fixed income - final

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Fixed Income SecuritiesCFA Level 1 (Dec 2010)

J K Shah Classes

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Study Session 15

Basic Concepts

60. Features of Debt Securities

61. Risks Associated with Investing in Bonds

62. Overview of Bond Sectors and Instruments

63. Understanding Yield Spreads

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61. Features of Debt Securities

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Bond Indentures

A bond indenture is a contract between borrower and lender which specifies

all the obligations of the borrower / issuer of a fixed income security and right of the lender.

It contains various Dos and Don’ts on the borrowers called covenants

Restrictions on asset sales Negative pledge of collateral Restrictions on additional borrowings

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Negative Covenants

Positive Covenants

Promises by the borrower to Maintain financial ratios

Timely payment of principal and interest

Don'ts

Dos

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Bond Features (Par Value, Market Value, Coupon Rate, Yield, Maturity)

Bond Terms: Face value/ par value/ maturity value, Quoted Price/Market Price, Dollar Price

Coupon rate: Annual % of par value

Par Value Quoted Price(in % term)

Dollar Price Coupon Payment

$ 1,000, 6% bond

.905 905 60

$ 5,000, 6.5% bond

1.0275 5137.5 325

$ 10,000, 6.25% bond

.7063 7063 625

$ 1,00,000, 5.95% bond

1.1334 113,343.75 5950

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Bond Features

Maturity: No of years remaining prior to final payment of the bond or no of years for which the debt is outstanding. The day on which the bond will cease to exist.1-5 yrs -- Short term5-12 yrs – Intermediate Term12 + -- Long term

Maturity is very important for bond for 3 reasons

Indicates the time over which the interest would be paid and principal amount paid by the issuer

Yield offered depends on the maturity Price of the bond will fluctuate over the life of the bond

J K Shah Classes

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Coupon Structures

Zero-coupon bondsThese bonds are that do not pay periodic interest. They pay the par value at maturity and the interest results from the fact that zero coupon bonds are initially sold at a

price below par value

Deferred coupon They carry coupons but initial coupon payments are deferred for some period. The coupon payments accrue at a compound rate over the deferred period and are paid as a lump sum at the end of that period J K Shah Classes

Issue Price of $ 100 bond

Whether a case of Zero coupon bond

$ 110 No as Issue Price not < face value

$ 95 Yes as Issue Price < face value

$ 100 No as Issue Price not < face value

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Step-up notesCoupon rate increase over time at a specified rate. The increase may take place one or more time during the life of the issue.

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Coupon Structures

Deferred coupon They carry coupons but initial coupon payments are deferred for some period. The coupon payments accrue at a compound rate over the deferred period and are paid as a lump sum at the end of that period or in some cases the increased rate of coupon is paid after deferment period.

J K Shah Classes

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Floating-Rate Securities

It’s a security with variable coupon payment over the maturity period. They are bonds for which the coupon interest payments over the life of the security vary based on a specified interest rate or index.

Coupon formula Reference rate + margin

e.g., LIBOR + 1.5%, annualized rates Cap: Maximum on formula rate Floor: Minimum on formula rate

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T(0)Jan 10

T(1)Jul 10

T(2)Jan 11

T(3)Jul 11

T(4)Jan 12

6 month LIBOR rate as on

Interest Rate

Jan 10 6

Jul 10 6.5

Jan 11 4

Jul 11 7

Jan 12 9

Coupon Reset

Formula= 6 month LIBOR on reset date +

1%

Cap Rate = 9%

Floor Rate = 6%

3.5%

3.75%

3% 4% 4.5%

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Duration!!!?

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Accrued Interest 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

Paid to a bond seller Portion of the next coupon interest payment already earned by the seller Full price = clean price + accrued interest

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AI = Days Between CouponX Coupon Payments

Days Since Last Coupon

Full Price / Dirty Price = Clean Price + AI

Clean Price = Full Price / Dirty Price - AI

or

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Redemption / Retirement of Bonds - Amortizing and Non amortizing Bonds

Nonamortizing securities pay only interest until maturity, then the par value is repaid. Its also called Bullet Maturity.

Coupon Treasury bonds Most corporate bonds

Date of

IssueJan 10

Maturity Date

Dec 14$ 10 $ 10 $ 10

$10+

$ 100

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15J K Shah Classes

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Prepayment Option On an amortizing security, such as a mortgage

Prepayments are repayment of principal in excess of scheduled principal payments

Its beneficial for issuer / borrower

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Call Provisions Issuer can repay principal prior to maturity

Call protection for some period

Call prices typically decrease over time (e.g., 15-year bond: callable after 5 years @ 102 and callable after 10 years @ par)

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Refunding

Refunding is calling (redeeming) a bond using the proceeds of a lower cost issue

Bond can be callable but not refundable ; such bonds are more beneficial for the lender than pure callable bond.

