CFA 1 Fixed Income II

8/13/2019 CFA 1 Fixed Income II

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Fixed Income

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Securities - II

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Mapping to Curriculum

Reading 56: Understanding Yi Reading 58: Yield Measures,

Reading 59: Introduction to M

eev Knowledge Management Pristine

Expect around 15 questions in th

eld Spreads pot Rates and Forward Rates

asurement of Interest Rate Risk

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e exam from todays lecture



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Key Concepts

Interest Rate Policy

Yield Curve Shapes

Theories Of Term Structure Of

LIBOR

Yield Measures

Reinvestment Risk

Bootstrapping

Nominal Spread, Zero-volatilit


Option-adjusted Spread

Forward Rates

Duration, Convexity, PVBP

Interest Rates

Spread,

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Agenda

Features of Debt Securities

Risks Associated with Investin

Overview of Bond Sectors and

Understanding Yield Spreads


in Bonds

Instruments

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Key Issues In Understanding Yield Sp


Yield Curve

Theories of Term structure of Interest Rates

Spot Rate

Yield Spread measures

Credit S read


Embedded options affect on yield spread

Liquidity affect on yield spread

After-tax Yield

LIBOR

reads

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To implement the Feds monetary policy, the Fed

Discount rate: is the rate at which banks borrow fro

Open Market Operations: refers to purchase and s

Bank Reserve requirements: refers to the percenta

Pursuation: refers to the Fed asking banks to alter

Lowering the discount rate and/or engaging in ope

rates in the market.


ses the following four interest rate tools:

m the Fed.

le of Treasury Securities in the open market.

ge of deposits the bank must keep with itself.

heir lending policies.

n market operations decrease the overall interest

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Yield Curve And Its Shapes

ield Curve: Shows the relationship between Yield

It can be: Upward Sloping - Normal

Downward Sloping - Inverted

Flat

Humped


nd Maturity

Rising Declining

Imp

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Flat Humped



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Theories Of Term Structure Of Interes

Pure Expectations Theory:

States that the future value of interest rates is eq

short-term rates are expected to rise then the yield

Shape of Term Structure Implication AcTheory

Upward sloping (normal) Rates expecte

Downward slo in inverted Rates ex ecte


Liquidity Preference Theory:

States that investors are risk-averse and will dem

Yield curve can be normal, inverted or flat as long

maturity.

Flat Rates not exp

Rates

al to the summation of market expectations. If

curve will be upward sloping

ording to Pure Expectations

d to rise

d to decline

Imp

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nd a premium for securities with longer maturities

s yield premium for interest rate risk increases with

cted to changeThis files has expired at 30-Jun-13


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Theories Of Term Structure Of Interes

Market Segmentation Theory:

States that most investors have set preferences

Example: a bank having large amount of short ter

An offshoot to above theory is that an investor can

preference, if they are compensated for taking on


range. This is known as the Preferred Habitat Th

Rates

regarding the length of maturities they will invest in

liabilities will prefer to invest in short term securities.

be induced to invest outside their term of

that additional risk by moving out of their preferred

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oryThis files has expired at 30-Jun-13


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Spot Rate

The discount rate of a zero coupon bond is called

In the case of a treasury security, its called the tre

The relationship between maturity an d treasury s

rates.

This is different from the treasury yield curve.


1

he spot rate for that maturity.

sury spot rate.

ot rates is called the term structure of interest

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0



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Yield Curve

Spot rate: The rate of return earned on a zero-co

Forward rate: The yield on a zero-coupon securitsecurities have not been issued yet, we can nevershort, a graph of forward rates is a graph of interesecurities in the future. (Forward rates are typicall

Yield curve: A graph that shows the yield earnedthe relationshi between short-term and lon -term


1

upon bond, if held to maturity.

y issued at some point in the future. Since the observe a forward rate, we can only estimate it. In

t rates that are expected to be paid on short-term estimated for 6-month Treasury bills.)

on bonds of various maturities. In short, it shows interest rates.

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1



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Yield Spread Measures

Yield Spread Measures: Yield Spread is the diffe

Absolute Yield Spread = (Yield on the subject bond

Relative Yield Spread = (Absolute Yield Spread/Yi

Yield Ratio = (Subject Bond Yield/Benchmark Bon

eev Knowledge Management Pristine 1

ence between the yield on two bonds

- Yield on benchmark bond)

ld on benchmark bond)

Yield)

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Credit Spread

Credit Spread: It is the spread between non - Tre

all respects except forthe credit rating

In an expanding economy, credit spreads becom

In a contracting economy, credit spreads widen.

