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BondsBondsA bond is a contract that requires the borrower to A bond is a contract that requires the borrower to
pay the interest income to the lender.pay the interest income to the lender.This specific rate of interest is known as coupon This specific rate of interest is known as coupon
rate.rate.Generally stocks are considered risky but bonds are Generally stocks are considered risky but bonds are
not.not.But this is not fully correct.But this is not fully correct.Bonds do have risk.Bonds do have risk.But the nature & types of risks may be different.But the nature & types of risks may be different.So we can discuss the nature of bonds with respect So we can discuss the nature of bonds with respect
to risk.to risk. SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
BONDS NATUREBONDS NATURE
1.1. Interest Rate Risk:Interest Rate Risk:• Variability in the return from the debt instrument to investor is Variability in the return from the debt instrument to investor is
caused by the changes in the market interest rate.caused by the changes in the market interest rate.• It is due to the relationship between coupon rate & market rate.It is due to the relationship between coupon rate & market rate.
2.2. Default Risk:Default Risk:• The failure to pay the agreed value of the debt instrument by the The failure to pay the agreed value of the debt instrument by the
issuer.issuer.
3.3. Marketability Risk:Marketability Risk:• Variation in return causes difficulty in selling the bonds quickly Variation in return causes difficulty in selling the bonds quickly
without having any substantial reason for price concession.without having any substantial reason for price concession.
4.4. Call-ability Risk:Call-ability Risk:• There is always an uncertainty regarding the maturity period, There is always an uncertainty regarding the maturity period,
because issuer can call the bond any time by redeeming it.because issuer can call the bond any time by redeeming it.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Bond BasicsTwo basic yield measures for a bond are its coupon rate and its current yield.
Coupon Rate =
Current Yield =
Annual Coupon
Par Value
Annual Coupon
Bond Price
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
The Bond Pricing Formula Recall: The price of a bond is found by adding together
two components1. The present value of the bond’s coupon payments and 2. The present value of the bond’s face value. The formula is:
Bond Price =
Where,C represents the annual coupon payments (in Rs),FV is the face value of the bond (in Rs), and M is the maturity of the bond, measured in years.
CYTM
1(1+YTM/
2)2M
+FV
(1+YTM/2)2M
1-
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Example:What is the price of a straight bond with:
Rs.1,000 face value, coupon rate of 5%, YTM of 6%, and a maturity of 10 years?
Bond Price =
=
= (833.33 X 0.44632) + 553.68= Rs.925.61
CYTM 1-
1(1+YTM/
2)2M
+FV
(1+YTM/2)2M
500.06 1-
1(1+0.06/2)2X10 +
1000(1+0.06/2)2X10
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Premium and Discount Bonds Bonds are given names according to the
relationship between the bond’s selling price and its par value.
Premium bonds: price > par value YTM < coupon rate
Discount bonds: price < par value YTM > coupon rate
Par bonds: price = par value YTM = coupon rate
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Calculating Yield to Maturity It is a single discount factor that makes
present value of future cash flows from a bond equal to the current price of the bond.
or YTM is the rate of return, which an investor
can expect to earn if the bond is held till maturity.
To find out YTM the present value technique is adopted i.e.
Present value =
Where,y=YTM
Coupon1
(1+y)1
+Coupo
n2
(1+y)2
+Couponn +
M.V.(1+y)n
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Example A 10 years bond with 4.5% coupon rate &
maturity value Rs. 1000/- selling at Rs.900/-. What is its YTM?
Solution Using Alternative formula:
YTM =
=
=
Annual coupon interest rate + (Discount/Years to maturity)(current price + par price)/2
45+(100/10)(900+1000)/2
45+(100/10)(900+1000)/2
45+10950
= 55950
= 5.79%=
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Calculating Yield to Maturity Suppose we know the current price of a bond,
its coupon rate, and its time to maturity. How do we calculate the YTM?
We can use the straight bond formula, trying different yields until we come across the one that produces the current price of the bond.
1083.17=
This is tedious. So, to speed up the calculation, financial calculators and spreadsheets are often used.
