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1 Bond Markets (FIN 601) - FSS 2015 - Prof. Dr. Erik Theissen - Chapter 3

Chapter 3: The Term Structure of Interest Rates

Introduction

Spot Rates

Forward Rates

Theories of the Term Structure

Estimating the Term Structure

Swaps and Swap Rates

2

The Term Structure

Introduction

Interest rates for different times to maturity are often different

Example (Frankfurter Allgemeine Zeitung, April 14, 2000)

Typical term structure

patterns:

flat (US)

normal (Germany)

inverse (UK)

Bond Markets (FIN 601) - FSS 2015 - Prof. Dr. Erik Theissen - Chapter 3

Term structure described

by:

level

slope (long minus short)

curvature

3

The Term Structure

Introduction

How best to describe the term structure?

The term structure is typically derived for a homogeneous

(with respect to default risk and liquidity) class of bonds, e.g.

AAA-rated corporate bonds

Which interest rates to use to describe the term structure?

Using yield to maturity:

Yields to maturity of coupon bonds with the same time to

maturity may be different even when all bonds are fairly

priced (see below) which ones to use to describe the

term structure?

To avoid the ambiguity we use spot rates. Spot rates are

the yields to maturity of zero bonds


4

The Term Structure

Spot Rates

Spot rates:

Consider a zero bond (i.e. a bond which makes only one

payment at time T, there are no interest payments)

The yield to maturity on such a bond is called the T-period

spot rate and is given by

Spot rates are the discount rates to use when valuing bonds

1T

T

T

CPV

r


5

The Term Structure

Spot Rates

Spot rates:

Note: Every coupon bond can be interpreted (and valued) as

a portfolio of zero bonds

- Stripping of government bonds (STRIPS = Separate

Trading of Registered Interest and Principal of Securities)

This is essentially what we did in chapter 2

Advantage of spot rates: Because they are derived from

zero bonds, there are no problems with intermediate

payments that require reinvesting


6

The Term Structure

Spot Rates

Spot rates and yield to maturity:

Assume the following spot rates

and two 2-year bonds, one with a 4% and one with a 8%

coupon and with PVs

Assume further that the bonds are fairly priced (i.e., price =

PV)

year 1 2

spot rate 5% 7%

2

2

4 10494 6472

1 05 1 07

8 108101 9504

1 05 1 07

., ,

,, ,


7

The Term Structure

Spot Rates

The yields to maturity are

Both bonds are fairly priced. Why does the 8% bond have

lower yield to maturity?

The 8% bonds has shorter economic time to maturity. At

the same time the term structure is upward-sloping -

instruments with shorter maturity offer lower yield

Thus: Comparing yields to maturity may be misleading

2

2

4 10494 647 0 0 06958

1 1

8 108101 95 0 0 06922

1 1

A

A A

B

B B

. y .y y

. y .y y


8

The Term Structure

Spot Rates

Source: Elton et al. (2007), figures 21.5 and 21.6

Illustration of the coupon effect on yields to maturity:


9

The Term Structure

Forward Rates

Forward rates

Forward rate agreement: you agree today to lend money at

time t1 which is to be paid back at time t2

The interest rate is fixed today

Such an interest rate is called a forward rate and denoted

ft,T

t0

terms of

contract fixed

t1

money

invested

t2

money

repaid

rate f1,2


10

The Term Structure

Forward Rates

Forward and spot rates:

There are two ways to invest money for two periods:

1: Lend money for two periods today at the two-period spot

rate

2: Lend money today for one period at the one-year spot

rate and enter into a forward-rate agreement for period two

t0 t1 t2

(1+r1)(1+f1,2)

(1+r2)2


11

The Term Structure

Forward Rates

Forward and spot rates (contd.):

Absence of arbitrage requires

Similarly one can use the 2-year and 3-year spot rate to

calculate f2,3 etc.

In general:

There is thus a correspondence between spot and forward

rates

An open question: What is the relation between forward

rates and future (expected) spot rates?

2

2 2

2 1 1 2 1 2

1

11 1 1 1

1, ,

rr r f f

r

0 0

11 1 1 1

1

TT T T

T,T , ,T ,T

rr r f f

r


12

The Term Structure

Forward Rates


Assume spot rates r1 = 5% and r2 = 6%. Two zero bonds

with one year and two years to maturity are traded at

Now assume the following investment:

Note:

1 2 2

100 10095 2381 88 9996

1 05 1 06PV . ; PV .

