (16)
where Ni is equal to the notional value of swap i, R0,1 is the 1-year zero rate at initiation
of the swap and 1 + R∗
0,1 a changed 1-year zero rate. The value for the fixed lag can be
24

26. Type of swap Notional
50 year swap e 1,798,291,327
40 year swap e 2,897,848,567
30 year swap e 1,985,280,281
25 year swap e 2,012,594,985
20 year swap e 1,799,456,998
15 year swap e 964,027,765
12 year swap e 563,908,799
10 year swap e 260,112,126
9 year swap e 239,942,027
8 year swap e 221,294,451
7 year swap e 201,625,799
6 year swap e 181,530,921
5 year swap e 160,582,818
4 year swap e 141,515,198
3 year swap e 123,024,321
2 year swap e 108,294,781
1 year swap e 102,598,780
Table 4.3: This table provides all notional values of each swap that is used in the hedge portfolio
of the fictive pension fund.
calculated with equation 17,
VFixed =
T
t=1
(Ni ∗ Ki)
(1 + Rt)t
, (17)
where Ni equals the notional value of swap i, Ki equals the swap rate of swap i, Rt equals the
t-year zero rate T is the maturity of the swap. So, if the yield curve changes, all zero rates
will change, and therefore the value of the swap will change. An upward shift of the yield
curve will cause a decreased value of the fixed leg. The value of the swap can be calculated
with equation 18
VSwap = VFixed − VFloating. (18)
25

27. 4.2.2. Yield curve forecasting
Diebold and Li (2006) provide in their paper a new method to forecast the yield curve.
This study tries to evaluate the interest rate swap by changes to the yield curve, so this
method should be applicable. They use the Nelson-Siegel model as basis and expand this
model into a model that can evolve dynamically over time, period-by-period. Diebold and
Li state in their introduction that interest rate point forecasting is crucial for bond portfolio
management (2006). The formula for the yield curve stated by Diebold and Li is expressed
as equation 19.
yt(τ) = β1t + β2t
1 − e−λtτ
λtτ
+ β3t
1 − e−λtτ
λtτ
− e−λtτ
, (19)
where β1t is the level of the yield curve, β2t is the slope of the yield curve, β3t is the curvature
of the yield curve, λt the exponential decay rate, for each maturity τ (in months).
A low value of λt produces a better fit of the curve for long maturities while large values of
λt better fit a curve for short maturities. Diebold and Li (2006) use a fixed number for this
parameter λt = 0.0609.
For each time t, the yield curve can be estimated using the three-factor model, described
in equation 19. This study uses the same value for lambda as stated above, When the
parameter λt equals 0.0609, the factor loadings can be calculated for each maturity. So
for each maturity, two regressors are available. By use of simple ordinary least squares
regression, the three different parameters {β1, β2, β3} can be estimated for each month t,
26

28. Figure 4.3: This figure shows the yield curve based on data (actual) and the yield curve based
on the model (fitted). The solid is the actual yield curve while the dotted line is the fitted yield
curve. The data that is used for this figure is of 2014:04.
which provides a time series of estimates of {β1, β2, β3}. Figure 4.3 shows an example of a
fitted yield curve. Table 4.4 provides descriptive statistics of the estimated factors.
Mean S.E. mean St. Dev. Minimum Maximum Obs
Level 3.9489 0.06857 0.7666 2.4716 5.5807 125
Slope -1.9003 0.13343 1.4918 -5.0100 0.4811 125
Curvature -1.9695 0.34735 3.8835 -10.1631 5.9818 125
Table 4.4: This table provides descriptive statistics of the estimated factors of the three-factor
model described above. The sample consists of data ranging from 2003:12-2014:04. Level represents
β1, Slope represents β2 and Curvature represents β3.
Based on the obtained estimations of {β1, β2, β3}, a Vector Autoregressive model (VAR) can
be estimated. To determine the amount of lags included in the VAR model, two different
information criteria are used. The first one is the Akaike Information Criterion (AIC),
developed by Akaike (1974). The value of AIC can be calculated with equation 20
AIC = 2k − 2 · ln(L). (20)
27

29. In equation 20 is L the maximum value for the likelihood function of the model and k is the
number of parameters in the model. The preferred model is the model with the lowest value
for the AIC.
The second information criterion is the Bayesian Information criterion, developed by
Schwarz (1978). The value of BIC can be calculated with equation 21
BIC = k · ln(n) − 2 · ln(L), (21)
where L is the maximum value for the likelihood function of the model, k is the number of
parameters in the model and n is the number of data points. The preferred model is again
the model with the lowest value for the BIC. Equation 20 as well as 21 show that a better
fitted model is rewarded, while there is a penalty for more parameters in the model. Based
on the two described information criteria, 2 lags are included. A vector autoregression based
on three variables with two lags can be written as equation 22.








β1t
β2t
β3t








=








C1
C2
C3








+








ϕ1
11 ϕ1
12 ϕ1
13
ϕ1
21 ϕ1
22 ϕ1
23
ϕ1
31 ϕ1
32 ϕ1
33








∗








β1t−1
β2t−1
β3t−1








+








ϕ1
11 ϕ1
12 ϕ1
13
ϕ1
21 ϕ1
22 ϕ1
23
ϕ1
31 ϕ1
32 ϕ1
33








∗








β1t−2
β2t−2
β3t−2








+








1t
2t
3t








, (22)
where the vector t ∼ N(0, Ω) and Ω =






σ2
1 σ1,2 σ1,3
σ1,2 σ2
2 σ2,3
σ1,3 σ2,3 σ2
3






.
28

30. As discussed by Lutkepohl (2005), matrix φ1 and matrix φ2 can be estimated by performing
linear regression on each equation. This produces the maximum likelihood estimates of the
coefficients, which can be used to calculate the residuals. The residuals can be used estimate
the cross-equation error variance covariance matrix Ω.
To use this model for simulation and forecasting, the error term has to be shocked.
Because of multiple correlated variables, the matrix Ω has to be decomposed by use of the
Cholesky decomposition. Matrix Ω is in such a way decomposed that the new matrix L is
a lower triangular matrix (all elements above the diagonal are equal to zero) and satisfies
equation 23
L · L = Ω. (23)
If the matrix xt consists of independent normals then this matrix is xt ∼ N(0, I). By
multiplying the decomposed matrix L with xt generates an variable t ∼ N(0, LIL ) which
is equal to t ∼ N(0, Ω).
4.2.3. Stationarity
This study tries to examine the influence of yield curve changes on the funding ratio of a
pension fund. To increase strength of the conclusion, a conclusion based on a model that
is estimated on stationary processes exceeds a conclusion that is based on a model that is
estimated on non-stationary processes. A stationary process is defined as where a time series
whose statistical properties such as variance, mean, autocorrelation, etc. are all constant
over time, i.e. the distribution of the values remains the same as time progresses.
29

