For these notes I will forego the algebraic adjustment accounting for semi-annual
compounding. The final result will be multiplied by 2.
zi = ½ x spot rate
ifj = ½ x forward rate
From the definition of a forward rate:
(1  z 2 ) 2  (1  z1 )  (11f1 )
and
(1  z3 ) 3  (1  z 2 ) 2  (1 2 f1 )
Substituting from the first expression into the second
(1  z3 ) 3  (1  z1 )  (11f1 )  (1 2 f1 )
Solving for the rate z3:


z3  2  3 (1  z1)  (11f1)  (1 2 f1) 1
Another way to write the same thing

z3  2  (1  z1)  (11f1)  (1 2 f1)
1
3

1
Long rates (z3) are a geometric average of the short rate (z1) and forward rates (1f1 , 2f1).
Lending Strategies (a similar discussion concerning borrowing strategies is also
possible):
Consider the following setting:
Semi
1
Spot
z1
2
z2
Rate forward period
3%
1f1 = 5%
4%
In this circumstance there are two possible lending strategies for a two period loan.
1
Strategy #1: Buy two period zero coupon bond
Total return form Strategy #1 is 4%. This equivalent to earning z1 = 3% over the first
semi-annual period and 1f1 = 5% during the forward period. There is no interest rate risk
in Strategy #1.
Strategy #2 Sequential lending
i.e. purchase one-period zero coupon bond planning to roll over proceeds into a second
one-period bond at the end of the first period. The rate to be earned in the second period
is a random variable. Label this rate 1 ~
z1 . Interest rate risk is present in Strategy #2. The
realized rate during the second period could be less than 5%.
Suppose the decision maker expects that the one-period rate during the forward period
will be 6%, i.e. E0 (1~
z1 )  6% .
Which Strategy will the decision maker choose?
How will the decision maker’s choice be reflected in the price of zero coupon bonds?
If E(z)s equal forward rates then total return is equal for all investment strategies
spanning an investment horizon.
If E(z)s are not equal forward rates then total returns are not equal for all investment
strategies spanning an investment horizon.
Conditioned on E(z)s and decision maker’s risk tolerance, a decision maker may select a
strategy that produces an interest rate risk exposure if the expected return from the
investment strategy or expected cost from borrowing strategy is sufficient to compensate
for the additional interest rate risk exposure.
Term structure theories attempt to answer the question
“What determines the “shape” of the term structure?”
Ultimately the term structure is determined by the factors that affect the supply and
demand for fixed income securities of different maturities. The factors determine the
relative price of securities of different maturities and hence the relative yields.
Term structure theories are important in that they help market participants anticipate how
economic phenomena will affect the level and slope of the term structure.
Transacting in securities with time to maturity different than the participant’s investment
(borrowing) horizon exposes the market participant to either price or reinvestment risk.
2
The term structure theories discussed in the text (pages 111-116) are predicated on
different assumptions concerning market participant’s interest rate risk tolerance.
Pure expectations theory;
 Assumption – market participants are indifferent to the risk of trading in securities
whose maturity is different than the participant’s investment (borrowing) horizon.
 Outcome – Zero coupon bonds prices (spot rates) will adjust until forward rates are
equal to participant’s expectations of spot rates in future periods.
 Implication 1 – The shape of the term structure is determined by market participant’s
expectations of short rates in future periods.
jfk
= E0(jzk)
Implication 2 - The term structure will change level and/or shape when market
participant’s expectations of short rates change.
Implication 3 – Long-term spot rates are geometric averages of the shortest term spot
rate and market participants expectations of spot rates.


