2.
YIELD TO MATURITY
• CALCULATING YIELD TO MATURITY
EXAMPLE
– Imagine three risk-free returns based on three
Treasury bonds:
Bond A,B are pure discount types;
mature in one year
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3.
Bond C coupon pays $50/year;
matures in two years
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4.
YIELD TO MATURITY
Bond Market Prices:
Bond A $934.58
Bond B $857.34
Bond C $946.93
WHAT IS THE YIELD-TO-MATURITY
OF THE THREE BONDS?
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5.
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
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6.
YIELD TO MATURITY
• CALCULATING YTM:
– BOND A
– Solving for rA
(1 + rA) x $934.58 = $1000
rA = 7%
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7.
YIELD TO MATURITY
• CALCULATING YTM:
– BOND B
– Solving for rB
(1 + rB) x $857.34 = $1000
rB = 8%
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8.
YIELD TO MATURITY
• CALCULATING YTM:
– BOND C
– Solving for rC
(1 + rC)+{[(1+ rC)x$946.93]-$50 =
$1000
rC = 7.975%
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9.
SPOT RATE
• DEFINITION: Measured at a given point
in time as the YTM on a pure discount
security
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10.
SPOT RATE
• SPOT RATE EQUATION:
Mt
Pt
1 st
where Pt = the current market price of a
pure discount bond maturing
in t years;
Mt = the maturity value
st = the spot rate
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11.
DISCOUNT FACTORS
• EQUATION:
Let dt = the discount factor
1
dt
1 st
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12.
DISCOUNT FACTORS
• EVALUATING A RISK FREE BOND:
– EQUATION
n
PV d t ct
t 1
where ct = the promised cash payments
n = the number of payments
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13.
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
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14.
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
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15.
FORWARD RATE
• EXAMPLE OF A FORWARD RATE
Let us assume that $1 paid in two years at a
spot rate of 7% has a
1
(1 f1, 2 )
PV $.8573
(1 .07)
f1, 2 9.01%
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16.
FORWARD RATE
f1,2 is the forward rate from year 1 to year 2
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17.
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 s1 ) (1 s2 ) 2
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FORWARD RATE
such that
(1 s1 )
1 f1, 2
(1 s2 )
or
2
(1 s1 )(1 f1, 2 ) (1 s2 )
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19.
FORWARD RATE
• More generally for the link between years t-
1 and t:
t
(1 st )
(1 f1, 2 ) t 1
(1 st ,1 )
• or
t 1 t
(1 st 1 ) (1 ft 1,t ) (1 st )
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FORWARD RATES AND
DISCOUNT FACTORS
• ASSUMPTION:
– given a set of spot rates, it is possible to
determine a market discount function
– equation 1
dt
(1 st 1 )t 1 (1 ft 1,t )
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21.
YIELD CURVES
• DEFINITION: a graph that shows the
YTM for Treasury securities of various
terms (maturities) on a particular date
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22.
YIELD CURVES
• TREASURY SECURITIES PRICES
– priced in accord with the existing set of spot
rates and
– associated discount factors
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23.
YIELD CURVES
• SPOT RATES FOR TREASURIES
– One year is less than two year;
– Two year is less than three-year, etc.
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24.
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 changes
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25.
TERM STRUCTURE THEORIES
• THE FOUR THEORIES
1.THE UNBIASED EXPECTATION THEORY
2. THE LIQUIDITY PREFERENCE THEORY
3. MARKET SEGMENTATION THEORY
4. PREFERRED HABITAT THEORY
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26.
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.
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27.
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
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28.
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
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29.
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
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30.
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
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31.
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
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32.
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
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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
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34.
TERM STRUCTURE THEORY: Liquidity
Preference
• BASIC NOTION OF THE THEORY
– investors primarily interested in purchasing
short-term securities to reduce interest rate risk
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35.
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
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36.
TERM STRUCTURE THEORY: Liquidity
Preference
• BASIC NOTION OF THE THEORY
– Liquidity Premium
• DEFINITION: the difference between the forward
rate and the expected future rate
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37.
TERM STRUCTURE THEORY: Liquidity
Preference
• BASIC NOTION OF THE THEORY
– Liquidity Premium Equation
L = es1,2 - f1,2
where L is the liquidity premium
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38.
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 )
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39.
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
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40.
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+s1)(1 +es1,2)<$1(1 + s2)2
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TERM STRUCTURE THEORY: Liquidity
Preference
• SHAPES OF THE YIELD CURVE:
– a downward-sloping curve
• means the market believes interest rates are going to
decline
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42.
TERM STRUCTURE THEORY: Liquidity
Preference
• SHAPES OF THE YIELD CURVE:
– a flat yield curve means the market expects
interest rates to decline
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43.
TERM STRUCTURE THEORY: Liquidity
Preference
• SHAPES OF THE YIELD CURVE:
– an upward-sloping curve means rates are
expected to increase
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44.
TERM STRUCTURE THEORY: Market
Segmentation
• BASIC NOTION OF THE THEORY
– various investors and borrowers are restricted
by law, preference or custom to certain
securities
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45.
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
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46.
TERM STRUCTURE THEORY: Preferred
Habitat
• BASIC NOTION OF THE THEORY:
– Investors and borrowers have segments of the
market in which they prefer to operate
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47.
TERM STRUCTURE THEORY: Preferred
Habitat
– When significant differences in yields exist
between market segments, investors are willing
to leave their desired maturity segment
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48.
TERM STRUCTURE THEORY: Preferred
Habitat
– Yield differences determined by the supply and
demand conditions within the segment
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49.
TERM STRUCTURE THEORY: Preferred
Habitat
– This theory reflects both
• expectations of future spot rates
• expectations of a liquidity premium
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