This is because in a contracting economy, compan

making it more difficult for corporate issuers to ser


deteriorates, and investors sell corporates and buy

1

sury and Treasury securities that are identical in

narrow

ies experience decline in revenues and cash flows

ice their debt obligations. Thus, credit quality

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treasuries. Thus, widening the spreads.

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Embedded Options

mbedded Options Effect on Yield Spread:

Call Provision:

Grants the issuer the right to retire the debt, fully o

From an investors point of view, a non-callable bon

Investors require a higher yield on the Callable bon

Put Provision/Conversion Provision


A putable-bond is more preferred to a plain vanilla

lower yield spread

The higher spread on an MBS is due to prepayme

1

partially, before the scheduled maturity date.

d is preferred against a Callable bond.

d and the yield spread is also larger for such bonds.

Imp

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bond from the investors point of view and will have a

t risk.

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After-tax Yield

The difference in yield between tax-exempt securit

in terms of absolute yield spread but as a yield rati

One should compare the after-tax yield to arrive at

yieldExemptTax

taxMarginal1*YieldTaxableYieldTaxAfter


ratetaxMarginal1

ies and treasury securities is typically measured not

o.

an investment decision

ate

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LIBOR

IBOR: It stands for London Inter bank Offered Rate

Is the rate paid on Negotiable CDs by banks locate

Determined by the British Bank Association (BBA)

It is quoted in many currencies:

Has become the most important reference rate ove

Is important because the fluctuations in LIBOR will


borrows to make an investments) will be able to bo

1

d in London

r time

impact the rate at which the funded investor (one who

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rrow funds

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Questions

1. The pure expectation theory can be used to expl

most likelyA. Incorrect; The market segmentation theory can b

B. Incorrect; The liquidity preference theory can be

C. Correct; The pure expectation theory explains an

2. With respect to the term structure of interest rate

A. An increase in demand for long term borrowings c


.yield curve

C. The yield curve reflects the maturity demands of fi

. The tool most commonly used by Fed is:

A. Open Market OperationsB. Bank reserve requirement

C. Discount rate

1

in any shape of the yield curve. This statement is

e used to explain any shape of the yield curve

used to explain any shape of the yield curve

y shape of the yield curve

s, the market segmentation theory holds that :

ould lead to an inverted yield curve

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nancial institutions and investors

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Questions (Cont...)

. As per the Liquidity Preference Theory :

A. Investors will demand a premium for shorter matB. Investors will demand a premium for longer matu

C. Investors will not demand any premium.

. As per the Preference habitat Theory :

A. Investors are will not move out of their preferenc

B. Investors demand a premium to invest outside th


.

. The impact of an expanding economy on the yiel

A. To increase the yield spread

B. To decrease the yield spread

C. Will not effect the yield spread

. Which of the following will have the least Yield Sp

A. Callable Bond

B. Putable Bond

C. A plain Fixed Coupon Bond

1

rity securities. rity securities.

habitat

eir preference range

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spread is:

read:

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Solutions

1. A. The market segmentation theory asserts that t

maturity sectors of the yield curve determine the i

. C. The correct answer is the yield curve reflects t

investors.

. A. Open Market Operations

. B. Investors will demand a premium for longer m


. B. Investors demand a premium to invest outside

. B. To decrease the yield spread

. B. Puttable Bond

1

e supply and demand for funds within the different

nterest rate for that sector.

e maturity demands of financial institutions and

turity securities

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their preference range

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Agenda

Introduction to the Valuation of

Yield Measures, Spot Rates, an

Introduction to Measurement of


ebt Securities

Forward Rates

Interest Rate Risk

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Traditional Yield Measures


Current Yield: the annnual interest income from

Current Yield = Annual Coup

Bond Price

The current yield is simply the coupon payment (C

Current yield = C/ P0.


Drawbacks :

Only Considers coupon interest

Capital Gains/Losses not taken into account

No consideration for reinvestment income

2

the bond

n interest received

) as a percentage of the (current) bond price (P).

