basic bond analysis2

7/31/2019 Basic Bond Analysis2

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BOND PRICE AND RISK DYAMICS

Golaka C Nath

1. Introduction

A student of Finance needs to understand few basic concepts on Bond Theory in order to

understand the relationship between price and yield, to design bond trading strategies, to

interpret yield curves movements, look at arbitrage opportunities, etc. A trader can deal

with various types of bonds. Bonds are issued by issuers with specific need and hence have

different risks associated with them. Rating agencies play a very important role in bond

issuances. Bond investment has many facets some regulatory and some pure investment.

Banks invest in Sovereign Bonds in many countries as a regulatory requirement known as

Statutory Liquidity Ration. This write-up helps to understand what kind of information one

can deduce from different bonds traded in the market, yield curves properties and theory

behind yield curves as well various properties of bonds.

2. Bond Pricing Mechanism

A bond is either a zero coupon bond or a coupon bearing bond regular inflow of future

stream of cash flows both equal and unequal. A zero coupon paper is a bullet payment by

the investor and repayment by the issuer. A coupon bearing bond is combination of many

zero coupon bonds as it a combination of many future cash flows. Hence, the price of a

bond is nothing but the present value of its expected future cash flows.

The present value (PV) of a bond will be lower than its future value primarily because having

100,000 three months from now is less valuable than having 100,000 at hand now. As we

move to the future, the value of a bond may deplete as the possibility of default increase,

inflation can also eat away part of the value, future interest rate scenario changes, etc.

There are many classes of bonds traded in the market. Sovereign bonds contribute the

largest chunk followed by corporate bonds. Sovereign bonds do not assume credit risk or

default risk while corporate bonds include default risk and rating information on such bonds

is very important to price the bond. An AAA rated bond will have a lesser chance of default

vis--vis a BB rated bond and accordingly credit spread is charged for the bonds. However,


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sovereign bonds provide the starting basis to price a bond with credit risk. Most bond prices

are quoted in decimals (upto 4 decimals). However, the US market quotes in 32nds.

2.1 Pricing a Single Cash Flow

While calculating the future value of an investment, we require a rate of return for the term

of our investment. If we want a 10% p.a. return for our investment, then 100 invested now

will become 110 in one years time.

Future Value (FV) = 100 * (1 + 10/100) = 110

Alternatively, we can also write the same as: where PV is thepresent value ris the rate of return. In general, we can write,

where n is the number of periods invested. If we want to calculate the price of a bond as a

function of its future value, we can rewrite this equation as

where P is the price of the bond which is nothing but the present value. And FV is the

future cash flow i.e. the repayment at redemption n periods ahead.

2.2 Discount Rate

The r is referred to as the yield or the discount rate, i.e. the rate used to discount all the

future cash flows in order to ascertain the current price or present value of the investment.

is the value of the discount function at period n. Multiplying the discount functionat period n by the cash flow expected to be received at period n gives the present value

of the cash flow today.

Pricing a bond with a semi-annual coupon follows the same principles as that of an annual

coupon. A 5 year bond with a FV of 100 and semi-annual coupons will have 10 periods, each

of six months maturity; and the price equation will be:

where c = annual coupon value and r is the redemption yield or discount rate (semi-annual

yield * 2 to make it annualized).


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In practice, most bonds will have multiple cash flows before their maturity and therefore

each cash flow needs to be discounted in order to find the present value (current price). The

general price is given as follows:

The P here is the dirty price (inclusive of all interests accrued from the last coupon

payment date). The common practice in bond markets is to discount using a redemption

yield and all future cash flows are discounted using this rate. The yields are derived from the

yield curve in the market for a particular type of debt. The clean price is used by traders to

quote a bond in the market. The accrued interest is deducted from the dirty price to get the

quoted price of a bond.

In the bond price equation, r is the redemption yield the yield that an investor is likely to

get if he holds the security till maturity. This assumption may have problems as many fund

managers may have different portfolio holding periods and the same may vary dynamically

depending on market condition and movement of yield curve. In theory, each investor will

have a slightly different view of the rate of return required on the basis of opportunity cost

facing each of them. Their perception on future inflation, appetite for risk, nature of

liabilities, investment time horizon etc. will vary. The required yield for an investor needs to

reflect the above considerations. While dealing in the market, investors will figure out what

they consider to be a fair yield to enthuse them to invest. They can then compute the

corresponding price using the yield and compare this to the market price before deciding

whether and how much to buy or sell.

In general, the bond mathematics notation for expressing the price of a bond is given by

where PV(cft) is the present value of the cash flow at time t.

In the above example, we have used the same redemption yield to price the bond. However,

Time Value of Money makes us to consider different interest rates applicable to each cash

flow in terms of its arrival time. So using a single r to price a bond, we may consider

different series of rs to estimate the price. These yields are known as the spot yields or

zero coupon yields. Typically these yields are not observed in the market (unless you have


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many Zeros trading in the market) and have to be estimated using various theories of yield

curve. In that case the bond price equation will be:

Here we have used different rs appropriate for each term.

2.3 Dirty Prices and Clean Prices

A bond has multiple coupon payment dates which starts with the issue of the bond and

ends with the maturity of the bond. On coupon payment dates, bonds are not typically

traded as it may go for a Shut period in order to make payment of the coupon to the last

recorded holder of the Bond. This is so as a bond is sold with accrued interest even if no

interest has been received by the seller at the time of selling the Bond. Interest will be paid

out by the issuer on the next designated coupon payment date (for example 8.79% GOI

2021 bond will pay coupon on 08-May and 08-Nov every year till 08-Nov-2021). Hence these

factors need to be taken into account while pricing a bond. The traders quote clean price

while trading a bond but the back office calculates the full price of the bond for invoicing to

the counterparty. If all other parameters like maturity, coupon and yield are constant, the

clean price remains more or less stable. That is the reason why clean price is important for a

trader and why the same is used while trading the bond.

When a bond is bought or sold during a coupon cycle (between last coupon date and next

coupon date), a certain amount of coupon interest will have accrued on the bond. Because

he may not have held the bond throughout the coupon period, he is required to pay the

previous holder some compensation for the amount of interest which accrued during his

ownership. In order to calculate the accrued interest, we need the basis of calculating the

days between coupon cycles. There are markets which follow Actual / 365 while some

markets follow 30/360 (each month is considered 30 days and year is considered 360 days)

and many other types of Basis is used in markets depending on the type of assets we value.

