Fixed Income

Basic concepts

Features Fixed income securities promises to pay

a stream of semiannual payments for a given number of years and then repay the loan amount at maturity date

The contract between the borrower and the lender (the indenture) can be designed to have any payment stream or pattern that the parties agree to

Bond indenture Defines the obligations of and restrictions

on the borrower and forms the basis for all future transactions between the bondholder and the issuer

Contract provision covenants:Negative prohibitions on the borrowerAffirmative actions that the borrower

promises to perform

Negative covenants Restrictions on asset sales the company

cant sell assets that have been pledged as collateral

Negative pledge of collateral the company cant claim that the same assets back several debt issues in the same time

Restrictions on additional borrowing the company cant borrow additional money unless certain financial conditions are met

Affirmative covenants

Maintenance of certain financial ratios and the timely payment of principal and interest

If the values of the agreed ratios are not maintained, then the bonds could be considered in technical default

Straight (option free) bond

Treasury bond Coupon 6% Maturity 5Y Notional (face value) USD 1000 Annual interest paid in two semiannual

installments Stream of payments fixed

Coupon rate structures

Zero-coupon bonds:Do not pay periodic interest Initially sold at a price below par value

(discount to par value)Pay the par value at maturity

Coupon rate structures

Step-up notesHave coupon rates that increase over time at

a specified rate Increase may take place one or more times

during the life of the issue

Deferred coupon bonds Carry coupons, but the initial coupon

payments are deferred for some period The coupon payments accrue, at a

compound rate, over the deferral period and are paid as a lump sum at the end of that period

After the initial deferment period has passed, pay regular coupon interest for the rest of the life of the issue

Floating rate securities

Bonds for which the coupon interest payments over the life of the security vary based on a specified interest rate or index

They have coupons that are reset periodically (normally every 3, 6 or 12 months) based on prevailing market interest rates

Floating rate securities

New coupon rate =reference rate +/- quoted margin

Reference rate: LIBOR, EURIBOR, ROBORQuoted margin may vary over time according to a schedule that is stated in the indenture

Inverse floater

Floating rate security with a coupon formula that actually increases the coupon rate when a reference interest rate decreases and vice versa

EgCoupon = 12% - reference rate

Inflation-indexed bonds

Coupon formula based on inflation EgCoupon = 3% + annual change in the Consumer Price Index (CPI)

Protection against extreme fluctuations Placing upper and lower limits on the

coupon rate:Upper limit cap puts a maximum on the

interest rate paid by the borrower/issuerLower limit floor puts a lower limit on the

periodic coupon interest payments received by the lender

Both limits collar Eg. floater with a coupon at issuance of

5%, a 7% cap and a 3% floor

Clean and dirty price When a bond trades between coupon

dates, the seller is entitled to receive any interest earned from the previous coupon date through the date of the sale accrued interest

Calculated as a fraction of the coupon period that has passed times the coupon

Full (dirty) price = clean price + accrued interest

Redemption of bonds Redemption provisions refer to how, when

and under what circumstances the principal will be repaid

Nonamotizing bullet bond or bullet maturity at maturity the entire par or face value is repaid

Amortizing make periodic payments of interest and principal

Redemption of bonds - options Prepayment options give the issuer/borrower

the right to accelerate the principal repayment on a loan

Call provisions give the issuer the right (but not the obligation) to retire all or a part of an issue prior to maturity

Call protection a period of years after issuance during which the bonds cannot be called

Call schedule specify when the bonds can be called and at what price (declining)

Nonrefundable vs noncallable Nonrefundable bonds prohibit the call of

an issue using the proceeds from a lower coupon bond issue

A bond may be callable but not refundable A bond that is noncallable has absolute

protection against a call prior to maturity A callable but not refundable bond can be

called for any reason other than refunding

Sinking fund Provides for the repayment of principal through

a series of payments over the life of the issue, which can be accomplished via cash or delivery Cash retire the applicable portion of bonds by using a

selection method such as lottery Delivery of securities: purchase bonds (at market

price) with a total par value equal to the amount that is to be retired in that year in the market

Accelerated sinking fund choice of retiring more than amount specified in sinking fund

Embedded options

Integral part of the bond contract and are not a separate security

Some are exercisable at the option of the issuer and some at the option of the purchaser of the bond

Security owner options Option granted to the security holder and

gives additional value to security

Conversion options convert bond into a fixed number of securities

Put options right to sell the bond to the issuer at a special price prior to maturity

Floors set a minimum on the coupon rate for a floating-rate bond

Security issuer options (1) Exercisable at the option of the issuer of the

fixed income security and gives lower value to security

Call provision gives the issuer the right to redeem the issue prior to maturity

Prepayment option gives the issuer the right to prepay the loan balance prior to maturity in whole or in part without penalty

Security issuer options (2) Accelerated sinking fund provisions are

embedded options held by the issuer that allow the issuer to (annually) retire a larger proportion of the issue than is required by the sinking fund provision, up to a specified limit.

Caps set a maximum on the coupon rate for a floating rate note

Exercises

Exercise

A 10 year bond pays no interest for three years, then pays USD 229.25, followed by payments of USD 35 for seven years and additional USD 1000 at maturity. This is a:

a) Step-up bondb) Zero coupon bondc) Deferred-coupon bond

Exercise Consider a USD 1 Mio semi-annual pay,

floating rate issue where the rate is reset on Jan. 1 and Jul. 1 each year. The reference rate is 6M Libor and the stated margin is + 1.25%. If 6M Libor is 6.5% on Jul. 1 what will be the next semi-annual coupon on this issue?

A) 38,750B) 65,000C) 77,500

Exercise An investor paid a full price of USD

1,059.04 each for 100 bonds. The purchase was between coupon dates, and accrued interest was USD 23.54 per bond. What was the bond clean price?

A. 1000.00 B. 1035.50C. 1082.58

Exercise

Consider a USD 1 Mio par value, 10Y, 6.5% coupon bond issued on Jan. 1 2005. The bonds are callable and there is a sinking fund provision. The market rate for similar bonds is currently 5.7%. The main points of the prospectus are summarized as follows:

Call dates and prices: 2005 through 2009: 103 After Jan. 1, 2010: 102

Exercise additional info The bonds are non-refundable The sinking fund provision requires that

the company redeem USD 0.1 Mio of the principal amount each year. Bonds called under the terms of the sinking fund provision will be redeemed at par

The credit rating of the bonds is currently the same as at issuance

Questions Using only the preceding information, an

analyst should conclude thatA. The bonds do not have call protectionB. The bonds were issued and currently

trade at a premiumC. Given current rates, the bonds will likely

be called and new bonds issued

Questions Which of the following statements about the

sinking fund provisions for these bonds is most accurate?

