# Chapter 6_3_ by fanzhongqing

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```									       Yield Measures, Spot
Rates, and Forward Rates
by Frank J. Fabozzi

PowerPoint Slides by
David S. Krause, Ph.D., Marquette University

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Chapter 6
Yield Measures, Spot Rates, and
Forward Rates
• Major learning outcomes:
– Yield and yield spread measures, which help investors gauge
relative bond value.
– Computing the spot rates from on-the-run Treasury issues.

– Understanding the limitations of the nominal spread measure.
• Explaining the two measures that overcome the limitation:

– Computing forward rates.
Key Learning Outcomes
•   Explain the sources of return from investing in a bond (coupon interest
payments. Capital gain/loss, and reinvestment income).

•   Compute the traditional yield measures for fixed-rate bonds (current yield,
yield to maturity, yield to first call, yield to first par call date, yield to
refunding, yield to put, yield to worst, and cash flow yield).

•   Explain the assumptions underlying traditional yield measures and the
limitations of the traditional yield measures.

•   Calculate the reinvestment income required to generate the yield computed
at the time of purchase.

•   Explain the factors affecting reinvestment risk.

•   Calculate the spread for life and discount margin measure for a floating-rate
security and explain the limitations of both.
Key Learning Outcomes
•   Calculate the yield on a discount basis for a Treasury bill and explain its
limitations.

•   Compute using the method of bootstrapping the theoretical Treasury spot
rate curve given the Treasury yield curve derived from the on-the-run
Treasury issues.

•   Explain the limitations of the nominal spread.

•   Describe and compute a zero-volatility spread given a spot rate curve.

•   Explain why the zero-volatility spread will diverge from, and is superior to,

•   Explain an option-adjusted spread for a bond with an embedded option and
explain what is meant by the option cost.
Key Learning Outcomes
•   Explain why the nominal spread hides the option risk for bonds with
embedded options.

•   Define a forward rate and compute forward rates from spot rates.

•   Demonstrate the relationship between short-term forward rates and spot
rates.

•   Explain why valuing a bond using spot rates and forward rates produces the
same value.

•   Calculate the forward discount factor from forward rates.

•   Calculate the value of a bond given forward rates.
Sources of Bond Return
• A bond investor receives a return from
one or more of the following:
– The coupon interest payments made by the
issuer;
– Any capital gain or loss when the bond
matures or is sold;
– Income from reinvestment of the interim cash
flows (interest/and or principal payments)

• Yield calculations should consider all
three.
Sources of Bond Return
• Coupon interest – periodic income, usually
semi-annually payments.

• Capital gain or loss – the profit or loss of market
value (selling / maturity price minus purchase
price).

• Reinvestment income is the interest income
generated by reinvesting coupon interest
payments and any principal payments from the
time of receipt to the bond’s maturity.
Bond Yields
• The current yield relates the annual dollar coupon interest to the
market price and fails to recognize any capital gain or loss and
reinvestment income.

• The yield to maturity is the interest rate that will make the present
value of the cash flows from a bond equal to the price plus accrued
interest.

• The market convention to annualize a semiannual yield is to
double it and the resulting annual yield is referred to as a bond-
equivalent yield.

• When market participants refer to a yield or return measure as
computed on a bond-equivalent basis it means that a semiannual
yield or return is doubled.
Current Yield
• Current yield is greater (less) than the coupon
rate, when the bond sells at a discount (premium)

• The current yield is a weak measure because it
does not measure any cash flows other than the
coupon payments:
– No consideration is given to the capital gain or loss
– Reinvestment income is also not considered
– Time value of money concepts are not followed
Yield to Maturity
• YTM is the most popular measure
– It is the interest rate that will make the present
value of the bond’s cash flow equal to its
market price plus accrued income.
– It is the same as the Internal Rate of Return
(IRR)
– Calculations:
• Trial and error / approximations
• Calculator
YTM Computation
• It’s simply a present value problem (solve for y):

C         C        C                 C       M
P                            ...            
1  y  1  y  1  y 
2        3
1  y  1  y 
N        N

–   P is the bond price
–   C is the periodic coupon payment
–   N is the number of years to maturity
–   M is the (face value) payment at maturity
–   y is the “risk-adjusted discount rate” (or yield to
maturity, or IRR)
Yield to Maturity
• For bonds that pay interest semi-annually,
the market convention adopted is to
annualize the semi-annual yield to maturity
by doubling it (multiplying by 2). This is the
annual YTM.
– This is also referred to as the: bond-equivalent
yield (BEY)

• For a bond selling at a discount (premium),
the coupon rate is < (>) the current yield,
and < (>) the yield to maturity.
Yield to Maturity
• The yield to maturity takes into account all three sources
of return but assumes that the coupon payments and any
principal repayments can be reinvested at an interest rate
equal to the yield to maturity.

