Financial Market Analsis (FMAx) Module 4 erm Structure of Interest Rates his training material is the propert of the International Monetar Fund (IMF) and is intended for use in IMF Institute for Capacit Development (ICD) courses. An reuse requires the permission of the ICD.
he Relevance to You You might be An investment manager A debt manager An economic analst/forecaster
Before We Begin he erm Structure of Interest Rates: Provide useful insight about how the market thinks about future interest rate movements. Allows central bankers to gauge market expectation about inflation, growth and risks Allows ou to price an asset
Simplifing Assumption o simplif the discussion, we assume the following: he face value of all bonds, M, is $00 Each period is a ear Coupon paments, c, are made annuall Consider onl clean bond pricing
Defining the erm Structure of Interest Rates Question: What is the term structure of interest rate? It is the relationship between the ield-to-maturit of zero-coupon bonds and their respective maturit It is often called the spot curve It can be derived mathematicall
Defining the erm Structure of Interest Rates he bond pricing formula for a t-ear zero coupon bond. ø ö ç ç è æ t t t P M ( ) M P t t t M DF P t t
Defining the erm Structure of Interest Rates 3 A graphical representation of a term structure of interest rates
Defining the Zero Coupon Bond Wh use zero-coupon bond? Consider the bond pricing equation of a coupon bond he YM of a coupon bond defines implicitl in the following equation P t c c ~ ~! t ( ) ( ) t t c M ~ t
Common Shapes of the Yield Curve
Common Shapes of the Yield Curve Remark On the shapes: he shapes showed here are not the onl possible shapes of a ield curve he shape of a ield curve could be more complicated than those On the interpretations: he interpretation given to each shape is not the onl correct interpretation Competing theories to explain the relationship between the shape of the ield curve and interest rates movements.
he Yield Curves for Indonesian Sovereign Bond Example: Zero Coupon Yield derived from Indonesian Sovereign Bond on Jan/ nd (blue), Apr/ st (Red) and Jul/ st (Green). 9.0 8.5 8.0 7.5 7.0 6.5 6.0 9.0 8.5 8.0 7.5 7.0 6.5 6.0 0 3 4 5 6 7 8 9 0 Maturit (in ears) Q Q3 Q Source: Bloomberg
he Spot Curves for Brazil Sovereign Bond Example: Zero Coupon Yield derived from Brazil Sovereign Bond on Jan/ nd (blue), Apr/ st (Red) and Jul/ st (Green). 05. 6 5 4 3 0 6 5 4 3 0 0 3 4 5 6 7 8 9 0 Maturit (in ears) Q Q Q3 Source: Bloomberg
Bond Pricing with the Yield Curve A stream of cash flow: he Price of Strateg A: P(3)
Bond Pricing with the Yield Curve A stream of cash flow: he Price of Strateg B: ( 00 5) ~ 5 5 P0 (3) º P0 () P0 () P 00 00 00 0 (3)
Defining the Spot Rate he No-Arbitrage Condition: Given the Law of One Price, the price of alternatives A and B should be identical. P(3) ~ P 0 (3) P0 () P0 () 5 5 00 00 5 5 ( 00 5) ( ) ( ) 3 05 3 P0 (3) 00
Defining the Spot Rate Let us generalize the formula to maturit : ( ) ( ) t t t t M c P t P M c P P å å ) ( ) ( ) ( ~ ) ( 0 0 0
Defining the Forward Interest Rate Forward Loan: Agreement toda to borrow/lend on some future date at an interest rate that is determined on toda. Forward Rate (or forward interest rate ): Interest rate on a forward loan. he forward rate is not necessaril equal to spot short rate that will prevail in the future.
Defining the Forward Interest Rate Forward rates are tightl related to the spot rates b the no-arbitrage condition. Gross Return: ( ) Gross Return: ( )( f ),
Defining the Forward Interest Rate 3 he no-arbitrage condition ensures that ( ) ( )( f ), f ( ) ( ),
Defining the Forward Interest Rate 4 We can generalize the idea 0, f ( ) ( ) ( ) i i i i i f, ( ) ( ), ø ö ç ç è æ i i i i i f B definition:
Defining the Forward Interest Rate 5 It is important to notice and remember that Forward lending/borrowing agreement ma not exist or ma not be allowed in realit in some countries However, forward interest rates can still be calculated as long as the ield curve exists!! f ( ) æ ö i ç i, i ç ( ) i i è ø
he Relationship between Spot and Forward Rates Forward interest rates are tightl related to spot interest rates. he are also tightl related to he discount factors (DF) he prices of the corresponding zero-coupon bonds (P)
he Relationship between Spot and Forward Rates Let us recall that ( ) ( ), ø ö ç ç è æ ø ö ç ç è æ ø ö ç ç è æ ø ö ç ç è æ i i i i i i i i i i i P P M M P P DF DF f
he Relationship between Spot and Forward Rates 3 Spot rates can also be rewritten as a (geometric) average of forward rates. ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) [ ] f f f f f f f f f f f f f f,,, 0,,,, 0,,,,,,,!!!