A. An investor would benefit from having his bonds called under the provision of the sinking fund

B. An investor would receive a premium if the bond is redeemed prior to maturity under the provision of the sinking fund

C. The bonds do not have an accelerated sinking fund provision

Risks associated with investing in bonds

Interest rate risk

Effect of changes in the prevailing market rate of interest on bond values

When interest rates rise, bond values fall. This is the source of interest rate risk which is approximated by a measure called duration.

Price yield relation

Bonds characteristics vs interest rate risk

Characteristic Interest Rate Risk Duration

Maturity up Interest rate risk up Duration up

Coupon up Interest rate risk down Duration down

Add a call Interest rate risk down Duration down

Add a put Interest rate risk down Duration down

Example of the coupon effect

Consider the durations of a 5-year and 20-year bond with varying coupon rates (semi-annual coupon payments):

5 year bond 20 year bondZero coupon 5 206% coupon 4.39 11.909% coupon 4.19 10.98

Impact of embedded options A call feature limits the upside price movement

of a bond when interest rates decline. Hence the value of a callable bond will be less sensitive to interest rate changes than an otherwise identical option-free bond.

A put feature limits the downside price movement of a bond when interest rates rise. Hence the value of a putable bond will be less sensitive to interest rate changes than an otherwise identical option-free bond

Price - yield callable bond

Callable bond value = Value of an option-free bond value of embedded call option

Interest rate risk in a floating rate security The objective of the resetting mechanism is to bring the

coupon rate in line with the current market yield so the bond sells at or near its par value. This will make the price of a floating-rate security much less sensitive to changes in market yields than a fixed-coupon bond of equal maturity

Between coupon dates, there is a time lag between any change in market yield and a change in the coupon rate (which happens on the next reset date)

The longer the time period between the two dates, t he greater t he amount of potential bond price fluctuation. Hence the longer (shorter) the reset period, the greater (less) the interest rate risk of a floating-rate security at any reset date

Interest rate risk in a floating rate security

Presence of a cap (maximum coupon rate) can increase the interest rate risk of a floating-rate security

If the reference rate increases enough that the cap rate is reached, further increases in market yields will decrease the floater's price

Duration Is a measure of the price sensitivity of a

security to changes in yield It can be interpreted as an approximation

of the percentage change in the security price for a 1% change in yield

Also can be interpreted as the ratio of the percentage change in price to the change in yield in percent

Duration - examples

If a bond has a duration of 5 and the yield increases from 7% to 8%, calculate the approximate percentage change in the bond price.

A bond has a duration of 7.2. If the yield decreases from 8.3% to 7. 9%, calculate the approximate percentage change in the bond price.

Dollar duration Sometimes the interest rate risk of a bond

or portfolio is expressed as its dollar duration, which is simply the approximate price change in dollars in response to a change in yield of 100 basis points (1%).

Another measure is Basis Point Value BPV which is the approximate price change in dollars in response to a change in yield of 1 basis point.

Duration examples If a bond's yield rises from 7% to 8% and

its price falls 5%, calculate the duration. If a bond's yield decreases by 0.1% and its

price increases by 1.5%, calculate its duration.

A bond is currently trading at $1,034.50, has a yield of 7.38%, and has a duration of 8.5. If the yield rises to 7.77%, calculate the new price of the bond.

Yield curve risk Arises from the possibility of changes in

the shape of the yield curve (which shows the relation between bond yields and maturity).

While duration is a useful measure of interest rate risk for equal changes in yield at every maturity (parallel changes in the yield curve), changes in the shape of the yield curve mean that yields change by different amounts for bonds with different maturities.

Yield curve shifts

Duration for a bond portfolio Computed as a weighted average based on

individual bond durations and the proportions of the total portfolio value invested in each bond

Is an approximation of the price sensitivity of a portfolio to parallel shifts of the yield curve

For a non-parallel shift in the yield curve, the yields on different bonds in a portfolio can change by different amounts, and duration alone cannot capture the effect of a yield change on the value of the portfolio.

Key rate durations

To estimate the impact of non-parallel shifts, bond portfolio managers calculate key rate durations, which measure the sensitivity of the portfolio's value for changes in yields for specific maturities (or portions of the yield curve)

Call risk When interest rates fall, a callable bond

investor's principal may be returned and must be reinvested at the new lower rates.

When interest rates are more volatile, callable bonds have relatively more call risk because of an increased probability of yields falling to a level where the bonds will be called.

Prepayment risk Prepayments are principal repayments in

excess of those required on amortizing loans

If rates fall, causing prepayments to increase, an investor must reinvest these prepayments at the new lower rate

As with call risk, an increase in interest rate volatility increases prepayment risk

Reinvestment risk When market rates fall, the cash flows (both

interest and principal) from fixed-income securities must be reinvested at lower rates, reducing the returns an investor will earn.

Reinvestment risk is related to call risk and prepayment risk.

Coupon bonds are also subject to reinvestment risk, because the coupon interest payments must be reinvested as they are received

Reinvestment risk

A security has more reinvestment risk under the following conditions:The coupon is higher so that interest cash

flows are higher It has a call feature It is an amortizing security It contains a prepayment option

Credit risk

Is the risk that the creditworthiness of a fixed-income security's issuer will deteriorate, increasing the required return and decreasing the security's value

It is reflected by the credit rating of the issuance

Rating

A bond's rating is used to indicate its (relative) probability of default, which is the probability of its issuer not making timely interest and principal payments as promised in the bond indenture

Rating agencies Rate specific debt issues The ratings are issued fo indicate the

relative probability that all promised payments on the debt will be made over the life of the security and, therefore, must be forward looking.

Ratings on long-term bonds will consider factors that may come into play over at least one full economic cycle.

Firm specific factors considered in rating

Past repayment history Quality of management, ability to adapt to changing

conditions The industry outlook and firm strategy Overall debt level of the firm Operating cash flow, ability to service debt Other sources of liquidity (cash, salable assets) Competitive position, regulatory environment, and union

contracts/history Financial management and controls. Susceptibility to event risk and political risk

Bond RatingsBond Ratings by Agency

Moody's S&P Fitch DBRS DCR DefinitionsAaa AAA AAA AAA AAA Prime. Maximum SafetyAa1 AA+ AA+ AA+ AA+ High Grade High QualityAa2 AA AA AA AAAa3 AA- AA- AA- AA-A1 A+ A+ A+ A+ Upper Medium GradeA2 A A A AA3 A- A- A- A-Baa1 BBB+ BBB+ BBB+ BBB+ Lower Medium GradeBaa2 BBB BBB BBB BBBBaa3 BBB- BBB- BBB- BBB-Ba1 BB+ BB+ BB+ BB+ Non Investment GradeBa2 BB BB BB BB SpeculativeBa3 BB- BB- BB- BB-B1 B+ B+ B+ B+ Highly SpeculativeB2 B B B BB3 B- B- B- B-Caa1 CCC+ CCC CCC+ CCC Substantial RiskCaa2 CCC - CCC - In Poor StandingCaa3 CCC- - CCC- -Ca - - - - Extremely Speculative

C - - - - May be in Default- - DDD D - Default- - DD - DD- D D - -- - - - DPSource: http://www.bondsonline.com/asp/research/bondratings.asp

Bond Ratings There is virtually no risk of default within 1 year, and very little over

longer periods, if investing in investment grade securities. Once go below investment grade, however, the risk of default rises

dramatically.