• The yield to maturity will only be realized if the interim
cash flows can be reinvested at the yield to maturity and
the bond is held to maturity.

• Reinvestment risk is the risk an investor faces that future
reinvestment rates will be less than the yield to maturity
at the time a bond is purchased.
Yield to Maturity Relationship
• The following relationships between the price
of a bond, coupon rate, current yield, and yield
to maturity hold:
Bond-Equivalent Yield Convention
• The market convention to annualize a
semiannual yield is to double it and the
resulting annual yield is referred to as a
bond-equivalent yield.

• When market participants refer to a yield or
return measure as computed on a bond-
equivalent basis it means that a semiannual
yield or return is doubled.
Bond-Equivalent Yield Convention
• The idea of simply doubling the semi-annual yield to get the
YTM should be somewhat troubling to those who
understand the concept of the time value of money. You
would think that computing the effective rate would be
proper.

– However, it is still done this way by many because “that’s the way it
always has been done.” This is the market convention, much the
same as the Dow Jones Industrial Average is used to express how
the overall stock market is performing.
• This is okay as long as all bond yields are calculated this way and will

– Even though the BEY is not the true effective annual yield, we’ll soon
see that the use of yields has many problems – the least of which
involves the doubling of the semi-annual rate to obtain a YTM
estimate.
Limitations of YTM
• YTM is better than current yield because it
considers the coupon, capital gain/loss, and
reinvestment income. It is also based on the time
value of money concepts.
– However, it assumes that the coupon payments from a
bond can be reinvested at an interest rate equal to the
yield to maturity.

• This assumption can be highly misleading because
of reinvestment risk
– We have seen that a term structure of interest rates
exists, which can result in different yields at different
points in time.
Example of the Limitation of YTM
• Suppose an investor has a 15-year 8%
semi-annual coupon bond purchased at par
(\$100).
– The YTM is 8%
• Translated into total future dollars this is:
– \$100 x (1.04)30 = \$324.34
– Decomposed this is \$100 of principal return and \$224.34 of
total dollar return

• Without reinvestment income, the dollar return would
be:
– \$120 of coupon income and \$0 capital gain (because the
bond is purchased at par)
Example of the Limitation of
YTM (continued)
• The dollar return shortfall is \$224.34 - \$120 = \$104.34
– This shortfall is made up if the coupon payments are reinvested at
a yield of 8% (the interest rate at the time the bond was purchased)
– For this bond, the reinvestment income is 46.5% (\$104.34/224.34)
of the total dollar return needed to produce a yield of 8%

• The investor will only realize the YTM of 8% if:
– The coupon payments can be reinvested at the YTM of 8% (this is
reinvestment risk)
– The bond is held to maturity (if the bond is not held to maturity, the
investor faces the risk of selling for less than the purchase price
which is known as interest rate risk)
– These are large and questionable assumptions
Factors Affecting
Reinvestment Risk
• Reinvestment risk is the risk an investor faces that
future reinvestment rates will be less than the yield
to maturity at the time a bond is purchased.

• Interest rate risk is the risk that if a bond is not
held to maturity, an investor may have to sell it for
less than the purchase price.

• The longer the maturity and the higher the coupon
rate, the more a bond’s return is dependent on
reinvestment income to realize the yield to
maturity at the time of purchase.
Reinvestment Risk
The two factors affecting reinvestment risk are:
For a given YTM and a given non-zero coupon rate, the
longer the maturity, the more the bond’s total dollar
return depends on reinvestment income to realize the
YTM at the time of purchase. That is, the greater the
reinvestment risk.

The implication is that the YTM for a long-term, high coupon bond
may have a large amount of the total dollar return as
reinvestment income (which is more risky).
Reinvestment Risk (continued)
The two factors affecting reinvestment risk are:

2. For a coupon bond, for a given YTM and maturity, the
higher the coupon rate, the more dependent the bond’s
total dollar return will be on the reinvestment of the
coupon payments in order to produce the YTM at the time
of the purchase.

–   The implication is that bonds selling at a premium will be more
dependent on reinvestment income than bonds selling at par. This is
because the reinvestment income has to make up the capital loss
due to amortizing the price premium when holding the bond to
maturity.

