1

Bonds and Related Issues

kicheon.chang@yahoo.co.kr

2

Contents

Topics Contents

Numerical Method 보간법행렬연산SimulationSolver/Optimization

Bond Price/Duration/ConvexityImmunizationFinding YTM

Term structure Term structure 이론BootstrappingFitting the yield curve

Bond Futures 채권선도 (FRA)국고채선물 이론가 계산

3

Preface• “TheoryExcelVBA” Type

– Memorize it in your hand rather than in your head!!

• Engineering Approach– 수학적 엄밀성보다는 직관적 이해– (ex)

• A bird in the hand is worth two in the bush– Do not hesitate to raise your hand whenever questionable– Use the break time, e-mail etc..

• Two way vs One way

4

Numerical Methods

5

Numerical Method & Finance

6

Numerical Method/Simultaneous equations

• Why simultaneous equations?

– (ex) (Polynomial, Spline) Interpolation, Finite Difference Method, Bootstrapping…

• How to do?

7

Numerical Method/Cholesky Decomposition

• Why?– We know– What is

• How?

• Where(application)?– Generating correlated random variables

1.414213562

?43

21

8

Numerical Method/Linear Interpolation

Linear interpolation Log interpolation

Exponential interpolation

9

note

10

Numerical Method/Polynomial interpolation

Polynomial interpolation

00 , yx

11, yx

nn yx ,

N+1 equationsN+1 unknowns

11

Numerical Method/Polynomial Interpolation

Example

12

Numerical Method/Polynomial interpolation

n10 BFBFBF xf

Lagrange polynomial interpolation

00 , yx

11, yx

nn yx ,

Base function(0)

00 , yx

11, yx

nn yx ,

00 , yx

11, yx

nn yx ,

Base function(1)

xLxf iiiBF

13

Numerical Method/Polynomial interpolation

Lagrange polynomial interpolation

14

Numerical Method/Polynomial interpolation

the problem

15

Numerical Method/Spline interpolation

16

Numerical Method/spline interpolation2 차 스플라인 보간법 3 차 스플라인 보간법

17

Numerical Method/2D interpolation

18

Numerical Method/Optimization: Solver

19

Numerical Method/Bisection method

20

Numerical Method/Newton method

1 2 3 4 5 6 7 8 9

10 11 12 13 14

A B C D E F G =E4 =B4^3+2*B4+1 =3*B4^2+2

=B4-C4/D4i x(i) f(x) f'(x) x(i+1) epsilon

0 10 1021 302 6.61920531 6.6192053 304.25147 133.44164 4.3391713 716.74853 =ABS(C5-C4)2 4.33917129 91.378028 58.485222 2.776759 212.873443 2.77675904 27.963415 25.131172 1.6640606 63.4146124 1.66406064 8.9360679 10.307293 0.7970951 19.0273475 0.79709512 3.1006331 3.9060819 0.0032989 5.83543486 0.00329889 1.0065978 2.0000326 -0.499992 2.09403537 -0.4999918 -0.124977 2.7499754 -0.454545 1.13157538 -0.45454505 -0.003004 2.6198336 -0.453398 0.12197339 -0.45339834 -1.79E-06 2.6167102 -0.453398 0.0030024

10 -0.453398 -6.38E-13 2.6167083 -0.453398 1.792E-06

Algorithm

21

Caution!

• There is no panacea!

– Try possible Initial values– 함수의 전반적 형태파악

• 단조증가 / 감소함소

22

Numerical Method/Simulation/Uniform Random Variable

RAND

0 1

1 1

10 xp a

dxxp0

RND

23

Numerical Method/Simulation/Uniform RV

Examples

Calculating PI

24

Numerical Method/Simulation/Transforming RV

• 12 난수법– Definition:

– Why 12 RVs?

– Central Limit Theorem:

– Does CLT really work?