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Sinking Fund

Sinking fund redemptions are calls of a portion of an outstanding bond issue, typically at par

Premium bonds: Cash paid to trustee, bonds to be retired chosen by lottery

Discount bonds: Bonds can be purchased and delivered to trustee to be retired

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Redemption Prices

Call prices are regular redemption prices

Sinking fund redemptions and redemptions under other provisions are special redemption prices

(e.g., redemptions due to forced asset sales)

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Embedded Options

Benefiting Issuer /

Borrower

Benefiting Lender

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Call Provision

Accelerated Sinking Fund

Caps

Prepayment Option

Put Provision Conversion Option Floors

Embedded Options

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Call Provision Issuer/Borrower

Prepayment Option Issuer/Borrower

Put Provision Buyer

Caps Issuer/Borrower

BuyerFloors

BuyerConversion Option

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Margin Buying and Repurchase Agreements

Margin buying: Borrowing funds to purchase securities. The securities are the collateral for the margin loan

Repurchase agreement: An institution sells a security with a commitment to buy it back at a specified higher price

Repo rate: The interest rate implied by the two prices Overnight repo: Repurchase agreement for one day Term repo: Agreement covering a longer period Most bond-dealer financing is achieved through

repurchase agreements rather than margin loans

J K Shah Classes

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Risks Associated with

Investing in Bonds

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Bond Risks

1. Interest rate risk

2. Yield curve risk

3. Call risk

4. Prepayment risk

5. Reinvestment risk

6. Credit riskJ K Shah Classes

7. Liquidity risk

8. Exchange-rate risk

9. Inflation risk

10. Volatility risk

11. Event risk

12. Sovereign risk

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Bond Discounts and Premiums

Yield = coupon rate → bond price at par

Yield < coupon rate → bond price over par bond priced at a premium

Yield > coupon rate → bond price under par bond priced at a discount

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Market Yield vs. Bond for an 8% Coupon Bond

BOND VALUE

PREMIUM TO PAR

PAR VALUE

DISCOUNT TO PAR

6% 7% 8% 9% 10% MARKET YIELD

Coupon Mkt Yield

At the time of issue

8% 8%

After 30 days 8% <8%

After 60 days 8% >8%

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Factors affecting Interest Rate Risk

Maturity

Coupon & Yield

Call Option & Put Option

Higher Maturity Higher Interest Rate Risk

Lower Maturity Lower Interest Rate

RiskLower Coupon Higher Interest

Rate RiskHigher Coupon Lower Interest

Rate Risk

Add any option Interest Rate Risk down

Remove Option Interest Rate Risk up

Price – Yield Curves for Callable and Noncallable Bond

Price

CALL OPTION VALUE

CALLABLE BOND VALUE

Yield

Price of Option Free BondCALL PRICE

8%

$100

$110

$105

ISSUE PRICE

$95

7%

6%

9%

Coupon Payment : 8%Yield at the time of issue : 8%Call Price : 105

As the yield falls, the value of embedded call increases

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Floating-Rate Securities

Coupon is periodically reset based on a reference rate (plus a fixed margin)

Has interest rate risk between reset dates

Price may differ from par at reset if: Credit quality of issuer changes after issuance Margin over reference rate no longer

appropriate

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Measure Interest Rate Risk With Duration

Duration is the approximate percentage price change for a 1% change in yield.

Duration =

% change in bond price

yield change in %-

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Price Impact of Yield Changes

Based on the duration of 4.29: If the yield goes up 0.25%, price goes

down by 4.29(0.25%) = 1.0725% For a bond valued at $2.5 million, a yield

change of 0.25% leads to an approximate change in value of 1.0725% (2.5 mil) = $26,812.50

Dollar duration of a bond is approximate change in value for a 1% change in yield, 0.0429 (2.5 mil) = $107,250

J K Shah Classes 33

Duration and Yield Curve Risk

Portfolio duration is an approximation of the price sensitivity of a portfolio to a parallel shift of the yield curve (yields on all the bonds change by the same percent)

For a non-parallel shift in the yield curve, the yields on different bonds in a portfolio can change by different amounts

Yield curve risk: The interest rate risk of a portfolio of bonds that is not captured by the duration measure

J K Shah Classes 34

Yield Curve Risk

YIELD A NON PARALLEL SHIFT

A PARREL SHIFT INITIAL YILED

CURVE

YIELD CURVE

MATURITY

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Callable and Pre payable Securities

Callable securities are likely to be called when interest rates are low

Principal repayment on pre payable securities is faster when interest rates are low

Investors must reinvest principal when rates are low

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Factors Affecting Reinvestment Risk

Reinvestment risk is higher when:

1. Coupon is higher

2. Bond has a call feature

3. A security is amortizing

4. A security contains a prepayment option

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Forms of Credit Risk

Bond ratings indicate relative probability of default

Downgrade risk: Probability of ratings decrease

Default risk: Probability of default Credit spread risk: Risk of increase in spread

to Treasuries to compensate for given default risk (bond rating)

The higher the rating (e.g., AA vs. A), the lower the market yield.

