CFA® Preparation www.dbf-.com

Reading Reading Title Study Session Number

32 The Term Structure and Dynamics 12 33 The -Free Valuation Framework

Valuation and Analysis of Bonds with Embedded 34 Options

35 Models 13

36 Credit Default Swaps

Luis M. de Alfonso CFA® Preparation FIXED INCOME www.dbf-finance.com

The Term Structure and Interest Rate Dynamics

Study Session 12

Reading Number 32

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.a: Describe relationships among spot rates, forward rates, to , expected and realized returns on bonds, and the shape of the

SPOT RATES Ø Annualized market interest rate for a simple paymentto be received in the future Ø Normally we use spot ratesfor government securities to generatethespot rate curve Ø Spot ratescan be interpreted as the yieds on zero- bonds (sometimes are referredas zero-couponrates)

P = discount factor (price today of a 1$ par zero-coupon ) Price of a zero-coupon bond P = S = spot rate () (discount factor) T = maturity

The termstructure of spot rates(graphof the spot rate S versus thematurity T) is known as the spot yield curve or spot curve

Shape of spot curve changescontinously with the market prices of the bonds

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve

FORWARD RATES Ø Annualized interest rate on a loan to be initiated at a future period Ø The termstructure of forward ratesis called the forward curve Ø Forward curves and spot curves are mathematically related(we can derive one from the other)

f (j,k) = the annualizedinterest rate applicableon a k-year loan starting in j years

k

F(,) = t = 0 t = j t = j + k (,)

F(,) = of a $1 par zero-coupon bond maturing at time j+k delivered at time j (discountfactor associatedwiththeforward rate)

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve

THE SPOT RATE FOR A GIVEN MATURITY CAN BE EXPRESSED AS A GEOMETRIC AVERAGE OF THE ONE PERIOD SPOT RATE AND A SERIES OF ONE PERIOD FORWARD RATES

S

S f (1, 1) f (2, 1) f (3, 1) f (� − 1, 1) …......

t = 0 t = 1 t = 2 t = 3 t = 4 t = k - 1 t = k

S = S x f (1,1) x f (2, 1) x f (3,1) x …...... x f (k − 1, 1)

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve

YIELD TO MATURITY (YTM) Ø Is the yield to maturity of a bond purchased at market price Ø If the spot rate curve is not flat, YTM will not be the same as the spot rate

EXPECTED RETURN ON BOND Ø Ex-ante holding period return that a bond investor expect to earn Ø Will be equal to the bond´s yield only when:

1. The bond is held to maturity 2. All payments (coupon and principal) are made in time and in full 3. All coupons are reinvested at the original YTM

REALIZED RETURN ON BOND Ø Actual return that the investor experiences over the investment´s holding period

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.a: Describe relationships among spot rates, forward rates, yield to maturity, expected and realized returns on bonds, and the shape of the yield curve

If spot rate curve would be flat (� = �= �)

YTM = Spot Rate

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.b: Describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models

FORWARD PRICING MODEL Forward pricing model values forward contractsbased on arbitrage-free pricing

Knowing the spot curve, we can If there is no () P P x F F calculate any forward price arbitrage then () = (,) (,) = (considering no arbitrage)

P()

t = 0 t = j t = j + k

P F(,)

P = Remember: and F(,) = (,)

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.b: Describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models

P()

P = t = 0 t = 2 t = 5

P F(,)

() F(,) =

F(,) =

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.b: Describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models

FORWARD RATE MODEL Relates forward and spot rates

() This equations derive () 1 + S (1 + S ) 1 + � () from the equations of () = x (,) 1 + �(,) = () the previous slide

S()

t = 0 t = j t = j + k

S �(,)

If the yield curve is upward sloping S() > S �(,) > S()

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.b: Describe the forward pricing and forward rate models and calculate forward and spot prices and rates using those models

S

t = 0 t = 2 t = 5

S f(,)

() () 1 + �(,) = ()

1 + �(,) = ( )

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.c: Describe how zero-coupon rates (spot rates) may be obtained from the par curve by