Coupon rate is already known at the time of issuing a bond. In most bond markets, accrued

interest is calculated on the following basis:


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Prices in the market are usually quoted on a clean basis but settled on a dirty basis or full

price. For 8.79% GOI 2021 maturing on 08-Nov-2021, the AI on 01-Jan-2012 will be =

(8.79/360)*53 days = 1.2941. Since the settlement is on 01-Jan-2012, the legal title of the

bond passes to the buyer from this date and he is eligible for all accruals from this date. The

previous holder of the bod is eligible for compensation till 31-Dec-2011 from 08-Nov-2011

(last coupon payment date) 23 days in November + 30 days in December making total 53

days.

The Dirty price of this bond on January 1, 2012 will be 102.5066. How this price is arrived?

The bond has been traded at an yield of 8.6021% by the trader. The computation of the

price on 01-Jan-2012 is given below (Table 1):

Table 1: Bond Price using Cash Flow Method

Settlement

Date

Cash flow

Date Cash flow Period

Period in

Years

Discount

Factors

Discounted

Cash Flow

Value Date

(The previous

holder ceases

and new one

takes over

legal

ownership)

Coupon

Payment

Dates

Money

Received

Days

from

Value

Date to

Next

Coupon

date

Period in

Years

(30/360

Basis)

R=8.6021%

DF * Cash

Flow

01-Jan-12 08-May-12 4.3950 127 0.3528 0.9707 4.2663

08-Nov-12 4.3950 307 0.8528 0.9307 4.090408-May-13 4.3950 487 1.3528 0.8923 3.9217

08-Nov-13 4.3950 667 1.8528 0.8555 3.7600

08-May-14 4.3950 847 2.3528 0.8202 3.6050

08-Nov-14 4.3950 1027 2.8528 0.7864 3.4563

08-May-15 4.3950 1207 3.3528 0.7540 3.3138

08-Nov-15 4.3950 1387 3.8528 0.7229 3.1771

08-May-16 4.3950 1567 4.3528 0.6931 3.0461

08-Nov-16 4.3950 1747 4.8528 0.6645 2.9205

08-May-17 4.3950 1927 5.3528 0.6371 2.8001

08-Nov-17 4.3950 2107 5.8528 0.6108 2.684608-May-18 4.3950 2287 6.3528 0.5856 2.5739

08-Nov-18 4.3950 2467 6.8528 0.5615 2.4678

08-May-19 4.3950 2647 7.3528 0.5383 2.3660

08-Nov-19 4.3950 2827 7.8528 0.5161 2.2684

08-May-20 4.3950 3007 8.3528 0.4949 2.1749

08-Nov-20 4.3950 3187 8.8528 0.4744 2.0852

08-May-21 4.3950 3367 9.3528 0.4549 1.9992

08-Nov-21 104.3950 3547 9.8528 0.4361 45.5293

Dirty Price 102.5066

AI 1.2941Clean price 101.2125


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The dirty price and clean price have different behavioral pattern. The clean price is stable

over time (a very decline in value as the time passes) if we have the maturity, yield and

coupon is held constant. The dirty price is maximum on one day before coupon payment

dates as on this day, the accrual is highest on this day. On coupon payment date, the dirty

price and clean price is same as there is no accrued interest on this date. Further, on coupon

payment dates, the price of the bond is equal to the FV (100) if yield and coupon is same.

But on any other day if we price the bond between two coupon payment dates), the said

rule does not hold good. For our own example, the theoretical value of the bond on 01-Jan-

2012 (if we make the yield as 8.79%), the price comes to 99.9804 and not 100 (Table 2).

Table 2: Bond Price using Cash Flow Method when Coupon equals Yield

Settlement

Date

Cash flow

Date Cash flow Period

Period in

Years

Discount

Factors

Discounted

Cash Flow

01-Jan-12 08-May-12 4.3950 127 0.3528 0.9701 4.2636

08-Nov-12 4.3950 307 0.8528 0.9293 4.0841

08-May-13 4.3950 487 1.3528 0.8901 3.9122

08-Nov-13 4.3950 667 1.8528 0.8527 3.7475

08-May-14 4.3950 847 2.3528 0.8168 3.5897

08-Nov-14 4.3950 1027 2.8528 0.7824 3.4386

08-May-15 4.3950 1207 3.3528 0.7494 3.2938

08-Nov-15 4.3950 1387 3.8528 0.7179 3.1552

08-May-16 4.3950 1567 4.3528 0.6877 3.022308-Nov-16 4.3950 1747 4.8528 0.6587 2.8951

08-May-17 4.3950 1927 5.3528 0.6310 2.7732

08-Nov-17 4.3950 2107 5.8528 0.6044 2.6565

08-May-18 4.3950 2287 6.3528 0.5790 2.5446

08-Nov-18 4.3950 2467 6.8528 0.5546 2.4375

08-May-19 4.3950 2647 7.3528 0.5313 2.3349

08-Nov-19 4.3950 2827 7.8528 0.5089 2.2366

08-May-20 4.3950 3007 8.3528 0.4875 2.1424

08-Nov-20 4.3950 3187 8.8528 0.4669 2.0522

08-May-21 4.3950 3367 9.3528 0.4473 1.9658

08-Nov-21 104.3950 3547 9.8528 0.4285 44.7287

Dirty Price 101.2745

AI 1.2941

Clean price 99.9804

The clean price is stable over period in time if we keep the parameters like yield, coupon

and maturity constant over time. When clean prices change, it is more for an economic

reason, for instance a change in interest rates or in the bond issuer's credit quality. Dirty

prices, on the other hand, change day to day depending on where the current date is inrelation to the coupon dates, in addition to any economic reasons. In our example, the clean


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price of 8.79% GOI 2021 maturing on 08-Nov-2021 and trading at an yield of 8.6021% on

various days between 01-Jan-2012 and 31-Mar-2012 is plotted below (Chart -1). The chart

clears shows that the clean price change is about only 0.016 over a three month period.

The dirty price changes day to day as interest are accrued every day (Chart 2).

The bond price equation we used above can also be written as:

[ ] The above equation has been derived from the original bond price equation. If we have

semi-annual bond, then the rate will be

and time will be 2*t. The above formula gives us

an approximate clean price of the bond. For our example, the bond will be priced as

101.194

101.199

101.204

101.209

30-Dec-11 19-Jan-12 08-Feb-12 28-Feb-12 19-Mar-12

Chart - 1: Clean Price

Clean Price

99.0000

100.0000

101.0000

102.0000

103.0000

104.0000

105.0000

106.0000

107.0000

23-Dec-11 06-May-13 18-Sep-14 31-Jan-16 14-Jun-17 27-Oct-18

Price

Settlement Date

Chart - 2: Dirty Price vs Clean Price

Dirty Price Clean Price


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= 102.1844*0.5639+43.6125

= 101.2317

Price of the bond can also be found out by finding out the value the bond as of the last

coupon payment date and then taking that value forward to the current point in time

(settlement date). This is the total price that the buyer will actually pay. To get the quoted

or clean price, subtract the accrued interest. For example, the price for the bond in our

example on last coupon payment date (08-Nov-2011) is =102.1844*.5692+43.0751 =

101.2434. To take that value forward till settlement date, we need to divide the same by the

DF till settlement date

. This makes the Dirty Price =

101.2434/0.9877 = 102.5066. If we deduct the AI of 1.2941, we get the Clean Price of

101.2125 which is the CP we derived earlier.

