0- Course Introduction - AFM 475

Course Introduction

Outline

General Background

Features of Bonds

Risks of Investing in Bonds

Time Value of Money Concepts

Supplemental Reading from Course Reserves:Fabozzi, Chapter 1Sundaresan, Chapter 1Tuckman & Serrat, pp. 1-19

AFM 475 - W2015 Course Introduction: Outline 1

General Backgroundfixed income securities are hard to define precisely since thereare many different types

in general, a fixed income security is a financial contract witha value that directly depends on one or more interest rates

general examples include government bonds, corporate bonds,mortgage-backed securities, interest rate futures, etc.

in many cases, cash flows specified in the contract may bedetermined by interest ratesfixed income securities do not receive as much attention in thefinancial press as equities, but in aggregate fixed incomemarkets are much larger than equity markets

as of December 2013, the global market value of equities wasabout $60 trillion USDthe global amount of debt securities outstanding exceeded $90trillion USD

AFM 475 - W2015 Course Introduction: General Background 2

Global Debt Marketstotal debt securities outstanding by residence of issuer forselected countries, Dec. 2013:

USD

Trillion

s

0

10

20

30US

Eurozone

Japan

UKFrance

Germ

any

China

Italy

Spain

Netherland

sCa

nada

Australia

Ireland

Denmark


Composition of Debt Marketsbreakdown of debt securities outstanding by issuer andresidence, Dec. 2013:

USD

Trillion

s

0

10

20

30US

Eurozone

Japan

UK

Governments

Financial institutions

Corporations


Global Derivative Markets

interest rate contracts are by far the largest component ofglobal derivative marketsnotional amounts outstanding as of Dec. 2013 for derivativefinancial instruments traded on organized exchanges, in USDbillions:

Futures OptionsInterest rate 24,191.2 32,793.8Currency 244.1 142.6Equity index 1,492.1 5,750.7

notional amounts vastly overstate market values, but the mainpoint is the predominant position of interest rate contracts inthese markets


Global Derivative Markets (Contd)most derivatives dont actually trade on organized exchanges,but rather in the over-the-counter (OTC) market whereinterest rate contracts are again the dominant typeOTC derivatives as of Dec. 2013, in USD billions:

Notional GrossAmount Market

Outstanding ValueInterest rate contracts 584,364 14,039Foreign exchange contracts 70,553 2,284Equity-linked contracts 6,560 700Commodity contracts 2,206 264Credit default swaps 21,020 653

breaking down interest rate contracts into components bynotional amount outstanding: forward rate agreements:73,819; options: 49,264; interest rate swaps: 461,281


U.S. Bond Market Sectorsthe U.S. bond market is typically divided into six sectors(amounts outstanding below are for 2013:Q4 and daily volumefigures are average trading volumes per day for 2013):

Treasury sector (issued by U.S. federal government):$11.85 trillion outstanding; daily volume: $551 billion

agency sector (issued by federally-related institutions andgovernment sponsored enterprises):

$2.06 trillion outstanding; daily volume: $7.1 billionmunicipal sector (issued by state and local governments):

$3.67 trillion outstanding; daily volume: $11.5 billioncorporate sector (issued by U.S. corporations or in the U.S. byforeign corporations):

$9.62 trillion outstanding; daily volume: $18.2 billionasset-backed securities sector (a corporate issuer pools loans orreceivables and uses this as collateral for a security):

$1.28 trillion outstanding; daily volume: $1.4 billionmortgage sector (securities backed by mortgage loans):

$8.80 trillion outstanding; daily volume: $247.3 billion


Types of Interest Ratesthere is a wide variety of interest rates in fixed income marketshere is a small part of a large list:

the Federal funds rate is the rate at which banks trade theirbalances held at the Federal ReserveTreasury rates are rates earned by investors in Treasury bills,notes, and bonds

generally assumed to be free of default riskEurodollar deposit rates are interest rates on USD deposits innon-U.S. banksrepo rates are rates for collateralized borrowingLIBOR stands for London Interbank Offer Rate: a referenceinterest rate reflecting the average rate at which banksestimate they can borrow from other banks

similar to Eurodollar deposit rates, and one of the mostimportant benchmark rates in fixed income markets since theyare widely used in the market for derivatives such as swapsmanipulation of LIBOR rates by various banks has been amajor recent legal and regulatory issue


Overview of Bond Featuresthe formal bond contract is known as the indenture, and thissets forth all of the obligations of the issuertype of issuer: there are three issuers in the U.S. market: thefederal government and its agencies, municipal governments,and corporations (domestic and foreign)principal and coupon are the cash flows specified in thecontract which the issuer is responsible for paying

coupon payments can be annual, semi-annual, quarterly,monthly, etc.floating rate bonds (a.k.a. floaters) are issues where couponsare reset periodically based on some reference interest rate(e.g. LIBOR)inverse floaters have coupons that move in the oppositedirection as the reference rate

amortization refers to whether the principal is paid entirely atthe maturity date (non-amortizing) or throughout the life ofthe bond (amortizing)

mortgage-based contracts are amortizingAFM 475 - W2015 Course Introduction: Features of Bonds 9

