finance - notes - topic 7

7/31/2019 Finance - Notes - Topic 7

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TopicTopic 77-- FixedFixed IncomeIncomeSecuritiesSecurities

David Moreno (Spanish version)

Beatriz Mariano (English version)

Universidad Carlos III

Financial EconomicsTeacher: Beatriz Mariano


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TopicTopic 77-- FixedFixed IncomeIncome SecuritiesSecurities

OutlineOutlinePART IPART I

1.1. VALUATION OF FIXEDVALUATION OF FIXEDINCOME SECURITIESINCOME SECURITIES

Valuation of fixed incomesecurities with periodical couponpayments

Valuation of strips Computation of accrued interest

2.2. THE TERM STRUCTURE OFTHE TERM STRUCTURE OFINTEREST RATESINTEREST RATES

Spot interest rates

The yield curve

PART IIPART II

1.1. FORWARD INTEREST RATESFORWARD INTEREST RATESAND THEORIES THATAND THEORIES THAT

EXPLAIN THE TERMEXPLAIN THE TERM

STRUCTURESTRUCTURE

Forward interest rates

The expectations hypothesis theory

The liquidity-preference theory

The inflation premium theory

2.2. RISK MANAGEMENTRISK MANAGEMENT

Default risk

Interest rate risk: Duration andImmunization


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TopicTopic 77-- FixedFixed IncomeIncome SecuritiesSecurities

ObjectivesObjectivesCompute the price of different fixed income securities andunderstand how it varies with different variables.

Understand how the interest rate varies with maturity and howthis is translated in the term structure of interest rates.

Distinguish between spot and forward rates and learn how to

compute them.

Learn the different theories that explain the term structure ofinterest rates.

Learn the types of risk that affect fixed income securities: interestrate risk and default risk. Learn how to measure interest rate riskusing duration and how to eliminate it (immunization).


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PartPart II


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11--VALUATION OF FIXED INCOME SECURITIESVALUATION OF FIXED INCOME SECURITIES

Fixed income securities are securities that promise theirholders the future payment of predetermined cashflows upto their maturity date.

Fixed income securities represent a loan that the firm thatissues them receives from the investors that buy them.

Most important elements of fixed income securities: Face value : Principal amount of the loan ( or security) used to calculate the

future regular payments (coupons).

Coupon: the interests payments paid at regular intervals (month, quarter, year,etc.) which are set at the time of the issue.

Maturity date: Time in the future at which the loan (or security) ceases to existand the principal repayment takes place.

Principal repayment or amortization: Repayment of the principal at thematurity date.


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Other variables that characterize bonds: Based on the issue price:

Par Bond

Discount bond (below par) Premium bond (above par)

Based on repayment value:

At par Below par Above par

Types of fixed income securities: Bonds Inflation-indexed bonds (tips) Pure discount or zero-coupon bonds (strips)

Pay a stated coupon at

regular intervals


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B.B. CONSTANT COUPON AMOUNT (C)CONSTANT COUPON AMOUNT (C)

Assume that a fixed income security pays regularly at each period(at t=1, 2, 3.n) a coupon that is equal to C and the bond is repaidat par. The cash-flows are represented as follows

The value of the bond is calculated as follows:

N

N

N

tt

t

N

N r

FV

r

C

r

NC

r

C

r

CP

)1()1()1(...

)1()1( 12

21

0+

++

=+

+++

++

+=

=

tnMaturity date

tn-1t2 t1Today (t0)

C C C C + FV


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C.C. ZERO COUPON BONDSZERO COUPON BONDS

Fixed income securities that do not pay any coupon over their lifetime,the only cash-flow that their holders receive is the principal amount at the

maturity date (it has no explicit return).

If the repayment amount exceeds the amount at which it was issued itsholder has an implicit return.

This are very important securities, which are going to be used to obtainthe spot interest rates and the Yield Curve.

