Sources of Return Yield Measures for Fixed-Rate Bonds Yield to Call Yield to Put Yield to Worst Cash Flow Yield Yield Measures for Floating Rate Notes Yield Measures for Money Market Instruments Theoretical Spot rates (Bootstrapping) Derivation of Forward Rates Yield Spreads Riding the Yield Curve

1. YIELD MEASURES

2. Flow of Presentation
A. Sources of Return
B. Yield Measures for Fixed-Rate Bonds
C. Yield to Call
D. Yield to Put
E. Yield to Worst
F. Cash Flow Yield
G. Yield Measures for Floating Rate Notes
H. Yield Measures for Money Market Instruments
I. Theoretical Spot rates (Bootstrapping)
J. Derivation of Forward Rates
K. Yield Spreads
L. Riding the Yield Curve

3. A. Sources of
Return
1. Coupon Interest Payments
2. Capital Gain when the security matures
3. Reinvestment income from interim cash flows

4. B. Yield Measures for Fixed-Rate Bonds
• Investors use standardized yield measures to allow for
comparison between bonds with varying maturities.
• An annualized and compounded yield on a fixed-rate bond
depends on the periodicity of the annual rate.
• The periodicity of the annual market discount rate for a zero-
coupon bond is arbitrary because there are no coupon payments.
• The effective annual rate helps to overcome the problem of
varying periodicity. It assumes there is just one compounding
period per year.
For bonds maturing in more
than one year:
• An annualized and compounded
yield-to-maturity is used.
For money market instruments
of less than one year to
maturity:
• These are annualized but not
compounded.

5. • Another way to overcome a problem of varying periodicities is
to calculate a semiannual bond equivalent yield (i.e., a YTM
based on a periodicity of two). For example, if a bond yield is 2%
per semiannual period, its annual yield is 4% when stated on a
semiannual bond basis.
General formula to convert yields based on different periodicities:
1 +
APR𝑚
𝑚
𝑚
= 1 +
APR𝑛
𝑛
𝑛
where APR is the annual percentage rate and m and n are the number
of payments/compounding periods per year, respectively.
• For example, converting a YTM of 4.96% from a semiannual
periodicity to a quarterly periodicity gives a YTM of 4.93%:
1 +
0.0496
2
2
= 1 +
𝐴𝑃𝑅4
4
4
, 𝐴𝑃𝑅4 = 0.0493

6. Limitations of Yield To Maturity
• It assumes that the coupon payments can be reinvested at an interest
rate equal to the yield to maturity.
• The bond is held to maturity

7. Other yield measures
Street
convention
yield-to-
maturity:
The internal
rate of return
on the cash
flows,
assuming the
payments are
made on the
scheduled
dates (no
weekends or
holidays)
True yield-to-
maturity:
The internal
rate of return
on the cash
flows using
the actual
calendar of
weekends and
bank holidays
Government
equivalent
yield:
Restatement
of a yield-to-
maturity
based on a
30/360 day-
count to one
based on
actual/actual
Current yield:
The sum of
coupon
payments
received over
the year
divided by the
flat price
Simple yield:
The sum of
coupon
payments plus
the straight-
line amortized
share of the
gain or loss,
divided by the
flat price

8. Current Yield
i. Current Yield
Current Yield related the annual dollar coupon
interest to a bond’s market price. The
formula for the current yield is :
Current yield = Annual Coupon
Interest/Market Price of Bond.

9. Current Yield
• What will be the current yield for a 7% 8-year
bond whose price is $94.17???
• Current Yield> Coupon Rate; if bonds sells at
discount
• Current Yield< Coupon Rate; if bond sells at
premium
• Drawback: takes into account only the coupon
interest and no other source of investor return
in considered.

10. Example
1. Consider a 20-year $1000 par value, 6% semi-annual pay bond that
is currently trading at a flat price of $802.07. Calculate the current
yield.
2. A 3-year 8% coupon semi-annual pay bond is priced at 90.165.
Calculate the simple yield.

11. Example
1. Current yield =
60
802.07
= 0.0748 𝑜𝑟 7.48%
2. Simple yield =
8+3.278
90.165
= 12.51%
The discount from value is 100-90.165 = 9.835. Annual straight-line
amortization of the discount is 9.835/3 =3.278

12. Limitations of Yield To Maturity
• Calculate the total future dollars returns for a 8-year 8% certificate of
deposit issued by a bank and its current price $94.17.
• Calculate the total future dollar returns for 7% coupon bond that
matures in 8 years and is available at the current price of $94.17.
• Refer to the excel sheet for solution.

13. C. Yield to Call
• Yield to call assumes the issuer will call a bond
on some assumed call date and the call price is
the price specified in the call schedule.
• Investors calculate ‘yield to first call or yield to
next call’, ‘yield to second call’ and ‘yield to
worst’.

14. Yield to Call
Consider a 10-year semi-annual pay 6% corporate bond(par value
$1000) trading at 102% of par value on January 1,2014. The bond is
callable according to the following schedule:
• Callable at $102 on or after January 1,2019
• Callable at $100 on or after January 1, 2022.
Calculate the bond’s YTM, yield-to-first call, yield-to-first par call and
yield to worst.

