THE VALUATION OF RISKLESS SECURITIES CHAPTER FIVE THE VALUATION OF RISKLESS SECURITIES

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

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).

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

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 ?

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 (1 + rA) x $934.58 = $1000 rA = 7% CALCULATING YTM: BOND A Solving for rA (1 + rA) x $934.58 = $1000 rA = 7%

YIELD TO MATURITY (1 + rB) x $857.34 = $1000 rB = 8% CALCULATING YTM: BOND B Solving for rB (1 + rB) x $857.34 = $1000 rB = 8%

YIELD TO MATURITY (1 + rC)+{[(1+ rC)x$946.93]-$50 = $1000 rC = 7.975% 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 that 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 change

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

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 Liquidity Premium DEFINITION: the difference between the forward rate and the expected future rate

TERM STRUCTURE THEORY: Liquidity Preference BASIC NOTION OF THE THEORY Liquidity Premium Equation 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

END OF CHAPTER 5