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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
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 slopingand 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 thatif deflation expected, the term structure
and yield curve are downward slopingif 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 bothexpectations of future spot ratesexpectations of a liquidity premium
END OF CHAPTER 5