90YTM 1- 1
(1+YTM/2)2X5+
1000(1+YTM/
2)2X5
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Yield to Call Yield to call (YTC) is a yield measure that
assumes a bond will be called at its earliest possible call date.
The formula to price a callable bond is:
Where,C is the annual coupon (in Rs),CP is the call price of the bond,T is the time (in years) to the earliest possible call
date,YTC is the yield to call, with semi-annual coupons.
CYTC
1(1+YTC/2)2T
+ CP(1+YTC/2)2T
1-Callable Bond Price=
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Calculating the Price of a Coupon Bond A Bond traded on 1 March 2008 matures in 20 years on
1 March 2028. Assuming an 8 percent coupon rate and 7% yield to maturity. What is the price of this Bond?
Solution using excel function PRICE =PRICE("3/1/2008", "3/1/2028",0.08,0.07,100,2,3)
Answer = 110.6775For a bond with Rs.1000 face value multiply the price by 10 to get Rs.1106.78
This function uses the following:=PRICE (Now, Maturity, Coupon, YTM,100,2,3)Where,100 indicates the redemption value as a percentage of
face value2 indicates semi annual coupons.3 specifies an actual day count with 365 days per year.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Calculating Yield to maturity of a Bond A Bond traded on 1 March 2008 matures in 8 years
on 1 March 2016. Assuming an 8 percent coupon rate and a price of Rs.110. What is the yield to maturity of this Bond?
Solution using excel function YIELD=YIELD(“3/1/2008",“3/1/2016",0.08,110,100,2,3)
Answer = 6.38%
This function uses the following:=yield(Now, Maturity, Coupon, Price,100,2,3)Where,
Price is entered as a percentage of face value100 indicates the redemption value as a percentage of face value2 indicates semi annual coupons.3 specifies an actual day count with 365 days per year.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Calculating Yield to call of a Bond A Bond traded on 1 March 2008 matures in 15 years on
1 March 2023. And may be called any time after 1 March 2013 at a call price of Rs.105. The Bond pays an 8.5% coupon and currently trades at par. What are the yield to maturity & yield to call for this bond?
Yield to maturity is based on 2023 maturity and current price of Rs.100
=YIELD(“3/1/2008",“3/1/2023",0.085,100,100,2,3) Answer = 8.5%
Yield to call is based on 2013 maturity and current price of Rs.100
=YIELD(“3/1/2008",“3/1/2013",0.085,100,105,2,3) Answer = 9.308%
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Bond Theorems
Theorem 1:If the market price of the bond increases,
the yield would decline and vice versa
Bond B
Rs.1000
10%
2 Years
Rs.1035.66
Bond A
Rs.1000
10%
2 Years
Rs.874.75
Example
Par Value
Coupon rate
Maturity period
Market Price
Yield18% 8%
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Theorem 2: If bond’s yield remains the same over its life, the
discount or premium depends on the maturity period.
Thus the Bond’s with short term to maturity sells at a lower discount than the bond with a long term to maturity.
Example
Par Value
Coupon rate
Yield
Maturity period
Market Price
Discount
Bond B
Rs.1000
10%
15%
3 Years
Bond A
Rs.1000
10%
15%
2 YearsRs.918.71
Rs.885.86
Rs.81.29 Rs.114.14
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Theorem 3:If bond’s yield remains constant over
its life.The discount or premium decrease at
an increasing rate as its life gets shorter.
It happens due to the concept of time value of money.
For example:If Investor get a rupee at T+5 period it would value lesser if he get he get it at T period
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Theorem 4:A rise in the bond’s price for a decline in the
bond’s yield is greater than the fall in the bond’s price for a raise in the yield.
For Example: A bond of 10% coupon rate, maturity period of five
years with face value of Rs.1000.
IF the yield declines by 2%, that is to 8% then the bond price will be Rs.1079.87. (using bond pricing formula, slide-2)
If the yield increases by 2% the, the bond price will be Rs.927.88. (using bond pricing formula, slide-2)
Now fall in yield has resulted in raise of Rs.79.87 but raise in the yield caused a variation of Rs.72.22 in the price. SANDEEP KAPOORSANDEEP KAPOOR
MIET, MEERUTMIET, MEERUT
Theorem 5: The change in the price will be lesser for a
percentage change in bond’s yield if its coupon rate is higher.
