Yield curve.pdf

Yield Curves and
Term Structure Theory

Yield curve
The plot of yield on bonds of the same credit quality and
liquidity against maturity is called a yield curve.
Remark The most typical shape of a yield curve has a upward
slope.
The relationship between yields on otherwise comparable
securities with different maturities is called the term
structure of interest rates.
Year to maturity
Yield

Ideally, yield curve should be plotted for bonds that are alike in
all respects other than the maturity; but this is extremely
difficult in practice. Bonds that have similar risks of default
may be different in coupon rates, marketability, callability, etc.
Benchmark interest rate or base interest rate
Yield curve on US Treasury bond instruments is used to serve
as a benchmark for pricing bonds and to set yields in other
sectors of the debt market. This is because the US Treasury
bonds are viewed as default free and they have the highest
liquidity.

Yield spread and risk premium
On Sept 19, 1997, the yield on the Wal-Mart Stores bonds
(rated AA) with 10 years to maturity was 6.476%. On the
same date, the yield on the 10 year most recently issued
Treasury was 6.086%.
Yield spread = 6.476% − 6.086% = 0.39%.
This spread, called a risk premium, reflects the additional risks
the investor faces by acquiring a security that is not issued by
the US Government.
Term structure theory addresses how interest rates are charged
depends on the length of time that the funds are held.

Spot rate
Spot rate is the yield on a zero-coupon Treasury security with
the same maturity.
Any bond can be viewed as a package of zero-coupon
instruments. It is not appropriate to use the same interest rate
to discount all cash flows arising from the bond. Each cash
flow should be discounted at a unique interest rate that is
appropriate for the time period in which the cash flow will be
received. That rate is the spot rate.

Example
A bank offers to depositors one-year spot rate of 4.5% and two-
year spot rate of 5%.
That is, if you deposit $100 today, you receive
(i) $104.5 in the one-year deposit one year later;
(ii) (1.05)2 × $100 = $110.25 in the two-year deposit two years
later.

Example
Given the spot rate curve for Treasury securities, find the fair
price of a Treasury bond.
An 8% bond maturing in 10 years (par value = $100).
Each cash flow is discounted by the discount factor for its
time. For example, the discount factor of the coupon paid 8
years later is
Year 1 2 3 4 5 6 7 8 9 10
Total
present value
spot rate
(%) 5.571 6.088 6.555 6.978 7.361 7.707 8.020 8.304 8.561 8.793
discount
factor 0.947 0.889 0.827 0.764 0.701 0.641 0.583 0.528 0.477 0.431
cash flow 8 8 8 8 8 8 8 8 8 108
present
value 7.58 7.11 6.61 6.11 5.61 5.12 4.66 4.23 3.82 46.50 97.34
.
528
.
0
)
08304
.
1
(
1
8
≈

Example (construction of a zero-coupon instrument)
Bond A: 10-year bond with 10% coupon; PA = $98.72.
Bond B: 10-year bond with 8% coupon; PB = $85.89.
Both bonds have the same par of $100.
Construct a portfolio of –0.8 unit of bond A and 1 unit of bond
B. Resulting face value is $20, and price is PA − 0.8 PB = 6.914.
The coupon payments cancel, so this is a zero coupon portfolio.
The 10-year spot rate is given by
(1 + S10)10 × $6.914 = $20
giving S10 = 11.2%.

Construction of spot rate curve
• The obvious way to determine a sport rate curve is to find the
prices of a series of zero-coupon bonds with various maturity
dates. However, “zero” with long maturities are rare.
• The spot rate curve can be determined from the prices of
coupon-bearing bonds by beginning with short maturities and
working forward toward longer maturities.

Example
Consider a two-year bond with coupon payments of
amount C at the end of each year. The price is P2 and the
par value is F. Since the price should equal to the
discounted value of the cash flow stream
where S1 and S2 are the spot rates for one-year period and
two-year period, respectively.
2
2
1
2
)
1
(
1 S
F
C
S
C
P
+
+
+
+
=

First, we determine S1 by direct observation of one-year zero-
coupon Treasury bill rate; then solve for S2 algebraically from
the above equation. The procedure is repeated with bonds of
longer maturities, say,
• Note that Treasury bonds (considered to be default free)
are used to construct the benchmark spot rates.
.
)
1
(
)
1
(
1 3
3
2
2
1
3
S
F
C
S
C
S
C
P
+
+
+
+
+
+
=

Forward rates
Forward rates are interest rates for money to be borrowed
between two dates in the future but under terms agreed upon
today.
Assume that the one-year and two-year spot rates, S1 and S2,
are known.
1. Buy a two-year bond
$1 in a 2-year account will grow to $(1 + S2)2 at the end of
2 years.
2. Buy a one-year bond and when it matures in one year
from now, buy another one-year bond for another year.