The value of such bond is calculated as follows:tN

CN

Today(t0)

t1 tN-1.


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ExampleExample:: Compute the value of a zero-coupon bond with aface value of 10.000$ issued at 1-1-2004 that matures at 1-1-2009 if the bonds amortization is at 120%. Suppose that thespot interest rate at 5 years is equal to 4.5%.

SolutionSolution::


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COUPON RATE:COUPON RATE:

The interests payments paid at regular intervals.

ExampleExample:: Suppose that a bond issued by Telefnica paysan annual coupon of 3.5% and has a face value of 6.000

CURRENT YIELD OR INTEREST YIELD:CURRENT YIELD OR INTEREST YIELD:

Annual coupon payment divided by the bonds market price.

ExampleExample:: Suppose the market price of the Telefnica

bond is 5650.


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YIELD TO MATURITY (IRRYIELD TO MATURITY (IRRororYTM):YTM):

Measures the return obtained by someone that buys the bond todayand holds it up to maturity.

COMPUTATION: It is the single interest rate that equates thepresent value of the bonds cash-flows to its price: it is the IRR orthe yield to maturity

Interpretation:

It is the compounded interest rate that would have been

obtained over the lifetime of the bond under the assumption

that the COUPON PAYMENTS ARE REINVESTED at the

same interest rate and the bond is held until maturity.

nIRR

Coupon

IRRIRR )1(

Ppal...

)1(

Coupon

)1(

CoupontodayPrice

21+

+++

++

+=


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The Yield to Maturity is a very complex and complete measure of thereturn of an individual security. It is affected by:

1. Issue price of the security: at par, below par or above par.

2. Principal Repayment: at par, below par or above par.

3. Coupons: coupon amount, payment frequency monthly, semi-annually or annually.

ExampleExample:: Compute the yield to maturity of a two year bond with facevalue of 1.000 and coupon rate of 4%, whose market price is equal to

963.69. Principal repayment at par. SolutionSolution::


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SolutionSolution::

We are looking for the discount rate or the average interest ratethat equates the price to the PV of the CF:

The IRR can be obtained by trial and error. In the previous case it canbe solved using the formula for the solution of a quadratic equation.


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ExampleExample usingusingExcelExcel

The function TIR in Excelcomputes the Internal Rate ofReturn for the cash-flows paidout at regular intervals

(monthly, semi-annually,annually)

This function gives the IRRexpressed in terms of this timeinterval.

When the cash-flows are notpaid out at regular intervals useTIR.NO.PER


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FLAT PRICE:FLAT PRICE: It is the quoted price of a bond which does notinclude the part of the next coupon that is due at a particularpoint in time (accrued interest).

The buyer of a bond pays to the seller of the bond

Flat price + accrued interest.

This is know as the full price

ACCRUED INTERESTACCRUED INTEREST

It is calculated as follows:

Coupon()*paymentscouponeconsecutivobetween twTime

bondtheheldsellerthetimeofAmount=AI


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ExampleExample:: Today 1-9-2004 you want to buy a (Spanish) Treasurybond which matures at 31-12-2009. The bond pays an annualcoupon of 7%. Moreover, the quoted price of the bond is 946.88

euros. How much do you pay the seller?a. 946.88

b. More than 946.88

SolutionSolution::


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22--THE TERM STRUCTURE OF INTEREST RATESTHE TERM STRUCTURE OF INTEREST RATES

Next we are going to see the definition and how to construct the termstructure of interest rates.

Before doing so we need to define the following concept: SPOTSPOT interestinterest raterate forfor thethe timetimeperiodperiod [0,t]:[0,t]: The internal rate of

return of a bond of the highest credit quality whose repayment

takes place at t. It is denominated as 0Rt Therefore, 0Rt can be obtained using the market prices of strips

(stripped bonds).

ExampleExample:: The spot interest rate 0Rt is obtained using the priceof a strip issued at t0 that matures at tt.

