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Lecture 9 (Chapters 15 & 16)
Term Structure of Interest Rates
Yield curve
Term structure refers to yields at different
maturities (terms)
Observe bond prices (thus yields) at
different maturities
Construct the relationship between YTM
and time to maturity  the yield curve
Extract information on future short-term
interest rates from the yield curve
Forward rate and future interest rate
Term structure theories
Measure interest rate risk – Duration
Manage interest rate risk - immunization
1
3
The Pure Yield Curve
Observed prices and yields of zeros:
Yield Curve and
Term Structure of Interest
Rate
Maturity
Price
1
2
3
4
$980.39 $942.60 $899.33 $854.80
YTM
2%
3%
3.6%
4%
YTM
4%
3%
2%
1
2
2
3
4
Years to
Maturity
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Yield Curve and Credit Risk
YTM
Inverted Yield Curve
Term spread < 0.
Expectations of weaker
economic growth, lower
inflation and interest
rates in the long run.
Not very often.
A strong predictor of
economic recession.
CCC
B
BB
BBB
A
AA
AAA
Maturity
Everything else equal, greater credit risk requires
greater YTM.
YTM
Inverted
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Normal/Rising Yield Curve
Term spread: (YTM of
10-yr T-bond) – (YTM of
90-day T-bill) > 0
Longer maturity, greater
risk, greater required
yield.
Expectations of strong
economic growth, higher
inflation and interest
rates in the long run.
Maturity
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Term Spread
Inverted yield curve, i.e. negative term
spread, is associated with recession.
YTM
Normal/Rising
Maturity
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Humped or Flat Yield Curves
Humped: rising then
declining YTM.
Flat: little difference in
YTM for short- and longterm bonds.
Expectation of slowing
down of the economy.
Not very often.
Transition between rising
and inverted yield curves.
Yield Curve and Coupon Bonds
Coupon bonds with the same maturity (term)
may have different prices and yields:
YTM
Price of a 3-yr 4% coupon bond = $1012.23
Humped
Maturity
YTM
YTM(3-yr 4% coupon) = 3.563%
Price of a 3-yr 12% coupon bond = $1238.012
YTM(3-yr 12% coupon) = 3.504%
Flat
Maturity
Can we construct an average yield curve from
coupon bonds? Yes
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Determinants of the Yield Curve
11
Yield Curve and Coupon Bonds
YTM reflects expectations about the future
Let dt=1/(1+yt)t, t = 1, 2, 3.
YTM vs realized yield
Short-term yields are heavily influenced by
expectations on monetary policy.
Long-term yields are affected by expectations
on economic growth, inflation, demand for
and supply of long (maturity) bonds.
The shape of the yield curve reflects the
aggregate market expectation on future
economic conditions and interest rates.
10
Bond price = d1CF1+d2CF2+d3CF3
Collect the prices of many 3-yr coupon
bonds and run the following regression:
Pi = d1CF1,i+d2CF2,i+d3CF3,i+ei, i=1,…,N
yt is calculated from the estimated dt, t=
1,2,3.
Not required for exam
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Yield Curve and Future Int Rates
Future Interest Rates
Spot rate yt, t = 1,…,T:
If there were no uncertainty, all investments
much have the same risk-free return.
1-year zero: Y1 = 2%; 2-year zero: Y2 = 3%
Short rates rt: unknown future interest rate
Forward rate ft: estimate of rt from {yt,yt-1,…}
Is ft a good estimate of rt?
Buy 1-yr zero = buy 2-yr zero and sell after 1 year
2-yr zero: P0 = Par/(1+y2)2, P1 = Par/(1+r2)
Equal return: y1=
−1
If rt were known, what relation must it satisfy?
y1 = (1+y2)2/(1+r2) – 1
If rt is unknown, what relation must it satisfy?
r2=(1+y2)2/(1+y1)-1=1.032/1.02-1=4.01%
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Future Interest Rates
Yield Curve and Future Int Rates
Short
rates
1
2
3
4
r1=2%
r2=?
r3=?
r4=?
Year
Future int rates
f2=?
f3=?
f4=?