PresenterPresentation NotesSource: Historical Default Rates of Corporate Bond Issuers, 1920 to 1999, Moodys Investor Research, January 2000, p. 20.

Bond RatingsDefault Rate by S&P Bond Rating

(15 Years)

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

S&P Bond Rating

Def

ault

Rat

e

Default Rate 0.52% 1.31% 2.32% 6.64% 19.52% 35.76% 54.38%

AAA AA A BBB BB B CCC

PresenterPresentation NotesSource: "The Credit-Raters: How They Work and How They Might Work Better", Business Week (8 April 2002), p. 40

Bond Ratings

Who Rates Bonds?

Fitch14%

Other6%

Moody's38%

Standard & Poor's42%

Each company's share of the total global revenue in 2001 for credit rating agencies

Source: Wall Street Journal, 6 January 2003, p. C1 and Moody's

Transition matrix (S&P)Initial rating

End of year rating

AAA AA A BBB BB B CCC D No ratingAAA 89.37 6.04 0.44 0.14 0.05 0.00 0.00 0.00 3.97AA 0.57 87.76 7.30 0.59 0.06 0.11 0.02 0.01 3.58A 0.05 2.01 87.62 5.37 0.45 0.18 0.04 0.05 4.22

BBB 0.03 0.21 4.15 84.44 4.39 0.89 0.26 0.37 5.26BB 0.03 0.08 0.40 5.50 76.44 7.14 1.11 1.38 7.92B 0.00 0.07 0.26 0.36 4.74 74.12 4.37 6.20 9.87

CCC 0.09 0.00 0.28 0.56 1.39 8.80 49.72 27.87 11.30

Source: Standard & Poors (Special Report: Ratings Performance 2002, 2003)

percent

Transition matrix (Moodys)percent*

Initial Rating

End of year rating

Aaa Aa A Baa Ba B Caa-C Faliment Rating retras

Aaa86.34 8.21 0.19 0.00 0.00 0.00 0.00 0.00 5.26

87.69 6.13 0.42 0.00 0.08 0.00 0.00 0.00 5.68

Aa0.76 86.71 9.13 0.10 0.00 0.00 0.00 0.00 3.30

0.72 85.21 8.75 0.45 0.12 0.02 0.00 0.00 4.74

A0.00 5.05 84.80 3.63 0.10 0.02 0.00 0.02 6.39

0.08 2.32 87.15 5.34 0.64 0.24 0.03 0.02 4.18

Baa0.74 0.25 4.82 78.83 2.86 1.16 0.04 0.00 11.31

0.07 0.30 5.55 83.01 4.54 0.99 0.08 0.18 5.28

Ba0.00 0.00 0.64 10.52 71.40 9.29 0.68 0.25 7.22

0.03 0.04 0.65 5.18 73.90 8.57 0.47 1.45 9.71

B0.00 0.00 0.33 1.03 9.40 65.52 8.28 3.29 12.17

0.01 0.06 0.23 0.64 5.06 73.94 3.84 7.18 9.04

Caa-C0.00 0.00 0.00 0.00 0.00 22.41 48.58 14.53 14.47

0.00 0.00 0.00 1.18 1.66 5.18 59.51 21.75 10.72

First line European companies, second line US companiesSource: Moodys 2002 (Default and Recovery Rates of European Corporate bond Issuers, 1985-2001)

Credit risk events ISDA (1999): Bankruptcy Rating downgradeMerger/acquisitionRestructuringAccelerating of obligationBankruptcy of a related entity Default on coupon/interestDebt repudiation

Downgrade risk The risk that a credit rating agency will

lower a bond's rating The resulting increase in the yield required

by investors will lead to a decrease in the price of the bond

A rating increase upgrade - will have the opposite effect, decreasing the required yield and increasing the price

Credit spread

The difference between the yield on a Treasury security, which is assumed to be default risk-free, and the yield on a similar maturity bond with a lower rating

Yield on a risky bond = Yield in a default free bond + credit spread

Credit spread risk

Refers to the fact that the default risk premium required in the market for a given rating can increase, even while the yield on Treasury securities of similar maturity remains unchanged

An increase in this credit spread increases the required yield and decreases the price of a bond

Corporate (credit) bond spreads over Treasuries

Bond yields (spreads over equivalent Treasuries) increase as credit ratings decline

Spreads widen as maturity increases

Corporate (Industrials) Spreads over Treasuries (in basis points)Rating 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 30 yr

Aaa/AAA 35 40 45 55 69 81 92Aa1/AA+ 40 45 55 65 79 91 102Aa2/AA 45 55 60 70 85 101 112Aa3/AA- 50 60 65 80 95 111 123A1/A+ 60 70 85 100 116 132 148A2/A 70 80 100 115 136 155 171A3/A- 80 95 110 125 152 170 193Baa1/BBB+ 100 115 130 145 167 190 208Baa2/BBB 120 135 150 165 183 200 228Baa3/BBB- 140 150 160 175 195 215 248Ba1/BB+ 275 300 325 350 375 425 475Ba2/BB 300 325 350 400 450 525 600Ba3/BB- 350 400 425 475 525 575 750B1/B+ 450 475 500 575 650 700 825B2/B 525 575 625 700 750 825 975B3/B- 600 650 750 850 975 1075 1200Caa/CCC 850 900 1050 1150 1250 1400 1600Source: http://www.bondsonline.com/asp/corp/spreadind.html on 20 Nov 2001

Credit Default Swap - CDS Captures the credit risk of an issuer Bilateral agreement in which periodical fixed

payments are made to protection seller in exchange of a single payment the protection buyer will make in case if a credit event (specified in the CDS contract occurs).

Flows payoff: Periodical payments (premium leg): basis points

applied to the notional of the CDS agreement Single payment if credit event occurs (protection leg):

par value of the bond [100 bonds market price after the credit event occurred]

CDS evolution

Source: International Swaps and Derivatives Association, Bank of International Settlements

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

2001 S1 2002 S1 2003 S1 2004 S1 2005 S2 2006 S2

USD bill.

CDS utilization

Credit risk hedging (credit risk transfer) Taking exposure on credit risk Structured products (credit linked note) Informational content

CDS - Valuation Market value determined by supply and demand of

such instruments. Cannot be available for some issuers.