Example
Coupon Rate
Yield
Maturity period
Price
Face Value
Yield Raise (YR)
Price after YR
% change in Price
Bond A
10%
8%
3 Years
Rs.105.15
Rs.100
Rs.100 Rs.100
Bond B
8%
8%
3 Years
1% 1%
Rs.102.53
Rs.97.47
2.4% 2.53%SANDEEP KAPOORSANDEEP KAPOOR
MIET, MEERUTMIET, MEERUT
Term Structure of Interest Term Structure of Interest RateRateThe relationship between the yield and The relationship between the yield and
time is called term structure.time is called term structure. It is also known as yield curve.It is also known as yield curve. In analyzing the effect of maturity on yield In analyzing the effect of maturity on yield
all other influences are held constant.all other influences are held constant.The maturity dates for bonds are The maturity dates for bonds are
different.different.But the risks, tax liabilities & redemption But the risks, tax liabilities & redemption
possibilities are similar.possibilities are similar.There are some theories that explains the There are some theories that explains the
term structure of interest rates.term structure of interest rates.SANDEEP KAPOORSANDEEP KAPOOR
MIET, MEERUTMIET, MEERUT
Theories of term structure of Interest RatesTheories of term structure of Interest Rates
1.1. Expectation TheoryExpectation Theory
2.2. Liquidity Preference TheoryLiquidity Preference Theory
3.3. Segmentation TheorySegmentation Theory
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Expectation TheoryExpectation TheoryThis theory is based on the This theory is based on the
expectations of the investors.expectations of the investors.Under this theory the shape of yield Under this theory the shape of yield
curve is studied.curve is studied.As it gives an idea of future interest As it gives an idea of future interest
rate change & economic activity.rate change & economic activity.There are three main types of yield There are three main types of yield
curve shapes i.e.curve shapes i.e.NormalNormalFlatFlatInvertedInverted SANDEEP KAPOORSANDEEP KAPOOR
MIET, MEERUTMIET, MEERUT
Normal Yield CurveNormal Yield Curve If investor expects that there would be a If investor expects that there would be a
continuous rise in market interest rates.continuous rise in market interest rates.Then the bond’s price will decrease.Then the bond’s price will decrease.Thus the yield will increase.Thus the yield will increase.Graphical presentationGraphical presentation
Years to Maturity
Yie
ld t
o
matu
rity
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Flat Yield CurveFlat Yield Curve If investor expects that there would be no If investor expects that there would be no
change market interest rates.change market interest rates.Then the bond’s price will remain constant.Then the bond’s price will remain constant.Thus the yield will also remain constant.Thus the yield will also remain constant.Graphical presentationGraphical presentation
Years to Maturity
Yie
ld t
o
matu
rity
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Inverted/Falling yield CurveInverted/Falling yield Curve
Years to Maturity
Yie
ld t
o
matu
rity
If investor expects that there would be If investor expects that there would be a decline in market interest rates.a decline in market interest rates.
Then the bond’s price will increase.Then the bond’s price will increase.Thus the yield will decrease.Thus the yield will decrease.Graphical presentationGraphical presentation
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Conclusion of Expectation Conclusion of Expectation TheoryTheory
There are only three There are only three types of returns i.e.types of returns i.e.