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Yield to Maturity(YTM): YTM is the IRR of the bon

Yield Measure Relationships:

Bond Selling at: Relationship

Par Coupon rate = C

Discount Coupon rate < C

12

Y T M1

C


dvantages:

Considers both coupon income and capital gain/lo

Considers the timing of cashflows

Limitations

It considers the reinvestment income; the interim cthe YTM.

2

Premium Coupon rate > C

. It is the annualised rate of return on the bond

urrent Yield = Yield to Maturity

urrent Yield < Yield to Maturity

2N2

2

Y T M1

P arC.....

2

Y T M

C

Imp

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s if held to maturity.

oupon payments are reinvested at a rate equal to

4

urrent Yield > Yield to MaturityThis files has expired at 30-Jun-13


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YTM of Annual Coupon Bond:

A 10 year, $1000 par value bond has a coupon of

PV = -920; N=10; FV=1000; PMT=70

I/Y = 8.20%

YTM for zero cou on bond:


The price of a 5-year Treasury bond is $804. Calc

YTM.

Semiannual-pay YTM =

Annual-pay YTM =

2

41.42*1804

1000 101

%46.41804

1000 51

7%. If it is priced at $920 what is the YTM?

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late the semiannual-pay YTM and annual-pay

5



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Bond Equivalent Yield: Doubling the semiannual

Yield to Call: yield on callable bonds (bonds can

premium. The calculation is the same as for norm

price and the total period is substituted with the pe

Yield to Put: yield on puttable bonds that are selli

Yield to Worst:A yield can be calculated for ever


these YTMs is called Yield to Worst.

Cash Flow Yield: used for Amortisinfg Securities.

prepayment rates may differ from those assumed

Yield to maturity (YTM): most popular yield meas

measure is that it assumes that cash flows are rei

2

yield to maturity.

e called before maturity) that are selling at a

l bonds. The par value is substitued with the call

riod upto the call date

g at a discount

possible call date and put date. The lowest of

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The limitation with this measure is that the actual

or calculation purposes.

ure of all the above. The limitation with this

vested at the YTM and the bond is held till maturity

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Reinvestment Income

If the reinvestment rate is less than the YTM then

How to calculate the Reinvestment Income earned

20-year Treasury bond purchased at par, 7% cou

generated to earn a YTM of 7%?

Total Value generated in 20 years = 100(1.035)40

Reinvestment income required = 395.9260 100


Factors Affecting:

Higher the coupon rate higher the reinvestment ris

Longer the maturity higher the reinvestment risk

If the above problem was for a 10 year bond with

would have been $13.8616 as compared to $ 155

2

he actual yield realised will be less than YTM

???

on rate, how much reinvestment income should be

395.9260

40*3.50 = 155.9260

Imp

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coupon of 5%, the reinvestment income required

.9260

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Computing Theoretical Treasury Spot

Bootstrapping: It is the method of calculating the

spot rate is used to calculate the spot rate for the n

used for calculating the next spot rate

Spot Rate Curve:

Theoretical Spot Rate Curve


0%

1%

2%

3%

4%

5%

6%

7%

0 0.5 1 1.5 2 2.5

(Term Structure of interest rates)

Rate

spot rates using the prices of coupon bonds. One

ext period. The two consecutive spot rates are

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3

Rate



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Bootstrapping

Yield Curve

Yields are bond-specific; given a boncoupons, the yieldis the rate that all cash flows are discandfuture values the same.

The spot curve diagrams what pure d

Summarizing the curves


po urve

app es o anycash flow at each maturity point. It isAlso called the zero curve.

Forward

Curve

This is a plot of what the market char6 month

period starting at certain future datesNote that forward curves could be m(i.e. 1 year forwards, 3 month forwar

's market price and

ounted at to make present

iscount rate the market

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not bond specific.

ges to borrow money for a

. de for any borrowing term

s, etc.)



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Bootstrapping Example:

Consider 3 treasury securities with their maturities

Using the method of bootstrapping, find the theore

Maturity Market Rate

6 months 3%

12 months 4%

18 months 5%


Solution:

The bond with six months left to maturity has a seannual bond equivalent yield (BEY) basis.

Since the bond will only make a single payment of

for cash flows to be received six months from now.

3

and market rates given in the table below:

tical Treasury spot rates.