–   Conversely, a bond selling at a discount will be less dependent on
reinvestment income.
Reinvestment Risk (continued)
•   Exhibit 1 shows an example of three
bonds with different coupons (and selling
prices)
– Based on the years to maturity and the
coupon rate (selling price) it shows:
•   Reinvestment risk is greatest for longer maturity
bonds
•   Reinvestment risk is greatest for higher coupon
bonds (those selling at a premium)
Reinvestment Risk
Semi-Annual versus Annual
Coupon Paying Bonds
•       Some bonds (especially those issued outside the
U.S.) pay interest annually rather than semi-
annually

•       The bond-equivalent yield for an annual paying
bond is computed as follows:
2 x [(1+ yield on annual-pay bond)0.5 -1]

–     The term in the square brackets involves determining what semi-
annual yield, when compounded, produces the yield on an
annual-paying bond.
–     Doubling this semi-annual yield gives the bond-equivalent yield.
Semi-Annual versus Annual
Coupon Paying Bonds Example
•   Suppose the YTM on an annual-pay bond
is 6%. Then the bond-equivalent yield is:
2 x [(1+ .06)0.5 -1] = 5.91%
–   The bond-equivalent yield will always be less than
the annual-paying bond’s YTM
Semi-Annual versus Annual
Coupon Paying Bonds Example
It is also possible to convert the bond-equivalent
yield to an annual-pay basis using the
following:
[(1+ (yield on bond-equivalent basis)/2)2 -1]

The yield on an annual-pay basis is always greater
than the yield on a bond-equivalent basis because
of compounding.
Yield to Call
•   The yield to call is the interest rate that will make the
present value of the expected cash flows to the assumed
call date equal to the price plus accrued interest.

•   Yield measures for callable bonds include yield to first call,
yield to next call, yield to first par call, and yield to
refunding.

•   The yield to call considers all three sources of potential
return but assumes that all cash flows can be reinvested at
the yield to call until the assumed call date, the investor will
hold the bond to the assumed call date, and the issuer will
call the bond on the assumed call date.
Yield to Call
•   For callable bonds, the practice has been to
calculate a YTM and a yield to call (YTC)

•   The YTC assumes the issuer will call the bond on
the call date at the specified call price
–   Investors calculate the yield to first call, yield to next
call, yield to first par call, and a yield to refunding.
•   Yield to refunding is used when bonds are currently callable but
have some restrictions on the source of the funds used to buy
back the debt when a call is exercised.
Yield to Call
•    Calculating the YTC is the same as that for
any YTM calculation.
– In the case of yield to first call, the expected
cash flows are the coupon payments to the
first call date and the call price.
– For the yield to first par call, the expected
cash flows are the coupon payments to the
first par call date and the par value.

•    Exhibit 2 contains an example of a YTC
under varying interest rate levels.
Yield to Call
Yield to Call
•   The YTC considers all three sources of
potential return from owning a bond;
however, like YTM it is assumed that all
cash flows can be reinvested at the YTC
until the assumed call date.
– As we have learned previously, this may be an
inappropriate assumption.

•   Other problems with YTC occur when
comparisons are made with YTM because
of the different maturity/call dates.
Yield to Put
•   When a bond is putable, the yield to the put date
is computed.
–   The yield to put is computed assuming that the issue
will be put on the first put date.

•   The yield to put is the interest rate that will make
the present value of the cash flows to the first
put date equal to the price plus accrued interest.

•   Like other yield measures, the yield to put
assumes that any interim coupon payments can
be reinvested at the yield calculated.
Yield to Worst
•   The yield to worst is the lowest yield
from among all possible yield to calls,
yield to puts, and the yield to maturity.
Yield to Worst
•   A yield can be computed for every possible
call and put date, as well as the YTM.
–   The lowest of all the yields is called the yield to
worst.

•   The yield to worst has little meaning as a
measure of potential return because
–   It does not identify the potential return over some
investment horizon
–   It does not recognize that each yield calculation
has different exposures to reinvestment risk.
Cash Flow Yield
•       Mortgage- and asset-backed securities (which are
secured with pools of loans or receivables) have
periodic cash flows that include interest and principal.
These are typically monthly.

•       Because the borrowers can prepay, the timing of the
principal payments are not known with certainty.

•       It is necessary to make an assumption about the rate at
which principal prepayments occur. This is referred to
as:
–     Prepayment rate, or
–     Prepayment speed (this is measured in units of PSA).

•       Given an assumed payment schedule, a cash flow yield
can be computed.
Cash Flow Yield
•   For mortgage-backed and asset-backed securities, the
cash flow yield based on some prepayment rate is the
interest rate that equates the present value of the
projected principal and interest payments to the price
plus accrued interest.

•   The cash flow yield assumes that all cash flows
(principal and interest payments) can be reinvested at
the calculated yield and that the assumed prepayment
rate will be realized over the security’s life.