PAR RATE Ø Yield to maturity of a bond trading at par (by definition, par rate = coupon rate of the bond) Ø Par ratesfor bonds with different maturities make up the par rate curve

By bootstrapping, spots rates (or zero-coupon rates) can be derived from the par curve

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.c: Describe how zero-coupon rates (spot rates) may be obtained from the par curve by bootstrapping

1.- The one year spot rate = one year par rate �� = 1,00%

, , 2.- We can value the 2 years bond using par rates: 100 = + (,) (, )

Alternatively, we can alsovalue the two years , , � � 100 = + introducing = 1,00% we get � = 1,252% bond using spot rates () ()

, , , � � � 3.- Similarlyforthe three years bond: 100 = + + introducing and we get � = 1,51% () () ()

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.d: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management

RELATIONSHIP BETWEEN SPOT AND FORWARD RATES

Upwardsloping spot curve Forward rates rises as j increases Ex. Spot and forward curves as of July 2013 Downward sloping spot curve Forward rates declines as j increases

Upwardsloping spot curve Forward curve will be above spot curve Downward sloping spot curve Forward curve will be below spot curve

From the forward rate model:

1 + S = (1 + S ) x (1 + f (1,1)) x (1 + f (2,1)) x …...... x (1 + f (T − 1, 1))

Spot rate for a long maturity equal the geometric mean of the 1+ S one period spot rate and a series of one-year forward rates

1+S (1 + f (1,1)) (1 + f (2,1)) (1 + f (T− 1,1)) …......

0 1 2 3 T - 1 T

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.d: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management

FORWARD PRICE EVOLUTION

Ø If the future spot ratesactuallyevolve as forecasted by theforward curve, the forward price will remain unchanged Ø Therefore, a change in the forward price, indicatesthat the future spot rate(s) did not conform the forward curve

When spot ratesturn out to be than implied by the forward curve forward price will

A trader expecting lower future spot rates (than impliedby the current forward rates) would purchase the forward contract to profit from its operation An active portfolio manager will try to outperform the overall by predicting how the future spot rates will differ from those predicted by the current forward curve

“For a bond investor, the return of a bond over one year horizon (independently from the maturity of the bond) is alwaysequal to the one year risk free rate if the spot ratesevolve as predicted by today´s forward curve. If not, the returnover the one year period will differ depending on the bond´s maturity”

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.d: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.d: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management

A) We calculate the predicted forward rates �(�,�) and �(�,�) based on the actual spot rates (values of the forward curve)

, 1 + �(,) = = �( , ) = 0,0501 () (, ) As we get the same values for the predicted forward () () rates than the spot rates after one year, we can 1 + �(,) = () conclude that the spot rates have evolved as , predicted by forward rates 1 + �(,) = = �( , ) = 0,0601 ( ) ( , )

B) Now we calculate the one year holding period return of each bond

1.- The price of a one year zero coupon bond given by the one yearspot rate of 3% is P = = = 0,9709 ,

After one year the bond is at maturity and pays $1, then the holding period return = - 1 = 3% ,

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.d: Describe the assumptions concerning the evolution of spot rates in relation to forward rates implicit in active bond portfolio management

B) Now we calculate the one year holding period return of each bond (continue)

2.- The price of a two years zero coupon bond given by the two years spot rate of 4% is P = = = 0,9246 ,

After one year the bond will have one year remaining to maturity, and based on a one year expected spot rate of 5,01%, the bond´s price will be = 0,9523 ,

, Then, the holding period return = - 1 = 3% ,

3.- The price of a three years zero coupon bond given by the three years spot rate of 5% is P = = = 0,8638 ,

After one year the bond will have two years remaining to maturity, and based on a two years expected spot rate of 6,01%, the bond´s price will be = 0,8898 ,

, Then, the holding period return = - 1 = 3% ,

Hence, regardless of the maturity of the bond, the one year holding period return will be the one year spot rate as the spot rates have evolved consistent withthe forward curve (as it existed when the trade was initiated)

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.e: Describe the strategy of riding the yield curve

