Interpolation of Yield curves

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School of Education, Culture and Communication

Division of Applied Mathematics

MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Interpolation of Yield curves

by

Abdulhamid Iebesh

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School of Education, Culture and Communication

Division of Applied Mathematics

Master thesis in mathematics / applied mathematics

Date:

2020-06-05

Project name:

Interpolation of Yield curves

Author: Abdulhamid Iebesh Supervisor(s): Jan Röman Co-supervisor: Ying Ni Reviewer: Marko Dimitrov Examiner: Anatoliy Malyarenko Comprising: 30 ECTS credits

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Abstract

In this thesis we survey several interpolation methods that are used to construct the yield curves. We also review the bootstrapping and show that the bootstrap is closely connected to the interpolation in the case of bootstrapping yield curve. The most effort is dedicated, in this thesis, on the monotone convex method and on investigation of the difficulties to get accurate yield curves.

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Dedication

To my dear father, the real motivation for my

ambition-Walid Iebesh

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Acknowledgements

I would like to express my sincere gratitude to my supervisor Jan Röman who super-vised me, introduced me to this research idea and helped me to get the market data. My special gratitude to Dr. Ying Ni and all our teachers.

My special gratitude to Prof.Anatoliy Malyarenko.

My heart-felt gratitude and appreciation goes to my family members for support and continuous encouragement.

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List of Figures

1 The Swedish Treasury zero coupon rates at 2016-09-09 . . . 15

2 Explaining example by Hagan and West[2008] about the piecewise linear for-ward method. . . 24

3 The function g . . . 29

4 The reformulated possibilities for g . . . 30

5 Forward curve by Cubic spline on rates in Table 2. . . 36

6 Spot and forward rate curves by the monotone convex method to the rates in Table 2 . . . 37

7 Bootstrapping with linear interpolation . . . 38

8 Bootstrapping with Cubic spline interpolation . . . 39

9 Bootstrapping with Monotone Convex interpolation . . . 39

10 Spot rate curves from three different interpolations . . . 40

11 Forward rate curves from three different interpolations . . . 40

12 Bootstrapping bonds with linear interpolation . . . 41

13 Bootstrapping bonds with monotone convex interpolation . . . 41

14 Spot rate curves from two different interpolations . . . 42

15 Forward rate curves from two different interpolations . . . 42

16 Original and “blipped” spot rate curves % obtained by applying monotone convex method of interpolation to the rates in Table 2. “Blipped” curve ob-tained by changing the input at t = 4, from 4.4% to 5.4%. . . 43

17 Iteration of a swap curve bootstrap . . . 51

18 Results of a swap curve bootstrap . . . 52

19 Iteration of a Swedish bonds curve bootstrap . . . 56

20 the spot and forward rate curves that were obtained by applying linear inter-polation on r(t), natural cubic spline on r(t) monotone convex interinter-polation to the EURIBOR Data (Table 3,Table 4 and Table 5). . . 57

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List of Tables

1 Quotes of Swedish Government securities . . . 14

2 Table used by Hagan and West [2006] . . . 36

3 Cash deposit with maturity S/W, 1M, 3M . . . 46

4 FRA 3M . . . 46

5 Swaps from EURIBOR IRS 3M . . . 47

6 Swedish Treasury Bonds from finansportalen.se . . . 48

7 Bootstrapped deposits and FRA . . . 50

8 Bootstrap Swedish Bonds with linear interpolation . . . 55

9 Bootstrap Swedish Bonds with Monotone Convex Interpolation Method . . . 56

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1 Introduction

Yield curves are used in the financial industry to generate and discount cash-flows to find the present value. These yield curves are generated from traded and/or quoted instruments. The two most common yield curves are government curves and inter-bank curves. The government curve is based on bonds issued by the government and the inter-bank curves are based on instrument traded or quoted between banks and other financial institutes.

When a yield curve is estimated, the curves are based on a limited number of instruments. To get a continuous curve, interpolation must be used between the maturities of these instruments. These curves are called zero-coupon curves. To generate cash-flows we also need forward curves representing the interest rates between different dates in the future. One problem with this interpolation is that we also want the market to become free of arbitrage. Another problem is to calculate the risk. Usually, the risk is calculated by shifting nodes on the yield curves. The most common interpolation methods cannot be used to estimate the risk in a time bucket (i.e. between two times in the future). This is obvious if we use cubic splines. A yield curve with cubic splines are estimated so that the first and second order derivative are the same in the left and the right polynomial. If one node in such a curve is shifted, the risk is distributed on all nodes on the curve, also beyond the maturity of an instrument valued by the curve. This thesis will study different interpolation methods in curve construction and investigate the difficulties to get accurate measures of risk. When using bootstrapping to estimate a yield curve the interpolation method must be incorporated in the calculation. This means you cannot bootstrap a curve and do the interpolation afterwards.

2 Literature Review

We survey a number of spline-based models available of yield curves construction and show the weaknesses and strengths of each model.

Under spline-based models, the individual segments are joined together at knot points (spe-cific points in time) continuously which form piecewise polynomials where ultimately these polynomials form the required yield curve.

Spline-based yield curve models is grounded on minimising the following function

min h(t) n

∑

i=1 (Pi− ¯Pi)2 ! ,

where n is the number of inputs securities, Pi are the market prices, and h(t) is the chosen

interpolation method (the spline function) used to calculate the new prices ¯P_i. h(t) might be defined in terms of the discount function, the spot rate, or the forward rate function.

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2.1 McCulloch (1975)

In 1975 McCulloch introduced the cubic spline to value the discount function to be smooth and continuous at each knot point [8]

P= ai+ bi(t − ti) + ci(t − ti)2+ di(t − ti)3,

for ti≤ t ≤ ti+1, i = 1, 2, ..., n and where ai, bi, ciand diare parameters to be estimated.

But there is an objection for this method by Vasicek and Fong that the use of polynomial splines results unstable forward rates [13]. Other objection by McCulloch and Kochin that be-yond the longest observed maturity there is no reasonable way to extrapolate a cubic discount factor [9].

2.2 Vasicek and Fong (1982)

The idea of Vasicek and Fong is to value the discount function with exponential splines [13]. The discount function

P(t) = D −1

α log(1 − x)

= G(x), where the variable x = 1 − e−αt. Also, Vasicek and Fong define

G(x) =

k

∑

i=1

βi· gi(x),

where gi(x) are a set of k polynomial functions for 0 ≤ x ≤ 1, i = 1, 2, ..., k. The objection for

this model comes by Shea that the gi(x) are non linear in α and this model doesn’t bring any

practical benefit [12]. Moreover, Shea notes that the resulting values with Vasicek and Fong method are similar nearly to that resulting with polynomial spline [12].

2.3 Hagan and West (2006)

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4. Until convergence is obtained, repeat steps (2) and (3).

The bootstrap with Hagan and West (2006) method is the ideal model and the result yield curve with this bootstrap technique has the ability of repricing exactly all input financial securities while that is unlikely with other spline-based models. In this thesis, we will consider only this spline-based method.

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3 Financial Instruments

We review some basic financial terms that we use in this thesis.

3.1 Zero Coupon Pricing

The market prices are not always available (or not reliable) when pricing OTC (over the counter) instruments and also standardized instruments. The zero coupon pricing is impor-tant to evaluate all cash flows to find the present value. That generates the yield curve which all risk management techniques need to calculate the risk over all types of instruments [10].

3.2 Discount Function

The present value at time t0of a cash flow at future date t is extracted by the discount function

which is monotonically decreasing, which coincides a positive interest rates.

The mathematical relationship between the annually compounded yield and the discount func-tion is written [10]:

P(0,t) =_(1+r1

1(t))t

where P(0,t) is the discount function at time t and r1(t) is the spot yield curve at time t where

t is measured in years.

3.3 Interest Rates

An interest rate is the amount of interest that a borrower pays to the lender, as a proportion of the magnitude lent, for using the money. However, in the market there are many kinds of interest rates; here we mention some of these rates.

Risk Free Rate:

The rate by taking a risk-less position is a risk-free rate and it is used to discount expected cash flows. It can be based on treasury bonds or swap rate. Some uses OIS (Overnight Indexed Swaps) rate as risk free rate.

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where t means the time to maturity and BN represents the total amount with interest.

