Description
This is a presentation explaining bond valuation and term structure.
FUNDAMENTALS OF BOND VALUATION
1
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
2
Bond C coupon pays $50/year;
matures in two years
3
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?
4
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
5
YIELD TO MATURITY
• CALCULATING YTM:
– BOND A – Solving for rA
(1 + rA) x $934.58 = $1000 rA = 7%
6
YIELD TO MATURITY
• CALCULATING YTM:
– BOND B – Solving for rB
(1 + rB) x $857.34 = $1000 rB = 8%
7
YIELD TO MATURITY
• CALCULATING YTM:
– BOND C – Solving for rC
(1 + rC)+{[(1+ rC)x$946.93]-$50 = $1000 rC = 7.975%
8
SPOT RATE
• DEFINITION: Measured at a given point in time as the YTM on a pure discount security
9
SPOT RATE
• SPOT RATE EQUATION:
Mt P = t (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
10
DISCOUNT FACTORS
• EQUATION: Let dt = the discount factor
1 dt = (1 +st )
11
DISCOUNT FACTORS
• EVALUATING A RISK FREE BOND:
– EQUATION
PV = ?d t ct
t =1
n
where ct = the promised cash payments n = the number of payments
12
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
13
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
14
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 ) = $. 8573
f1, 2 = 9.01%
15
FORWARD RATE
f1,2 is the forward rate from year 1 to year 2
16
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
17
FORWARD RATE
such that
1 + f1, 2
or
(1 +s1 ) = (1 +s2 )
2
(1 + s1 )(1 + f1, 2 ) = (1 + s2 )
18
FORWARD RATE
• More generally for the link between years t1 and t:
(1 +st ) (1 + f1, 2 ) = 1 (1 +st ,1 ) t ?
t
• or
(1 + st ?1 ) (1 + f t ?1,t ) = (1 + st )
t ?1
t
19
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 + f t ?1,t )
20
YIELD CURVES
• DEFINITION: a graph that shows the YTM for Treasury securities of various terms (maturities) on a particular date
21
YIELD CURVES
• TREASURY SECURITIES PRICES
– priced in accord with the existing set of spot rates and – associated discount factors
22
YIELD CURVES
• SPOT RATES FOR TREASURIES
– One year is less than two year; – Two year is less than three-year, etc.
23
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
24
TERM STRUCTURE THEORIES
• THE FOUR THEORIES 1. THE UNBIASED EXPECTATION THEORY 2. THE LIQUIDITY PREFERENCE THEORY 3. MARKET SEGMENTATION THEORY 4. PREFERRED HABITAT THEORY
25
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.
26
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
27
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
28
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
29
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
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
31
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
32
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
33
TERM STRUCTURE THEORY: Liquidity Preference
• BASIC NOTION OF THE THEORY
– investors primarily interested in purchasing short-term securities to reduce interest rate risk
34
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
35
TERM STRUCTURE THEORY: Liquidity Preference
• BASIC NOTION OF THE THEORY
– Liquidity Premium
• DEFINITION: the difference between the forward rate and the expected future rate
36
TERM STRUCTURE THEORY: Liquidity Preference
• BASIC NOTION OF THE THEORY
– Liquidity Premium Equation
L
= es1,2 - f1,2
where L is the liquidity premium
37
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 )
38
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
39
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
40
TERM STRUCTURE THEORY: Liquidity Preference
• SHAPES OF THE YIELD CURVE:
– a downward-sloping curve
• means the market believes interest rates are going to decline
41
TERM STRUCTURE THEORY: Liquidity Preference
• SHAPES OF THE YIELD CURVE:
– a flat yield curve means the market expects interest rates to decline
42
TERM STRUCTURE THEORY: Liquidity Preference
• SHAPES OF THE YIELD CURVE:
– an upward-sloping curve means rates are expected to increase
43
TERM STRUCTURE THEORY: Market Segmentation
• BASIC NOTION OF THE THEORY
– various investors and borrowers are restricted by law, preference or custom to certain securities
44
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 longerterm funds or a relative greater supply of shorter-term funds
45
TERM STRUCTURE THEORY: Preferred Habitat
• BASIC NOTION OF THE THEORY:
– Investors and borrowers have segments of the market in which they prefer to operate
46
TERM STRUCTURE THEORY: Preferred Habitat – When significant differences in yields exist between market segments, investors are willing to leave their desired maturity segment
47
TERM STRUCTURE THEORY: Preferred Habitat – Yield differences determined by the supply and demand conditions within the segment
48
TERM STRUCTURE THEORY: Preferred Habitat – This theory reflects both
• expectations of future spot rates • expectations of a liquidity premium
49
doc_432255909.ppt
This is a presentation explaining bond valuation and term structure.