he Relationship between Spot and Forward Rates 4 [( f )( f )!( f )( f )] 0, ( ) [ ln( f ) ln( f )! ln( f ) ln( f )] ln Since interest rates are generall of a small magnitude 0,,, he spot interest rate is approximatel equal to the arithmetic average of forward interest rate.,,» 0,,!, [ f f f ],,
he Relationship between Spot and Forward Rates 5 his expression has an economic interpretation: he spot interest rate at maturit ear is the simple average -ear borrowing cost over ears. he x-ear forward -ear interest rate f x, is the marginal borrowing cost when the loan is extended b one ear.
he Relationship between Spot and Forward Rates 6 For example: A 3-ear zero coupon bond is trading at YM of 3% A -ear zero coupon bond is trading at YM of % ( ) 3 æ 0.03 ö f, ç ç ( 0.0) è ø 5.03% he interest rate for the additional ear is 5.03% (after rounding).
he Relationship between Spot and Forward Rates 7 Just as we did with spot rates, we can define a forward ield curve. 7 6 Brazil: he spot curve on Jan//05 7 6 Brazil: -ear forward -ear curve on Jan//05 5 5 4 4 3 3 0 0 4 6 8 0 Maturit (in ears) Source: Bloomber g 0 0 4 6 8 0 Years from now
he Relationship between Spot and Forward Rates 8 Because the spot interest rate is the average of forward interest rates: spot curve is upward-sloping Forward curve is above the spot curve spot curve is downward-sloping Forward curve is below the spot curve spot curve is flat Forward curve is equal to spot curve.
he Par-Yield Defining the par-ield: Measures the coupon rate (in percentage term) at which a coupon bond would be traded at par. he par-ield is often used as a reference for pricing new issues. It is NO the spot interest rate but is closel related to it. hus, ou cannot use it directl for bond pricing or discount future cash-flow
he Par-Yield Given the term-structure of interest rates, the par-ield of maturit, implicitl as * c M M M å ( ) t ( ) t t c ( ) DF å ( ) å t DFt * t t t * c, is defined
he Par-Yield 3 Similar to the relationship between the spot curve and the forward curve, the parield he par-ield can be viewed as a kind of average of the spot interest rate. he par-ield curve is flatter than both the spot curve and the forward curve.
Bootstrapping Defining bootstrapping: If ou fall into a well and no one is around, ou use our bootstrap to help ourself climb up from a well Use available data on coupon bonds to construct the spot curve. he method relies on the assumption that no-arbitrage condition holds.
Bootstrapping Recall that the bond price equation, under the assumption of no-arbitrage å ( ) ( ) t t t M c P ) ( ( ) ) ( ø ö ç ç ç ç è æ å t t t c P M c
Bootstrapping 3 Suppose ou are given the following: Maturit Coupon Price 5.5 0.5 4.37 99.8
Bootstrapping 4 he -ear spot interest rate: æ ç ç ç ç è P( ) c M å t c ( ) t t ö ø he -ear spot interest rate: he -ear spot interest rate is then: 00 5.5 0.5 3.69% æ ö ç ç 00 4.7 ç 4.7 ç 99.8 è ø 4.39%
Bootstrapping 5 Suppose we have coupon bonds and the maximum maturit among these bond is ears. P( P( P( ) ) ) " CF CF CF DF DF DF CF CF CF DF DF DF! CF! CF! CF DF DF DF
Bootstrapping 6 Stack the equations in matrix form é P( ) CF & CF DF ê ê ê ë $ P( ) #%"%! P ù ú ú ú û é ù é ù ê ú ê ú $ ' $ $ ê ú ê ú ê CF & CF ú ê DF ú #% ë %% "%%%! û #"! ë û CF DF
Bootstrapping 7 Pre- (Left-) multipl the inverse of cash-flow matrix to the price vector P é DF ù ê ú % ê ú ê DF ú $#" ë û DF inv é CF & CF ù ê ú ê % ' % ú ê CF & CF ú $! ë!!#!!!" û Inverse of CF, CF - é ê ê ê ë P( ) % P( ) $!#!" P ù ú ú ú û
Regression Approach More generall, suppose we have N coupon bonds and the maximum maturit is. é P( ) CF & CF DF ê ê ê ë $ P( ) #%"%! N P ù ú ú ú û é ù é ù ê ú ê ú $ ' $ $ ê ú ê ú ê CF N & CF N ú ê DF ú #% ë %% "%%%! û #"! ë û CF DF
Regression Approach If N, then regression approach reduce to the bootstrap approach. If N<, then too man discount factors satisf the sstem. If N>, then unique discount factors with pricing error.
Regression Approach 3 We can estimate the vector DF b the method of Ordinar-Least-Square (OLS). he pricing error: e Find the DF such that P N å i e i CF DF æ ö ç P( ) å i i CF tdft è t ø is minimized ( ) ( ) OLS estimated discount factor: DF CF' CF CF' P
Parametric Yield Curve Models he resulting estimated ield curve is usuall not smooth. Interpolation is often used to calculate the discount factor/spot rate for maturit that we do not have.