2.4 Relationship between price and yield

There is a direct relationship between the price of a bond and its yield. Bond price and yield

move inversely. The price is the amount the investor will pay for the future cash flows; the

yield is a measure of return on those future cash flows. The price-yield relationship of a

standard non-callable coupon bearing bond shows a convex shape (Chart 3 and 3A).

When the required yield decreases, the discount factor also decreases and the price of the

bond rises. Hence the price increases as yield decreases.

60.0000

80.0000

100.0000

120.0000

140.0000

160.0000

180.0000

1.00% 5.00% 9.00% 13.00% 17.00%

Price

Yield

Chart - 3: Price and Yield Relationship

Dirty Price Clean Price


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3. Yields and yield curves

For Bonds, prices are derived for convenience of transferring value from buyer to sell. Since

we have bonds with different parameters like maturity and coupon, they can be compared

by means of their relative attractiveness which is found by comparing the yield of the bond.

It is not possible for an investor to compare the relative value of bonds by simply looking at

their traded prices as the different maturities and coupons will affect the traded price.

Hence to figure out the relative value of bonds, we need to compare bond yields. Yields are

usually quoted on an annual basis (semi-annual *2). In order to convert to a semi-annual

basis (and vice versa), we can apply the following formulae:

1

In general, the formula applied to convert from an annual to other period yield is

1) * n.3.1 Money market yields

Money market yields are quoted on a different basis (Act/365) and therefore in order to

compare short-term bonds and money market instruments it is necessary to look at them

on a comparable basis. The price of a 91-day T-Bill is found out as follows:

96

98

100

102

104

106

108

110

112

114

7.10% 7.60% 8.10% 8.60% 9.10%

AxisTitle

Axis Title

Chart - 3A: Price Yield Relationship

CP DP


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If a 91-day TB is trading at 8.15%, then its price will be 98.0085.

3.2 Uses of Yield Curves and Yield Curve Theories

A yield curve (Chart 4) is a graphical representation of the term structure of yields for a

given market. It depicts how yields for a set of bonds with similar characteristics except

maturity vary with maturity. Yield curves are therefore constructed from a homogeneous

group of bonds. Bonds with different credit quality will have different yield curves.

Sovereign yield curves represent the yield on Government securities traded in a market.

Yield curves help us to price bonds whether they are traded or not. Yield curves represent a

continuous time domain.

Yield curve is the Bible for Bond market participants and have different uses for different

people. Sovereign yield curves tell us market participants expectation about the future.

Sovereign yield curves demonstrate the tightness (and expected tightness) of monetarypolicy; it allows allow cross-country comparisons; assist pricing of new issues; assess relative

value between bonds; allow one to derive implied forward rates; and help traders/investors

understand risk; help in pricing illiquid and non-traded bonds. Different yield curves

(depending upon the class of bonds used) are used for different purposes.

Theories of the yield curve attempt to explain the shape of the curve, depending on

preferences/views of the market participants. The major theories behind yield curve

8.2000

8.3000

8.4000

8.5000

8.6000

8.7000

8.8000

8.9000

0.00 5.00 10.00 15.00 20.00 25.00 30.00

Yield%

Maturity

Chart - 4: Yield Curve

Indian_23-04-2012


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construction are - Liquidity Preference Theory (risk premia increase with time so, other

things being equal, one would expect to see a rising yield curve); Pure Expectations

Hypothesis (forward rates govern the curve - these are simply expectations of future spot

rates and do not take into account risk premia); Segmented MarketsHypothesis (the yield

curve depends on supply and demand in different sectors and each sector of the yield curve

is only loosely connected to others); Preferred Habitat (again investors have a maturity

preference, but will shift from their preferred maturity if the increase in yield is deemed

sufficient compensation to do so).

Sovereign yield curves are very important tools for monetary policy. The shape and slop of

the curve helps to frame monetary policy as future expectations are inbuilt in the curve.

Central Banks around the world plan Open Market Operations (OMO) to moderate the

shape of the curve by buying and selling bonds through designated windows.

3.3 Flat Yield

This is the simplest measure of yield (also known as current yield, interest yield, income

yield or running yield). Before the onset of computers and Financial Calculators, buyers and

sellers used this form of yield to compare the bonds. It does not talk about time value of

money or compounding. It is a very crude measure of yield. It just maps the holding cost of a

Bond and compares bonds across time horizon. It just assumes one period holding till next

coupon date. It does not take into account accrued interest. It assumes that price is not

going to change during holding period. It is given by:

Flat Yield = Coupon Rate/Clean price

For a bond 8.79% GOI 2021 maturing on 08-Nov-2021 and trading at an yield of 8.6021% for

value date as on 01-Jan-2012, the clean price is 101.2125. The Flat Yield will be =

8.79/101.2125 = 8.6847%

It can only sensibly be used as a measure of value when the term to maturity is very long (as

coupon income will be more dominant in the total return than capital gain/loss).

3.4 Simple Yield


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Simple Yield is a slightly more sophisticated measure of return than flat yield. This takes into

account the capital gain. But it assumes that capital gain accrues in a linear fashion over the

life of the bond. However, it does not allow for compounding of interest; nor does it take

into account accrued interest. It uses the clean price in the calculation.

For our Bond, the same will be (8.79 +

)/( ) = 8.8594%When a bond reaches near its last coupon period is, in terms of its cash flows, it directly

comparable with a money market instrument as there is no intervening coupon. In this case

simple interest yield calculations are used.

3.5 Redemption Yield (Yield to Maturity)

A redemption yield or yield to maturity is that rate of interest (current interest rate

prevailing in the market for similar kind of bonds) at which the total discounted values of

future payments of income and capital equate to its price in the market.

where P =dirty price; C =coupon, R =redemption payment (typically 100); and n =no of

periods; r =redemption yield.

The redemption yield is also referred to as the Internal Rate of Return or the Yield to

Maturity (YTM). It assumes that investor is going to hold this bond till maturity. When

quoting a yield for a bond, it is the redemption yield that is normally used by traders. Yield

curve captures all the factors contributing to the expectation of return on investment in a

single number. The redemption yield takes into account the time value of money by using

the discount function: each cash flow is discounted to give its net present value. Obviously a

bond with near maturity is expected to be traded at a lower yield than a long maturity bond.

