Analysis under Analysis under CertaintyCertaintyThe one investment certainty is The one investment certainty is that we are all frequently wrongthat we are all frequently wrong

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Up to nowUp to now

Financial markets and Financial markets and instrumentsinstruments– Specifics of stocks, bonds, and Specifics of stocks, bonds, and

derivativesderivatives– Trading processTrading process– Financial intermediariesFinancial intermediaries

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PlanPlan

Analysis under certaintyAnalysis under certainty– Term structure of interest ratesTerm structure of interest rates– Fixed income instrumentsFixed income instruments

• PricingPricing• RisksRisks

– Capital budgetingCapital budgeting

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Definitions of ratesDefinitions of rates

Reinvestment:Reinvestment:– Simple vs compound interestSimple vs compound interest

Frequency of compounding:Frequency of compounding:– Nominal (coupon) rate vs effective Nominal (coupon) rate vs effective

(annual) rate(annual) rate Continuous compounding: Continuous compounding:

– Log-returnLog-return

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Definitions of rates (2)Definitions of rates (2)

Yield to maturity / internal yield / bond Yield to maturity / internal yield / bond yieldyield– Rate that equates cash flows on the bond with Rate that equates cash flows on the bond with

its market valueits market value– Return earned from holding a bond to maturityReturn earned from holding a bond to maturity

• Assuming reinvestment at same rateAssuming reinvestment at same rate Par yieldPar yield

– Coupon rate that causes the bond price to Coupon rate that causes the bond price to equal its face valueequal its face value

Current yieldCurrent yield– Annual coupon payment divided by the bond’s Annual coupon payment divided by the bond’s

priceprice– Often quoted but uselessOften quoted but useless

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Definitions of rates (3)Definitions of rates (3)

Zero rateZero rate– YTM of a zero-coupon bondYTM of a zero-coupon bond– How to get zero rates from coupon bond How to get zero rates from coupon bond

prices?prices?– Bootstrapping method: coupon bond as a ptf Bootstrapping method: coupon bond as a ptf

of zero-coupon bondsof zero-coupon bonds Spot rateSpot rate

– One-period zero rateOne-period zero rate Forward rateForward rate

– Rate on a one-period credit from T to T+1Rate on a one-period credit from T to T+1

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Term structure of Term structure of interest ratesinterest rates Relationship between yields and maturitiesRelationship between yields and maturities

– For bonds of a uniform quality (risks and taxes)For bonds of a uniform quality (risks and taxes)– E.g., Treasury / BaaE.g., Treasury / Baa

Equivalent ways to describe TSIR:Equivalent ways to describe TSIR:– Prices of zero-coupon bonds: Prices of zero-coupon bonds: P(t,T), with P(t,T), with

P(T,T)=1P(T,T)=1– Zero rates: Zero rates: y(t, T)y(t, T)– Forward rates:Forward rates: f(t, T)f(t, T)

Upward sloping yield curve: Upward sloping yield curve: – Fwd Rate > Zero Rate > Par YieldFwd Rate > Zero Rate > Par Yield

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Theories of the term Theories of the term structurestructure Expectations theory: Expectations theory:

– Unbiased expectations hypothesis: f(t, T) = Unbiased expectations hypothesis: f(t, T) = EEtt[r(T)][r(T)]

– Term structure is explained by expected spot Term structure is explained by expected spot ratesrates

• Upward sloping yield curve: signal that spot rate will Upward sloping yield curve: signal that spot rate will increaseincrease

Liquidity preference theory: Liquidity preference theory: – Investors demand a premium for bonds with Investors demand a premium for bonds with

higher riskhigher risk• Long-term bonds require a liquidity premiumLong-term bonds require a liquidity premium

– Upward sloping yield curve: forward rates Upward sloping yield curve: forward rates higher than expected future zero rateshigher than expected future zero rates

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Theories of the term Theories of the term structurestructure (2)(2) Preferred habitat: Preferred habitat:

– Investors try to match the life of their assets Investors try to match the life of their assets with liabilitieswith liabilities

– There is a premium for maturities with There is a premium for maturities with insufficient demandinsufficient demand

Market segmentation: Market segmentation: – Different rates determined independently of Different rates determined independently of

each othereach other• SR%: D – corporations financing sr obligations, S – SR%: D – corporations financing sr obligations, S –

banks banks • LR%: D – corporations financing lr inv projects, S – LR%: D – corporations financing lr inv projects, S –

insurance co-s, pension fundsinsurance co-s, pension funds– Investors don’t react to yield differentials Investors don’t react to yield differentials

between the maturitiesbetween the maturities

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Empirical estimation of Empirical estimation of TSIRTSIR Discrete rates:Discrete rates:

– Regression Regression P = cDP = cD11 + cD + cD22 + … + … + + (c+F)D(c+F)DTT

where Dwhere Dtt = 1/P(0,t) = = 1/P(0,t) = 1/y(0,t)1/y(0,t)tt

Continuous rates:Continuous rates:– Regression P = ΣRegression P = Σt=1:Tt=1:T c ctt (a (a00+a+a11t+at+a22tt22+…)+…)

– P = aP = a00[Σ[Σt=1:Tt=1:Tcctt]+a]+a11[Σ[Σt=1:Tt=1:Ttctctt]+a]+a22[Σ[Σt=1:Tt=1:Ttt22cctt]+]+……

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Modeling changes in Modeling changes in bond pricesbond prices Due to passage of time:Due to passage of time:

– E.g., flat yield curve: ΔP = r PE.g., flat yield curve: ΔP = r P00

Unanticipated shift in the TSIR: Unanticipated shift in the TSIR: – Need to approximate the function P Need to approximate the function P

= f(y)= f(y)– Duration: sensitivity of a bond’s Duration: sensitivity of a bond’s

price to the change in the interest price to the change in the interest ratesrates

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Macaulay’s durationMacaulay’s duration

Wtd-avg maturity of bond paymentsWtd-avg maturity of bond payments– Generalized maturity for coupon bonds, D Generalized maturity for coupon bonds, D

≤ T ≤ T Elasticity of a bond’s price wrt Elasticity of a bond’s price wrt ytmytm

– The larger the duration, the riskier is the The larger the duration, the riskier is the bondbond

For small changes in %: For small changes in %:

ΔP ≈ -D P Δy/y = -[D/y] P ΔyΔP ≈ -D P Δy/y = -[D/y] P Δy– D* = D/y: modified durationD* = D/y: modified duration

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Macaulay’s durationMacaulay’s duration (2)(2) Properties:Properties:

– C, coupon: C, coupon: – – – Y, %:Y, %: ––– T, maturity: T, maturity: ++

Limitations:Limitations:– Assumes horizontal TSIRAssumes horizontal TSIR– Applies only to small changes in Applies only to small changes in

%%

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Duration modificationsDuration modifications

Convexity Convexity Fisher-Weil durationFisher-Weil duration

– For parallel shifts of (non-horizontal) For parallel shifts of (non-horizontal) TSIRTSIR

Non-parallel shifts: Non-parallel shifts: – Two types: LR% usually more stable than SR%Two types: LR% usually more stable than SR%– Analytical approach: Analytical approach:

• E.g., assume d ln y(t,T) = KT-t+1 d ln r(t)E.g., assume d ln y(t,T) = KT-t+1 d ln r(t)

– Empirical approach: Empirical approach: • Separate estimation of duration for sr and lr %Separate estimation of duration for sr and lr %

ConclusionsConclusions