chapter five 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 $934.58Bond B $857.34Bond 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

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

tt

t s

MP

1

DISCOUNT FACTORS

EQUATION:Let dt = the discount factor

tt sd

1

1

DISCOUNT FACTORS

EVALUATING A RISK FREE BOND:•EQUATION

where ct = the promised cash payments

n = the number of payments

n

tttcdPV

1

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 RATELet us assume that $1 paid in one

year at a spot rate of 7% has

9346$.07.1

1PV

FORWARD RATE

EXAMPLE OF A FORWARD RATELet 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

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

221

2,1

)1(

1$

)1(

11$

ss

f

FORWARD RATE

such that

or

)1(

)1(1

2

12,1 s

sf

222,11 )1()1)(1( sfs

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

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

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 THEORY4. 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


THE THEORY STATES:•The longer the term, the higher the

spot rate, and

•If investors expect higher rates ,then the yield curve is upward slopingand vice-versa


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



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



or fall in the future?the future spot rate is greater than

current rates due to expectations of inflation


•Current conditions influence the shape of the yield curve, such thatif deflation expected, the term structure

and yield curve are downward slopingif inflation expected, the term structure

and yield curve are upward sloping


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


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


BASIC NOTION OF THE THEORY•Liquidity Premium

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


BASIC NOTION OF THE THEORY•Liquidity Premium Equation

L = es1,2 - f1,2

where L is the liquidity premium


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 )


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


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


SHAPES OF THE YIELD CURVE:•a downward-sloping curve

means the market believes interest rates are going to decline


SHAPES OF THE YIELD CURVE:•a flat yield curve means the market

expects interest rates to decline


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


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


•When significant differences in yields exist between market segments, investors are willing to leave their desired maturity segment


•Yield differences determined by the supply and demand conditions within the segment


•This theory reflects bothexpectations of future spot ratesexpectations of a liquidity premium

END OF CHAPTER 5

chapter five the valuation of riskless securities

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