This measure gives only the potentialreturn as an investor can sell of the bond before its

maturity or she may have her own investment period.

The limitations of using the YTM yield to discount future cash flows are:


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It assumes that a bond is held to maturity. (i.e. the YTM is only achieved if a bond is

held to maturity);

It discounts each cash flow at the same rate irrespective of its time of arrival in the

hands of the investor and hence against the principle of Time Value of Money;

Since we discount using the same YTM throughout its life, YTM assumes a

bondholder can reinvest all coupons received at the same rate (i.e. assumes a flat

yield curve), whereas in reality coupons will be reinvested at the market rate

prevailing at the time they are received (it assumes no reinvestment risk);

The discount rate used for a cash flow in, say, three years time from a 10 year bond

will be different from the rate used to discount the three year payment on a 30 year

bond.

The YTM curve suffers from these above limitations. We use the curve for simple analysis,

and it can also be used when there are insufficient bonds available to construct a more

sophisticated yield curve. In an illiquid market, this is the best we can have. Approximate

YTM is found out by using IRR function (Table 3). Here we need to use only the clean price

as first Cash outflow and all coupons and Principal as inflows in future dates. For our Bond,

the same is working out as 8.6067% (this is little higher than actual yield).

Table 3: Yield Using IRR

SettlementCash flow

Date

Maturity to

CFYears Cash flow DF DCF

-101.2125

01-Jan-12 08-May-12 127 0.35 4.395 0.9707 4.2663

08-Nov-12 307 0.85 4.395 0.9307 4.0904

08-May-13 487 1.35 4.395 0.8923 3.9217

08-Nov-13 667 1.85 4.395 0.8555 3.7600

08-May-14 847 2.35 4.395 0.8202 3.6050

08-Nov-14 1027 2.85 4.395 0.7864 3.4563

08-May-15 1207 3.35 4.395 0.7540 3.3138

08-Nov-15 1387 3.85 4.395 0.7229 3.1771

08-May-16 1567 4.35 4.395 0.6931 3.0461

08-Nov-16 1747 4.85 4.395 0.6645 2.9205

08-May-17 1927 5.35 4.395 0.6371 2.8001

08-Nov-17 2107 5.85 4.395 0.6108 2.6846

08-May-18 2287 6.35 4.395 0.5856 2.5739

08-Nov-18 2467 6.85 4.395 0.5615 2.467808-May-19 2647 7.35 4.395 0.5383 2.3660


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08-Nov-19 2827 7.85 4.395 0.5161 2.2684

08-May-20 3007 8.35 4.395 0.4949 2.1749

08-Nov-20 3187 8.85 4.395 0.4744 2.0852

08-May-21 3367 9.35 4.395 0.4549 1.9992

08-Nov-21 3547 9.85 104.395 0.4361 45.5293

DP 102.5066

AI 1.2941

CP 101.2125

IRR 8.6067%

The Bond price equation used in this section looks at gross returns, but bond investors are

likely to be subject to tax: possibly both on income and capital gain. The net yield, if taxed

on both coupon and redemption payments, is given by:

P = Dirty price; C = Coupon; R = Redemption payment; r = net redemption yield and t =

applicable Tax rate

The above equation changes and becomes more complicated if withholding tax is imposed

(as a percentage of tax will be imposed at source with the remainder being accounted for

after the payment has been received). As tax rules can materially affect the price of bonds,

their effects need to be taken into account in any yield curve 14odeling process in order to

avoid distortions in the estimated yield curve.

Yield can also be approximated using the following formula:

For our Bond, we can approximate the YTM as

3.6 Spot Rate and the Zero Coupon Curve


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Given the limitations of the YTM we discussed above, it would seem more logical to

discount each cash flow by a discount rate appropriate to its maturity; that is, using a spot

rate or a zero rate.

where P =Price (dirty); C =Coupons; n =Number of periods; ri=Spot rate for period I; and R =

Redemption payment

Each spot rate above is the specific zero coupon yield related to that maturity. Hence, it will

give a more accurate rate of discount at that maturity than the usual YTM we used in earlier

equations. Reinvestment risk is taken care of in the above equation. Zero rates take into

account current spot rates, expectations of future spot rates, expected inflation, liquidity

premia and risk premia. The zero coupon yield curve or spot yield curve is also referred to as

the Term Structure of Interest Rates; the plot of appropriate spot rates of varying

maturities against those maturities. The spot yield curve gives us an unambiguous

relationship between yield and maturity. This curve is used for pricing all kinds of products.

A spot yield coupon curve can be estimated using the coupon bearing traded bonds in the

market or by using actual zero coupon papers like STRIPS in the market. If we know the spot

rate (r1) of a 6-month bond, then we can determine the one-year spot rate (r2). Then, r3,

the third period spot rate, can be found from looking at a 3 year bond. We can use the

following series of equation to solve for different rates:

We need to plug in the rates found in previous periods to find the spot rate for next period.

This method is known as the Bootstrapping method of extracting spot rates from the bond

price information. However, this assumes to have traded bonds at every 6-month period to

bootstrap the future spot rates. This may not be the case.


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Further, an YTM curve can also be used to derive the spot yield curve. Assuming the

sovereign yield curve to be the most representative and efficiently estimated, it will

represent the market participants expectation of all economic variables . If suppose

Government comes to market to borrow money by issuing new bonds in all time spectrum

(6-month, 12-month, 1.5years,say upto 30 years), then it will issue at Par (FV=100).

Hence all coupons on those bonds will be nothing but the YTMs applicable to the respective

maturities. Assuming yield as coupon, we can bootstrap the spot rates for each term.

However, traders may choose to use more sophisticated models like Nelson Siegel, Nelson

Siegel and Svensson, Splines, etc. to create the zero curve. However, evidence suggests that

the zero curve constructed from bootstrapping and the zero curve constructed from a more

sophisticated model are very similar.

The main uses of the zero coupon curve are finding relatively mis-priced bonds, valuing

swap portfolios and valuing new bonds at auction. The advantage of this curve is that it

discounts all payments at the appropriate rate, provides accurate present values and does

not need to make reinvestment rate assumptions.

3.7 Forward Zero Coupon Yield

Forward spot yields indicate the expected spot yield at some date in the future. These can

be derived simply from spot rates:

The 6 month spot rate is the rate available now for investing for 6-month (rate = r1 )

The 1-year spot rate is the rate available now for investing for one years (rate = r2 )

The two year spot rate is the rate available now for investing for n years (rate = rn )

Hence, there is a rate implied for investing for a one-year period in one year s time (f12,).