15. Yield to Call
• YTM of the bond is calculated as:
N=20; PMT =30; FV =1000; PV= -1020; CPT : I/Y = 2.867%
2*2.867% = 5.734% =YTM
Yield-to-First Call is calculated as:
N=10; PMT=30; FV= 1020; PV= -1020; CPT: I/Y = 2.941*2
Yield to First Call = 5.882%
Yield-to-First Par Call/Second Call is calculated as:
N=16; PMT =30; FV=1000; PV= -1020; CPT : I/Y = 2.843*2
Yield –to-First Par Call = 5.686%
As the lowest Yield is 5.686%, so the yield to worst is 5.686%.

16. Yield to Call
Consider the following bond:
Coupon rate = 11%
Maturity = 18 years
Par Value = $1000
Market Price = $1169
Calculate the yield-to-first call, yield-to-first par call and yield to worst if
the bond can be:
i. Called at $1055 in 8 years
ii. Called at $1000 in 13 years

17. D. Yield to Put
• Yield to Put is the interest rate that will make the present value of the
cash flows to the first put date equal to price plus accrued interest.
• Yield to put assumes that any interim coupon payments can be
reinvested at the yield calculated.

18. Yield to Put
• Compute the yield to put for a 6.2% coupon bond maturing in 8 years,
if it is put in three years at par. The current price of the bond is
$102.19.

19. E. Yield to Worst
• A yield can be calculated for every possible call date and put date, in
addition to the yield to maturity.
• The lowest of all these possible yields is called the ‘yield to worst’.
• For example, there are four possible call dates for a callable bond and
the yield to call assuming each possible call date is 6%, 6.2%, 5.8%
and 5.7% and the yield to maturity is 7.5%.
• In this case the yield to worst is 5.7%.

20. G. Yield Measures for Floating-Rate Notes
The interest
payments on a
floating-rate note
vary from period
to period
depending on the
current level of a
reference interest
rate.
The principal on the floater is typically non-amortizing
and is redeemed in full at maturity.
The reference rate is determined at the beginning of
the period, and the interest payment is made at the
end of the period. This payment structure is called “in
arrears”.
The most common day-count conventions for
calculating accrued interest on floaters are actual/360
and actual/365.

21. Simplified FRN pricing model:
PV =
Index+𝑄𝑀 ×FV
𝑚
1+
Index+DM
𝑚
1 +
Index+QM ×FV
𝑚
1+
Index+DM
𝑚
2 +…
Index+QM ×FV
𝑚
+FV
1+
Index+DM
𝑚
𝑁
where PV is the present value/price of the FRN, Index is the annual
reference rate, QM is the quoted margin (annualized), FV is the value at
maturity, m is the periodicity of the FRN, DM is the annualized discount
margin, and N is the number of evenly spaced periods to maturity.
The specified yield spread
over the reference rate is
called the “quoted margin”
on the FRN.
The required margin (i.e.,
discount margin) is the yield
spread over, or under, the
reference rate such that the
FRN is priced at par value on
a rate reset date.

22. Pricing of FRN
• Suppose that a two-year FRN pays six-month Libor plus 0.50%.
Currently six-month Libor is 1.25% and the yield spread required by
investors is 40bps. Compute the value of the bond with $ 100 of par
value.

23. Pricing of FRN
• Coupon payment =
𝐼𝑛𝑑𝑒𝑥+𝑄𝑀 𝐹𝑉
𝑀
=
0.0125+0.005 100
2
= 0.875
• Required rate of return =
𝐼𝑛𝑑𝑒𝑥+𝐷𝑀
𝑚
=
0.0125+0.004
2
= 0.00825
•
0.875
(1+0.00825)1 +
0.875
(1+0.00825)2 +
0.875
(1+0.00825)3 +
0.875+100
(1+0.00825)4 = 100.196

24. Example.
Suppose that a five-year FRN pays three-month SIBOR
(Singapore Interbank offered rate) plus 0.75% on a
quarterly basis. Currently, three-month Sibor is 1.10%.
The price of the floater is 95.50 per 100 of par value.
Calculate the discount margin.

25. Example. Suppose that a five-year FRN pays three-month SIBOR (Singapore
Interbank offered rate) plus 0.75% on a quarterly basis. Currently, three-month
Libor is 1.10%. The price of the floater is 95.50 per 100 of par value. Calculate the
discount margin:
95.50 =
0.011+0.0075 ×100
4
1+
0.011+𝐷𝑀
4
1 +
0.011+0.0075 ×100
4
1+
0.011+𝐷𝑀
4
2 +… +
0.011+0.0075 ×100
4
+100
1+
0.011+𝐷𝑀
4
20
Solving for DM, DM = 1.718%, or 171.8 bps

26. Example
A four-year French floating rate note pays three-month Euribor (Euro
Interbank Offered Rate) plus 1.25%. The floater is priced at 98 per 100
of par value. Calculate the discount margin for the floater assuming
that three-month Euribor is constant at 2%. Assume the 30/360 day-
count convention and evenly spaced periods.

