Fixed Income CFA Level II PDF

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Fixed Income CFA Level II...

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SS12 Fixed Income: The Term Structure & Interest rate Dynamics: Spot Rate  Annualized market interest rates for a single payment to be received in the future. It can be interpreted as the yields on 0-Coupon Bonds. Where: PT = Discount Factor (price today) & ST = Spot Interest Forward Rate  Interest rate for a loan to be made at some future date. Where: f(j,k) = Annualized interest rate applicable on a k-year loan in j years F(j,k) = Forward price of a bond maturing at time j + k delivered in j Yield to Maturity  It is the spot interest rate for a maturity of T. However, for a coupon bond, if the spot rate curve is not flat, the YTM will not be the same as the spot rate.

Given N Annual Spot What is the maximum number of Forward that can be derived? You have to select all forward F (j,k) such that (j + k)  N

Expected Return  Ex-ante holding period return that a bond investor expects to earn. The expected return will be equal to the bond’s yield only when all 3 of the following are true: - The bond is held to maturity - All payments are made on time & in full - All coupons are reinvested at the original YTM The second requirement implies that the bond is Option-free & there is No default risk. Reinvesting coupons at the YTM is the least realistic assumption. Forward Pricing Model  Values forward contract based on arbitrage-free pricing. Where: P (j + k) = Price of a bond maturing in j + k years F (j,k) = Forward price of a bond maturing at time j + k delivered in j

Forward Rate Model  Relates forward & Spot rates as follow: It illustrates how forward rates & spot rates are interrelated. It suggests that the Forward rate should make investors indifferent between buying a 5-year zero coupon bond versus buying a 2 year zero coupon bond & at maturity reinvesting the principal for 3 additional years.

Forward Rate Model: (1+ST)T = (1+S1)(1+f(1,T-1))(T-1)

A Par Rate  Yield to maturity of a bond trading at par. It will be equal to the coupon rate on the bond. By using Bootstrapping, spot rates or 0-coupon rates can be derived from the par curve. Bootstrapping: It involves using the output of 1 step as an input to the next step.

Forward Price Evolution: A change in the Forward Price indicates that the future spot rates did not confirm to the forward curve. When spot rates turn out to be lower than implied by the forward curve, the forward price will increase. For a bond investor, the return on a bond over 1-year horizon is equal to the 1-year risk free rate if the spot rates evolve as predicted by today’s forward curve.

Short-term rates are more volatile than long-term rates!

Locked-in-rate  Forward rate are interpreted as the rate than can be “locked in” for some future period. Related to the Pure Expectations Theory. Break Even Rate  Forward rate that make an investor indifferent between investing for the full investment horizon or part of it & then rolling over the proceeds for the balance of the horizon @ the forward rate. Pure Expectations Theory

For an Upward-sloping curve, the Forward curve will be above the Spot Curve! When the Spot Curve is Downward sloping, the Forward curve will be below it!

Riding the Yield Curve Strategy  When an investor will purchase bonds with maturities longer than his investment horizon. In an upward-sloping yield curve, shorter maturity bonds have lower yields than longer maturities bonds. As the bond approaches to maturity, it is valued using successively lower yields & therefore at higher prices. Also called “Rolling Down the yield Curve”  Related to Liquidity Preference Theory Swap Fixed Rate  In a plain vanilla interest rate swap, 1 party makes payments based on a fixed rate & the other party makes payments based on a floating rate. Market participants prefer the swap rate curve as a benchmark interest rate curve than a government bond yield curve for the following reasons: -

Swap rates reflect the Credit risk of Commercial banks rather than the credit risk of governments. Swap Rate is NOT affected by Technical Factors The Swap Market is NOT regulated by any government The Swap curve has yield quotes at Many Maturities.

Banks that manages interest rate risk with swap contracts are more likely to use Swap curve. Retail Banks are more likely to use Government bond yield curve.

or SFRT (Pt + PT) + PT = 1

Swap Spreads  It refers to the amount by which the swap rate exceeds the yield of a government bond with the same maturity. Information about Demand & Supply! Swap Spread t = (Swap Rate t) – (Treasury Yield t) Swap Spreads are always positive, reflecting the lower credit risk of government compared to the credit risk of surveyed banks that determines the Swap rate. Libor is the most used. I-Spread  It is the amount by which the yield on the risky bond exceeds the swap rate for the same maturity. I-spread only reflects compensation for credit & liquidity risks. The higher the I-Spread, the higher the compensation for liquidity & credit risk.