. .

t0 t1 t2

buy bond 1 -95.2381 +100 0

sell bond 2 +95.2381 0 -107.0095

sum 0 100 -107.0095

95.2381100 107.0095

88.9996


13

The Term Structure

Forward Rates


The return on this (net) investment is 7,0095%

This is equivalent to the forward rate:

Thus the forward rate is contractable by forming long-short

portfolios of traded bonds

21.061 0.070095

1.05


14

The Term Structure

Forward Rates

There are various interest rate futures contracts

Futures contracts are standardized, exchange-traded

contracts

Fed Fund Futures

- underlying: the monthly average (!) of the effective

overnight fed funds rate

Eurodollar Futures

- underlying: the 3-months Libor (London Interbank Offered

Rate; an interbank rate for Eurodollar deposits)

Exchange-traded futures contracts on government bonds

(e.g. the Eurex Bund Futures Contracts)

- often physically settled which entails valuation problems

(the cheapest-to-deliver option) Bond Markets (FIN 601) - FSS 2015 - Prof. Dr. Erik Theissen - Chapter 3

15

The Term Structure


Approaches:

Expectations Hypothesis

Liquidity Preference Hypothesis

others (not covered)

- Preferred Habitat

- Market Segmentation Hypothesis


16

The Term Structure


Investment horizon and risk:

Consider an investor with a t-period investment horizon

She invests in a (default-free) zero bond

- Maturity = t: riskless

- Maturity < t: reinvestment risk

- Maturity > t: price risk

Risk-neutral investors dont care about the risk, they do not

require a risk premium

Risk-averse investors should require a risk premium

If the majority of investors has a short horizon (a preference

for liquidity), long-term bonds must offer a premium over

repeated short term investment


17

The Term Structure


The Expectations (or Risk-Neutrality) Hypothesis:

Forward rates are equal to expected future spot rates

Consequence: The expected return of a long-term invest-

ment and a repeated short-term investment are equal

Implication: There is no liquidity premium

This implies risk neutrality - there is no premium for bearing

the risk associated with a mismatch between investment

horizon and term to maturity of a bond

T T

T

0,T 0,1 t 1,t 0,1 0 t 1,t

t 2 t 2

1 r 1 r 1 f 1 r 1 E r

long-term repeated short-term


18

The Term Structure


The Expectations Hypothesis (contd.):

The term-structure is entirely driven by expectations on

future interest rates

These depend on expectations on a) real rates and b)

inflation

On average the term structure should be flat


19

The Term Structure


The Liquidity Preference (=Risk Aversion) Hypothesis:

Basic assumption: Investors (on average) have a preference

for liquidity (i.e. for investments with short term to maturity)

while issuers (on average) have a preference for long time

to maturity

Forward rates are larger than expected future spot rates

because they incorporate a liquidity premium

The expected return of a long-term investment is larger than

the expected return on repeated short-term investments

,T 0 ,T ,T 0 ,Tf E r L E r

T

T

0,T 0,1 0 t 1,t

t 2

1 r 1 r 1 E r

long-term repeated short-term


20

The Term Structure


The Liquidity Preference Hypothesis (contd.):

The term structure depends on

- expected future real rates

- expected inflation

- the liquidity premium

The liquidity premium may depend on the time to maturity

( term structure of liquidity premia), it may change over

time, and it is unobservable

We cannot derive expectations on future spot rates from

todays term structure

E.g. a normal term structure can be caused by a)

expectations of increasing spot rates and/or b) by an

increasing liquidity premium Bond Markets (FIN 601) - FSS 2015 - Prof. Dr. Erik Theissen - Chapter 3

21

The Term Structure


How to determine spot rates?

There are not too many zero bonds

Stripped coupon payments are difficult to use

- lower liquidity liquidity premium

- possibly different tax treatments (interest payments on a

coupon bond may be taxed differently than capital gains

from a zero bond)

- usually only available for government bonds

We thus often need to rely on coupon bonds to estimate the

spot rates

But remember: a coupon bond is a portfolio of zero bonds


22

The Term Structure


Illustration (Elton et al. 2007, p. 514)

Assume there are the following two bonds

Bond A is already a zero bond. We thus have r1 = 0.06

Now we construct a portfolio

Bond t0 (price today) t1 t2

A -100 106

B -96.54 6 106

Bond t0 (price today) t1 t2

B -96.54 6 106

A 5.6604 -6

Portfolio -90.8796 106


23

The Term Structure


Illustration (contd.)