1)1(

/1

0

0

0

0

=

+=

t

ttt

t

t

P

CR

R

CP


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HOW TO INTERPRET THE SPOT INTEREST RATE:HOW TO INTERPRET THE SPOT INTEREST RATE:

if one period=one year

The spot interest rate is like an average annual interestinterestraterate thatthat isis obtainedobtained betweenbetween [0,t][0,t]..

As an average ANNUALaverage ANNUAL returnreturn correspondingcorresponding toto thetheperiodperiod [0,t][0,t] given that it is an IRR.


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ExampleExample:: Suppose that thefollowing zero coupon bondsof the (Spanish) Treasury have

current quoted prices asfollows (face value of the bond1.000)

Compute the 1-year, 2-year, 3-year, 4-year and 5-year spotinterest rates.

SolutionSolution:: Bond 1

PaymentSchedule(years)

Repayment Quoted Price

1 100% 97,561%2 105% 98,018%

3 110% 97,790%

4 120% 101,013%5 125% 99,115%


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Bond 2:

Bond 3:

Bond 4:

Bond 5:


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If we plot in a graph the spot interest rates computed before for thecorresponding maturities (1, 2, 3, 4 and 5 years), we obtain the yield curve forthis market.

In the following section we will see why this happens and its

implications but before we need to define forward interest rates.

2,5%

3,5%

4,0%

4,4%4,75%

0,0%

0,5%

1,0%

1,5%

2,0%

2,5%

3,0%

3,5%

4,0%

4,5%

5,0%

1 2 3 4 5

Spoti

nterestrates

Maturity (years))

Yield curve of spot rates


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CONCLUSIONCONCLUSION: This relationship between spot interest rates and maturitydates is called yield curve.

The shape of the yield curve as of a particular point in time can be verydiverse and it changes over time.

i

Matur.

i

Matur.

i

Matur.

i

Matur.


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FORWARD INTEREST RATEFORWARD INTEREST RATEThe forwardforward interestinterest raterate atat time ttime t00 forfor thetheperiodperiod [t[t11,t,t22]] needs tosatisfy the following equation:

ExampleExample:: For the simple case with two periods the expression usedto compute the forward interest rate is the following:

Interpretation of the interest rate 0F1,2:

0F1,2 is the interest rate that should take place between periods 1 and 2,such that the return from investing in long-term bonds (buying a bondthat matures in 2 years) is equal to the return of buying a short-term

bond (buying a bond that matures in 1 year) and then reinvesting themoney (buying another 1-year bond).

)(

,

)()( 12

210

01

10

02

20)1()1()1(

tt

ttt

tt

tt

tt

tt FRR

++=+


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This can be analyzed graphically.The forward interest rate equates the returns of bothinvestment strategies:

We will see later that the forward interest rates are very important toexplain the term structure of interest rates and the shape of the yieldcurve.

(1+0R2)2

(1+0R1) (1+0F1,2)

(1+0R1)(1+0F1,2)

Buy 2-year bond

Buy 1-year bondReinvest in another 1-

year bond


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REMEMBER THAT :REMEMBER THAT :

InIn anan ENVIRONMENT WITH CERTAINTY:ENVIRONMENT WITH CERTAINTY: Theforward interests rate are going to coincide with interest rates

in the future, in order to eliminate arbitrage opportunities. For example 0F1,2=1R1

InIn anan ENVIRONMENT WITH UNCERTAINTY :ENVIRONMENT WITH UNCERTAINTY : Thisdoes not have to be the case

Then, the future spot interest rates (1R1, 2R1,) are not

known. However, the forward interest rates for the same time

intervals (0F1,2; 0F2,3; ) are known today as they arecalculated using the term structure of interest rates.


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ExampleExample:: Suppose that you read in todays newspaper that the1-year and the 2-year spot interest rates are equal to 3% and4% respectively. Compute the forward interest rate for the

time interval [1,2].