31. One of the drawbacks of vector autoregressive models is that if one wishes to examine the
statistical significance of the estimated coefficients, it is essential that all of the components in
the VAR are stationary (Brooks, 2013, p. 292). Therefore, it is very important to determine
whether a time series is stationary or not, since this can influence the estimated vector
autoregressive coefficients. A non-stationary process is defined as a series that contains a
unit root. A non-stationary process can consists of a random walk, a trend, a cycle or
combinations of the three and therefore cannot, by definition, be modeled or forecasted.
For a stationary process, the order of integration is determined as 0, which is denoted
as I(0). The order of integration for non-stationary processes is denoted as I(d), where d is
the number of differences required to obtain a stationary series. An I(1) series contains one
unit root, which can wander a long way from its mean while an I(0) crosses its mean value
frequently (see figure 4.4). By differencing a I(1) series once, creates a new variable which
has no unit root (Brooks, 2013, p. 326).
Testing for unit root
Dickey and Fuller (1979) developed a procedure to detect a unit root in an autoregressive
model. This procedure tries to answer on the question whether the true data generating
process contains a unit root given the data sample. When the parameters from equation 19
are estimated, the presence of a unit root can be tested for the three time series.
The Dickey-Fuller test states under the H0 hypothesis that the series contains a unit root
versus H1 that the series is stationary. However, this test can only test for an autoregresive
model with 1 lag. The augmented Dickey-Fuller test can test for multiple lags. Therefore,
30

32. an augmented Dickey-Fuller test will be examined for the obtained estimations of {β1, β2,
β3}. Figure 4.4 shows a graphical representation of estimated parameters. Table 4.5 shows
the results of the augmented Dickey-Fuller tests up to 4 included lags. The hypotheses for
the test are:
H0: There is a unit root
H1: There is no unit root
For all parameters {β1, β2, β3}, H0 for can not be rejected. This indicates that there is no
evidence that the three time series are stationary. A Dickey-Fuller test is also carried out for
the differenced time series. New variables are created, {∆β1, ∆β2 and ∆β3} with equations
24, 25 and 26. The lower part of table 4.5 shows the results of the augmented Dickey-Fuller
tests for the differenced parameters.
∆β1t = β1t − β1t−1 (24)
∆β2t = β2t − β2t−1 (25)
∆β3t = β3t − β3t−1 (26)
For all parameters {∆β1, ∆β2, ∆β3}, H0 for can be rejected, which means that there is
evidence that the differenced time series are stationary.
31

33. Figure 4.4: This figure shows the time series of the parameters, {β1, β2, β3} and differenced
parameters {∆β1, ∆β2 and ∆β3} from 2003:12-2014:04. (a), (c) and (e) are all I(1) processes and
(b), (d) and (f) are all I(0) processes.
(a) β1 (b) ∆β1
(c) β2 (d) ∆β2
(e) β3 (f) ∆β3
32

34. 0 1 2 3 4
β1 0.2521 0.2217 0.2000 0.1624 0.1662
β2 0.6041 0.3286 0.3274 0.2167 0.2692
β3 0.8615 0.8105 0.7327 0.6806 0.8494
∆β1 0.0000 0.0000 0.0000 0.0000 0.0001
∆β2 0.0000 0.0000 0.0001 0.0000 0.0003
∆β3 0.0000 0.0000 0.0001 0.0002 0.0001
Table 4.5: This table provides p-values of the performed augmented Dickey-Fuller tests on the
parameters {β1, β2, β3} and the differenced parameters {∆β1, ∆β2, ∆β3}
Cointegration
In finance, many time series are non-stationary, but move together. Two variables that share
a common stochastic drift, are cointegrated variables. If the variables are all non-stationary,
a test for cointegration has to be carried out. If the time series are not cointegrated, a VAR
can be estimated based on the differenced time series. When cointegration exists between
the time series, error correction terms have to be included into the VAR model, which makes
it a restricted model. To test for cointegration, Johansen (1991) developed a procedure for
testing cointegration between several I(1) time series. There are two types of Johansen tests.
One based on a trace statistic and one based on the eigenvalue.
The H0 for the method based on the trace statistic is that r ≤ n where n is the number
of cointegration relations and H0 for the method based on the eigenvalue is r = n. If the
null hypothesis for r ≤ 0 is rejected, it is proven that there are no cointegration relationships
between the time series. For both the trace as the eigenvalue method, the H0 can be rejected
if the test statistic ≤ critical value.
There are three parameters, so a maximum of 2 cointegration relationships can be present.
33

35. Table 4.6 presents the results of the Johansen test. Both tests show that there is no evidence
for cointegration between the parameters and therefore, a vector autoregressive model based
on differenced time series can be estimated. To determine the number of lags included in the
new difference based (∆β1, ∆β2, ∆β3) VAR model, information criteria (equations 20 and
21) are used. Based on the information criteria, 1 lag will be included in the VAR model.
The new tri-VAR(1) model is described in equation 27.
r Trace statistic 5% Critical value 1% Critical value
0 17.1510* 29.68 35.65
1 6.9701 15.41 20.04
2 0.8975 3.76 6.65
r Max statistic 5% Critical value 1% Critical value
0 10.1809 20.97 25.52
1 6.0726 14.07 18.63
2 0.8975 3.76 6.65
Table 4.6: This table presents the results of the Johansen tests. The upper part of the table
presents the trace statistic procedure results while the lower part of the table presents the eigenvalue
procedure results








∆β1t
∆β2t
∆β3t








=








C1
C2
C3








+








ϕ1
11 ϕ1
12 ϕ1
13
ϕ1
21 ϕ1
22 ϕ1
23
ϕ1
31 ϕ1
32 ϕ1
33








∗








∆β1t−1
∆β2t−1
∆β3t−1








+








1t
2t
3t








, (27)
where the vector t ∼ N(0, Ω) and Ω =






σ2
1 σ1,2 σ1,3
σ1,2 σ2
2 σ2,3
σ1,3 σ2,3 σ2
3






.
34

36. 4.2.4. Ultimate Forward Rate
As noticed in section (2.4.2), the Ultimate Forward Rate was introduced in September 2012.
The introduction of the UFR has no effect on the zero rates for maturities up to 20 years, but
has effect for zero rates with maturities of 21 years or more. The zero rate will be adjusted
by extrapolating the underlying 1-year forward rate. This will be done until the ultimate
forward rate is reached. The set UFR of 4.2% is based on a long-term inflation expectation
of 2% and an expected long-term real rate of 2.2% (AG, 2012). The forward rate can be
calculated by equation 11. The adjusted 1-year forward rates F∗
t−1,t can be calculated by
equation 28.
F∗
t−1,t =



Ft−1,t 1 ≤ t ≤ 20
(1 − wt) · Ft−1,t + wt · UFR 21 ≤ t ≤ 60
UFR 61 ≤ t
, (28)
where Ft−1,t is the relevant forward rate, UFR is equal to 4.2% and wt is equal to the
weight at time t. These weights are based on the Smith-Wilson method (Smith and Wilson,
2001), proposed in Solvency II. The weights are fixed and the determination of the weights
is explained in appendix B. The new zero rates can be calculated with equation 29
(1 + z∗
t )t
=
t
j=1
(1 + f∗
j−1,j), (29)
where z∗
t is the adjusted zero rate.
35

37. 5. Interest rate simulation
In section 4.2, all relevant methodology is explained. This chapter will provide the outcomes
of the estimation of the different models. Section 5.1 provides the basic model. Section 5.2
will comment on the influence of lambda on the model. Section 5.3 provides a model based
on data that is modified according the UFR method.
5.1. Basic model
The tri-VAR(2) model expressed in equation 22 can be estimated by use of the data provided
by the DNB. Because of the presence of a unit root in the time series of the estimated
parameters, differences are calculated. Based on the differenced parameters, a tri-VAR(1)
model expressed in equation 27 can be estimated .
Equation 30 provides the estimated model will all factor coefficients. This VAR model
is estimated for λ = 0.0609, data ranging from 2003:12-2014:04 and the zero rates that are
included are maturities ranging from 1-60.