12

1

~~i

ixz

?~ zE ?~ zV

21,0~ Nz CLT

FREQUENCY(samples, 구간 )

25

Numerical Method/Simulation/Transforming RV

• Transform method– Idea

– Functions: NormSDist(x), NormSInv(p)

– test

z0

1

pxN

pNx 1

0

10

20

30

40

50

60

70

0

20

40

60

80

100

120

140

Transform

26

Multi-variable case

• From uncorrelated RVs to correlated RVs

0, 21 xx

2

1transform

2

1

y

y

x

x 9.0, 21 yy

2

1

2

121

01

y

y

x

x

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

A B C D E F G H I J K L rho= 0.99

=rho*A4+SQRT(1-rho^2)*B4x1 x2 y2

0.562829417 0.759027 0.6642750.745888995 0.587151 0.8212580.616898146 0.768017 0.7190710.90212717 0.002266 0.893425

0.461556773 0.160399 0.4795680.492802736 0.026995 0.4916830.331733922 0.170176 0.3524230.505072821 0.5315 0.5749990.073725059 0.265133 0.1103890.084891145 0.927261 0.2148480.841372506 0.330709 0.8796110.100927256 0.417909 0.1588710.495363841 0.17961 0.5157470.159663535 0.266904 0.1957180.471160815 0.481691 0.53440.897593149 0.473643 0.9554330.860525513 0.432385 0.912916

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

27

Bonds

28

Bonds/Market

채권시장

발행시장

유통시장 장내시장

장외시장

국채전문유통시장

일반채권거래시장

대고객상대매매시장

Inter Dealer Broker

사모발행

공모발행 직접발행

간접발행

매출발행

공모입찰발행

위탁모집

인수모집 잔액인수

총액인수

Conventional

Dutch

29

Bonds/Market

자료 :www.ksdabond.or.kr

30

Bonds/Market

자료 :www.ksdabond.or.kr

31

Bond/price

32

Bond/Price

Example2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A B C D E F G H I issue 97-12-27 1.calcuate "A"mat 00-12-27 date i cf df pvfreq 4 98-03-27 0 300 1 300cpn 12% 98-06-27 1 300 0.953516 286.0548yield 19.50% 98-09-27 2 300 0.909193 272.7579notional 10,000 98-12-27 3 300 0.86693 260.079trade 98-01-20 99-03-27 4 300 0.826632 247.9895daycount 30/360 99-06-27 5 300 0.788207 236.462

99-09-27 6 300 0.751568 225.4703price(함수) 8351.597 99-12-27 7 300 0.716632 214.9896

00-03-27 8 300 0.68332 204.99600-06-27 9 300 0.651557 195.46700-09-27 10 300 0.62127 186.380900-12-27 11 10300 0.592391 6101.625

8732.272let's try 98-03-01 2.caculate "B"

98-01-20 A set B setDSC 66 67 dirty priceE 90 90 8432.723 A setdf 0.965696 0.965186 8428.264 B set

3.calcuate Accrual Interest clean priceAccrual 80 76.66667 8348.264 A set

8351.597 B set

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Bonds/Excel functions정리 그밖에 채권 관련된 함수

함수명 : 설명

YIELD : 정기적으로 이자를 지급하는 유가증권의 수익률

YIELDDISC : 할인된 유가증권의 연수익률

YIELDMAT : 만기 시 이자를 지급하는 유가증권의 연 수익률

COUPDAYSNC : 결산일부터 다음 이자 지급일까지의 날짜 수

COUPDAYBS : 이자 지급기간의 시작일부터 결산일까지의 날짜 수

COUPDAYS : 결산일이 들어 있는 이자 지급 기간의 날짜 수

COUPNCD : 결산일 다음 첫 번째 이자 지급일

COUPPCD : 결산일 바로 전 이자 지급일

COUPNUM: 결산일과 만기일 사이의 이자지급횟수 ( 정수로 반올림 ) 

참고 이자기간함수간의 관계COUPDAYS = COUPDAYSNC+COUPDAYBS = COUPNCD-COUPPCD

34

Bonds/Terminology

• Conventional price

• Theoretical price

• Dirty price– Cash price– Invoice price

• Clean price– Quoted price

35

Bonds/note

• Conventional VS Theretical

• Convert between c-price and t-price using Excel Function “PRICE”