Ø The most straightforward strategy for a bond investor is Maturity Matching purchasing bond with the same maturity to the investor´s investment horizon Ø However, with an upward sloping interest rate termstructure, investorscould obtain superior return by “riding the curve yield” (also known as rolling down the curve yield)

Under this strategy, investor purchase bonds with longer maturitiesthan the investment horizon

If the yield curve remains unchanged during theinvestment period riding the yield curve will produce higher returns via two effects:

1.- getting more yield by purchasing a bond with longer maturity (longer maturity bonds have higher yield when term structure is upward sloping)

2.- when selling the bond, we also obtain a capital profit

However, this strategy increasesinterest risk

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.e: Describe the strategy of riding the yield curve

Int rate Example of riding the yield curve

7% Given this yield curve, and investor withone year horizon could: 6% 1.- Purchase a one year maturity bond and get a return of 6% (maturitymatching)

0 1 2 Years 2.- Follow a riding the yield curve strategyby purchasing a 2 years maturity bond and selling it after one year. If we do that, the total return will be:

(a) 7% return for the first year + (b) the benefit of selling the bond after one year

Supposing the of the bond is $1000:

(a) 7% return for the first year = $70 (7%) Total return = $79,43 (7,943%) (b) the benefit of selling the bond after one year (*) = 1.009,43 - 1.000 = $9,43

(*) After one year (considering that the yield curve remains the same), the value of the bond will be: = 1.009,43 (after one year the interest rate for a one year bond will be 6% while mybond has a yield of 7%) ,

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.f: Explain the rate curve and why and how market participants use it in valuation

• One party makes payments on a fixed rate Ø In a plain vanilla • Counterparty makes payments based on floating rate

Ø The fixedrate in an interest rate swap is called swap fixedrate or swap rate

Ø Swap rates versus maturities swap rate curve

Ø Market participants prefer swap rate curve as a benchmark interest rate curve rather than a yield curve forthe following reasons: • Swaps rates reflect credit risk of commercial banks rather than the credit risk of governments • Swap market is not regulated by governments, which makes swap rates in different countries more comparable • Swap curve has yield qoutes at many maturities SFR 1 Ø The SFR (swap fixedrate for tenor T) can be computed using relevant ( normally) spot rate curve as: + = 1 (1 + S) (1 + S )

In the equation, SFR can be thought of as the coupon rate of a $1 par value bond given the underlying spot rate curve

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.f: Explain the swap rate curve and why and how market participants use it in valuation

SFR 1 + = 1 (1 + S) (1+ S) Ø For a one year swap contract SFR is 3% Ø For a two years swap contract SFR is 3,98% Ø For a three years swap contract SFR is 4,93% SFR can be thought as the coupon of a $1 par value bond given the underlying spot rate curve (LIBOR normally) The SFR is the fixedinterest rate that one party must pay, while the counterparty pays the LIBOR (underlying) interest rate

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.g: Calculate and interpret the swap spread for a given maturity

Ø SWAP SPREAD Amount by which the swap rate exceedsthe yield of a government bond withthe same maturity

swap spread = swap rate - Treasury yield

The LIBOR swap curve is the most commonly used interest rate curve, it roughly reflects the default risk of a commercial bank

Ø I - SPREAD I – spread for a credit-risky bond is the amount by which the yield of the risky bond exceeds the swap rate for the same maturity

I − spread = yield of the risky bond - swap rate

Investors use I-spread to separate the time value portion of a bond yield from the risk premia for credit and liquidity (i.e. While a bond yield reflects time value as well as compensation for credit and liquidity risk, I-spread only reflects compensation for credit and liquidity risk)

The higher the I-spread, the higher the compensation for credit and liquidity risk Fora default free bond, I-spread provides an indication of liquidity risk

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.h: Describe the Z-spread

Ø Z - SPREAD The spread that when added to each spot rate on the yield curve makesthe present value of a bond´s cash flows equal to the bond´s market price

Example

• One year spot rate S = 4% • Twoyears spot rate S = 5% $ $ $104,12 = + Z = 0,008 • Market price of a two year bond with (,) (,) annual coupon of 8% = $104,12