J K Shah Classes 38

Liquidity Risk

The bid-ask spread indicates the liquidity of the market for a security

A decrease in liquidity will increase the bid-ask spread, lead to a lower sale price, and decrease the returns on the position

Even if an investor plans to hold the security until maturity, marking the security prices to market will result in lower returns when liquidity decreases (bids fall)

J K Shah Classes 39

Exchange Rate Risk

If an investor buys a security denominated in a foreign (to the investor) currency

Depreciation of the foreign currency reduces the returns to a dollar-based investor

Exchange rate risk: Actual cash flows from the investment may be worth more or less than was expected when the bond was purchased

J K Shah Classes 40

Inflation Risk

Inflation (purchasing power) risk: Prices of goods and services increase more than expected

An increasing price level decreases the amount of real goods and services that bond payments will purchase

When expected inflation increases, nominal yields rise, values of debt securities fall

J K Shah Classes 41

Effects of Yield Volatility

Increase in yield volatility increases option values

Increases value of putable bond =

(option-free bond value + put value↑)

Decreases value of callable bond = (option-free bond value – call value↑)

J K Shah Classes 42

Event Risk

Disasters (e.g., hurricanes, earthquakes, or industrial accidents) can impair the ability of a corporation to meet its debt obligations

Corporate restructurings may result in bond-rating downgrades

Regulatory issues may cause large cash expenditures to meet new regulations

New regulations prohibiting financial institutions from holding a certain type of security can lead to a volume of sales that decreases prices for the whole sector

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Overview of Bond Sectors and Instruments

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Government Securities

Sovereign debt: Bonds issued by central governments; domestic, foreign, or Eurobond.

U.S. Treasury securities considered essentially free of default risk.

Sovereign (non US) debts of other countries are considered to have varying degrees of credit risk.

Local Currency Debt Rating:

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Sovereign Debt Issuance Methods

Regular cycle auction—single price: Highest price (lowest yield) at which the entire issue can be sold awarded to all bidders (e.g., U.S. Treasury debt)

Regular cycle auction—multiple price: Winning bidders receive the bonds at the prices they bid

Ad hoc auction system: Government auctions new securities when market conditions are advantageous

Tap system: Auction of bonds identical to previously issued bonds, periodically, no regular cycle

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U.S. Treasury Securities – Issued by US dept of treasury

Fixed Principal Treasuries

TIPs Treasuries Tips

T Bonds

T Notes

T Bill Zero Coupon Less than 12 months maturityCash Management Bills 1 to 10 yrs maturity

Semi Annual Coupon

> 10 yrs maturity Semi Annual Coupon

Coupon Strip

Principal Strip

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Quotation of T. Bond / Notes in the secondary market

T Bonds and notes are quoted in percentage and 32nd of 1% of face value

e.g. a quote of 102-5 (or 102:5) for a $ 1,000 bond means

= 102% + 5/32% of par (*1000)

= 1.0215625 * $1,000

= 1021.5625

Now Calculate for 103-7 and 97:6

103-7 = $1000 * (103+ 7/32) % = $1032.187597:6 = $1000 * (97+ 6/32) % = $97.1875

J K Shah Classes 48

U.S. Treasury Securities

Treasury Inflation Protected Securities (TIPS) Coupon rate is fixed Par value is adjusted for inflation

semiannual payment = ½ coupon rate × inflation adjusted par value

2010 H1FV: $1,000Coupon Rate : 3%Annual Inflation Rate : 4%

Adjusted Par value : $ 1,000 * 1.02Coupon payment : $ 1,020 * 1.5% = 15.3

Find out the 2010 H2 coupon payment?

Inflation Rate : 1%

Adjusted Par value : $ 1,020 * 1.01Coupon payment : $ 1,030.2 * 1.5% = 15.453

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TIPS

On the date of maturity

Holder to get par value, if adjusted value is lower than par value

Holder to get such adjusted value if the adjusted value if higher than the par value

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On- and Off-the-Run Treasuries

On-the-run issues: Most recent auction issues, most liquid, actively

traded

Off-the-run issues: Older issues (replaced by more recent issues)

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Stripped Treasury Securities

A type of zero-coupon bond created from Treasury notes and bonds; the pieces (coupon payments and the principal payment) are separated

Coupon strips are denoted ci Principal strips are denoted pi

STRIPS (zeros) are taxed on implicit interest

J K Shah Classes 52

Securities Issued by Federal Agencies

Federally related institutions (e.g., GNMA, TVA) Exempt from SEC Registration Backed by US Govt.; hence risk free

Government-sponsored enterprises (Sallie Mae, Freddie Mac, Fannie Mae) Although privately help, created by US congress; a little credit risk

Agency securities, very little credit risk

Debentures: Securities not backed by collateral, unsecured bonds

J K Shah Classes 53

Mortgage-Backed Securities

Passthrough Securities

CMOs STRIPS

Securitization :

to increase debt attractiveness, increasing fund availability Decrease yield as they are backed by pool of mortgages

Cash flow in the form of Interest payment , principal payment and prepayments

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Passthrough Securities

CMOs STRIPs

Proportionate payment to all the security holder

Some diversification due to pool of mortgage

Derivative Mortgage Product

More complex Structure

Different Tranches (slices) with different claim

Some diversification due to pool of mortgage

Redistribution of prepayment risk and maturity

Same over all risk

Principal and Interest Strips

PO to benefit from prepayment

IO to lose in case of prepayments

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CMO Tranche example – Sequential Tranches

Tranche I receives interest on its outstanding principal and all principal payments until the tranche is completely paid off

Tranche II receives interest on its outstanding principal and begins receiving all principal payments when Tranche I is paid off

Tranche III receives only interest until Tranches I and II are paid off, then receives all remaining principal payments

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Prepayment Risk

Risk of receiving principal repayment in excess of scheduled principal payments

May lead to more funds to be reinvested when rates for reinvestment are low—reinvestment risk

When rates increase, prepayments slow

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Motivation for Creating a CMO

Alter maturity range and redistribute prepayment risk to make securities attractive to different institutional investors

Creating a CMO does not alter the overall risk of prepayment

Goal is lower overall cost of funds

J K Shah Classes 58

Special Types of Municipal Bonds

Insured bonds Backed by insurance policies in the event of defaults, insured for life of

issue, lowers yield, increases liquidity

Prerefunded bonds Collateralized with escrow of Treasury securities which will support bond

payments

Normally coupon payment is exempt & capital gain is taxable at federal level

Instate bonds are tax free at state and federal level; where as out of state bonds are taxable at state level and federally exempt.

