part-08

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Part-08

Prices & Yields: Advanced Perspectives

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Valuation in between Coupon Dates While valuing a bond we assumed

that we were standing on a coupon payment date.

This is a significant assumption because it implies that the next coupon is exactly one period away.

What should be the procedure if the valuation date is in between two coupon payment dates?

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The Procedure for Treasury Bonds

Calculate the actual number of days between the date of valuation and the next coupon date.

Include the next coupon date. But do not include the starting

date. Let us call this interval N1.

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Treasury Bonds (Cont…)

Calculate the actual number of days between the coupon date preceding the valuation date and the following coupon date.

Once again include the ending date but exclude the starting date.

Let us call this time interval as N2.

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The next coupon is then k periods away where

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Illustration There is a Treasury bond with a face

value of $1,000. The coupon rate is 8% per annum,

paid on a semi-annual basis. The coupon dates are 15 July and 15

January. The maturity date is 15 January

2022. Today is 15 September 2002.

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No. of Days Till the Next Coupon Date

Month No. of Days

September 15

October 31

November 30

December 31

January 15

TOTAL 122

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No. of Days between Coupon Dates

Month No. of Days

July 16

August 31

September 30

October 31

November 30

December 31

January 15

TOTAL 184

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K = 122/184 = .6630 This method is called the

Actual/Actual method and is often pronounced as the Ack/Ack method.

It is the method used for Treasury bonds in the U.S.

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The Valuation Equation

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

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Valuation

In our example

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The Treasury Method

There is a difference between the Wall Street approach and the approach used by the Treasury to value T-bonds. The difference is that the Treasury

uses a simple interest approach for the fractional first period.

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The Treasury Method (Cont…) The Treasury will thus use the

following equation.

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The Treasury Method (Cont…)

The Treasury approach will always give a lower price because for a fractional period the simple interest approach will always give a larger discount factor than the compound interest approach.

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The 30/360 PSA Approach The Actual/Actual method is applicable

for Treasury bonds in the U.S. For corporate bonds in the U.S. we use

what is called the 30/360 PSA method. In this method the number of days

between successive coupon dates is always taken to be 180.

That is each month is considered to be of 30 days.

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The 30/360 Approach (Cont…)

The number of days from the date of valuation till the next coupon date is calculated as follows.

The start date is defined as D1 = (month1, day1,year1) The ending date is defined as D2 = (month2,day2,year2)

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The 30/360 Approach (Cont…)

The number of days is then calculated as

360(year2 – year1) + 30(month2 – month1) + (day2 – day1)

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Additional Rules

If day1 = 31 then set day1 = 30 If day1 is the last day of February,

then set day1 = 30 If day1 = 30 or has been set equal

to 30, then if day2 = 31, set day2 = 30

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Examples of Calculations

Start Date

End Date Actual Days

Days Based on 30/360

Jan-01-86 Feb-01-86 31 30

Jan-15-86 Feb-01-86 17 16

Feb-15-86 Apr-01-86 45 46

Jul-15-86 Sep-15-86 62 60

Nov-01-86 Mar-01-87 120 120

Dec-15-86 Dec-31-86 16 16

Dec-31-86 Feb-01-87 31 31

Feb-01-88 Mar-01-88 29 30

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Pricing of A Corporate Bond

Let us assume that the bond considered earlier was a corporate bond rather than a Treasury bond.

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Pricing (Cont…)

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30/360 ISDA

The difference between 30/360 PSA and 30/360 ISDA is that the additional rule pertaining to the last day of February is not applicable.

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30/360 SIA

The additional rules for this convention are the following. If day1 = 31, then set day1 = 30. If day1 is the last day of February and

the bond pays a coupon on the last day of February then set day1 = 30.

If day1 = 30 or has been set equal to 30, then if day2 = 31, set day2 = 30.

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30/360 European Convention

In this convention, if day2 = 31, then it is always set equal to 30.

So the additional rules are: If day1 = 31 then set day1 = 30 If day2 = 31 then set day2 = 30

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Examples of Calculations

Start Date

End Date Actual Days

Days Based on 30/360E

Mar-31-86

Dec-31-86

275 270

Dec-15-86

Dec-31-86

16 15

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Actual/365 Convention

The difference between this and the Actual/Actual method is that the denominator in this convention will consist of 365 even in leap years.

