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Bangalore

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Bond Markets: Advanced PerspectivesPart-03A

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We have assumed earlier that we are on acoupon payment date.

It implies that the next coupon is exactly oneperiod away.

How do we value a bond between twocoupon dates?

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Calculate the actual no. of days between thedate of valuation and the next coupon date.

Include the next coupon date.

But do not include the starting date.Or vice versaLet us call this interval N 1.

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Calculate the actual no. of days between thepreceding coupon date and the next coupondate.

Include the ending date but exclude the startingdate or vice versaLet us call this time interval as N 2.

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The next coupon is then k periods awaywhere

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Month No. of Days

September 15October 31

November 30

December 31January 15TOTAL 122

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Month No. of DaysJuly 16

August 31September 30

October 31

November 30December 31

January 15

TOTAL 1848/28/2014 10

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K = 122/184 = .6630This is called the Actual/Actual method andis often pronounced as the Ack/Ack method.

It is the method used for Treasury bonds in theU.S.It is the method used for Corporate bonds inIndia.

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Wall Street professionals will then price thebond using the following equation.

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In our example

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The approach used by the Treasury to valueT-bonds is different.

The Treasury uses a simple interest approach forthe fractional first period.

This is known as the Moosmuller yieldcalculation method.

It is also used in some German markets

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The Treasury will use the followingequation.

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The Treasury approach will always give alower price

For a fractional period simple interest will alwaysgive a larger discount factor

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The Actual/Actual method is applicablefor Treasury bonds in the U.S.For corporate bonds in the U.S. we use

what is called the 30/360 NASDmethod.The number of days between successive coupondates is always taken to be 180.

Each month is considered to be of 30 days.

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The # of days from the valuation date till thenext coupon date is calculated as:The start date is defined as

D1 = (month 1, day 1,year 1)The ending date is defined asD2 = (month 2,day 2,year 2)

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The number of days is calculated as360(year 2 year 1) + 30(month 2 month 1) +(day 2 day1)

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If day 1 = 31 then set day 1 = 30If day 1 is the last day of February, then setday 1 = 30

If day 1 = 30 or has been set equal to 30, thenif day 2 = 31, set day 2 = 30

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Assume that the bond considered earlier wasa corporate bond and not a Treasury bond.

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In this convention, if day 2 = 31, then it isalways set equal to 30.So the additional rules are:

If day 1 = 31 then set day 1 = 30If day 2 = 31 then set day 2 = 30This is the convention used in India forGovernment securities

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The denominator in this convention willconsist of 365 even in leap years.

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This is used for Japanese Government Bonds(JGBs)

It is similar to the Actual/365 method.

The difference is that in this case, the extra dayin February is ignored in leap yearsWhile calculating both the numerator and thedenominator.

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It is identical to Actual/365 for a coupon periodthat does not include days within a leap year.However for a period that includes days fallingwithin a leap year, the day count is given by:

#of days falling within the leap year ______________________________ +366

#of days not falling within the leap year _________________________________

365

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This is a simple variant of Actual/365.This is the convention used for money marketinstruments in most countries.

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A bond with one coupon remaining (atmaturity) is akin to a ZCBThus these bonds are usually discounted on a

simple interest basis for the fractional periodthat remains

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A T-bond has a face value of $1,000A coupon of 8% payable semi-annuallyThe maturity date is 15 January 2014

Today is 15 September 2013The quoted price is 99-12The day-count convention is Actual/Actual

The fractional period remaining = 122/184 =0.6630The AI = 40x62/184 = 13.4783The dirty price is 1007.2283

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If we compute the YTM as1007.2283 = 1040/(1+i) 0.6630

i = 4.9478

Thus the annual YTM is 9.8956%However the market in the US wouldcompute as follows.1007.2283 = 1040/(1+i x 0.663/2)Thus i = 9.8149%

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Price of a bond is the PV of all the cashflows that the buyer will receive

The seller is compensated for all the cash flowsthat he is parting with.This includes the amount due to the seller forparting with the entire next coupon

Although he has held it for a part of the currentcoupon period.

