CHAPTER TWENTY

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CHAPTER TWENTY. FUNDAMENTALS OF BOND VALUATION. 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 . Bond C coupon pays \$50/year;. matures in two years.

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### CHAPTER TWENTY

FUNDAMENTALS OF BOND VALUATION

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

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-MATURITY 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:

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

DISCOUNT FACTORS
• EVALUATING A RISK FREE BOND:
• EQUATION

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

FORWARD RATE
• EXAMPLE OF A FORWARD RATE

Let us assume that \$1 paid in two years at a spot rate of 7% has a

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
FORWARD RATE

such that

or

FORWARD RATE
• More generally for the link between years t-1 and t:
• or
FORWARD RATES AND DISCOUNT FACTORS
• ASSUMPTION:
• given a set of spot rates, it is possible to determine a market discount function
• equation
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 than 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 changes
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

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?
• 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
• 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
• DEFINITION: the difference between the forward rate and the expected future rate
TERM STRUCTURE THEORY: Liquidity Preference
• BASIC NOTION OF THE THEORY

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: The Inequality holds

\$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