β1t
β2t
β3t








=








−0.01415
0.01806
−0.02855








+








0.1645 0.1194 0.0506
0.2697 0.4214 0.0236
0.7441 0.0622 0.1650








∗








β1t−1
β2t−1
β3t−1








+








1t
2t
3t








, (30)
where the vector ∼ N(0, Ω) and Ω =






0.0380 −0.0140 −0.0808
−0.0140 0.0992 −0.1557
−0.0808 −0.1557 0.8409






.
36

38. The notional values presented in table 4.3 are based upon the observed yield curve provided
by the Dutch Central Bank. If these notional values are used, and {β1t+1, β2t+1, β3t+1}
are equal to {β1t, β2t and β3t}, the present value of the liabilities is changed and the swap
portfolio has generated value, because the yield curve based upon {β1t+1, β2t+1, β3t+1} is
different than the real yield curve in time t.
If the notional values of the swaps are determined based upon the fitted curve of time t,
and there is no difference between {β1t, β2t, β3t} and {β1t+1, β2t+1, β3t+1}, the present value
of the liabilities will be the same and the swap portfolio has not generated value. If {β1t, β2t,
β3t} and {β1t+1, β2t+1, β3t+1} are equal, when the notional values based upon the real data
is used, the funding ratio of the portfolio with swaps is 100.5 % and the funding ratio of the
portfolio without swaps is then 101.5 %. This generates already an error at initiation, and
causes the funding ratio to be structurally too high. Therefore, for simulation, the notional
values based upon the fitted curve will be used (appendix C).
5.2. Influence of lambda
Diebold and Li (2006) use a fixed value for the parameter λ = 0.0609. This section tries to
reveal whether this lambda is optimal for this dataset. Diebold and Li arise in their paper
the question what a proper value for λ is. They state that λ determines the maturity at
which the loading on the medium-term factor, or curvature, reaches its maximum.
They use in their study bonds with maturities ranging from 3 months up to 10 years.
Diebold and Li state that the medium-term is defined as 2 to 3 years to maturity. This ends
37

39. Figure 5.1: This figure shows all adjusted R2 values of the regression yt(τ) = β1t +β2t
1−e−λtτ
λtτ +
β3t
1−e−λtτ
λtτ − e−λtτ , for λ ranging from 0.001-0.1 with steps of 0.001 and with maturities τ =
1-60.
up with a decay factor (λ) equal to 0.0609. This study uses maturities ranging from 1-60
years. Therefore, the medium-term is likely to be different from Diebold and Li (2006) and
thus the decay factor. The optimal lambda should be different from λ = 0.0609.
Figure 5.1 shows all adjusted R2
values for the regression yt(τ) = β1t + β2t
1−e−λtτ
λtτ
+
β3t
1−e−λtτ
λtτ
− e−λtτ
, for λ ranging from 0.001-0.1 with steps of 0.001 and with maturities
τ = 1-60. As can be seen in the graph, the adjusted R2
has the highest value if λ = 0.01 for
almost every sample in the time period 2003:12-2014:04.
Figure 5.2 shows the difference between the estimated yield curves based on different
lambda’s. Because {β1, β2 and β3} based on λ = 0.01 are different than {β1, β2 and β3}
based on λ = 0.0609 for each time time t, a different yield curve will be produced. The kink
38

40. Figure 5.2: This figure shows the estimated yield curve based on λ = 0.01 (dashed line), the
estimated yield curve based on λ = 0.0609 (dotted line), and the observed yield curve of 2014:04
(solid line).
in the beginning of the estimated yield curve with λ = 0.0609 does not longer exists in the
estimated yield curve based on λ = 0.01. The difference of the adjusted R2
values can be
explained by figure 5.3. As can be seen in the graph, the factor loading of the curvature
reaches it maximum when t = 15. This is 25% of 60 which is the same as in the Diebold
and Li paper, namely 2.5 years of 10 years (highest maturity used).
When a different lambda is used, the explanatory variables are different, but the controlled
variable will be the same (observed yield curve does not change). Therefore, the estimated
{β1t, β2t, and β3t} will be different. Table 5.1 provides descriptive statistics of the estimated
factors based on λ = 0.01. Because {β1t, β2t, and β3t} are different, a new model with
different coefficients in the tri-VAR(1) model will be estimated.
39

41. Mean S.E. mean St. Dev. Minimum Maximum Obs
Level 3.4589 0.0870 0.9732 1.4176 5.0653 125
Slope -1.6643 0.1138 1.2720 -3.7419 1.0147 125
Curvature 3.4318 0.2173 2.4300 -0.2617 8.8278 125
Table 5.1: This table contains descriptive statistics of the estimated factors of the three-factor
model described in equation 19. The sample consists of data ranging from 2003:12-2014:04. Level
represents β1, Slope represents β2 and Curvature represents β3.
The augmented Dickey-Fuller test is used test for stationarity in the new estimated parameters.
Again, the hypotheses for the test are:
H0: There is no unit root
H1: There is a unit root
Table 5.2 presents the outcome of the augmented Dickey-Fuller test. For all parameters {β1,
β2, β3}, H0 for can not be rejected, regardless the lags included. This indicates that there is
no evidence that there is no unit root present in the three time series. To create stationary
time series, the obtained parameters will be differenced once. The lower part of table 5.2
shows that H0 can be rejected and that the differenced time series are stationary.
0 1 2 3 4
β1 0.3211 0.3089 0.2649 0.1733 0.2070
β2 0.7899 0.7607 0.5798 0.3666 0.6822
β3 0.7291 0.5921 0.5794 0.3711 0.4380
∆β1 0.0000 0.0000 0.0000 0.0000 0.0001
∆β2 0.0000 0.0000 0.0001 0.0000 0.0003
∆β3 0.0000 0.0000 0.0001 0.0002 0.0001
Table 5.2: This table provides p-values of the performed augmented Dickey-Fuller tests on the
parameters {β1, β2, β3} and the differenced parameters {∆β1, ∆β2, ∆β3}
40