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iannual discount rate of 0.03/2 = 0.015 or 3.0% on an

101.50 in six months, the market rate is the spot rate

2



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Solution

100

)2

1(

102

015.1

2

21

S

The one-year bond will make two payments, one incan solve for the one-year spot rate in the equation

where S1.0 is the annualized 1-year spot rate. Solvi

Using the 6-month and 1-year spot rates, we can urate from the equation


100

)2

1(

.

)02.1(

.

015.1

.

35.12

S

where S1.0 is the annualized 18-month spot rate. Sol

six months of 2 and one in one year of 102. We :

g we get: S1.0 = 4.01 %.

se the same approach to find the 18-month spot

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ving we get: S1.5 = 5.03%.



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Nominal Spread, Zero-volatility Sprea

Option-adjusted Spread

Nominal Spread: is the YTM of a bond minus the

Zero-Volatility Spread: is the constant spread tha

POINT on the Treasury curve where a cash flow is

equal to the present value of its cash flows. Each

appropriate Treasury spot rate plus the Z-spread.

YSpreadN om in a l


PV of Bond(for a two year annual pay security)

Z-spread Vs Nominal:

A nominal spread uses one point on the Treasury yie

will equal the present value of the security's cash flow

Option Adjusted Spread: is the spread without th

options.

3

Sp o1 y r1

C ouP r i ce

SAdjustedOption

,

YTM of a Treasury security of similar maturity

t is to be added to the spot rate yield at EACH

received that will make the price of a security

ash flow of the security is discounted at the

It is also known as "static spread"

Trea suryB ond Y TMM

Imp

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ld curve to determine the spread at a single point that

s to its price

e affect of the option for a bond with embedded

4

21 21

C oupo n

ZSra tet

p on

ZSrateSpotyr

CostOptionSpread-Zpread



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Option Cost In A Bond

Option Cost in % = Z-spread Option Adjusted S

In case of a callble bond the OAS < Z-spread as o

In case of putable options the OAS > Z-spread

SpreadMeasure

Benchmark Reflects Co

Nominal Treasury YieldCurve

Credit Risk, LO tion Risk


Zero-Volatility Treasury SpotRate Curve

Credit Risk, LOption Risk

Option-Adjusted

Treasury SpotRate Curve

Credit Risk, L

read(OAS)

ne needs to be compensated for the call feature

pensation for

iquidity Risk,

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iquidity Risk,

iquidity Risk



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Forward Rates

Forward Rates: rates of interest implied by the cu

For example, 6-Month Forward in 6 Months is e

Amount for 6 month after 6 months from today

The same is represented as:

S = f = Current S ot rate


1f1 = is the rate for a 1-year loan to be made on

1f2 = is the rate for a 1-year loan to be made tw

Relating the above terminology:

3

2111013

3 1111 fffS

rrent zero rates for a period of time in the future

uivalent to borrowing or lending the Notional

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year from now

years from now

6



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Forward Rates

For example: if we have the zero rates for year 4 a

time between year 4 and year 5 would be known aYear 4

F4,5F4= 4%


The 5-year spot rate is 10.50% and the 4-year spofour years from no?

7.02%

7.55%

8.35%

Solution: (1+z5)5 = (1+z4)

4*(1+f1) = (1.105)5 = (1.1

(1+f1) = 1.0755

f1 = 7.55%

3

nd year 5 then the forward rate for the period of

s the forward rate for that time period of 1 year.

Year 5

F5= 5%

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t rate is 11.25%. What is the one year forward rate

125)4*(1+f1)

7



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Questions

1. Karen invests in an 8% 5-year semi-annual calla

callable bond is 150bps. The option cost is 56 bpA. 100 bps

B. 94 bps

C. 206 bps

. Reinvestment income is least effected by:

A. The time to maturity.


B. The size of the debt issue.

C. The Coupon rate.

. The z-spread of a callable bond is 340 basis poin

A. Greater than 340 basis points

B. Lesser than 340 basis points

C. Equal to 340 basis points

3

le bond on 5th January 2010. The Z-spread for the

. The OAS is closest to

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s. The OAS of the bond is most likely to be:

8



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Questions (Cont....)