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Actual/365 Japanese

This is used for Japanese Government Bonds (JGBs)

It is similar to the Actual/365 method. The only difference is that in this

case, the extra day in February is ignored in leap years, while calculating both the numerator and the denominator.

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Actual/365 ISDA This day count convention is identical to the

Actual/365 convention for a coupon period that does not include days falling within a leap year.

However for a coupon period that includes days falling within 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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Actual/360 Convention

This is a simple variant of Actual/365.

This is the convention used for money market instruments in most countries.

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Global ConventionsCountry Security Convention

Japan T-bills Act/365 Japanese

Japan JGBs Act/365 Japanese

Japan Other Bonds Act/365 Japanese

UK Fixed rate gilts Act/Act

UK Index linked gilts Act/Act

UK Strips Act/Act

US T-bills Act/360

US T-notes and T-bonds

Act/Act

US Other bonds 30/360 PSA

India Government bonds

30E/360

India Corporate bonds Act/365

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Accrued Interest The price of a bond is the present value of

all the cash flows that the buyer will receive when he buys the bond.

Thus the seller is compensated for all the cash flows that he is parting with.

This compensation includes the amount due for the fact that the seller is parting with the entire next coupon, although he has held it for a part of the current coupon period.

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Accrued Interest (Cont…)

This compensation is called Accrued Interest.

Let us denote the sale date by t; the previous coupon date by t1; and the following coupon date by t2

The accrued interest is given by

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Accrued Interest (Cont…)

Both the numerator and the denominator are calculated according to the conventions discussed above.

That is for U.S. Treasury bonds the Actual/Actual method is used, whereas for U.S. corporate bonds the 30/360 method is used.

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Why Accrued Interest?

Why should we calculate the accrued interest if it is already included in the price calculation?

The answer is that the quoted bond price does not include accrued interest.

That is, quoted prices are net of accrued interest.

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Why Accrued Interest? (Cont…) The rationale is as follows. On July 15 the price of the Treasury

bond using a YTM of 10% is $829.83. On September 15 the price using a

yield of 10% is $843.5906. Since the required yield on both the

days is the same, the increase in price is entirely due to the accrued interest.

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Why Accrued Interest (Cont…)

On July 15 the accrued interest is zero.

This is true because on a coupon payment date, the accrued interest has to be zero.

On September 15 the accrued interest is

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Why Accrued Interest? (Cont…) The price net of accrued interest is $843.5906 - $13.4783 = $830.1123$, which

is very close to the price of $829.83 that was observed on July 15.

We know that as the required yield changes, so will the price.

If the accrued interest is not subtracted from the price before being quoted, then we would be unsure as to whether the observed price change is due to a change in the market yield or is entirely due to accrued interest.

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Why Accrued Interest? (Cont…)

However if prices are reported net of accrued interest, then in the short run, observed price changes will be entirely due to changes in the market yield.

Consequently bond prices are always reported after subtracting the accrued interest.

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Clean versus Dirty Prices

Quoted bond prices are called clean or add-interest prices.

When a bond is purchased in addition to the quoted price, the accrued interest has also to be paid.

The total price that is paid is called the dirty price or the full price.

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Negative Accrued Interest One logical question is

Can the accrued interest be negative? That is, can there be cases where the seller of

the bond has to pay accrued interest to the buyer.

The answer is yes. In markets where bonds trade ex-dividend the

dirty price will fall by the present value of the next coupon on the ex-dividend date and the dirty price will be less than the clean price.

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Example Take a T-bond that matures on 15

July 2021. It pays a 9% coupon semi-annually

on 15 January and 15 July every year.

The face value is 1000 and the YTM is 8%.

Assume that we are on 5 January 2002 which is the ex-dividend date.

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Example (Cont…) Using the Actual/Actual convention we

can calculate k to be 0.0543.

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Example (Cont…)

The moment the bond goes ex-dividend the dirty price will fall by the present value of the forthcoming coupon, because the buyer will be no longer entitled to it.

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Example (Cont…) Thus the ex-dividend dirty price is

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Example (Cont…) This is the amount payable by the

person who buys the bond an instant after it goes ex-dividend.

