Transcript Document

The Term Structure
of Interest Rates




The relationship between yield to maturity
and maturity.
Yield curve - a graph of the yields on bonds
relative to the number of years to maturity
Information on expected future short term
rates can be implied from yield curve.
Three major theories are proposed to explain
the observed yield curve.


Measure of rate of return that accounts for
both current income and the price increase
over the life
YTM is the discount rate that makes the
present value of a bond’s payments equal to
its price
◦ Proxy for average return
Solve the bond formula for r
ParValue
T
C
t
PB  

T
t
(1 r )
t 1 (1 r )
T

Yields on different maturity bonds are
not all equal
◦ Need to consider each bond cash flow as a standalone zero-coupon bond when valuing coupon
bonds
◦ Example: 1-year maturity T-bond paying
semiannual coupons can be split into a 6-month
maturity zero and a 12-month zero. If each cash
flow can be sold off as a separate security, the
value of the whole bond should be the same as the
value of its cash flows bought piece by piece.
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Bond stripping and bond reconstitution offer
opportunities for arbitrage
Law of one price, identical cash flow bundles
must sell for identical prices
Value each stripped cash flow
◦ discount by using the yield appropriate to its
particular maturity

Treat each of the bond’s payments as a standalone zero-coupon security
(bond stripping : portfolio of three zeros)
P

100

100
1  5% 1  6%
1
2

1100
1  7%
3
 1082.17
Pure yield curve
◦ Relationship between YTM and time to maturity for zerocoupon bonds

On-the-run yield curve
◦ Plot of yield as a function of maturity for recently issued
coupon bonds selling at or near par value
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If interest rates are certain
Considering two strategies
◦ Buying the 2-year zero, YTM=6%, hold until maturity.
Price=890
◦ Invest 890 in a 1-year zero, YTM=5%. Reinvest the
proceeds in another 1-year bond
◦ The proceeds after 2 years to either strategy must
be equal
890(1  6%)  890(1  0.5)  (1  r2 )
2
r2  7.01%

Spot rate
◦ The rate that prevails today for a time period
corresponding to the zero’s maturity
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Short rate
◦ For a given time interval (e.g. 1 year) refers to the
interest rate for that interval available at different
points in time
2-year spot rate is an average (geometric) of
today’s short rate and next year’s short rate
(1  y2 ) 2  (1  r1 )  (1  r2 )
1  y2   (1  r1 )  (1  r2 ) 
1
2

An upward sloping yield curve is evidence
that short-term rates are going to be
higher next year
2
(1  y2 )  (1  r1 )  (1  r2 )
1  y2   (1  r1 )  (1  r2 ) 

1
2
When next year’s short rate (r2=7.01%) is
greater than this year’s short rate, the
average of the two rates is higher than
today’s rate
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When interest rate with certainty, all bonds
must offer identical rates of return over any
holding period
Calculate HPR for 1-year maturity zerocoupon bond (YTM=5%)
The first 1-year HPR for 2-year maturity
zero-coupon bond (YTM=6%)

For 1-year maturity bond
◦ Rate of return=(1000-952.38)/952.38=5%
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For 2-year maturity bond
◦ Price of today=890
◦ One year later, when next year’s interest
rate=7.01%, sell it for 1000/1.0701=934.49
◦ Rate of return=(934.49-890)/890=5%
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No access to short-term interest rate
quotations for coming years---infer future
short rates from yield curve of zeros
Two alternatives get same final payoff
◦ 3-year zero
100*(1  1.0966)3  131.87
◦ 2-year zero, reinvest in 1-year bond
100*(11.08995) *1  r3 
2
100*(1  y3 )  100* 1  y2  * 1  f3 
3
2
(1  yn )
(1  f n ) 
n 1
(1  yn 1 )
n

fn = one-year forward rate for period n

yn = yield for a security with a maturity of n
n 1
(1  yn )  (1  yn1 ) (1  f n )
n


future short rates are uncertain
Forward interest rate
◦ defined as the break-even interest rate that equates
the return on an n-period zero-coupon bond to
that of an (n-1)-period zero-coupon bond rolled
over into a 1-year bond in year n
◦ Calculated from today’s data, interest rate that
actually will prevail in the future need not equal the
forward rate.
4 yr = 8.00%
3yr = 7.00%
(1.08)4 = (1.07)3 (1+fn)
(1.3605) / (1.2250) = (1+fn)
fn = .1106 or 11.06%
fn = ?
12.3
Interest Rate Uncertainty and
Forward Rates
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What can we say when future interest rates are not
known today
Suppose that today’s rate is 5% and the expected
short rate for the following year is E(r2) = 6% then:
(1  y2 ) 2  (1  r1 )  [1  E (r2 )]  1.05 1.06
1000
p
 898.47
1.05 1.06
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The rate of return on the 2-year bond is risky for if
next year’s interest rate turns out to be above
expectations, the price will lower and vice versa
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Short-term-horizon investors
◦ If invest only for 1 year
certain return=(1000-952.38)/952.38=5%
◦ If invest for 2-year zero, if expect the 1-year rate
be 6% at the end of the first year
 the price will be 1000/1.06=943.4
 the first-year’s expected rate of return also is
5%=(943.4-898.47)/898.47
 but the 2nd year’s rate is risky, 943.4 is not certain
 If >6%, bond price<943.4
 If <6%, bond price>943.4
 2-year bond must offer an expected rate of return
greater than riskless 5% return , sell at price lower than
898.47
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If the investors will hold the bond when it falls
to 881.83
◦ Expected holding period return for the first year
=(943.4-881.83)/881.83=7%
◦ risk premium=7%-5%=2%
◦ Forward rate:
1000
881.83 
1  5%  (1  f 2 )
f 2  8%  E  r 2   6%