We can write:

( ) i.e. the forward rate, is such that an investor will be indifferent to investing for twoyears or investing for one year and then rolling over the proceeds for a further year. For

example, if the investor has 100 to invest and he has two options: (a) investing for 6 months

@10% p.a. or (b) invest for first 3 months @9% p.a. and then agreeing to reinvest the


17/33


proceeds for 3 more months from the date of maturity of first 3 months @10.76%. The

implied forward in this case is 10.76%p.a. The same is calculated as follows if we invest for

3 months we get 102.25 which needs to be reinvested for next 3 months to yield 2.75 more

to make it 105 which we would have received if we had agreed to invest for 6 months at the

beginning. The investor is indifferent between two options given to him and the implied

forward rate is risk neutral arbitrage free one. Now the trader has to apply all his

expectation about future to quote the rate in the market. The forward zero rate curve is

thus derived from the spot yield curve by calculating the implied one period forward rates.

Expressing the price of the bond in terms of these rates gives:

= fi is the I period forward rate for one further period (i.e. the one-year rate in I years time)

All forward yield curves can be calculated in this way. However, the above formula assumes

the expectations hypothesis i.e. the implied forward rate equals the spot rate that prevails

in the future. However, the liquidity premium hypothesis suggests that the implied forward

rate equals the expected future spot rate plus a risk premium.

3.8 Par Yield

The par yield is a hypothetical yield typically used to figure out the coupon of a new security

going to be auctioned. The spot yields are used to find out the coupon of a Bond is it has to

be sold at Par (100). We use the same equation to figure out the Coupon.

Suppose we have the following spot yields and we would like to find out the expected

coupon of a new bond to be issued at Par.

Year 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Spot

Rate 8.15% 8.26% 8.35% 8.42% 8.48% 8.56% 8.62% 8.71% 8.74% 8.80%DF 0.9608 0.9222 0.8845 0.8479 0.8125 0.7777 0.7442 0.7110 0.6805 0.6501


18/33


Then the expected Coupon will be:

The par yield curve is used for determining the coupon to set on new bond issuances, and

for assessing relative value.

3.9 STRIPS

A STRIP is a zero coupon bond which has been created by separating the coupons and

principal of a coupon bearing bond into different zeros and trade the same in the market as

zero coupon papers. For example, a 10-year bond with an semi-annual coupon could be

separated into 21 zero-coupon bonds, 20 representing coupons and one relating to the

principal repayment. If STRIPS market is liquid, then spot yield curve can be easily

constructed using simple bootstrapping techniques. The STRPS will be priced and traded

using the simple formula:

A trader can compare the STRIPS traded with the bond and see if it is valuable to buy all

STRIPS parts to construct the bond.

For example, we are on 09-Sep-2011 and dealing with a Bond 7.80% GOI 2021 issued on 11-

Apr-2011 for 10 years maturity that has been permitted for stripping (Table 4). The bond is

trading at 96.35. The spot rates for the appropriate coupon maturity dates are as follows:


19/33


Table 4: STRIP PRICING USING SPOT YIELD

Date: Cash Flows Zeros Rates

STRIPS Rate

(%) PV Cash Flow:

11-10-11 3.9 0.078 7.8 3.7536

11-04-12 3.9 0.0792 7.92 3.7256

11-10-12 3.9 0.0795 7.95 3.5822

11-04-13 3.9 0.0799 7.99 3.4435

11-10-13 3.9 0.0802 8.02 3.3088

11-04-14 3.9 0.0807 8.07 3.1777

11-10-14 3.9 0.0811 8.11 3.0505

11-04-15 3.9 0.0816 8.16 2.9269

11-10-15 3.9 0.0819 8.19 2.8085

11-04-16 3.9 0.082 8.2 2.6965

11-10-16 3.9 0.0823 8.23 2.5862

11-04-17 3.9 0.0825 8.25 2.481611-10-17 3.9 0.0826 8.26 2.3817

11-04-18 3.9 0.0829 8.29 2.2831

11-10-18 3.9 0.0832 8.32 2.1875

11-04-19 3.9 0.0827 8.27 2.1081

11-10-19 3.9 0.0829 8.29 2.0210

11-04-20 3.9 0.0836 8.36 1.9292

11-10-20 3.9 0.0839 8.39 1.8467

11-04-21 103.9 0.0842 8.42 47.0925

Dirty Price 99.3912

If we buy all the parts and use the zero rates applicable for the day, we get a value of

99.3912 (zero rates have been derived from the market trades on the said date). The bond

has an accrued interest of 3.2067. Hence the dirty price using trading information works out

to be 99.5567 while the STRIPS give us a dirty price of 99.3912. STRIPS are cheaper than the

Bond. Can we buy the bond and sell the STRIPS? This is a common situation: The On-the-Run

Bonds sell at a premium to its "intrinsic value" measured by STRIPS, or even other Treasury

papers. Whether this creates an arbitrage opportunity is another question, as that must

include the costs of shorting the overvalued paper. It is common for recently issued papers

to trade on special in the repo market, which means the costs of borrowing the paper to

short it may well offset the apparent price differential.

4. Bond Risk Measures

Bond risk measure concentrates on Duration and Convexity concepts. Duration is a

sophisticated measure of finding out the payback period of a bond as it takes into account


20/33


all cash flows and the time value of money.The duration of a bond is a measure of how long,

on average, the holder of the bond has to wait before receiving cash payments. A zero-

coupon bond that matures in n years duration of n years. However, a coupon-bond

maturing in n years will have a duration of less than n years. This is because some of the

cash payments are received by the holder prior to year n. Duration is therefore the

weighted average of the net present values of the cash flows where the weights, w t, are the

present values of the payments in each period. It allows us to compare the riskiness of

bonds with different maturities, coupons etc. For coupon bonds, duration is less than time

to maturity because some of the coupons are received in years prior to maturity of the

bond. For zero-coupon bonds, duration equals time to maturity.

For estimating duration, we not only consider the maturity over which cash flows are

received but also the time pattern of interim cash flows. Hence the of a bond without

embedded options is a measure of the sensitivity of its market price to a change in interest

rates. Bond having higher duration would mean a higher sensitivity to changes in interest

rate.

We can write the bond price equation as

A specific formula for computing duration may be obtained by taking the derivative of price

with respect to r. If we differentiate both sides of above equation:

Multiplying the above by (1 + r), we get:

Finally, divide the above equation by P and get:

n

t

t

trCP

1

)1(

n

t

t

t rtCdr

dP

1

1)1(

n

t

t

trtC

dr

dPr

1

)1()1(

DP

rC

trdr

PdP n

t

t

t

1

)1(

)1/(

/


21/33


where the expression in the square bracket is defined as weight. This is the Macauley

duration (D):

Duration is therefore the

weighted average of the net present values of the cash flows. It allows us to compare the

riskiness of bonds with different maturities, coupons etc. we can rewrite is as

From the above equation, the negative of duration, -D, can be interpreted as the percentage

change in the bond price, dP/P, induced by a change in the bond's yield, dr, scaled by

1/(1+r). In other words, we can say that duration not only measures the weighted average

pay back period for the bond; it also approximates the elasticity of the value (price, or P) of

the bond with respect to a change in one plus the bonds yield to maturity. Macaulays

duration is approximately equal to the negative of the elasticity with respect to a change in

one plus the internal yield to maturity.

