2. Spot Rate
Yield Curve Should Not Be Used In Pricing B
The price of a bond is the sum of the present values of its
cash flows. In discounting the cash flows the discount rate
used should be the yield on a Treasury security with the
same maturity plus a spread or margin that is appropriate
with the risk. However, there is a problem with using
Treasury yield curve to determine the appropriate yield or
discount rate. The following example illustrates the
problem:
The two hypothetical 5-year Treasury bonds A and B have
coupon rates of 12% and 3% respectively. Therefore, the
semiannual cash flows are as follows:
Bond A: 1 – 9 time periods. CF = 6 Time period 10 CF=106
Bond B: 1 – 9 time periods. CF=1.5 Time period 10 CF=101.5
3. Spot Rate
Yield Curve Should Not Be Used In Pricing B
Because the cash flows are occurring at different points
in time, as such, it is incorrect to use the same interest
rate for discounting all the cash flows. Instead, each cash
flow should be discounted by a unique interest rate
appropriate for the time in which the cash flow is
occurring. The correct approach is to consider the bonds
A and B as packages of cash flows i.e. packages of zero-
coupon instruments. Therefore, the amount of interest is
the difference between maturity value and the price paid.
Bond A can be viewed as 10 zero-coupon bonds one with
a maturity value of 6 maturing 6 months from now, a
second with a maturity value of 6 maturing 12 months or
1 year from now, a third with a maturity value of 6
maturing 18 months or 1½ years from now & so on. The
4. Spot Rate
same is the case with Bond B. Therefore, the value of
the bonds should equal the total value of all the
component zero-coupon bonds. Otherwise, arbitrage
profit can be made.
To determine the value of each zero-coupon bond it is
necessary to know the yield on a zero-coupon Treasury
with the same maturity. This yield is called the spot rate
and the graphical depiction of the relationship between
the spot rate and maturity is called the spot rate curve.
As there are no zero-coupon Treasury debt issues with
a maturity greater than one year, it is not possible to
construct such a curve solely from observations of
market activity on Treasury securities. Rather it is
necessary to derive the curve from theoretical
considerations as applied to the yields of the actually
5. Spot Rate
traded Treasury debt securities. Such a curve is called a
theoretical spot rate curve and is the graphical
depiction of the term structure of interest rate.
Development of Theoretical Spot Rate Curve
for Treasuries
A default-free theoretical spot rate curve can be
constructed from the yield on Treasury securities. The
Treasury issues that can be considered are:
i) on-the-run Treasury issues
ii) on-the-run Treasury issues and selected off-the-
run Treasury issues
iii) all Treasury coupon securities and bills
iv) Treasury coupon strips
6. Spot Rate
The methodology of constructing the theoretical spot rate
curve varies on the basis of the type of securities included
for construction. The following two methods are used for
on-the-run Treasury issues:
On-the-Run Treasury Issues
These are the most recently auctioned issues of given
maturity. These issues include 3-month and 6-month
Treasury bills, 2-year, 5-year, and 10-year Treasury notes
and the 30-year Treasury bond. Treasury bills are zero-
coupon instruments and the notes and bonds are coupon
securities. There is an observed yield for each of the on-
the-run issues. For the coupon securities, these yields are
not the yields used in the analysis when the security is not
trading at par. Rather, for each on-the-run coupon
security, the estimated yield necessary to make the issue
7. Spot Rate
trade on par is used. The resulting on-the-run yield curve
is called the par coupon curve.
The objective is to develop a theoretical spot rate curve
with 60 semiannual spot rates: 6-month rate to 30-year
rate. Excluding the 3-month bill, there are only 6
maturity points available when only on-the-run issues
are used. The 54 missing points are interpolated from the
surrounding maturity points on the par yield curve. The
simplest and most commonly used interpolation method
is the linear extrapolation. On the basis of the yields at
two maturity points on the par coupon curve, the
following is calculated:
(yield at higher maturity – yield at lower maturity)
Number of semiannual periods between two maturity points + 1
The yield for all intermediate semiannual maturity points
is found by adding to the yield at the lower maturity the
8. Spot Rate
the amount computed from the formula. For example,
the yield from the par coupon curve for the 2-year and 5-
year on-the-run issues are 6% and 6.6% respectively.