42. Johansen’s test is used to detect possible cointegration between the estimated paratmeters.
Based on the results of Johansen’s test (table 5.3) can be concluded that there is no
cointegration between the differenced parameters. Therefore, no error correction terms are
required in the vector autoregressive model. The model with the estimated coefficients based
on λ = 0.010 is expressed in equation 31.
r Trace statistic 5% Critical value 1% Critical value
0 17.9504* 29.68 35.65
1 8.0130 15.41 20.04
2 0.9658 3.76 6.65
r Max statistic 5% Critical value 1% Critical value
0 9.9374 20.97 25.52
1 7.0472 14.07 18.63
2 0.9658 3.76 6.65
Table 5.3: This table presents the results of the Johansen tests. The upper part of the table
presents the trace statistic procedure results while the lower part of the table presents the eigenvalue
procedure results








β1t
β2t
β3t








=








0.00859
−0.00034
−0.02867








+








0.1043 0.1580 −0.0291
0.3363 0.2425 0.0038
−0.0224 −0.1278 0.1839








∗








β1t−1
β2t−1
β3t−1








+








1t
2t
3t








(31)
where the vector t ∼ N(0, Ω) and Ω =






0.0673 −0.0496 −0.0740
−0.0496 0.0723 0.0199
−0.0740 0.0199 0.3051






.
41

43. Figure 5.3: This figure shows the factor loadings for the model yt(τ) = β1t + β2t
1−e−λtτ
λtτ +
β3t
1−e−λtτ
λtτ − e−λtτ . Part (a) shows the factor loadings for λ = 0.0609, while part (b) shows the
factor loadings for λ = 0.010. The factor loadings are 1, 1−e−λtτ
λtτ and 1−e−λtτ
λtτ − e−λtτ for β1t,
β2t and β3t, respectively.
(a) factor loadings with λ = 0, 010 (b) factor loadings with λ = 0, 010
5.3. Modified data
The introduction of the UFR in 2012 amended the yield curve from 2012:09. This section
estimates a new model, based on amended yield curves from 2003:12. All yield curves will
be adjusted according the method described in paragraph 4.2.4 and with weights determined
in appendix B. Assumed is, that because of equal maturities involved, the influence of λ is
equal to the previous paragraph (4.2.3). Therefore, for this model is λ, as well as the model
in paragraph 4.2.3, equal to 0.01. The data modified ranges from 2003:12-2012:08. This is
because the DNB adjusted the yield curves beyond 2012:09 already. Appendix D provides
the descriptive statistics of the modified data. The zero rates of maturities 21-60 show a
higher average but a smaller standard deviation.
42

44. Figure 5.4: This figure shows the observed yield curve (solid line), the amended yield curve
(dotted line) and the estimated yield curve (dashed line), for time sample 2012:08.
Based on the new yield curves, {β1t, β2t, β3t} can be estimated for each time t. Now the
controlled variables will be different, but the explanatory variables will be the same as in
paragraph 5.2. Table 5.4 provides descriptive statistics of the estimated β1t, β2t and β3t.
Mean S.E. mean St. Dev. Minimum Maximum Obs
Level 3.4589 0.0870 0.9732 1.4176 5.0653 125
Slope -1.6643 0.1138 1.2720 -3.7419 1.0147 125
Curvature 3.4318 0.2173 2.4300 -0.2617 8.8278 125
Table 5.4: This table contains descriptive statistics of the estimated factors of the three-factor
model described in equation 19. The sample consists of modified data ranging from 2003:12-2014:04.
Level represents β1, Slope represents β2 and Curvature represents β3.
Figure 5.4 presents the estimated yield curve as well as both the observed yield curve and
the amended yield curve. As can be seen, the UFR induces the yields for higher maturities
to increase. The estimated yield curve based on the new yield curve fits the new yield curve
better than the yield curve without the modification. Again, these series are likely to contain
43

45. a unit root. Therefore the augmented Dickey-Fuller test is used to test for stationarity, with
hypotheses:
H0: There is a unit root
H0: There is no unit root
Table 5.5 presents the outcome of the augmented Dickey-Fuller test. For all parameters {β1,
β2, β3}, H0 for can not be rejected. Therefore, there is no evidence that there is no unit
root in the three time series. Again, the obtained parameters are differenced once. These
time series are also tested for stationarity and based on the lower part of table 5.5 can be
concluded that the differenced parameters are stationary.
0 1 2 3 4
β1 0.6077 0.4683 0.4292 0.3383 0.3634
β2 0.9128 0.7864 0.6257 0.4889 0.7966
β3 0.2211 0.1526 0.1473 0.1041 0.1175
∆β1 0.0000 0.0000 0.0000 0.0001 0.0000
∆β2 0.0000 0.0000 0.0005 0.0000 0.0003
∆β3 0.0000 0.0000 0.0000 0.0000 0.0001
Table 5.5: This table provides p-values of the performed augmented Dickey-Fuller tests on the
parameters {β1, β2, β3} and the differenced parameters {∆β1, ∆β2, ∆β3}
To test for cointegration relationships between the three parameters, Johansen’s test is used.
Table 5.6 presents the results of the test and there can be concluded that there are not
cointegration relationships between the parameters.
44

46. r Trace statistic 5% Critical value 1% Critical value
0 21.8578* 29.68 35.65
1 9.6440 15.41 20.04
2 1.6696 3.76 6.65
r Max statistic 5% Critical value 1% Critical value
0 12.2138 20.97 25.52
1 7.9744 14.07 18.63
2 1.6696 3.76 6.65
Table 5.6: This table presents the results of the Johansen tests. The upper part of the table
presents the trace statistic procedure results while the lower part of the table presents the eigenvalue
procedure results
The estimated model based on the modified data is expressed as equation 32,








β1t
β2t
β3t








=








−0.0028
−0.0026
−0.0446








+








0.1913 0.1829 −0.0092
0.9898 0.3142 0.0236
−0.9798 −0.2277 0.0787








∗








β1t−1
β2t−1
β3t−1








+








1t
2t
3t








(32)
where the vector t ∼ N(0, Ω) and Ω =






0.0041 −0.0003 −0.0059
−0.0003 0.0331 −0.0342
−0.0059 −0.0342 0.3252






.
45

47. 6. Results
The purpose of the interest rate swap is to provide a stable funding ratio. Two test can
be done to prove this argument. The first test is a two-sided t-test for the mean (Berenson
et al., 2011, chap. 9). If the calculated tSTAT (equation 33) exceeds the critical tSTAT , the
H0 hypothesis can be rejected.
tSTAT =
¯X − µ
S√
n
, (33)
where the tSTAT test statistic follows a t distribution having n − 1 degrees of freedom, ¯X
equals the average funding ratio of the portfolio with the swaps included, µ equals the
population mean (H0: µ = 100%), S is the standard deviation of the sample and n is the
number of simulations.
The second test is a paired t-test for the mean difference (Berenson et al., 2011, chap.
10). If the calculated tSTAT (equation 34) exceeds the critical tSTAT , the H0 hypothesis can
be rejected.
tSTAT =
¯D − µD
SD√
n
, (34)
where the tSTAT test statistic follows a t distribution having n − 1 degrees of freedom, ¯D
is the average absolute difference between the funding ratio of the portfolio with swaps and
the funding ratio of the portfolio without swaps (equation 35), µ equals the difference of the
means (H0: µD = 0). S is the standard deviation of Di and n is the number of simulations.
46