•   For amortizing securities, reinvestment risk is greater
than for standard coupon nonamortizing securities
because payments are typically made monthly and
include principal as well as interest payments.
Bond-Equivalent Cash Flow Yield
•       The interest rate that will make the present value of the
projected principal and interest payments equal to the
market price plus accrued interest is a monthly yield that
gets annualized as follows:
•     The semi-annual effective yield is computed from monthly yield
by compounding it for six months: (1 + monthly yield)6 – 1

•     Effective semi-annual yield is doubled to get the annual cash flow
yield on a bond-equivalent basis: 2 x effective semi-annual yield

•       Again, it seems odd to be doubling up the semi-annual
rate by multiplying by 2, but this is the market convention
(“that’s the way it has always been done”).

•       Remember, too, that this is a minor issue compared to
the other problems with using yield as a relative
measure.
Limitations of Cash Flow Yield
•       The shortcomings of cash flow yield are:
–     The projected cash flows are assumed to be reinvested at the
cash flow yield
–     The mortgage- and asset-backed securities are assumed to be
held until the final payoff of all loans (or based on some
assumed prepayment rate)

•       The reinvestment risk is especially significant since the
payments are monthly and include principal payments.

•       If the actual prepayment differs significantly from the
actual principal payment rate, the cash flow yield will not
be close to the actual yield.
Floating Rate Bonds
•   For floating-rate securities, instead of a
yield measure, margin measures (i.e.,
spread above the reference rate) are
computed.

•   Two margin measures commonly used are
spread for life and discount margin.
Floating Rate Bonds
•       The coupon rate for a floating (variable) rate bond
(floater) changes periodically according to a reference
rate (i.e. LIBOR).

•       Since the future reference rate is not known, it is not
possible to determine the cash flows – meaning that YTM
cannot be calculated.
–     Margin measures are used instead of YTM for floaters.
–     Margin is simply some spread above the floater’s reference rate.

•       There are several spread or margin measures:
–     Discount margin
•   When a floater is selling at a premium
(discount) as another source (reduction) of
the return.
– Spread for life (or simple margin) is a measure
of potential return that accounts for the
(discount) as well as the constant quoted
margin over the bond’s remaining life.
•       Spread for life is calculated as follows:
•      [(100(100 – Price))/Maturity + Quoted Margin] x (100/Price)

–         Where
•      Price = market price per \$100 of par value
•      Maturity = number of years to maturity
•      Quoted Margin = quoted margin in the coupon reset formula
measured in basis points

–         Example: a floater with a quoted margin of 80 basis points is
selling for 99.3098 and matures in 6 years:
•      [(100(100 – 99.3098))/6 + 80] x (100/99.3098) = 92.14 bps

•       This is a simple calculation, but has the limitations that it considers
only the accretion / amortization of the discount/premium over the
floater’s remaining term to maturity and does not consider the level
of the coupon rate or the time value of money.
Discount Margin
•   The discount margin assumes that the
reference rate will not change over the
life of the security and that there is no
cap or floor restriction on the coupon
rate.

•   The discount margin estimates the
average margin over the reference rate
that the investor can expect to earn over
the life of the security.
Discount Margin
•   The discount margin is calculated as follows:
–   Determine the cash flows assuming the reference
rate does not change over the life of the security
–   Select a margin
–   Discount the cash flows by the current value of the
reference rate plus the margin
–   Compare the present value of the cash flows to the
price plus accrued interest.
•   If the present value is equal to the price plus accrued
interest, the discount margin is the same as the selected
margin
•   If the present value is not equal, then use a different margin
to equate the two.
Discount Margin (continued)
•    Exhibit 3 shows the ‘trail and error’ approach to
calculating the discount margin for a floating
rate bond

•    The drawbacks of the discount margin as a
measure of potential return from investing in a
floating rate bond are as follows:
–   The measure assumes that the reference rate will
not change over the life of the security

–   If the floating-rate security has a cap or floor, is not
taken into consideration
Discount Margin
Yield on Treasury Bills
•   Because T-bills have a maturity of one year or
less, the standard convention is to compute a
yield on a discount basis. There are two
variables:
–   The settlement price per \$1 of maturity value (p)
–   The number of days to maturity which is calculated as
the number of days between the settlement date and
the maturity date (NSM)
•   The yield on a discount basis is compute as:
Discount = (1 – p) x (360/NSM)
Example of Yield on Treasury Bills
•   Settlement date of 8/6/06 with maturity of 1/8/07 and a
price of 0.9776922. The number of days from settlement
to maturity is 155.

•   The yield on a discount basis is:
5.18% = (1 – 0.9776922) x (360/155)

•   For a given yield on a discount basis, the price of a bill
(per \$1 maturity value) is computed as:
p = 1 – (d x (NSM/360))

p = 1 – (.0518 x (155/360)) = 0.97769722
Yield on Treasury Bills
•   The quoted yield on a discount basis is not a
meaningful measure of return because:
–   The measure is based on a maturity investment value
rather than on the actual dollar amount
–   The yield is annualized according to a 360-day year
rather than a 365-day year, making it difficult to
compare yield on T-bills with T-notes and bonds which
pay interest based on the actual number of days in a
year

•   Market participants recognize these limitations of
yield on discount basis and make adjustments to
make the yield on a T-bill comparable to other
Treasury investments.
Theoretical Spot Rates
•   The theoretical spot rate is the interest rate that should be used to
discount a default-free cash flow.

