Transcript The Structure of Interest Rates

The Term Structure of Interest
Rates
I. Yield Curve
• Yield: The single interest rate that equates the
present value of a bond’s payments to the
bond’s price.
• Yield to Maturity: A measure of the average
rate of return that will be earned on a bond if
held to maturity.
• Yield Curve:A curve showing the relationship
between the yield to maturity and the maturity.
– Ascending Curve
– Descending Curve
(Normal Curve)
Yield
(Inverse Curve)
Yield
Maturity
Maturity
– Flat Curve
– Humped Curve
Yield
Yield
Maturity
Maturity
II. Term Structure
• Short Rate: The interest rate for a given
time interval.
• Spot Rate: The interest rate appropriate for
a given maturity.
• Spot Rate Curve: The graphical depiction
of the relationship between the spot rate and
its maturity.
• Theoretical Spot Rate:The theoretical
interest rate appropriate for a given
maturity.
• Theoretical Spot Rate Curve: The
graphical depiction of the relationship
between the theoretical spot rate and its
maturity.
• Forward Rate: The interest rate for a future
period.
• Implicit Forward Rate (Implied Forward
Rate): The interest rate for a future period,
computed on the basis of theoretical spot
rates.
0
1
2
3 . . .n-1
Short
Short
Short
Short
Rate1
Rate2
Rate3
Raten
Spot Rate2
n
Forward Rate3
Spot Raten-1
Forward Raten
year
Time Line
III. Theoretical Spot Rate
• On-the-Run Treasury Issues: The most
recently auctioned Treasury securities of a
given maturity.
• The Principle Underlying Bootstrapping:
The value of the Treasury coupon security
should be equal to the value of the package
of zero-coupon Treasury securities that
duplicates the coupon bond’s cash flow.
Maturity
0.5
1.0
1.5
2.0
2.5
3.0
Coupon
Rate
0.000
0.000
0.085
0.090
0.110
0.095
Yield to
Maturity
0.080
0.083
0.089
0.092
0.094
0.097
Price
$96.15
92.19
99.45
99.64
103.49
99.49
Cash flow for the 1.5-year Treasury:
Maturity value = $100.00
Coupon rate = 8.5%
Maturity
0.5
1.0
1.5
Cash Flow
$100 x 0.5 x 8.5% = $4.25
$100 x 0.5 x 8.5% = $4.25
$100 x 0.5 x 8.5% + $100 = $104.25
The present value of the cash flow:
PV = t[CFt / (1 + zt)t],
where
PV = present value,
CFt = cash flow of period t,
zt = actual or theoretical spot rate for
period t.
Theoretical 1.5-year spot rate:
0.5-year spot rate = 0.080 / 2 (given)
1.0-year spot rate = 0.083/ 2 (given)
99.45 = [4.25 / (1 + 0.04)] + [4.25 / (1 +
0.0415)2] + [104.25 / (1 + z3)3].
z3 = 0.04465.
Cash flow for the 2-year Treasury:
Maturity value = $100.00
Coupon rate = 9.0%
Maturity
0.5
1.0
1.5
2.0
Cash Flow
$100 x 0.5 x 9.0% = $4.5
$100 x 0.5 x 9.0 % = $4.5
$100 x 0.5 x 9.0 % = $4.5
$100 x 0.5 x 9.0 % + $100 = $104.5
Theoretical 2-year spot rate:
0.5-year spot rate = 0.080 / 2 (given)
1.0-year spot rate = 0.083 / 2 (given)
1.5-year spot rate = 0.04465 (derived)
99.64 = [4.5 / (1 + 0.04)] + [4.5 / (1 +
0.415)2] + [4.5 / (1 + 0.04465)3] +
[104.25 / (1 + z4)4].
z4 = 0.046235.
IV. Implicit Forward Rate
(1+ zn)n (1+ nft)t = (1+ zn+t)n+t ,
or
(1+ nft)t = (1+ zn+t )n+t / (1+ zn)n,
where
zn = theoretical spot rate,
nft = the forward rate, n periods from now, at will
be carried by loans with maturities of t
periods,
n, t = 1, 2, 3, … .
V. Theory of The Term Structure
A. The Expectations Theory
The expectations theory holds that the shape
of the yield curve is determined by the
investors’ expectations of future interest
movements, and that changes I these
expectations change the shape of the yield
curve.
The forward rate equals the market consensus
expectation of the future short rate, or
if1=E(ri). The yield to maturity would thus be
determined solely by current and expected
future one-period rates (r1 , if1).
1+ yn = [(1+ r1)(1+ 2f1) ... (1+nf1)]1/n.
The long-term rate is the geometric mean of the
short-term rates.
An upward-sloping yield curve would be an
indication of higher expected forward rates over
time.
B. The Liquidity Preference Theory
Short-term investors dominate the market so
that the forward rate exceeds the expected short
rate. The excess of fi over E(ri) is known as the
liquidity premium.
Yield
Constant liquidity premium
Constant forward rate
Yield curve
Constant expected short rate
Maturity
Yield
Rising liquidity premium
Rising forward rate
Rising yield
curve
Rising expected short rate
Maturity
Yield
Constant liquidity premium
Falling forward rate
Humped yield
curve
Falling expected short rate
Maturity
C. The Market Segmentation Theory
(Institutional Theory)
The market segmentation theory maintains that
market participants have strong preferences for
securities of a particular maturity and holds that
they buy and sell securities consistent with
these maturity preferences.
Long- and short-maturity bonds are traded in
essentially distinct or segmented markets, each
of which finds its own equilibrium
independently.
Yield
Dm
Sl
Dl
Sm
Ss
Ds
Years to maturity
D. The Preferred Habitat Theory
The preferred habitat theory asserts that investors
will not hold debt securities outside of their
preferred habitat (maturity preference) without an
additional reward in the form of a risk premium.
That is, investors prefer specific maturity ranges
but can be induced to switch if premiums are
sufficient.
VI. Economic Implications of the
Yield Curve
A. Yield Curve and Business Cycle
• Interest rates and the business cycle are
procyclical.Increasing interest rates imply that
market participants expect a period of economic
expansion. Descending yield curve are common
near the final phase of a period of economic
expansion.
• As the spread between long-term and shortterm rates narrows, the market consensus is
that the rate of economic expansion will be
slowing.
Interest rate
Time
Interest rate
Maturity
Interest rate
Maturity
Interest rate
Maturity
B. Yield Curve and Financial Intermediaries
The more steeply the yield curve slopes
upward, the wider the spread earned by
depository institutions.
Interest rate
Spread
Maturity