CHAPTER 5 POWERPOINT PRESENTATION

CHAPTER FIVE

THE VALUATION OF

RISKLESS SECURITIES

INTEREST RATES



NOMINAL V. REAL INTEREST RATES

•

Nominal interest rates:



represent the rate at which consumer can trade present money for future money

INTEREST RATES



NOMINAL V. REAL INTEREST RATES

•

real interest rate



the rate of return from a financial asset expressed in terms of its purchasing power (adjusted for price changes).

YIELD TO MATURITY



CALCULATING YIELD TO MATURITY : AN EXAMPLE

•

Suppose three risk free returns based on three Treasury bonds:

Bond A,B are pure discount types;

mature in one year

Bond C coupon pays $50/year;

matures in two years

YIELD TO MATURITY

Bond Market Prices:

Bond A $934.58 Bond B $857.34 Bond C $946.93

WHAT IS THE YIELD-TO-MATURIYTY

OF THE THREE BONDS ?

YIELD TO MATURITY

 YIELD-TO-MATURITY (YTM)

•

Definition: the single interest rate* that would enable investor to obtain all payments promised by the security.

•

very similar to the internal rate of return (IRR) measure

* with interest compounded at some specified interval

YIELD TO MATURITY



CALCULATING YTM:

•

^{BOND A}

•

Solving for r_A

(1 + r

_A

) x $934.58 = $1000

r

A

= 7%

YIELD TO MATURITY

 CALCULATING YTM:

•^{BOND B}

•Solving for r_B

(1 + r_B) x $857.34 = $1000 rB = 8%

YIELD TO MATURITY



CALCULATING YTM:

•

^{BOND C}

•

Solving for r_C

(1 + r

_C

)+{[(1+ r

_C

)x$946.93]-$50

= $1000

r

C

= 7.975%

SPOT RATE



DEFINITION: Measured at a given

point in time as the YTM on a pure

discount security

SPOT RATE

 SPOT RATE EQUATION:

where P_t = the current market price of a

pure discount bond maturing in t years;

M_t = the maturity value s_t = the spot rate



^t _t



t s

P M

 

1

DISCOUNT FACTORS



EQUATION:

Let d

_t

= the discount factor



_t



t s

d   1

1

DISCOUNT FACTORS



EVALUATING A RISK FREE BOND:

•

^EQUATION

where c_t = the promised cash payments n = the number of payments







ⁿ

t

t t

c d PV

1

FORWARD RATE



DEFINITION: the interest rate today that will be paid on money to be

•

borrowed at some specific future date and

•

to be repaid at a specific more distant future date

FORWARD RATE



EXAMPLE OF A FORWARD RATE

Let us assume that $1 paid in one year at a spot rate of 7% has

9346 07 $.

.

1 1 



PV

FORWARD RATE



EXAMPLE OF A FORWARD RATE

Let us assume that $1 paid in TWO yearS at a spot rate of 7% has a

8573 ) $.

07 . 1 (

) 1

(

1

2 ,

1 



  f PV

% 01 .

2 9

,

1 

f

FORWARD RATE

f

_1,2

is the forward rate from year 1 to year

2

FORWARD RATE



To show the link between the spot rate in year 1 and the spot rate in year 2

and the forward rate from year 1 to year 2

2 2 1

2 , 1

) 1

(

1

$ )

1 ( 1

1

$

s s

f

 





FORWARD RATE

such that

or

) 1

(

) 1

1 (

2 2 1

,

1

s

f s



 



2 2 2

, 1

1

)( 1 ) ( 1 )

1

(  s  f   s

FORWARD RATE



More generally for the link between years t-1 and t:



or

1 1

, 2

,

1

( 1 )

) 1

) ( 1

(

_



 



_t

t

t t

s f s

t t t

t t

t

f s

s ) ( 1 ) ( 1 )

1

( 

_1 ^1



_1,

 

FORWARD RATES AND DISCOUNT FACTORS



ASSUMPTION:

•

given a set of spot rates, it is possible to determine a market discount function

•

^equation

) 1

( )

1 (

1

, 1 1

1 t t

t t

t s f

d

 

 

 

YIELD CURVES



DEFINITION: a graph that shows the

YTM for Treasury securities of various

terms (maturities) on a particular date

YIELD CURVES



TREASURY SECURITIES PRICES

•

priced in accord with the existing set of spot rates and

•

associated discount factors

YIELD CURVES



SPOT RATES FOR TREASURIES

•

One year is less that two year;

•

Two year is less than three-year, etc.

YIELD CURVES



YIELD CURVES AND TERM STRUCTURE

•

yield curve provides an estimate of



the current TERM STRUCTURE OF INTEREST RATES



yields change daily as YTM change

TERM STRUCTURE THEORIES



THE FOUR THEORIES

1. THE UNBIASED EXPECTATION THEORY 2. THE LIQUIDITY PREFERENCE THEORY 3. MARKET SEGMENTATION THEORY

4. PREFERRED HABITAT THEORY

TERM STRUCTURE THEORIES



THEORY 1: UNBIASED EXPECTATIONS

•

Basic Theory: the forward rate represents the average opinion of the expected future spot rate for the period in question

•

in other words, the forward rate is an

unbiased estimate of the future spot rate.