Portfolio managers use the measure of duration which is simply the percentage change in

bond price, dP/P, induced by a change in yield, dr, rather than to think of it as an elasticity

measure. To find this expression, we divide the equation by (1 + r), and get:

Dm is called modified duration and is used to measure interest risk. It is simply the Macaulay

duration as defined divided by (1 + r). If we have semi-annual cash flows from coupon, then

the same will be (1+r/2) instead of (1+r). For our Bond 8.79% GS 2021 maturing on 08-Nov-

2021 and trading at an yield of 8.6021% on January 1, 2012, the Duration (6.7262)

calculation is as follow (Table 5):

,

)1(

1

n

t

t

t

P

rC

tD

.)1/(

/

1

Dtwrdr

PdP n

t

t

.)1()1(

)1(/

1

m

n

t

t

t Dr

DrP

rCt

dr

PdP


22/33


The Duration of the bond is 6.7262 and the Modified Duration of the bond is 6.4488 which is

nothing but The modified duration computed in this example predicts that if interest rates increase by

100 basis points, the bond price will drop by 6.4488%. Conversely, a decrease in yield of 100

basis points implies a 6.4488% increase in the bond price. This shows that the percentage

change in a bond price is equal to its modified duration multiplied by the size of the parallel

shift in the yield curve. From the duration equation, we can also say that the change in price

of a bond (dP) is equal to negative of Modified Duration multiplied by Price of the Bond and

Table 5: BOND DURATION

Settlement

Date

Cash flow

DateCash flow Period

Period in

Years

Discount

Factors

Discounted

Cash Flow Duration

Value Date

(The

previous

holderceases and

new one

takes over

legal

ownership)

CouponPayment

Dates

Money Receivedon each Coupon

Date

Days from

Value Date

to NextCoupon

date

(investment

time)

Period inYears (30/360

Basis)

r=8.6021%DF * Cash

Flow

(DCF*

Price)/

Time

01-Jan-12 08-May-12 4.395 127 0.3528 0.9707 4.2663 0.0147

08-Nov-12 4.395 307 0.8528 0.9307 4.0904 0.0340

08-May-13 4.395 487 1.3528 0.8923 3.9217 0.0518

08-Nov-13 4.395 667 1.8528 0.8555 3.76 0.0680

08-May-14 4.395 847 2.3528 0.8202 3.605 0.0827

08-Nov-14 4.395 1027 2.8528 0.7864 3.4563 0.0962

08-May-15 4.395 1207 3.3528 0.754 3.3138 0.1084

08-Nov-15 4.395 1387 3.8528 0.7229 3.1771 0.1194

08-May-16 4.395 1567 4.3528 0.6931 3.0461 0.1293

08-Nov-16 4.395 1747 4.8528 0.6645 2.9205 0.1383

08-May-17 4.395 1927 5.3528 0.6371 2.8001 0.1462

08-Nov-17 4.395 2107 5.8528 0.6108 2.6846 0.1533

08-May-18 4.395 2287 6.3528 0.5856 2.5739 0.1595

08-Nov-18 4.395 2467 6.8528 0.5615 2.4678 0.1650

08-May-19 4.395 2647 7.3528 0.5383 2.366 0.1697

08-Nov-19 4.395 2827 7.8528 0.5161 2.2684 0.1738

08-May-20 4.395 3007 8.3528 0.4949 2.1749 0.1772

08-Nov-20 4.395 3187 8.8528 0.4744 2.0852 0.1801

08-May-21 4.395 3367 9.3528 0.4549 1.9992 0.1824

08-Nov-21 104.395 3547 9.8528 0.4361 45.5293 4.3762

Dirty Price 102.5066 6.7262

AI 1.2941 6.4488

Clean price 101.2125


23/33


change in interest rate. For this bond, if our expected change in interest rate is 1bps

increase, the expected change in price will be -6.4488*102.5066*.01% = 0.0661in opposite

direction (fall). This equation enables a hedger to assess the sensitivity of a bond to

infinitesimally small changes in its yield (discount or interest rate).

The duration price sensitivity or elasticity depends on the maturity, coupon, and yield

to maturity. The longer maturity bonds likely to have greater the duration, ceteris paribus

(Chart 5).

Higher coupon bonds are likely to have smaller duration (Chart 6) as larger part of the cash

flows will be received in early stages. Coupon payments cause weight to be put on the early

years in the duration formula.

0.0000

2.0000

4.0000

6.0000

8.0000

10.0000

12.0000

0.00 5.00 10.00 15.00 20.00 25.00

DURATION

Maturity

Chart - 5: Duration and Maturity

Duration

6.0000

6.5000

7.0000

7.5000

8.0000

8.5000

9.0000

9.5000

0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00%

DURATION

Coupon

Chart - 6: Duration and Coupon

Duration


24/33


Third, duration decreases with increases in Yield to Maturity (Chart 7). This occurs

because an increase in the yield (interest rate) has a greater damping effect on the present

value of a distant coupon than on the present value of a nearby coupon.

For a standard bond, duration is not infinite and the duration for a 30-year bond and a

perpetual will be similar. This can be understood by looking again at the mathematics: the

net present value of cash flows 30 years away are small, and the longer out, will get smaller.

So, in calculating the duration, these terms will become negligible. The maximum duration

for a standard bond depends on the yield environment.

PV01 of a bond is defined as the price change in the bond if the yield changes by 1bps. This

can be approximated by

. For our bond, the same is equal to (6.4488*102.5066)/10000 = -0.0661.

Duration is used as a hedging tool. If we do a Principal Component Analysis using the

historical yield curves (CCIL yield curves from 01-Jan-2004 to 27-Apr-2012), we find that at

least 52% of shifts in the curve is parallel, while 30% shifts are slope related and only 7%

shifts are curvature related ones. Parallel shifts in the term structure imply that the change

in yield is the same for all maturities.

Duration is derived from the first derivative of the Bond price equation. Mathematically,

duration is a first approximation of the price/yield relationship. Modified duration is an

approximation of the percentage change in bond price for a given change in yield. In fact, it

4.5000

5.0000

5.5000

6.0000

6.5000

7.0000

7.5000

8.0000

0.00% 5.00% 10.00% 15.00% 20.00% 25.00%

DURATION

Yield

Chart - 7: Duration and Yield

Duration


25/33


is accurate only for very small and parallel shifts in the yield curve. Duration can

approximate price changes if the yield changes are small but when the yield changes are

large, the duration does not approximate the price changes accurately. This is so because of

bonds have different degree of convex shapes. When considering moderate - or large -

changes in interest rates, a factor known as convexity is important. That is, duration

attempts to estimate a convex relationship with a straight line (the tangent line). The above

can be explained with the example of our Bond.