This compensation is termed asAccrued Interest

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The fraction of the next coupon that ispayable is calculated as per the day-countconvention

That is for U.S. Treasury bonds the Actual/Actualmethod is used Whereas for U.S. corporate bonds the 30/360NASD method is used.

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Why calculate the accrued interest if it isalready included in the price calculation?

The quoted bond price does not include accruedinterest.That is, quoted prices are net of accruedinterest.

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The rationale is as follows.On July 15 the price of the Treasury bondusing a YTM of 10% is $829.83.On September 15 the price using a yield of10% is $843.5906.

The yield on both the days is the same Hence the increase in price is entirely due to theaccrued interest.

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On July 15 the accrued interest is zero.On a coupon payment date, the accrued interesthas to be zero .

On September 15 the accrued interest is

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However if prices are reported net ofaccrued interest, then in the short run

observed price changes will be entirely due tochanges in the yield.Consequently bond prices are always reportedafter subtracting the accrued interest.

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Quoted bond prices are called clean or add-interest prices.When a bond is purchased quoted price plus

the accrued interest has to be paid.The total price that is paid is called the dirtyprice or the full price or all-in price.

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Can the accrued interest be negative?Can there be cases where the seller of the bondhas to pay accrued interest to the buyer.

The answer is yes.In markets where bonds trade ex-dividendThe dirty price will fall by the present value ofthe next coupon on the ex-dividend date

And the dirty price will be less than the cleanprice.

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It is actually a misnomer for bonds do notpay dividendsTill this date the buyer will get the

forthcoming couponStarting from this date, the next coupon willgo to the seller.

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Take a T-bond that matures on 15 July 2032.It pays a 9% coupon semi-annually on 15January and 15 July every year.The face value is 1000 and the YTM is 8%.Assume that we are on 5 January 2013 whichis the ex-dividend date.

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Using the Actual/Actual conventionwe can calculate k to be 0.0543.

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The moment the bond goes ex-dividend thedirty price will fall

By the present value of the forthcoming couponBecause the buyer will be no longer entitled to it.

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Thus the ex-dividend dirty price is

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This is the amount payable an instant after itgoes ex-dividend.The accrued interest an instant before thebond goes ex-dividend is:0.09x1000 174

________ x ____ = $ 42.55432 184

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The clean price at the time of the bondgoing ex-dividend is1140.4910 42.5543 = $1097.9367

The clean price is greater than the ex-dividend dirty price.

This represents the fact that the seller has tocompensate the buyer

While the buyer is entitled to his share of the nextcoupon the entire amount will be received by the seller.

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The fraction of the next coupon that ispayable to the buyer is0.09x1000 10

_________ x ____ = $2.44572 184

Hence the buyer has to pay1097.9367 2.4457 = $1095.4910

Which is the ex-dividend dirty price.

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The yield or the rate of return from a bondcan be computed in various ways.Various yield measures have their merits and

demerits.

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This is very commonly reported.Although it is technically very unsatisfactory.

It relates the annual coupon payment to the

current market price.

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Also known asFlat YieldInterest YieldRunning Yield

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A 15 year 15% coupon bond is currentlyselling for $800.The face value is $1,000

The current yield is given by

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If this bond is bought for $800 and held forone year it will earn an interest of $150.

So the interest yield is 18.75% for a trader with aone-year horizon

However, if it is sold after one year thetrader will make a Capital Gain or Loss.

The current yield does not take such gains andlosses into account.

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The current yield is used to estimate the costof or profit from holding the bond.It is a Rough & Ready interest ratecalculationIf short-term rates are higher than thecurrent yield

The bond is said to involve a running cost.

This is known as negative carry or negativefunding.