SolutionSolution::


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ExampleExample:: Suppose that you read in todays newspaper that the1-year, 2-year and 3-year spot interest rates are equal to 3%, 4%and 4.5% respectively. Compute the current forward interest rate

for the period between years 2 and 3?

SolutionSolution::


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33--THEORIES THAT EXPLAIN THE TERMTHEORIES THAT EXPLAIN THE TERM

STRUCTURESTRUCTURE

The definition of forward interest rate is going to be usedto explain the shape of the yield curve.

Each theory assumes that the forward interest rate dependson some variables.

To simplify, assume there are only two periods.

Remember that:

(1+0R2)2 =(1+0R1)(1+0F1,2)


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STRUCTURESTRUCTURE

A.A. THE EXPECTATIONS HYPOTHESIS THEORYTHE EXPECTATIONS HYPOTHESIS THEORY

Forward interest rates are a function of the expected

future interest rates by all the agents in the market

Therefore, according to this theory:

i. If the agents expect interest rates to increase in the

future, then the yield curve ispositively sloped.ii. If the agents expect interest rates to decrease in the

future, then the yield curve is negatively sloped.

0F1,2 = E0[1R1]


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STRUCTURESTRUCTURE

B.B. THE LIQUIDITY PREFERENCE THEORYTHE LIQUIDITY PREFERENCE THEORY

Assumes that there is a risk premium (L) for bonds with longermaturity therefore, the forward interest rates depend on the

expected interest rates for the future and of this liquidity premium.

C.C. THE INFLATION PREMIUM THEORYTHE INFLATION PREMIUM THEORY

There is a premium associated to inflation risk (), and the bonds

issuers have to compensate bondholders for this risk.

0F1,2 = E0[1R1]+L1

0F1,2 = E0[1R1]+1

3 T O S T AT A T TTHEORIES THAT EXPLAIN THE TERM


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STRUCTURESTRUCTURE

These theories help us to derive the short term rates expectedby the market for future dates (E0(1R1); E0(2R1);)

ExampleExample:: Suppose that current 1-year, 2-year and 3-yearspot rates are equal to 4.5%, 4% and 3.8% respectively. There

is a liquidity premium of 1.2% and an inflation premium of0.5%. Compute the one-year interest rate expected for nextyear.

AnswerAnswer::


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44-- INTEREST RATE RISK MANAGEMENTINTEREST RATE RISK MANAGEMENT

There is a general impression that there is no risk associated with fixedincome securities. We are going to see that this is not true. Fixedincome securities are affected by: Default Risk and Interest Rate

Risk.

DEFAULT RISK:DEFAULT RISK:

It refers to the possibility that the issuer fails to make the payments onthe bond/loan.

In other words, that the issuer fails to pay the coupon or fails to repaythe principal at maturity.

It is obvious that this risk is higher the lower the credit quality of theissuer. For this reason, investors demand a higher return for investingin securities issued by entities with a lower credit quality (as can beobserved from the following picture).


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The credit quality of bondscan be measured by a creditrating that is issued by a

rating agency, for example: Moodys

Standard and Poors.

Moodys gives a triple A rating tothe highest quality bonds.

The bonds with a Baa or morerating are known as investment

grade bonds and many institutionalinvestors (banks) can only invest inbonds which have been rated.

The bonds with a rating lower than

Baa are known as junk bonds.

There is a close relationshpbetween the rating and the returnof the bond.


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The bonds issued with maturities longer than one year receive a rating between Aaa and C: from the highest quality ratingto the lowest quality rating. Intermediate ratingsabove Baa are known as "investment grade." Those below Ba are known as"speculative grade."

These are the possible ratings:

Aaa

Highest quality and lowest risk bonds. Are known as gilt edged. Thecoupon payments, as well as the principal payment, are guaranteed bya wide and stable margin.