Forward rates
y1=2%
Spot
rates
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If there were no uncertainty, short rates and
spot rates must satisfy
rt=(1+yt)t/(1+yt-1)t-1-1
Return of (t-1)-year zero = (1+yt-1)t-1-1
Return of t-year zero sold after t-1 years =
(1+yt)t/(1+rt)-1
y2=3%
r3=(1+y3)3/(1+y2)2-1=1.0363/1.032-1=4.81%
y3=3.6%
r4=(1+y4)4/(1+y3)3-1=1.044/1.0373-1=5.21%
y4=4%
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The Forward Rate
When Spot Rates Are Uncertain
The forward rate, ft, is an “estimate” of the
future interest rate in year t.
The forward rate is calculated from the
yield curve of zero-coupon bonds:
ft = (1+yt
)t/(1+y
We can still calculate ft. But is rt = ft?
Short
rates
t-1
t-1) -1
1
2
3
4
r1=2%
r2=?
r3=?
r4=?
f2=4.01% f3=4.81% f4=5.21%
Year
Future int rates
Forward rates
y1=2%
ft is the 1-year rate after (t-1) years.
Spot
rates
If there were no uncertainty, rt = ft.
y2=3%
y3=3.6%
y4=4%
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19
Interest Rate Risk
Yield Curve and Forward Curve
A 1-yr investor can
5.2%
Forward Curve
buy a 1-yr zero with return r1 = y1, or
Yield Curve
buy a 2-yr zero and sell after 1 year: return =
P1/P0 – 1 with P0 = Par/(1+y2)2, P1 = Par/(1+r2)
4.8%
4%
3%
2%
1
2
3
4
P1 is uncertain because r2 is uncertain.
Years to
Maturity
When future interest rates are uncertain, she
needs a premium to invest in 2-year zeros:
Yield curve: expected annual yield at maturity.
Forward curve: expected interest rate in a future year.
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P1/P0 – 1 > y1 → (1+y2)2/[1+ E(r2)]-1 > y1
E(r2) < (1+y2)2/(1+y1)-1 = f2 = 4.01%
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Interest Rate Risk
Expectations Hypothesis
A 2-year investor can
(1+y2)2-1
Buy a 2-year zero: eff annual return =
Buy a 1-year zero in each of the 2 years: return
= (1+y1)[1+ E(r2)]-1
When future interest rates are uncertain, she
needs a premium to invest in 1-year zeros:
The yield curve reflects the market’s
expectations of future interest rates
The market has equal number of short-term (1year) and long-term (2-year) investors.
The forward rate is an unbiased estimate of the
future short rate: E(rt) = ft.
An upward (downward) sloping yield curve
implies that the market believes interest rates
will rise (fall).
(1+y1)[1+ E(r2)]-1 > (1+y2)2-1
E(r2) > (1+y2)2/(1+y1)-1 = f2 = 4.01%
So E(r2) < f2 or E(r2) > f2?
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Theories of the Term Structure
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Liquidity Preference
There are more short-term investors than
long-term investors due to higher risk in
long-term bonds.
Long-term bonds must offer higher returns
to compensate the higher risk: f2 > E(r2)
The forward rate is the expected future
short rate plus a liquidity premium
ft = E(rt) + liquidity premium
Term structure theories examine what
determines yields at different maturities
and implications for future short rates
The expectations hypothesis
Liquidity preference hypothesis
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Liquidity Premium
Yield Curve and Future Short Rates
Extremely difficult to estimate/measure
Likely to be positive: people need to be
“induced” to hold long-term bonds.
Likely to rise with maturity
When the yield curve is normal, future short
rates may also be lower.
Forward rates
Liquidity premium
Yield curve
Liquidity Premium
Expected short rates
=Ft - LPt
Years to
Maturity
Maturity
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Yield Curve and Future Short Rates
When the yield curve is normal, future short
rates may be higher.
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Yield Curve and Future Short Rates
When the yield curve is inverted, future short
rates will be lower.
Forward rates
Yield curve
Expected short rates
=Ft - LPt
Yield curve
Forward rates
Liquidity premium
Liquidity premium
Maturity
Expected short rates
=Ft - LPt
Maturity
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Macaulay Duration
Initial cash outflow: P0 (=∑
Duration and Interest Rate
Risk Management
Weight of CFt: wt=
⁄(
)
(
)
with ∑
)
w =1.
Macaulay duration = weighted average
time to cash flows (in years):
D=∑
t×w =∑
t×
⁄(
)
For zeros: wT = 1 and D = T.