1.1.NormalNormal
2.2.Flat Flat
3.3. InvertedInvertedAs indicated by the As indicated by the
graphgraphYears to Maturity
Yie
ld t
o
matu
rity
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Liquidity Preference TheoryLiquidity Preference TheoryAccording to this theory Investor prefers the According to this theory Investor prefers the
liquidity.liquidity.So, he prefers short term bonds over long term So, he prefers short term bonds over long term
bonds due to liquidity.bonds due to liquidity.If no premium exists for holding the long term bondIf no premium exists for holding the long term bondInvestor would prefer to hold short term bonds.Investor would prefer to hold short term bonds.Thus they must be motivated to buy the long term Thus they must be motivated to buy the long term
bond by some sort of premium.bond by some sort of premium.As the forward rates are actually higher than the As the forward rates are actually higher than the
projected interest rate.projected interest rate.SANDEEP KAPOORSANDEEP KAPOOR
MIET, MEERUTMIET, MEERUT
Segmentation TheorySegmentation Theory According to this theory liquidity can not be the main According to this theory liquidity can not be the main
consideration for all classes of investors.consideration for all classes of investors. For exampleFor example
Insurance companiesInsurance companiesPension FundsPension FundsRetired PersonsRetired Persons
All of above prefer the long term rather than short term All of above prefer the long term rather than short term securities to avoid the possible fluctuations in the interest rate.securities to avoid the possible fluctuations in the interest rate.
On the other hand there are corporate who prefers liquidity.On the other hand there are corporate who prefers liquidity. They prefer short term bondsThey prefer short term bonds Supply and demand for fund are segmented in sub-markets Supply and demand for fund are segmented in sub-markets
because of the preferred habitats of the individuals.because of the preferred habitats of the individuals. Thus yield is determined by the demand and supply of the funds.Thus yield is determined by the demand and supply of the funds.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
DurationDurationThe term duration has a special The term duration has a special
meaning in the context of bonds.meaning in the context of bonds. It is a measurement of how long (in It is a measurement of how long (in
years), it takes for the price of a bond years), it takes for the price of a bond to be repaid by its internal cash flow.to be repaid by its internal cash flow.
It is important to considerIt is important to considerAs bonds with higher durations carry As bonds with higher durations carry
more risk and have higher price volatility more risk and have higher price volatility than bonds with lower duration.than bonds with lower duration.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Types of Bonds w.r.t. Types of Bonds w.r.t. DurationDuration
For each of the two basic types of For each of the two basic types of bonds the duration is following:bonds the duration is following:
Zero coupon Bond:-Zero coupon Bond:-
Duration is equal to its time to Duration is equal to its time to maturitymaturity
Vanilla Bond:-Vanilla Bond:-
Duration will always be less than its Duration will always be less than its time to maturity.time to maturity.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Types DurationTypes DurationThere are four main types of duration There are four main types of duration
calculations.calculations.Each of which differ in the way they Each of which differ in the way they
account for factors such asaccount for factors such asInterest rate changesInterest rate changesBonds redemption featureBonds redemption feature
The four types of durations are:The four types of durations are:1.1.Macaulay durationMacaulay duration
2.2.Modified durationModified duration
3.3.Effective durationEffective duration
4.4.Key rate durationKey rate duration SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Macaulay DurationMacaulay Duration
The formula of Macaulay duration was The formula of Macaulay duration was created by Frederick Macaulay in created by Frederick Macaulay in 1938.1938.
It is calculated by adding the results of It is calculated by adding the results of multiplying the present value of each multiplying the present value of each cash flow by the time it is received and cash flow by the time it is received and dividing by the total price of security.dividing by the total price of security.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Macaulay DurationMacaulay Duration
Macaulay Duration=Macaulay Duration=
n Ƹ t X C n X Mt=1 (1+i)t (1+i)n
Price of the Bond
+
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Recall Recall Bond priceBond price
Bond price =Bond price =CYTM
1(1+YTM/
2)2M
+FV
(1+YTM/2)2M
1-
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Thus Macaulay Duration (MD)Thus Macaulay Duration (MD)
MD=MD=
n Ƹ t X C n X Mt=1 (1+i)t (1+i)n
+
CYTM
1(1+YTM/
2)2M
+FV
(1+YTM/2)2M
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
For ExampleFor Example
Mr. B holds a five year bond with a Mr. B holds a five year bond with a par value of Rs. 1000 and coupon rate par value of Rs. 1000 and coupon rate of 5%. For simplicity, let’s assume of 5%. For simplicity, let’s assume that the coupon is paid annually and that the coupon is paid annually and that interest rates are 5%. What is that interest rates are 5%. What is the Macaulay Duration of the Bond.the Macaulay Duration of the Bond.
