fundamentals of bond valuation
TRANSCRIPT
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FUNDAMENTALS OF BOND VALUATION
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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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Bond C coupon pays $50/year;
matures in two years
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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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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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YIELD TO MATURITY
• CALCULATING YTM:– BOND A
– Solving for rA
(1 + rA) x $934.58 = $1000
rA = 7%
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YIELD TO MATURITY
• CALCULATING YTM:– BOND B– Solving for rB
(1 + rB) x $857.34 = $1000
rB = 8%
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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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SPOT RATE
• DEFINITION: Measured at a given point in time as the YTM on a pure discount security
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SPOT RATE
• SPOT RATE EQUATION:
where Pt = the current market price of a
pure discount bond maturing in t years;
Mt = the maturity value
st = the spot rate
tt
t s
MP
1
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DISCOUNT FACTORS
• EQUATION:
Let dt = the discount factor
tt sd
1
1
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DISCOUNT FACTORS
• EVALUATING A RISK FREE BOND:– EQUATION
where ct = the promised cash payments
n = the number of payments
n
tttcdPV
1
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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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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
1PV
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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
%01.92,1 f
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FORWARD RATE
f1,2 is the forward rate from year 1 to year 2
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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
221
2,1
)1(
1$
)1(
11$
ss
f
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FORWARD RATE
such that
or
)1(
)1(1
2
12,1 s
sf
222,11 )1()1)(1( sfs
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FORWARD RATE
• More generally for the link between years t-1 and t:
• or
11,
2,1 )1(
)1()1(
tt
tt
s
sf
tttt
tt sfs )1()1()1( ,1
11
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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()1(
1
,11
1 ttt
tt fsd
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YIELD CURVES
• DEFINITION: a graph that shows the YTM for Treasury securities of various terms (maturities) on a particular date
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YIELD CURVES
• TREASURY SECURITIES PRICES– priced in accord with the existing set of spot
rates and– associated discount factors
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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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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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TERM STRUCTURE THEORIES
• THE FOUR THEORIES1.THE UNBIASED EXPECTATION THEORY
2. THE LIQUIDITY PREFERENCE THEORY
3. MARKET SEGMENTATION THEORY
4. PREFERRED HABITAT THEORY
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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TERM STRUCTURE THEORY: Liquidity Preference
• SHAPES OF THE YIELD CURVE:– an upward-sloping curve means rates are
expected to increase
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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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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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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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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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TERM STRUCTURE THEORY: Preferred Habitat
– Yield differences determined by the supply and demand conditions within the segment
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TERM STRUCTURE THEORY: Preferred Habitat
– This theory reflects both• expectations of future spot rates
• expectations of a liquidity premium