Part-08Prices & Yields: Advanced Perspectives

1

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?2

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 N1.3

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 N2.4

Treasury Bonds (Cont)

The next coupon is then k periods away where

5

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 2002.6

No. of Days Till the Next Coupon DateMonth September October November December January TOTAL No. of Days 15 31 30 31 15 1227

No. of Days between Coupon DatesMonth July August September October November December January TOTAL No. of Days 16 31 30 31 30 31 15 184

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Treasury Bonds (Cont)

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.9

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.15

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)16

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 = 3018

Examples of CalculationsStart Date Jan-01-86 Jan-15-86 Feb-15-86 Jul-15-86 Nov-01-86 Dec-15-86 Dec-31-86 Feb-01-88 End Date Feb-01-86 Feb-01-86 Apr-01-86 Sep-15-86 Mar-01-87 Dec-31-86 Feb-01-87 Mar-01-88 Actual Days 31 17 45 62 120 16 31 29 Days Based on 30/360 30 16 46 60 120 16 31 3019

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 = 30.23

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 CalculationsStart Date Mar-3186 Dec-1586 End Date Actual Days Dec-3186 Dec-3186 275 Days Based on 30/360E 270

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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. 27

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 _________________________________ 36528

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 Japan Japan Japan UK UK UK US US US India India Security T-bills JGBs Other Bonds Fixed rate gilts Index linked gilts Strips T-bills T-notes and Tbonds bonds Other Government bonds Corporate bonds Convention Act/365 Japanese Act/365 Japanese Act/365 Japanese Act/Act Act/Act Act/Act Act/360 Act/Act 30/360 PSA 30E/360 Act/36530

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 by32

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.33

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.34

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 35 accrued interest.

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 is36

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 37

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.38

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.39

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. 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.

The answer is yes.

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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 41 date.

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 exdividend 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 18445

Example (Cont)

Thus the clean price at the time of the bond going ex-dividend is 1140.4910 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.46

Example (Cont)

The fraction of the next coupon that is payable to the buyer is 0.09x1000 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. 47

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

50

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.51

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.52

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.53

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.54

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.55

Simple YTM (Cont)

The formula is: Simple YTM = C M-P __ + ____ P PXN/2

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Simple YTM (Cont)

For the 15 year bond that we considered earlierSimple YTM = 150 1000-800 _____+_________ = 20.42% 800 15 x 800

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Simple YTM (Cont)

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.58

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. 59

YTM (Cont)

Consider a bond that makes an annual coupon of C on a semiannual 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. 62

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 65

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. 66

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.67

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 68 intermediate coupons are

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 69 promised.

The Re-investment Assumption

Consider a bond that pays a semiannual coupon of $C/2. Let r be the annual rate of interest at which these coupons can be reinvested. 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. 70

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 reinvested at a rate of r/2 per six 71

Reinvestment (Cont)

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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Reinvestment (Cont)

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 is73

Re-investment (Cont)

The interest on interest is therefore

The YTM Calculation assumes that r/2 = y/2.74

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 885.295.75

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 = 100076

Reinvestment in Action (Cont)

Interest on interest gotten by reinvesting the coupons

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Reinvestment in Action (Cont)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 885.295.78

Reinvestment in Action (Cont)

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

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Reinvestment in Action (Cont)

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 investors interest on interest income would be higher, and the return on investment, i, would be greater 81

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. 82

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 83

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. 84

Reinvestment Risk (Cont)

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 85

Reinvestment Risk (Cont)

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 87

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.88

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. 89

Illustration (Cont)

The interest on interest

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Illustration (Cont)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 by91

Illustration (Cont)

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. 92

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. 93

Illustration (Cont)

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.95

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 96

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 97

Illustration (Cont)

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.98

Illustration (Cont)

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

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Illustration (Cont)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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Illustration (Cont)

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.102

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 years coupon as a Call Premium at the time of recall. If so, M* = M + C104

Illustration (Cont)

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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Illustration (Cont)

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, 107 and that therefore there is a

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.108

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 semiannually at the rate of 10% per annum. The YTM is 12% per annum.111

Illustration (Cont)

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?112

Illustration (Cont)

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 113 1,754.035

The Cash Flow TablePeriod 0 1 2 3 4 5 6 7 8 9 10 Investmen Inflow t from (1754.035) TELCO 50 50 50 50 50 50 50 50 50 1050 Inflow from Ranbaxy 50 50 50 50 50 50 50 1050 Total (1754.035) 100 100 100 100 100 100 100 1100 50 1050114

Illustration (Cont)

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

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