The above formula is very complex. The above formula is very complex. Alternatively we can use the following Alternatively we can use the following table to find out Macaulay Duration.table to find out Macaulay Duration.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
SolutionSolution
Years (t)
1
2
3
4
5
Inflow
50
50
50
50
1050
PVF @5%
0.953
0.907
0.863
0.822
0.784
Inflow (t)
50
100
150
200
5250
P.V. of inflow (t)
47.6
90.7
129.45
164.4
4116
4569.15
Price of the Bond
(column 2 X4)47.645.3543.1541.1823.21000
Duration =
P.V. of inflow (t)Price of the Bond
4569.151000
=4.56 YearsSANDEEP KAPOORSANDEEP KAPOOR
MIET, MEERUTMIET, MEERUT
Modified DurationModified Duration It is a modified version of the Macaulay It is a modified version of the Macaulay
model that accounts for changing model that accounts for changing interest rates.interest rates.
As interest rate affect the yield.As interest rate affect the yield.The fluctuating interest rates will affect The fluctuating interest rates will affect
duration.duration.This modified formula shows that:This modified formula shows that:
How much the duration changes for each How much the duration changes for each percentage change in the yield.percentage change in the yield.
So there is an inverse relationship between So there is an inverse relationship between modified duration and change in yield.modified duration and change in yield.
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
FormulaFormula
Modified Duration =Modified Duration =
Modified Duration =Modified Duration =
Macaulay DurationYield to Maturity
Number of coupon periods per Year
1+
Macaulay DurationYTM
n1+
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
ExampleExample Let’s consider the example of Mr. B’s bond and run Let’s consider the example of Mr. B’s bond and run
through the calculation of his modified duration.through the calculation of his modified duration. Currently his bond is selling at Rs.1000/- or parCurrently his bond is selling at Rs.1000/- or par Which translates to yield to maturity of 5%.Which translates to yield to maturity of 5%. Recall, we calculated a Macaulay duration of 4.56Recall, we calculated a Macaulay duration of 4.56
Modified Duration =Modified Duration =
=4.33 years=4.33 yearsModified duration will always be lower than Macaulay Duration.Modified duration will always be lower than Macaulay Duration.
4.560.05
11+
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT
Effective DurationEffective Duration Modified duration assumes that the expected Modified duration assumes that the expected
cash flows will remain constant even if prevailing cash flows will remain constant even if prevailing interest rates change. Such as:interest rates change. Such as:
Option free fixed income bondsOption free fixed income bonds Effective duration is used when expected cash Effective duration is used when expected cash
flows also changes with a change in interest flows also changes with a change in interest rates.rates.
Effective duration requires the use of binomial Effective duration requires the use of binomial trees to calculate the option adjusted spread trees to calculate the option adjusted spread (OAS)(OAS)
There are entire courses build around just those There are entire courses build around just those two topics.two topics.
So calculations involved for effective duration are So calculations involved for effective duration are beyond the scope of our syllabusbeyond the scope of our syllabusSANDEEP KAPOORSANDEEP KAPOOR
MIET, MEERUTMIET, MEERUT
Key Rate DurationKey Rate Duration It is used for portfolios which consists of fixed It is used for portfolios which consists of fixed
income securities with differing maturities.income securities with differing maturities. It allows the duration of a portfolio to be calculated It allows the duration of a portfolio to be calculated
for one basis point change in interest rates.for one basis point change in interest rates. It calculates the spot durations of each of 11 key It calculates the spot durations of each of 11 key
maturities i.e.maturities i.e. 3 months, 1, 2, 3, 5, 7, 10, 15, 20, 25, and 30 years3 months, 1, 2, 3, 5, 7, 10, 15, 20, 25, and 30 years The formula for key rate duration is as follows:The formula for key rate duration is as follows:
The sum of the key rate duration is equal to the The sum of the key rate duration is equal to the effective duration.effective duration.
Price of security after 1% decrease in yield - Price of security after 1% increase in yield
2 X (Initial price of security) 1%
SANDEEP KAPOORSANDEEP KAPOORMIET, MEERUTMIET, MEERUT