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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
Bond C
are pure discount types;
mature in one year
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 rA
(1 + rA) x $934.58 = $1000
rA = 7%
YIELD TO MATURITY

CALCULATING YTM:
• BOND B
• Solving for rB
(1 + rB) x $857.34 = $1000
rB = 8%
YIELD TO MATURITY

CALCULATING YTM:
• BOND C
• Solving for rC
(1 + rC)+{[(1+ rC)x$946.93]-$50
= $1000
rC = 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:
Mt
Pt 
1  s t 
where Pt = the current market price of a
pure discount bond maturing
in t years;
Mt = the maturity value
st = the spot rate
DISCOUNT FACTORS

EQUATION:
Let dt = the discount factor
d
t
1

1  s t

DISCOUNT FACTORS

EVALUATING A RISK FREE BOND:
• EQUATION
PV 
n
dc
t 1
t
t
where ct = the promised cash payments
n = the number of payments
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

1
PV 
 $. 9346
1 .07
FORWARD RATE
EXAMPLE OF A FORWARD RATE
Let us assume that $1 paid in TWO yearS
at a spot rate of 7% has a

PV 
1
(1  f1, 2 )
(1  .07 )
f1, 2  9.01%
 $. 8573
FORWARD RATE
f1,2 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
$1
1  f1,2
$1

(1  s 1 )
(1  s 2 ) 2
FORWARD RATE
such that
1  f1,2
(1  s 1 )

(1  s 2 )
or
(1  s1 )(1  f1, 2 )  (1  s2 )
2
FORWARD RATE

More generally for the link between
years t-1 and t:
(1  s t )
(1  f 1 , 2 ) 
t 1
(1  s t , 1 )
t

or
t 1
(1  st 1 ) (1  ft 1,t )  (1  st )
t
FORWARD RATES AND
DISCOUNT FACTORS

ASSUMPTION:
• given a set of spot rates, it is possible to
determine a market discount function
• equation
dt 
1
(1  s t 1 ) t 1 (1  f t 1, t )
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.
2.
3.
4.
THE UNBIASED EXPECTATION THEORY
THE LIQUIDITY PREFERENCE THEORY
MARKET SEGMENTATION THEORY
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
es1,2 = f1,2
where es1,2 = the expected future spot
f1,2 = 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 = es1,2 - f1,2
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 + es1,2 )
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:
holds
The Inequality
$1(1+s1)(1 +es1,2)<$1(1 + s2)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