Lesson 2 Financial Instruments Bonds Institute of Economic Studies Faculty of Social Sciences...

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 bonds‘ 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 bond‘s maturity

1 2 T

%


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 risk𝑃0=

𝑀(1+𝑌𝑇𝑀 )𝑡

Incomplete structure of zero-coupon bonds needed for the construction of zero yield curve can be solved by bootstrapping or synthesization

𝑀

𝑃0

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)

𝑌𝑇𝑀=𝑧 𝑡=(𝑀𝑃0)1 /𝑡

−1


Bondstripping (unbundling)

𝑃1=𝑀

(1+𝑟 1 )=

𝑀(1+𝒛𝟏)

Identical present values for identical cash flows𝑃2=

𝐶2

(1+𝑟2 )1+𝐶2+𝑀

(1+𝑟 2 )2=

𝐶2

(1+𝑧 1 )1+𝐶2+𝑀

(1+ 𝒛𝟐 )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 bond𝑪𝟏 𝑪𝟐 𝑪𝑻 𝑴

𝑪𝟏 𝑪𝟐 𝑪𝑻 𝑴 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 … yield to maturity of theselected t-year coupon bond … yield to maturity of the bootstrapped t-year zero bond … market price and coupon of the t-year coupon bond

Zero bonds • • •• • •

Bootstrapping

𝒛𝑻=(𝑀 𝑠

𝑃𝑠 )1 /𝑇

−1


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

𝑀 𝑠

𝑃𝑠

Synthesization


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

… 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)𝑓 𝑡+𝑝𝑡

❑

𝑧𝑝=𝑧𝑝0= 𝑓 𝑝0

❑

, … forward yield curve expected in one year’s time, … forward yield curve expected in two year’s time

Do not confuse forward rates with future spot rates… p-year zero rate that will exist at the time t

𝟎 𝟏 𝟐 t +1t t +p

Spot interest rate is the rate charged for immediate borrowing and lendingTT−1

𝑧𝑝𝑡


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

𝑓 2❑=

(1+𝑧 2)2

(1+𝑧 1 )−1

1

❑

𝑓 𝑡+𝑝❑ =[ (1+𝑧 𝑡+𝑝)𝑡+𝑝

(1+𝑧 𝑡 )𝑡 ]

1 /𝑝

−1𝑡

❑

General formula(1+𝑧 𝑡+𝑝 )𝑡+𝑝=(1+ 𝑧𝑡 )𝑡× (1+ 𝒇 𝒕+𝒑

❑𝒕

❑ )𝑝


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

𝐿=+𝑀× (1+ 𝑧𝑡 )𝑡

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:𝑅=−𝑀 × (1+𝑧 𝑡+𝑝 )𝑡+𝑝

𝑓 𝑡+𝑝❑ =[ −𝑅𝐿 ]

1 /𝑝

−1=[ (1+𝑧𝑡+𝑝)𝑡+𝑝

(1+𝑧 𝑡 )𝑡 ]

1 /𝑝

−1𝑡

❑

𝑀−𝑀=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


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

𝑃 𝐹𝑅𝑁=𝑧1𝑀

(1+𝑧 1 )+

𝑓 2❑

1❑ 𝑀

(1+𝑧2 )2+

𝑓 3❑

2❑ 𝑀

(1+𝑧 3 )3+. . .+

𝑓 𝑇❑

𝑇 −1❑ 𝑀

(1+𝑧𝑇 )𝑇+ 𝑀

(1+𝑧𝑇 )𝑇

¿𝑧 1𝑀

(1+𝑧 1 )+ 𝑀

(1+𝑧 2 )2 [ (1+𝑧2 )2

(1+𝑧 1 )−1]+ . ..+ 𝑀

(1+𝑧𝑇 )𝑇 [ (1+𝑧𝑇 )𝑇

(1+𝑧𝑇− 1 )𝑇− 1−1]+ 𝑀

(1+ 𝑧𝑇 )𝑇=𝑀

Synthetic floater1 2 . . . t . . . T

𝑀 (1+𝑧 11)

𝑀

𝑀 (1+𝑧 1𝑡)

𝑀

𝑃𝑇 −1=𝑀 (1+𝑧1

𝑇 )1+𝑧1

𝑇 =𝑀

𝑃0=𝑀

𝑀 (1+𝑧 1𝑇)

. . .𝑃0=𝑃1+𝑀 𝑧1

0

1+ 𝑧10 =𝑀𝑃𝑇 −2=

𝑃𝑇 −1+𝑀𝑧 1𝑇− 1

1+𝑧 1𝑇− 1 =

𝑀 (1+𝑧 1𝑇− 1)

1+𝑧1𝑇−1 =𝑀


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 × 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)

… real coupon rate, … annual rate of inflation, … price index

Use of outdated price index as a result of: i) statistical data processing (delay up to 2 months)


Nominal and real yield to maturity

Break-even inflation Break-even inflation can be found using the Fisher equation

𝑃=𝛾𝑀 (1+𝜋 )

(1+𝑟 𝑇 )+𝛾𝑀 (1+𝜋 )2

(1+𝑟𝑇 )2+ .. .+

(𝛾𝑀+𝑀 ) (1+𝜋 )𝑇

(1+𝑟𝑇 )𝑇

Nominal (money) YTM for T-year inflation-linked bond Real YTM for T-year inflation-linked bondreal yield [Fisher equation: (1+r)=(1+ρ)×(1+π); r=ρ+π]

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)


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 bonds𝑃0=

𝑟𝑇 𝑀

(1+𝑟𝑇 )+𝑟𝑇𝑀

(1+𝑟𝑇 )2+…+

𝑟𝑇 𝑀+𝑀

(1+𝑟𝑇 )𝑇=𝑀

Par yields are used to determine the required coupon on new bonds that are to be issued at par𝑀=

𝑟𝑇𝑀

(1+𝑧 1 )+𝑟𝑇𝑀

(1+𝑧2 )2+…+

𝑟 𝑇𝑀+𝑀

(1+ 𝑧𝑇 )𝑇

𝑑𝑡=1

(1+𝑧 𝑡 )𝑡 ,𝑡=1 ,…,𝑇

… coupon rate of T-year par bond … t-year discount factor𝑟1=

1−𝑑1𝑑1,𝑟2=

1−𝑑2𝑑1+𝑑2

, . . .,𝑟𝑇=1−𝑑𝑇

𝑑1+𝑑2+…+𝑑𝑇

Position property 𝑟 𝑡<𝑧 𝑡

𝑑𝑡=1−𝑟 𝑡× (𝑑1+𝑑2+…+𝑑𝑡 −1 )

1+𝑟 𝑡

Upward sloping zero yield curve 𝑟 𝑡>𝑧 𝑡 Downward sloping zero yield curve 𝑡=1 ,…,𝑇

Lesson 2 Financial Instruments Bonds Institute of Economic Studies Faculty of Social Sciences...

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