•   Because there are a limited number of on-the-run Treasury
securities traded in the market, interpolation is required to obtain the
yield for interim maturities; hence, the yield for most maturities used
to construct the Treasury yield curve are interpolated yields rather
than observed yields.

•   Default-free spot rates can be derived from the Treasury yield curve
by a method called bootstrapping.

•   The basic principle underlying the bootstrapping method is that the
value of a Treasury coupon security is equal to the value of the
package of zero-coupon Treasury securities that duplicates the
coupon bond’s cash flows.
Theoretical Spot Rates
“Bootstrapping”
•   In order to value default-free cash flows, the
theoretical spot rate for Treasury securities must
be determined. (This was given in Chapter 5 –
now we calculate it).

•   The default-free theoretical spot rate curve is
constructed from the observed Treasury yield
curve.

•   Several techniques are used to create the yield
curve; however, the most commonly employed
method is called “bootstrapping”
Bootstrapping Spot Rates
•   Bootstrapping uses the yield for the on-the-run
Treasury issues (since there are no credit or
liquidity risks).
–   A problem exists because there may be an insufficient
number of data points for on-the-run issues to construct
a yield curve.

•   Issuance of Treasury securities
–   3-month, 6-month, 2-year, 3-year, 5-year, and 10-year
notes (the 30-year has recently been reissued)
–   This leaves gaps in the yield curve which can be filled
in with simple linear interpolation
Bootstrapping Spot Rates
•   To fill in the gap for each missing one year maturity, it is
the highest maturity with the following formula:

(yield at higher maturity – yield at lower maturity)
Number of years between two observed maturity points

•   The estimated on-the-run yield for all intermediate whole-
year maturities is found by adding the amount computed
to the yield at the lower maturity.
Bootstrapping Spot Rates
Example: 2-year 4.52%, 5-year 4.66%, 10-year 4.80%, 30-year 5.03%

Using the above information, to bootstrap the 3- and 4-year Treasury rates,
the following interpolation of .0466% was computed as follows:
(4.66% – 4.52%)
3 years

Then the interpolated 3-year rate would be:
4.52% + .0466% = 4.567%
The interpolated 4-year rate would be:
4.567% + .0466% = 4.614%

Therefore, when a yield curve is shown, many of the points are only
approximations. Exhibits 4 and 5 show an interpolated “bootstrapped”
Treasury yield curve.

This method produces only a ‘crude approximation’
Bootstrapping Spot Rates
Bootstrapping Spot Rates
Theoretical Spot Rates
•   The basic principle is that the value of a Treasury
coupon series should be equal to the value of a
package of zero-coupon Treasuries that
duplicates the coupon bond’s cash flows.

•   Using the arbitrage-free method, it is possible to
compute the approximate yield of bonds (spot
rates) over any maturity range (including months)
and going forward in time.
–   These will be more precise that the linear interpolated
results from bootstrapping.
–   Exhibit 6 shows the plot of the theoretical spot rates
and the par value Treasury yield curve.
Theoretical Spot Rates
Method of Bootstrapping Spot
Rates from the Par Yield Curve
1. Begin with the 6-month spot rate.
2. Set the value of the 1-year bond equal to the
present value of the cash flows with the 1-year
spot rate divided by 2 as the only unknown.
3. Solve for the 1-year spot rate.
4. Use the 6-month and 1-yar spot rates and
equate the present value of the cash flows of
the 1.5 year bond equal to its price, with the
1.5 year spot rate as the only unknown.
5. Solve for the 1.5 year spot rate.
Example
• Consider the yields on coupon Treasury bonds
trading at par (given in the table).
• YTM for the bonds is expressed as a bond
equivalent yield (semi-annual YTM).