정리 관행적 계산법 vs 이론적 계산법(1) 관행적 계산법 : 2 단계 할인할 때 단리 할인법 적용(2) 이론적 계산법 : 2 단계 할인할 때 복리 할인법 적용

accrual

41

100

41

4100

41

11

11

N

N

kk

E

daysyldyld

rate

yldp

1pA

발행일 거래일 이자지급일 1 이자지급일 2 이자지급일 3 …… 만기일

1 단계 할인2 단계 할인

36

Bonds/Day count conventionCountDay RateInterest Amount PrincipalAmountInterest

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Bonds/User Defined Function

Test Numbers

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Bonds/Duration

• History of bond sensitivity– Maturity– CF Weighted Average Term to Maturity– PV Weighted Average Term to Maturity

• Meaning of Macauley Duration– Investment Horizon → Immunization– Sensitivity → Modified Duration

P

dctwtD ii

n

iii

n

iiMAC

11

DurationMacauley

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Bonds/Modified Duration

• What we want to know is…– What if the yield moves up 1%p?

• Mac. Duration does not give the answer

• Modified Duration

y

y

P

P

y

y

P

P

A

BDmac

)1(

)1(

Py

Py

P

P

y

DD mac 1

1

40

Bonds/Convexity

• If we hedged the bond’s duration, then what happens to the value of our portfolio when yield moves?

• Convexity

– 듀레이션 헤지된 포트폴리오의 손익변화 측정에 사용– See the Taylor Expansion

P

dy

Pd

C

2

2

)(

41

Bonds/Numerical Convexity

21111 2111

Y

PPP

PY

PP

Y

PP

YP

dY

dP

dY

d

PdY

Pd

PC

112

2

Difference approximation

42

note

(Excel2007 에서는 위 설정없이 “ Application.price” 로 사용가능 )

43

Bonds/Taylor Expansion

• We want to decompose(analyse) any function to “polynomial functions”

• How to find the coefficients– Above equation should hold when x=0

– Differentiating and inputting x=0 still hold equality

– and so on

...2210 xaxaaxf

00 fa

01 fa

...000 2!2

1 xfxffxf

44

Bonds/Taylor expansion

• Taylor series expansion of bond price

• The relationship

– 수익률의 1 차 변화분에 의한 채권가격의 변화율은 듀레이션에 의해서 설명되며 ,

– 수익률의 2 차 변화분에 의한 채권의 가격변화율은 컨벡시티에 의해 설명된다 .

...)()(2

1 2

2

2

01 ydy

Pdy

dy

dPPPP

2

2

1yCyD

P

P

45

Bonds/Summary

46

Bonds/Summary

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A B C D E F G H I J K L

/ /채권가격 듀레이션 컨벡시티 =1/(1+$B$4)^D4

n 3 Price Duration Convexityyield 4.51% i df cf pv w(i) w*i i*(i+1) c/(1+y)^(i+2)coupon 5% 1 0.9568462 0.05 0.0478423 0.0472066 0.0472066 2 0.043802254 0.0876045

2 0.9155547 0.05 0.0457777 0.0451694 0.0903388 6 0.041912022 0.2514721

거래일 07-02-22 3 0.8760451 1.05 0.9198473 0.907624 2.722872 12 0.842170569 10.106047

만기일 10-02-22 sum 1.0134674 2.8604175 10.445123convexity= 10.306324

price 101.346739 =price(B7,B8,B5,B4,100,1,0)duration 2.86041745convexity 10.3063244 =DURATION(B7,B8,B5,B4,1,0)

Convexity 손으로 계산하기 =Convexity(B7,B8,B5,B4,1,0)p0 101.374482p1 101.346739p2 101.319005dYld 0.0001d^2P/dy^2 1044.51237 =(B17-2*B16+B15)/(B18)^2Convexity 10.3063244 =B19/B10Convexity 검증dy 0.001P0 101.346739P1 101.069877 =B23-B11*B23/(1+B4)*B22+0.5*B19*B22^2

