Lecture 1

Introduction and Bond Pricing

Lecturer

• Teachers – The first part of the course is taught by Joakim

Bang • Consultation hours Mondays 11 to 1 in ASB 311

– You can reach me (in preferred order) on: • The blackboard discussion forum

• Email: j.bang@unsw.edu.au

– The second part of the course is taught by Sid Sahgal • Consultation hours and contact details TBA

Textbook

• We use Bodie, Kane and Marcus Investments, 9th edition

• You may or may not get by with an earlier edition of the book

• Everything will be covered in the lectures

• If you want to do additional exercises, you may want to buy the solution manual associated with the book

Lectures

• Each lecture will have a corresponding thread on the blackboard discussion forum

• For each lecture, the course outline lists essential readings (which you should really try to read before the lecture) and recommended readings (which is good to read before the lecture)

Tutorials

• There are weekly tutorials

• Exercises for each tutorial will be posted on blackboard

• You should try to solve the questions yourself before watching the tutorial

Online Quizzes

• Every lecture comes with an online quiz that counts towards your final grade

• You get a maximum of three attempts, with the highest score counting

• There’s a total of 11 quizzes, counting for a total of 11 % of your final grade

• The quiz deadlines are given in the course outline • Please make sure to meet the deadlines.

Extensions will only be given in exceptional circumstances.

Computer assignment

• The assignment is to be submitted via blackboard on May 6

• You should also email me a copy of your solution (as a backup in case something goes wrong with your blackboard submission)

• The assignment is solved in groups of up to three students. You may form smaller groups, but the total workload remains the same

• The assignment counts for 10 % of the final grade

Exams

• The midterm counts for 37% of the grade

– It’s given on the weekend after week 7

• The final exam counts for 37% of the grade

– It’s given on UNSW campus in the summer term examination period

Top Hat Monocle

• We’re trying out some software to submit anonymous feedback during the lecture

• You can do this via your pad or phone

• Please read the student instructions posted on blackboard on how to do this

First topic: Bond pricing

• We will approach this topic a bit differently than the textbook

• Specifically, we will be explicit about how arbitrage pricing allows use discounting to price a bond

• The lecture notes will give you a good understanding about what is relevant for the midterm

• Some of the extensions in the textbook will be covered in the tutorial sessions

What is a bond?

• A claim on some fixed future cash flow(s), CF.

• The bond matures at the time of its last cash flow, T.

• Typically a “large” cash flow at maturity. We call this the par value or face value (FV).

• There may be a series of smaller cash flows before maturity. We call these coupons.

• There may be zero, one or more coupons in a given year.

What is a bond?

• The sum of the annual coupons are often expressed as a fraction of the FV, e.g. 5 %. We call this the coupon rate (C). Let’s denote the actual coupon, e.g. $5, with ct, where t is the period in which we get the coupon.

• A bond with no coupons is called a zero-coupon bond

Cash flows of a bond

• This figure illustrates the cash flows of a bond with a FV of 100 and a yearly coupon of 5

c1 c2

FV

-P

5 5

100

-91.3

Default risk

• That somebody promises to pay you some money doesn’t necessarily mean they will

• The risk that you will be unable to collect your cash flows is called default risk

• This is very important in practice, but we will generally ignore it in this course

Other frequent assumptions

• No transaction costs

• Constant interest rates

• Complete markets

• These are all true within our model

– Compare this to the assumption of vacuum in classical mechanics

Two approaches to pricing

• Fundamental pricing – Prices are set in a supply-demand equilibrium – The properties of an asset tell us what that price is

likely to be – We will use this approach when pricing stocks later in

the course

• Arbitrage pricing – Take some price as given and price other assets

relative to that – We will use this approach when pricing bonds and

derivatives

What is arbitrage?