27. Example. A four-year French floating rate note pays three-month
Euribor (Euro Interbank Offered Rate) plus 1.25%. The floater is priced
at 98 per 100 of par value. Calculate the discount margin for the floater
assuming that three-month Euribor is constant at 2%. Assume the
30/360 day-count convention and evenly spaced periods.
98 =
0.02+0.0125 ×100
4
1+
0.02+𝐷𝑀
4
1 +
0.02+0.0125 ×100
4
1+
0.02+𝐷𝑀
4
2 +… +
0.02+0.0125 ×100
4
+100
1+
0.02+𝐷𝑀
4
16
Solving for DM, DM = 1.791%, or 179.1 bps

28. H. Yield Measures for Money Market Instruments
• Money market instruments are short-term debt securities. They range in time-to-
maturity from overnight sale and repurchase agreement (repos) to one-year bank
certificates of deposits.
• Money market instruments also include commercial paper, government issues of less
than one year, bankers’ acceptance and time deposits based on indices such as Libor
and Euribor.
• There are several important differences in yield measures between the money market
and the bond market:
The rate of return
on a money market
instrument is stated
on a simple interest
basis.
Money market
instruments often are
quoted using nonstandard
interest rates and require
different pricing equations
than those used for bonds.
Money market
instruments having
different times-to-
maturity have
different periodicities
for the annual rate.

29. Yield Measures for Money Market Instruments
• In general, quoted money market rates are either discount rates or
add-on rates.
• Although money market conventions vary around the world,
commercial paper, Treasury bills and bankers’ acceptances are often
quoted on a discount rate basis.
• Bank certificates of deposits, repos and such indices as Libor and
Euribor are quoted on add-on rate basis.

30. where Days is the number of days between settlement and maturity;
Year is the number of days in a year (365 or 360); DR is the discount
rate, stated as an annual percentage rate; and AOR is the add-on rate,
stated as an annual percentage rate.
• PV = FV × 1 −
Days
Year
× DR
Pricing formula for money
market instruments quoted
on a discount rate basis:
• PV =
FV
1+
Days
Year
× AOR
Pricing formula for money
market instruments quoted
on an add-on rate basis:

31. The first term for both formulas, Year/Days, is the periodicity of the
annual rate.
The second term for the add-on rate is the interest earned, FV – PV,
divided by PV, the amount invested.
However, for the discount rate, the denominator in the second term is
FV, not PV. Therefore, by design, a money market discount rate
understates the rate of return to the investor.
• DR =
Year
Days
×
FV−PV
FV
The discount rate is
calculated using the
formula:
• AOR =
Year
Days
×
FV−𝑃𝑉
PV
The add-on rate is
calculated using the
formula:

32. Example:
a. Suppose that a 91-day US Treasury bill (T-bill) with a face value of USD10 million
is quoted at a discount rate of 2.25% for an assumed 360-day year.
Find the price of the T-bill.
b. Suppose that a Canadian pension fund buys an 180-day bank certificate of
deposit with a quoted add-on rate of 4.38% for a 365-day year. If the initial
principal amount is CAD10 million, Calculate the redemption amount due at
maturity
c. Suppose that after 45 days, the pension fund sells the certificate of deposit to a
dealer. At that time, the quoted add-on rate for a 135-day BA is 4.17%. Calculate
the sale price of certificate of deposit.

33. Example:
a. Suppose that a 91-day US Treasury bill (T-bill) with a face value of USD10 million is quoted at a
discount rate of 2.25% for an assumed 360-day year. Enter FV = 10,000,000, Days = 91, Year = 360,
and DR = 0.0225. Find the price of the T-bill:
𝑷𝑽 = 𝟏𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎 × 𝟏 −
𝟗𝟏
𝟑𝟔𝟎
× 𝟎. 𝟎𝟐𝟐𝟓 = 𝐔𝐒𝐃𝟗, 𝟗𝟒𝟑, 𝟏𝟐𝟓
b. Suppose that a Canadian pension fund buys an 180-day bank certificate of deposit with a quoted
add-on rate of 4.38% for a 365-day year. If the initial principal amount is CAD10 million, calculate the
redemption amount due at maturity:
𝑭𝑽 = 𝟏𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎 + 𝟏𝟎, 𝟎𝟎𝟎, 𝟎𝟎𝟎 × 𝟏 +
𝟏𝟖𝟎
𝟑𝟔𝟓
× 𝟎. 𝟎𝟒𝟑𝟖 = 𝐂𝐀𝐃𝟏𝟎, 𝟐𝟏𝟔, 𝟎𝟎𝟎
c. Suppose that after 45 days, the pension fund sells the certificate of deposit to a dealer. At that
time, the quoted add-on rate for a 135-day certificate of deposit is 4.17%. Calculate the sale price of
banks certificate of deposit.
𝑷𝑽 =
𝟏𝟎, 𝟐𝟏𝟔, 𝟎𝟎𝟎
(𝟏 +
𝟏𝟑𝟓
𝟑𝟔𝟓
∗ 𝟎. 𝟎𝟒𝟏𝟕)
= 𝐂𝐀𝐃 𝟏𝟎, 𝟎𝟔𝟎, 𝟖𝟐𝟗

34. Example
Suppose that an investor is comparing two money market instruments:
(A) 90-day commercial paper quoted at a discount of 5.76% for a 360-
day year and
(B) 90-day bank time deposit quoted at an add-on rate of 5.90% for a
365-day year
Which offers the higher expected rate of return assuming that the credit
risks are the same?