State Municipal Bonds must meet certain federal criteria to be exempt from federal tax

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Special Types of Municipal Bonds

Revenue BondTax Backed bonds / General Obligation

(GO)bonds

Backed by the taxing capacity of the issues

Limited Tax GOUnlimited Tax GODouble barreled bondAppropriation backed bonds

Supported only by the revenues from the projects covered

Obligation to pay interest / principal only from the project revenueMore risky than GOs

J K Shah Classes 60

Corporate Bonds

Rating

Secured bonds: first claim against specific collateral (mortgage debt, collateral trust bonds)

Debenture bonds: unsecured bonds, no specific collateral (debentures)

Credit enhancements bonds: Third Party guarantee, Letter of credit, Bond Insurance; Ensure the strength of the party enhancing credit

Bonds have a priority of claims over both preferred and common stockholders in the event of bankruptcy

J K Shah Classes 61

Corporate Debt Securities

Structured Notes

MTN combined with derivative, “rule busters”Create a security to some institutional investors

Corporate Bonds are normally

Sold at onceSold on firm commitment basisSingle coupon and maturity

Medium-Term Notes (MTN)

Continuously offered by agent Buyers can customize 9 months to 30+ years Fixed, floating, or structured

Corporate Deposit

Short term (<270 days) UnsecuredLike Zero CouponNo SEC registration required

J K Shah Classes 62

Corporate Debt Securities Commercial paper

2 to 270 days Pure discount Not liquid Sold through dealers or by the company itself

Directly placed paperSold to large investors without going through an agent Select group of regular commercial paper buyers

Dealer placed paperSold to purchase through commercial paper dealerLarge investments firms have commercial paper desk

J K Shah Classes 63

Debt Securities Issued by Banks

Negotiable CDs Days to 5 years Secondary market Domestic (U.S.) and Eurodollar Issued primarily in London – LIBOR

Bankers acceptances Created to guarantee payment for shipped goods Short-term Pure discount Few dealers, liquidity risk

J K Shah Classes 64

Asset-Backed Securities (ABS)

Debt securities backed by financial assets (e.g., mortgages, auto loans, credit card receivables)

Firm sells assets to Special Purpose Vehicle Separate entity, bankruptcy remote SPV issues securities Can have better rating (lower yield) than firm’s debt Reduce funding costs

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Sources of External Credit Enhancements Corporate guarantees: Which may be obtained from a bank fee

Bank letters of credit: Which may be obtained from a bank for a

fee

Bond insurance (insurance wrap): Which may be obtained from an insurance company or a provider specializing in underwriting such structures.

Asset-Backed Securities (ABS)

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Collateralized Debt Obligations Tranches Created based on seniority of claims to cash flows from collateral Collateral is a pool of other debt obligations –e.g. business loans,

mortgages, asset-backed securities, other CDOs etc.

Arbitrage CDOs Profit from cash flow spread

Balance Sheet CDOs To reduce loans on balance sheet (banks)

J K Shah Classes 67

Primary and Secondary Markets

Primary market: Newly created debt securities

Firm commitment: Investment banker purchases the entire issue and resells it

Best efforts basis: Investment banker agrees to sell all of the issue that they can

Private placement (Rule 144A offering): Sold to a small number of

investors, issue is not registered for sale to the public

Secondary market: Sales of existing securities through exchanges, OTC (dealer) markets, or electronic trading networks

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Understanding Yield Spreads

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Federal Reserve’s Interest Rate Policy Tools (Monetary Policy)

Discount rate

Open market operations (most common)

Bank reserve requirements

Persuading banks to tighten or loosen their credit policies

LOS 63.a

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Yield Curve Shapes

YIELD YIELD

NORMAL

FLAT

YIELD TERM TO MATURITY YIELD TERM TO MATURITY

HUMPED

INVERTED TERM TO MATURITY TERM TO MATURITY

LOS 63.b

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Term Structure Theories

Pure Expectations Theory Yield curve shape determined by expectations about future

short-term rates

Liquidity Preference (Premium) Theory In addition to above expectations, investors also require a risk premium for holding long term maturity bonds

Greater premium (yield) required for longer maturities; may take any shape;

Market Segmentation Theory Supply and demand for specific maturity ranges determines interest rates; any shape

LOS 63.c

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Liquidity Premium

YIELD YIELD CURVE WIT H LIQUIDITY

PREFERENCES

LIQUIDITY PREMIUM

YIELD CURVE WITHOUT LIQUIDITY PREFERENCES (PURE EXPECTATIONS)

MATURITY

LOS 63.c

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Liquidity Premium Added to Decreasing Expected Rate

YIELD

LIQUIDITY PREFERENCES YIELD CURVE

LIQUIDITY PREMIUM

PURE EXPECTATIONS YIELD

CURVE (SHORT-TERM RATES EXPECTED TO

DECLINE)