Par Yields for Three Semiannual-Pay Bonds

Maturity      YTM       Coupon       Price
6 months      5.00%       5.00%      \$100.00
1 year      6.00%       6.00%      \$100.00
18 months     7.00%       7.00%      \$100.00
1. Begin with the 6-month spot rate.
• The bond with six months left to maturity has a
semi-annual discount rate of 5%/2 = 2.5% or 5%
on a bond equivalent yield basis.
• Since this bond will only make one payment of
\$102.50 in six months, the YTM is the spot rate
for cash flows to be received six months from
now.
• The bootstrapping process proceeds from this
point using the fact that the 6-month annualized
spot rate is 5%.
2. Set the value of the 1-year bond equal to the
present value of the cash flows with the 1-
year spot rate divided by 2 as the only
unknown. Solve for the 1-year spot rate.
The 1-year bond will make two payments, one in six
months of \$3.0 and one in one year of \$103.0,
and that the appropriate spot rate to discount the
coupon payment (which comes 6 months from
now), is written as:
\$3.0/(1.025)1 + \$103.0/(1+z2/2)2 = \$100
where z2 is the annualized 1-year spot rate
3. Solve for the 1-year spot rate.
\$3.0/(1.025)1 + \$103.0/(1+z2/2)2 = \$100
where z2 is the annualized 1-year spot rate.

Solve for z2/2 as: \$103.0/(1+z2/2)2 = \$100 - \$3/1.025
= \$100 - \$2.927 = \$97.073
or:
\$103.0/\$97.073 = (1+z2/2)2

So: sq. root of (\$103.0/\$97.073) -1 = z2/2 = 3.0076%

1-yr Spot rate (z2) = 3.0076% times 2 = 6.0152%
4. Use the 6-month and 1-year spot rates and
equate the present value of the cash flows
of the 1.5 year bond equal to its price, with
the 1.5 year spot rate as the unknown.
• Now that we have the 6-month and 1-year spot rates,
this information can be used to price the 18-month bond.
• Set the bond price equal to the value of the bond’s cash
flows as:
\$3.5/(1.025)1 + \$3.5/(1.030076)2 +
\$103.5/(1+z3/2)3 = \$100
where z3 is the annualized 1.5-year spot rate.
5. Solve for the 1.5 year spot rate.
\$3.0/(1.025)1 + \$3.5/(1.030076)2 + \$103.5/(1+z3/2)3 = \$100
where z3 is the annualized 1.5-year spot rate.

Solve for z3/2 as: \$103.5/(1+z3/2)2 = \$100 - \$3.5/1.025 -
\$3.5/(1.030076)2 = \$100 - \$3.415 - \$3.30 =\$93.285
or:
\$103.5/\$93.285 = (1+z3/2)2

So: cube root of (\$103.5/\$93.285) -1 = z3/2 = 3.5244%

1.5-yr Spot rate (z3) = 3.5244% times 2 = 7.0488%
Yield Spreads Relative to the Spot
Rate Curve
•       The nominal spread is the difference between a
non-Treasury bond’s yield and the YTM for a
benchmark Treasury coupon security.
–     The nominal yield spread measures the compensation for the
additional credit risk, option risk, and liquidity risk an investor is
exposed to by investing in a non-Treasury security with the same
maturity.

•       The problems with the nominal spread measure are:
–     For both bonds, the yield fails to take into consideration the term
structure of spot rates
–     In the case of a call or put bond, expected interest rate volatility
may alter the cash flows of the non-Treasury bond

•   The nominal spread is the difference
between the yield for a non-Treasury bond
and a comparable-maturity Treasury
coupon security.

•   The nominal spread fails to consider the
term structure of the spot rates and the fact
that, for bonds with embedded options,
future interest rate volatility may alter its
cash flows.

measure of the spread that the investor will
realize over the entire Treasury spot rate curve if
the bond is held to maturity, thereby recognizing
the term structure of interest rates.

spread off one point on the Treasury yield curve
but is a spread over the entire spot rate curve.
•       The zero-volatility or Z- spread is a measure of
the spread the investor would realize over the
entire Treasury spot rate curve if the bond is held
to maturity.
–     It is not the spread off of one point on the Treasury yield curve
(nominal spread), it is an average over all spot rates.

•       The Z-spread is also called a static spread – and is
calculated as the spread which will make the present
value of the cash flows from the non-Treasury bond, when
discounted at the Treasury spot rate plus the spread,
equal to the non-Treasury bond’s price.
–     Trial and error is used to determine the Z-spread.
•   Exhibits 8 and 9 show how trial and error
is used to compute the Z-spread

•   The Z-spread is measured relative to the
Treasury spot rate curve and represents a
spread to compensate for the non-
Treasury bond’s credit risk, liquidity risk,
and any embedded option risk.
•    For bullet bonds, unless the yield curve is
very steep, the nominal spread will not
differ significantly from the Z-spread; for
securities where principal is paid over
time rather than just at maturity there can
be a significant difference, particularly in a
steep yield curve environment.
•       The Z- and nominal spreads will not differ much for standard coupon-
paying bullet bonds. They will diverge when:

–      The slope of the term structure is steep
–      The principal is paid off before maturity (i.e. mortgage- and asset-back
bonds)

•       The Z-spread can be calculated to any benchmark spot rate curve.
–      When used for the same issuer, it is possible to isolate liquidity risk
•   The option-adjusted spread (OAS) converts the cheapness or richness of a
bond into a spread over the future possible spot rate curves.