. For a 7% 3-year semi-annual option-free bond. T

at par. Calculate the no-arbitrage price for the boto

Maturity (months) Yield

6 5.2%

12 5.5%

18 5.8%

24 6.0%


30 6.2%

36 6.5%

5. The yield on a Bond Equivalent basis of an annuA. 4.16%

B. 8.33%C. 6.43%

6. The annual-pay yield to maturity of a 8.50% coupA. 17.72%B. 8.68%C. 13.43%

e Treasury spot rates are given below. The bond is

d. If the market price is $104.5 the BEY is closest

No-ArbitragePrice

BEY

A 102.34 5.45%

B 101.48 5.36%

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C 104.50 5.25%

l-pay 8.50% coupon bond prices at par is:

on semi-annual pay bond is:



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Solutions

1. B. OAS = Z-spread option cost = 150 -56 = 94

. B. The size of the debt issue

. B. Lesser than 340 basis points

. B. No-arbitrage price is calculated by discounting

M o n th s Y ie ld P V F a c to r

6 5 . 2 0 % 0 . 9 7 4 7

1 2 5 . 5 0 % 0 . 9 4 7 2

1 8 5 . 8 0 % 0 . 9 1 7 8


he bond equivalent yield can be calculated by usingInput 6 cash flows for coupon payment and one princIRR = 2.68% BEY

. B. 8.33%

. B. 8.68%

4

2 4 6 . 0 0 % 0 . 8 8 8 5

3 0 6 . 2 0 % 0 . 8 5 8 4

3 6 6 . 5 0 % 0 . 8 2 5 4

bps

all the cash flows by the spot rates

C a s h F lo w P V o f C F

3 . 5 3 . 4 1 1 3

3 . 5 3 . 3 1 5 2

3 . 5 3 . 2 1 2 3

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the CF function ipal payment cash flow. CF0 = 104.5 CPT IRR. = 2* IRR = 5.36%

0

3 . 5 3 . 1 0 9 7

3 . 5 3 . 0 0 4 5

1 0 3 . 5 8 5 . 4 2 8 0

1 0 1 . 4 8 0 9 8 0 9



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Agenda

Introduction to the Valuation of Debt

Yield Measures, Spot Rates, and For

Introduction to Measurement of Intere


ecurities

ard Rates

st Rate Risk

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K I I I t d ti T Th M t Of I t t


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Key Issues In Introduction To The Me

Rate Risk

Measuring Interest Rate Risk

Price Volatility

Convexity

Effective Duration

Alternative definitions of Duration

Duration of a ortfolio


Convexity measure of a bond

Modified and Effective Convexity

Price Value of a Basis Point(PVBP)

4

surement Of Interest

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43/66

Measuring Interest Rate Risk

Interest rate risk can be measured by two method

Full Valuation Method:

This is referred to as scenario analysis.

Under this method the normal pricing technique

options

When the interest rates change the entire portfo


The two values are compared to arrive at the im

Calculation gets complicated when there are a l

Duration/Convexity Method:

This gives an approximate result of the sensitivit

It is much simpler compared to the full valuation

4

:

are used to value a bond or a bond with embedded

lio is re-evaluated by the same method

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pact of change in interest rate

rge number of bonds in the portfolio.

y of the bond.

method.

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Disadvantages Of A Callable Bond

From the investors perspective the disadvantages of

Cash flow pattern is not known with certainity

Investor exposed to reinvestment risk

Price appreciation potential will be decreased relati

Negative convexity


an embedded call option is:

ve to an otherwise comparable option-free bond.

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Price Volatility And Convexity

We have already seen that the price-yield curve is

to as convex.

Price


Properties concerning the price volatility of an option

Percentage price change per change in interest ra

For either small increases or decreases in yield, pthe same.

For a given large change in yield, the percentagedecrease.

4

a negatively sloped and is a curve. This is referred

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free bond:

es is not the same for all bonds

rcentage change in price for given bond is roughly

rice increase is greater than the percentage price

5



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Price Volatility And Convexity

The curve of a Puttable bond exhibits Positive Co

of a security as a result of increase in the yield is li

Pr

ice

Putable Bond


Coupon

Value of

Yield

nvexity. This is because the decrease in the price

mited to the put price. See the below graph:

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PutThis files has expired at 30-Jun-13


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Effective Duration

Duration is the measure of how long on an average th

payments on the bond. A coupon paying bonds dura

his payments in the form of coupons before n years

In simple words, duration of a bond is sensitivity of

Effective duration is calculated as:

(BonDurationEffective


ercentage change in Bond Price = -Effective Duration *

Example: Consider a bond trading at 96.54 with durat

B = - 96.54* 4.5 y

B = -434.43 y

If there is 10 basis points increase ( + y) in the yi

B = -434.43 * ( 0.001) = -.43443

Hence, B = 96.54- .43443 = 96.10

4

e holder of the bond has to wait before he receives his

tion would be lower than n as the holder gets some of

ond price to change in its interest rate

decimals)inyieldin(Change*Price)(Initial*2

rises)yieldwhenpriceBondfallsyieldwhenprice

Imp

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Change in yield in percent. (y)

ion of 4.5 years. In this case

eld then the bond price would change by:

8



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Percentage Change In Price Using Du

Approximate percentage price change = - Duratio

For example, you hold a bond that has a durationCalculate the approximate percentage price chan

Answer: Approximate percentage price change = -


For large changes in yield, convexity should also binaccurate with only taking duration into account.

4

ration

* y * 100

f 7.8 years. The interest rates fell by 25 bps. e.

Duration * y * 100

= -7.8 *(- .0025) * 100

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= 1.95%

e used. Percentage change in price becomes

9



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Alternative Definitions Of Duration

Macaulay Duration: is the weighted average of

weights are a ratio of the coupon paid at time t toMacaulay duration is also used to measure how schanges in interest rates.

where:

t = Respective time period

C= Periodic Coupon payments ; y =Periodic yield :

C uDura t ionM acaulay

n

1t


Calculating Macaulay Duration:

Note that this is 3.77 six-month periods, which is a

5

.9643636

54.964

4

05.1

10403

05.1

402

05.1

401

05.1

40432

D

0

-964.54

he times when the payments are made. And the

the present bond price nsitive a bond or a bond portfolio's price is to

n = Total number of periods

Pr iceBondrrent

y)(1

M*n

y)(1

C*tnt

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bout 1.89 years

0

77.35476.

1 2 3 4

0

,

40 40 40



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Change In Bond Price With Change In

Modified Duration

The modified duration is equal to the percentage c

Example:

The current price of a bond is 98.75. Its mo

yModDVVyV

..

V1-M odD


. .

Solution:

V = -98.75 * 5.2 * 0.005

= -2.57

The new price of the bond is 96.18

5

Discount Rate

ange in price for a given change in yield.

ified duration is 5.2 years. The YTM of the bond is

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1



52/66

Alternative Definitions Of Duration

Modified Duration: is derived from Macaulay Dur

takes into account the current YTM.

Effective Duration calculations explicitly take into

yperpaymentsinterestofno

YTM(1

DurationMacaulayDurationModified


embedded options. The other methods of calcula

In summary duration is,

The first derivative of the price-yield function

The slope of the price-yield curve.

A weighted average of the time till the cash flow

The approximate percentage change in price fo

5

tion. It is better than Macaulay Duration as it

account the a bonds option provisions such as

)ear

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tion ignore the option provision

s willl be received.(Macaulay Duration)

a 1% change in yield.(Effective Duration)

2



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Duration Of A Portfolio

Duration of a portfolio is the weighted average of t

portfolio.

Portfolio Duration =

The problem with the above equation is that it hol

This is because securities with different maturities

2211 ...DWDW


5

e duration of the individual securities in the

s good only for a parallel shift in the yield curve.

may have different changes in yield.

NNDW......

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3



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Price Value Of A Basis Point (PVBP)

This is a measure of interest rate risk.

This is also known as the dollar value of an 01 (D

PVBP It is the absolute value of the change in t

yield.

changesyieldwhenPrice-PriceInitialPVBP


The PVBP is the same for both increase and decr

The PVBP is a special case of dollar duration.

5

ValueBond*0.01%*DurationPVBP

01)

he price of a bond for a 1 basis point change in

pointbasis1by

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ase (because change in yield is small)

5



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Yield Volatility

Price Yield Relationship

As seen in the graph, the when the yield level is hi


large change in price.

However, when yields are low, changes in interest

Interest Rate Risk can be decomposed into:

Duration risk

Yield Volatility

Yield volatility explains why junk bonds have highegiven by the standard deviation of yield changes

5

gh, a change in interest rates does not produce a

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rates produces a large change in price.

r interest rate risk than treasuries. Yield Volatility is

6



57/66

Questions

1. A 5 year bond paying 8% annual pay coupon is currecalculate the effective duration of the bond given 25

Given : V- = 1033.88, V+ = $1013.29

A. 5.03%

B. 4.02%

C. 4.56%

. Calculate the duration of the portfolio of two bonds ADuration of bond A is 7.9 and duration of bond B is 6.