Annual Compounding Rate:

It can be expressed as percentage of the principal invested amount based on effect of com-pounding on a year [10]. If we have f periods ( f the number of payments in the year) for example we get (1 + rannual)t = (1 + rf f )f·t, (1 + rannual)t= (1 + rquarterly 4 )4·t.

The semi annual rate

(1 + rannual)t= (1 +r2₂(t))2·t.

The continuous compounding rate

B_N= B0· er(t)·t.

Forward Rate:

The forward rate is the rate that made today to represent the interest rate between two dates in the future, t1and t2. We cover this rate in more details later in next chapter.

Swap Rate:

The swap rate is a fixed rate that prices a swap to zero value and sometimes the swap rate is used as risk-free rate [10].

3.4 Interest Rate Instruments

3.4.1 Bonds

Bonds, bills and notes are financial debt instruments issued by a company or a government. A bond is a loan that a government or company obtains from the holder in exchange of interest payment and the repayment of the loan in future where most bonds pay coupons to the holder at the end of their life. The owner receives cash flows and the present value of these cash flows represents the theoretical price of a bond [6].

The price of bond is given by the following formula

P= N (1 + ytm)T + n

∑

i=1 C (1 + ytm)ti ,

where P is the price of bond, N is the face value, C is the coupon payments, ytm is the yield to maturity, T is the time to maturity and tiis the times for individual coupons.

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3.4.2 FRA-Forward Rate Agreement

The forward rate agreement (FRA) is a forward-starting loan with exchanges only the dif-ference in interest rate and without exchanges of principal. The FRA is an over-the-counter agreement to lend or borrow a certain principal for a specified period of time at FRA rate (pre-specified fixed interest rate when FRA is traded) [10].

The buyer of an FRA (the borrower of a certain principal) will protect hem-self of rising rates in this period and the seller also will be protected of falling interest rates in the same period of FRA traded. If the rates goes up the seller pays the difference otherwise if the rates falls then the buyer pays the difference between the rate when FRA was traded.

3.4.3 Swaps

The Swap is a contractual agreement in which making periodic payments between two parties to each other under two different indices [10]. First party makes fixed rate on the face value of swap while the other party makes floating rate payments to the first one based on the same face value of the swap for the same period of time.

At starting date the net value is zero. To calculate the par swap rates we use the formula

r_Tpar= DT/N− DT ∑T_t=1−1Yt· Dt+YT· DT , and thus D_T = DT/N− r par T ∑ T−1 t=1 Yt· Dt 1 +YT · r_Tpar ,

where D_T/N is the discount tomorrow next, DT is the discount swap, r_Tparis the par swap rate

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4 Constructing Yield Curves

4.1 The Yield Curve

All cash flows need to be evaluated systematically within one evaluation framework as if they were zero coupon bonds. That is why we want to have a risk free yield curve or, otherwise, a discount function (zero coupon bond) [10]. The contract which ensure the holder 1 (SEK, USD,...) to be paid at maturity date T , is called ’zero coupon bond’, also called discount bond or T-bond, as it is denoted in [10].

The yield curve is the function that relates the current spot yields for any set of bonds to their maturities. These bonds can be bills, notes or bonds which also are defined by the same credit rating or same issuer. We have the yield curves for corporate, mortgage and governments as Jan Röman clarifies in his book [10].

Geiger introduces that the zero coupon yield curve is a continuous function of interest rates varied in time to maturity which means for any small step on the curve, there is a accurately priced zero bond but in practice, it is not possible to monitor a adequate amount of zero coupon bonds in the financial markets. For that reason, the yield curve needs to be fitted [3]. There are two approaches to fit the yield curve either smoothing techniques or structural model. The most popular method among central banks and market participants is the smoothing technique since it allows to get the forward rates [3].

We have in the Table 1 the Swedish Government (bills and bonds) 2016-09-09. Table 1: Quotes of Swedish Government securities

Securities Issued date Maturity Coupon Price

STB 21 Sep 16 2016-03-11 2016-09-21 0 100.02 STB 21 Sep 16 2016-03-11 2016-09-21 0 100.02 STB 19 Oct 16 2016-07-01 2016-10-19 0 100.09 STB 16 Nov 16 2016-08-05 2016-11-16 0 100.15 STB 21 Dec 16 2016-06-03 2016-12-21 0 100.23 STB 15 Mar 17 2016-09-02 2017-03-15 0 100.42 SGB 1051 3.75% 12 Aug 17 2006-09-15 2017-08-12 3.75 104.45 SGB 1052 4.25% 12 Mar 19 2007-11-21 2019-03-12 4.25 114.47 SGB 1047 5.00% 1 Dec 20 2004-01-28 2020-12-01 5.00 127.47 SGB 1054 3.50% 1 Jun 22 2011-02-09 2022-06-01 3.50 123.18 SGB 1057 1.50% 13 Nov 23 2012-10-22 2023-11-13 1.50 113.68 SGB 1058 2.50% 12 May 25 2014-02-03 2025-05-12 2.50 123.27 SGB 1059 1.00% 12 Nov 26 2015-05-22 2026-11-12 1.00 109.57 SGB 1056 2.25% 1 Jun 32 2012-03-20 2032-06-01 2.25 124.84 SGB 1053 3.50% 30 Mar 39 2009-03-30 2039-03-30 3.50 153.64

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Figure 1: The Swedish Treasury zero coupon rates at 2016-09-09

The yield curve has different shapes, upwards sloping, downwards sloping (inverted) and mixed one. Moreover, the capitalisation factor is the inverse of the decreasing function zero coupon bond. As result of the decreasing function, the interest rate is always positive [10]. There is a relationship between the discount function and the spot yield curve which com-monly works in term of continuously compounded risk free rate:

C(0,t) = exp(r(t) · t), thus,

P(0,t) = exp(−r(t) · t), (1)

and from (1) we get

r(t) = −1

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4.2 Forward Rates

The rate that is applied between two future dates is called the forward rate [10]. A forecast of future interest rate between two dates calculated from either a yield curve or spot rate, is given by the following formula which is based on an arbitrage argument:

(1 + r_tspot₁ )t1· (1 + rf orward t2−t1 ) t2−t1 _{= (1 + r}spot t2 ) t2 _=⇒ r_tf orward₂_−t₁ = ((1+r spot t2 )t2 (1+rspot_t1 )t1) 1 t2−t1_{− 1,}

and easy via discount factor:

P(0,t1) · P(t1,t2) = P(0,t2),

where P(0,t) zero coupon bond at time 0 with maturity t and P(0;t1,t2) is the forward discount

factor at time 0, to ensure no arbitrage, between two dates in the future from t1to t2. Then

P(0;t1,t2) = e− f (t2,t1)·(t2−t1). (3)

As we know that the discount function is decreasing hence, it arises that the forward rates are positive. We have f(t2,t1) = − ln P(0,t2) − ln P(0,t1) t₂− t1 (4) = r2· t2− r1· t1 t₂− t1 . (5)

In addition, the instantaneous forward rate: f(t) = −d dt ln (P(t)) (6) = d dt(r(t) · t), (7) this gives f(t) = r(t) + r0(t) · t, here we have two cases,

• the yield curve is normal (upwards sloping), then the forward rates will exist above the curve.

• the yield curve is inverted (downwards sloping), then the forward rates lie below the curve. By integrating we have, t R 0 f(s) ds = t R 0 r(s) ds + t R 0 r0(s) s ds,

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t R 0 f(s) ds = t R 0 r(s) ds + [r(s) · s]t₀− t R 0 r(s) ds = [r(s) · s]t₀, by simplifying r(t) · t = t Z 0 f(s) ds, (8) thus, P(t) = e (− t R 0 f(s) ds) . We have also, r_i· ti− ri−1· ti−1= ti R 0 f(s) ds − ti−1 R 0 f(s) ds = ti R ti−1 f(s) ds =⇒ ri· ti− ri−1· ti−1 t_i− ti−1 = 1 t_i− ti−1 ti Z ti−1 f(s) ds, (9)

which presents the parity between the discrete forward rate on [ti−1,ti] and the average of the

instantaneous forward rate over any intervals [ti−1,ti],

r(t) · t = ri−1· ti−1+ t

Z

ti−1

f(s) ds, t∈ [ti−1,ti], (10)

easily by given the forward function in the interpolation formula (10) we get the risk free rate function [5].