MATURITY

LOS 63.c

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Treasury Spot Rates

Treasury spot rates : Appropriate discount rate for single payments of various maturities from Treasury securities.3

And the spot rates for different time periods that correctly value the cash flows from treasury bond are called arbitrage free Treasury Spot rate Curve

YTM is the single disc rate which makes the PV of the bond’s promised cash flow equal to its market price

LOS 63.d

Actually the discount rate for cash flow which come at different time periods are typically not same

Spot rate curve is not horizontal

J K Shah Classes 75

Treasury Spot RatesLOS 63.d

FV : 100$Coupon : 10%

Spot rates:

1 yr = 8%2yrs = 9%3yrs = 10%

Compute the value of the bond….

Price = 1100

1.1031.092

100100

1.08

+ +

FV : 1000$Coupon : 8%

Spot rates:

1 yr = 9%2yrs = 10%3yrs = 11%


Yield Spread Measures

76J K Shah Classes

Absolute spread = Yield of higher yield bond – yield of lower yield bond 6.75 – 6.5 = .25 bsp

Relative Yield Spread = Absolute Spread / yield on the benchmark bond .25/6.5 = 3.8%

Yield ratio = Bond yield / Benchmark bond yield 6.75 / 6.5 = 1.038

Yield of Bond X : 6.5%

Yield of Bond Y : 6.75%

X is the benchmark yield

LOS 63.e

Yield of Bond X : 6.5%

Yield of Bond Y : 6.75%

X is the benchmark yield

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Credit Spreads

Difference between yields of bonds that differ only in credit rating

Often quoted as a spread to Treasuries

Credit spreads narrow during expansions and widen during contractions/recessions

LOS 63.f

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Embedded Options and Spreads

Including a put, conversion, or exchange option with a corporate bond reduces required yield and decreases yield spread relative to Treasuries

Including a call option increases required yield and increases yield spread relative to Treasuries

LOS 63.g

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Liquidity and Yield

Investors prefer more liquidity so less liquid issues have greater required yields and greater yield spreads relative to Treasuries, which are very liquid

Larger issues typically have more liquidity and therefore, lower yields and lower yield spreads than otherwise identical smaller issues

LOS 63.h

J K Shah Classes 80

After-Tax and Taxable Equivalent Yields

The after – tax yield on a taxable security can be calculated as:

After – tax yield = taxable yield x (1 – marginal tax rate)

Taxable equivalent yield = tax- free yield /(1- marginal tax

rate)

LOS 63.i

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After-Tax and Taxable Equivalent Yields Example

LOS 63.j

Bond A : tax free rate of 4.5% Bond B: taxable rate of 6.75% Investor marginal tax rate is 35%

Tax Equivalent Yield = 4.5/(1-.35) = 6.92%After tax return =.0675*(1-0.35) = 4.39%

Either approach gives the same answer she should buy the tax free bond.

4.50%>4.39% & 6.92% < 6.75%

J K Shah Classes 82

LIBOR and Funded Investors

LIBOR – London Interbank Offer Rate Most important reference rate for floating-rate

securities

A funded investor borrows short term (typically at LIBOR) to finance an investment position

Profits depend on funding costs

LOS 63.c

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Introduction to the Valuation of Debt Securities

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Study Session 16

64. Introduction to the Valuation of Debt Securities

65. Yield Measures, Spot Rates, and Forward Rates

66. Introduction to the Measurement of Interest Rate Risk

J K Shah Classes 85

Introduction to the Valuation of Debt Securities

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3-Step Bond Valuation Process

Bond value = present value of future cash flows, coupons and principal repayment

Step 1 : Estimate cash flows

Step 2 : Determine the appropriate discount rate The risk factors of uncertainty about the receipt of cash

flow . Liquidity risk, interest rate risk, call/prepayment risk, credit risk/default risk, etc.

Step 3 : Calculate present values of promised cash flows

87

Difficulties in Estimating the Cash Flow Stream

The principal repayment stream is not known with certainty (for e.g. lower rates will increase prepayments of mortgage pass-through securities, and principal will be repaid earlier)

The coupon payments are not known with certainty. With floating securities, future coupon payments' depends on the path of interest rates

The bond is convertible or exchangeable into another security

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Valuing an Annual-Pay Bond Using a Single Discount Rate

Term to maturity = 3 years

Par = $1,000

Coupon = 10% annual coupon Discount rate 12%

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8% Annual-Pay Bond Cash Flows

Year Year Year Year

0 1 2 3

100 100 100

1,000

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LOS 63.d

FV : 1000$

Coupon : 10%

Yield : 12%


Price = 1100

1.1231.122

100100

1.12

+ +

Bond Value: 10% Coupon, 12% Yield

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Same (8% 3-yr.) Bond With a Semiannual-Pay Coupon

PMT = coupon / 2 = $80 / 2 = $40 N = 2 × # of years to maturity = 3 × 2 = 6 I/Y = discount rate / 2 = 12 / 2 = 6% FV = par = $1,000