48. ¯D =
n
i=1 Di
n
, (35)
where Di is calculated with equation 36 and n is the number of simulations.
Di = |FRwithout − 1| − |FRwith − 1| (36)
where FRwithout is the funding ratio of simulation i without swap in the portfolio and FRwith
is the funding ratio of simulation i with swaps in the portfolio.
Each model produced 10,000 simulations. Simulation i changes the yield curve which
creates a value for the swap portfolio and changes the present value of the liabilities. The
funding ratio of the portfolio with swaps is determined by the value of cash, the value of
the swap portfolio and the value of the liabilities. The funding ratio of the portfolio without
swaps, which is determined by the value of cash, divided by the value of liabilities. The
absolute difference is then determined by equation 36.
Sections 6.1, 6.2 and 6.3 provide the results of the basis model, the model based on an
adjusted λ, and the results of a model based on modified yield curves, respectively. For all
models, the initial value of the assets is equal to the present value of the liabilities and the
value of the swap portfolio is equal to 0.
47

49. 6.1. Basis model
Section 5.1 states that at initiation an error is included if the model will use a swap portfolio
that is based on the real yield curve. With a separate-variance t-test (equation 37) is tried
to prove that this is a correct assumption.
tSTAT =
FR
E
− FR
R
− (µE − µR)
SE
2
nE
+ SR
2
nR
, (37)
where FR
E
equals the average funding ratio based on a model where the notional values are
determined by the estimated yield curve and FR
R
equals the average funding ratio based
on a model where the notional values are determined by the real yield curve. (µE − µR) is
equal to 0 and SE
2
and SR
2
is the variance of the sample, respectively. nE and NR are equal
to 10,000. The hypotheses for this test are:
H0: FRE
− FRR
≥ 0
H1: FRE
− FRR
< 0
The critical tSTAT for the 1 % significance level is equal to 3.0906. The calculated tSTAT from
equation 37 is 223.776. Therefore, the null hypotheses can be rejected. There is evidence
that FRE
− FRR
< 0. Therefore, further results will concentrate on simulations based on
notional values determined by the estimated yield curve.
The model as in equation 30 is used for simulation of the basis model. A three month
forecast is produced with each simulation, where the funding ratio is stored for all three
48

50. months. In each simulation, the simulated {β1t+1, β2t+1, β3t+1} are used as input for the
next month in equation 30. Table 6.1 provides descriptive statistics of 10,000 simulations for
the 1-month, 2-month and 3-month forecast when notional values are based on the estimated
yield curve.
Mean Standard Error Standard Deviation Minimum Maximum Obs
F R1
1 100.04% 0.00002 0.0016 99.45% 100.69% 10000
F R1
2 99.79% 0.00027 0.0271 89.12% 110.07% 10000
D1
2.05% 0.00016 0.0162 -0.40% 10.72% 10000
F R2
1 100.07% 0.00003 0.0029 98.83% 101.21% 10000
F R2
2 99.69% 0.00041 0.0407 83.54% 114.53% 10000
D2
3.02% 0.00024 0.0243 -0.79% 16.10% 10000
F R3
1 100.08% 0.00004 0.0042 98.53% 101.90% 10000
F R3
2 99.57% 0.00052 0.0520 80.97% 120.77% 10000
D3
3.82% 0.00031 0.0312 -1.05% 18.87% 10000
Table 6.1: This table presents descriptive statistics of the funding ratio with and without swaps
and of the absolute difference between the portfolios simulated by the basis model. FRt
1 equals
the funding ratio of the portfolio with swaps of time t, and FRt
2 for the portfolio without swaps of
time t and Dt equals the difference of time t (equation 36).
All funding ratios in the 1st
percentile came from interest rates declines in combination with
the short term interest rate was below zero. The negative absolute differences in the first
percentile were produced by very asymmetric shocks, where zero rates with short maturities
faced declines and became negative while zero rates with long maturities were increased, or
vice versa.
49

51. Swap performance
To test whether the portfolio with swaps has on average a funding ratio of 100% a t-test for
the mean is executed (equation 33). The hypotheses for this test are:
H0: µ = 100%
H1: µ = 100%
µ equals the funding ratio. The calculated tSTAT is 25.774. For the 1% significance level is the
upper critical tSTAT equal to 2.808 and the lower critical tSTAT equal to -2.808. Therefore,
the H0 hypothesis can be rejected. There is evidence that the funding ratio is different
from 100 % for a portfolio with swaps. The sign of the tSTAT says that the funding ratio
determined by the simulation procedure is above 100 %. Equation 34 is used to test whether
the funding ratio of a portfolio with swaps is closer to 100 % than a portfolio without swaps.
The hypotheses for this test are:
H0: µD = 0
H1: µD = 0
The calculated tSTAT equals 127.136. For the 1% significance level is the upper critical tSTAT
equal to 2.808 and the lower critical tSTAT equal to -2.808. Therefore, H0 can be rejected,
because the calculated tSTAT exceeds the critical value a lot. There is significant evidence
that the portfolio with swaps has a funding ratio closer to 100% than the portfolio without
swaps. The high positive tSTAT says that µD is strong positive.
50

52. This means by definition that the funding ratio of a portfolio with swaps is closer to 100%
than a portfolio without swaps. A similar test can be performed with different hypotheses:
H0: µD ≤ 2%
H1: µD > 2%
This makes it a one-tailed t-test. The critical tSTAT is now 2.576. The calculated tSTAT is
equal to 3.3654. This is above the critical value and therefore, H0 can be rejected. There
is significant evidence that the portfolio with swaps has an absolute difference with the
portfolio without swaps of at least 2%.
6.2. Adjusted-lambda model
This paragraph provides the results for the model based on λ = 0.01. In section 6.1 is stated
that the simulation where the notional values are determined on the estimated yield curve
instead of the real observed yield curve. Therefore, this section will adopt this statement.
When the notional values are determined (appendix C) the swap values can be determined
with the different simulated yield curve changes, and thus the funding ratios. Again, a three
month forecast is executed. Table 6.2 presents the descriptive statistics of 10,000 simulated
funding ratio’s of t + 1, t + 2 and t + 3.
The model produces still negative funding ratios although the portfolio contains swaps.
The same effect as in the basis model can be observed. The funding ratios in the 1st
percentile
are due to very asymmetric shocks to the yield curve. Zero rates with short maturities faced
declines and zero rates with long maturities were increased, or vice versa.
51