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4.3 Bootstrap and Interpolation

What we need to emphasize is that the bootstrap is closely connected to the interpolation in the case of bootstrapping yield curve. The interpolation scheme completes the incomplete information that occurs by bootstrap.

Suppose we want to bootstrap a yield curve by using a set of bonds. The choice of bonds is not trivial since excluding a large number of bonds may lead to a risk of eliminating the information of market, while including a large number of bonds might result implausible yield curve. The selection includes some known rates at short end i.e some zero coupon bonds will be known. In case of not enough liquidity at short end, in some markets, we will use some inter-bank money market rates [5].

As Hagan and West [2008] introduced that the next relationship must be achieved for each bond and curve,

[A] = n

∑

i=0 s_iP(0;t_settle,t_i), where

• s0, s1, ..., sn are the cash flows related to bond (usually s0= e₂c, si= ₂c, 1 ≤ i < n, sn=

1 +₂c where c is the annual coupon and e is the cum-ex switch1). • t0,t1, ...,tncash flow dates.

• t_settle is the delivered date for a brought bond. • A the price of bond.

For more convenient form, we rewrite the formula [A] · P(0;t_settle) =

n

∑

i=0

siP(0,ti). (11)

If we determine the risk free rates then we determine the discount factors. So we assume the risk free rates at t0,t1, ...,tn−1 are determined and thus, the value of P(0,tn) is

P(0,tn) = 1 sn " [A] · P(0;t_settle) − n−1

∑

i=0 s_iP(0,ti) # . Instead of the discount factor we rewrite the formula in risk-free rates form,

rn= 1 t_n " ln sn− ln "

[A] · e−rsettletsettle₋

n−1

∑

i=0 sie−riti ## . (12)

1_{Cum coupon means that when the selling in secondary market, the buyer gets the current coupon of the bond.} Ex coupon means that the seller holds the coupon before the buyer receives the bond.

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The formula (12) introduces algorithm of an iterative solution:

1. We determine rsettle and ri for i = r0, r1, ..., rn−1 by inserting the known rates from e.g

money market, into our interpolation algorithm.

2. We solve the equation (12) for rnby inserting all these rates what we get from first step.

3. The new guess is taken for this bond, of course also for all other bonds, and once more apply the algorithm of interpolation.

We repeat this process 4 or 5 iterations until we reach a fixed point, then we get the yield curve [5].

We now consider swap curves. For the fixed payments at the time t1,t2, ...,tn where the

time is measured in years, the par swap rate is Rn= 1−P(tn)

∑ni=1αi·P(ti),

where P(ti) is the discount factor for each i = 1, 2, ..., n and αi is the time from ti−1 to ti .

Moreover, if the Rn is known and the discount factor P(ti) is known for i = 1, 2, ..., n − 1 then

we get

P(tn) =

1 − Rn∑n−1_i=1αi· P(ti)

1 + Rn· αn

. (13)

In the case of lack of liquidity one approach is to interpolate the input swap rates to get the swap rates which are not quoted and then continue with complete information. We can rewrite the formula (13) above as

r_n= −1 t_n ln " 1 − Rn∑n−1_j=1αj· P(t,tj) 1 + Rn· αn # . (14)

The formula (14) introduces algorithm of an iterative solution:

1. We estimate initial rates rnfor quoted expiries and perform our chosen method

interpo-lation to find the missing rj.

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4.4 Quality Criteria of Curve Construction

To judge the curve construction and interpolation there are some criteria that should be used: 1. The forward rates should be positive since the curve must be free of arbitrage. Today the

short forward rates are negative due to the market situation. But normally they should be positive3.

2. The forward rates should be continuous since the forward rates are 1m or 3m the same as instantaneous rates.

3. Locality of interpolation method. We want the interpolation preserve locality where the change in inputs affects only nearby.

4. Are the forward rates stable?

For a given some basis point change in one input in the forward curve we can measure the degree of stability by quantify the maximum basis point change.

3_{This can be seen if we look at the discount factor that should always be lower than one. But for short time,} it is above 1 today in some currencies such as EUR and SEK.

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5 Traditional Interpolation Methods

Suppose we have some known data x1, x2, ..., xnand t1,t2, ...,tnwhere x(ti) = xiis a function of

time, then we want to predict what other data points between them. The interpolation methods do that. Ken Adams discusses the smooth interpolation of zero curves and refers to that the best choice of various interpolation methods should require knowledge of data points [1]. According to Adams this way excludes Lagrange polynomials and Hermitian interpolation, thus the considered interpolation is only piecewise spline curves [1, p. 14].

A polynomial spline is a piecewise function which is in every term a polynomial. The spline corresponds the input data by organizing the coefficients in good order to ensure the continuity.

5.1 Linear methods

To form a complete yield curve, the points of market data are connected by a straight line by linear interpolation.

5.1.1 Linear on Rates

The interpolation formula for ti−1< t < tiis

r(t) = t− ti−1 t_i− ti−1 r_i+ ti− t t_i− ti−1 r_i−1. (15) From f (t) = −_dtd ln P(t) =_dtdr(t)t we get f(t) = 2t − ti−1 t_i− ti−1 ri+ ti− 2t t_i− ti−1 ri−1. (16)

The function r(t)t is clearly not differentiable at ti so that the function f is undefined there

which changes the slope frequently, therefore it is not suitable for yield curves [11]. Further, the ri−1 has been reduced to zero when t reaches tiand the forward has a jump at ti since the

left and right limits are not equal. Moreover, this choice of interpolation allows the negative forward rates as Hagan and West [2008] explain in the next example: if we create two points

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5.1.2 Linear on the Log of Rates

The interpolation formula for t_i−1≤ t ≤ t_i is ln(r(t)) = t−ti−1 ti−ti−1ln(ri) + ti−t ti−ti−1ln(ri−1), thus, r(t) = r t_−ti−1 ti−ti−1 i · r ti−t ti−ti−1 i−1 . (17)

According to the logarithm function properties, we notice that the formula above prevents negative interest rates and as before the forward has jumps at nodes for the same argument as linear interpolation on rates (also the function P is not decreasing) [5].

5.1.3 Linear on Discount Factors

The interpolation formula for t_i−1≤ t ≤ ti is

P(t) = t−ti−1 ti−ti−1Pi+ ti−t ti−ti−1Pi−1, thus, r(t) =−1 t ln t − ti−1 t_i− t_i−1e −riti₊ ti− t t_i− t_i−1r −ri−1ti−1 . (18)

There are jumps at each node for the forward rate and the property of being the P function decreasing might not be satisfied [5].

5.1.4 Raw Interpolation (Linear on the Log of Discount Factors)

The models developer of yield curve use this method as starting point to find mistakes in the new developed methods through comparing the new results with Raw method. This method is easy to implement.

The instantaneous forward rates are constant in this method on each interval ti−1 < t < ti.

As long as the instantaneous forward rate is constant, from (9) we get f (t) = riti−ri−1ti−1

ti−ti−1 for

t_i−1< t < tiand then from (10) we obtain

r(t)t = ri−1ti−1+ (t − ti−1)riti−r_t_i_−ti−1_i−1ti−1,

by simplifying the formula above we get r(t)t = (t − ti−1)

(t_i− t_i−1)riti+ t_i− t

t_i− t_i−1ri−1ti−1. (19)

The important thing in this method is that it ensures the positivity of instantaneous forwards since it equals to the corresponding discrete forward on original interval. Otherwise, the in-stantaneous forward is undefined at the points t1,t2, ...,tn which means that the function has

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5.1.5 Piecewise Linear Forward

There is one defect in previous attractive Raw method that being the forwards piecewise con-stant. By remedying that to be a piecewise continuous linear function, it leads to not desired behaviour. Hagan and West [2008] explain that through the example.

First, assume we have zero coupon rates (a curve with these inputs data): • r(t) = 5% for t = 1, 2, ..., 5.

• r(t) = 6% for t = 6, 7, ..., 10. and we should have:

• f (t) = r(1) for t ≤ 1.

• f (t) = r(t) for t ≤ 5 to ensure continuity. The discrete forward rate for the interval [5, 6] is:

f(t2,t1) = r2t_t₂2−r_−t₁1t1 −→ f (5, 6) = 0.06×6−0.05×5₆₋₅ = 11%

and it must have f (6) = 17% to get the average of piecewise linear forward function on the interval [5, 6] to be 11% (i.e 0.17+0.05₂ = 11%).