FUNDAMENTALS OF BOND VALUATION
1
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
2
Bond C coupon pays $50/year;
matures in two years
3
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?
4
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
5
YIELD TO MATURITY
• CALCULATING YTM:
– BOND A – Solving for rA
(1 + rA) x $934.58 = $1000 rA = 7%
6
YIELD TO MATURITY
• CALCULATING YTM:
– BOND B – Solving for rB
(1 + rB) x $857.34 = $1000 rB = 8%
7
YIELD TO MATURITY
• CALCULATING YTM:
– BOND C – Solving for rC
(1 + rC)+{[(1+ rC)x$946.93]-$50 = $1000 rC = 7.975%
8
SPOT RATE
• DEFINITION: Measured at a given point in time as the YTM on a pure discount security
9
SPOT RATE
• SPOT RATE EQUATION:
Mt P = t (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
10
DISCOUNT FACTORS
• EQUATION: Let dt = the discount factor
1 dt = (1 +st )
11
DISCOUNT FACTORS
• EVALUATING A RISK FREE BOND:
– EQUATION
PV = ?d t ct
t =1
n
where ct = the promised cash payments n = the number of payments
12
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
13
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
14
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 ) = $. 8573
f1, 2 = 9.01%
15
FORWARD RATE
f1,2 is the forward rate from year 1 to year 2
16
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
17
FORWARD RATE
such that
1 + f1, 2
or
(1 +s1 ) = (1 +s2 )
2
(1 + s1 )(1 + f1, 2 ) = (1 + s2 )
18
FORWARD RATE
• More generally for the link between years t1 and t:
(1 +st ) (1 + f1, 2 ) = 1 (1 +st ,1 ) t ?
t
• or
(1 + st ?1 ) (1 + f t ?1,t ) = (1 + st )
t ?1
t
19
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 + f t ?1,t )
20
YIELD CURVES
• DEFINITION: a graph that shows the YTM for Treasury securities of various terms (maturities) on a particular date
21
YIELD CURVES
• TREASURY SECURITIES PRICES
– priced in accord with the existing set of spot rates and – associated discount factors
22
YIELD CURVES
• SPOT RATES FOR TREASURIES
– One year is less than two year; – Two year is less than three-year, etc.
23
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
24
TERM STRUCTURE THEORIES
• THE FOUR THEORIES 1. THE UNBIASED EXPECTATION THEORY 2. THE LIQUIDITY PREFERENCE THEORY 3. MARKET SEGMENTATION THEORY 4. PREFERRED HABITAT THEORY
25
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.
26
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
27
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
28
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
29
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
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
31
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
32
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
33
TERM STRUCTURE THEORY: Liquidity Preference
• BASIC NOTION OF THE THEORY
– investors primarily interested in purchasing short-term securities to reduce interest rate risk
34
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
35
TERM STRUCTURE THEORY: Liquidity Preference
• BASIC NOTION OF THE THEORY
– Liquidity Premium
• DEFINITION: the difference between the forward rate and the expected future rate
36
TERM STRUCTURE THEORY: Liquidity Preference
• BASIC NOTION OF THE THEORY
– Liquidity Premium Equation
L
= es1,2 - f1,2
where L is the liquidity premium
37
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 )
38
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
39
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
40
TERM STRUCTURE THEORY: Liquidity Preference
• SHAPES OF THE YIELD CURVE:
– a downward-sloping curve
• means the market believes interest rates are going to decline
41
TERM STRUCTURE THEORY: Liquidity Preference
• SHAPES OF THE YIELD CURVE:
– a flat yield curve means the market expects interest rates to decline
42
TERM STRUCTURE THEORY: Liquidity Preference
• SHAPES OF THE YIELD CURVE:
– an upward-sloping curve means rates are expected to increase
43
TERM STRUCTURE THEORY: Market Segmentation
• BASIC NOTION OF THE THEORY
– various investors and borrowers are restricted by law, preference or custom to certain securities
44
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 longerterm funds or a relative greater supply of shorter-term funds
45
TERM STRUCTURE THEORY: Preferred Habitat
• BASIC NOTION OF THE THEORY:
– Investors and borrowers have segments of the market in which they prefer to operate
46
TERM STRUCTURE THEORY: Preferred Habitat – When significant differences in yields exist between market segments, investors are willing to leave their desired maturity segment
47
TERM STRUCTURE THEORY: Preferred Habitat – Yield differences determined by the supply and demand conditions within the segment
48
TERM STRUCTURE THEORY: Preferred Habitat – This theory reflects both
• expectations of future spot rates • expectations of a liquidity premium
49
doc_432255909.ppt