A. 7.64


B. 7.42

C. 7.24

. A bond has a convexity of 63.80. The convexity effec

A. 0.41%

B. 0.35%

C. 0.54%. A bond has a duration of 9.75 and a convexity of 105

100 basis fall in the yield:

A. 10.25%

B. 9.75%

C. 10.80%

5

ntly trading for $1023.56 and having YTM of 7.42%, asis point in YTM.

and B having weights of 60% and 40% respectively. 7.

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t if the yield decreases by 80 basis points is:

.80. What is the change in the price of the bond for a

7



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Questions (Cont...)

. The most accurate measure for arriving at the effect

A. Duration ApproachB. Full valuation approach

C. PVBP

. A bond manager has collected the following inforinvestments which have a par value of $10mn. Tis 5.2 the most likelyestimate of the price change

A. 1.3% of $10mn


. . .

C. 2.1% of $11.25mn

. A portfolio manager notices the following in his pwill be the change in the portfolio if the interest ra

A. $ 28,280

B. $ 14,250C. $ 27,100

5

IssueA

B

C

of duration is?

mation regarding a portfolio of fixed income e current market price is $11.25mn. If the duration

for the bond issue for a 25 bps change is

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rtfolio has a portfolio duration of 4.35. How much e declines by 25 bps

8

Maturity Market Value2 $8.5mn

5 $4.6mn

10 $12.9mn



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Solutions

1. B. V = $1023.56, V- = 1033.88, V+ = $1013.29,

So effective duration is = ($1033.88 - $1013.29)/

. B. The portfolio duration is =0.6 * 7.9 + 0.4 *6.7 =

. A. 0.41%

. C. 10.80%

. B. Full valuation a roach


. B. The estimated change = 5.2*0.25 = 1.3%. (Th

candidate. Par value never changes. Current val

. A. 26mn * 4.35 * (0.25)% = $ 28,280

5

hange in yield is = 25 bps = 0.0025

* $1023.88 *0.0025 = 4.02

7.42

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par value of $10mn is given to confuse the

e of $11.25mn is more important)

9



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Extra-Quiz Questions

1. What is least likely to be true regarding Macaulay and

A. Both are calculated from the bonds expected cascash flows

B. For bonds with no options, modified duration is si

C. Macaulay duration takes into consideration embe

. A fixed income analyst makes the following two state

Statement 1: YTM assumes that coupon payment

reinvested at the rate e ual to the cash flow ield.


Statement 2: The bond is assumed to be held

till maturity.

. Consider the following two statements:

Statement 1: The static spread is the spread

over the Treasury spot rate that makes the PV

of all the cash flows from a non-Treasury security

equal to its price.

Statement 2: The Z-spread ignores the interest rat

volatility and assumes it to be zero.

6

modified duration

h flows with no adjustments for embedded options on

ilar to effective duration

ded options in the bond

ents:

are

Statement 1 Statement 2

A Correct Correct

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e

0

orrec ncorrec

C Incorrect Correct

Statement 1 Statement 2A Correct Correct

B Correct Incorrect

C Incorrect Correct



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. Sally states that there are a number of yield measureleast likely yield measure that is used

A. Yield to call

B. Yield to worst

C. Yield to settlement

. Duration is not a good measure for large changes inshift in a parallel fashion. The statements are most lik

A. Both statements are correct.


B. Only one statement is correct.

C. Both the statements are incorrect.

. An 8% coupon bond is valued at 104.35. When the yito 103.44. The PVBP for the bond is closest to

A. $0.0455B. $0.0512

C. $0.0519

6

that are used traditionally in the bond market. The

ield. Duration also assumes that the yield curve will ely

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ld increases by 20 bps the price of the bond declines

1



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. Which of the following 10-year fixed-coupon bonds haa coupon rate of:

A. 6.50%

B. 5.00%

C. 8.00%

. Carl manages the following portfolio

he value for the portfolio duration is

losest to

Coupon

8%


A. 5.833

B. 4.351

C. 4.555

6

11%

9.75%

10.25%

s the least price volatility? All else equal, the bond with

Maturity Par ValueMarketValue

Duration

5 years $ 5 mn $ 4 mn 4.87

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2

7 years $ 10 mn $ 11.4 mn 5.72

10 years $ 15 mn$ 14.5

mn8.50

5 years $ 20 mn$ 21.2

mn4.25



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Solutions

1. C.

. A.