Overview of Bond Features (Contd)

embedded options: many bond contracts contain features thatallow either the issuer or the investor to alter the contract,such as:

call provisionsput provisionsconversion provisions

time to maturity:at the maturity date, the debt will cease to existin general, bonds with maturities of up to 5 years areconsidered short term, maturities between 5 and 12 years areintermediate term, and maturities of more than 12 years arelong termtime to maturity is very important because changes in interestrates affect values of long-term bonds more than short-termbonds

AFM 475 - W2015 Course Introduction: Features of Bonds 10

Risks of Investing in Bondsbond investors are potentially exposed to several types of riskinterest rate risk arises if an investor has to sell a bond beforethe maturity date because changes in interest rates canproduce capital gains and/or capital lossesreinvestment risk is the risk that the interest rate at whichinterim cash flows can be reinvested will changecall risk refers to the possibility that the issuer will shorten thematurity of bond that is callable if interest rates declinecredit risk refers to risk related to not receiving promisedpayments

default riskdowngrade riskcredit spread riskcounterparty risk

inflation risk arises because interest and principal paymentsare normally specified in nominal terms, and the real values ofthese cash flows are eroded by inflation over time

AFM 475 - W2015 Course Introduction: Risks of Investing in Bonds 11

Risks of Investing in Bonds (Contd)exchange-rate risk comes about because issues denominatedin foreign currencies have uncertain cash flows in terms ofdomestic currencyliquidity risk depends on how easy it is to sell a bond (withoutsubstantially discounting the price)volatility risk affects bonds with embedded options: if thevolatility of interest rates changes, the values of these optionswill change and so will the value of the bond itselfevent risk refers to the risk that holders of corporate bondscan suffer losses in the event of a corporate reorganizationsuch as a leveraged buyouttax risk can arise because bonds may be issued subject to acertain tax treatment, which can subsequently changerisk risk

AFM 475 - W2015 Course Introduction: Risks of Investing in Bonds 12

Time Value of Money Conceptsthe time value of money is important for fixed incomesecurities for at least two reasons:

fixed income investments pay cash flows at various futurepoints in time, and it is necessary to calculate the values ofthese cash flows today (i.e. the present value of the future cashflows) to determine if an investment is fairly pricedcalculating the rate of return earned by a fixed incomeinvestment involves determining the total amount of cashgenerated at a future date by that investment (i.e. the futurevalue of the cash flows)

consider a contract which promises to pay a single future cashflow of C at time T , and let the value of this contract todaybe PV0

if investors are willing to pay d(T ) today in return for apromised payment of one unit of money (e.g. one dollar) attime T , then PV0 = C d(T )

we will generally refer to money in terms of dollars, but thesame ideas apply to euros, yen, pounds, etc.

AFM 475 - W2015 Course Introduction: Time Value of Money Concepts 13

Discount Factors

d(T ) is a discount factor: it represents the value today ofreceiving one unit of money at time Tdiscount factors can vary according to the nature of thecontractsome factors which could affect d(T ):

length of time until Tprobability of receiving full payment at Tliquidity (ability to sell contract to a different investor)optionality (ability to change terms of contract)

if just a single future cash flow is involved, we can easilydetermine d(T ) from the price that investors are willing topay for the contractmoreover, if we know d(T1), d(T2), etc. (i.e. discount factorsfor a series of future payment dates), we can value a contractwhich promises to make payments at those dates


Discount Factors (Contd)for example, suppose a contract is to pay Cj at Tj , forj = 1, . . . ,Nthe value of this contract today is

PV0 = C1 d(T1) + C2 d(T2) + + CN d(TN)

=Nj=1

Cj d(Tj)

although they are of fundamental importance, discount factorsare not very intuitiveinvestors are more accustomed to thinking about the timevalue of money in terms of interest rateshowever, interest rates can be quoted in any of severaldifferent ways (most commonly according to differentcompounding frequencies such as annually or quarterly), andit is often necessary to convert from one quoting conventionto a different one


Interest Rate Conversionsan annual percentage rate of APR that is compounded mtimes per year means a rate of APR /m over a period of 1/myearse.g. an APR of 4% compounded 4 times per year means 1%per quarter, or 1.014 1 = 4.0604% per yearlet rm denote an annual percentage rate which is compoundedm times per yearan investment of C today will grow to

C [1+ rmm

]mTafter T yearsin the limit as m, we have continuous compoundingunder which an investment of C today will grow to CerTafter T years


Interest Rate Conversions (Contd)

we will typically just use r to denote a continuouslycompounded rate, i.e. r = r , and rm to denote an annualrate that is compounded a finite number of times m per yearconverting with continuously compounded rates:

example #1: an interest rate is quoted as an APR of 6%compounded semi-annually. What is the equivalentcontinuously compounded rate?

example #2: a non-amortizing loan of $1 million withquarterly interest payments is made at an interest rate of 6%compounded continuously. What is the size of each interestpayment?