There are 5 semiannual time periods between the two
maturity points. The extrapolated yield for the 2.5, 3,
3.5, 4, and 4.5 is computed as follows:
(6.6% – 6%)
6
= 0.1%
=
Therefore, 2.5-year yield = 6.0%+.1% = 6.1%
3.0-year yield = 6.1%+.1% = 6.2%
3.5-year yield = 6.2%+.1% = 6.3%
4.0-year yield = 6.3%+.1% = 6.4%
4.5-year yield = 6.4%+.1% = 6.5%
9. Spot Rate
There are two problems with using just the on-the-run
issues. First, there is a large gap between some of the
maturity points which may result in misleading yields
for those maturity points when estimated using the linear
extrapolation method. The problem is more prevalent in
case of gap between 5 and 10-year maturity points and
in case of 10 and 30-year maturity points. Second
problem is that as the true yields are different from the
quoted yields in the market, the yields of the on-the-run
issues themselves may be misleading.
The par yield curve can be converted to theoretical spot
rate curve by using bootstrapping. For example, for
computing theoretical spot rate curve for 10 years, 20
semiannual spot rates have to be computed. The
hypothetical par yield is shown in the Table where the
annualized yield (YTM), price, maturity, and computed
10. Spot Rate
spot rates of 20 Treasury securities are stated. The
coupon rate and YTM of the issues are same, as such,
their price is equal to par except for 6-month and 1-year
issues. It should be noted that all the analysis have been
done keeping in view the basic principle that the value of
treasury coupon security should be equal to the total
value of the package of zero-coupon Treasury securities
that copies or duplicates cash flows of the coupon bond.
The 6-month Treasury bill is a zero-coupon issue, as
such, its annualized yield 5.25% is equal to the spot rate.
Similarly, 1-year Treasury has a rate of 5.5% which is
equal to the 1-year spot rate. Considering these rates,
11. Spot Rate
the spot rate for the 1.5-year Treasury can be computed.
The price of theoretical 1.5-year zero-coupon Treasury
should equal to the total present value of three cash
flows from an actual 1.5-year coupon Treasury where
the yield used for discounting is the spot rate
corresponding to the cash flow. The Table shows coupon
rate for 1.5-year Treasury is 5.75% and price equal to
100 as it is trading at par.
Therefore, the semiannual coupon will be (5.75%
÷2)*100 = 2.875
Table shows coupon rate for 2-year Treasury is 6.0%
and as such, the semiannual coupon will be 3.0
12.
13.
14.
15. Forward Rate
It has been demonstrated how the theoretical spot rates
can be extrapolated from the yield curve. Similarly, the
market consensus future interest rates can also be
developed. The following example illustrates the
significance of the market consensus future interest
rates:
An investor with one-year investment horizon faces two
investment alternatives.
Alternative I: Buy a one-year instrument
Alternative II: Buy a 6-month instrument and when it matures
buy another 6-month instrument
16. Forward Rate
In case of the first alternative, the investor will realize
the 1-year spot rate with certainty. Whereas, with the
second alternative, the investor will realize the 6-month
spot rate for sure, but the 6-month rate 6 months from
now is unknown. As such, with the alternative II, the
rate that will be earned over one year time period is not
known with certainty. The following graph shows this.
17.
18. Forward Rate
The investor will be indifferent between the two
investment alternatives if they produce the same amount
of return over the 1-year investment horizon. Given the
1-year spot rate, there is some rate on a 6-month
instrument 6 months from now which will make the
investor indifferent between the two alternatives. That
rate is denoted by f. If the 1-year and 6-month spot rates
are known, then the rate f can be determined readily. If
100 is invested in 1-year instrument, then after one year
the amount will be 100*(1+z2)2 where z2 is the 1-year
spot rate. If 100 is invested in 6-month instrument then
after 6 months the amount will be 100*(1+z1) where z1
is the 6month spot rate.If this amount is reinvested at the
19. Forward Rate
6-month rate 6 months from now i.e. at rate f, the total
amount at the end of one year will be 100*(1+z1)*(1+f).
The investor will be indifferent if 100*(1+z1)*(1+f) =
100*(1+z2)2. If solved for f then
f = [(1+z2)2 ÷ (1+z1)] – 1
If the 6-month rate 6 months from now is less than the
computed forward rate then the investor will get more
return from alternative I, otherwise, the alternative II
will provide higher return to the investor. If the above
two rates are equal then the investor will be indifferent
between the two alternatives.
20. Forward Rate
The rate determined for f is called market’s consensus
for the 6-month rate 6 months from now. A future
interest rate calculated either from spot rate or the yield
curve is called the forward rate.
The notation that is used to indicate 6-month forward
rates is 1fm where the subscript 1 indicates a 1-period (6-
month) rate and the subscript m indicates the period
beginning m periods from now. When m is equal to zero,
this means the current rate. Thus, the first 6-month
forward rate is simply the current 6-month spot rate.