The portfolio has the cash flow structure of a zero bond and

can be used to infer r2:

In a similar way we can obtain r3 from a three-year coupon

bond and so on

This procedure is called bootstrapping

From the derived spot rates we can also derive prices of

hypothetical zero bonds (implied zeros)

Bootstrapping can be done more conveniently using matrix

notation (see Veronesi 2010, p. 66 and the end-of-chapter

problems)

2

2 290.8796 1 r 106 r 0.08


24

The Term Structure


Problem of the approach:

Different two-year bonds may result in different estimates of

the two-year spot rate

Differences may occur because of differences in default risk,

liquidity risk, because of non-synchronous trading, because

of the bid-ask-spread and because of tax effects (different

taxation of interest income and capital gains; relevant when

comparing bonds with different coupons)


25

The Term Structure


A regression approach:

Define

where t may be a fraction. Z(0,t) is the discount factor, and is

also the price of a zero bond that pays 1 at maturity in t

Then we have

which can be estimated using OLS or more advanced

techniques (e.g. spline regressions)

Spot rates for maturities not covered by observations can be

obtained by interpolation

t

t

1Z 0,t

1 r

i i,t iP Z 0;t C


26

The Term Structure


Regression approach (contd.):

Note: Each discount factor (= each maturity date) is one

estimated parameter (one unknown)

We need more observations than we estimate parameters

Thus: the regression approach only works when there are

more bonds than maturity dates

This is typical for shorter maturities (e.g. up to 5 years)

see Veronesi (2010, p. 67)


27

The Term Structure


Estimating a continuous function:

The regression only gives us discrete points (those covered

by payment dates in the data set)


28

The Term Structure


Estimating a continuous function (contd.):

We can assume a functional form for the term structure, e.g.

a quadratic equation

We now rewrite our earlier regression as

and estimate the parameters 0, 1, 2

With these estimates we can estimate any spot rate via

20 1 2Z 0,t t t

2i 0 1 2 i,t iP t t C

20 1 2 Z 0,


29

The Term Structure


Estimating a continuous function (contd.):

A quadratic function is not able to accommodate all shapes

of the term structure observed in reality (e.g. it has no

inflection points)

We can use different, more complex functional forms (and

more advanced estimation techniques)

In general

where f(.) is a functional form that is flexible enough to

accommodate the shapes of the term structure observed in

reality

r 0,t f t


30

The Term Structure


Parametric approach: Nelson/Siegel (1987)

: level, long-term interest rate (t )

: instantaneous interest rate (t 0)

: slope factor: short-term long-term interest rate

: shape factor, drives medium-term yields

Estimation: minimize sum of squared differences between

model prices and observed data points

0 1 2 21 exp t /

r 0,t exp t /t /

0

0 1

1

2


31

The Term Structure


Parametric approach: Svensson (1994)

Extension of Nelson/Siegel (1987), allows for additional

turning point

Sufficiently flexible: monotonous, U-, inverse U- or S-shaped

Common approach of Deutsche Bundesbank and ECB; see

e.g. the technical note available at https://www.ecb.europa.eu/stats/money/yc/html/technical_notes.pdf

1

0 1 2 2 1

1

2

3 2

2

1 exp t /r 0;t exp t /

t /

1 exp t /exp t /

t /


32

The Term Structure


Parametric approach: Svensson (1994) (contd.)


33

The Term Structure


Definition of an interest rate swap:

A swap in general: Two parties exchange two cash flow

streams over a pre-specified period of time

Interest rate swap: One party pays a floating rate, the other

party (usually) pays a fixed rate

The floating rate is typically a benchmark rate such as Libor

or Euribor

The swap rate is the fixed interest rate which is equivalent

to the floating rate

Notional value = amount on which the interest is calculated


34

The Term Structure


Party A Party B Swap

Dealer

3-months

Libor

rT+20 bps rT+30 bps

A swap dealer with a matched book:

(rT: Treasury bond rate corresponding to the term of the swap)

3-months

Libor

Source: Sundaresan (2010), p. 326


35

The Term Structure


Complications and extensions:

Caps and floors can be added to the floating rate leg of the

swap (a floor will increase and a cap decrease the fixed

rate)

Swaption: Option to enter into a swap at predetermined

conditions


36

The Term Structure


Valuation:

At the inception, the terms of the swap are usually fixed

such that its value is zero

When interest rates change the value of the swap will

change. Example:

- assume you pay Libor and receive a fixed rate of 5%

- now assume interest rates increase

- the floating rates you pay increase while the rate you

receive is fixed - your position now has a negative value


37

The Term Structure


The swap rate curve:

Swaps for different maturities are commonly traded

We can use the swap rates for different maturities to

estimate the term structure: swap (rate) curve or Libor curve

Advantages:

- Swap rates are observed for many maturities

- The swap market is very liquid

- There is no on the run / off the run problem

- Swap rate curves can be compared across countries

(country ratings differ, while the same banks offer swap

rates in different currencies)


38

The Term Structure


The swap rate curve (contd.):

Disadvantages:

- Swap rates are not default-free - they reflect the credit

risk of the contract parties


39

The Term Structure


Source: Brsenzeitung, 12.2.2015


40

Reading List

Required Reading:

Bodie, Z., A. Kane and A. Marcus (2009): Investments, 8th edition,

McGraw Hill, sections 15.3.-15.6.

Elton, E., M. Gruber, St. Brown and W. Goetzmann (2007): Modern

Portfolio Theory and Investment Analysis, 7th edition. Wiley, Chapter

20.

Please note: This chapter assumes bi-annual interest payments.

Fabozzi, F. (2010): Bond Markets, Analysis and Strategies, 7th edition,

Pearson, chapter 5.

Sundaresan, S. (2009): Fixed Income Markets and Their Derivatives,

3rd edition, Academic Press, chapter 8 and section 1 of chapter16.

Veronesi, P. (2010): Fixed Interest Securities, Wiley, chapter 2 (including

the appendix).

(please note: some of the contents of chapter 2 of the book has been

covered in chapter 2 of the lecture)


41

Study Questions


Question 1

Assume that the one-year spot rate is 5% and the two-year spot rate is 6%. Assume

further that the expectations hypothesis holds (i.e., the expected one-year spot rate

one year from now is the forward rate implied by today's term structure).

a) What is the price of a one-year zero bond and the price of a two-year zero bond

today?

b) What is the expected price of the two year zero-bond one year from now?

c) What is the expected return from holding the two-year zero bond in the first year?

Question 2

There are three bonds, a zero bond, a 4% coupon bond and a 6% coupon bond. All

bonds mature in exactly three years. The term structure in two scenarios is given in

the following table.

Assume that all bonds are fairly priced.

a) Calculate the YtM of the three bonds under scenario A and under scenario B.

b) Interpret your result.

t1 t2 t3

Scenario A 4.0% 5.0% 6.0%

Scenario B 6.0% 5.0% 4.0%

42

Study Questions


Question 3

The prices and future cash flows of three coupon bonds are given as follows:

Use this information to obtain the one-, two- and three-year spot rates as well as the

one-period forward rates at time 1 and 2.

Question 4

Now consider the following bonds (see next slide). Use this information to obtain the

spot rates.

Note: You may want to reformulate the problem using matrix notation and then use

the matrix operators in Excel.

Bond Price Year 1 Year 2 Year 3

A 99.50 105

B 101.25 6 106

C 100.25 7 7 107

43

Study Questions


Price Time to maturity

(years)

Coupon rate

100,976 1 3.5%

104,333 2 5.0%

102,866 3 4.0%

102,377 4 3.75%

112,668 5 6.0%

108,696 6 5.5%

109,626 7 5.25%

105,803 8 4.75%

103,642 9 4.50%

98,12 10 4.0%

44

Study Questions


Question 5

Consider the following investment alternatives shown below.

a) Calculate the YtM of all investments.

b) Calculate the weighted average yield to maturity of the components of the three

portfolios. Use the proportions invested in each bond as weights (for portfolio "A

and C" this would be 100/192 for bond A and 92/192 for bond C).

c) Compare your results from a and b.

t=0 t=1 t=2 t=3

Bond A -100 15 15 115

Bond B -100 6 106 ---

Bond C -92 9 9 109

A and B -200 21 121 115

B and C -192 15 115 109

A and C -192 24 24 224

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