35. Example: Suppose that an investor is comparing two money market instruments: (A) 90-day commercial
paper quoted at a discount of 5.76% for a 360-day year and (B) 90-day bank time deposit quoted at an
add-on rate of 5.90% for a 365-day year. Which offers the higher expected rate of return assuming that
the credit risks are the same?
Solution:
Step 1: Calculate the present value of the 360-day commercial paper:
𝑃𝑉 = 100 ∗ 1 −
90
360
∗ 0.0576 = 98.560
Step 2: Given the PV of 360-day commercial paper, determine the AOR for 365 days in order to make the
yield of commercial paper comparable with bank time deposit.
𝐴𝑂𝑅 =
365
90
∗
100 − 98.560
98.560
= 0.05925
Step 3: the 90-day commercial paper discount rate of 5.76% when converted to an add-on rate gave yield
of 5.925%. This yield is also known as ‘bond equivalent yield’. A bond equivalent yield is a money market
rate stated on a 365-day add-on rate basis.
If the risks are the same, the commercial paper offers 2.5 bps more in annual return than the bank time
deposit.

36. Example:
The bond equivalent yield (i.e. 365 days AOR) of a 180-day banker’s
acceptance quoted at a discount rate of 4.25% for a 360-day year is
closest to:
(a) 4.31%
(b) 4.34%
(c) 4.40%

37. Example:
The bond equivalent yield (i.e. 365 days AOR) of a 180-day banker’s
acceptance quoted at a discount rate of 4.25% for a 360-day year is closest
to ?
Solution:
Step 1: Calculate the PV of 180-day banker’s acceptance:
𝑃𝑉 = 100 ∗ 1 −
𝐷𝑎𝑦𝑠
𝑌𝑒𝑎𝑟
∗ 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑅𝑎𝑡𝑒
𝑃𝑉 = 100 ∗ 1 −
180
360
0.0425 = 97.875
Step 2: Convert the discount rate of 4.25% for 180 days into Add-on rate for
365 days:
𝐴𝑂𝑅 =
365
180
∗ (
100−97.875
97.875
)= 2.02778*2 = 4.40% Approx.

38. On-the-Run &
Off-the-Run
Treasury
• On-the-run Treasuries are the most recently issued
U.S. Treasury bonds or notes of a particular maturity.
• "On-the-run" Treasuries are the opposite of "off-the-run"
Treasuries, which refer to Treasury securities that have
been issued before the most recent issue and are still
outstanding.
• On-the-run government bond is the most actively traded
security and has a coupon rate closest to the current
market discount rate for that maturity (this implies that
they are priced close to par value).
• On-the-run bonds typically trade at slightly lower yields-
to maturity than off –the-run bonds having the same or
similar times-to-maturity because of difference in the
demand for the securities.

39. I. Theoretical Spot Rates(Bootstrapping)
• In US, the U.S. Department of the treasury currently issues 3-month,
6-months and 12 months Treasury bills and 2-year, 3-year, 5-year, 7-
year and 10-year notes and 20-year and 30-year bonds.
• Treasury bills are zero-coupon instruments and Treasury notes are
coupon-paying instruments.
• However, there are not many data points from which to construct a
Treasury yield curve, particularly after 2 and 5 years.
• Linear interpolation of the results will lead to misleading results, thus
the method of bootstrapping is considered more robust.

40. Theoretical Spot Rates(Bootstrapping)
• A spot rate curve, also known as a zero curve refers to the yield curve
constructed using the spot rates such as Treasury spot rates instead of the
yields.
• A spot rate Treasury curve is more suitable to price bonds because most
bonds provide multiple cash flows (coupons) to the bond holders at
different points in time, and it is better to use the spot rates as the
discount rates for different time periods rather than using a single discount
rate.
• A spot rate curve can be constructed using on-the-run treasuries, off-the-
run treasuries, or both.
• In the bootstrapping technique one repetitively applies a no-arbitrage
implied forward rate equation to yields on the estimated Treasury par yield
curve and the result of one calculation is used in the subsequent one.

41. Theoretical Spot Rates(Bootstrapping)
Step 1: Decide on the Instrument for Yield Curve
The spot curve can be obtained by using on-the-run Treasury
securities, off-the-run treasury securities, or a combination of
both, or Treasury coupon strips. The instrument selected should
not have credit risk, liquidity risk, embedded options, or any
pricing anomalies.
Step 2: Select the Par Yield Curve
Typically, you will find Treasury securities for only a few
maturities such as 3-month, 6-month, 2-year, 5-year, 10-year,
and 30 years.
Step 3: Calculate Spot Rates Using Treasury Yields
In this step, apply the bootstrapping method to calculate the
spot rates.