N = 6; I/Y = 6; PMT = 40; FV = 1,000; CPT PV = –901.65

J K Shah Classes 92

9% 3-Year Bond With Semiannual Coupon Payments

40 40 40 40 40 1040

+ + + + +

1.06 1.06 1.06 1.06 1.06 1.06

=901.65

J K Shah Classes 93

Price-Yield RelationshipSemiannual-Pay 8% 3-yr. Bond

At 4%: I/Y = 2% N = 6 FV = 1,000 PMT = 40 CPT PV = $1,112.03

At 8%:I/Y = 4% N = 6 FV = 1,000 PMT =

40 CPT PV = $1,000.00

At 12%:I/Y = 6% N = 6 FV = 1,000 PMT = 40 CPT PV = $901.65

J K Shah Classes 94

Price Change as Maturity Approaches

Bond Value ($) A premium bond (e.g. a 6% bond trading at YTM of 3%)1,085.458

A par value bond( e.g. a 6% bond trading at YTM of 6%)1,000.00 A discount bond ( e.g. a 6% bond trading at YTM of 12%)

852.480

Time

As maturity nears, the bond value reaches FV.

J K Shah Classes 95

Value Change as Time Passes – Problem

6% 10-year semiannual coupon bond is priced at $864.10 to yield 8%

N = 20, PMT = –30, FV = –1,000, I/Y = 4% PV = 864.10

1. What is the value after 1 year if the yield does not change? N = 18, PMT = –30, FV = –1,000, I/Y = 4% PV = 873.41

2. What is the value after 2 years if the yield does not change? N = 16, PMT = –30, FV = –1,000, I/Y = 4% PV = 883.480

J K Shah Classes 96

Calculate a Zero-Coupon Bond Price

$1,000 par value zero-coupon bond matures in 3 years and with a discount rate of 8%

TVM Keys: N = 3 × 2 = 6, PMT = 0, FV = 1,000, I/Y = 8 / 2 = 4 CPT PV = –790.31

Mathematically: 1000 =$790.91 (1.04)6

J K Shah Classes 97

Arbitrage-Free Bond Prices

Dealers can separate a coupon Treasury security into separate cash flows (i.e., strip it)

If the total value of the individual pieces based on the arbitrage-free rate curve (spot rates) is greater or less than the market price of the bond, there is an opportunity for arbitrage

The present value of the bond’s cash flows (pieces) calculated with spot rates is the arbitrage-free value

98

Arbitrage-Free Pricing Example

Market Price of a 1.5-year 6% Treasury note is $984 Value cash flows using (annual) spot rates of 6 months = 5%, 1-yr. = 6%, 1.5 yr. = 7%

Maturity Annual Spot Rate

Semi-Annual spot Rate

Cash Flow per ($1,000)

0.5years 5% 2.5% $30

1.0years 6% 3.0% $30

1.5years 7% 3.5% $1050

J K Shah Classes 99

Valuing the Pieces Using Spot Rates

6 mo. 12 mo. 18 mo.

30 30 30 + + =986.55 1.025 (1.03)2 (1.035)3

Buy the bond for $984, strip it, sell the pieces for a total of $986.55, keep the arbitrage profit = $2.55

J K Shah Classes 100

Arbitrage Process

Dealers can strip a T-bond into its individual cash flows or combine the individual cash flows into a bond

If the bond is priced less than the arbitrage free value: Buy the bond, sell the pieces

If the bond is priced higher than the arbitrage-free value: Buy the pieces, make a bond, sell the bond


Yield Measures, Spot Rates, and Forward Rates


Sources of Bond Return

1. Coupon interest

2. Capital gain or loss when principal is repaid

3. Income from reinvestment of cash flows


Traditional Measures of Yield

Nominal yield (stated coupon rate) Current yield Yield to maturity Yield to call Yield to refunding IRR-based

yields Yield to put Yield to worst Cash flow yield


YTM for an Annual-Pay Bond

Consider a 6% , 3years annual pay bond priced at 943$

60 60 1060943 = + + (1 +YTM) (1+YTM)2

(1+YTM)3

TVM functions: N = 3, PMT = 60, FV = 1,000,

PV = –943, CPT I/Y = 8.22% Priced at a discount → YTM > coupon rate


YTM for a Semiannual-Pay Bond

With semiannual coupon payments, YTM is 2 × the semiannual IRR

COUPON 1 COUPON 2 COUPON N + PAR VALUE

PRICE = + +…+ (1+YTM/2) (1+YTM/2)2 (1+YTM/2)N


Semiannual-Pay YTM Example

A 3-year 5% Treasury note is priced at $1,028

N = 6 PMT = 25 FV = 1000 PV = –1,028 CPT I/Y = 2% YTM = 2 × 2% = 4%

The YTM for a semiannual-pay bond is called a Bond Equivalent Yield (BEY)

Note: BEY for short-term securities in Corporate Finance reading is different.


Current Yield (Ignores Movement Toward Par Value)

annual coupon payment

CURRENT YIELD = current price

For an 8%, 3- years (semiannual pay) bond price 901.65

80 CURRENT YIELD= = 8.873

YTM = 12% 901.65


Yield to First Call or Refunding

For YTFC substitute the call price at the first call date for par and number of periods to the first call date for N

Use yield to refunding when bond is currently callable but has refunding protection

Yield to worst is the lowest of YTM and the YTCs for all the call dates and prices


Yield to Call – Problem

Consider a 10-year, 5% bond priced at $1,028

N = 20 PMT = 25 FV = 1,000 PV = –1,028 CPT → I/Y = 2.323% × 2 = 4.646% = YTM

If it is callable in two years at 101, what is the YTC?