The discrete forward rate for the interval [6, 7] is:

f(t2,t1) =r2t_t₂2−r_−t₁1t1 −→ f (6, 7) =0.06×7−0.05×6₇₋₆ = 6%

and it must have f (7) = −5% to get the average of piecewise linear forward function on interval [6,7] to be 6% (i.e 0.17−0.05₂ = 6%). The shape takes zig-zag frequently. Moreover the actual yield curve has implausible shape.

Second, by adding a new rate (node) r(6.5) = 6%. Clearly, the reversed zig-zag arises since the new information has an effect in the bootstrap curve. Hence, the locality in this method is very poor.

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Figure 2: Explaining example by Hagan and West[2008] about the piecewise linear forward method.

5.2 Cubic Splines

The best spline interpolation that ensures the smooth yield curve , according to Adams [1], is a cubic-spline interpolation.

The Cubic spline interpolation guarantees not only differentiability but also the continuity to the second derivative [2, p. 145].

Suppose we have some known data r1, r2, ..., rn and t1,t2, ...,tn where r(ti) = ri, for given

coefficients (ai, bi, ci, di) for 1 ≤ i ≤ n − 1 at any time t the function value will be

r(t) = ai+ bi(t − ti) + ci(t − ti)2+ di(t − ti)3 ti≤ t ≤ ti+1. (20) Thus, r0(t) = b_i+ 2c_i(t − t_i) + 3d_i(t − t_i)2, t_i< t < ti+1, r00(t) = 2ci+ 6di(t − ti), ti< t < ti+1, r000(t) = 6di(t − ti), ti< t < ti+1, let hi= ti+1− ti.

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• the given data meet the interpolating function, ai= rifor i = 1, 2, ..., n − 1 (n − 1

condi-tions).

and an−1+ bn−1hn−1+ cn−1h_n−12 + dn−1h3_n−1= rn:= an(1 condition).

• continuity of the entire function ai+ bihi+ cih2i + dih3_i = ai+1, i = 1, 2, ..., n − 2.

• differentiability of the entire function bi+ 2cihi+ 3dih2i = bi+1, i = 1, 2, ..., n − 2.

The system above of 3n − 4 equations with 4(n − 1) unknown coefficients and there are still another n linear conditions to find. Clearly, we have

f(t) =_dtd(r(t) · t),

f(t) = ai+ bi(2t − ti) + ci(t − ti)(3t − ti) + di(t − ti)2(4t − ti), ti≤ t ≤ ti+1. (21)

We define the derivative of the interpolating function as

b_n:= bn−1+ 2cn−1hn−1+ 3dn−1h2n−1, (22)

and moreover, in general case, if we specify b1, b2, ..., bn, then this will be equivalent to the

specification of n linear conditions (that is because we want another n linear conditions), we get 4n − 4 equations [4].

5.2.1 Natural Cubic Spline

It is the most common of the Cubic spline interpolation and it is required that the second derivative to be zero at endpoints [1]. Moreover, it is also required to be twice difierentiable which adds another n − 2 conditions [5].

5.2.2 Financial Cubic Spline

The twice difierentiable of the function is required where at right hand end its derivative zero (the function is horizontal ) and at left hand end its second derivative is also zero (the function is linear) [1].

5.2.3 Bessel (Hermit) Method

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5.2.4 Quadratic Natural Spline

The function refers to a cubic spline in which is applied to the function r(t)t and also required to be twice differentiable. It is constructed so that for endpoints to be quadratic on the left and linear on the right as proposed in McCulloch and Kochin [9].

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6 Monotone Convex Method of Interpolation

The monotone convex method to interpolate yield curve data was introduced by Hagan and West [2006]. Indeed, many ideas by Hyman [7] was developed by Hagan and West [2006] to appropriate the financial problem (yield curve data) by treatment the financial lack. Moreover, none of the commonly interpolation methods are aware to solve a financial problem. The need of continuous and positive forward rates arises in the monotone convex method. Thus, we use this interpolation method to bootstrap the yield curve.

In this chapter, we will review the monotone convex method of Hagan and West [2006] and Hagan and West [2008], and investigate whether we have the same difficulties associated to “traditional” interpolation methods or not.

6.1 Proper Forward Rates

The monotone convex method interpolation is performed on the discrete forwards as inputs, and not on the interest rate curve itself. Hence, if we have a set of interest rate r1, r2, ..., rnfor

periods t1,t2, ...,tnthen we start to calculate

f_id= riti− ri−1ti−1

t_i− t_i−1 , (23)

for 1 ≤ i ≤ n, provided r0= 0 at t0. Here we check fid if positive, it means arbitrage free and

the curve is legal. As such, the forward curve will be bootstrapped by the monotone convex interpolation method, and the risk free rates will be recovered by

r(t)t =

t

Z

0

f(s)ds. (24)

The f_idbelongs to the generic interval [ti−1,ti] and the shortest rate usual might be an overnight

rate. The instantaneous forward rates fiwe choose as follows:

f_i= ti− ti−1 t − ti−1

f_i+1d + ti+1− ti t − ti−1

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6.2 The Basic Interpolator

The attempts to find an interpolating function f , f (ti) = fi for i = 0, 1, ..., n, on [0,tn] that

satisfies the following conditions: i) f_id =_t 1

i−ti−1

ti R

ti−1

f(t)dt, so the method reproduces the zero coupon rates, as in (9). ii) f is positive.

iii) f is continuous.

iv) f (t) is increasing on the interval [ti−1,ti] if

f_i−1d < f_id< f_i+1d , and f (t) is decreasing on the interval [ti−1,ti] if

f_i−1d > f_id> f_i+1d .

Firstly, as seen in Hagan and West [2008] that seeking a function g on the interval [0,1] (more precisely each gicorresponds [ti−1,ti] one at a time):

g(x) = f (t_i−1+ (ti− ti−1)x) − fid, (28)

where g is continuous, thus (iii) is satisfied, and the function g is a piecewise quadratic in a way that (i) is satisfied by construction. The choice of being g quadratic, is to analyse where the maximum or minimum happens which results the ability to modify g to guarantee satisfying (iv) while (iii) and (i) is still satisfied.

Also, for (ii) will be satisfied, as we will see later, if the forward values of fi had met certain

restrictions.

The interpolation function f should be positive and to guarantee that, we do some adjustments to fi. We recall some details here that we have some information about g:

• g(0) = fi−1− fid. • g(1) = fi− f_id. • 1 R 0 g(x)dx = 0.

We assume g(x) = K + Lx + Mx2having three unknown and three equation, thus K= g(0), K+ L + M = g(1), K+1 2L+ 1 3M= 0, thus,

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  1 0 0 1 1 1 1 1₂ 1₃     K L M  =   g(0) g(1) 0  

by solving the system above we get,

K= g(0),

L= −2g(1) − 4g(0), M= 3[g(0) + g(1)], thus,

g(x) = g(0)[1 − 4x + 3x2] + g(1)[−2x + 3x2]. (29)

Figure 3: The function g

6.3 Monotonicity on f (t)

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We see that Hagan and West[2006] examine the monotonicity of the g(x) by studying the behavior of g on [0, 1]. Moreover, in case of g0(0) = g0(1) = 0 coincide to two lines g(1) = −2g(0) and g(0) = −2g(1) which divide the g(0)/g(1) plane into eight sectors (it is labelled into four groups).

We study where g0(0) and g0(1) are positive or negative.

Figure 4: The reformulated possibilities for g The four groups defined as follows:

• (i) if g(0) > 0,−1₂ g(0) ≥ g(1) ≥ −2g(0) or if g(0) < 0,−1₂ g(0) ≤ g(1) ≤ −2g(0). • (ii) if g(0) < 0, g(1) > −2g(0) or if g(0) > 0, g(1) < −2g(0).

• (iii) if g(0) > 0, 0 > g(1) >−1₂ g(0) or if g(0) < 0, 0 < g(1) < −1₂ g(0). • (iv) if g(0) ≥ 0, g(1) ≥ 0 or if g(0) ≤ 0, g(1) ≤ 0.

