bond market maths

8/4/2019 Bond Market Maths

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Group Members

Swarana Biyani ---- 01

Sanket Desai ------02

Rohan Jadhav -----08

Shama Lonare ---- 14

Bond Market Maths

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No Topic Slide Number

1 Features Of Debt Securities. 3---8

2 Bond Sector and Instruments. 9 -- 23

3 Convexity and Duration. 24 -- 37

4 Yield Spread. 38 -- 65

5 Yield Measures. 65 -- 76

22

Index


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Bond Indenture:- It is the contract which specifies allthe rights and obligation of Issuer and holder of the debt

security.

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Straight (Option free bond):- Consider a treasury bond

with 6% coupon and matures in 5 years in amount of 1000 Rs.

It will make 4 payments if 60 Rs annually for 4 years and a

payment of 1060 at the end of 5th year.

Zero coupon Bond:- They do not pay periodic interest, insteadthey are issued at discount to par value and pay the par values at the

end of maturity.

Eg A treasury bond issued at 98.5$ with a par value of 100% willpay 100$ at the end of maturity with effective return of 2.5%.

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Step Up Notes :- A bond that pays an initial coupon rate for the

first period, and then a higher coupon rate for the following periods.

For example a five-year bond may pay a 4% coupon for the first twoyears of its life and a 6% coupon for the final three years.

Deferred coupon bond:- The initial interest/coupon payments aredeferred for a period. The coupon payments accumulate at compound rate

till deferred period after which they are paid as lump sum. After deferredperiod the bond start paying regular coupon payments.

Eg A bond with face value of 100Rs and coupon rate of 5%, deferred

period of 5 years will pay 34.01Rs at the end of 5 years and will startmaking annual payments if 5Rs thereafter. 55


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Caps and Floors.:- The upper limit which is a cap puts

maximum limit on the interest rate paid by the issuer. Floor is thelower limit which puts minimum on the interest rate received by theowner.

Prepayment option:- This option gives the issuer the right topre pay the principle and hence in effect constitutes a pre paymentrisk to the holder.

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Put Option:- It gives the right to the bond holders to sell the

bond to the issuer at a specified price. If the bond price fallsbelow certain level the bond holder can force the issuer to buyback the bond at a specified price.

As the put option gives the holder an advantage the yields onsuch bonds are lower than straight bonds.

Eg 6.72%GS2012 bond issued in 2002 with the put optionexercisable after every 2.5 years.

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Call option:- This option gives the right to the issuer to buy back

the bond at a specific price. If the bond prices rise above certain levelthe issuer can buy back the bonds at a specified price.

Since this option gives the issuer an advantage the yields on suchcallable bonds are greater than straight bonds.

The same 6.72%GS2012 bond had a call option whereby the Govtcould call in the bonds after every 2.5 years.

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Click to edit Master subtitle style

BOND SECTOR

&INSTRUMENTS

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Corporate Debt Securities

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UNSECURED DEBT:

They are not backed by any collateral.

Referred as debentures

If pledged assets generate excess funds they areused for unsecured debt.

CREDIT ENHANCEMENT :

Guarantee given that the corporate debt will berepaid.

Third party

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Fixed rate bonds have a coupon that remainsconstant throughout the life of the bond.

Floating rate notes (FRNs)

Zero-coupon bonds

Inflation linked bonds

Asset-backed securities

Registered bond

Types Of Bonds

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Subordinate bonds

Perpetual bonds

Bearer bonds

Treasury bonds

Lottery bonds

Municipal bonds

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V l ti Of D bt


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Basic steps

1) Estimate the cash flow

2) The interest rate

3) Present value) Difficulties involved:

1) Principal repayment in unknown2) Coupon payments

Valuation Of DebtSecurities

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Valuation of bond when1) single coupon rate

PV= FV/(1+r)n

E.g. cash inflow= Rs100

N= 10 years

Future value= Rs 1000

Discount rate= 8%

100/(1.08)+ 100/(1.08)2+ 100/(1.08)3.

+100/(1.08)10 =Rs(1134.20)

Computation

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2) Zero coupon bonds

Bond value= maturity value/(1+i)n*2

E.G. N=10 years (semi annual)

YTM =8%

Face value= Rs1000

Zero coupon Bond

1000/(1+.08/2)10*2=1000/(1.04)20=Rs456.39

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Securitization is the process ofconversion of existing assets or future

cash flows into marketable securities.