Table 6: Price Change and Modified Duration

Interest Rate

shock

Actual Price

Change

Price Change

implied by MD Difference (Error)

0.10% -0.6421 -0.6610 0.0189

-0.10% 0.6477 0.6610 0.01340.20% -1.2787 -1.3221 0.0434

-0.20% 1.3009 1.3221 0.0211

0.30% -1.9099 -1.9831 0.0733

-0.30% 1.9599 1.9831 0.0232

0.40% -2.5356 -2.6442 0.1085

-0.40% 2.6246 2.6442 0.0196

0.50% -3.1560 -3.3052 0.1492

-0.50% 3.2950 3.3052 0.0102

0.60% -3.7712 -3.9663 0.1951

-0.60% 3.9713 3.9663 -0.00500.70% -4.3810 -4.6273 0.2463

-0.70% 4.6534 4.6273 -0.0261

0.80% -4.9857 -5.2884 0.3027

-0.80% 5.3416 5.2884 -0.0532

0.90% -5.5852 -5.9494 0.3642

-0.90% 6.0357 5.9494 -0.0863

1.00% -6.1797 -6.6105 0.4308

-1.00% 6.7359 6.6105 -0.1254

1.50% -9.0771 -9.9157 0.8385

-1.50% 10.3298 9.9157 -0.41422.00% -11.8543 -13.2209 1.3666

-2.00% 14.0846 13.2209 -0.8637

The error in approximating the bond price changes using modified duration gets larger as

the interest shocks become larger. Further, the increase in interest rate has a relatively

lesser impact on bond price changes that the fall in interest rate. Hence, duration

underestimates the price change in case of interest rate fall and over estimates the price

change in case of an increase in interest rate (Chart 8).


26/33


The actual price change curve looks more convex vis--vis the linear line suggested by

modified duration. Hence, we need to look at the effect of convexity on the price change to

figure out better precision.

Further, effective duration can also be approximated using a uniform change in interest rate

on both sides (positive and negative) and generally given by

In our example, if we shock the present yield of 8.6021% by 20bps in either direction and

use their resultant prices, we can approximate duration. The same is:

In our actual calculation, we arrived at -6.4488 for modified duration (effective convexity).This is so as the above was an approximation.

60.0000

65.0000

70.0000

75.0000

80.0000

85.0000

90.0000

95.0000

100.0000

105.0000

110.0000

115.0000

120.0000

125.0000

130.0000

5.00% 7.00% 9.00% 11.00% 13.00% 15.00%

Price

Yield

Chart - 8: Price Change & Duration

Price Price suggested by Duration


27/33


Greater precision in measuring the bond's sensitivity to yield changes can be achieved by

taking into account the bond's convexity. To understand convexity, we have to remember

that a fundamental property of calculus is that a mathematical function can be

approximated by a Taylor (or Maclaurin) series. The more terms of a Taylor series used, the

better the approximation. We expand the bond price equation used earlier into a Taylor

series (using only the first two terms):

The error term recognizes the fact that we have used only the first two terms of the Taylor

series expansion. (the number two in second part of the equation is in fact 2!, i.e., two

factorial.) Now,

ignoring the error

term and divide both

sides by P.

05-08-12

05-08-14

05-08-16

05-08-18

05-08-20

05-08-22

05-08-24

05-08-26

05-08-28

05-08-30

05-08-32

05-08-34

05-08-36

05-08-38

05-08-40

0

5

10

15

20

25

30

0

0.

025

0.

05

0.

075

0.

1Maturity Date

Modified Duration

Coupon Rate

Chart - 9: Effects of Coupon and Maturity on Duration of High-Yield Bond

2

2

2

)(1

2

1)( dr

Pdr

Pddr

dr

PdP

P

dP


28/33


The fact that duration is the first term of the Taylor series can be seen by comparing the first

right-hand side of equation above with other equations. The second term of the Taylor

series expansion requires the calculation of the second derivative of the bond price

function. Using the above analogies, we can define convexity as

The convexity measure

is an approximation of the curvature. Hence, using both duration and convexity, we can

approximate the price change in the bond as:

dP/P = - Dm * dr + Convexity * (dr)2

We obtain the value of the second derivative of the bond price (dP) with respect to a change

in yield (dr) by differentiating bond price equation again:

Convexity varies with maturity. A longer bond is likely to be more convex than a short

duration bond (Chart 10).

50.0000

60.0000

70.0000

80.0000

90.0000

100.0000

110.0000

120.0000

6.00% 7.00% 8.00% 9.00% 10.00% 11.00% 12.00% 13.00% 14.00% 15.00% 16.00%

PRICE

YIELD

Chart - 10: Convexity of Bonds

PRICE2017 PRICE2020 PRICE2023 PRICE2027

PRICE2030 PRICE2034 PRICE2037 PRICE2040

Pdr

PdConvexity

1

!2

12

2

.)1(

)1(2

12

2

t

tn

t r

Ctt

dr

Pd


29/33


For our Bond 8.79% GS 2021, the Convexity works out to be 55.5712 as detailed below

(Table 7):

If we use the approximation, effective convexity will be written as

Using our example, the same works out to

To complete our illustration, we now use convexity in conjunction with modified duration to

arrive at a more accurate estimate of the percentage change in bond price attributable to a

100 basis point (1%) fall in yield.

dP/P = - Dm * dr + .5* Convexity * (dr)2

= - 6.4488 * (-0.01) + .5 *55. 5712* (-0.01)^2 = 6.727 %

Table 7: Bond Convexity

SDATE

Next

CouponDate Coupon Years*2

DiscountFunction DCF

Share

of CF inPrice DUR

Share*(t(t+1))