We analyse the previous groups as follows:

• (i) In these regions g(0) and g(1) are of opposite sign while their derivatives g0(0) and g0(1) are of the same sign. Thus, g monotone as required and g not need to be modified. • (ii) The function of interpolation must be modified in (ii) because the function g is not monotone in these sectors. We see g(0) and g(1) are of opposite sign while their derivatives g0(0) and g0(1) are of opposite sign. Also, we mention that on the boundary A4_{we have g(x) = g(0)[1 − 3x}2_{] and on the boundary}_{B we have g(x) = g(0)[1 − 3x +} 3

2x 2_].

4_{The region boundaries marked}_{A, B, C and D. Moreover, A represents the line g(1) = −2g(0) and B} represents the line g(0) = −2g(1) and so on.

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• (iii) We see g(0) and g(1) are of opposite sign while their derivatives g0(0) and g0(1) are also of opposite sign, thus the function g is not monotone in these regions.

• (iv) We see in these regions g(0) and g(1) are of the same sign and it is not required from g to be monotone in these sectors but the formulas for (iv) and (iii) need to agree onD and also for (iv) and (ii) on C.

The treatment for sectors (ii),(iii) and (iv) will be as in Hagan and West [2006] as follows: 1. The areas including (ii), the function g(x) has modified to become as follows:

g(x) = g(0) 0 ≤ x ≤ η g(0) + (g(1) − g(0))(_1−ηx−η)2 η < x ≤ 1 (30) where η = 1 + 3 g(0) g(1) − g(0)= g(1) + 2g(0) g(1) − g(0) , (31)

and η −→ 0 as g(1) −→ −2g(0) which means g(x) = g(0)(1 − 3x2) at A. This guarantees the monotonicity of g.

2. The areas including (iii), the function g(x) is modified to become as follows: g(x) = g(1) + (g(0) − g(1))( η −x η ) 2 _{0 ≤ x < η} g(1) η ≤ x ≤ 1 (32) where η = 3 g(1) g(1) − g(0), (33)

and η −→ 1 as g(1) −→−1₂ g(0) which means g(x) = g(0)(1 − 3x +3₂x2) atB. This guarantees the monotonicity of g.

3. The areas including (iv). We seek for a formula of an interpolation function that leads to the same formulas defined in (ii) atC and in (iii) at D.

Hagan and West [2006] do the following: 

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1 R 0 g(x)dx =2₃A+1₃g(0)η +1₃g(1)(1 − η), 2 3A+ 1 3g(0)η + 1 3g(1)(1 − η) = 0, hence, we obtain A=−1₂ [ηg(0) + (1 − η)g(1)]. By simple choice of η = g(1) g(1) + g(0), (35)

we get the value of A:

A= −g(0)g(1)

g(0) + g(1). (36)

We notice that

• if g(1) = 0 −→ η = A = 0 first line of g(x) satisfies (iii). • if g(0) = 0 −→ η = 1, A = 0 second line of g(x) satisfies (ii).

6.4 Integrating the g Function

From (29) we have,

f(t) = g t − ti−1 ti− ti−1

+ f_id.

Suppose G0= g where the function g is defined on the interval [t_i−1,t_i] and assume we need to calculateRβ α f(s)ds on ti−1≤ α < β ≤ ti. Then Z β α f(s)ds = (β − α) f_id+ (ti− ti−1) G β − ti−1 t_i− ti−1 − G α − ti−1 t_i− ti−1 . The second evaluation of G should be 0 since our usual integral likewise as in (10)

r(t)t = r(ti−1)ti−1+

Z t

ti−1

f(s)ds, (37)

which implies that we need only one evaluation of G to calculate each r. Clearly, we have many regions for g and the evaluation of G relies on which region exercises. By integrating we get

(i)

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(ii) G(x) = ( g(0)x 0 < x < η g(0)x +1₃(g(1) − g(0))(x−η)_(1−η)32 η ≤ x < 1 (39) (iii) G(x) = ( g(1)x +1₃(g(0) − g(1))[η −(η−x)3 η2 ] 0 < x < η g(1)x +1₃(g(0) − g(1))η η ≤ x < 1 (40) (iv) G(x) =    Ax+1₃(g(0) − A)[η −(η−x)3 η2 ] 0 < x < η Ax+1₃(g(0) − A)η +1₃(g(1) − A)(η−x)_(1−η)32 η ≤ x < 1 (41)

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6.5 Ensuring the Positivity

We have from (28) the function g is defined g(x) = f (t) − f_id =⇒ f (t) = g(x) + f_id. We want f positive everywhere and to achieve that, it is enough for x in [0, 1] that:

g(x) > − f_id. (42)

At the endpoints we get that;

g(0) = fi−1− f_id > − f_id,

and

g(1) = fi− f_id> − f_id,

since the forward function is positive in endpoints. The function g is monotone in regions (i), (ii) and (iii), thus the inequality is satisfied in those regions. It remains the (iv) region where the function g is not monotone. As seen at endpoints the function g is positive and at x = η has a minimum of A. So, we need to prove that A = −_g(0)+g(1)g(0)g(1) > − f_id. Thereby, the sufficient requirement of f to be positive is:

g(0)g(1) g(0) + g(1) < f

d i .

To prove that we see in case 0 < g(0), g(1) < 2 f_id then g(0) + g(1) g(0)g(1) = 1 g(0)+ 1 g(1)> 1 2 f_id + 1 2 f_id = 1 f_id, g(0)g(1) g(0) + g(1) < f d i =⇒ A = − g(0)g(1) g(0) + g(1)> − f d i .

The previous condition 0 < g(0), g(1) < 2 f_idis equivalent to the case that fi−1, fi< 3 fid:

• 0 < g(0) < 2 fd

i −→ fi−1− fid < 2 fid−→ fi−1< 3 fid.

• 0 < g(1) < 2 fd

i −→ fi− fid< 2 fid−→ fi< 3 fid.

The forwards function will be positive also under stricter condition fi−1, fi< 2 fid [4].

6.6 Amelioration

We make the interpolated curve smoother by shifting the forwards fiwhich makes the method

less local (i.e when a yield curve is shifted, the risk will distribute to more parts in the curve than before amelioration). For details, turn to Hagan and West[2006].

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6.7 The Monotone Convex Algorithm

The monotone Convex algorithm is 1. From input data determine the f_id.

2. State fifor i = 1, 2, ..., n as in (25), (26) and (27).

3. If the forward rate f is required to be positive then we should do the following (a) collar f0between 0and 2 f₁d.

(b) collar fibetween 0 and 2 min( f_id, f_i+1d ) for i = 1, 2, ..., n − 1.

(c) collar fnbetween 0and 2 fnd.

Omit this step if it is not required of the forward rate to be positive everywhere. 4. By considering to which of the four sectors we are in, construct the function g. 5. State f as in (29).

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7 Numerical Results

In this chapter we implement the linear, cubic spline and monotone convex methods of inter-polation to bootstrap the EURIBOR swap curve and the Government treasury bonds curve.

7.1 Particular example by Hagan and West

Hagan and West (2006) had used the Table 2 to illustrate the deficiencies of various interpola-tion methods.

Table 2: Table used by Hagan and West [2006] t_i r_i(%) Capitalization factor Discrete forward

0.1 8.1 % 1.00813289 1 7 % 1.07250818 6.88 % 4 5 % 1.22140276 4.33 % 9 7 % 1.87761058 8.60 % 20 4 % 2.22554093 1.55 % 30 3 % 2.45960311 1.00 %

The next figure is derived by the Natural Cubic on r(t) and Bessel method on r(t) for the rates in Table 2 which Hagan and West [2006] had used to illustrate the deficiencies of different interpolation methods.

Figure 5: Forward curve by Cubic spline on rates in Table 2.

On the rates in Table 2, we notice that the Cubic spline method did not preserve the pos-itivity of the forward rates as arises in the figure above while we will see later in Monotone Convex method the forward rate will be positive.

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In other hand, the next figure is derived by the monotone convex method for the rates in Table 2 which Hagan and West [2006] had used to illustrate the deficiencies of different interpolation methods. By this method, for this example, we get both positive and continuous forward rate curve.

Figure 6: Spot and forward rate curves by the monotone convex method to the rates in Table 2 We notice that the Monotone Convex interpolation by Hagan and West [2006] and Hagan and West [2008] method produces a positive and continuous forward rate curve to the rates in Table 2.

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7.2 Bootstrapping EURIBOR Swap Curve

This an example that illustrates the spot curve and the forward curve obtained by bootstrapping the EURIBOR swap curve under the Linear, and the monotone convex methods and cubic spline.