53. Mean Standard Error Standard Deviation Minimum Maximum Obs
F R1
1 100.00% 0.00002 0.0019 99.33% 100.72% 10000
F R1
2 99.67% 0.00027 0.0266 89.80% 109.88% 10000
D1
1.99% 0.00016 0.0161 -0.42% 9.95% 10000
F R2
1 100.00% 0.00003 0.0032 98.84% 101.57% 10000
F R2
2 99.45% 0.00040 0.0404 85.38% 116.94% 10000
D2
3.02% 0.00024 0.0240 -0.84% 16.25% 10000
F R3
1 100.00% 0.00004 0.0044 98.18% 102.02% 10000
F R3
2 99.22% 0.00051 0.0511 79.58% 121.00% 10000
D3
3.78% 0.00031 0.0307 -0.92% 20.35% 10000
Table 6.2: This table presents descriptive statistics of the funding ratio with and without swaps
and of the absolute difference between the portfolios simulated by the model based on λ = 0, 01 (in
equation 19 is λ = 0.01 instead of λ = 0.0609). FRt
1 equals the funding ratio of the portfolio with
swaps of time t, and FRt
2 for the portfolio without swaps of time t and Dt equals the difference of
time t.
Swap performance
A t-test (equation 33) is used to test whether the portfolio with swaps generates an average
funding ratio of 100%. The hypotheses for this test are:
H0: µ = 100%
H1: µ = 100%
µ equals the average funding ratio of the 10,000 simulated portfolios with swaps. The
calculated tSTAT is 0.27749. For the 1% significance level is the upper critical tSTAT equal
to 2.808 and the lower critical tSTAT equal to -2.808. The calculated tSTAT is lower than the
upper critical value and higher than the lower critical value, therefore the H0 can not be
rejected.
There are simulations where the funding ratio differs from 100%, but there is no significant
evidence that the funding ratio of a portfolio with swaps is not equal to 100 %.
52

54. Equation 34 is used to test for differences between the funding ratio of a portfolio with and
without swaps. The hypotheses for this test are:
H0: µD = 0%
H1: µD = 0%
The calculated tSTAT is equal to 123.240 where the upper critical tSTAT is equal to 2.808 and
the lower critical tSTAT equals -2.808. The calculated tSTAT is higher than the upper critical
tSTAT and therefore, the H0 can be rejected. There is significant evidence that the absolute
difference between the average funding ratio a portfolio with swaps differs from the average
funding ratio of a portfolio without swaps. A similar test can be performed with different
hypotheses:
H0: µD ≤ 2%
H1: µD > 2%
Again this is a one-tailed t-test. The critical tSTAT is now 2.576. The calculated tSTAT is
equal to -0.6656. This is below the critical value and therefore, H0 cannot be rejected. There
is no significant evidence that the portfolio with swaps has an absolute difference with the
portfolio without swaps of 2%.
Stability of the funding ratio
Because there is no evidence that funding ratio of portfolio with swaps differs from 100%,
the stability of the funding ratio will be tested. There are more than two means involved
(µt+1, µt+2 and µt+3), so the technique of a paired t-test cannot be used. To deal with more
than two means, an Analysis of Variance (ANOVA) will be used. This is an extension of the
53

55. t-test and can test for equality between more than two means. This procedure produces an
F-statistic that follows an F-distribution of K −1, N −K, where K is equal to the number of
groups and N is equal to the number of observations. Again, if the calculated FSTAT exceeds
the critical FSTAT , the H0 can be rejected. The hypotheses for this test are:
H0: µ1 = µ2 = µ3
H1: Not all means are equal
where µt equals the average funding ratio of time t. The critical FSTAT is equal to 4.60588.
The calculated FSTAT is equal to 0.35229, which is less than the critical FSTAT . Therefore,
H0 cannot be rejected. There is no significant evidence that one of the average funding ratio
of time t + 1, t + 2 or t + 3 differs from the others for at least three months.
The second test that is carried out is on the stability of the average funding ratio of the
portfolio without swaps. The hypotheses for this test are:
H0: µ1 = µ2 = µ3
H1: Not all means are equal
where µt equals the average funding ratio of time t. The critical FSTAT is equal to 4.60588.
The calculated FSTAT is equal to 31.5359, which is more than the critical FSTAT . Therefore,
H0 can be rejected. There is significant evidence that the funding ratio of the portfolio
without swaps is not constant over time. The conclusion of these tests is that the model
produces an average funding ratio of the portfolio without swaps to differ, while there is no
evidence that the funding ratio of a portfolio with swaps will differ.
54

56. Model performance
To test the performance difference between the models, a test as in equation 37 is executed,
where FR
E
equals the average funding ratio of the portfolio with swaps generated by the
basis model, FR
R
equals the average funding ratio of the portfolio with swaps generated by
the model based on λ = 0.01, µE −µR is equal to 0 (H0: µE ≤ µR), SE
2
is the variance of the
simulated funding ratios produced by the basis model, SR
2
is the variance of the simulated
funding ratios produced by the model based on λ = 0.01, and nE and nR are equal to 10,000.
The test as in equation 34 can not be used because the output is not generated by the
same model. The hypotheses for this test are:
H0: µE ≤ µR
H1: µE > µR
The calculated tSTAT is equal to 16.1628 and exceeds the critical tSTAT of 2.576. Therefore,
there is significant evidence that the average funding ratio produced by the model based
on λ = 0.01 is different than the average funding ratio produced by the model based on
λ = 0.0609. The sign of the tSTAT reveals that this produces a lower funding ratio than the
model based on λ = 0.0609.
This is in line with the previous test, because in paragraph 6.1 is stated that there was a
significant difference between the funding ratio equal to 100% and in this paragraph was the
outcome that there was not a significant difference between the funding ratio of a portfolio
with swaps equal to 100%.
55

57. 6.3. Modified-data model
This section provides the results of the model that is estimated based on yield curves that
are modified with the UFR method. Again, the notional values are determined with the
estimated yield curve instead of the observed yield curve. Table 6.3 provides descriptive
statistics of 10,000 simulations of time t + 1, t + 2 and t + 3.
Mean Standard Error Standard Deviation Minimum Maximum Obs
F R1
1 100.01% 0.00002 0.0019 99.31% 100.70% 10000
F R1
2 99.81% 0.00023 0.0231 90.97% 108.32% 10000
D1
1.70% 0.00014 0.0136 -0.47% 8.62% 10000
F R2
1 100.01% 0.00003 0.0034 98.83% 101.44% 10000
F R2
2 99.64% 0.00035 0.0352 86.37% 115.15% 10000
D2
2.55% 0.00021 0.0207 -0.70% 14.08% 10000
F R3
1 100.00% 0.00005 0.0046 98.44% 101.94% 10000
F R3
2 99.41% 0.00045 0.0446 84.09% 119.22% 10000
D3
3.23% 0.00026 0.0260 -0.97% 18.66% 10000
Table 6.3: This table presents descriptive statistics of the funding ratio with and without swaps
and of the absolute difference between the portfolios simulated by the model based on modified
data (in equation 19 is λ = 0.01 and the controlled variables are the observed yield curves). FRt
1
equals the funding ratio of the portfolio with swaps of time t, and FRt
2 for the portfolio without
swaps of time t and Dt equals the difference of time t.
Model performance
Because of the amended yield curves, the model behaves different. This section tries to
test the performance difference between the models. A test as in equation 37 is executed,
where FR
E
equals the average funding ratio of the portfolio with swaps generated by the
model based on λ = 0.01, FR
R
equals the average funding ratio of the portfolio with swaps
generated by the model based on modified data, µE − µR is equal to 0 (H0: µE = µR), SE
2
is the variance of the simulated funding ratios produced by the model based on λ = 0.01,
56