§ Z-spread is a spread over the entire spot curve § Z (comes from zerovolatility) Z-spread assumes interest rate volatility is zero § Z-spread is notappropiate to use to valuebondswith embedded options(as without any interest rate volatility, options are meaningless)

Z-spread is a constant, and measures the spread that an investor will receive over the entirely of the Treasure yield curve it gives a realistic valuation of a

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.h: Describe the Z-spread

() Note that for calculating spot rates we use the formula from the forward rate model (seen before) 1 + S() = (1 + S) x 1 + �(,)

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.i: Describe the TED and LIBOR-OIS spreads

Ø TED SPREAD Amount by which the interest rate on loans between banks (formally, three month LIBOR) exceeds theinterest rate on short term US government debt

TED spread = (3-month LIBOR rate) - (3-month T-bill rate)

T-bills are considered risk free while LIBOR reflectsthe risk of lending to commercial banks

TED spread is seen as an indication of the risk of interbak loans

Ø LIBOR-OIS SPREAD Amount by which the LIBOR rate (which includes credit risk) exceeds the OIS rate (which includes only minimal credit risk)

OIS ( rate) rate roughly reflectsthe and includes minimal counterparty risk

LIBOR-OIS spread is a useful measure of credit risk and of the overallwellbeing of the banking system

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.j: Explain traditional theories of the term structure of interest rates and describe the implications of each theory forforward rates and the shape of the yield curve

1. Unbiased Expectations Theory

2. Local Expectations Theory

3. Liquidity Preference Theory

4. Segmented Markets Theory

5. Preferred Habitat Theory

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.j: Explain traditional theories of the term structure of interest rates and describe the implications of each theory forforward rates and the shape of the yield curve

1.- Unbiased Expectations Theory

§ Investor expectations determine the shape of the interest rate termstructure

§ Also known as unbiased expectationstheory or pure expectationstheory

§ Forward ratesare an unbiased predictor of future spot rates

§ Under this theory invest on a 5 year bond = invest in a 2 years bond and then invest in a 3 years bond

§ The underlying principle of this theory is risk neutrality. “investors do not demand a risk premium for maturity strategies that differ from their investment horizon”

Ex 5% 9% Unbiased expectations Theory says that if one yearspot rate is 5% and two years spot rate is 7% then the one year forward rate in one yearmust be 9% 7% 7% Itwould be the same to invest in a two years bond than invest first in a one year bond an then in another year bond (risk neutrality) Today 1 year 2 years

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.j: Explain traditional theories of the term structure of interest rates and describe the implications of each theory forforward rates and the shape of the yield curve

2.- Local Expectations Theory

§ Equal to unbiased expectationstheory but only in the short term: “risk neutrality is assumed only for short holding periods but risk premiums should exist for long termperiods”

§ This implies that over short time periods, every bond should earn the risk free rate

§ This Theory does not hold as short-holding-period returns of long-maturity bonds can be shown to be higher than short- holding-period returnon short-maturity bonds

3.- Liquidity Preference Theory

§ Investors demand a liquidity premium that is positively relatedto a bond´s maturity (to compensate investrors for exposure to interest rate risk)

§ Forward ratesreflectsinvestor expectations of future spot ratesplus a liquidity premium § Liquidity preference theory states that forward ratesarebiased estimatesof the market’s expectationof future rates because they include a liquidity premium

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.j: Explain traditional theories of the term structure of interest rates and describe the implications of each theory forforward rates and the shape of the yield curve

4.- Segment Market Theory

§ The yield curve is determined by the preferences of borrowers and lenders

§ Yield at each maturity is independent of the yields of other maturities

§ Yield is determined by supply and demand

§ Different market participants only deal with securities at different maturities

5.- Preferred Habitat Theory

§ Forward rates = expectedspot rate + premium (not directly relatedwith maturity)

§ An imbalance between supply and demand for funds in a given maturity range will induce lenders and borrowers to shift from their preferred habitats (maturity range) to other if they are compensated adequately