We find that the spot curves more smoother than the forward curves whatever the interpolation method is used. There are no difference in first 12 years for both the forward and spot curves in three interpolation methods. The forward curves are above the zero curves from 3 to around 21 years and then the forward curves start to go down and meet the spot curves when the spot curves start downwards sloping around 21 years. After that the forward curves in all interpolation methods stay below the spot curves.

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Figure 10: Spot rate curves from three different interpolations

Figure 11: Forward rate curves from three different interpolations

7.3 Bootstrapping Government Swedish bonds

This an example that illustrates the spot curve and the forward curve obtained by bootstrapping the Swedish bonds to find the zero-coupon curve under the linear and the monotone convex interpolation methods.

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We find that the spot curves are more smoother than the forward curves whatever the inter-polation method is used. Almost no difference in first 11 years for the spot curves in two interpolation methods and the forward curves shows nearly same. The forward curves are above the zero curves from 3 to around 20 years methods while the forward curves are below the zero curves from beginning to around 3 years for both interpolation.

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vex method has lager values than the one in linear interpolation method after 11 years. For both interpolation methods the forward curves start to go down after 12 years but they remain above the spot curves. Moreover, we find that also for spot rate curves after 11 years, the values with the monotone convex method become larger than that with linear method.

Figure 14: Spot rate curves from two different interpolations

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7.4 Shifting the zero coupon Yield Curve

In risk system, the zero coupon yield curve is used to calculate the risk by shifting the nodes. what does it happen to the yield curve if we shift the nodes?

Suppose we change the value at tiof an input and we would like to know the interval [ti−l,ti+u]

values change in the yield curve. Hagan and West (2006) use l and u as locality indices to display which an interpolation method is local.

The locality indices l = u = 2 for the monotone convex method while the indices are l = i − 1, u= n − i for the natural cubic spline [4]. Moreover, the indices are l = 1, u = 1 for the linear on rates which means that the linear interpolation is the most local.

We consider rates in Table 2 to show the influence of locality. The following figure shows the change in the spot rate curves under the considered method, when changing the input at t = 4, from 4.4% to 5.4%.

Figure 16: Original and “blipped” spot rate curves % obtained by applying monotone convex method of interpolation to the rates in Table 2. “Blipped” curve obtained by changing the input at t = 4, from 4.4% to 5.4%.

We see that with cubic splines by shifting a part of the yield curve the risk is spread out in all other time buckets, thus the whole yield curve will be influenced and therefore this method is unattractive. However, by shifting with monotone convex method a part of the yield curve,

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We notice that Hagan and West (2006) and Hagan and West (2008) the monotone convex method is the only method that satisfy all the criteria5aforementioned in section [4.4].

5_{The short forward rates today are negative due to the market situation. But normally they should be positive.} This can be seen if we look at the discount factor that should always be lower than one. But for short time, it is above 1 today in some currencies such as EUR and SEK.

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8 Conclusions

We conclude that whatever the interpolation methods is used to interpolate the yield curve data, when we have negative interest rates, all methods show weaknesses; they fail to get the positive forward curves everywhere. However, we notice that the monotone convex method is trying to get close to make a positive forward rates and it is close to succeed in the nodes 30, 40 and 50 years.

Hagan and West (2006) and Hagan and West (2008) suggested an ideal spline-based model in comparison with other spline-based models because of its accuracy and simplicity. The boot-strap method is a simple iterative procedure which yields a curve that is capable of repricing all input securities exactly (except after long term i.e after the node 11 years in case of bond curve and we think that belongs to the choice of the last two bonds).

When a Yield curve with the monotone convex interpolation method is shifted, the risk is not distributed on all nodes on the curve. Thus, it preserves the locality where the change in inputs affects only nearby.

We test the effectiveness of the monotone convex method with negative interest rates com-pared to other traditional interpolation methods and this method shows good results.

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A

Appendix

A.1 Instruments selection

In bootstrapping, it is important to select reasonable instruments and these instruments should have different maturities. Our choice base on choosing several similar instruments that have same underlying which can help with the required information to bootstrap the yield curve. In our thesis we focus on EURIBOR market (The Euro Interbank Offered Rate), published by the European Money Markets Institute (EMMI) and the Swedish treasury bonds.

A.2 EURIBOR Data

The rates are chosen to be up 50 years and the start date 4/16/2020. We select money market instruments S/W spot corporate week cash deposit for one week and the short term 1M and 3M months of the curve. We also use the FRA (1 × 4, 2 × 5, 3 × 6, 4 × 7) because it is more liquid than short term.

Finally, we choose the swaps (1Y, 15M, 18M, 21M, 2Y, ..., 12Y, 15Y, 20Y, 25Y, 30Y, 40Y, 50Y )

Table 3: Cash deposit with maturity S/W, 1M, 3M

Name Quote(%) To date Days

EURIBOR S/W -0.518 4/23/2020 7

EURIBOR 1M -0.404 5/18/2020 32

EURIBOR 3M -0.248 7/16/2020 91

Table 4: FRA 3M

EUR 1X4 FRA -0.27 8/17/2020 123

EUR 2X5 FRA -0.299 9/16/2020 153

EUR 3X6 FRA -0.322 10/16/2020 183

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Table 5: Swaps from EURIBOR IRS 3M

EUR 1Y IRS 3M -0.347 4/16/2021 365 EUR 15M IRS 3M -0.363 7/16/2021 456 EUR 18M IRS 3M -0.373 10/18/2021 550 EUR 21M IRS 3M -0.377 1/17/2021 641 EUR 2Y IRS 3M -0.38 4/18/2022 732 EUR 3Y IRS 3M -0.371 4/17/2023 1096 EUR 4Y IRS 3M -0.346 4/16/2024 1461 EUR 5Y IRS 3M -0.31 4/16/2025 1826 EUR 6Y IRS 3M -0.271 4/16/2026 2191 EUR 7Y IRS 3M -0.229 4/16/2027 2556 EUR 8Y IRS 3M -0.184 4/16/2028 2923 EUR 9Y IRS 3M -0.139 4/16/2029 3287

EUR 10Y IRS 3M -0.094 4/16/2030 3652

EUR 11Y IRS 3M -0.05 4/16/2031 4017

EUR 12Y IRS 3M -0.008 4/16/2032 4383

EUR 15Y IRS 3M 0.094 4/16/2035 5478

EUR 20Y IRS 3M 0.157 4/16/2040 7305

EUR 25Y IRS 3M 0.014 4/16/2045 9132

EUR 30Y IRS 3M 0.092 4/16/2050 10959

EUR 40Y IRS 3M 0.016 4/16/2060 14610

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A.3 Swedish Bonds Data

The data is taken from www.finansportalen.se at 15 May 2020 (Table 6) The day count con-vention: 30/360 and the Swedish bonds are quoted in ytm yield to maturity. There is one coupon per year for bonds.

Table 6: Swedish Treasury Bonds from finansportalen.se

Name YTM(%) Coupon To date Days

RGKT 2005 -0.10 2020-05-15 5 RGKT 2006 -0.10 2020-06-17 32 RGKT 2007 -0.11 2020-07-15 60 RGKT 2008 -0.14 2020-08-19 94 RGKT 2009 -0.15 2020-09-16 121 RGKT 2010 -0.23 2020-10-21 156 RGKT 1047 -0.23 5 2020-12-1 196 RGKT 1054 -0.38 3.5 2022-06-1 736 RGKT 1057 -0.35 1.5 2023-11-13 1258 RGKT 1058 -0.31 2.5 2025-05-12 1797 RGKT 1059 -0.25 1.0 2026-11-12 2337 RGKT 1060 -0.18 0.75 2028-05-12 2877 RGKT 1061 -0.09 0.75 2029-11-12 3417 RGKT 1062 0.06 0.13 2031-05-12 3957 RGKT 1056 0.13 2.25 2032-06-01 4336 RGKT 1053 0.34 3.5 2039-03-30 6795

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A.4 Bootstrap Swap curve procedures

The instruments that we will bootstrap are EURIBOR-type instruments, FRAs, and swaps. The assumption is that the swaps expire after the EURIBOR instruments and FRAs. We will show only the bootstrap with Monotone Convex interpolation method because it has different processing and we will use the iteration way.