58. SR
2
is the variance of the simulated funding ratios produced by the model based on modified
data, and nE and nR are equal to 10,000. The hypotheses for this test are:
H0: µE = µR
H1: µE = µR
The calculated tSTAT is equal to -2.9007 and exceeds the lower critical tSTAT of 2.808.
Therefore, there is significant evidence that the average funding ratio produced by the model
based on modified data is different from the average funding ratio produced by the model
based on λ = 0.01. The sign of the tSTAT reveals that the model based on λ = 0.01 produces
a higher funding ratio than the model based on modified data.
This contradicts possibly that the model improves the performance. A t-test is used to
test whether the model based on modified data produces an average funding ratio of the
portfolio with swaps that is equal to 100%. The hypotheses are:
H0: µ = 100%
H1: µ = 100%
The calculated tSTAT is equal to 4.317 which exceeds the upper critical value of 2.808.
Therefore, the H0 can be rejected. There is evidence that the model based on modified data
does not produce an average funding ratio of the portfolio with swaps of 100%, but slightly
above 100%. This is in line with the expectations. There is evidence that the model based on
modified data produces an average funding ratio above 100%, where there was no evidence
for the model based on λ = 0.01 to produce an average funding ratio different from 100%.
57

59. Because of the amended yield curves, one would expect a reduction of the variance in the
yield curves. This induces that the present value of the liabilities is less volatile. Therefore,
the variance of the funding ratio of the portfolio without swaps should be lower and the
variance of the absolute difference should be lower.
To test the for a difference between two variances of two simulations, an F-test is executed.
The hypotheses for the test between the variance of funding ratio without swaps in the
portfolio are:
H0: σ2
1 ≤ σ22
H1: σ2
1 > σ2
2
The hypotheses for the test between the variance of absolute differences are: The Variance
of the absolute difference, with hypotheses:
H0: σ2
3 ≤ σ2
4
H1: σ2
3 > σ2
4
σ1 and σ3 equal the standard deviation of the respective factor produced by the model based
on λ = 0.01 and σ2 and σ4 equal the standard deviation of the respective factor produced by
the model based on modified data. The critical value for both tests is equal to 1.0477. The
calculated values are equal to 1.328 and 1.399, respectively. Therefore H0 can be rejected
for both tests. There is evidence that the variance of the funding ratio without swaps, and
the absolute difference, produced by the model based on λ = 0.01, is higher than the output
produced by the model based on modified data.
58

60. 7. Conclusion
This study addresses the questions whether an interest rate swap portfolio performs as is
should perform, and tries to evaluate this financial product by use of yield curve simulation.
The research question for this study was: What is the effectiveness of the use of interest rate
swaps for pension funds to manage the risk of interest rate changes, evaluated by interest
rate simulation?
The health of a pension fund is expressed as the funding ratio, where the funding ratio
is determined by the total value of the assets of a pension fund divided by the present value
of the liabilities. Dutch pension funds face interest rate risk since they have to determine
the present value of their liabilities with interest rates observed in the market. The Dutch
Central Bank (DNB) provides a monthly updated zero yield curve, which pension funds are
obliged to use. Interest rate sensitivity is expressed as the duration. Interest rate risk is
present if the duration of the assets is different from the duration of the liabilities.
After a fictive pension fund was created, zero rates provided by DNB are used to
determine the notional values of the swaps. Swaps with maturities of 1-10, 12, 15, 20,
25, 30, 40 and 50 year are used.
Diebold and Li (2006) provide a model that can be used for yield curve simulation. Based
on this model, three different vector autoregressive models are estimated. On the basis of
the simulation results with the basis model can be concluded that the funding ratio of a
portfolio with swaps is not equal to 100%. However, there is significant evidence that a
59

61. portfolio with swaps has a funding ratio closer to 100% than a portfolio without swaps. This
can possibly be caused by a kink in the estimated yield curve. Therefore, a second model is
estimated, based on a different λ.
This model provides better fitted estimated yield curves. The simulation results with
the model based on a different λ show that there is no significant evidence that the funding
ratio is not equal to 100%. A second test provides no evidence of an unstable funding ratio
for at least three months.
All yield curves up to 2012:08 are modified according the UFR method for the third
model. This should reduce the volatility in the zero rates with high maturities, but have no
effect on the funding ratio with swaps. However, there is evidence that the model without
the modified data produces a different funding ratio of the portfolio with swaps than the
model based on the model with the modified data.
The deviation in the funding ratio’s of the portfolio with swaps is caused by not using
a swap for each liability with the same maturity. But by looking at the simulation results,
shocks to the yield curve are fairly well absorbed, with the exception when the shock is very
asymmetric.
Adding the interest rate swap to a portfolio reduces the volatility of the funding ratio.
If the interest rate swap used properly, as a hedge instrument, the interest rate swap is
performing properly. However, because an interest rate swap generates value as a stand
alone instrument when the yield curve changes, it can also be used for speculation purposes.
60

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63. Diebold, F. X. and Li, C. (2006). Forecasting the term structure of government bond yields.
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64. Smith, A. and Wilson, T. (2001). Fitting yield curves with long term constraints. Research
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63

65. Appendices
64

66. A. Liabilities
Table A.1 shows the values of the liabilities with maturities of 0-60 years that are used to
create a fictive pension fund.
Maturity Cash flow Maturity Cash flow
0 438,523 31 520,843
1 438,569 32 496,717
2 443,856 33 471,885
3 458,088 34 446,191
4 475,852 35 420,116
5 493,816 36 393,829
6 513,189 37 367,593
7 531,145 38 341,635
8 548,072 39 316,152
9 563,334 40 291,331
10 579,466 41 267,315
11 596,281 42 244,200
12 611,336 43 222,083
13 625,458 44 201,031
14 636,607 45 181,093
15 646,133 46 162,302
16 652,643 47 144,680
17 657,146 48 128,240
18 659,925 49 112,982
19 660,639 50 98,901
20 660,115 51 85,984
21 659,269 52 74,212
22 656,482 53 63,562
23 651,165 54 54,002
24 643,288 55 45,493
25 632,601 56 37,987
26 618,771 57 31,429
27 603,154 58 25,753
28 585,275 59 20,890
29 565,612 60 16,766
30 544,064
Table A.1: This table contains all liability values for maturities 0-60 years. All cash flows are
divided by 1,000.
65

67. B. Description of weights determination
The weights in formula 29 are determined in September 2012. This was done in four steps:
Step 1: The first step was to calculate the 3 month arithmetic average of the past 3 months
zero coupon swap rates.
Step 2:With these zero rates, the 1-year forward rate for all maturities of 1 to 60 years were
derived (see equation 11).
Step 3: The Smith-Wilson techniques is applied to the yield curve based on the following
parameters:
T1 = 20, T1 = 60, UFR = 4.2%, α = 0.1
Step 4: The fixed weights are then determined with equation 38
wt =
FSW
t−1,t − F19,20
FSW
60,61 − F19,20
, (38)
for t = 21, ..., 60, where FSW
t−1,t is the 1-year rate t-year forward (calculated by step 3), and
F19,20 is the 1-year rate 19 years forward (calculated in step 2). Table B.1 presents the fixed
weights for maturities ranging from 21 years up to 60 years.
66