N = 4 PMT = 25 FV = 1,010 CPT → I/Y = 2.007% × 2 = 4.014% = YTC PV = –1,028


Yield to Put and Cash Flow Yield

For YTP substitute the put price at the first put date for par and number of periods to the put date for N

Cash flow yield is a monthly IRR based on the expected cash flows of an amortizing (mortgage) security


Assumptions and Limitations of Traditional Yield Measures

1. Assumes held to maturity (call, put, refunding, etc.)

2. Assumes no default

3. Assumes cash flows can be reinvested at the computed yield

4. Assumes flat yield curve (term structure)


Required Reinvestment Income

6% 10-year T-bond priced at $928 so YTM = 7%

1st: Calculate total ending value for a semiannual compound yield of 7%, $928 × (1.035)20 = $1,847

2nd: Subtract total coupon and principal payments to get required reinvestment income

$1,847 – (20 × $30) – $1,000 = $247


Factors That Affect Reinvestment Risk

Other things being equal, a coupon bond’s reinvestment risk will increase with:

Higher coupons—more cash flow to reinvest Longer maturities—more of the value

of the investment is in the coupon cash flows and interest on coupon cash flows


Annual-Pay YTM to Semiannual-Pay YTM

Annual-pay YTM is 8%, what is the equivalent semiannual-pay YTM (i.e., BEY)?

( 1.08 – 1) x 2 = 7.846%


Semiannual-Pay YTM to Annual-Pay YTM

Semiannual-pay YTM (BEY) is 8%, what is the annual-pay equivalent?

Semiannual yield is 8/2 = 4%. Annual-pay equivalent (EAY) is:

0.08 2 1+ - 1 = 8.16% 2


Theoretical Treasury Spot Rates

Begin with prices for 6-month, 1-year, and 18-month Treasuries:

6-month T-bill price is 98.30, 6-month discount rate is 1.73% BEY = 2 × 1.73 = 3.46%

1-year 4% T-note is priced at 99.50

20 1020 20 1020 + =995 995 - = 975.34

?


Theoretical Treasury Spot Rates

Begin with prices for 6-month, 1-year, and 18-month Treasuries:

1.5-year 4.5% T-note is priced at 98.60

By “bootstrapping,” we calculated the 1-year spot rate = 4.52% and the 1.5-year spot rate = 5.52%


Valuing a Bond With Spot Rates

Use the spot rates we calculated to value a 5% 18-month Treasury Note.

25 25 1025 + + = 993.09 (1.0173) (1.0226)2 (1.0276)3


Nominal and Zero-Volatility Spreads

Nominal spreads are just differences in YTMs

Zero-volatility (ZV) spreads are the

(parallel) spread to Treasury spot-rate curve to get PV = market price

Equal amounts added to each spot rate to get PV = market price


Option-Adjusted Spreads

Option-adjusted spreads (OAS) are spreads that take out the effect of embedded options on yield, reflect yield differences for differences in risk and liquidity

Option cost in yield% = ZV spread% – OAS% Option cost > 0 for callable, < 0 for

putable

Must use OAS for debt with embedded options


Forward Rates

Forward rates are N-period rates for borrowing/lending at some date in the future

Notation for one-period forward rates:

1F0 is the current one-period rate S1

1F1 is the one-period rate, one period from now

1F2 is the one-period rate, two periods from now

2F1 is the two-period rate, one period from now


Spot Rates and Forward Rates

(1+ S3)3 = (1+ S1) (1+ 1F1) (1+ 1F2)

(1+ S3)3 = (1+ S1) (1+ 1F2)2

(1+ S3)3 = (1+ S2) (1+ 1F2 )

Cost of borrowing for 3 yr. at S3 should equal cost of:

Borrowing for 1 yr. at S1, 1 yr. at 1F1, and 1 yr. at 1F2

Borrowing for 1 year at S1 and for 2 years at 1F2

Borrowing for 2 years at S2 and for 1 year at 1F2


Forward Rates From Spot Rates

S2 = 4% , S4 = 5%, CALCULATE 1F 2

(1+S3)3 (1.05)3

- 1 = 1F2 so, - 1 = 7.03%

(1+S2)2 (1.04)2

Approximation: 3 × 5% – 2 × 4% = 15% – 8% = 7%


Forward Rates From Spot Rates

S2 = 4% , S4 = 5%, CALCULATE 2 F 2

(1+S4)4 (1.05)4

- 1 = 2F2 SO - 1 = 6.01%

(1+S2)2 (1.04)2

Approximation: 4 × 5% – 2 × 4% = 20% – 8% = 12% 12% / 2 = 6%

2F2 is an annual rate, so we take the square root above and divide by two for the approximation


Spot Rates From Forward Rates

Spot rate is geometric mean of forward rates (1 + S1) (1+ 1F1) (1+ 1F2)1/3 – 1 = S3)

Example: S1 = 4%, 1F1 = 5%, 1F2 = 5.5%

3-period spot rate = (1.04) (1.05)(1.055) 1/3 – 1 = S3=

4.8314%

Approximation: (4+5+5.5) = 4.833

3


Valuing a Bond With Forward Rates

1-year rate is 3%, 1F1 = 3.5%, 1F2 = 4% Value a 4%, 3-year annual-pay bond

40 40 1040 + +

= 1014.40 (1.30) (1.30)(1.035) (1.30)(1.035)(1.04)