Z - Spread  It is the spread that when added to each spot rate on the default-free spot curve, makes the PV of the bond’s CF = Bond’s Market price. It is a Spread over the entire spot rate curve. NOT appropriate to use to value bonds with Embedded Options Coupon (Coupon+Face Value) Market Price = + 2 (1+1− year Spot Rate+ Z) ( 1+2− year Spot Rate +Z ) TED Spread  It is the amount by which the interest rate on loans between banks exceeds the interest rate on short-term US government debt. Risk in the Banking system. The TED is seen as an indication of the risk of interbank loans. A rising TED indicates that market participants believe banks are increasingly likely to default on loans & that risk-free T-Bills are becoming more valuable. TED Spread = (LIBOR Rate in %) – (T-Bill Rate in %) OIS (Overnight Index Swap) Rate  It reflects the Federal Funds rate & includes minimal counterparty risks. LIBOR-OIS Spread  Amount by which the LIBOR rate (include credit risk) exceeds the OIS rate (include only minimal credit risk). Overall well being of the banking system. A low LIBOR-OIS spread is a sign of High market liquidity. Unbiased Expectations Theory & Pure Expectations Theory  Investor’s expectations determine the shape of the interest rate term structure. Forward Rates are solely a function of expected spot rates & every maturity strategy has the same expected return over a given investment horizon. Market expects short-term rates to rise in the future The underlying principle behind the Pure Expectation Theory is Risk Neutrality: Investors do not demand a risk premium for maturity strategies that differ from their investment horizon: - If the yield curve is Upward sloping, the short-term rates are expected to Rise - If the yield curve is Downward sloping, the short-term rates are expected to Fall - A flat yield curve implies that the market expects short-term rates to remain Constant.

Local Expectations Theory  Similar to the unbiased expectations theory with one major difference: it preserves the risk-neutrality assumption only for Short Holding Periods. Short-Term holding period return of Long-maturity bonds > Short-Term holding period returns of Short-maturity bonds Liquidity Preference Theory  Forward rates reflect investors’ expectations of future spot rates + a liquidity premium to compensate investors for exposure to interest rate risk. Liquidity premium is positively related to maturity. A positive-sloping yield curve indicate: - The market expects future interest rates to rise (yield curve may have any shape) - Rates are expected to remain constant, but the addition of the liquidity premium results in a positive slope. Incorporates expectations of short-term rates The size of the liquidity premiums need not be constant over time. Segmented Markets Theory  Yields are not determined by liquidity premiums & expected spot rates. The shape of the yield curve is determined by the preferences of borrowers & lenders which drives the balance between supply of & demand for loans of different maturities. It supposes that various market participants only deal in securities of a particular maturity because they are prevented from operating at different maturities. Preferred Habitat Theory  Forward rates represent expected future spot rates + a premium. It does not support the view that this premium is directly related to maturity. The existence of an imbalance between the supply & demand for funds in a given maturity range will induce lenders & borrowers to shift their preferred habitats to one that has the opposite balance. The investors must be offered an incentive to compensate for the exposure to price. Borrowers require cost savings & lenders require a yield premium. Equilibrium Term Structure Model  It attempts to describe changes in the term structure through the use of Fundamental economic variables that drive interest rates. Such as: Cox-Ingersoll-Ross Model  Based on the idea that interest rate movements are driven by individuals choosing between consumption today versus investing & consuming at a later time. The single factor is the Short-term Interest Rate.

Volatility in Short-term rates is linked to Monetary Policy (interest rates) Volatility in Long-term rates is linked to uncertainty of the Real Economy & Inflation. The Vasicek Model  It suggests that interest rates are Mean reverting to some Long-run value. Volatility does not increase as the level of interest rates increase in this model. The main disadvantage is that the model does not force interest rates to be Non-Negative

Arbitrage-Free Models  Assumption that bonds trading in the market are correctly priced & the model is calibrated to value such bonds consistent with their market price. These model do not try to justify the current yield curve. The Ability to calibrate arbitrage-free models to match current market prices is 1 advantage of arbitrage-free models over the equilibrium. Such as: The Ho-Lee Model  The model assumes that changes in the yield curve are consistent with a No-arbitrage condition. It is calibrated by using Current Market prices to find the timedependent drift term t that generates the current term structure. Symmetrical Normal Distribution

Yield Curve Risk  It refers to risk to the value of a bond portfolio due to unexpected changes in the yield curve. Yield curve sensitivity can be generally measured by effective duration using key rate duration. Effective Duration  Measures Price sensitivity to small parallel shift in the yield curve. It is not an accurate measure of interest rate sensitivity to non-parallel shift in the yield curve. Shaping Risk  It refers to changes in portfolio value due to changes in the shape of the benchmark yield curve. Key Rate Duration  It is the sensitivity of the value of a security to changes in a single par rate, holding all other spot rates constant. Compared to effective duration, it is superior for measuring the impact of Nonparallel yield curve shifts. It is the approximate % change in the value of a bond portfolio in response to a 100 basis point change. D1 = 0,7 D5 = 3.5 D25 = 9.5

Exposure = Somme des Key Rate Duration  Sensibilité

Decreasing the Key Rates at the Short end of the yield curve makes an upward-sloping yield curve steeper! Key rate duration  changing the yield of Specific Maturity Changes in the Shape of Yield curve is explained by (in order of importance): -

Level XL -> A parallel increase or decrease of interest rates Steepness XS -> Long-term interest rates increase while short-terms rates decreases Curvature XC -> Increase curvature means short & long term interest rates increase while intermediate rates do not change.