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This tries rectify shortcomings of the currentyield by considering capital gains and losses.

The assumption made is that capital gains andlosses accrue evenly over the life of the bond.

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The formula is:Simple YTM = C M-P

__ + ______

P PXN/2

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For the 15 year bond that we consideredearlierSimple YTM = 150 1000-800

_____+ _________ = 20.42%800 15 x 800

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The simple YTM does not take into accountthe compound interest that can be earned

By reinvesting the coupons.This will obviously increase the overall returnfrom the bond.

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The YTM is the rate that equates the presentvalue of the cash flows from the bond to itsprice

Assuming that the bond is held to maturityIt is analogous to the concept of the IRRused in project valuation.

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Consider a bond that pays an annual couponof C on a semi-annual basis.The face value is M, the price is P, and thenumber of coupons remaining is N.

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The YTM is the value of y that satisfies thefollowing equation.

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The YTM is a solution to a non-linearequation.

We generally require a financial calculator ora computer to calculate it.However it is fairly simple in two cases

If we have a coupon paying bond with exactly twoperiods to maturity.In such a case it is simply a solution to a quadraticequation.

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The YTM is easy to compute in the case ofzero coupon bonds.

Consider a ZCB with a face value of $1,000,maturing after 5 years.The current price is $500.The YTM is the solution to

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A bondholder can expect to receive incomefrom the following sources.

There are coupon payments which are typicallypaid every six months.There will be a capital gain/loss when a bond isredeemed

It may mature or may be called or may be sold beforematurity.

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The YTM calculation assumes that the bond isheld to maturity.

Finally when a coupon is received it willhave to be reinvested till redemption

Once again the YTM calculation assumes that the bond isheld till maturity.The reinvestment income is interest on interest.

A satisfactory yield measure should take into

account all the three sources

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The YTM calculation takes into account allthe three sources of income.However it makes two key assumptions.

Firstly it assumes that the bond is held tillmaturity.Secondly it assumes that all intermediatecoupons are reinvested at the YTM itself .

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The latter assumption is built in to themathematics of the calculation.

The YTM is called a Promised Yield.It is Promised because in order to realize it the

holder has to satisfy both the above conditions.If either of the two conditions is violated she may notget what was promised.

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Consider a bond that pays a semi-annualcoupon of $C/2.

Let r be the annual rate of interest at whichthese coupons can be re-invested.

r would be dependent on the market conditionsprevailing when the coupon is receivedIt need not be equal to y, the YTM, or c, thecoupon rate.

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For ease of exposition we will assume that r is aconstant for the life of the bond.In practice, each coupon may have to bereinvested at a different rate of interest.

Thus each coupon can be re-invested at a rate ofr/2 per six monthly period.

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The coupon stream is an annuity.The final payoff from re-investment is thefuture value of this annuity.

The future value is

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The future value is the sum of all thecoupons which are reinvested

Which in this case is the principalPlus the interest from re-investment.

The total value of coupons that arereinvested is

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The interest on interest is therefore

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The YTM Calculation assumes that r/2 = y/2.

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Consider an L&T bond with 10 years tomaturity.

The face value is Rs 1,000.It pays a semi-annual coupon at the rate of 10%per annum.The YTM is 12% per annum.

Price can be calculated to be Rs 885.295.

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Assume that the coupons can be reinvestedat a six monthly rate of 6%

Which corresponds to a nominal annual rate of12%.

The total coupon income = 50 x 20 = 1000

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Interest on interest gotten by reinvesting thecoupons

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In the end the holder will get back the facevalue of Rs 1,000.So the total cash flow at the end

= 1000 + 839.3 + 1000 = 2839.3To get this income, the bondholder has tomake an initial investment of 885.295.

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So what is the effective rate of return?It is the value of i that satisfies the followingequation

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So the rate of return is 6% on a semi-annualbasis or 12% on a nominal annual basis

Which is exactly the same as the YTM.So how was this return achieved?