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.k: Describe modern term structure models and how they are used

“Modern termstructure models attempt to capture the statistical propertiesof interest rate movements”

1) EquilibriumTerm Estructure Models

1.1) The Cox-Ingersoll-Ross Model

1.2) The

2) Arbitrage-Free Models

2.1) Ho-Lee Model

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.k: Describe modern term structure models and how they are used

1) Equilibrium Term Estructure Models

Attempt to describe changes in the term structure through the use of fundamental economic variables that drive interest rates

1.1) The Cox-Ingersoll-Ross Model Interest rates are driven by individuals choosing between consumption today versus (CIR model) investing and consuming at a later time Assumes that the economy has a long-run interest rate (b) to which the short-term rate converges

• a(b-r)dt : this term forces the interest rate to mean revert toward the long-run value (b) at a speed dr = a(b-r)dt + � � dz determined by the parameter (a) • � �dz : indicates that volatility increases with interest rate (at high interest rates, the amount of period- over-period fluctuation in rates is also high)

1.2) The Vasicek Model Like CIR, Vacisek model suggest that interest rates are mean reverting to some long-run value

• Volatility in this model does not increase as the level of interest rates increases (no interest rate (r) dr = a(b-r)dt + � dz term appears in the second term σ dz) • The main disadvantage of this model is that the model does not force interest rate to be non-negative

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.k: Describe modern term structure models and how they are used

2) Arbitrage-Free Models

Begins withobserved market prices and the assumption that securities are correctly priced, and the model is calibrated to value such securities consistent withtheir market price This model does not try to justify the curve,rather, they take this curve as given

2.1) The Ho-Lee Model The model assumes that changes in the yield curve are consistent with a no-arbitrage condition

• �: time dependent drift (deviation) term

d� = �dt + �dz • This model uses the market prices to find the time depent drift term (�) that generates the current term structure

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.l: Explain how a bond´s exposure to each of the factors driving the yield curve can be measured and how these exposures can be used to manage yield curve risks

Yield curve risk • Risk to the value of a bond portfolio due to unexpected changesin the yield curve • Yield curve sensitivity can be measured by effective duration, key rate duration or a threefactor model (level, steepness and curvature)

Ø EffectiveDuration Measures the sensitivity of a bond´s price to parallel shifts in the yield curve

Ø Key Rate Duration Measures bond price sensitivity to a change in a specific spot rate keeping everithing else constant

Numerically is defined as the change in the value of a bond portfolio in response of a 1% change in thecorresponding key rate, holding all other ratesconstant

∆ D = key rate duration formaturity i (are data givenby the model to ≈ - D ∆r - D ∆r - D ∆r… predict changes in portfolio price P while changes in key rates r, r,…)

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.l: Explain how a bond´s exposure to each of the factors driving the yield curve can be measured and how these exposures can be used to manage yield curve risks

Ø Sensitiveto parallel, steepnessand Measures sensitivity to threedistinct categoriesof changesin theshape curvature movements of thebenchmark curve

§ Level (∆�) A parallel increase or decrease of interest rates

§ Steepness (∆�) Long term interest rates increase while short term rates decrease

§ Curvature (∆�) Short and long term interest rates increase while intermediate rates do not change

∆ D, D, D : are the portfolio sensitivities to changes We can model the change in the value of the portfolio: ≈ - D ∆X - D ∆X - D ∆X in the yield curve´s level, steepness and curvature

Luis M. de Alfonso CFA® Preparation FI – The Term Structure and Interest Rate Dynamics www.dbf-finance.com

LOS 32.m: Explain the maturity structure of yield volatilities and their effect on price volatility

The term structureof interest rate volatility isthe graphof yield volatility versus maturity It provides an indication of yield curve risk

Ø Short term interest ratesare generallymore volatile than are long term rates Ø Volatility at long terminterest ratesis relatedto uncertainty regarding thereal economy and inflation Ø Volatility at short terminterest ratesreflects risks regarding monetary policy Ø Fixed income instruments with embedded options can be specially sensitive to interest rate volatility

Luis M. de Alfonso