The accrued interest an instant before the bond goes ex-dividend is:0.09x1000 174________ x ____ = $ 42.5543

2 184

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Example (Cont…) Thus the clean price at the time of the

bond going ex-dividend is1140.4910 – 42.5543 = $1097.9367

The clean price is therefore greater than the ex-dividend dirty price. This represents the fact that the seller has to

compensate the buyer because while the buyer is entitled to his share of the next coupon the entire amount will be received by the seller.

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Example (Cont…) The fraction of the next coupon

that is payable to the buyer is0.09x1000 10_________ x ____ = $2.4457

2 184 Hence the buyer has to pay

1097.9367 – 2.4457 = $1095.4910 which is the ex-dividend dirty price.

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Yield Measures

The yield or the rate of return from a bond can and is computed in various ways.

We will discuss various yield measures and their relative merits and demerits.

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The Current Yield

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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Example of the Current Yield

A 15 year 15% coupon bond is currently selling for $800.

The current yield is given by

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Current Yield (Cont…)

If you buy this bond for $800 and hold it for one year you will earn an interest income of $150.

So your interest yield is 18.75% However, if you sell it after one

year you will either make a Capital Gain or a Capital Loss.

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Current Yield (Cont…) What is a Capital Gain? If the price at the time of sale is

higher than the price at which the bond was bought, the profit is termed as a Capital Gain.

Else if there is a loss, it is termed a Capital Loss.

The current yield does not take such gains and losses into account.

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Current Yield (Cont…) One question is:

Should the current yield be based on the dirty price or the clean price

The advantage of using the clean price is that the current yield will stay constant till the yield changes.

However if the dirty price is used the current yield will be higher in the period between the ex-dividend date and the coupon date when the dirty price is less than the clean price and will be lower between the coupon date and the subsequent ex-dividend date when the dirty price will be more than the clean price.

This gives rise to a sawtooth pattern.

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Current Yield (Cont…)

The current yield is used to estimate the cost of or profit from holding the bond.

If short-term rates are higher than the current yield, the bond is said to involve a running cost. This is known as negative carry or

negative funding.

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Simple YTM

This yield measure attempts to rectify the shortcomings of the current yield by taking into account capital gains and losses.

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

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Simple YTM (Cont…)

The formula is:Simple YTM = C M-P

__ + ____ P PXN/2

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

_____+_________ = 20.42%

800 15 x 800

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The problem with the simple YTM is that it does not take into account the compound interest that can be earned by reinvesting the coupons.

This will obviously increase the overall return from the bond.

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Yield to Maturity (YTM)

The YTM is the interest rate that equates the present value of the cash flows from the bond (assuming that the bond is held to maturity), to the price of the bond.

It is exactly analogous to the concept of the Internal Rate of Return (IRR) used in project valuation.

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YTM (Cont…)

Consider a bond that makes an annual coupon of C on a semi-annual basis.

The face value is M, the price is P, and the number of coupons remaining is N.

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YTM (Cont…)

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

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YTM (Cont…) The YTM is a solution to a non-linear

equation. We generally require a financial

calculator or a computer to calculate it. However it is fairly simple to compute the

YTM in the case of a coupon paying bond with exactly two periods to maturity.

In such a case it is simply a solution to a quadratic equation.

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YTM for a Zero Coupon Bond

The YTM is easy to compute in the case of zero 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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Features of YTM

The YTM calculation takes into account all the coupon payments, as well as any capital gains/losses that accrue to an investor who buys and holds a bond to maturity.

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Sources of Returns From a Bond

A bondholder can expect to receive income from the following sources.

Firstly there are coupon payments which are typically paid every six months.

There will be a capital gain/loss when a bond matures or is called before maturity or is sold before maturity.

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Returns From a Bond (Cont…) The YTM calculation assumes that the

bond is held to maturity. Finally when a coupon is received it will

have to be reinvested till the time the bond eventually matures or is sold or is called.

Once again the YTM calculation assumes that the bond is held till maturity.

The reinvestment income is nothing but interest on interest.

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YTM

A satisfactory measure of the yield should take into account all the three sources of income.

The current yield measure considers only the coupon for the first year.

All the other factors are totally ignored.