Parametric Yield Curve Models Parametric function to model the discount factor or the spot curve (or the forward curve) can give ou the desired smoothness. Pros Give the desired smoothness Alleviate the difficult of not enough number of coupon bonds relative to the maximum maturit Cons At the potential expense of higher pricing error. Ma requires non-linear method to estimate.
Parametric Yield Curve Models 3 For example he Polnominal Yield Curve. he spot curve can be modeled as a polnominal equation with an order t t b t b t 3 3 a b
Parametric Yield Curve Models 4 t 3 0.05 0.0t 0.003t 0.0005t 8% 6% 4% % 0% 8% 6% 4% % 0% 0 4 6 8 0 Maturit in ears (t)
Parametric Yield Curve Models 5 wo other common models: Nelson-Siegel Svennson i é æ i ö ù t ì ï é æ i ö ù t æ i ö ü ï a b ê expç ú b í ê expç ú expç ý ë è t ø û i ï î ë è t ø û i è t ø ï þ t ( i) é æ i ö ù t ï ì é æ ö ù æ ö ï ü ï ì é æ ö ù æ ê ç ç i t í ê ç ç i i t ú ç ç ý í ê ç ç i a b ú ú ç ç exp b exp exp b 3 exp exp ë è t ø û i ï î ë è t ø û i è t ø ï þ ï î ë è t ø û i è t ö ï ü ý ø ï þ
he Pure Expectation Hpothesis he strong form of pure expectation hpothesis Forward rates reflect toda s expectation of what spot rates will be in the future. Investors are ASSUMED to be risk-neutral. Investors care about the return onl. Risk is not a concern.
he Pure Expectation Hpothesis Consider the three following strategies: A. Bu a -ear zero-coupon bond on toda. ( ) B. Bu a -ear zero-coupon bond and then roll-over with a one-ear forward -ear zero-coupon bond on toda. ( )( f, ) C. Bu a -ear zero-coupon bond on toda and then roll-over the proceed with another -ear zero-coupon one ear later ( )( E[, ])
he Pure Expectation Hpothesis 3 Recall that no-arbitrage condition ensures that ( ) ( )( f, ) Pure expectation hpothesis: Investors will be indifferent between all three strategies if the expected return is the same: ( )( f,) ( )( E[, ]) f E[ ],,
he Pure Expectation Hpothesis 4 Moreover, this implies that ( ) (, )( E[, ])» E[ ], Extending the logic to the -ear case:» E[, ]! E[, ]
he Pure Expectation Hpothesis 5 Forward rates correspond to toda s expectations of future spot rates [ > > Þ E, ] If i.e., market participants are expecting an increase in the spot rate in the future. An upward-sloping spot curve suggests that the market is expecting rates to rise.
he Pure Expectation Hpothesis 6.4.4.. % 0.8 0.6 0.8 0.6 0.4 0.4 0. 0. 0 Jul-3 Jan-4 Jul-4 Jan-5 Axis itle 0 -r spot rate Expected one-r forward -ear spot
erm Premium here must be something else that helps explain. Suppose that E[, ] tp, and ASSUME that E[, ] hen tp, tp > 0 If, then > (even though the market does not expect rates to rise.),
erm Premium tp, Question: What is? It is often called the term premium --- he premium that investors require to hold a bond with a particular maturit over another maturit (We usuall consider the term premium of longer-bond over short-term bond!) Alternativel, the no-arbitrage condition suggests that we can define it as tp, f,, E[ ] (he excess of the forward rate over pure expectation of future interest rate.)
erm Premium 3 he term-premium of longer-term bond over shorter-term bond is usuall (but not alwas). Positive: Investors are assumed to be risk-averse in order to justif the positive term-premium Increasing: Longer the maturit, higher the price sensitivit.
erm Premium 4 he term-premium can then be decomposed into two components. [Price] risk premium (increasing with the maturit; First-order): Higher the duration, more sensitive is the price to a change in interest rates; Convexit premium (decreasing with the maturit Second-order): Higher the convexit, for the same amount of interest rate Drop è larger the price increases Increase è smaller the price decreases he term-premium can be positive or negative depending on which component dominates.
Market Segmentation 5 Market Segmentation or Preferred Habitat Hpothesis Bonds of a given maturit are mainl traded b a particular group of investors. Longer-term bond ó Pension fund he suppl and demand conditions of a bond with a given maturit are independent to the suppl and demand conditions of bonds of other maturit. Arbitrage opportunit across maturities is missing. his explanation is less popular nowadas.
Module Wrap-Up he erm Structure of Interest Rates Relates the zero coupon ields to different maturities. Zero coupon ields are often not readil available. Need to construct the curve using coupon bonds and their prices. Bootstrap and Regression Parametric Models provide smoother ields
Module Wrap-Up If the Pure Expectation Hpothesis holds he ield curve essentiall reflects the market forecast of future interest rate movements. he presence of a term-premium, necessitates a theor to model the termpremium.