Let f denote the forward rate between one year and two years
agreed upon now. The investment will grow to $(1 + S1)(1 + f)
at the end of two years.
By no arbitrage principle, these two investments should have
the same returns (if otherwise, one can long the higher return
investment and sell short the lower return one).
Hence,
(1 + S1)(1 + f) = (1 + S2)2
giving
This forward rate f1, 2 is implied by the two spot rates S1 and S2.
.
1
1
)
1
(
1
2
2
2
,
1 −
+
+
=
S
S
f

Forward rate formulas
The implied forward rate between times t1 and t2 (t2 > t1) is the rate of
interest between those times that is consistent with a given spot rate
curve.
(1) Yearly compounding
giving
(2) Continuous compounding
so that
i
j
f
S
S i
j
j
i
i
i
j
j >
+
+
=
+ −
,
)
1
(
)
1
(
)
1
( ,
.
1
)
1
(
)
1
(
)
/(
1
, −








+
+
=
−i
j
i
i
j
j
j
i
S
S
f
)
( 1
2
2
,
1
1
1
2
2
t
t
f
t
S
t
S t
t
t
t
e
e
e
−
=
.
1
2
1
2 1
2
2
1
t
t
t
S
t
S
f
t
t
t
t
−
−
=

Most spot rate curves slope rapidly upward at short maturities
and continue to slope upward but more gradually as maturities
lengthen.
Three theories are proposed to explain the evolution of spot rate
curveS:
1. Expectations;
2. Liquidity preference;
3. Market Segmentation.
Determinants of term structure of interest rates
Spot rate
Years

Expectations theory
From the spot rates S1,…., Sn for the next n years, we can
deduce a set of forward rates f1,2 ,.., f1,n. According to the
expectations theory, these forward rates define the expected
spot rate curves for the next year.
For example, suppose S1 = 7%, S2 = 8%, then
Then this value of 9.01% is the market’s expected value of next
year’s one-year spot rate .
1
1
1
1 ,
, −
n
S
S !
%.
01
.
9
1
07
.
1
)
08
.
1
( 2
2
,
1 =
−
=
f
'
1
S

Turn the view around: The expectation of next years curve
determines what the current spot rate curve must be. That is,
expectations about future rates are part of today’s market.
Weakness
According to this hypothesis, then the market expects rates to
increase whenever the spot rate curve slopes upward.
Unfortunately, rates do not go up as often as expectations would
imply.

Liquidity preference
• For bank deposits, depositors usually prefer short-term
deposits over long-term deposits since they do not like to tie
up capital (liquid rather than tied up). Hence, long-term
deposits should demand high rates.
• For bonds, long-term bonds are more sensitive to interest rate
changes. Hence, investors who anticipate to sell bonds
shortly would prefer short-term bonds.

Market segmentation
The market for fixed income securities is segmented by
maturity dates.
• To the extreme, all points on the spot rate curves are
mutually independent. Each is determined by the forces of
supply and demand.
• A modification to the extreme view is that adjacent rates
cannot become grossly out of line with each other.

Expectations Dynamics
The expectations implied by the current spot rate curve will
actually be fulfilled.
• To predict next year’s spot rate curve from the current
one under the above assumption. Given S1,…,Sn as the current
spot rates, how to estimate next year’s spot rates
Recall that the current forward rate f1,j can be regarded as the
expectation of what the interest rate will be next year, that is,
.
,
,
, '
1
'
2
'
1 −
n
S
S
S !
.
1
1
)
1
(
)
1
/(
1
1
,
1
'
1 −








+
+
=
=
−
−
j
j
j
j
j
S
S
f
S

Example
S1 S2 S3 S4 S5 S6 S7
current 6.00 6.45 6.80 7.10 7.36 7.56 7.77
forecast 6.90 7.20 7.47 7.70 7.88 8.06
072
.
0
1
06
.
1
)
068
.
1
(
069
.
0
1
06
.
1
)
0645
.
1
(
2
/
1
3
3
,
1
2
2
,
1
=
−






=
=
−
=
f
f

Invariance theorem
Suppose that interest rates evolve according to the
expectation dynamics. Then a sum of money invested in
the interest rate market for n years will grow by a factor
(1 + Sn)n, independent of the investment and reinvestment
strategy (so long as all funds are fully invested).
This is not surprising since every investment earns the
relevant short rates over the period of investment (short
rates do not change under the expectations dynamics).

To understand the theorem, take n = 2.
1. Invest in a 2-year zero-coupon bonus;
2. Invest in a 1-year bond, then reinvest the proceed at the
end of the year.
The second strategy would lead as a growth of
the same growth as that of the first strategy.
;
)
1
(
1
)
1
(
)
1
(
)
1
)(
1
( 2
2
1
2
2
1
2
,
1
1 S
S
S
S
f
S +
=






+
+
+
=
+
+

Discount factors between two times
Let dj, k denote the discount factor used to discount cash
received at time k back to an equivalent amount of cash at
time j (j < k).
We then have
and these discount factors observe the compounding rule:
di,k = di, j · dj, k.
j
k
k
j
k
j
f
d
−








+
=
,
,
1
1

Short rates
Short rates are the forward rates spanning a single time
period. The short rate at time k is rk = fk, k+1.
The spot rate Sk and the short rates r0, …, rk−1 are related by
(1 + Sk)k = (1 + r0) (1 + r1) … (1 + rk−1)
In general,
(1 + fi, j)j−i = (1 + ri) (1 + ri+1) … (1 + rj−1).
The short rate for a specific year does not change (in the
context of expectations dynamics).

Yield curve.pdf

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Yield curve.pdf