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Lesson 2
Analysisof yield curve
Financial Instruments
Bonds
Institute of Economic StudiesFaculty of Social SciencesCharles University in Prague
Analysis of yield curve 2
Concept of yield curve
Practical considerations
Definition
Yield curve must be constructed from a homogenous group of bonds: same economic segment, same credit risk, same degree of liquidity The most frequent measure used in the construction of yield curve is the yield to maturity (YTM) Practical problems: gaps in existing maturity structures, more bondsβ yields observed for some maturities, short-term price anomalies
time to maturity
rate of returnfalling (inverted) YChorizontal (flat) YCrising YC
humped YC
Yield curve (term structure of interest rates) is the relationship between a particular yield measure and a bondβs maturity
1 2 T
%
Analysis of yield curve 3
Zero yield curve
1 2 . . . T
-year zero rate Properties of zero-coupon bonds
Definition Zero yield curve is the yield curve which is constructed from yields of zero-coupon bonds Zero-coupon bonds are bonds that make no coupon payment and the only cashflow is the price paid and the principal amount received at maturity Zero-coupon bonds have convenient analytical properties and do not suffer from reinvestment riskπ0=
π(1+πππ )π‘
Incomplete structure of zero-coupon bonds needed for the construction of zero yield curve can be solved by bootstrapping or synthesization
π
π0
Ambiguous correspondence between the market returns and extracted zero rate yields (different sets of bonds with the same YTM may generate different zero yield curves)
πππ=π§ π‘=(ππ0)1 /π‘
β1
Analysis of yield curve 4
Bondstripping (unbundling)
π1=π
(1+π 1 )=
π(1+ππ)
Identical present values for identical cash flowsπ2=
πΆ2
(1+π2 )1+πΆ2+π
(1+π 2 )2=
πΆ2
(1+π§ 1 )1+πΆ2+π
(1+ ππ )2
Technique of financial engineering that breaks down a coupon-bearing bond into a set of zero-coupon bonds with the same combined cash flowCoupon bondπͺπ πͺπ πͺπ» π΄
πͺπ πͺπ πͺπ» π΄ Indifference between the two identical cash flows received at the same point in time The process works iteratively from the lowest to the highest maturities β¦ yield to maturity of theselected t-year coupon bond β¦ yield to maturity of the bootstrapped t-year zero bond β¦ market price and coupon of the t-year coupon bond
Zero bonds β’ β’ β’β’ β’ β’
Bootstrapping
ππ»=(π π
ππ )1 /π
β1
Analysis of yield curve 5
1 2 3 T-2
T-1 T
cash flow from T-year coupon bond
Synthetic security A bundle of securities whose combined cash flow is the same as the cash flow of the imitated security Synthetic zero bond is a set of coupon bonds whose combined cash flow creates the cash flow of the zero-coupon bond Sketch of synthesization
cash flow from (T-1)-year coupon bonds that clear the flows at the end of year T-1 cash flow from (T-2)-year coupon bonds that clear the flows at the end of year T-2
π π
ππ
Synthesization
Analysis of yield curve 6
Forward rates
Definitions Forward interest rate (forward-forward) is the rate negotiated now for future borrowing and lending
Spot rates can be seen as a special case of forward rates Forward yield curves
1 2 . . . t+1 . . . t+p . . . T
β¦ p-year interest rate related to the future period which starts at the time t (the beginning of the period t +1) and ends at the time t +p (the end of the period t +p)π π‘+ππ‘
β
π§π=π§π0= π π0
β
, β¦ forward yield curve expected in one yearβs time, β¦ forward yield curve expected in two yearβs time
Do not confuse forward rates with future spot rates⦠p-year zero rate that will exist at the time t
π π π t +1t t +p
Spot interest rate is the rate charged for immediate borrowing and lendingTTβ1
π§ππ‘
Analysis of yield curve 7
Implied forward rates
Definition Implied forward zero rates are forward rates that are consistent with the observed zero yield curve in efficient financial markets Derivation
Yield from purchasing two-year zero-coupon bonds Yield from purchasing one-year zero-coupon bonds and rolling over the investment for another year by buying again zero-coupon bonds Consistency condition
π 2β=
(1+π§ 2)2
(1+π§ 1 )β1
1
β
π π‘+πβ =[ (1+π§ π‘+π)π‘+π
(1+π§ π‘ )π‘ ]
1 /π
β1π‘
β
General formula(1+π§ π‘+π )π‘+π=(1+ π§π‘ )π‘Γ (1+ π π+π
βπ
β )π
Analysis of yield curve 8
Synthetic forward rates