101.069876 =price(B7,B8,B5,B4+B22,100,1,0)

47

Bonds/Immunization

Alternatives

Request

48

Bond/ImmunizationExample

Check “Macauley duration”

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

A B C D E F G H I J

투자채권 명세목표투자기간 5.5만기 15 현재 05-11-01표면금리 6.00% 거래일 11-05-01이자횟수 2 만기일 20-11-01Price 100투자원금 100 =price($E$12,$E$13,$B$12,A20,100,2,3)*$B$15/$B$14

=$B$15*$B$12*$B$10*$B$15/$B$14시나리오 =-FV(A20/2,$B$10*2,$B$15*$B$12/$B$13,0,0)*$B$15/$B$14

=IF(ISERROR(H20),100,H20) =D20+C20Yield.chged 원래이자 복리 채권가격 만기회수액 유효수익률 =RATE($B$10*$B$13,,-$B$15,E20,0)*2

8.50% 33 40.98731 83.92578 124.91309 4.09% 83.925788.00% 33 40.45905 86.86606 127.32511 4.44% 86.866067.50% 33 39.93865 89.93702 129.87566 4.81% 89.937027.00% 33 39.42598 93.14508 132.57106 5.19% 93.145086.50% 33 38.92093 96.49703 135.41796 5.59% 96.497036.00% 33 38.42339 100 138.42339 6.00% 1005.50% 33 37.93325 103.6615 141.59475 6.42% 103.66155.00% 33 37.4504 107.4894 144.93984 6.86% 107.48944.50% 33 36.97473 111.4922 148.46691 7.32% 111.49224.00% 33 36.50615 115.6785 152.18461 7.78% 115.67853.50% 33 36.04453 120.0576 156.1021 8.26% 120.0576

Scenario Analysis

49

Bond/Finding YTM(1)

• Why numerical method?

• Bisection MethodMARKETPPE

Pfy 1 yfP ???

y

LL Ey ,

HH Ey ,

mm Ey ,

50

Bond/Finding YTMVBA code

51

Bond/Finding YTM(2)

2 3 4 5 6 7 8 9

10 11 12 13 14 15

A B C D E F G H I J K L find YTM using Newton method =PRICE($B$4,$B$5,$B$6,E7,$B$8,$B$9,$B$10)bond spec Newton method =$B$11-F7settlement 08-07-01 target f= f(y)=p-p(y) =F8/(1+E8)

maturity 11-07-01 =I8*H8rate 5% y p(y) f p/(1+y) duration f' y(i+1)yield 6% unknown 0.0000% 115 -17.673 115 2.869565 330 5.3555% =E8-G8/J8redemption 100 5.3555% 99.03841 -1.71142 94.00406 2.858678 268.7274 5.9923%frequency 1 5.9923% 97.34714 -0.02015 91.84358 2.857363 262.4305 6.0000%basis 0 6.0000% 97.32699 -2.9E-06 91.81792 2.857347 262.3557 6.0000%price 97.32699 knownduration 2.857347 =DURATION($B$4,$B$6,$B$7,E8,$B$10,$B$11)P/(1+y) 91.81791dP/dy 262.3557

yield

yPPyf market

iy 1iy

52

Term structure of Interest rates

53

TS/Terminology

• Zero rate– Definition:

• Par yield– Definition: such that

• Forward rate(Implied forward rate, Forward rate agreement)– see FRA for pricing– Relationship between spot rate and forward rate

• Discount factor– _

*y

TyP

*

MARKET

1

1

TyeP

*MARKET

nn

ii

i

yy

c*

1* 1

100

1100

54

Term structure of interest rates

55

Term structure of interest rates

• Which one?(in terms of modeling yield curve)– Yield of zero coupon bond

– Zero price• Cubic functions

– Forward rate• Nelson-Seigel function

tsdd ts

0f

0r is not enough

56

Term Structure of Interest rates

57

TS/Why yields differ?