It deals with the conversion of assetswhich are not marketable into

marketable ones.

Securitization

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The SPV is a separate entity formed

exclusively for the facilitation of thesecuritization process and providing fundsto the originator.

The SPV will act as an intermediary whichdivides the assets of the originator intomarketable securities.

These securities issued by the SPV to theinvestors and are known as pass-through-certificates (PTCs)

Special Purpose Vehicle

2020


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Assemble an entire portfolio of credit riskexposures, segment that exposure intotranches with unique risk/return/maturity

profiles, which are then transferred or soldto investors.

Securitization issues backed by debt

obligations are called CDO CDOs reference (underlying) portfolio can

be assembled with physical cash flowassets such as bonds, loans, MBS, ABS etc

o a era ze eObligation

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M t B k d


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Securitization issues backed bymortgages are called MBS

Mortgage BackedSecurities (MBS)

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A t B k d S iti


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Securitization issues backed byconsumer-backed products - car loans,

consumer loans and credit cards,among others are called ABS

Assets Backed Securities

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Interest Rate Risk

- Uncertainty about bond prices due to change

in market interest rates

Call Risk

-The risk that the bond will be called prior to

maturity under the terms of call provision and thefunds must be reinvested at the then current yield.

Prepayment Risk

-The uncertainty about the amount of bond

principal that will be repaid prior to maturity

Risk Associated

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Liquidity Risk

-The risk that an immediate sale will result in a pricebelow fair value

Exchange Rate Risk-The risk that the domestic currency value of bond

payments in a foreign currency will decrease

Event Risk

-The risk of decrease in a security value fromdisasters, corporate restructuring, or regulatory changesthat negatively affect the firm

Sovereign Risk

Risk Associated

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Bonds Price Relative To


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Bonds Price Relative ToPar

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Characteristic Interest Rate Risk Duration

Maturity Up Interest Rate Risk Up Duration Up

Coupon Up Interest Rate RiskDown

Duration Down

Add a Call Interest Rate RiskDown

Duration Down

Add a Put Interest Rate RiskDown

Duration Down

Bond Characteristic & Interest RateRisk

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The term duration has a special meaning in the

context of bonds. It is a measurement of how long,in years, it takes for the price of a bond to berepaid by its internal cash flows.

For each of the two basic types of bonds theduration is the following:

1. Zero-Coupon Bond Duration is equal toits time to maturity.

2. Vanilla Bond - Duration will always be lessthan its time to maturity.

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Duration

Duration of a Zero


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Duration of a ZeroCoupon Bond

The entire cash flow of a zero-coupon bond occursat maturity, so the fulcrum is located directlybelow this one payment.

Duration of a Vanilla


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Consider a vanilla bond that pays couponsannually and matures in five years.

The straight bond pays coupon payments

throughout its life and therefore repays the full3030

Duration of a Vanillaor Straight Bond

Factors Affecting


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Factors AffectingDuration

Duration is decreasing as time moves closer tomaturity

But duration also increases momentarily on the

Duration: Other


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Other factors that affect abond's duration: the couponrate and its yield

Bonds with high couponrates and, in turn, highyields will tend to have

lower durations than bondsthat pay low coupon rates oroffer low yields.

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Duration: Otherfactors


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Types of durations are :-

Macaulay duration

Modified duration

Effective duration

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Types of Duration
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The formula usually used to calculate a bond's

basic duration is the Macaulay duration, which wascreated by Frederick Macaulay in 1938

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Macaulay Duration

n = number of cash flows

t = time to maturityC = cash flowi = required yieldM = maturity (par) valueP = bond price


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Example 1: Betty holds a five-year bond with

a par value of $1,000 and coupon rate of 5%.For simplicity, let's assume that the couponis paid annually and that interest rates are

5%. What is the Macaulay duration of thebond?

3535

Example

= 4.55 years


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Modified duration is a modified version of the

Macaulay model that accounts for changinginterest rates.

Modified formula shows how much the duration

changes for each percentage change in yield

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Modified Duration


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Betty's bond and run through the calculation of her

modified duration. Currently her bond is selling at$1,000 or par, which translates to a yield tomaturity of 5%.

We calculated a Macaulay duration of4.55 Years.