Theoretical value based on: Probability of default of the issuer (implied by the credit rating) Recovery rate (1 Loss Given Default) Coupon/interest rate of the bond Maturity Market interest rates

CDS - Romania

Source: Bloomberg

CDS term structure Romania

Source: Bloomberg

CDS informational content

Source: Bloomberg

Debt crisis 2010

Source: Bloomberg

Liquidity risk

Risk that the sale of a fixed-income security must be made at a price less than fair market value because of a lack of liquidity for a particular issue

Since investors prefer more liquidity to less, a decrease in a security's liquidity will decrease its price, as the required yield will be higher

Bid-ask spread The difference between the price that

dealers are willing to pay for a security (the bid) and the price at which dealers are willing to sell a security (the ask)

If trading activity in a particular security declines, the bid-ask spread will widen (increase), and the issue is considered to be less liquid

Exchange-rate risk

Arises from the uncertainty about the value of foreign currency cash flows for an investor in terms of his home-country currency

Inflation risk

Unexpected inflation risk or purchasing-power risk

Uncertainty about the amount of goods and services that a security's cash flows will purchase

Volatility risk

Is present for fixed-income securities that have embedded options, such as call options, prepayment options, or put options.

Changes in interest rate volatility affect the value of these options and, thus, affect the values of securities with embedded options

Volatility risk Value of a callable bond =Value of an option-free bond value of a call Value of a putable bond =Value of an option-free bond + value of a put

Volatility risk for callable bonds is the risk that volatility will increase, and volatility risk for putable bonds is the risk that volatility will decrease

Event risk

The risks outside the risks of financial markets, such as the risks posed by natural disasters, regulatory changes and corporate restructurings

Sovereign risk

The credit risk of a sovereign bond issued by a country other than the investor's home country

Law under which the bond is issued

Exercises

Exercise

A bond with a 7.3% yield has a duration of 5.4 and is trading at $985. If the yield decreases to 7.1%, the new bond price is closest to:

A. $974.40B. $995.60C. $1, 091.40

Exercise

The current price of a bond is 102.50. If interest rates change by 0.5%, the value of the bond price changes by 2.50. What is the duration of the bond?

A. 2.44.B. 2.50.C. 4.88.

Question

Which of the following bonds has the greatest interest rate risk?

A. 5% 1 0-year callable bondB. 5% 1 0-year putable bondC. 5% 1 0-year option-free bond

Question

A floating-rate security will have the greatest duration:

A. the day before the reset date.B. the day after the reset date.C. Never - floating-rate securities have a duration of zero

Exercise A straight 5% bond has two years remaining to

maturity and is priced at $981.67. A callable bond that is the same in every respect as the straight bond, except for the call feature, is priced at $917.60. With the yield curve flat at 6%, what is the value of the embedded call option?

A. $45.80B. $64.07C. $101.00

Question

Which of the following statements about the risks of bond investing is most accurate?

A. A bond rated AAA has no credit riskB. A bond with call protection has volatility riskC. A U.S. Treasury bond has no reinvestment risk

Bond sectors and instruments

Sovereign bonds Bonds issued by a countrys central government Largest market sovereign debt of the US

Government which consists of US Treasury and considered to be essentially free of default risk

Sovereign debt of other countries is considered to have varying degrees of credit risk

Can be issued on own domestic market, another countrys foreign bond market or in the Eurobond market

Issued in own currency but also in other currencies

U.S. Treasury securities Bills matures in one year or less, issued

at a discount Notes matures between 2-10 years,

issued as a coupon security Bonds maturities longer than 10 years Treasury inflation protection securities

(TIPS) principal is indexed to CPI with real rate being fixed

T-bills

Maturities of less than 1Y (29, 91 and 182 days)

Do not make explicit interest payments, paying only the face value at the maturity date

Issued at discount

Bonds quotation Treasury bond and note prices in the

secondary market are quoted in percent and 32nds of 1% of face value.

A quote of 102-5 (sometimes 102:5) is 102% plus 5/32% of par, which for a $100,000 face value T-bond, translates to a price of:

Bond quotations

Bonds can be quoted also in yield in format BID ASK

Example: 5.00 4.50

TIPS Maturities: 5, 10 and 20Y Make semi-annual coupon payments at a rate

fixed at issuance The par value begins at USD 1000 and is

adjusted semi-annually for changes in CPI The fixed coupon rate is paid semiannually as a

percentage of the inflation adjusted par value

Any increase in the par value taxed as income

TIPS - example For example, consider a $100,000 par value

TIPS with a 3% coupon rate, set at issuance. Six months later annual rate of inflation measured by CPI is 4%. The par value will be increased by one half of 4% and will be 1.02 x 100,000 = $102,000.

The first semi-annual coupon will be one half of 3% coupon rate times the inflation adjusted par value: 1.5% x 102,000 = 1,530

Any percentage change in the CPI over the next 6M period will used to adjust the par value from 102,000

Stripped Treasury Securities Several major brokerages have created an

investment vehicle from Treasury securities. They purchase these securities, deposit them in a bank custody account and then separate out each coupon payment and principal. Then a receipt is issued to investors representing an ownership in the account. In essence, the security is stripped.

STRIPSSTRIPS U.S. Treasury program issues these direct obligations of the U.S. government, ending trademark and generic receiptsTreasury strips - zero-coupons or stripped Treasury securities: Treasury coupon strips created from the future

coupon Treasury principal strips - created from the

principal payment at maturity

Agency bonds Debt securities issued by various agencies and

organizations of the US Government as Federally related institutions as Government

National Mortgage Association (Ginnie Mae) which are owned by US Government

Government sponsored enterprises, as Federal National Mortgage Association (Fannie Mae), Federal Home Loan Bank Corporation (Freddie Mac), which are created by the US Congress, but privately owned. They issue debentures securities not backed by collateral (unsecured)

Mortgage-backed securities - MBS Backed (secured) by pools of mortgage

loans, which not only provide collateral but also the cash flows to service the debt.

Security where the collateral for the issued security is a pool of mortgages.

The Government National Mortgage Association (GNMA), the Federal National Mortgage Association (FNMA), and the Federal Home Loan Mortgage Corporation (FHLMC) all issue mortgage-backed securities.

Securitisation

Process of combining many similar debt obligations as the collateral for issuing securities

Primary reason for mortgage securitization is to increase the debt's attractiveness to investors and to decrease investor required rates of return, increasing the availability of funds for home mortgages

Cash flows from mortgages

Periodic interest, Scheduled repayments of principal Principal repayments in excess of

scheduled principal payments

Prepayment risk Because the borrower can accelerate

principal repayment, the owner of a mortgage has prepayment risk.

Prepayment risk is similar to call risk except that prepayments may be part of or all of the outstanding principal amount.