A.4.1 Deposits

We Find the NACC rates corresponding to the EURIBOR instruments: The capitalization factor for the first instrument,

C(7d) = (1 + −0.00518 ×₃₆₅7 ) = 0.999901, thus, the zero rate is,

Z(7d) = 100 × ln(C(7d)) = −0.518025731%. The capitalization factor for the 1M instrument,

C(1M) = (1 + (−0.00404) ×₃₆₅32 ) = 0.999646, thus, the zero rate is,

Z(1M) = 100 × ln(C(1M)) = −0.404071564%. The capitalization factor for the 2M instrument,

C(2M) = 0.999517, ( Monotone convex interpolation), thus, the zero rate is,

Z(2M) = 100 × ln(C(2M)) = −0.289050000%. The capitalization factor for the 3M instrument,

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A.4.2 FRA

We Find the NACC rates corresponding to the FRA instruments: The capitalization factor for the FRA1X 4 instrument,

C(FRA1X 4) = C(1M) × (1 + (−0.0027) ×₃₆₅91) = 0.998973,

and the zero rate,

Z(FRA1X 4) = 100 × ln(C(FRA1X 4)) ×365₁₂₃ = −0.304947670%.

The capitalization factor for the FRA2X 5 instrument,

C(FRA_{2X 5}) = C(2M) × (1 + (−0.00299) ×₃₆₅92 ) = 0.998753, and the zero rate,

Z(FRA_{2X 5}) = 100 × ln(C(FRA_{2X 5})) ×365₁₅₃ = −0.29759978%. The capitalization factor for the FRA3X 6 instrument,

C(FRA3X 6) = C(3M) × (1 + (−0.00322) ×₃₆₅92 ) = 0.998571,

thus, the zero rate

Z(FRA3X 6) = 100 × ln(C(FRA3X 6)) ×365₁₈₃ = −0.285306054%.

The capitalization factor for the FRA4X 7 instrument,

C(FRA4X 7) = C(FRA1X 4) × (1 + (−0.00347) ×₃₆₅91 ) = 0.998109,

thus the zero rate,

Z(FRA_{4X 7}) = 100 × ln(C(FRA_{4X 7})) ×365₂₁₄ = −0.322893599%. From above we get the FRA periods are of length 91,92,92,91 days.

Table 7: Bootstrapped deposits and FRA

Name Zero rates(%)

EURIBOR S/W -0.518025731 (%) EURIBOR 1M -0.404071564 (%) EURIBOR 2M -0.289050000 (%) EURIBOR 3M -0.248076701 (%) EUR 1X4 FRA -0.304947670 (%) EUR 2X5 FRA -0.297599781 (%) EUR 3X6 FRA -0.285306054 (%) EUR 4X7 FRA -0.322893599 (%)

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A.4.3 Swaps

First, we guess the rates at the nodes as NACQ rate. Then we use Monontone Convex interpo-lation quarterly for missing nodes between the given nodes. We obtain columns0rAnd then

we find next column1r.

We find the values for 1Y, 15M, 18M, 21M, 2Y to12Y, 15Y, 20Y, 25Y, 30Y, 40Y and 50Y by ap-plying the following formula;

1r(tn·m) = −1 tn·m ln" 1 − Rn·m∑ n−1 j=1αj·mexp(−t3 jm· (0r3 j·m)) 1 + Rn·mαn·m # ,

where n is the number of months, this gives us the required iterative formula for bootstrap where all terms in the right are found by interpolation on the estimated curve. While the term in the left is used for next iteration. By doing this for all instruments 8 iterations in our data until convergence.

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We then get the following rates,

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A.5 Bootstrap Swedish bonds curve procedures

The price of bond with no coupon can be calculated using the given formula.

P= 100

(1 + ytm ·₃₆₀d ), where d is the number of days and ytm is the yield to maturity.

The price of bond paying coupons is calculated using the next formula,

P= N (1 + ytm)T + n

∑

i=1 C (1 + ytm)ti,

where P is the price of bond, N is the face value, C is the coupon payments, ytm is the yield to maturity, T is the time to maturity and tiis the times for individual coupons. Also, the present

value of all the coupons

PV = C

(1 + r(d))360d

. The zero-Coupon Price

ZCP= P − PV,

where, P is the price of the bond, PV is the present value of the coupons and C represent the coupons. We have bootstrapped a zero-coupon curve using Excel. The data for the Swedish Government bonds and Treasury Bills, is taken from the website www.finansportalen.se /mark-nadsrantor/ and it is collected on 15th May 2020. We have used the data to calculate the bond price, present value of coupons, spot rate and the forward rate.

A.5.1 Bootstrap with Linear Interpolation

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r(196) = −0.23.

1) First bond RGKB1054: We start bootstrap the bond RGKB 1054 which has coupon rate c= 3.5% and ytm = −0.38. Time to maturity is ((2022-06-01)–(2020-05-15) = 2y16d = 360+360+16 = 736 days. We have two coupons payment of 3.5% at (376 day, 16 days from now 2020-05-15). We start calculating the present values of the coupons using interpolation and extrapolation:

r(16) = −0.10 , r(376) = −0.23. The present value of the coupons is

PV = 3.5

(1 + (−0.001))36016

+ 3.5

(1 + (−0.0023))376360

= 7.0085.

The price of the bond is

PV = 3.5 (1 + (−0.0038))36016 + 3.5 (1 + (−0.0038))376360 + 103.5 (1 + (−0.0038))736360 = 111.3233.

Now we have calculated the zero-coupon price with maturity T= 736 days have the present value

ZCP= P − PV = 111.3233 − 7.0085 = 104.3147. This gives the zero-coupon rate for T=736 days:

104.3147 = 103.5 (1 + r)736360 , r(736) = ( 103.5 104.3147) 360 736− 1 = −0.00383 = −0.383%.

this is close to (ytm = −0.38%) which shows that the result is correct.

We repeated the same process as above for each bond. We calculated zero coupon rate, Present Value of Coupons, Bond Price and Zero-coupon rate which are given in the table below. We get the following results of bootstrapping the Swedish government bonds with linear in-terpolation,

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Table 8: Bootstrap Swedish Bonds with linear interpolation

Name YTM(%) Coupon To date df linear ZCP Linear(%) FW Linear

RGKT 2005 -0.10 2020-05-15 1.000013 -0.1 -0.1 RGKT 2006 -0.10 2020-06-17 1.000088 -0.1 -0.1 RGKT 2007 -0.11 2020-07-15 1.000183 -0.11 -0.1214 RGKT 2008 -0.14 2020-08-19 1.000365 -0.14 -0.1929 RGKT 2009 -0.15 2020-09-16 1.000504 -0.15 -0.1848 RGKT 2010 -0.23 2020-10-21 1.000997 -0.23 -0.5065 RGKT 1047 -0.23 5 2020-12-1 1.001253 -0.23 -0.23 RGKT 1054 -0.38 3.5 2022-06-1 1.007860 -0.383 -0.4385 RGKT 1057 -0.35 1.5 2023-11-13 1.012305 -0.35 -0.3035 RGKT 1058 -0.31 2.5 2025-05-12 1.015493 -0.308 -0.2099 RGKT 1059 -0.25 1.0 2026-11-12 1.016216 -0.2478 -0.0475 RGKT 1060 -0.18 0.75 2028-05-12 1.014255 -0.1771 0.1287 RGKT 1061 -0.09 0.75 2029-11-12 1.008071 -0.0847 0.4076 RGKT 1062 0.06 0.13 2031-05-12 0.993241 0.0617 0.9881 RGKT 1056 0.13 2.25 2032-06-01 0.980322 0.165 1.2435 RGKT 1053 0.34 3.5 2039-03-30 0.931138 0.378 0.7536

A.5.2 Bootstrap with Monotone Convex Method Interpolation We start with bills which directly give us the zero-coupon yields

r(5) = −0.10, r(32) = −0.10, r(60) = −0.11, r(94) = −0.14, r(121) = −0.15,

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2. We solve the equation (12) for rnby inserting all these rates what we get from first step.

3. The new guess is taken for this bond, of course also for all other bonds, and once more apply the algorithm of interpolation.

We repeat this process 15 iterations until we reach a fixed point, then we get the yield curve.