68. Maturity Weight Maturity Weight
21 0.086 41 0.903
22 0.186 42 0.914
23 0.274 43 0.923
24 0.351 44 0.932
25 0.42 45 0.94
26 0.481 46 0.947
27 0.536 47 0.954
28 0.584 48 0.96
29 0.628 49 0.965
30 0.666 50 0.97
31 0.701 51 0.974
32 0.732 52 0.978
33 0.76 53 0.982
34 0.785 54 0.985
35 0.808 55 0.988
36 0.828 56 0.99
37 0.846 57 0.993
38 0.863 58 0.995
39 0.878 59 0.997
40 0.891 60 0.998
Table B.1: This table presents for all maturities of 21-60 years the relevant value of the weight
for equation 28.
67

69. C. Notional values
Notional values for the swap portfolio are determined based upon the liability distribution
and on the zero yield curve. When the zero yield curve is different, other notional values
are optimal. Table C.1 shows the notional values for different yield curves. The fitted yield
curve is estimated by yt(τ) = β1t + β2t
1−e−λtτ
λtτ
+ β3t
1−e−λtτ
λtτ
− e−λtτ
.
Real λ = 0.0609 λ = 0.01
50 year swap e 1,798,291,327 e 1,769,284,046 e 1,781,606,244
40 year swap e 2,897,848,567 e 2,836,546,502 e 2,870,480,426
30 year swap e 1,985,280,281 e 1,976,908,537 e 1,986,369,423
25 year swap e 2,012,594,985 e 2,004,529,583 e 2,018,990,395
20 year swap e 1,799,456,998 e 1,787,119,084 e 1,806,322,876
15 year swap e 964,027,765 e 948,696,590 e 961,857,424
12 year swap e 563,908,799 e 552,826,573 e 563,454,579
10 year swap e 260,112,126 e 254,217,024 e 260,526,387
9 year swap e 239,942,027 e 233,890,678 e 240,480,255
8 year swap e 221,294,451 e 215,130,368 e 221,867,548
7 year swap e 201,625,799 e 195,424,069 e 202,150,441
6 year swap e 181,530,921 e 175,407,091 e 181,940,897
5 year swap e 160,582,818 e 154,667,733 e 160,827,964
4 year swap e 141,515,198 e 135,938,762 e 141,576,460
3 year swap e 123,024,321 e 117,855,520 e 122,927,777
2 year swap e 108,294,781 e 103,449,777 e 108,143,182
1 year swap e 102,598,780 e 97,537,878 e 102,566,288
Table C.1: This table provides all notional values for each swap. The real column is when the
real yield curve is used, the λ = 0.0609 column is when the yield curve estimated by yt(τ) =
β1t + β2t
1−e−λtτ
λtτ + β3t
1−e−λtτ
λtτ − e−λtτ and λ equals 0.0609 and the λ = 0.01 column is when
the yield curve is estimated based on λ equals 0.01.
68

70. D. Descriptive statistics modified data
This appendix provides descriptive statistics of the modified data as explained in paragraph
5.3 (table D.1).
Maturity Mean St. Dev. Min Max Obs Maturity Mean St. Dev. Min Max Obs
1 2.21% 0.0143 0.34% 5.36% 125 31 3.88% 0.0075 2.54% 5.14% 125
2 2.34% 0.0136 0.41% 5.36% 125 32 3.88% 0.0074 2.57% 5.12% 125
3 2.51% 0.0130 0.52% 5.28% 125 33 3.88% 0.0072 2.61% 5.10% 125
4 2.68% 0.0123 0.68% 5.19% 125 34 3.89% 0.0071 2.65% 5.08% 125
5 2.84% 0.0116 0.85% 5.11% 125 35 3.89% 0.0069 2.68% 5.07% 125
6 2.99% 0.0110 1.04% 5.06% 125 36 3.90% 0.0067 2.71% 5.05% 125
7 3.12% 0.0105 1.20% 5.02% 125 37 3.90% 0.0066 2.75% 5.03% 125
8 3.24% 0.0101 1.37% 5.02% 125 38 3.91% 0.0064 2.78% 5.01% 125
9 3.34% 0.0098 1.52% 5.02% 125 39 3.91% 0.0063 2.81% 4.99% 125
10 3.44% 0.0095 1.66% 5.03% 125 40 3.92% 0.0062 2.84% 4.98% 125
11 3.52% 0.0093 1.79% 5.05% 125 41 3.92% 0.0060 2.87% 4.96% 125
12 3.60% 0.0091 1.90% 5.06% 125 42 3.93% 0.0059 2.90% 4.95% 125
13 3.66% 0.0090 2.00% 5.08% 125 43 3.93% 0.0058 2.92% 4.93% 125
14 3.72% 0.0088 2.09% 5.09% 125 44 3.94% 0.0057 2.95% 4.91% 125
15 3.76% 0.0087 2.16% 5.10% 125 45 3.94% 0.0055 2.98% 4.90% 125
16 3.79% 0.0088 2.20% 5.09% 125 46 3.95% 0.0054 3.00% 4.89% 125
17 3.81% 0.0088 2.23% 5.11% 125 47 3.95% 0.0053 3.02% 4.87% 125
18 3.83% 0.0088 2.23% 5.16% 125 48 3.96% 0.0052 3.05% 4.86% 125
19 3.85% 0.0088 2.24% 5.20% 125 49 3.96% 0.0051 3.07% 4.85% 125
20 3.87% 0.0088 2.25% 5.24% 125 50 3.97% 0.0050 3.09% 4.83% 125
21 3.86% 0.0088 2.25% 5.25% 125 51 3.97% 0.0049 3.11% 4.82% 125
22 3.86% 0.0088 2.27% 5.26% 125 52 3.97% 0.0048 3.13% 4.81% 125
23 3.86% 0.0087 2.29% 5.26% 125 53 3.98% 0.0047 3.15% 4.80% 125
24 3.87% 0.0086 2.32% 5.26% 125 54 3.98% 0.0047 3.17% 4.79% 125
25 3.87% 0.0085 2.34% 5.25% 125 55 3.99% 0.0046 3.19% 4.78% 125
26 3.87% 0.0083 2.37% 5.23% 125 56 3.99% 0.0045 3.21% 4.77% 125
27 3.87% 0.0082 2.40% 5.22% 125 57 3.99% 0.0044 3.22% 4.76% 125
28 3.87% 0.0080 2.43% 5.20% 125 58 4.00% 0.0043 3.24% 4.75% 125
29 3.87% 0.0079 2.47% 5.18% 125 59 4.00% 0.0043 3.26% 4.74% 125
30 3.87% 0.0077 2.50% 5.16% 125 60 4.00% 0.0042 3.27% 4.73% 125
Table D.1: This table provides descriptive statistics of the zero rates for different maturities which
are modified by the UFR method. The sample period is 2003:12-2014:04.
69