1 /(1+y%/2)^2 CONVX

1-Jan-

12 8-May-12 4.395 0.710.9707 4.2663

4.16%0.0294

0.0501 0.9192 0.0460

8-Nov-12 4.395 1.71 0.9307 4.0904 3.99% 0.0681 0.1841 0.9192 0.1693

8-May-13 4.395 2.71 0.8923 3.9217 3.83% 0.1035 0.3836 0.9192 0.3526

8-Nov-13 4.395 3.71 0.8555 3.7600 3.67% 0.1359 0.6396 0.9192 0.5879

8-May-14 4.395 4.71 0.8202 3.6050 3.52% 0.1655 0.9442 0.9192 0.8679

8-Nov-14 4.395 5.71 0.7864 3.4563 3.37% 0.1924 1.2900 0.9192 1.1858

8-May-15 4.395 6.71 0.7540 3.3138 3.23% 0.2168 1.6704 0.9192 1.5354

8-Nov-15 4.395 7.71 0.7229 3.1771 3.10% 0.2388 2.0791 0.9192 1.9112

8-May-16 4.395 8.71 0.6931 3.0461 2.97% 0.2587 2.5108 0.9192 2.3080

8-Nov-16 4.395 9.71 0.6645 2.9205 2.85% 0.2765 2.9603 0.9192 2.72128-May-17 4.395 10.71 0.6371 2.8001 2.73% 0.2924 3.4231 0.9192 3.1466

8-Nov-17 4.395 11.71 0.6108 2.6846 2.62% 0.3066 3.8951 0.9192 3.5804

8-May-18 4.395 12.71 0.5856 2.5739 2.51% 0.3190 4.3725 0.9192 4.0193

8-Nov-18 4.395 13.71 0.5615 2.4678 2.41% 0.3299 4.8521 0.9192 4.4602

8-May-19 4.395 14.71 0.5383 2.3660 2.31% 0.3394 5.3309 0.9192 4.9003

8-Nov-19 4.395 15.71 0.5161 2.2684 2.21% 0.3476 5.8061 0.9192 5.3372

8-May-20 4.395 16.71 0.4949 2.1749 2.12% 0.3544 6.2756 0.9192 5.7687

8-Nov-20 4.395 17.71 0.4744 2.0852 2.03% 0.3602 6.7371 0.9192 6.1930

8-May-21 4.395 18.71 0.4549 1.9992 1.95% 0.3648 7.1890 0.9192 6.6083

8-Nov-21 104.395 19.71 0.4361 45.5293 44.42% 8.7524 181.2236 0.9192 166.5856

DP 102.5066 13.4523

Semi-annual

Convexity 222.285

AI 1.294 DUR 6.73 A Convexity 55.5712

CP 101.213 MDUR 6.4488


30/33


The actual price change for the fall of 1% interest for our Bond works out to be 6.7359%

which is very close to the actual change. Hence Convexity helps in accurate approximation

of price change in a bond. The error gets reduced when we use convexity along with the

duration. If the interest rate rises by 1%, then the bond price will fall by

= - 6.4488 * (0.01) + .5 *55. 5712* (-0.01)^2 = 6.171 %

In the above example, the contribution from convexity in price change is about 0.28%. Since

convexity is positive for a non-callable bond, it always helps the trader as it either reduces

the risk or increases the return. After considering the convexity factor, the price change

approximation is more accurate (Chart 11).

For a long position in bond portfolios, a high-convexity bond portfolio with a certain

duration is always more attractive than a low-convexity portfolio with the same duration.

Investors use duration and convexity for hedging their exposure to possible adverse changesin interest rates. In many cases, institutional investors endeavor to immunize their bond

portfolios such that their duration is equal to the duration of their liabilities. Such portfolios

tend to be devoid of interest rate risk and they should be able to provide guaranteed return

over their horizon interval. It is true that portfolios should be re-immunized with each

change in interest rates, because as interest rates change, durations change as well.

60.000065.0000

70.000075.000080.000085.000090.000095.0000

100.0000105.0000110.0000115.0000120.0000125.0000130.0000

5.0000% 7.0000% 9.0000% 11.0000% 13.0000% 15.0000%

Price

Yield

Chart - 11: Price Change _ Duration and Convexity

Price Price suggested by Duration Price suggested by D and C


31/33


For bonds having embedded options like Call and Put features, the behavior of duration and

convexity will be different. In fact for a bond having call option, the convexity may be

negative. Suppose our bond in the example has a Call option when the price reaches (dirty

price for simpilicity) 103.50 (yield falls so that the issues can replace the bond with a lower

interest rate bond). Our duration and convexity calculation becomes:

and

For this callable bond, convexity is not the friend of the trader anymore as the negative

convexity will reduce the profit for the trader or increase his loss when interest rate

changes. This happens when yields drop and prices increase above par: that is, when prices

approach or exceed the redemption price of a security, the price-yield relationship becomes

non-convex so although the price of a callable bond will still increase in response to a fall

in yields, it will do so at a decreasing rate. This is because the chances that the security will

be called increases.

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0.045

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0.115

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0.185

Coupon Rate

Convexity

Yield

Chart - 12: Effects of Coupon Rate and Yield on Convexity


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Convexity is generally thought of as desirable and traders may give up yield to have

convexity. This is because, in general, a more convex bond will outperform in an

environment of falling yields, and will underperform less as yields rise. Furthermore, the

convexity effect is asymmetric; for a specified fall in yields, the price rise on account of

convexity will be greater (in magnitude) than the price fall related to the convexity factor for

the same rise in yields. Convexity can therefore be associated with a bonds potential to

outperform. However, convexity is a good thing if the yield change is sufficient that the

greater convexity leads to outperformance.

Duration and Convexity are used for immunizing portfolios by fund managers. Suppose we

have the following bonds of 1 year maturity in the portfolio:

Bonds Coupon Yield Price Duration Convexity FV Investment

Bond 1 0% 5% 95.2381 1 1.8141 16832 -16030

Bond 2 4.939% 5% 99.9419 0.9639 1.4048 100000 99942

Bond 3 5.939% 5% 100.9058 0.9616 1.4004 83158 83911

We buy 9.939% bond and sell the other two bonds with 16.04% of zero coupon bond and

83.96% of 5.939% bond. If the interest rate moves by 50bps (rise) after our purchase, the

portfolio is immunized to a very large extent:

Bonds New price New Investment Value

% change by

Duration and

Convexity

change by

Duration and

Convexity Actual change

Bond 1 94.7867 -15955 0.50% 79.79 -75.97

Bond 2 99.4620 99462 0.48% -479.92 479.87

Bond 3 100.4224 -83509 0.48% 401.98 -401.99

0.00

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70.0080.00

1.13 2.13 3.13 4.13 5.13 6.13 7.13 8.13 9.13 10.13 11.13

Convexity

Maturity

Chart - 13: Duration and Convexity

CONVEXITY MDURATION


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Risk in sovereign bond markets is predominantly interest rate risk. Duration matching is

used to reduce this risk. Since daily interest rate changes are not large, duration works fine

for short horizon investments. Of course, there are many other risks that an investor will

take into account when buying a bond - in liquidity risk. Since investors have different risk

appetite, it helps create a liquid market.

5. Concluding Remarks

The document has concentrated on various issues on Bond price and yield and its practical

implications for the traders. The document has been written to explain the Indian market

conventions.

basic bond analysis2

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