Only by reinvesting all the coupons at an annualrate of 12%, compounded on a semi-annual basis.

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The reinvestment rate affects only theinterest on interest income.The other two sources are unaffected.

If r > y, then the investors interest oninterest income would be higherAnd the return on investment, i, would begreater than the YTM, y.

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If r < y, then the interest on interestincome would be lower

And the rate of return, i, would be less than theYTM, y.

So if one buys a bond by paying a pricecorresponds to a YTM

The holder will realize that YTM only if

The bond is held till maturityHe is able to reinvest all the intermediate coupons atthe YTM.

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Future reinvestment rates may be less thanthe rate prevailing at the outset

This risk is called Reinvestment Risk.The degree of this risk depends on the time to

maturity as well as the quantum of the coupon.

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For a bond with a given YTM, and a couponrate, the greater the time to maturity

The more dependent is the total return from thebond on the reinvestment income.

Thus everything else remaining constant, thelonger the term to maturity

The greater is the reinvestment risk.

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Thus premium bonds will be more vulnerableto such risks than bonds selling at par.Correspondingly, discount bonds will be lessvulnerable than bonds selling at par.

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If a zero coupon bond is held to maturity,there will be no reinvestment risk

Because there are no coupons to reinvest.Thus if a ZCB is held to maturity, the actual rateof return will be equal to the promised YTM.

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This explains the popularity of ZCBs for manyinvestors, in particular, long-term investors

They can lock in a compounded rate of return ifthey hold the bond till maturityWithout worrying about the need to reinvest cashflows periodicallyThis explains the popularity of securities such asSTRIPS

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We will continue with the assumption thatthe bond is held till maturity.But we will explicitly assume the rate at

which the coupons can be reinvested.That is we will no longer take it forgranted

That intermediate cash flows can be reinvested

at the YTM.

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Let us reconsider the L&T bond.Assume that intermediate coupons can bereinvested at 7% for six months

That is a nominal annual rate of 14%.The total coupon income and the final facevalue payment will remain the same

But the reinvestment income will change.

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The interest on interest

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So the final amount received= 1000 + 1049.75 + 1000 = 3049.75

The initial investment is once again 885.295

Therefore, the rate of return is given by

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This is the return for six months.The nominal annual return is 6.38 x 2 = 12.76%Which is greater than the YTM of 12%.

The RCY is greater than the YTM, because weassumed

That the reinvestment rate was greater than theYTM.

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Had we assumed the reinvestment rate to beless than the YTM

The RCY would have turned out to be less thanthe YTM.

The RCY can be an ex-ante or an ex-postmeasure.Ex-ante means that we assume a reinvestmentrate and calculate the RCY.Ex-post means that we take into account theactual reinvestment

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We will now relax both the assumptionswhich were used to calculate the YTM.

Firstly the investor need not hold the bond untilmaturity.

Secondly he may not be able to reinvest thecoupons at the YTM.

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Now the return will depend on three sources the coupons receivedthe reinvestment income and the price at which the bond is sold prior tomaturity.

The sale price would depend inversely on theprevailing yield at that point in time

And need not equal the face value .

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Assume that an investor with a 7 yearinvestment horizon buys the L&T bond

He will get coupons for 14 periods (not 20).The total coupon income will be

50x14 = 700Assume that the reinvestment rate isexpected to be 7% per six monthly period.

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We will also assume that the investor expectsthe YTM after 7 years to be 12% per annum.The first step is to calculate the expectedprice at the time of sale.

At that point in time the bond will have 3 yearsto maturity.

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The price using a YTM of 12% can be shownto be Rs 950.865.The interest on interest

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The total terminal cash flow= 700 + 427.50 + 950.865 = 2,078.365

The initial investment as before is

885.295

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The nominal annual rate of return is6.29x2 = 12.58%

This is the Horizon Yield .It is also known as the Holding Period Yield

Once again, it can be calculated ex-post orex-ante.