58

TS/Issues

• Finding the current term structure of interest rates– Fitting Yield Curve

– To price illiquid bonds

• Estimating the future term structure of interest rates– Economics/Econometrics

– To trade bonds

• Modeling the future term structure of interest rates– Finance

– To price Fixed Income Derivatives

59

Finding Yield Curve

• Bootstrapping and Interpolation– 다양한 만기의 이자율 상품의 가격이 고시되는

경우– Interest Rate Swap Market

• Functional Approach– Function types

• Cubic function

• Piece-wise cubic function

• Nelson-Seigel function

60

TS/Bootstrapping

61

TS/Bootstrapping

Example

행렬을 이용하는 방법

62

TS/BootstrappingMore considerations

Available prices Bootstrapping formula

25.0d 5.0d 75.0d 0.1d

63

TS/Bootstrapping/more consideration

Example

64

TS/Bootstrapping/more consideration

Example

65

TS/Bootstrapping/summary

66

TS/Fitting the yield curve

67

TS/Functional Forms

68

Polynomial Model/Cubic Function

• Yield curve function– Model:

– No Arbitrage condition

• Bond price

• Find coefficients

nni tdctdctdcP 1211

33

2210 ttttd

m

MARKETii PP

1i

Min

0

t

d

032 s.t. 2321 tt

10 d

69

Polynomial Model/Cubic Function/Data

자료 : Bloomberg

70

Polynomial Model/Cubic Function

Example2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

A B C D E F G H I J K L M N fitting Yield Curve.xls coefficients

b0 1b1 -0.08896b2 0.04839b3 -0.01544

0.5 0.965686=PRICE($B$10,D14,E14,C14,100,2,0)/100 =cubicF(K13,$J$3,$J$4,$J$5,$J$6)

today 08-09-07 =SUMPRODUCT($I$12:$N$12,I14:N14)

Market Data market model t 0.980542 0.965686 0.953983 0.943987 0.923322 0.89211t yield maturity coupon price price cf 0.25 0.5 0.75 1 1.5 2

0.25 5.32 0.0532 08-12-07 0.0532 0.999825 1.006625 1.02660.5 5.37 0.0537 09-03-08 0.0537 0.999998 0.991615 0 1.02685

0.75 5.41 0.0541 09-06-07 0.0541 0.99991 1.006312 0.02705 0 1.027051 5.45 0.0545 09-09-07 0.0545 1 0.996025 0.02725 1.02725

1.5 5.53 0.0553 10-03-08 0.0553 0.999998 1.001654 0.02765 0.02765 1.027652 5.77 0.0577 10-09-07 0.0577 1 0.99958 0.02885 0.02885 0.02885 1.02885

Function cubicF(t, b0, b1, b2, b3)cubicF = b0 + b1 * t + b2 * t ^ 2 + b3 * t ^ 3End Function

21 22 23 24 25 26 27 28 29 30 31

E F G H I J K L M N t_function

4.62E-05 =(F14-G14)^27.03E-05

4.1E-051.58E-052.74E-061.77E-07

sum 0.000176 =SUM(F22:F27)

71

Spline: Piece-wise polynomial

• More freedomMore accurate one

• Bond price

• Continuity Condition+1st & 2nd differential condition• Find coefficients

31,3

21,21,11,01 ttttd

32,3

22,22,12,02 ttttd

3,3

2,2,1,0 ttttd iiiii

m

MARKETii PP

1i

Min

T

Zero price

n

jjij

m

jjj

l

jjji tdctdctdcP

112

11

72

Polynomial Model/Nelson-Siegel

Instantaneous forward rate discount factor

73

note

74

Polynomial Model/Nelson-Seigel

Example

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

A B C D E F G H I J K L M N fitting Yield Curve NelsonSeigel.xls coefficients

a 0.047973b -0.02019c 0.010625alpha 0.526315

0 1

today 08-09-07

Market Data market model t 0.983716 0.968754 0.954867 0.941859 0.91786 0.895799t yield maturity coupon price price cf 0.25 0.5 0.75 1 1.5 2