42. Theoretical Spot Rates(Bootstrapping)
• Suppose the semi-annualized yield for 6-months and 1-year Treasury
bill is 1.5% and 1.65%.
• Thus, the annualized BEY is 1.5%*2= 3% and 1.65%*2 = 3.30%
• Given these two spot rates(i.e. par rates of T-bills), can you compute
the spot rate for a theoretical 1.5 year 3.5% Treasury?
• The value of a theoretical 1.5 year treasury should equal the present
value of the three cash flows from the 1.5 year coupon treasury,
where the yield used for discounting is the spot rate corresponding to
the time of receipt of each six-month cash flow.

43. Theoretical Spot Rates(Bootstrapping)
• The present value of the cash flows is :
100 =
1.75
(1 + 𝑧1)1
+
1.75
(1 + 𝑧2)2
+
101.75
(1 + 𝑧3)3
• 100 =
1.75
(1+0.015)1 +
1.75
(1+0.0165)2 +
101.75
(1+𝑧3)3
wherein, 𝑧1 is the 6-month theoretical spot rate, i.e. 3%/2 = 1.5%
𝑧2 is the six month 1-year theoretical spot rate, i.e. 3.30%/2= 1.65%
𝑧3 is the six-month 1.5 year theoretical spot rate

44. Theoretical Spot Rates(Bootstrapping)
• 100 =
1.75
(1+0.015)1 +
1.75
(1+0.0165)2 +
101.75
(1+𝑧3)3
• 100 = 1.7241 + 1.6936 +
101.75
(1+𝑧3)3
• 96.5822 =
101.75
(1+𝑧3)3
• (1 + 𝑧3)3 =
101.75
96.5822
• 𝑧3 = 1.7527%, thus BEY is 1.7527%*2 = 3.5053%

45. Theoretical Spot Rates(Bootstrapping)
• Given the theoretical, 1.5 spot rate, can you calculate the theoretical 2-year spot
rate of a 3.9% treasury.
• 100 =
1.95
(1+𝑧1)1 +
1.95
(1+𝑧2)2 +
1.95
(1+𝑧3)3 +
101.95
(1+𝑧4)4
• 100 =
1.95
(1+0.015)1 +
1.95
(1+0.0165)2 +
1.95
(1+0.0175)3 +
101.95
(1+𝑧4)4
• 94.3407 =
101.95
(1+𝑧4)4
• (1 + 𝑧4)4 =
101.95
94.3407
• 𝑧4 = 1.9582%, thus BEY is 1.9582*2 = 3.9164%

46. Spot Rates
• If spot rates are 3.2% for one year, 3.4% for two years and 3.5% for
three years, the price of a $1,00,000 face value, 3-year annual pay
bond with a coupon rate of 4% is closest to:
A) $1,01,420
B) $1,01,790
C) $1,08,230

47. Spot Rates
• 𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 =
4000
(1.032)1 +
4000
(1.034)2 +
104000
(1.035)3
• 𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 = $1,01,419.28

48. Spot Rates
• A 3-year bond offers a 10% coupon rate with interest paid annually. Assume the
following sequence of spot rates, the price of the bond is closest to:
a) 96.98
b) 101.46
c) 102.95
Time-to-Maturity Spot Rates
1 year 8%
2 year 9%
3 year 9.5%

49. J. Forward Rates
• Forward rates can be considered as market consensus of future interest
rates.
• Since the forward rates are implicitly extrapolated from the default-free
theoretical spot rate curve, these rates are sometimes referred to as
‘Implied Forward Rate’.
• An implied forward rate is a break-even reinvestment rate.
• The notation for forward rates commonly used is ‘2y5y’, i.e. pronounced as
“the two-year into five-year rate” or simply 2’s5’s.
• The first number (two years) refers to the length of the forward period in
years from today and the second number(five years) refers to the time-to-
maturity of the underlying bond.
• For example, 5.8372% is the ‘2y5y’. That means the 5.8372% is the five-
year yield two years into the future

50. Deriving 6-month forward rate
• Consider an investors who has a 1-year investment horizon and is faced with the following two
alternatives:
• Buy a 1 year Treasury bills
• Buy a 6-month Treasury bill and when it matures in six months buy another 6-month Treasury bill.
• The investor will be indifferent toward the two alternative if they produce the same return over
the 1-year investment horizon.
• Given the spot rates for the 6-month and 1-year Treasury bill, the forward rate on a 6-month
treasury bill is the rate that equalizes the dollar return between two alternatives:
• X(1+Z1)(1+f) = X(1+Z2)^2

51. Forward Rates
• The general formula for determining a n-year forward rate is:
• tfm =
(𝟏+ 𝒛𝒎+𝒕)𝒎+𝒕
(𝟏+ 𝒛𝒎)𝒎
𝟏/𝒕
− 𝟏
• The subscript before ‘f’ is “t” and is the tenor of the forward rate. The
subscript after ‘f’ is “m” and is when the forward rate begins.
• For example, 1f8 means 6-month forward rate four years (eight 6-month
periods) from now.