Liquidity premium compensates short-term
investors for the uncertainty about the price at
which they will be able to sell their long-term
bonds
liquidity premium  fn  E  rn 
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Short-term Investors require a risk
premium to hold a longer-term bond
This liquidity premium compensates
short-term investors for the uncertainty
about future prices
If most individuals are short-term
investors, bonds must have prices that
make f2 greater than E(r2)

Wish to invest a full 2-year period
◦ Purchase 2-year zeros at 841.75,
guaranteed YTM=8.995%
◦ If roll over two 1-year investments, an
investment of 841.75 grow in 2 years to
be
841.75*1.08*(1  r )
2
◦ The investor will require
1.08* 1  E  r2    (1.08995) 2  1.08*(1  f 2 )
E  r2   f 2
◦ Offered as a reward for bearing interest
rate risk

Wish to invest a full 2-year period
◦ Purchase 2-year zeros at 890, guaranteed
YTM=6%
◦ If roll over two 1-year investments, an
investment of 841.75 grow in 2 years to
be
890*1.05*(1  r2 )
◦ The investor will require
1.05* 1  E  r2    (1.06) 2  1.05*(1  f 2 )
E  r2   f 2
◦ Offered as a reward for bearing interest
rate risk
12.4
Theories of the Term Structure
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Expectations theories
Liquidity Preference theories
◦ Upward bias over expectations
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Market Segmentation
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Observed long-term rate is a function of
today’s short-term rate and expected future
short-term rates.
Forward rates that are calculated from the
yield on long-term securities are market
consensus expected future short-term rates.
E  r2   f2
◦ An upward-sloping yield curve if investors anticipate
increases in interest rates
◦ Upward slope means that the market is expecting
higher future short term rates
◦ Downward slope means that the market is expecting
lower future short term rates
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Short-term investors dominate the market
Forward rates contain a liquidity premium
and are not equal to expected future
short-term rates.
f2  E  r2 
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Investors will demand a premium for the
risk associated with long-term bonds.
The yield curve has an upward bias built
into the long-term rates because of the
risk premium.
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Short- and long-term bonds are traded
in distinct markets.
Trading in the distinct segments
determines the various rates.
Observed rates are not directly
influenced by expectations.

Expected One-Year Rates in Coming Years

Year

0 (today)
8%

1
10%

2
11%

3
11%

Interest Rate
8%
10%
11%
11%
1
PVn 
(1  r1 )(1  r2 )...(1  rn )

PVn = Present Value of $1 in n periods

r1 = One-year rate for period 1

r2 = One-year rate for period 2

rn = One-year rate for period n
Price of 1-year maturity bond
1000
PV1 
 925.93
(1  8%)
Price of 2-year maturity bond
1000
PV2 
 841.75
(1  8%)(1  10%)
Price of 3-year maturity bond
1000
PV3 
 758.33
(1  8%)(1  10%)(1  11%)
YTM is average rate that is applied to
discount all of the bond’s payments
1000
925.93 
(1  y1 )
1000
841.75 
(1  y2 )2
1000
758.33 
(1  y3 )3
Time to Maturity Price of Zero* Yield to Maturity
1
$925.93
8.00%
2
841.75
8.995
3
758.33
9.660
4
683.18
9.993
* $1,000 Par value zero
expected
Short rate
in each year
8%
10%
11%
y1=8%
YTM
for various
maturities
(Current spot rate)
y2=8.995%
y3=9.660%
y4=9.993%
11%

YTM, average of the interest rates in
each period (geometric)
1000
1000
 841.75 
(1  8%)(1  10%)
(1  y2 ) 2
(1  y2 )2  (1  8%)(1  10%)
1  y2  (1  r1 )(1  r2 )
1
2
1  y4  (1  r1 )(1  r2 )(1  r3 )(1  r4 )
1
4
12.4
Interpreting the Term Structure

Direct relationship between YTM and
forward rate
Under certainty

Uncertain


1  yn  (1  r1 )(1  r2 )
1  yn  (1  r1 )(1  f 2 )
(1  f n )
(1  rn )
1
n
The yield curve is upward sloping at any
maturity date, n, for which the forward rate
for the coming period is greater than the
yield at that maturity
1
n


If yield curve is rising,
Example
f n1 must exceed
◦ YTM on 3-year zero is 9%, YTM on 4-year zero
1  y4  (1  9%)3 (1  f4 ) 
◦ If
◦ If
f 4  9%, then
f 4  9%
, then
y4  9%  y3
y4  9%  y3
1
4
yn

Given an upward-sloping yield curve,
What account for the higher forward rate?
fn  E  rn   liquidity premium

Expectations of increases in E  rn  can
result in rising yield curve; converse is
not true .