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YTM (Cont…) The YTM calculation takes into account

all the three sources of income. However it makes two key assumptions. Firstly it assumes that the bond is held

till maturity. Secondly it assumes that all

intermediate coupons are reinvested at the YTM itself.

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YTM (Cont…) The latter assumption is built in to the

mathematics of the YTM calculation. The YTM is called a Promised Yield. It is Promised because in order to

realize it you have to satisfy both the above conditions.

If either of the two conditions is violated you may not get what was promised.

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The Re-investment Assumption Consider a bond that pays a semi-

annual coupon of $C/2. Let r be the annual rate of interest at

which these coupons can be re-invested.

r would be dependent on the market rate of interest that is prevailing when the coupon is received, and need not be equal to y, the YTM, or c, the coupon rate.

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Reinvestment (Cont…)

For ease of exposition we will assume that r is a constant for the life of the bond.

However, in practice, it is likely that each coupon may have to be reinvested at a different rate of interest.

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

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The coupon stream is an annuity. The final payoff from re-investment

is the future value of this annuity. The future value is

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The future value represents the sum of all the coupons which are reinvested (which in this case is the principal), plus the interest from re-investment.

The total value of coupons that are reinvested is

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Re-investment (Cont…)

The interest on interest is therefore

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

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Reinvestment in Action

Consider an L&T bond with 10 years to maturity.

The face value is Rs 1,000. It pay 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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Reinvestment in Action (Cont…)

Assume that the semi-annual interest payments can be reinvested at a six monthly rate of 6%, which corresponds to a nominal annual rate of 12%.

The total coupon income = 50 x 20 = 1000

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Interest on interest gotten by reinvesting the coupons

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Finally in the end you will get back the face value of Rs 1,000.

So the total cash flow at the end = 1000 + 839.3 + 1000 = 2839.3 To get this income, the bondholder

has to make 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 following equation

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So the rate of return is 6% on a semi-annual basis or 12% on a nominal annual basis, which is exactly the same as the YTM.

So how was this return achieved? Only by being able to reinvest all the

coupons at a nominal annual rate of 12%, compounded on a semi-annual basis.

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The Significance of the Reinvestment Rate The reinvestment rate affects only

the interest on interest income. The other two sources are unaffected. If r > y, that is if the reinvestment

rate were to be higher than the YTM, then the investor’s interest on interest income would be higher, and the return on investment, i, would be greater than the YTM, y.

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The Reinvestment Rate (Cont…) On the contrary, if r < y, that is, the

reinvestment rate is less than the YTM, then the interest on interest income would be lower, and the rate of return, i, would be less than the YTM, y.

So if you buy a bond by paying a price which corresponds to a given YTM, you will realize that YTM only if

You hold the bond till maturity You are able to reinvest all the intermediate

coupons at the YTM.

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Reinvestment Risk One of the risks faced by an investor

is that the future reinvestment rates may be less than the YTM which was in effect at the time the bond was purchased.

This risk is called Reinvestment Risk. The degree of reinvestment risk

depends on the time to maturity as well as the quantum of the coupon.

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Reinvestment Risk (Cont…)

For a bond with a given YTM, and a coupon rate, the greater the time to maturity, the more dependent is the total return from the bond on the reinvestment income.

Thus everything else remaining constant, the longer the term to maturity, the greater is the reinvestment risk.

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For a bond with a given maturity and YTM, the greater the quantum of the coupon, or in other words, the higher the coupon rate, the more dependent is the total return on the reinvestment income.

Thus everything else remaining the same, the larger the coupon rate, the greater is the reinvestment risk.

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Thus premium bonds will be more vulnerable to such risks than bonds selling at par.

Correspondingly, discount bonds will be less vulnerable than bonds selling at par.

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Zero Coupon Bonds and Reinvestment Risk 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 rate of return will be equal to the promised YTM.

If the risk is lower or absent, the return should also be less.

Thus a ZCB will command a higher price than an otherwise similar Plain Vanilla bond.

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The Realized Compound Yield We will continue with the assumption

that the bond is held till maturity. But we will make an explicit assumption

about the rate at which the coupons can be reinvested.

That is, unlike in the case of the YTM, we will no longer take it for granted that intermediate cash flows can be reinvested at the YTM.

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Illustration

Let us reconsider the L&T bond. Assume that intermediate coupons can

be reinvested at 7% for six months, or at a nominal annual rate of 14%.