52. How to read Forward Rates
Forward rate Notation(Method I) Notation (Method II)
6-Month forward rate
beginning 6 years from now 1f12
6y6m
1-year forward rate
beginning 4 years from now 2f8
4y1y
3-year forward rate
beginning 2 years from now 6f4
2y3y
4-year forward rate
beginning 5 years from now 8f10
5y4y

53. Example
Suppose the yield-to-maturity on three-year and four-year zero-coupon
bonds are 3.65% and 4.18% respectively, stated on a semiannual bond
basis.
Calculate the one year forward rate three years from now 3y1y, such
that the investor is indifferent between choosing to invest in three-year
bond and four-year bond.

54. Forward Rates (Example)
Suppose that the two-year and five-year semi-annual BEY of the
theoretical spot rate are 𝑆2= 4% and 𝑆5 = 6%.
Calculate the implied three-year forward rate for a loan starting two
years form now, i.e. 2y3y or 6f4

55. Forward Rates (Example)
Now, in this example;
“t” is the tenor of the forward rate, i.e. 6 periods or 3 years.
“m” is when the forward rate begins, in this case 2 years from now or 4 periods.
Thus, implied forward rate “6f4” is:
=
1+
0.06
2
4+6
1+
0.04
2
4
1
6
− 1
=
1.3439
1.0824
1
6
− 1
= 1.24159
1
6 − 1
= 1.036724 – 1
=0.036724*2
=7.3449 or 7.35%

56. Forward Rates (Example)
• Assume the following spot rates are quoted on a semi-annual bond basis.
a. What is the 6-month forward rate one year from now?
b. What is the 1-year forward rate one year from now?
c. What is the value of a 2-year, 4.5% coupon, semi-annual pay bond?
Years to Maturity Spot Rates
0.5 4.0%
1.0 4.4%
1.5 5.0%
2.0 5.4%

57. Forward Rates (Example)
a) To calculate implied 6-month forward rate one year from now,
we will take “t” as 1 and “m” as 2, then 1f2 can be computed as:
=
1+
0.05
2
1+2
1+
0.044
2
2
1
1
− 1
=
1.07689
1.04448
1
1
− 1
= 1.031029 -1
= 6.205 or 6.21%

58. Forward Rates (Example)
b) To calculate the 1 year forward rate one year from now, i.e. 2f2, that means “t” is 2
periods or 1 year and “m” is also 2 periods or 1 year.
2f2 = =
1+
0.054
2
2+2
1+
0.044
2
2
1
2
− 1
=
1.11245
1.044448
1
2
− 1
= 1.03202 -1
= 6.40%

59. Forward Rates(Example)
• Value of a 2-year 4.5% semi-annual pay bond is:
• 𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 =
2.25
(1+
0.04
2
)1
+
2.25
(1+
0.044
2
)2
+
2.25
(1+
0.05
2
)3
+
102.25
(1+
0.054
2
)4
=
2.25
(1.02)1 +
2.25
(1.0445)
+
2.25
(1.0769)
+
102.25
(1.1125)
= 98.36

60. Forward Rate(Example)
Suppose that an investor observes these prices and BEY on zero-
coupon government bonds:
The prices are per 100 of par value.
Compute the “1y1y” and “2y1y” implied forward rates, stated on semi-
annual bond basis.
Maturity Price Yield-to-Maturity
1 year 97.50 2.548%
2 year 94.25 2.983%
3 years 91.75 2.891%

61. Relationship between Spot rates & Forward rates
• Suppose an investor invests $X in a 3-year zero-coupon Treasury security. The total proceeds
three years (six periods) from now would be:
𝑿(𝟏 + 𝒛𝟔)𝟔
• The investor could instead buy a 6-month Treasury bill and reinvest the proceeds every six
months for three years. The future dollars or dollar returns will depend on the 6-month forward
rates.
• Suppose that an investor can actually reinvest the proceeds maturing every six months at the
calculated 6-month forward rates, then at the end of three years, an investment of X would
generate the following proceeds:
𝑋 1 + 𝑧1 1 + 1𝑓1 1 + 1f2 1 + 1f3 1 + 1f4 1 + 1f5 = X(1 + 𝑧6)6
𝑧6 = [ 1 + 𝑧1 1 + 1𝑓1 1 + 1𝑓2 1 + 1𝑓3 1 + 1𝑓4 1 + 1𝑓5 ]1/6
−1
This equation tells us that the 3-year spot rate depends on the current 6-month spot rate and the
five 6-month forward rates.

62. Relationship between Spot rates & Forward rates
• The relationship between a T-period spot rate the current 6-month
spot rate and 6-month forward rates is as follows:
• Thus, discounting at the forwards rates will give the same present
value as discounting at spot rates.