TERM STRUCTURE THEORY:

Unbiased Expectations

 THEORY 1: UNBIASED EXPECTATIONS

•

A Set of Rising Spot Rates



the market believes spot rates will rise in the future

– the expected future spot rate equals the forward rate

– in equilibrium

es

_1,2

= f

_1,2

where

es

_1,2

=

the expected future spot

f

_1,2

=

the forward rate

TERM STRUCTURE THEORY:

Unbiased Expectations



THE THEORY STATES:

•

The longer the term, the higher the spot rate, and

•

If investors expect higher rates ,



then the yield curve is upward sloping



and vice-versa

TERM STRUCTURE THEORY:

Unbiased Expectations



CHANGING SPOT RATES AND INFLATION

•

Why do investors expect rates to rise or fall in the future?



spot rates = nominal rates

– because we know that the nominal rate is the real rate plus the expected rate of inflation

TERM STRUCTURE THEORY:

Unbiased Expectations



CHANGING SPOT RATES AND INFLATION

•

Why do investors expect rates to rise or fall in the future?



if either the spot or the nominal rate is

expected to change in the future, the spot rate will change

TERM STRUCTURE THEORY:

Unbiased Expectations



CHANGING SPOT RATES AND INFLATION

•

Why do investors expect rates to rise or fall in the future?



the future spot rate is greater than current rates due to expectations of inflation

TERM STRUCTURE THEORY:

Unbiased Expectations

•

Current conditions influence the shape of the yield curve, such that



if deflation expected, the term structure and yield curve are downward sloping



if inflation expected, the term structure and yield curve are upward sloping

TERM STRUCTURE THEORY:

Unbiased Expectations



PROBLEMS WITH THIS THEORY:

•

upward-sloping yield curves occur more frequently

•

the majority of the time, investors expect spot rates to rise

•

not realistic position

TERM STRUCTURE THEORY:

Liquidity Preference



BASIC NOTION OF THE THEORY

•

investors primarily interested in purchasing short-term securities to reduce interest

rate risk

TERM STRUCTURE THEORY:

Liquidity Preference



BASIC NOTION OF THE THEORY

•

^{Price Risk}



maturity strategy is more risky than a rollover strategy



to convince investors to buy longer-term

securities, borrowers must pay a risk premium to the investor

TERM STRUCTURE THEORY:

Liquidity Preference



BASIC NOTION OF THE THEORY

•

Liquidity Premium



DEFINITION: the difference between the forward rate and the expected future rate

TERM STRUCTURE THEORY:

Liquidity Preference



BASIC NOTION OF THE THEORY

•

Liquidity Premium Equation

L = es

_1,2

-

f

_1,2

where

L

is the liquidity premium

TERM STRUCTURE THEORY:

Liquidity Preference



How does this theory explain the shape of the yield curve?

•

rollover strategy



at the end of 2 years $1 has an expected value of

$1 x (1 + s₁ ) (1 + es_1,2 )

TERM STRUCTURE THEORY:

Liquidity Preference



How does this theory explain the shape of the yield curve?

•

whereas a maturity strategy holds that

$1 x (1 + s₂ )²

•

which implies with a maturity strategy, you must have a higher rate of return

TERM STRUCTURE THEORY:

Liquidity Preference



How does this theory explain the shape of the yield curve?

•

Key Idea to the theory: The Inequality holds

$1(1+s

₁

)(1 +es

_1,2

)<$1(1 + s

₂

)

²

TERM STRUCTURE THEORY:

Liquidity Preference



SHAPES OF THE YIELD CURVE:

•

a downward-sloping curve



means the market believes interest rates are going to decline

TERM STRUCTURE THEORY:

Liquidity Preference



SHAPES OF THE YIELD CURVE:

•

a flat yield curve means the market expects interest rates to decline

TERM STRUCTURE THEORY:

Liquidity Preference



SHAPES OF THE YIELD CURVE:

•

an upward-sloping curve means rates are expected to increase

TERM STRUCTURE THEORY:

Market Segmentation



BASIC NOTION OF THE THEORY

•

various investors and borrowers are

restricted by law, preference or custom to certain securities

TERM STRUCTURE THEORY:

Liquidity Preference



WHAT EXPLAINS THE SHAPE OF THE YIELD CURVE?

•

Upward-sloping curves mean that supply and demand intersect for short-term is at a lower rate than longer-term funds

•

cause: relatively greater demand for longer-term funds or a relative greater supply of shorter-term funds

TERM STRUCTURE THEORY:

Preferred Habitat



BASIC NOTION OF THE THEORY:

•

Investors and borrowers have segments of the market in which they prefer to operate

TERM STRUCTURE THEORY:

Preferred Habitat

•

When significant differences in yields exist between market segments, investors are willing to leave their desired maturity

segment

TERM STRUCTURE THEORY:

Preferred Habitat

•

Yield differences determined by the supply and demand conditions within the segment

TERM STRUCTURE THEORY:

Preferred Habitat

•

This theory reflects both



expectations of future spot rates



expectations of a liquidity premium

END OF CHAPTER 5