The total coupon income and the final face value payment will remain the same, but the reinvestment income will change.

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Illustration (Cont…)

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 rate of 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 we assumed that the reinvestment rate was greater than the YTM.

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Illustration (Cont…) Had we assumed the reinvestment

rate to be less than the YTM, the RCY would have turned out to be less than the YTM.

The RCY can be an ex-ante or an ex-post measure.

Ex-ante means that we make an assumption about the reinvestment rate and then calculate the RCY.

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Ex-post means that we take into account the actual rate at which we have been able to reinvest and calculate the RCY.

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The Horizon Return

Let us now relax both the assumptions which were used to calculate the YTM.

Firstly the investor need not hold the bond until maturity.

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

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The Horizon Return (Cont…) Now the return will depend on three

sources – the coupons received, the reinvestment income, and the price at which the bond is sold prior to maturity.

The important issue is that the sale price of the bond would depend on the prevailing market yield at that point in time, and need not equal the face value.

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Illustration Assume that an investor with a 7 year

investment horizon buys the L&T bond that we discussed earlier.

He will get coupons for 14 periods (not 20).

The total coupon income will be 50x14 = 700 We will assume that the reinvestment

rate is expected to be 7% per six monthly period.

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We will also assume that the investor expects the YTM after 7 years to be 12% per annum.

The first step is to calculate the expected price at the time of sale.

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

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The price using a YTM of 12% can be shown to 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 is 6.29x2 = 12.58%

This is the Horizon Yield. Once again, it can be calculated

ex-post or ex-ante.

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Yield to Call (YTC) This measure of the rate of return

is used for callable bonds. The YTC is the yield that will make

the present value of the cash 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 call dates.

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YTC (Cont…)

In practice the cash flows are usually taken only till the first call date, although they can easily be taken to any subsequent call date.

The YTC is given by the equation

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YTC (Cont…) N* is the number of coupons till the call

date. M* is the price at which the bond is

expected to be recalled. M* need not equal the face value. In practice companies pay as much as

one year’s coupon as a Call Premium at the time of recall.

If so, M* = M + C

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Let us assume that the L&T bond is a callable bond and that the first call date is 7 years away.

Assume that a call premium of Rs 100 will be paid if the bond is recalled.

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The YTC is the solution to the following equation

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Illustration (Cont…) The solution comes out to be 6.74%. So the YTC on an annual basis is

13.48%. The YTC is very important for Premium

Bonds. The very fact that a bond is selling at a

premium, indicates that the coupon is greater than the yield, and that therefore there is a greater chance of recall.

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The Yield to Worst

In practice the investors compute the YTC for every possible call date.

They then compute the YTM as well.

The lowest of all possible values is called the Yield to Worst.

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Portfolio Yield

Consider a case where you hold a portfolio or a collection of bonds.

You cannot simply calculate the yield from the portfolio as a weighted average of the YTMs of the individual bonds.

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Portfolio Yield (Cont…)

You have to first compute the cash flows from the portfolio, and then find that interest rate which will make the present value of the cash flows equal to the sum of the prices of the component bonds.

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Illustration

Consider a person who buys a TELCO bond and a Ranbaxy bond.

The TELCO bond has a time to maturity of 5 years, face value of 1000, and pays coupons semi-annually at the rate of 10% per annum.

The YTM is 12% per annum.

112


The Ranbaxy bond has a face value of 1000, time to maturity of 4 years, and pays a coupon of 10% per annum semi-annually.

The YTM is 16% per annum. Consider a portfolio consisting of

one bond of each company. What is the portfolio yield?

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The first step is to calculate the two prices.

The price of the TELCO bond can be shown to be 926.405.

The price of the Ranbaxy bond can be shown to be 827.63.

The total initial investment is therefore 1,754.035

114

The Cash Flow TablePeriod Investmen

tInflow from TELCO

Inflow from Ranbaxy

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 50 100

7 50 50 100

8 50 1050 1100

9 50 50

10 1050 1050

115


Using a financial calculator or a spread sheet, the portfolio yield can be calculated to be 13.76%.

part-08

Documents

following coupon date

valuation date

date of valuation

treasury approach

treasury method cont

coupon period

day2 day1

ending date