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Duration
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Macaulay Duration
Duration is a measure of a bond’s sensitivity
to interest rate changes, i.e. the slope of
the price-yield curve.
It incorporates the effects of yield, coupon,
maturity, and possibly call features.
It is expressed as a number of years for the
price of a bond to be repaid by its internal
cash flows.
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For zeros:
For coupon bonds:
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Macaulay Duration
Other Duration Measures
The price impact of changes in YTM:
= −D ×
(
)
Modified duration
Bond return

D∗ ≡
= −D ×
= −D∗ Δy
Effective duration after observing P
Market-wide risk factor
D
Duration as sensitivity to interest rate changes
Analogous to a stock’s factor loading, 
ri = i + irM + ei
=−
×
Dollar duration: $D = ΔP = −D ×
×P
Duration calculation
Page 500: 16.1, 16.2, Page 504: 16.3
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Proof of D as Sensitivity to Δy
Present value of CFt: PV =
PVt sensitivity to y:
=
(
(
What Determines Duration?
Let c is the coupon rate:
)
×
=
)
D=
PV
= −D ×
⇒
(
)
Duration is inversely related to coupon rate.
∑
PV =
−
Ceteris paribus:
Bond price: P = ∑ PV
Price sensitivity to y:
=∑
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×
=
Duration is inversely related to YTM.
D
Duration generally increases with time to maturity.
≈ −D ×
Duration of a perpetuity is (1+y)/y.
Duration of a zero equals maturity.
Not required for exam
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Duration Determinants
Duration Determinants
Duration of zeros
= maturity
Higher coupon
lower duration
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Duration Determinants
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Duration Determinants
Duration generally increases with maturity,
Except for some deep-discount bonds.
Higher yield
lower duration
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Duration of a Bond Portfolio
Need for Immunization
If there are N bonds, each has the same
yield and duration Di, i = 1,…,N.
A bond portfolio is formed with weights wi, i
= 1,…,N, with ∑ w = 1.
The duration of the bond portfolio is the
weighted average of durations of individual
bonds: DP = ∑ w D .
For bonds with different yields, the portfolio
duration defined above can be used as an
approximation
Banks
Assets (loans) have maturities 5 to 30 years
Liabilities (deposits) have very short maturities,
i.e. very few term deposits.
If interest rate goes up, asset value drops more
than liability value.
Pension funds
Assets (bond funds) have shorter duration than
liabilities (pension payouts).
If interest rate goes down, liability value increases
more than asset value.
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Passive Bond Management
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Target Date Immunization
Bond index funds
Major global bond indices
Mimic the returns of major bond indices: accept
interest rate risk of the index
Indices cover bonds with maturity  1 year, and
exclude junk and convertible bonds
Stratification of bonds; Page 514
Sell a guaranteed investment contract (GIC)
for $10k, with 8% annual interest for 5
years, i.e. $14,693 in 5 years.
Invest in 8% annual coupon bond, 6 years
to maturity, selling at par: y = c = 8%
D=
−
(
)
=
.
.
−
.
.
.
.
= 4.993
Buy 10 bonds and sell after 5 years.
When interest rate changes, price variations
offset reinvestment variations.
Page 517: 16.4
Immunization: insulate the impact of
interest rate changes
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When Interest Rate Changes
Summary
Lower reinvestment value
Yield curve
Forward rate and future interest rate
Term structure theories
Measure interest rate risk – Duration
Higher bond price
Higher reinvestment value
Manage interest rate risk - immunization
Lower bond price
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Rebalancing Portfolios
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Reading and Exercises
Page 520: 16.4, 16.5
Current interest rate 10%
Payment of $19,487 in 7 years, with PV = $10,000
Buy 3-year zeros (D=3 years) and perpetuities
(D=1.10/0.10=11 years)
Duration matching
w×3 + (1-w)×11=7, w = 50%
$5000 in zeros and $5000 in perpetuities.
After one year: portfolio = $11,000
w×2 + (1-w)×11=6, w = 56%.
56%×$11,000 = $6111.11 in zeros.
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Reading:
Ch 15: 15.1 – 15.5
Ch 16: 16.1, 16.3 (up to page 518)
Recommended exercises
Ch 15: 1 – 6, 9, 14, 15, CFA: 2, 3, 4
Ch 16: 1 – 3, 6, 7, 9, 12, CFA: 2, 3 (a, b, c)
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