If the bond's yield changed from 5% to 6%, theduration of the bond

3737

Example

= 4.33years


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Cash flows from securities with embeddedoptions or redemption features will changewhen interest rates change.

For calculating the duration of these types ofbonds, effective duration is the most

appropriate.

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Effective Duration


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For any given bond, a graph of the

relationship between price and yield isconvex.

The degree to which the graph is curvedshows how much a bond's yield changes inresponse to a change in price.

Convexity

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Convexity and Duration

The exact point where thetwo lines touch representsMacaulay duration.

The yellow portions of thegraph show the ranges in

which using duration forestimating price would beinappropriate.

Convexity shows how much4040


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A bond with greater convexity is less affected

by interest rates than a bond with lessconvexity.

Bonds with greater convexity will have ahigher price than bonds with a lowerconvexity, regardless of whether interestrates rise or fall.

Properties of Convexity

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Graphical Illustrations

If two bonds offer the same duration and yield butone exhibits greater convexity, changes ininterest rates will affect each bond differently4242


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Kinds of Convexities Plain Vanilla bond

Positive convexity.

The price-yield curve

will increase as yielddecreases, and viceversa.

As market yields

decrease, the durationincreases .

Callable bond

Negative convexity atcertain price-yieldcombinations.

Negative convexitymeans that as marketyields decrease, durationdecreases as well.

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Convexity allows the


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To better comprehend the way in whichduration is best measured

how changes in interest rates affect theprices of both plain vanilla and callablebonds.

Convexity allows theinvestor :

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UnderstandingYield Spreads

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Yield Curve & Various


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Yield Curve:

Yield curve gives the relationship between the

maturity and the yield of the bond.

Various shapes of Yield Curve

1. Normal or Upward Sloping

2. Inverted or Downward Sloping

3. Flat

4. Humped

Yield Curve & VariousShapes of Yield Curve

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Theories of term


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Pure Expectation Theory

The yield for a particular maturity is an average

of the short-term rates that are expected in thefuture.

If short-term rates are expected to rise in future,interest rate yields on longer maturities will be

higher and the yield curve will be Upward sloping.

Theories of termstructure of interest rates

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Theories of term


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Liquidity Preference Theory

Investors require a risk premium for holding long

term bonds. Interest rate risk is greater for longermaturity bonds.

The size of the liquidity premium depends on howmuch additional compensation investor requires

to take on a greater risk for long term bonds.


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Theories of term


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Theories of term


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Market Segmentation Theory

Investors & borrowers have different pref. for

different maturity ranges.

The supply and demand determines equilibriumyields for various maturity ranges.

Eg: Life insurers & pension funds may prefer longmaturities due to their long term liablities.


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Theories of term


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Preferred Habitat Theory

This is somewhat weaker version ofMarket

Yields depends on the supply and demand

for various maturityranges but investors can be induced to

move from theirpreferred maturity ranges when yields are

sufficiently higher in


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Q. An annual-pay bond of 1000 with 10%coupon rate and 3 years to maturity.

Suppose the spot rates are:1 year = 8%

2 year = 9%

3 year = 10%

Find the value of bond?

Spot Rate

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Solution:

Value of bond =

100/1.08 + 100/(1.09) + 1100/(1.10)

= 1003.21Spot rate is the discount rate for individual

future payments.

Spot Rate

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Yield spread is the difference between theyields on two bonds or two types of bonds.

Three different yield measures are:

1. Absolute Yield Spread

2. Relative Yield Spread

3. Yield Ratio

Yield Spread

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i ld d


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Absolute Yield Spread

It is the difference between yields on two

bonds.

It is expressed in basis point (100th of 1%)

Absolute yield spread = yield on thehigher yield bond yield on the lower yieldbond

Yield Spread

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i ld S d


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Relative yield spread

It is absolute yield spread expressed as a

percentage of the yield on the benchmark bond.

Relative yield spread = absolute yield

spread----------------------------------

benchmark bond yield

Yield Spread

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Yi ld S d


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Yield ratio

It is the ratio of the yield on the subject bond to

the yield on the benchmark bond

Yield ratio = subject bond yield

-----------------------------------

benchmark bond yield

Yield Spread

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Yi ld S d


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Q. Consider two bonds X & Y. Their resp. yields are6.50% & 6.75%.

Using bond X as the benchmark bond,

Compute the absolute yield spread, relative yieldspread & yield ratio.