Aa High quality bonds. Together with the Aaa bonds are known as high-grade bonds.They are not as safe as the Aaa-rating bonds.

A Bonds of medium to high quality. The factors that guarantee thecoupon paymenst and the principal are good enough, but there areelements that could increase risk.

Baa Bonds known as medium-grade. The coupon and principal payment areadequatly guaranteed at present but there is some uncertainty in thelong run. Have speculative characteristics.

Ba Protection of coupon and principal payments is moderate. Havespeculative elements and are not safe in the future.

B Bonds which lack investment appeal. The guarantee of coupon andprincipal payment, as well as of not violating other terms of the debtcontract, are small in the long run.

Caa Poor quality and high risk bonds. Issuers are in danger of defaulting.

Ca Highly speculative bonds Issuers are usually in default.

C Bonds of insolvent issuers unlikely not to fail with their future debt

obligations.

Moodys

AAA

AA

A

BBB

BB

B

CCC

CC

C

S

tandard

and

Poors



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INTEREST RATE RISK:INTEREST RATE RISK:

The risk that arises from the possibility that the value of a portfolio offixed income securities (or of a single security) decreases due to the

increase of interest rates.

BOND PRICE-INTEREST RATE RELATIONSHIP.

Its is an inverse relationship.

The bond price is a decreasing function of the correspondinginterest rate.

P

i


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These changes in the price of a portfolio of bonds (orindividual bond) are known as its volatility. We are going toanalyse the factors that affect it.

We should not forget that interest risk only affects

positively or negatively the holder of a bond that has to

sell it prior to the maturity date.


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WhichWhich factorsfactors affectaffect thethe sensitivitysensitivity ofof thethe priceprice ofof aa bondbond toto

changeschanges inin thethe interestinterest raterate??

Until the 60s

It was believed that it was a function of the lifetime of the bond.

More sensitive, the farther from maturity.

However, people realized that bonds with similar maturity dates

could have different sensitivities to changes in interest rates, forexample, if coupons were different.

Since the 70s It is known that the sensitivity of the price of a bond to changes in

interest rates depends on the DURATION of the bond.


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Hopewell and Kaufmann (1973) ---- They compute the semi-elasticity of the value of a fixed income security with respectto its yield to maturity (y)

Where Duration is defined as:

DurationyPdi

dP

)1(

11

+=

+=

=

T

stt

ss

osy

Ctt

PD

1)(0

)1()(

1 MACAULAYMACAULAY

DurationDuration

P is the price of the security, theCs are the expected cash-flows


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DURATION:DURATION: It is the average maturity of a security (expressed in years) It is the weighted average of the maturities of the cash-flows

generated by the security, where the weights are the relative weight of

each cash-flow in the total value of the security. Bond with constant coupon payments

Zero coupon bond The duration coincides with the maturity.

ExampleExample:: Compute the duration of a strip issued 4 years ago whichhas 9 years and 6 months left to maturity.

+

+

+

=

=

T

sTs y

FVT

y

Cs

PD

1 )1()1(

1

D= ts-t0


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FACTORS THAT AFFECT DURATION:FACTORS THAT AFFECT DURATION: Time to maturity (T)

If the time to matutity increases, duration increases Coupon amount (C)

If the coupon amounts paid out increase, duration decreases. Interest rates (y)

If the interest rate or the IRR increases, duration decreases.

VOLATILITY :VOLATILITY :

It is the slope of the curve that relates Pbond and the interest rate. It is the same as the Modified Duration (DM)

Therefore, the bonds price sensitivity is equal to:

y

DDM

+=

1

yD

P

PM =

)(


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ExampleExample::

1. Compute the Duration and the Modified Duration of a SpanishTreasury (face value 1,000) with a 5% coupon rate which matures

in 5 years. Assume an yield to maturity of 4%.2. Determine the change in the bonds price if interest rates increase by

25 basis points.