Definition Synthetic zero forward rate is a forward rate that is locked in appropriate combinations of borrowing and lending at zero spot rates
πΏ=+πΓ (1+ π§π‘ )π‘
1 2 . . . t t+p . . .M [issuance of (t +p)-year zero bonds]M [purchase of t-year zero bonds] R [redemption of (t+p)-year zero bonds]L [redemption of t-year zero bonds]
time 0:π =βπ Γ (1+π§ π‘+π )π‘+π
π π‘+πβ =[ βπ πΏ ]
1 /π
β1=[ (1+π§π‘+π)π‘+π
(1+π§ π‘ )π‘ ]
1 /π
β1π‘
β
πβπ=0time t :time t +p :
Net cash flows
Locked-in p-year forward zero rate
9Analysis of yield curve
Waning property
Expectation hypothesis Current spot rates may change under the pressure of changing expectations
The more distant the time location of the forward yield curve, the shorter its range of values
Position property Upward sloping zero yield curve is below all forward yield curves
Implied forward rates are the best indicator of expected future interest rates
Downward sloping zero yield curve is above all forward yield curves
Properties of implied forward rates
Analysis of yield curve 10
Pricing of floaters
Definition Par property
Floating rate note (FRN, floater) is a type of bond whose coupon rate is fixed for a given period by reference to some short-term market interest rate and reset periodically on the coupon reset dates Fair price of a floater based on the expectation hypothesis
Cash flow from a floater is equivalent to investing in a short-term money market instrument and reinvesting the principal on a rolling basis
π πΉπ π=π§1π
(1+π§ 1 )+
π 2β
1β π
(1+π§2 )2+
π 3β
2β π
(1+π§ 3 )3+. . .+
π πβ
π β1β π
(1+π§π )π+ π
(1+π§π )π
ΒΏπ§ 1π
(1+π§ 1 )+ π
(1+π§ 2 )2 [ (1+π§2 )2
(1+π§ 1 )β1]+ . ..+ π
(1+π§π )π [ (1+π§π )π
(1+π§πβ 1 )πβ 1β1]+ π
(1+ π§π )π=π
Synthetic floater1 2 . . . t . . . T
π (1+π§ 11)
π
π (1+π§ 1π‘)
π
ππ β1=π (1+π§1
π )1+π§1
π =π
π0=π
π (1+π§ 1π)
. . .π0=π1+π π§1
0
1+ π§10 =πππ β2=
ππ β1+ππ§ 1πβ 1
1+π§ 1πβ 1 =
π (1+π§ 1πβ 1)
1+π§1πβ1 =π
Analysis of yield curve 11
Inflation-linked bond (1)
Definition Inflation-linked (inflation-indexed) bonds are bonds whose coupons and principal take into account the evolution of a particular price index with the aim to provide protection against inflation
repayment of face value = face value Γ Imperfect protection against inflation
Weighted composition of price index does not coincide with the collection of goods by investors who want to be protected against price changes
nominal value of coupon (paid at time t ) = real value of coupon (paid at time t )
ii) fixing the value of the price index at the beginning of the coupon period because of the accrued interest (delay up to 6 months)
β¦ real coupon rate, β¦ annual rate of inflation, β¦ price index
Use of outdated price index as a result of: i) statistical data processing (delay up to 2 months)
Analysis of yield curve 12
Nominal and real yield to maturity
Break-even inflation Break-even inflation can be found using the Fisher equation
π=πΎπ (1+π )
(1+π π )+πΎπ (1+π )2
(1+ππ )2+ .. .+
(πΎπ+π ) (1+π )π
(1+ππ )π
Nominal (money) YTM for T-year inflation-linked bond Real YTM for T-year inflation-linked bondreal yield [Fisher equation: (1+r)=(1+Ο)Γ(1+Ο); r=Ο+Ο]
Break-even inflation makes the nominal yield on an inflation-linked bond equal to the yield on a conventional bond of the same maturity Break-even inflation makes observable expected inflationexpected infl. is higher than break-even infl. investors prefer IL-bonds over conventional ones price of IL-bonds goes up real yield on IL-bonds goes down break-even infl. goes up and is closer to the expected infl.
Inflation-linked bond (2)
Analysis of yield curve 13
Par yield curve
Definition
Consistency between par yield and zero yield curves
Par yield curve is a plot of yields to maturity for bonds priced at par Par yields are equal to coupon rates for par bondsπ0=
ππ π
(1+ππ )+πππ
(1+ππ )2+β¦+
ππ π+π
(1+ππ )π=π
Par yields are used to determine the required coupon on new bonds that are to be issued at parπ=
πππ
(1+π§ 1 )+πππ
(1+π§2 )2+β¦+
π ππ+π
(1+ π§π )π
ππ‘=1
(1+π§ π‘ )π‘ ,π‘=1 ,β¦,π
β¦ coupon rate of T-year par bond β¦ t-year discount factorπ1=
1βπ1π1,π2=
1βπ2π1+π2
, . . .,ππ=1βππ
π1+π2+β¦+ππ
Position property π π‘<π§ π‘
ππ‘=1βπ π‘Γ (π1+π2+β¦+ππ‘ β1 )
1+π π‘
Upward sloping zero yield curve π π‘>π§ π‘ Downward sloping zero yield curve π‘=1 ,β¦,π
See you in the next lecture
14Analysis of yield curve
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