1+S1 ( 1+S2)2 (1+s3)3


Introduction to the Measurement of

Interest Rate Risk


Measuring Interest Rate Risk

Full valuation approach: Re-value every bond based on an interest rate change scenario

Good valuation models provide precise values

Can deal with parallel and non-parallel shifts

Time consuming; many different scenarios

Duration/convexity approach: Gives an approximate sensitivity of bond/portfolio values to changes in YTM Limited scenarios (parallel yield curve shifts) Provides a simple summary measure of interest rate risk


Option-Free Bond Price-Yield Curve

110.67

100.00

90.79

Price (% of Par)

For an option-free bondthe price-yield curve isconvex toward the origin.

Price falls at a decreasingrate as yields increase.

YTM7% 8% 9%


Callable Bond Value

Callable bond = option-free value – call option

Negative convexityoption-free bond

call optionvalue

callable bond

Yield

Price (% of Par)

y' Positive ConvexityNegative Convexity

102


Price-Yield for Putable Bond

Less interest rate sensitivity

option-free bondYield

Price

y'

putable bond

value of the put option


Computing Effective Duration

Price at YTM – ∆y Price at YTM + ∆y

v_ - v+ Duration = 2(V0) (∆y)

Current price Change in YTM


Computing Effective Duration

Example: 15-year option-free bond, annual 8% coupon,

trading at par, 100

Interest rates ↑ 50bp, new price is 95.848

Interest rates ↓ 50bp, new price is 104.414

Effective duration is:

104.414 95.8488.57

2 100 0.005

50 basis points

current price

V–

V+


Using Duration

Our 8% 15-year par bond has a duration of 8.57

Duration effect = –D × Δy

If YTM increases 0.3% or 30bp, bond price decreases by approximately:

–8.57 × 0.3% = –2.57%


Duration Measures

Macaulay duration is in years Duration of a 5-yr. zero-coupon bond is 5 1% change in yield, 5% change in price

Modified duration adjusts Macaulay duration for market yield, yield up → duration down

Effective duration allows for cash flow changes as yield changes, must be used for bonds with embedded options


Effective Duration

Both Macaulay duration and modified duration are based on the promised cash flows and ignore call, put, and prepayment options

Effective duration can be calculated using prices from a valuation model that includes the effects of embedded options (e.g., call feature)

For option-free bonds, effective duration is very close to modified duration

For bonds with embedded options, effective duration must be used


Duration Interpretation

1. PV-weighted average of the number of years until coupon and principal cash flows are to be received

2. Slope of the price-yield curve (i.e., first derivative of the price yield function with respect to yield)

3. Approximate percentage price change for a 1% change in YTM: The best interpretation!


Bond Portfolio Duration

Duration of a portfolio of bonds is a portfolio-value-weighted average of the durations of the individual bonds

DP = W1D1 + W2D2 +……+WnDn

Problems arise because the YTM does not change equally for every bond in the portfolio


Convexity Adjustment

Consider an 8% 20-year Treasury bond

priced at $908 so that it has a YTM of 9%

For a 50 bp increase in YTM, price = $866.80

For a 50 bp decrease in YTM, price = $952.30

Duration = (952.3 – 866.8)/(2x908x0.005)= 9.42


Convexity Adjustment

Using duration estimated price for a 1% decrease in YTM is $993.53 (908 x 1.0942)

Using duration estimated price for a 1% increase in YTM is $822.47 (908 x (1- 0.0942))

Actual price for a 1% decrease $1000 (YTM 8%)

Actual price for a 1% increase $828.41(YTM 10%)


The Convexity Adjustment

Duration-based estimates of new bond prices are below actual prices for option-free bonds

$1,000.00$993.53

$908.00

$828.41$822.47

Price

Actual price-yield curve

Prices based on durationare underestimates of actual prices.

YTM8% 9% 10%

Price estimates basedon a duration of 9.42


Convexity Effect

To adjust for the for the curvature of the bond price-yield relation, use the convexity effect:

+ Convexity (∆y)2

Assume convexity of the bond = 68.33 Convexity (∆y)2 = 68.33(0.01)2 = 0.006833 y=1.00% So our convexity adjustment is + 0.6833%

for a yield increase or for a yield decrease


Duration-Convexity Estimates

For a yield decrease of 1.0% we have:

–9.42 (–0.01) + 68.33 (–0.01)2 = +10.103% New Price $908 x 1.10103 = $999.74

For a yield increase of 1.0% we have:

–9.42 (0.01) + 68.33 (0.01)2 = –8.737% New Price $908 x (1- 0.08737) = $828.67

Convexity adjustment improved both estimates!


Modified and Effective Convexity

Like modified duration, modified convexity assumes expected cash flows do not change when yield changes

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


Price Value of a Basis Point

A measure of interest rate risk often used with portfolios is the price value of a basis point

PVBP is the change in $ value for a 0.01% change in yield

Duration × 0.0001 × portfolio value = PVBP

Example: A bond portfolio has a duration of 5.6 and value of $900,000

PVBP = 5.6 × 0.0001 × $900,000 = $504

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