0.25 5.32 0.0532 08-12-07 0.0532 0.999825 1.009883 1.02660.5 5.37 0.0537 09-03-08 0.0537 0.999998 0.994765 0 1.02685

0.75 5.41 0.0541 09-06-07 0.0541 0.99991 1.007306 0.02705 0 1.027051 5.45 0.0545 09-09-07 0.0545 1 0.993923 0.02725 1.02725

1.5 5.53 0.0553 10-03-08 0.0553 0.999998 0.996067 0.02765 0.02765 1.027652 5.77 0.0577 10-09-07 0.0577 1 1.003245 0.02885 0.02885 0.02885 1.02885

75

Polynomial Model/Nelson-Seigel

ExampleFunction df_NelsonSeigel(a_, b_, c_, alpha, t)Dim A As Double, B As Double, C As DoubleA = AA(b_, c_, alpha)B = BB(a_)C = CC(c_, alpha)df_NelsonSeigel = Exp(-A - B * t - (A + C * t) * Exp(-alpha * t))End Function

Function AA(b_, c_, alpha)AA = b_ / alpha + c_ / (alpha * alpha)End Function

Function BB(a_)BB = a_End Function

Function CC(c_, alpha)CC = c_ / alphaEnd Function

21 22 23 24 25 26 27 28 29 30

E F G H I J K L M t_function0.000101 =(F14-G14)^22.74E-055.47E-053.69E-051.55E-051.05E-05

sum 0.000246 =SUM(F22:F27)

76

Graphs: polynomial vs Nelson-Seigel

2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22

A B C D E F G H I J K coefficients coefficientsa 0.047973 b0 1b -0.02019 b1 -0.08896c 0.010625 b2 0.04839alpha 0.526315 b3 -0.01544=df_NelsonSeigel($B$3,$B$4,$B$5,$B$6,A11)

=-1/A11*LN(B11) =cubicF(A11,$D$3,$D$4,$D$5,$D$6)NS Polynomial

term df zero rate df zero rate0.25 0.983716 6.57% 0.980542 7.86%

0.5 0.968754 6.35% 0.965686 6.98%0.75 0.954867 6.16% 0.953983 6.28%

1 0.941859 5.99% 0.943987 5.76%1.25 0.929567 5.84% 0.934249 5.44%

1.5 0.91786 5.71% 0.923322 5.32%1.75 0.906633 5.60% 0.909758 5.40%

2 0.895799 5.50% 0.89211 5.71%2.25 0.885291 5.42% 0.868931 6.24%

2.5 0.875051 5.34% 0.838773 7.03%2.75 0.865036 5.27% 0.800187 8.11%

3 0.85521 5.21% 0.751728 9.51%

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

9.00%

10.00%

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3

NS

Polynomial

77

Bond Futures/Forward

78

Bond Futures/ 주식선도

79

Bond Futures/ 이자율 선도 (FRA)

80

Bond Futures/FRAExample

투자자는 몇 % 로 ( 선도 ) 예금계약을 체결해야할까 ? ( 단 , 투자자는 동일한 금리로 대출 / 투자가 가능하다 .)

81

Bond Futures/ 채권 선도이표가 없는 경우

이표가 1 회 있는 경우

82

Bond Futures/UDF

83

KTB Futures/Market

84

KTB Futures

Check

Check

85

KTB Futures/Market

86

KTB Futures/Market

3 년 국채선물의 최종결제 기준채권

자료 : www.krx.co.kr/상품안내/선물옵션상품/채권금리상품 /3 년국채선물

87

KTB Futures/ 이론가 계산

88

KTB Futures/functions

89

KTB Futures/functions

90

KTB Futures

Example

91

Note: Excel functionsDSC=coupdaysnc 기준일부터 다음 이자지급일까지의 날수 A=coupdaybs

E=coupdays N=coupnum

92

bonus

• Convexity adjustment

93

References

• 엑셀 VBA 를 이용한 금융공학실습 ( 서울경제경영 )

94

Thank you!