. A.

The Z-spread is also known as the static spread aspot rates to calculate the PV of cash flows of a bzero hence it is also known as the zero-volatility

. C.

Yield to settlement is not a traditional measure ofa ield to maturit b ield to call c ield to ut d


. A.

As the duration measure is not useful for measuriyield. The duration also assumes that yields chan

. A.

The PVBP = 104.35 103.44 / 20 = 0.0455

6

nd it is the spreads that should be added on top of nd. It also assumes the volatility of interest rates is AS.

ield. The yield measures that are generally used are ield to worst e current ield f cash flow ield.

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g changes in price when there are large changes in e is parallel across the entire yield curve.

3



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Solutions

. C.

If bonds are identical except for the coupon rate, thvolatility. This is because a bonds price is determibond pays more of its cash flows later (more of thehigher-coupon bond does. Longer-term cash flowscalculation. Another way to think about this: The re(all else equal) is inverse a greater coupon resultconfused on the examination, remember that a zerbecause it delivers all its cash flows at maturity. Sicoupon equates to high price volatility.


. A.

6

Issue Market ValueMV % of Portfolio

Value

A $ 4 mn 7.83%

B $ 11.4 mn 22.31%

C $ 14.5 mn 28.38%

D $ 21.2 mn 41.49%

Total $ 51.1 mn 100%

e one with the lowest coupon will exhibit the most price ed by discounting the cash flows. A lower-coupon cash flow is comprised of principal at maturity) than a

are discounted more heavily in the present value lationship between the coupon rate and price volatility

s in less price volatility. Examination tip: If you get -coupon bond has the highest interest rate risk ce a zero-coupon bond has a 0.00% coupon, a low

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4

DurationMV% * Duration

4.87 0.3813

5.72 1.2761

8.50 2.4123

4.25 1.7633

5.8330



65/66

Five Minute Recap

32 ......YTM)(1

C

YTM)(1

C

YTM)(1

CbondaofValue

Bond Selling at: Relationship

Par Coupon rate = Current Yield = Yield to Maturi

Discount Coupon rate < Current Yield < Yield to Maturi

Premium Coupon rate > Current Yield > Yield to Maturi


bondbenchmarkon theYield

spreadyieldAbsolute

spreadyieldRelative

yieldbondbenchmark

yieldbondSubjectRatioYield

BenchmarkonYield-BondonYieldSpreadYieldAbsolute

NYTM)1

PARC

y

y

y

1YTMAnnual1*2 21

BEY

1

2

BEY1

2

YTM

Price

Option-freebond Callable

bond

Value of

call

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BondYieldCoup

on

Pri

ce

Coupon

Putable

Bond

Valueof Put

Yield



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Five Minute Recap

preadeasure

Benchmark Reflects Compensation for

ominal Treasury YieldCurve

Credit Risk, Liquidity Risk,Option Risk

ero-Volatility Treasury SpotRate Curve

Credit Risk, Liquidity Risk,Option Risk

ption-djusted

Treasury SpotRate Curve

Credit Risk, Liquidity Risk

TreasuryBond Y TMY TMSpreadomina l

2121

C oupon

ZSrateSpo t1 y r1

C ouponce

ZSrateSpotyr

CostOptionSpread-ZSpreadAdjustedOption

(BondDurationEffective

whprice(BondConvexity

yModDVVy

V

..

V

1-M odD

yeaperpaymentsinterestofno

YTM

(1

DurationMacaulayDurationModified

ValueBond*0.01%*DurationPVBP

Theories Of Term Structure Of Interest Rates

2111013

3 1111 fffS

decimals)inyieldin(Change*Price)(Initial*2

rises)yieldwhenpriceBondfallsyieldwhenprice

2decimals)inyieldin(Change*Price)(Initial*2

PricBondInitial*2-risesyieldwhenpriceBondfallsyieldn

Pure Expectations Theory Liquidity Preference Theory Market Segmentation Theory


Documents

CFA 1 Fixed Income II