Figure 19: Iteration of a Swedish bonds curve bootstrap

We get the following results of bootstrapping the Swedish government bonds with the monotone convex interpolation

Table 9: Bootstrap Swedish Bonds with Monotone Convex Interpolation Method

Name YTM(%) Coupon To date Df MC ZCP MC(%) FW MC

RGKT 2005 -0.10 2020-05-15 1.00001 -0.1 -0.1 RGKT 2006 -0.10 2020-06-17 1.00009 -0.1 -0.1 RGKT 2007 -0.11 2020-07-15 1.00018 -0.11 -0.1214 RGKT 2008 -0.14 2020-08-19 1.00037 -0.14 -0.1929 RGKT 2009 -0.15 2020-09-16 1.00050 -0.15 -0.1848 RGKT 2010 -0.23 2020-10-21 1.00100 -0.23 -0.5065 RGKT 1047 -0.23 5 2020-12-1 1.00125 -0.23 -0.23 RGKT 1054 -0.38 3.5 2022-06-1 1.00819 -0.3991 -0.4605 RGKT 1057 -0.35 1.5 2023-11-13 1.01228 -0.3491 -0.2786 RGKT 1058 -0.31 2.5 2025-05-12 1.01485 -0.2953 -0.1698 RGKT 1059 -0.25 1.0 2026-11-12 1.01538 -0.2351 -0.0344 RGKT 1060 -0.18 0.75 2028-05-12 1.01274 -0.1584 0.17348 RGKT 1061 -0.09 0.75 2029-11-12 1.00486 -0.0511 0.5205 RGKT 1062 0.06 0.13 2031-05-12 0.99212 0.0720 0.8510 RGKT 1056 0.13 2.25 2032-06-01 0.94265 0.4904 4.8585 RGKT 1053 0.34 3.5 2039-03-30 0.83075 0.9824 1.8499

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B

Appendix

B.1 Arbitrage Potential

We explain how arbitrage opportunities may arise when we use different methods of interpo-lation where there is not agreed a unified method of interpointerpo-lation. By considering an example of the EURIBOR swap curve.

Figure 19 shows the spot and forward rate curves that were obtained by applying linear in-terpolation on r(t), natural cubic spline on r(t) and the monotone convex inin-terpolation to the EURIBOR Data (Table 3 Table 4 and Table 5).

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Table 10: Bootstrap Swaps rates from EURIBOR IRS 3M Swap Linear Natural Cubic Spline Monotone Convex

1y -0.352 -0.352 -0.347 2y -0.382 -0.382 -0.381 3y -0.373 -0.373 -0.371 4y -0.347 -0.347 -0.347 5y -0.311 -0.311 -0.312 6y -0.272 -0.272 -0.273 7y -0.230 -0.230 -0.232 8y -0.185 -0.185 -0.187 9y -0.140 -0.140 -0.142 10y -0.095 -0.095 -0.097

B.2 Discount function

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B.3 Code of Interpolation

A VBA class in Excel created by Hagan and West for the Monotone Convex interpolation method. The Excel file is available from the Graeme West’s website www.finmod.co.za.

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C

Appendix

C.1 Criteria for a Master thesis

In this section, we explain how thesis objectives satisfies the requirements by the Swedish National Agency for Higher Education for Master degree (2 years).

Objective 1: Knowledge and understanding.

In this thesis, we started with an introduction of the Yield (Government and inter-bank) curves that are used to generate the cash-flows to find the present value. We reviewed the literature on a number of spline-based models available of yield curves construction and showed the weaknesses and strengths of each model. Hagan and West (2006) method (the monotone con-vex method) is the ideal method which we reviewed in details and implemented to bootstrap the EURIBOR Swap and Swedish Government bond Curves. Also, we have surveyed several traditional interpolation methods.

Objective 2: Methodological knowledge.

In this thesis, we discussed in details the monotone convex interpolation method as a financial mathematical problem. Clearly, we described all the details on this method in chapter (6) and presented a numerical results in chapter (7) to bootstrap the EURIBOR Swap and Swedish Bond curves with monotone convex interpolation method. In appendices, we have described the procedures of bootstrapping processes.

Objective 3: Critically and Systematically Integrate Knowledge.

We used different sources to get the information to develop the main concept with supporting by supervisor Jan Röman. Also, many comparisons with alternative interpolation methods have done.

Objective 4: Ability to Critically, Independently and Creatively Identify and Carry out Advanced Tasks.

We tested the effectiveness of the monotone convex method with negative interest rates com-pared to other traditional interpolation methods. We showed a significant ability to evaluate the solutions and to present both algorithms and results of experiments. Moreover, identify and formulate questions, within a given time frame.

Objective 5: Ability in both national and international contexts, Present and Discuss Conclusions and Knowledge

We have described our results in the Numerical results and Conclusion sections in a simple way that any reader who has basic concepts in financial mathematics can understand.

Objective 6: Scientific, Social and Ethical Aspects.

This thesis will help the reader understand the areas of research in financial mathematics with respect to yield curves by comparing traditional interpolation methods with the monotone con-vex method . Our numerical results show that this method is more accurate and aware to solve a financial problem when compared to traditional interpolation methods. This method also opens the way to search for other new methods and developments.

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D

List of References

References

[1] Ken Adams. “Smooth interpolation of zero curves”. In: Algo Research Quarterly 4.1/2 (2001), pp. 11–22.

[2] Richard Burden and JD Faires. Numerical analysis. Cengage Learning, 2011.

[3] Felix Geiger. The yield curve and financial risk premia: Implications for monetary pol-icy. Vol. 654. Springer Science & Business Media, 2011.

[4] Patrick S Hagan and Graeme West. “Interpolation methods for curve construction”. In: Applied Mathematical Finance13.2 (2006), pp. 89–129.

[5] Patrick S Hagan and Graeme West. “Methods for constructing a yield curve”. In: Wilmott Magazine, May(2008), pp. 70–81.

[6] John Hull et al. Options, futures and other derivatives/John C. Hull. Upper Saddle River, NJ: Prentice Hall, 2009.

[7] James M Hyman. “Accurate monotonicity preserving cubic interpolation”. In: SIAM Journal on Scientific and Statistical Computing4.4 (1983), pp. 645–654.

[8] J Huston McCulloch. “The tax-adjusted yield curve”. In: The Journal of Finance 30.3 (1975), pp. 811–830.

[9] J Huston McCulloch, Levis Abraham Kochin, et al. The inflation premium implicit in the US real and nominal term structures of interest rates. Charles A. Dice Center for Research in Financial Economics, Fisher College . . ., 2000.

[10] Jan RM Röman. Analytical finance. Springer, 2017.

[11] Uri Ron et al. A practical guide to swap curve construction. Citeseer, 2000.

[12] Gary S Shea. “Interest rate term structure estimation with exponential splines: a note”. In: The Journal of Finance 40.1 (1985), pp. 319–325.

[13] Oldrich A Vasicek and H Gifford Fong. “Term structure modeling using exponential splines”. In: The Journal of Finance 37.2 (1982), pp. 339–348.

Interpolation of Yield curves

School of Education, Culture and Communication

Division of Applied Mathematics

MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Interpolation of Yield curves

by

Abdulhamid Iebesh

School of Education, Culture and Communication

Division of Applied Mathematics

Dedication

To my dear father, the real motivation for my

ambition-Walid Iebesh

Contents

List of Figures

List of Tables

1

Introduction

2

Literature Review

∑

2.1

McCulloch (1975)

2.2

Vasicek and Fong (1982)

∑

2.3

Hagan and West (2006)

3

Financial Instruments

3.1

Zero Coupon Pricing

3.2

Discount Function

3.3

Interest Rates

3.4

Interest Rate Instruments

∑

4

Constructing Yield Curves

4.1

The Yield Curve

4.2

Forward Rates

4.3

Bootstrap and Interpolation

∑

∑

∑

∑

4.4

Quality Criteria of Curve Construction

5

Traditional Interpolation Methods

5.1

Linear methods

5.2

Cubic Splines

6

Monotone Convex Method of Interpolation

6.1

Proper Forward Rates

6.2

The Basic Interpolator

6.3

Monotonicity on f (t)

6.4

Integrating the g Function

6.5

Ensuring the Positivity

6.6

Amelioration

6.7

The Monotone Convex Algorithm

7

Numerical Results

7.1

Particular example by Hagan and West

7.2

Bootstrapping EURIBOR Swap Curve