1
z3  2  (1  z1)  (1  E0 (1~
z2 ))  (1  E0 ( 2 ~
z3 ))  3  1
The Liquidity Theory, Preferred Habitat Theory and the Segmented Markets theory differ
in the assumed response of market participants to the risk of transacting in securities with
time to maturity different than the participant’s investment (borrowing) horizon.
These theories posit that market participants are averse to the risk of transacting in
securities with time to maturity that does not match their investment (borrowing) horizon.
The Liquidity Theory states that to get lenders to lend long term borrowers will have to
offer a higher expected return to overcome the lender’s loss of liquidity. Hence the price
of zero-coupon bonds reflects not only expected short rates as in the expectations
hypothesis, but also reflects the necessary premium required to entice lenders.
Liquidity theory;
 Assumption – market participants are averse to the risk of trading in securities whose
maturity is different than the participant’s investment (borrowing) horizon.
 Outcome – Zero coupon bond prices (spot rates) will adjust until forward rates reflect
participant’s expectations of short rates in future periods and the necessary liquidity
premiums.
 Implication – The shape of the term structure is determined by market participant’s
expectations of short rates in future periods and the necessary liquidity premiums.
jfk
= E0(jzk) +jk
3
The term structure will change level and/or shape when market participant’s
expectations of short rates change and/or when liquidity premiums change.
Preferred Habitat theory;
 Assumption – market participants are averse to the risk of trading in securities whose
maturity is different than the participant’s investment (borrowing) horizon.
 Outcome – Zero coupon bond prices (spot rates) will adjust until forward rates reflect
participant’s expectations of short rates in future periods and the necessary premiums
to eliminate excess supply/excess demand for each segment of the term structure.
 Implication – The shape of the term structure is determined by market participant’s
expectations of short rates and the necessary premiums. Premiums can be either
positive or negative and need not be uniformly increasing.
jfk
= E0(jzk) +jk
The term structure will change level and/or shape when market participant’s
expectations of short rates change and/or when premiums change.
Segmented Markets theory;
 Assumption – market participants are extremely averse to the risk of trading in
securities whose maturity is different than the participant’s investment (borrowing)
horizon. In fact market participants will only transact in securities that have time to
maturity that matches their investment (borrowing) horizon.
 Outcome – Zero coupon bond prices (spot rates) reflect only the supply and demand
for securities. The borrowers who have a horizon equal to the time to maturity
determine the supply for a given time to maturity. The lenders who have a horizon
equal to the time to maturity determine the demand for a given time to maturity.
Implication – The shape of the term structure is independent of expectations of future
interest rates and risk premiums. The term structure will change level and/or shape when
factors affecting either supply or demand in a maturity segment change.Yield Spreads:
4
RPj = (ytmj – ytmUS)
j = CORP, MUN, MBS, ABS, etc.
Yield spreads can be expressed as simple differences, proportional differences, or simple
ratios.
Credit Spreads reported by Reuter’s
http://www.bondsonline.com/asp/corp/spreadfin.html
Moody's/S&P
Aaa/AAA
Aa1/AA+
Aa2/AA
Aa3/AAA1/A+
A2/A
A3/ABaa1/BBB+
Baa2/BBB
Baa3/BBBBa1/BB+
Ba2/BB
Ba3/BBB1/B+
B2/B
B3/BCaa/CCC
BANK
14
23
25
26
44
47
51
62
65
72
185
195
205
280
290
300
415
One-year
FIN IND
13
5
41
10
43
15
45
20
53
30
68
40
73
50
87
60
92
68
97
78
165
220
175
250
185
275
290
430
300
460
310
660
410 1500
Five-years
UTIL BANK FIN IND
3
34
33 20
6
45
60 29
11
50
67 35
16
54
77 44
19
61
93 55
22
63
94 63
31
67
99 75
45
89 121 90
58
94 133 108
64
99 141 119
315
215 225 205
325
225 235 315
305
235 245 315
385
330 345 350
515
340 355 435
590
350 375 685
680
560 495 1400
Ten-years
UTIL BANK FIN IND
13
62
73 30
29
73
93 40
31
76
96 50
36
79
99 55
49
89 119 63
55
91 121 71
61
94 128 84
83 140 159 101
91 148 168 117
101 155 183 130
302 255 270 190
305 265 275 265
325 275 285 280
460 410 405 330
510 420 415 395
610 430 425 720
810 580 600 1375
UTIL
21
35
46
60
75
79
79
95
120
132
275
210
265
375
445
545
620
Tax treatement of interest
As a general rule interest paid by one government issuer (Federal, State, Municipal) is not
taxable by another.
For instance interest income earned from US Treasury securities is not taxable for the
purpose of State income taxes.
Interest earned from municipal securities is not taxable for the purpose of both State and
Federal income taxes.
after tax yield = ytm*(1-marginal tax rate)

equivalent taxable yield = tax-exempt yield ÷ (1 – marginal tax rate)
5
6