SolutionSolution::

First, compute the bonds price which is the denominator in theduration formula:


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Second, compute duration:

Knowing that

Now, compute the Modified Duration:

Finally, compute the price change due to an increase in interest rates of 25basis points:

++

+=

=

T

sTs y

NT

y

Cs

PD

1 )1()1(

1


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IMMUNIZATION STRATEGYIMMUNIZATION STRATEGY:THE PROBLEM OF AN INVESTOR WITH A

LIMITED INVESTMENT HORIZON An investor who does not want to be affected by interest rate risk

can immunize his portfolio of fixed income securities. In otherwords, if an investor has a limited investment horizon, and does

not wish that the value of his portfolio changes with interest ratesduring this time, he should immunize his portfolio.

The easiest immunization technique is to form a portfolio offixed income securities whose duration equals the investors

investment horizon.

Portfolio Duration =Investment HorizonPortfolio Duration =Investment Horizon


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versus

PRICE RISK

- If interest ratesincrease, bond price

decreases.

REINVESTMENTREINVESTMENT

RISKRISK

-- IfIfinterestinterest ratesrates

increaseincrease,,youyou havehave betterbetter

optionsoptions toto reinvestreinvest thethe

couponcoupon alreadyalready receivedreceived..

These go in opposite

directions and therefore

allowfor

IMMUNIZATIONIMMUNIZATION

T ST T S T


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EXAMPLE:EXAMPLE: Suppose that a family wishes their son toenroll in a prestigious masters programme in 10 years time.The cost of such programme is 35,000.

In order to make sure that they will have enough money tocover for this cost they decide to invest in bonds today suchthat the returns of those bonds during 10 years achieve afuture value (VT) of 35,000.

Lets show that in this case immunization happens whenthe bonds Duration=10. If the familiy invests in bonds that mature in 14 years

(Duration=13.5) If during this period interest rates go up

The family gains (VT) because the coupons that are received over time canbe reinvested at an higher interest rate.

The family losses (VT) at time T (in 10 years) when they sell the bonds inorder to obtain the money to pay for the masters, as when the interest ratesin the market go up the bond prices go down.

44 I TEREST RATE RISK MA AGEME TINTEREST RATE RISK MANAGEMENT


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If Duration=10 years, both movements (reinvestment ofcoupon payments and bond price at T) would be identical.In this case, even if interest rates change FV=35,000.

A very easy exampleof bonds with duration of 10 years is tobuy zero coupon bonds that mature in 10 years. In thiscase, there is no risk related to reinvestment of couponpayments, and with the sale of the bond before maturitygiven that the bond matures exactly when the money for

the masters is needed.But this can also be done with coupon bonds with durationequal to 10.

USEFUL WEBSITES


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USEFUL WEBSITES

MOODYS: httphttp://://www.moodys.comwww.moodys.com

DIRECICIN GENERAL DEL TESORO: httphttp://://www.tesoro.eswww.tesoro.es

AIAF MARKET:

httphttp://://www.aiaf.eswww.aiaf.es

MADRID STOCK EXCHANGE

httphttp://://www.bolsamadrid.eswww.bolsamadrid.es

ANALISTAS FINANCIEROS INTERNACIONALES

httphttp://://www.afi.eswww.afi.es

READINGS


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READINGS

Basic or general:Bodie, Z., Kane, A. and Marcus, A. J. (1999). Investments. McGraw Hill(Fouth Edition)

Chapters 9 and 10.

Brealey, R.A. and Myers, S.C. (2006). Principles of Corporate Finance. McGrawHill

Parts of chapters 23 and 24

Mascareas Prez-Iigo, J. (2002). Gestin de activos financieros de renta fija.Pirmide.

Chapters 4, 5 and 6.

Specialized:

Navarro, E. and Nave, J. (2001). Fundamentos de Matemticas Financieras. Antoni BoschEditor, SA. Chapters 5 y 6.

finance - notes - topic 7

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