This, in turn, subjects the mortgage holder to reinvestment risk, as principal may be repaid when yields for reinvestment are low

Securitization types

Mortgage pass-through security Collateralized mortgage obligations

(CMOs) Stripped mortgage-backed securities

Mortgage passthrough security

Passes the payments made on a pool of mortgages through proportionally to each security holder

A holder of a mortgage passthrough security that owns a 1% portion of the issue will receive a 1 % share of all the monthly cash flows from all the mortgages, after a percentage fee for administration is deducted

Mortgage passthrough security Each monthly payment consists of interest,

scheduled principal payments, and prepayments of principal in excess of the scheduled amount therefore prepayment risk

Since prepayments tend to accelerate when interest rates fall, due to the refinancing and early payoff of existing mortgage loans, security holders can expect to receive greater principal payments when mortgage rates have decreased since the mortgages in the pool were issued.

CMOs

Created from mortgage passthrough certificates and referred to as derivative mortgage-backed securities

A CMO issue has different tranches, each of which has a different type of claim to the cash flows from the pool of mortgages

Sequential CMO - example Tranche I (the short-term segment of the issue) receives

net interest on outstanding principal and all of the principal payments from the mortgage pool until it is completely paid off.

Tranche II ( the intermediate-term) receives its share of net interest and starts receiving all of t he principal payments after Tranche I has been completely paid off. Prior to that, it only receives interest payments.

Tranche III (the long-term) receives monthly net interest and starts receiving all principal repayments after Tranches I and II have been completely paid off. Prior of that, it only receives interest payments.

Stripped mortgage-backed securities

Are either the principal or interest portions of a mortgage passthrough security

The holder of a principal-only strip will gain from prepayments because the face value of the security is received sooner rather than later.

The holder of an interest-only strip will receive less total payments when prepayment rates are higher since interest is only paid on the outstanding principal amount, which is decreased by prepayments.

Municipal bonds Debt securities issued by state and local

governments in the United States are known as municipal bonds (or munis for short)

Municipal bonds are often referred to as tax-exempt or fax-free bonds, since the coupon interest is exempt from federal income taxes.

While interest income may be tax free, realized capital gains are not.

Secured debt Backed by the pledge of assets/collateral, which

can take the following forms: Personal property (e.g., machinery, vehicles,

patents) Real property (e.g., land and buildings) Financial assets (e.g., stocks, bonds, notes).

These assets are marked to market from time to time to monitor their liquidation values. Covenants may require a pledge of more assets if values are insufficient. Bonds backed by financial assets are called collateral trust bonds.

Unsecured debt

Is not backed by any pledge of specific collateral

Unsecured bonds are referred to as debentures

They represent a general claim on any assets of the issuer that have not been pledged to secure other debt

Credit enhancements The guarantees of others that the

corporate debt obligation will be paid in a timely manner:

Third-party guarantees that the debt obligations will be met. Often, parent companies guarantee the loans of their affiliates and subsidiaries.

Letters of credit are issued by banks and guarantee that the bank will advance the funds for service the corporation's debt.

Medium term notes - MTNs

Once registered, such securities can be "placed on the shelf" and sold in the market over time and at the discretion of the issuer.

MTNs are sold over time, with each sale satisfying some minimum dollar amount set by the issuer, typically $1 million and up.

MTNs Are issued in various maturities Can have fixed or floating-rate coupons Can be denominated i n any currency Can have special features, such as calls,

caps, floors, and non-interest rate indexed coupons

The notes issued can be combined with derivative instruments to create the special features that an investor requires

Structured notes A debt security created when the issuer

combines a typical bond or note with a derivative

Example: an issuer could create a structured note where the periodic coupon payments were based on the performance of an equity security or an equity index by combining a debt instrument with an equity swap

Types of structured MTNs Step-up notes - Coupon rate increases over time on a preset

schedule Inverse floaters - Coupon rate increases when the reference rate

decreases and decreases when the reference rate increases Deleveraged floaters- Coupon rate equals a fraction of the reference

rate plus a constant margin Dual-indexed floaters - Coupon rate is based on the difference

between two reference rates. Range notes - Coupon rate equals the reference rate if the

reference rate falls within a specified range, or zero if the reference rate falls outside that range.

Index amortizing notes - Coupon rate is fixed but some principal is repaid before maturity, with the amount of principal prepaid based on the level of the reference rate.

Commercial paper

Short-term, unsecured debt instrument used by corporations to borrow money at rates lower than bank rates

Is typically issued as a pure discount security and makes a single payment equal to the face value at maturity

Certificates of deposit - CDs Are issued by banks and sold to their

customers Negotiable CDs, permit the owner to sell

the CD in the secondary market at any time

Negotiable CDs have maturities ranging from days up to 5Y. The interest rate paid on them is called the London Interbank Offering Rate because they are primarily issued by banks' London branches.

Assetbacked securities - ABSs Securitization of credit card debt, auto loans,

bank loans, and corporate receivables The assets are transferred to a special purpose

entity SPV for bankruptcy protection External credit enhancements to increase the

rating Corporate guarantees, which may be provided by the

corporation creating the ABS or its parent Letters of credit, which may be obtained from a bank

for a fee

Collateralized debt obligation (CDO)

Is a debt instrument where the collateral for the promise to pay is an underlying pool of other debt obligations and even other CDOs

Underlying debt obligations can be business loans, mortgages, debt of developing countries, corporate bonds of various ratings, asset-backed securities, or even problem/ non-performing loan

Tranches of the CDO are created based on the seniority of the claims to the cash flows of the underlying assets, and these are given separate credit ratings depending on the seniority of the claim, as well as the creditworthiness of the underlying pool of debt securities.

Exercises

Exercise

A Treasury security is quoted at 97-17 and has a par value of $ 100,000. Which of the following is its quoted dollar price?

A. $97,170.00.B. $97,531.25.C. $100,000.00

Exercise An investor holds $100,000 (par value) worth of

Treasury Inflation Protected Securities (TIPS) that carry a 2.5% semiannual pay coupon. If t he annual inflation rate is 3%, what is the inflation-adjusted principal value of the bond after six months?

A. $101,500. B. $102,500. C. $103,000.

Question

A Treasury note (T-note) principal strip has six months remaining to maturity. How is its price likely to compare to a 6-month Treasury bill (T-bill) that has just been issued? The T-note price should be:

A. lowerB. higherC. the same

Yield curves

Yield curves

A plot of yields by years to maturity Shapes of yield curves: Normal or upward sloping Inverted or downward sloping. Flat Humped

Yield curve shapes

RON yield curve 2008

Source: Bloomberg

RON yield curve 2012 - 2013

Source: Bloomberg

EUR and RON yield curves

Source: Bloomberg

Term structure theories

Pure expectations theory Liquidity preference theory Market segmentation theory

Pure expectation theory Yield for a particular maturity is an average (not

a simple average) of the short-term rates that are expected in the future

If short-term rates are expected to rise in the future, interest rate yields on longer maturities will be higher than those on shorter maturities, and the yield curve will be upward sloping

If short-term rates are expected to fall over time, longer maturity bonds will be offered at lower yields

Liquidity preference theory In addition to expectations about future short-

term rates, investors require a risk premium for holding longer term bonds

This is consistent with the fact that interest rate risk is greater for longer maturity bonds

The size of the liquidity premium will depend on how much additional compensation investors require to induce them to take on the greater risk of longer maturity bonds or, alternatively, how strong their preference for the greater liquidity of shorter term debt is

Pure expectations vs. liquidity preference

Market segmentation theory

Is based on the idea that investors and borrowers have preferences tor different maturity ranges.