63. Spot Rates & Forward Rates
• Suppose the current forward one-year rates are:
• Calculate the theoretical spot rates from the given forward
rates.
• Show that a 4-year 3.75% annual coupon bond value is same
when calculated using either implied spot rate or forward rate.
Time Period Forward Rate
0y1y or 2f0 1.88%
1y1y or 2f2 2.77%
2y1y or 2f4 3.54%
3y1y or 2f6 4.12%

64. Spot Rates & Forward Rates
• 0y1y is the one-year spot rate
• Spot rates can be calculated as the geometric average of the forward
rates, thus:
(1.0188*1.0277) = (1+z2)^2
z2 = 2.3240%
Similarly, (1.0188)(1.0277)(1.0354) = (1+z3)^3
z3 = 2.727%
(1.0188)(1.0277)(1.0354)(1.0412) = (1+z4)^4
z4 = 3.074%

65. Spot Rates & Forward Rates
• 𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 =
3.75
(1.0188)1 +
3.75
(1.0234)2 +
3.75
(1.02727)3 +
103.75
(1.03074)4
• 𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 = 102.637
•
• 𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 =
3.75
(1.0188)1 +
3.75
(1.0188)(1.0277)
+
3.75
(1.0188)(1.0277)(1.0354)
+
103.75
(1.0188)(1.0277)(1.0354)(1.0412)
• 𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 = 102.637

66. The maturity structure of interest rates
The term structure of interest rates is the factor that explains the differences
between yields. It involves the analysis of yield curves, which are relationships
between yields-to-maturity and times-to-maturity.
• currency denomination • credit risk
• liquidity • tax status
• periodicity • varying time-to maturity
The difference between yields on two bonds might be due to
various reasons, such as:
Examples of yield curves
The (government bond) spot curve is
a sequence of yields-to-maturity on
zero-coupon (government) bonds.
The yield curve on coupon bonds is a
sequence of yields-to-maturity on
coupon paying (government) bonds.

67. K. Yield Spreads
The spread is the difference between the yield-to-
maturity and the benchmark.
The benchmark is often called the “risk-free rate of
return.” Fixed-rate bonds often use a government
benchmark (on-the-run) security with the same time-to-
maturity as, or the closest time-to-maturity to, the
specified bond.
A frequently used benchmark for floating-rate notes was
Libor. As a composite interbank rate, it is not a risk-free
rate.

68. Spread Risk Premium
“Risk-Free”
Rate of Return
Benchmark
Expected Real
Rate
Expected
Inflation Rate
Credit Risk
Liquidity
Taxation
Yield-to-Maturity Building Blocks

69. Yield Spread Measures
•Yield of non-Treasury Bond – Yield of Treasury Bond
•Compensation for the additional credit risk, liquidity risk and
option risk
Nominal Spread
• Also known as ‘Static Spread’. Z-spread is calculated as the spread that
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.
Z-Spread
•OAS = Z spread – option cost
•Option cost is positive, when the investor has sold an option to the issuer (Z
spread> OAS)
•Option cost is negative, when the investor has purchased an option from the
issuer. (Z spread< OAS)
Option-Adjusted
Spread
•A swap spread is the difference between the fixed rate
component of a given swap and the yield on a Treasury item
or other fixed-income investment with a similar maturity.
Swap spread

70. What is Nominal Spread?
• Traditional analysis of the yield spread for a non-Treasury bond involves calculating the difference between
the bond’s yield and the yield-to-maturity of a benchmark Treasury coupon security. The latter is obtained
from the Treasury yield curve.
• For example, consider the following 10-year bonds:
• The yield spread for these two bonds as traditionally computed is 140 basis points (7.4% minus 6%). This
traditional yield spread is referred to as – nominal spread.
• What is the nominal spread measuring?
• It is measuring the compensation for the additional credit risk, option risk (i.e. the risk associated with
embedded options) and liquidity risk an investor is exposed to by investing in a non-Treasury security rather
than a Treasury security with the same maturity.
Issue Coupon Price Yield to Maturity
Treasury 6% 100.00 6.00%
Non-Treasury 8% 104.19 7.40%

71. Drawbacks of
Nominal Spread
The drawbacks of nominal spread are:
1. The yield fails to take into consideration the term
structure of spot rates.
2. In the case of callable and/or putable bonds,
expected interest rate volatility may alter the cash
flows of the non-Treasury bond.
So, what are the alternatives to nominal spread ?
The alternatives to nominal spread are Zero-volatility
Spread(Z-spread) and Option-adjusted spread (OAS).

72. Zero-Volatility
Spread
(Z-Spread)
• The zero-volatility spread or Z-spread is a measure of the
spread that the investor would realize over the entire
Treasury spot rate curve if the bond is held to maturity.
• It is not a spread off one point on the Treasury yield curve,
as is the nominal spread.
• The Z-spread, also called the static spread, is calculated as
the spread that will make the present value of the cash flows
form the non-Treasury bond, when discounted at the
Treasury spot rate plus the spread, equal to the non-
Treasury bond’s price.
• A trial and error procedure is required to determine the Z-
spread.
• A clear distinction between Z-spread and nominal spread is
that a nominal spread is a spread off of one point on the
Treasury yield curve while the Z-spread is a spread over the
entire theoretical Treasury spot rate curve.