•   An OAS is said to be option adjusted because it allows for future interest rate
volatility to affect the cash flows.

•   The OAS is a product of a valuation model and, when comparing the OAS of
dealer firms, it is critical to check on the volatility assumption (and other
assumptions) employed in the valuation model.

•   The cost of the embedded option is measured as the difference between the

•   Investors should not rely on the nominal spread for bonds with embedded
options since it hides how the spread is split between the OAS and the option
cost.

•   OAS is used as a relative value measure to assist in the selection of bonds
with embedded options.
spot rate curve, has a problem in that it fails to take future
interest rate volatility into consideration – which could
change the cash flows for bonds with embedded options.

the dollar difference between the fair valuation and the
market price and convert it to a yield spread measure.
–      The OAS is used to reconcile the fair price (value) and the market
price by finding a return (spread) that will equate the two.
–      The spread is measured in basis points.
•       The OAS depends upon the valuation model employed.
–      OAS models primarily differ in how they forecast interest rate changes.

•       What are the key modeling differences?
–      Interest rate volatility is a crucial assumption. The higher the interest rate
volatility, the lower the OAS.
–      The OAS is a spread over the Treasury spot rate curve or the issuer’s
benchmark. In the model, the spot rate curve is the result of a series of
assumptions that allow for changes in interest rates.

•       The spread is referred to as “option adjusted” because the bond’s
embedded options can change the cash flows and therefore the
value of the security. Note: the Z-spread ignores how interest rate
changes can impact cash flows – which is why it is referred to as the
zero-volatility OAS.
Option Cost
•   The implied cost of the option embedded in a
bond can be obtained by calculating the
difference between the OAS at the assumed
interest rate or yield volatility and the Z-spread.

•   The Z-spread is the OAS plus the option cost.
Therefore, the option cost equals the Z-spread
minus the OAS.

•   Thus, the option cost is the difference between
the spread that would be earned in a static
interest rate environment (Z-spread) and the
Option Cost
•       For callable bonds and mortgage- and asset-backed
securities, the option cost is positive.
–     This is because the issuer’s ability to alter the cash flows will
result in an OAS that is less than the Z-spread.

•       In the case of a putable bond, the OAS is larger than the
Z-spread because of the investor’s ability to alter the
cash flows.

•       In general, when the option cost is positive (negative),
the investor has sold (bought) an option to the issuer or
borrower.
–     An investor that relies only on the nominal spread may not be
adequately compensated for taking on option risk – which is one
of the strengths of the OAS approach.

Reflects
Credit risk, option risk,
Nominal         Treasury yield curve           liquidity risk
Treasury spot rate     Credit risk, option risk,
Zero-volatility          curve                   liquidity risk
Treasury spot rate
Option-Adjusted           curve           Credit risk, liquidity risk
Forward Rates
•   Besides default-free theoretical spot rate curves
extrapolated from the Treasury yield curve, it is
possible to compute forward rates.

•   Since forward rates are extrapolated from the
default-free theoretical spot rate curve, these
rates are referred to as implied forward rates.

•   Besides using the Treasury yield curve, it is
possible to compute forward rates from other
interest rate curves (i.e. LIBOR).
Forward Rates
•   Using arbitrage arguments, forward rates
can be extrapolated from the Treasury
yield curve or the Treasury spot rate curve.

•   The spot rate for a given period is related
to the forward rates; specifically, the spot
rate is a geometric average of the current
6-month spot rate and the subsequent 6-
month forward rates.
Forward Rates
• Notation:       1f1

when issued         time to maturity

• Definition of forward rate: The implied
rate of return on a security to be issued
at some future date.

• Definition of spot rate: The rate of return
Spot and Forward Rates for Fixed
Income Securities
• A spot rate is a rate agreed upon today, for a
loan that is to be made today. (e.g. r1 = 5%
indicates that the current rate for a one-year
loan is 5%).

• A forward rate is a rate agreed upon today, for a
loan that is to be made in the future. (e.g. 2f1 =
7% indicates that we could contract today to
borrow money at 7% for one year, starting two
years from today).
Forward Rates
Forward rates of interest are implicit in the term structure of interest rates

t=0          1          2         3         4…

r1        1f2

r2             2f3

r3                 3f4

Note the notation: 3f4 means “the forward rate from period 3 to period 4.”

When the beginning subscript is omitted, it is understood that the forward
rate is for one period only: 3f4 = f4 .
General Formula for Forward Rates
(1  rn )n  (1 rn1 ) n1 (1  fn )1
• One-period forward rates:
imply ing that ...