Under this theory, the supply of bonds (desire to borrow) and the demand for bonds (desire to lend) determine equilibrium yields for the various maturity ranges

Market segmentation

Types of curves

Zero (spot) Yield to maturity (YTM) Par Forward

Zero Rates

A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T

ExampleMaturity(years)

Zero Rate(% cont comp)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8

Bond Pricing

To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate

In our example, the theoretical price of a two-year bond providing a 6% coupon semiannually is 3 3 3

103 98 39

0 05 0 5 0 058 1 0 0 064 1 5

0 068 2 0

e e ee

+ +

+ =

. . . . . .

. . .

Bond Yield The bond yield is the discount rate that

makes the present value of the cash flows on the bond equal to the market price of the bond

Suppose that the market price of the bond in our example equals its theoretical price of 98.39

The bond yield (continuously compounded) is given by solving

to get y=0.0676 or 6.76%.3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y + + + =. . . . .

Par Yield The par yield for a certain maturity is the

coupon rate that causes the bond price to equal its face value.

In our example we solve

g)compoundin s.a. (with get to 876

1002

100

2220.2068.0

5.1064.00.1058.05.005.0

.c=

ec

ececec

=

++

++

Sample Data

Bond Time to Annual Bond CashPrincipal Maturity Coupon Price(dollars) (years) (dollars) (dollars)

100 0.25 0 97.5

100 0.50 0 94.9

100 1.00 0 90.0

100 1.50 8 96.0

100 2.00 12 101.6

The Bootstrap Method

An amount 2.5 can be earned on 97.5 during 3 months.

The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding

This is 10.127% with continuous compounding Similarly the 6 month and 1 year rates are

10.469% and 10.536% with continuous compounding

The Bootstrap Method continued

To calculate the 1.5 year rate we solve

to get R = 0.10681 or 10.681%

Similarly the two-year rate is 10.808%

9610444 5.10.110536.05.010469.0 =++ Reee

Zero Curve Calculated from the Data

9

10

11

12

0 0.5 1 1.5 2 2.5

Zero Rate (%)

Maturity (yrs)

10.127

10.469 10.53610.681 10.808

Forward Rates

The forward rate is the future zero rate implied by todays term structure of interest rates

Calculation of Forward Rates

n-year Forward Ratezero rate for n th Year

Year (n ) (% per annum) (% per annum)

1 3.02 4.0 5.03 4.6 5.84 5.0 6.25 5.3 6.5

Formula for Forward Rates

Suppose that the zero rates for time periods T1and T2 are R1 and R2 with both rates continuously compounded.

The forward rate for the period between times T1and T2 is R T R T

T T2 2 1 1

2 1

Embedded options vs. yield

Investors will require a higher yield on a callable bond, compared to the same bond without the call feature

The inclusion of a put provision or a conversion option with a bond will have the opposite effect

Question

Under the pure expectations theory, an inverted yield curve is interpreted as evidence that:

a. demand for long-term bonds is fallingb. short-term rates are expected to fall in the futurec. investors have very little demand for liquidity

Question With respect to the term structure of interest

rates, the market segmentation theory holds that:

a. An increase in demand for long-term borrowings could lead to an inverted yield curve

b. Expectations about the future of short-term interest rates are the major determinants of the shape of the yield curve

c. the yield curve reflects the maturity demands of financial institutions and investors

Bond valuation

Bond valuation

The intrinsic value of a bond, like stocks, is the present value of its future cash flows.

Bonds, however, have much more predictable cash flows and a finite life.

The cash flows promised by a bond are:A series of (usually) constant interest

paymentsThe return of the face value of the bond at

maturity

Bond valuation The value of a bond is determined by four variables:

The Coupon Rate This is the promised annual rate of interest. It is normally fixed at issuance for the life of the bond. To determine the annual interest payment, multiply the coupon rate by the face value of the bond. Interest is normally paid semiannually, and the semiannual payment is one-half the annual total payment.

The Face Value This is nominally the amount of the loan to the issuer. It is to be paid back at maturity.

Term to Maturity This is the remaining life of the bond, and is determined by todays date and the maturity date. Do not confuse this with the original maturity which was the life of the bond at issuance.

Yield to Maturity This is the rate of return that will be earned on the bond if it is purchased at the current market price, held to maturity, and if all of the remaining coupons are reinvested at this same rate. This is the IRR of the bond.

Premium, par and discount

Price vs. yield

Yield to maturity A summary measure and is essentially an

internal rate of return based on a bond's cash flows and its market price

Assumes that all cash flows are reinvested at the YTM

Reinvestment risk

A coupon bond's reinvestment risk will increase with:Higher coupons - because there's more cash

flow to reinvestLonger maturities - because more of the total

value of the investment is in the coupon cash flows (and interest on coupon cash flows)

Arbitrage-free valuation Discount each cash flow using a discount

rate that is specific to the maturity of each cash flow.

These discount rates are the spot rates and can be thought of as the required rates of return on zero-coupon bonds maturing at various times in the future

If the market value of the bond different than the arbitrage-free valuation arbitrage opportunity

Arbitrage opportunities

If the bond is selling for more than the sum of the values of the pieces (individual cash flows), one could buy the pieces, package them to make a bond, and then sell the bond package to earn an arbitrage profit

If the bond is selling for less than the sum: buy the bond and sell the pieces

Arbitrage example

Consider a 6% Treasury note with 1.5 years to maturity.

Spot rates (expressed as yields to maturity) are: 6 months = 5%, 1 year = 6%, and 1.5 y ears = 7%.

If the note is selling for $992, compute the arbitrage profit, and explain how a dealer would perform the arbitrage

Yield to call

Used to calculate the yield on callable bonds that are selling at a premium to par

For bonds trading at a premium to par, the yield to call may be less than the yield to maturity

This can be the case when the call price is below the current market price

Yield to call - calculation Similar as the calculation of yield to maturity,

except that the call price is substituted for the par value in FV and the number of semiannual periods until the call date is substituted for periods to maturity, N

When a bond has a period of call protection, we calculate the yield to first call over the period until the bond may first be called, and use the first call price in the calculation as FV

In a similar manner, we can calculate the yield to any subsequent call date using the appropriate call price

Yield to call - example

Consider a 20-year, 10% semiannual-pay bond with a full price of 112 that can be called in five years at 102 and called at par in seven years.