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This measure of the rate of return is used forcallable bonds.It is the yield that will make the PV of thecash flows from the bond equal to the price

Assuming the bond is held till the call date.In principle a bond can have many possible calldates.Each date will give rise to a corresponding YTC.

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In practice the cash flows are usually takenonly till the first call date

Although they can be taken to any subsequentcall date.

The YTC is given by the equation

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N* is the number of coupons till the calldate.M* is the price at which the bond is

expected to be recalled.M* need not equal the face value.In practice many companies pay a CallPremium at the time of recall.

As much as one years couponIf so, M* = M + C

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Assume that the L&T bond is a callable bondand that the first call date is 7 years away.Assume that a call premium of Rs 100 will bepaid if the bond is recalled.

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The YTC is the solution to the followingequation

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The solution comes out to be 6.74%.So the YTC on an annual basis is 13.48%.

The YTC is very important for PremiumBonds.

The fact that a bond is at a premium, indicatesthat the coupon is greater than the yieldAnd that therefore there is a greater chance ofrecall.

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In practice the investors compute the YTCfor every possible call date.They then compute the YTM as well.The lowest of all possible values is called theYield to Worst .

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Consider a portfolio or a collection of bonds.We cannot calculate the portfolio YTM as aweighted average of the individual YTMs

Although in most cases it gives a very closeapproximation

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You have to compute the cash flows from theportfolio, and then find that interest rate

Which makes the PV of the cash flows equal tothe sum of the prices of the component bonds.

In other words we need to calculate the PortfolioIRR

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Consider a person who buys a TELCO bondand a Ranbaxy bond.The TELCO bond has a maturity of 5 years,face value of 1000, and pays a 10% coupon

The YTM is 12% per annum.

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The Ranbaxy bond has a face value of 1000,4 years to maturity and pays a coupon of 10%

The YTM is 16% per annum.Consider a portfolio consisting of one bond ofeach company.

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The price of the TELCO bond can be shown tobe 926.405.The price of the Ranbaxy bond can be shownto be 827.63.The total initial investment is therefore1,754.035

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Period Investment Inflow fromTELCO

Inflow f romRanbaxy

Total

0 (1754.035) (1754.035)

1 50 50 100

2 50 50 100

3 50 50 100

4 50 50 100

5 50 50 100

6 50 50 1007 50 50 100

8 50 1050 1100

9 50 50

10 1050 10508/28/2014 113

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Using a calculator or EXCEL, the portfolioyield can be calculated to be 13.76%.

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Certain bonds are tax exemptIn the US both federal and state governmentscan levy income taxTo compare a normal bond with a tax-freebond we need to compute the TEY of thetax-free bond

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In countries like the US, both federal andstate governments are empowered to levyincome taxGovernments in the US follow the guidelineof mutual reciprocity

Federal securities are exempt from state incometaxesState and local government securities areexempt from federal income taxesCertain securities are exempt from both taxesand are consequently tax free

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The method of calculation depends on theapplicable taxesLet us consider a Municipal bond which isyielding 6%

It is obviously exempt from Federal ITAssume the Federal tax rate is 25%TEY = 6.00/(1-0.25) = 8%

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The implication is the followingAn investor should be indifferent between ataxable bond yielding 8% and the Muni

If a taxable bond yields more than 8% go for itIf it yields less than 8% choose the Muni

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Sometimes a Muni may be exempt from bothFederal and state taxes.The net impact of federal and state taxes froan investor may be computed as follows.

Assume the federal tax rate is 25%And the state tax rate is 8%

State taxes may be claimed as an expensewhile computing federal taxes

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On $100, the income net of state taxes willbe $92. $23 will be deducted as state tax.

The post tax income is $69Thus the effective tax rate is 31%

So the TEY of a Muni that yields 6% is6 (1 0.31) = 8.70%