Interest Rate Conversions (Contd)

suppose we want to convert from compounding m1 times peryear to compounding m2 times per year, where both m1 andm2 are finitethen [

1+ rm1m1

]m1=[1+ rm2m2

]m2 rm2 =

([1+ rm1m1

]m1/m2 1

)m2

warning: do not try to memorize complicated-lookingformulas, concentrate on the intuition!example: given a rate of 6% compounded monthly, what isthe equivalent rate compounded semi-annually?


More Basic Concepts

in general, interest rates vary according to maturity

however, if we assume that rates are the same for allmaturities, then several simplifications are available

with discrete compounding, these simplifications require us towork in terms of periods rather than years

note that with continuous compounding, we will always workin terms of years

notation:cash flow at end of period j : Cjinterest rate per period: y = rm/m for discrete compounding,r (per year) for continuous compoundingpresent value today (at time 0): PV0future value after n periods: FVn


A Single Cash Flowsuppose there is a single cash flow Cn at the end of period nwith discrete compounding, the present value of this cash flowis

PV0 =Cn

(1+ y)n = Cn (1+ y)n

with continuous compounding, a single cash flow of CT thatis to be received after T years has a present value ofPV0 = CT erT (note: with continuous compounding, timewill always be measured in terms of years)similarly, let C0 be a single cash flow now (end of period 0)with discrete compounding, the future value of this cash flowat the end of period n is

FVn = C0 (1+ y)n

with continuous compounding after T years, the future valueis FVT = C0erT


An Example

consider a zero-coupon bond paying $1,000 after 5 years, withan annually compounded interest rate of r1 = 6% per year:


An Example (Contd)

what about using semi-annual, quarterly, or monthlycompounding to determine PV0?


Discount Factor Revisitedfor a single cash flow of M received after T yearsPV0 = M d(T )if we work with annually compounded interest rates,

PV0 =M

(1+ r1)T d(T ) = 1(1+ r1)T

if we work with quarterly compounded interest rates,

PV0 =M

(1+ r4/4)4T d(T ) = 1(1+ r4/4)4T

in general, for compounding m

Multiple Cash Flows

consider an annuity paying C at the end of each period for atotal of n periods: what is this worth when the last paymentis received? What is it worth today?


Multiple Cash Flows (Contd)annuities are the most common type of simple cash flowpattern in fixed income markets, but three other patternsworth remembering are:

perpetuity: the same cash flow C is paid every period forever

PV0 =Cy

growing perpetuity: cash flows paid every period forever,increasing at a rate of g per period after the first payment of C

PV0 =C

y g (assuming y > g)

growing annuity: cash flows paid every period for a fixednumber of periods n, increasing at a rate of g per period afterthe first payment of C

PV0 =C

y g (1

[1+ g1+ y

]n)(assuming y 6= g)


Multiple Cash Flows (Contd)consider again the formula for the present value of a growingannuity:

PV0 =C

y g (1

[1+ g1+ y

]n)this contains all of the other present value formulas above asspecial cases:

set g = 0 for the PV of an annuitytake the limit as n for the PV of a growing perpetuitytake the limit as n and set g = 0 for the PV of aperpetuity

these present value formulas implicitly assume that:the first cash flow of C is paid one period from nown is the length of the annuity (in periods), i.e. the totalnumber of paymentsy and g are expressed on a per period basis

what if the first cash flow is today, rather than after oneperiod? What if it is after k periods?


An Example

suppose a growing annuity pays $100 starting today for 3years. Payments are made quarterly (i.e. there is a total of 12payments, one at the start of each quarter). The paymentsincrease each quarter, with the rate of increase correspondingto an annual growth rate of 2%. What is the present valuetoday of this growing annuity, given a continuouslycompounded interest rate of r = 6.8% per year?


The General Contextin practice, the simple PV and FV formulas above areimportant when determining the yield to maturity for a bondor when calculating cash flows for some contracts, but forvaluation purposes they are just shortcuts that assume thatthe discount rate for all future time periods is the same andthat the pattern of cash flows is nice

consider arbitrary cash flows of C1, C2, . . . , CN at times T1,T2, . . . , TN respectively (measured in years)

PV0 =N

j=1 Cj d(Tj) still holds, but it cannot be simplifiedthe situation is further complicated by if interest rates vary bymaturity:

let rm(Tj) denote the rate applicable for a maturity of time Tj ,compounded m

The General Context (Contd)with compounding m

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0- Course Introduction - AFM 475