Calculate the YTM, YTC, and yield to first par call

Yield to call - example

YTM: N = 40; PV = -112; PMT = 5; FV = 100

Yield to first call: N = 10; PV = -112; PMT = 5; FV = 102

Yield to first par call: N = 14; PV = -112; PMT = 5; FV = 100

Yield to worst

Is the worst yield outcome of any that are possible given the call provisions of the bond

Yield to put Is used if a bond has a put feature and is

selling at a discount The yield to put will likely be higher than

the yield to maturity. The yield to put calculation is just like the

yield to maturity with the number of semiannual periods until the put date as N, and the put price as FV

Yield to put - example

Consider a 3-year, 6%, $1,000 semiannual-pay bond

The bond is selling for a full price of $925.40

The first put opportunity is at par in two years.

Calculate the YTM and the YTP

Yield to put - example

YTM: N = 6; PV = -925; PMT = 30; FV = 1000

Yield to first call: N = 4; PV = -925; PMT = 30; FV = 1000

Spread measures

Nominal spread Zero-volatility spread: Z-spread Option adjusted spread: OAS

Nominal spread

It is an issue's YTM minus the YTM of a Treasury security of similar maturity

Zero volatility spread

It is the equal amount that we must add to each rate on the Treasury spot yield curve in order to make the present value of the risky bond's cash flows equal to its market price

Option adjusted spread The measure is used when a bond has

embedded options The option-adjusted spread takes the

option yield component out of the Z-spread measure

The option-adjusted spread is the spread to the Treasury spot rate curve that the bond would have if it were option-free

Z-spread - OAS = option cost in percent

OAS

For embedded short calls (e.g., callable bonds): option cost > 0 (you receive compensation for writing the option to the issuer) - OAS < Z-spread. In other words, you require more yield on the callable bond than for the option-free bond.

For embedded puts (e.g., putable bonds), option cost < 0 (i.e., you must pay for the option) - OAS > Z-spread. In other words, you require less yield on the putable bond than for an option-free bond

Interest rate risk

Price vs. yield option free, 8%, 20Y

Callable bond - negative convexity

Putable bond

Duration Is the slope of the price-yield curve at the bond's

current YTM. Mathematically, the slope of the price-yield curve is the first derivative of the price-yield curve with respect to yield.

Is a weighted average of the time (in years) until each cash flow will be received. The weights are the proportions of the total bond value that each cash flow represents.

Is the approximate percentage change in price for a 1% change in yield. This interpretation, price sensitivity in response to a change in yield, is the preferred, and most intuitive, interpretation of duration.

Duration measures

Effective duration Macaulay duration Modified duration

Effective duration

The ratio of the percentage change in price to change in yield

Effective duration

Effective duration - example Consider a 20-year, semiannual-pay bond

with an 8% coupon that is currently priced at $908.00 to yield 9%.

If the yield declines by 50 basis points (to 8.5%), the price will increase to $952.30, and if the yield increases by 50 basis points (to 9.5%), the price will decline to $866.80.

Based on these price and yield changes, calculate the effective duration of this bond

Macaulay duration Macaulay duration is an estimate of a

bond's interest rate sensitivity based on the time, in years, until promised cash flows will arrive

Macaulay duration - examples A 5-year zero-coupon bond has only one

cash flow five years from today, its Macaulay duration i s five. The change in value in response to a 1% change in yield for a 5-year zero-coupon bond is approximately 5%.

A 5-year coupon bond has some cash flows that arrive earlier than five years from today (the coupons), so its Macaulay duration is less than five.

Macaulay duration Weighted average term to maturity

Measure of average maturity of the bonds promised cash flows Duration formula:

where:

t is measured in years

P)1/(

)(PV)(PV ttt

tyCF

BondCFw +==

Dm = t wt( )t =1

T

wt =1t =1

q

Macaulay duration

Dm = t wtt =1

T

= t PV(Ct )

PV(Bond)

t =1

T

=1 C1

(1 + y)1

+ 2 C2

(1+ y)2

+ ... + N CN

(1+ y)N

C1

(1+ y)1+

C2(1 + y)2

+ ... + CN(1+ y)N

Modified duration

Is derived from Macaulay duration and offers an improvement over Macaulay duration in that it takes the current YTM into account

For option-free bonds, effective duration (based on small changes in YTM) and modified duration will be very similar

Modified duration (D*m)

Direct measure of price sensitivity to interest rate changes

Can be used to estimate percentage price volatility of a bond

yDD mm +

=1

*

PP

= Dm* y

Derivation of modified duration

So D*m measures the sensitivity of the % change in bond price to changes in yield

yDD mm +

=1

*

P = Ct(1+ y)tt =1

N

Py

= 11 + y

t Ct(1 + y) t

t =1

N

Py

=Dm1 + y

P = Dm* P

1P

Py

= Dm*

Example Consider a 3-year 10% coupon bond selling at $107.87 to yield

7%. Coupon payments are made annually.

87.10779.8973.835.9bond of Price

79.89)07.1(

110)(

73.8)07.1(

10)(

35.9)07.1(

10)(

33

22

1

=++=

==

==

==

CFPV

CFPV

CFPV

7458.287.10779.89*3

87.07173.8*2

87.10735.9*1)(Duration

=

+

+

=mD

Example

Modified duration of this bond:

If yields increase to 7.10%, how does the bond price change? The percentage price change of this bond is given by:

= 2.5661 .0010 100= .2566

5661.207.1

7458.2* ==mD

PP

100 = Dm* y 100

Example What is the predicted change in dollar

terms?

New predicted price: $107.87 .2768 = $107.5932

Actual dollar price (using PV equation): $107.5966

P = .2566100

P

= .2566100

$107.87

= $.2768

Effective vs. modified duration Modified duration is calculated without any

adjustment to a bond's cash flows for embedded options.

Effective duration is appropriate for bonds with embedded options because the inputs (prices) were calculated under the assumption that the cash flows could vary at different yields because of the embedded options in the securities

Duration of a portfolio

Duration of a portfolio is the weighted average of the durations of the individual securities in the portfolio

Convexity

Is a measure of the curvature of the price-yield curve

The more curved the price-yield relation is, the greater the convexity

Price vs. yield - 8%, 20Y

Using duration and convexity By combining effective duration and

convexity, we can obtain a more accurate estimate of the percentage change in price of a bond, especially for relatively large changes in yield

Example - convexity

Consider an 8% Treasury bond with a current price of $908 and a YTM of 9%.

Calculate the percentage change in price of both a 1% increase and a 1% decrease in YTM based on a duration of 9.42 and a convexity of 6 8.33.

Example - convexity The duration effect, is 9.42 x 0.01 = 0.0942 = 9.42%.

The convexity effect is 68.33 x 0.012 x 100 = 0.00683 x 100 = 0.683%.

The total effect for a decrease in yield of 1% (from 9% to 8%) is 9.42% + 0.683% = + 10.103%, and the estimate of the new price of the bond is 1.10103 x 908 = 999.74.