• An arbitrage is a (set of) trades that generate zero cash flows in the future, but a positive and risk free cash flow today

• This is the proverbial “free lunch” or “money machine”

• A simple example exploits violations of the law of one price, e.g. an identical bond selling for two different prices – Simultaneously buying the cheap bond and selling the

expensive bond would be an arbitrage trade

• All arbitrage pricing is priced based on the same principle, but the trades are (slightly) more complex

Replicating portfolios

• We typically rely on a portfolio of assets that exactly mimic the cash flows of some other asset

• We call such portfolios replicating portfolios or synthetic assets

• Arbitrage pricing is all about constructing replicating portfolios using assets with known prices

Example: Pricing a zero-coupon bond

• How would you price the risk-free one-year zero-coupon bond below?

Bond A

100

Example: Pricing a zero-coupon bond

• You may already know how to discount the future cash-flow with some appropriate discount rate, y, to get the present value

• Assuming that r = 10% you’d get

• What is the economic logic behind this?

100 100

90.91 1.10

APy

Where does the discount rate come from?

• The appropriate discount rate, y, is the return we could have earned at some alternative investment with the same risk

• Let’s say there’s a bank where you can lend and borrow money at 10% interest

• Suppose the price of Bond A was actually $80.9, i.e. lower than what we found on the last slide

Constructing a replicating portfolio

• We know that the bond is mispriced. How do we exploit this?

• We want to make a synthetic version of the bond, i.e. some investments that mimic its cash flows exactly

• In this simple example we can just put some amount of money, M, in the bank.

• How large must M be? • After one year in the bank account earning 10% interest, it

should have grown to match the bonds cash flow of $100 • We must have

1.1 100

10090.9

1.1

M

M

Exploiting the mispricing

• The $90.9 bank deposit replicates the bonds cash flow (is a synthetic bond) but has a different price

• We buy the cheap instrument and sell the expensive (in this case the synthetic) instrument

• “Selling” a bank deposit means borrowing the money

– Short selling (or shorting) strategies or instruments in this way is often an important part of arbitrage pricing

What are our cash flows?

• Today we borrow $90.9 and buy the bond for $80.9. We are left with $10.

• In one year the bond pays us $100 which is exactly enough to repay the loan. We have zero net cash flow.

• Our “free” $10 is an arbitrage profit and the entire scheme is an arbitrage trade

Arbitrage pricing

• In practice smart people will identify arbitrage opportunities and trade on them

• This will increase the demand for the bond and raise its price until no further arbitrage trades are possible, i.e. until prices are in equilibrium

• In this course we are interested in finding those equilibria, e.g. arbitrage-free prices

• We can not say whether it was the bond price or the bank’s interest rate that was wrong

• We can only say (and only care) if the prices are internally consistent

How do we find the price of our bond?

• Strategy: Replicate the entire CF-stream we want to price

• For there to be no arbitrage the price of the CF-stream must be the same as the price of the replication

5 5

100

-P

How do we find the price of our bond?

• Think of the bond as two zero-coupon bonds

• Replicate each bond by depositing money in the bank, as before

• Together, our two deposits will form a replicating portfolio

• Note that the interest rate we get for a two year deposit may be different from that of a one year deposit – To indicate the maturity of an interest rate we

typically use a time index: yt

How do we find the price of our bond

• We want to replicate a cash flow of c at time T = 1 – Observe some interest rate, y1, that is valid over the time

[0,1] – Today, deposit M1 such that M1(1 + y1) = c – Find that M1 = c/(1 + y1)

• We want to replicate a cash flow of FV + c at time T = 2 – Observe some interest rate, y2, that is valid over the time

[0,2] – Today, deposit M2 such that M2(1 + y2)2 = FV + c – Find that M2 = (FV + c)/(1 + y2)2

• Our complete strategy costs M1 + M2 = c/(1 + y1) + (FV + c)/(1 + y2)2

Discounting

• We say that P is the present value, PV, of the future cash flows

• This process of calculating the PV of future CFs is called discounting

• The market determines the appropriate interest rates, y1 and y2

• We are typically not explicit with the entire arbitrage argument

Discounting and prices

• The price of a bond (or indeed any financial asset) is the sum of the present values of its future cash flows.