That is, 1f0 = z1.
21. Forward Rate
The general formula for determining a 6-month forward
rate is:
1fm = [(1 + zm+1)m+1 ÷ (1 + zm)m] −1
For example, assuming that the 6-month forward rate
four years (eight 6-month periods) from now is sought.
In terms of the notation, m is 8. The formula is then:
1f8 = [(1 + z9)9 ÷ (1 + z8)8] −1
From Exhibit 4, since the 4-year spot rate is 5.065% and
the 4.5-year spot rate is 5.1701%, z8 is 2.5325% and z9 is
2.58505%.
Then, 1f8 = [(1.0258505)9 ÷ (1.025325)8]− 1 = 3.0064%
22. Forward Rate
Using spot rates, any forward rate can be computed.
With the same arbitrage arguments as shown above to
derive the 6-month forward rates, any forward rate can
be obtained. There are two elements to the forward rate.
The first is when in the future the rate begins. The
second is the length of time for the rate. For example,
the 2-year forward rate 3 years from now means a rate
three years from now for a length of two years. The
notation used for a forward rate, f, will have two
subscripts—one before f and one after f i.e. t fm
The subscript before f is t and is the length of time that
the rate applies. The subscript after f is m and is when
the forward rate begins. That is, the length of time of the
forward rate f when the forward rate begins. The time
periods are 6-month periods.
23. Forward Rate
Given the above notation, here is what the following
mean:
Notation Interpretation for the forward rate
1f12 6-month (1-period) forward rate beginning 6 years
(12 periods) from now
2f8 1-year (2-period) forward rate beginning 4 years (8
periods) from now
6f4 3-year (6-period) forward rate beginning 2 years (4
periods) from now
8f10 4-year (8-period) forward rate beginning 5 years (10
periods) from now
24. Forward Rate
Derivation of The Forward Rate Formula
Consider the following two alternatives for an investor
who wants to invest for m + t time periods:
• buy a zero-coupon Treasury bond that matures in m + t
time periods, or
• buy a zero-coupon Treasury bond that matures in m
periods and invest the proceeds at the maturity date in a
zero-coupon Treasury bond that matures in t time
periods.
The investor will be indifferent between the two
alternatives if they produce the same return over the m +
t investment horizon. For example, if $100 is invested in
the first alternative, the proceeds for this investment at
the horizon date assuming that the semiannual rate is zm+t
is $100 (1 + z m+t )m+t
25. Forward Rate
Derivation of The Forward Rate Formula
For the second alternative, the proceeds for this
investment at the end of m periods assuming that the
semiannual rate is zm is $100 (1 + zm)m
When the proceeds are received in m periods, they are
reinvested at the forward rate, t fm, producing a value for
the investment at the end of m + t periods of $100 (1 +
zm)m(1 + t fm)t
For the investor to be indifferent to the two alternatives,
the following relationship must hold:
$100 (1 + zm+t )m+t = $100 (1 + zm)m(1 +tf m)t
Solving for tfm we get:
t fm = [(1 + zm+t )m+t ÷ (1 + z m)m]1/t − 1
26. Forward Rate
Derivation of The Forward Rate Formula
Notice that if t is equal to 1, the formula reduces to the
1-period (6-month) forward rate. To illustrate, for the
spot rates shown in Exhibit 4, suppose that an investor
wants to know the 2-year forward rate three years from
now. In terms of the notation, t is equal to 4 and m is
equal to 6. Substituting for t and m into the equation for
the forward rate we have:
4f6 = [(1 + z10)10 ÷ (1 + z6)6 ]¼ − 1
This means that the following two spot rates are needed:
z6 (the 3-year spot rate) and z10 (the 5-year spot rate).
From Exhibit 4 we know z6(the 3-year spot rate) =
4.752%/2 = 0.02376 z10(the 5-year spot rate) =
5.2772%/2 = 0.026386
Then 4 f6 = (1.026386)10− (1.02376)6 ¼ − 1 = 0.030338
27. Forward Rate
Derivation of The Forward Rate Formula
Therefore, 4 f6 is equal to 3.0338% and doubling this rate
gives 6.0675% the forward rate on a bond-equivalent
basis.
We can verify this result. Investing $100 for 10 periods
at the spot rate of 2.6386% will produce the following
value: $100 (1.026386)10 = $129.7499
Investing $100 for 6 periods at 2.376% and reinvesting
the proceeds for 4 periods at the forward rate of
3.030338% gives the same value:
$100 (1.02376)6(1.030338)4 = $129.75012