The total effect for an increase in yield of 1% (from 9% to 10 %) is -9.42% + 0.683% = -8.737%, and the estimate of the bond price is (1 - 0.08737)(908) = $828.67.

Convexity - determination

Second derivative of price with respect to yield divided by bond price

2P2 y

= 1(1 + y)2

CFt(1+ y)t

(t 2 + t)

t =1

N

Convexity = 1P

2 P2 y

Predicted percentage price change

Recall approximation using only duration:

The predicted percentage price change accounting for convexity is:

PP

100 = Dm* y 100

PP

100 = Dm* y 100( ) + 12 Convexity (y)

2 100

Example with convexity Consider a 20-year 9% coupon bond

selling at $134.6722 to yield 6%. Coupon payments are made semiannually.

Dm= 10.98

The convexity of the bond is 164.106.

66.10)2/06.0(1

98.10* =+

=mD

Example with convexity If yields increase instantaneously from 6% to 8%,

the percentage price change of this bond is given by: First approximation (Duration):

10.66 .02 100 = 21.32

Second approximation (Convexity)0.5 164.106 (.02)2 100 = +3.28

Total predicted % price change: 21.32 + 3.28 = 18.04%(Actual price change = 18.40%.)

Example with convexity What if yields fall by 2%? If yields decrease instantaneously from 6% to 4%,

the percentage price change of this bond is given by: First approximation (Duration):

10.66 .02 100 = 21.32

Second approximation (Convexity)0.5 164.106 (.02)2 100 = +3.28

Total predicted price change: 21.32 + 3.28 = 24.60%

Note that predicted change is NOT SYMMETRIC.

Effective vs. modified convexity

Effective convexity takes into account changes in cash flows due to embedded options, while modified convexity does not

The difference between modified convexity and effective convexity mirrors the difference between modified duration and effective duration

Effective convexity is the appropriate measure to use for bonds with embedded options, since it is based on bond values that incorporate the effect of embedded options on the bond's cash flows

Price value of a basis point Is the dollar change in the price/value of a

bond or a portfolio when the yield changes by one basis point, or 0.01%

PVBP can be calculated directly for a bond by changing the YTM by one basis point and computing the change in value

As a practical matter, duration can be used to calculate the price value of a basis point

PVBP = duration x 0.0001 x bond value

PVBP - example A bond has a market value of $100,000 a and a duration

of9.42. What is t he price value of a basis point? Using t he duration formula, the percentage change in

the bond's price for a change in yield of 0.01% is 0.01% x 9.42 = 0.0942%.

We can calculate 0.0942% of the original $100,000 portfolio value as 0.000942 x 100,000 = $94.20.

If the bond's yield increases (decreases) by one basis point, the portfolio value will fall (rise) by $94.20. $94.20 is the (duration-based) price value of a basis point for this bond.

We can ignore the convexity adjustment here because it is of very small magnitude

Fixed IncomeBasic conceptsFeaturesBond indenture Negative covenantsAffirmative covenantsStraight (option free) bondCoupon rate structuresCoupon rate structuresDeferred coupon bondsFloating rate securitiesFloating rate securitiesInverse floaterInflation-indexed bondsProtection against extreme fluctuationsClean and dirty priceRedemption of bondsRedemption of bonds - optionsNonrefundable vs noncallableSinking fundEmbedded optionsSecurity owner optionsSecurity issuer options (1)Security issuer options (2)ExercisesExerciseExerciseExerciseExerciseExercise additional infoQuestionsQuestionsRisks associated with investing in bondsInterest rate riskPrice yield relationBonds characteristics vs interest rate riskExample of the coupon effectImpact of embedded optionsPrice - yield callable bondInterest rate risk in a floating rate securityInterest rate risk in a floating rate securityDurationDuration - examples Dollar durationDuration examplesYield curve risk Yield curve shiftsDuration for a bond portfolioKey rate durationsCall riskPrepayment riskReinvestment riskReinvestment riskCredit riskRatingRating agenciesFirm specific factors considered in ratingBond RatingsBond RatingsBond RatingsBond RatingsTransition matrix (S&P)Transition matrix (Moodys)Credit risk eventsDowngrade riskCredit spreadCredit spread riskCorporate (credit) bond spreads over TreasuriesCredit Default Swap - CDSCDS evolution CDS utilization CDS - ValuationCDS - RomaniaCDS term structure RomaniaCDS informational contentDebt crisis 2010Liquidity riskBid-ask spreadExchange-rate risk Inflation risk Volatility riskVolatility riskEvent riskSovereign risk ExercisesExerciseExerciseQuestionQuestionExerciseQuestionBond sectors and instrumentsSovereign bondsU.S. Treasury securitiesT-billsBonds quotationBond quotationsTIPSTIPS - exampleStripped Treasury SecuritiesSTRIPSAgency bondsMortgage-backed securities - MBSSecuritisationCash flows from mortgagesPrepayment riskSecuritization typesMortgage passthrough security Mortgage passthrough security CMOsSequential CMO - exampleStripped mortgage-backed securitiesMunicipal bondsSecured debtUnsecured debtCredit enhancementsMedium term notes - MTNsMTNsStructured notesTypes of structured MTNsCommercial paperCertificates of deposit - CDsAssetbacked securities - ABSsCollateralized debt obligation (CDO)ExercisesExerciseExerciseQuestionYield curvesYield curvesYield curve shapesRON yield curve 2008RON yield curve 2012 - 2013EUR and RON yield curvesTerm structure theoriesPure expectation theoryLiquidity preference theoryPure expectations vs. liquidity preferenceMarket segmentation theoryMarket segmentationTypes of curvesZero RatesExampleBond PricingBond YieldPar YieldSample DataThe Bootstrap MethodThe Bootstrap Method continuedZero Curve Calculated from the DataForward RatesCalculation of Forward RatesFormula for Forward RatesEmbedded options vs. yieldQuestionQuestionBond valuationBond valuationBond valuationPremium, par and discountPrice vs. yieldYield to maturityReinvestment riskArbitrage-free valuationArbitrage opportunitiesArbitrage exampleYield to callYield to call - calculationYield to call - exampleYield to call - exampleYield to worstYield to putYield to put - exampleYield to put - exampleSpread measuresNominal spreadZero volatility spreadOption adjusted spreadOASInterest rate riskPrice vs. yield option free, 8%, 20Y Callable bond - negative convexityPutable bondDurationDuration measuresEffective durationEffective durationEffective duration - example Macaulay durationMacaulay duration - examplesMacaulay durationMacaulay durationModified durationModified duration (D*m)Derivation of modified durationExampleExampleExampleEffective vs. modified durationDuration of a portfolioConvexityPrice vs. yield - 8%, 20YUsing duration and convexityExample - convexityExample - convexityConvexity - determinationPredicted percentage price changeExample with convexityExample with convexityExample with convexityEffective vs. modified convexityPrice value of a basis pointPVBP - example