My BOnds

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


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-


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


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


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


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%=


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


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=


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.


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.


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%


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%


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


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


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


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


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


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


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


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


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


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


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


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.


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.


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.


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.


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

+


Recall Recall Bond priceBond price

Bond price =Bond price =CYTM

1(1+YTM/

2)2M

+FV

(1+YTM/2)2M

1-


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


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.


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


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.


FormulaFormula

Modified Duration =Modified Duration =


Macaulay DurationYield to Maturity

Number of coupon periods per Year

1+

Macaulay DurationYTM

n1+


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


=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+


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


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%


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