Decrease in Short Term & Long Term rates  Indication of change in Level of Interest rate If Intermediate Term change differently that Short & Long Term  Change in Curvature Spot Rates < Forward Rates  Bonds are Undervalued, they should be purchased.

The Arbitrage-Free Valuation Framework: (A Voir Absolument pour Level II) Arbitrage-Free Valuation  It values securities such that No market participant can earn an arbitrage profit in a trade involving that security. An arbitrage transaction involves No initial cash outlay but a positive riskless profit at some point in future. 2 types of arbitrages opportunities: 1) Value Additivity (when the value of whole differs from the sum of the values of parts) 2) Dominance (When 1 asset trades at a lower price than another asset with identical characteristics) If the principle of Value Additivity does not hold, arbitrage profits can be earned by Stripping or Reconstitution.

Arbitrage-Free Valuation of a Fixed-rate, option-free bond entails discounting each of the bond’s future CF using the corresponding Spot Rate.

We can value option-free bonds with a simple spot rate curve. For bonds with Embedded Options, changes in future rates will affect the probability of the option being exercised & the underlying future CF.

Binomial Interest Rate Tree Framework  It assumes that interest rates have an Equal probability of taking one of 2 possible values in the next period. It is a Lognormal Random Walk with 2 Desirable Properties: Forward Rate1,Low = Forward Rate1,Upper e-2 1) Higher volatility at higher rates where:  = % Volatility 2) Non-Negative interest rates Backward Induction  It refers to the process of valuing a bond using a binomial interest rate tree. The current value of the bond is determined by computing the bond’s possible values at Period N & working backwards to node 0. The appropriate discount rate is the Forward rate associated with the node.

The construction process for a binomial interest rate comports 3 rules: 1) The interest rate tree should generate Arbitrage-Free values for the benchmark security. The value of bonds produced by the interest rate tree must be equal to their Market price. 2) Adjacent forward rates are 2 Standard Deviations apart. 3) The Middle Forward Rate in a period is equal to the implied Spot rate one period forward rate for that period. (Cf. question 2)

Aide: PA = prob. (PUP + Coupon)/ 1 + % + prob. (PDown + Coupon)) / (1 + Rate)

For bonds with Embedded Options, the future CF are uncertain as they depend on whether the embedded option will be in the money or not. The underlying CF are also dependent on the same future interest rates. Hence, to value bonds with embedded options, we have to allow rates to fluctuate & to use the binomial interest rate tree. All Option values increase when the volatility of the underlying asset increases!!

Path Dependency  Prepayments on underlying residential mortgages affect the CF of a mortgage-backed security. Prepayment risk is similar to call risk in a callable bond. Prepayment risk is affected not only by the level of interest rate at a particular point in time, but also by the path rates tool to get there. Prepayment Risk is Path Dependent.

An important assumption of the binomial valuation process is that the value of the CF at a given point in time is independent of the path that interest rates followed up to that point. CF are NOT Path Dependent; they do not depend on the path rates took to get to that node. The Binomial tree backward induction process cannot be used to value such securities & instead use the Monte Carlo simulation to value mortgage-backed securities. Monte Carlo Forward Rate Simulation  It involves randomly generating a large number of interest paths, using a model that incorporates a volatility assumption & an assumed probability distribution. The underlying CF can be path dependent. For Mortgage Back Secu! A Monte Carlo simulation may impose upper & lower bounds on interest rates as part of the model generating the simulated paths. These bounds are based on the notion of mean reversion; rates tend to rise when they are too low & to fall when they are too high. Increasing the number of paths  Increase the Statistical Accuracy of the estimate

Rappel: Z Spread  Rate that make the PV of the Bond = Market Price of the Bond

For a 3-year, Semiannual coupon bond, there will be six nodal periods, resulting in 2(6-1) paths. OAS Spread = Z-Spread – Option Cost  Spread on a bond with an embedded option after the embedded option cost has been removed. Highest OAS  Bond is Underpriced! The OAS for a corporate bond must be calculated using a Binomial Interest rate model....