• The price of a bond (or indeed any financial asset) is the sum of the present values of its future cash flows. That’s worth repeating.

Pricing formula and yield-to-maturity

• When we have many CFs, discounting each one gets tedious

• It would be useful with a compact pricing formula

• To get one we will assume that interest rates are constant, i.e. yt = y for any t

• We also have to introduce the concept of perpetuities

Perpetuities

• A perpetuity is a never ending constant cash flow stream, e.g. an annual payment of c

• How do we value such a thing? Set up a replicating portfolio

• Let’s deposit some amount of money, M, in the bank and withdraw the interest every year

• How large would M have to be in order to give an interest of c?

y

cM

cMy

What do we want to replicate?

• One coupon stream (of c) from 1 to T • One large payment (of FV) at T • The present value of the FV is easy

– PV(FV) = FV/(1+y)T

• The coupon stream, CS, can be viewed as the difference between two perpetuities: – One perpetuity starting at time 1, X1

– One perpetuity starting at time T+1, X2

• Its PV is the difference in the PV of X1 and X2 – PV(CS) = PV(X1) - PV(X2)

Replicating the coupon stream

• The coupon stream, CS, can be viewed as the difference between two perpetuities:

– One perpetuity starting at time 1, X1

– One perpetuity starting at time T+1, X2

• Its PV is the difference in the PV of X1 and X2

T

T

yy

cy

y

c

y

cCSPV

XPVXPVCSPV

1

111)(

)()()( 21

Pricing formula and yield-to-maturity

• Adding the PV of the coupon stream and FV we get our pricing formula:

• Note that in practice interest rates are not constant

• Instead we take P as given and define y as whatever interest rate satisfies the equation above. Expressed on an annual basis, we call this interest rate the yield-to-maturity (YTM, although we often just denote it y).

• Each bond has its own YTM

TTy

FV

yy

cP

11

11

YTM and bond prices

• This graph shows the price of a 30-year bond with a FV of $100 and a coupon rate of 10 % for different YTMs

0

50

100

150

200

250

300

350

0% 5% 10% 15% 20% 25%

Pri

ce

YTM

YTM and bond prices

• The bond price decreases with the YTM

• The price is less sensitive to changes in the YTM when the YTM is high

• When YTM = C = 10 %, P = FV = $100 – When P = FV (C = YTM), the bond trades at par

– When P < FV (C < YTM), the bond trades at a discount

– When P > FV (C > YTM), the bond trades at a premium

Realized compound yield

• Suppose that the two bonds above have the same YTM

• For a given investment the time two cash flows will differ, since bond B pays a coupon at time 1

• That coupon will have to be reinvested to make the time two cash flows comparable

Bond A FVA

1

0

2

Bond B c

FVB

c

1

0

2

Realized compound yield

• If the coupon can be reinvested at an interest rate that equals the YTM, the time two cash flows will be equal

• The realized compound yield is a useful concept when the reinvestment rate is different from the YTM – Collect all cash flows at the maturity of the bond

– Solve for the annualized return by dividing by the price

Example: Realized compound yield

• A bond maturing in 2 years has a face value of $100 and pays an annual coupon of $10

• The price is $96.62, so the yield to maturity is 12% • Suppose we can reinvest the coupon at 10% • The resulting cash flow at time 2 would be CF2=100 +

10 + 10(1+0.1) = 121 • The realized compound yield would be

• We will be very interested in such reinvestments in the

next lecture

1211 11.9%

96.62y

An alternative interpretation of the YTM

• Suppose you could reinvest all coupons at an interest rate that equals the YTM

• The realized compound yield, i.e. your investment return at the maturity of the bond, would equal the YTM

• Although this is a common interpretation of the YTM, the concept does not make any assumptions on reinvestment rates

Current yield

• You may hear about a bond’s current yield

• This simply means the bond’s annual coupon divided by its price