CHAPTER FIVE

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CHAPTER FIVE

• THE VALUATION OF RISKLESS SECURITIES

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INTEREST RATES

• NOMINAL V. REAL INTEREST RATES
• Nominal interest rates
• represent the rate at which consumer can trade
  present money for future money

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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).

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YIELD TO MATURITY

• CALCULATING YIELD TO MATURITY AN EXAMPLE
• Suppose 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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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 ?

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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)x946.93-50 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

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DISCOUNT FACTORS

• EQUATION
• Let dt the discount factor

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DISCOUNT FACTORS

• EVALUATING A RISK FREE BOND
• EQUATION
• where ct the promised cash payments
• n the number of payments

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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

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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

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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

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FORWARD RATE

• such that
• or

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FORWARD RATE

• More generally for the link between years t-1 and
  t
• or

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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

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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 that 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 change

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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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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?
• the future spot rate is greater than current
  rates due to expectations of inflation

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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(1s1)(1 es1,2)lt1(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

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• END OF CHAPTER 5