Fundamentals Of Bond Valuation

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FUNDAMENTALS OF BOND VALUATION

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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

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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-MATURITY 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)x$946.93]-$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

tt

t s

MP

1

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DISCOUNT FACTORS

• EQUATION:

Let dt = the discount factor

tt sd

1

1

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DISCOUNT FACTORS

• EVALUATING A RISK FREE BOND:– EQUATION

where ct = the promised cash payments

n = the number of payments

n

tttcdPV

1

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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

9346$.07.1

1PV

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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

8573$.)07.1(

)1(1

2,1

f

PV

%01.92,1 f

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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

221

2,1

)1(

1$

)1(

11$

ss

f

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FORWARD RATE

such that

or

)1(

)1(1

2

12,1 s

sf

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

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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

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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

)1()1(

1

,11

1 ttt

tt fsd

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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 than 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 changes

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TERM STRUCTURE THEORIES

• THE FOUR THEORIES1.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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• 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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• 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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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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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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– 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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• 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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• 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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• BASIC NOTION OF THE THEORY– Liquidity Premium

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

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• BASIC NOTION OF THE THEORY– Liquidity Premium Equation

L = es1,2 - f1,2

where L is the liquidity premium

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• 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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• 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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• 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

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

• means the market believes interest rates are going to decline

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

interest rates to decline

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

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

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– This theory reflects both• expectations of future spot rates

• expectations of a liquidity premium

Fundamentals Of Bond Valuation

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