73. Difference
Between Z-
Spread &
Nominal Spread
• Typically, for standard coupon-paying bonds with a bullet
maturity, the Z-spread and the nominal spread will not
differ significantly.
• In general, the difference between the Z-spread and
nominal spread will be a function of (i) the shape of the
term structure of interest rates and (ii) the characteristics of
the security such as provision for prepayment, time to
maturity etc.
• Further, nominal spread assumes that the same yield is
used to discount each cash flow but in case of Z-spread the
entire Treasury spot rate curve is considered.
• The difference between the Z-spread and the nominal
spread is greater for issues in which the principal is repaid
over time rather than only at maturity. Thus, the difference
between the nominal spread and the Z-spread will be
considerably greater for mortgage-backed and asset-backed
securities in a steep yield curve environment.

74. G-spread & I-spread
• In the United Kingdom, the United States and Japan, the benchmark rate
for fixed-rate bonds is a government bond yield. The yield spread in basis
points over an actual or interpolated government bond is known as the ‘G-
spread’.
• The spread over a government bond is the return for bearing greater
credit, liquidity and other risks relative to the sovereign bond.
• The yield spread of a specific bod over the standard swap rate in that
currency of the same tenor is known as the I-spread or Interpolated spread
to the swap curve.
• This yield spread over Libor allows comparison of bonds with different
credit and liquidity risks against an interbank lending benchmark.

75. Z-Spread
(Question)
• The yield to maturity on a quoted semi-annual basis
on 6-month, 1 year and 1.5-year T-bills are 2.80%,
3.20% and 4.02%.
a. If a 1.5-year semi- annual pay corporate bond
with a 7% coupon is selling for 102.395, what is the
spread over the Treasury note(G-spread) for this bond,
if a 1.5-year, 4% Treasury note is selling at par?
b. Is the zero-volatility spread (in basis points) equal
to:
a) 127 bps
b) 130 bps
c) 133 bps

76. Z-Spread (Question)
• Compute the YTM on the corporate bond:
N= 3; PV= -102.395; PMT = 3.5; FV=100; CPT: I/Y = 2.6588*2 = 5.32%
G-spread = YTM(Bond) – YTM( Treasury)
= 5.32%-4% = 1.32%
• Solve for the zero-volatility spread by setting the present value of the bond’s cash flows equal to the bond’s price, discounting
each cash flow by the Treasury spot rate plus a fixed Z-spread.
102.295 =
3.5
(1 +
0.028 + 𝑍𝑆
2
)1
+
3.5
(1 +
0.032 + 𝑍𝑆
2
)2
+
103.5
(1 +
0.0402 + 𝑍𝑆
2
)3
• Substitute each given value (i.e. 127 bps, 130 bps and 133 bps) in the above equation and a Z-spread of 133 bps will produce a
value closest to the bond’s price

77. Option Adjusted
Spread
• The option adjusted spread (OAS) is a constant spread that when
added to the interest rates used to discount the cash flows
produces a theoretical value of the bond that is equal to the market
price of the bond.
• It is an alternative way of expressing the difference that lies
between the theoretical value and the observed market price, i.e. in
the form of a yield spread rather than a difference in prices.
• In general, when the option cost is positive, this means that the
investor has sold an option to the issuer or borrower (callable
bonds and mortgage-backed securities).
• A negative value for the option cost means that the investor has
purchased an option from the issuer (putable bond).
• Z-spread =OAS (Option adjusted Spread) + Option Cost

78. Yield Spread Measures
Spread Measures Benchmark Reflects
Compensation for:
Nominal Treasury Yield Curve Credit Risk. Option
Risk, Liquidity Risk
Zero-volatility Treasury Spot Curve Credit Risk. Option
Risk, Liquidity Risk
Option-adjusted Treasury Spot Rate
Curve
Credit risk, Liquidity
Risk

79. L. Riding the Yield Curve
• In an upward sloping interest rate structure, investors seeking superior returns may pursue a
strategy called ‘riding the yield curve’ or a.k.a. ‘rolling down the yield curve’.
• Under this strategy, an investor will purchase bonds with maturities longer than his investment
horizon.
• In an upward-sloping yield curve, shorter maturity bonds have lower yields than longer maturity
bonds.
• If the yield curve remains unchanged over the investment horizon, riding the yield curve strategy
will produce higher returns than a simple maturity matching strategy, increasing the total return
of a bond portfolio.
• The greater the difference between the forward rate and the spot rate, and the longer the
maturity of the bond, the higher the total return.

80. Riding the Yield Curve
• Suppose an investor is contemplating the
purchase of a 5-year bond. Now, he has two
options, either he can buy a 5-year par bond
with yield of 3% with no capital gains or he
can purchase a 30-year bond for 63.67, hold it
for five years and sell it for 71.81, earning an
additional return beyond the 3 % coupon over
the same period (assuming the yield curve
does not change over the investment horizon).
Price of a 3% Annual Pay Bond
Maturity Yield(%) Price
5 3 100
10 3.5 95.84
20 4.5 80.49
25 5 71.81
30 5.5 63.67