(1  rn ) n
fn              n1
1
(1 rn1 )

• n-period forward rates:   (1  rk n )k  n  (1  rk ) k (1         f k  n) n
k

imply ing that...
1
(1  r )k  n  n
fk  n        kn       1
k
 (1  rk ) 

k

Example: Forward Rates
• What one-year forward rates are implied by the
following spot rates?
Maturity Year             Spot Rate (rt)          Forward Rate (ft)

1                       4.0%                            –

2                       5.0%                      6.01%

3                       5.5%                     6.507%

(1  r2 )2  (1 r )(1 f2 )
1                        (1  r3 )3  (1  r2 )2 (1  f3 )
3            2
2
(1.05)  (1.04)(1  f2 )                 (1.055)  (1.05) (1  f3 )
f2  6.01%                                f3  6.507%
Implied Forward Rate Example
• Suppose the spot term structure of zero-
coupon yields is: {r1=0.08, r2=0.10, r3=0.13,
r4=0.14,…}

• If investors wish to invest \$1,000,000 for two
years. They can choose between:
– buying a 2-yr. discount bond, and
– buying a sequence of two 1-yr. bonds, i.e.,
one now and one in one year from now.
What Will the Investor Choose?
• The alternative that pays the higher
cumulative return over the 2-yr time horizon.

• Caveat: The rate of return on the bond
issued one year from now is uncertain.

• How do we estimate it?
– With the implied forward rate
Estimating the Implied Forward Rate

(1+ 1 f1 )(1+r1 ) = (1+r2 )2 
(1+r2 )2 (1.1)2
(1+ 1 f1 ) =          =        = 1.12   12%
(1+r1 )   (1.08)

• Underlying assumption: These must be
equal cumulative returns, with no arbitrage
possible.
Estimation of Implied Forward Rates
(using the spot term structure from a previous slide)
3f1

t= 0        1      2         3           4
1f2         2f2

1f3
(i)    f is given by:
3 1

(1+ 3 f1 )(1+r3 )3 = (1+r4 ) 4 
(1+r4 ) 4
(1+ 3 f1 )=           = 1.17 or 17%
(1+r3 ) 3
Estimation of Implied Forward
Rates (continued)
(ii) 1 f2 is defined by:
(1+ 1 f2 )2 (1+r1 ) = (1+r3 )3 
(1+r3 )3   (1.13)3
(1 + 1 f2 ) =          =         = 1.1558 or 15.58%
(1+r1 )    (1.08)
(iii) 1 f3 is defined by:
(1+ 1 f3 )3 (1+r1 ) = (1+r4 ) 4 
(1+r4 )4 1 3  (1.14) 4 1 3
(1+ 1 f3 ) = (          ) =(         ) = 1.1605 or 16.05%
(1+r1 )       (1.08)
General Formula for Implied
Forward Rates
1
 (1+ri+j )i+j         j
1+ i fj =           i 
 (1+ri ) 

• Note that implied fwd rates are internally
consistent, e.g.,
(1 1f 2 )2 (1 3f1)  (1 1f 3 )3 
1
(1 1f 3 )  (1 1f 2 ) (1 3f1)

2

3
Deriving a 6-Month Forward Rate
To compute a 6-month forward rate, it is necessary to utilize
a yield curve and the corresponding spot rate curve.

•       The following 2 investments should have the same
value:
–     1-year Treasury bill and
–     2 six-month Treasury bills (one purchased now and the other in
six months)

•       An investor should be indifferent since they should
produce the same investment income over the same
investment horizon.
Deriving a 6-Month Forward Rate
Although an investor does not know the interest rate of the
second 6-month T-bill, it is possible to compute it because
the “forward” rate must such that it equalizes the dollar
return between the two alternatives.

Exhibit 11 shows the timeline for the two investment
alternatives:
•   The value of first six-month T-bill is: X(1 + z1)
•   The value of the total investment following the second six-
month T-bill is: X(1 + z1)(1 + f)
–   Where z1 is one-half the bond-equivalent yield of the 6-month spot
rate and f is one-half the forward rate on a 6-month Treasury bill
available 6 months from now. X is the amount of the investment.
Deriving a 6-Month Forward Rate
Relationship Between Spot Rates and Short-
Term Forward Rates
•        The value of alternative investment (a 1-year T-bill) is computed as:
X(1 + z2)2

•        Because the two alternatives should generate identical returns:

X(1 + z1)(1 + f) = X(1 + z2)2
•        Solving for f   = [(1 + z2)2 / (1 + z1)] -1

–      Multiplying f by 2 to get the forward rate on a bond-equivalent yield
basis.

•        Forward rates can be computed on various combinations of short-
and longer-term interest rates. Exhibit 12 provides the six-month
forward rates for the entire yield curve. Exhibit 13 is a graph of the
forward rate curve.

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