Yield Spread

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Yi ld S d


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Solution:

Absolute yield spread = 6.75% - 6.50% = 0.25%

or 25 basis

Relative yield spread = 0.25% / 6.50% = 3.8%

Yield ratio = 6.75% / 6.50% = 1.038

Yield Spread

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Aft t i ld


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Computing after-tax yield on taxablesecurities

After-tax yield = taxable yield x (1- marginaltax rate)

Q. Calculate after-tax yield on a corporate bondwith yield of 10% for an investor with 40%marginal tax rate?

After-tax yield

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Aft t i ld


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Solution:

After-tax yield = 10% (1-40%)

= 10% (1-0.4)

= 6% after tax

After-tax yield

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T i l t i ld


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Taxable-equivalent yield:

It is a yield a particular investor must earn on a

taxable bond to have the same after-tax returnthe investor may receive from a tax-exempt bond.

Taxable-equi. yield = tax free yield--------------------------------

(1-marginal tax rate)

Tax-equivalent yield

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T i l t i ld


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Q. Municipal bond offers yield of 4.5%. If an

investor consider buying taxable Treasury securityoffering 6.75% yield. Should the investor buy the

Treasury security or municipal bond, given themarginal tax rate is 35%.


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T i l t i ld


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Solution:

Taxable-equivalent yield = 4.5%

-------------------------

(1 - 0.35)

= 6.92%

Therefore, municipal bond is preferred overtreasury security as taxable-equivalent onmunicipal bond is higher than that of treasurysecurity.


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Yi ld M


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Current yield is the simplest measure of valuing a bond

Current yield = annual cash interest payment / Bond Price

Consider a 20 year bond with a face value of $ 1000 which

pays 6% annually, the bond is trading at $ 802.07.

The current yield is 6% * 1000 / 802.07 = 7.48%.

Yield Measures

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Yi ld M


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Yield to Maturity : - It values the bond on the PV of thefuture payments.

Bond Price = ( CP (T) / ( 1 + YTM) ) + ( CP (2) / ( 1 + YTM )^2 ).

+( CP ( N ) + par / ( 1 + YTM )^ N).

YTM and price give the same information, given YTM wecan calculate price and given price we can calculate YTM.

Yield Measures.

6666

Yi ld M


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Consider an annual pay 20 year $ 1000par value bond with a 6% coupon ratetrading at a price of $ 802.07 Calculate

the YTM.6767

Yield Measures.

Yi ld M


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Yield to call : - It is used to measure yield on callablebonds which are trading at premium.

Typically callable bonds cant be called for a fixedperiod of time after issuance.

If within that period the bond price goes above thecall price yield to call is used.

Yield Measures.

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Yi ld M


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Consider a 10% semiannual pay bond

with a current price of $112 that can becalled in 5 years at 102. Calculate the

YTM and YTC.6969

Yield Measures.

Yi ld M


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Yield to put : - It measures the yield of bonds having putoption which are trading a discount.

Typically put option in a bond cant be invoked for a fixedperiod of time after issuance.

If within that period the bond price starts trading at

discount to the actual value of the bond then yield to put isused.

The yield to put is higher than YTM due to lower bondrice.

Yield Measures.

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Yield Meas res


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Consider a 3 year, 6%, $ 1000 semiannualpay bond. The bond is selling for a price of925.40. The opportunity for put is at par in2 years. Calculate YTM and YTP. 7171

Yield Measures.

Reinvestment Income


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If a bond holder holds a bond until maturity and

reinvests the interest payments then the totalamount generated by the bond over the life is

1. Bond Principle.

2. Interest payments.

3. Interest on reinvested Income.

. If the reinvestment income is less than YT, of thebond the total return generated will be less than

YTM.

Reinvestment Income.

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Reinvestment Income


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If you purchase a 6% 10 year bond how much

reinvestment income must be generated over its lifeto provide the investor with a compound return of 6%on annual basis. The par value of bond is Rs 100.

If the investment yields 6% compounded annualreturn the total value will be 100 * ( 1.06 ) ^ 10 =179.084.

The bond will make 10 annual payments of Rs 6 andan end payment of Rs 100.

There fore it will make total payment of 160RS

The required reinvestment Income is 179.084 160 =

Reinvestment Income.

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The End


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Thank You

The End

bond market maths

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