Stephen Wright, Birkbeck

MSc Finance

MSc Finance & Commodities

MSc Financial Engineering

Introduction to Finance

Lecture 1, 2012

1

Introduction to Finance

Course Description

Aims

This short compulsory module covers some key ideas and concepts in

finance. It is aimed primarily at students coming on to the MSc Finance

and MSc Financial Engineering programmes with little or no prior

training in finance or economics, but should also be helpful as a revision

session.

2

Objectives

By the end of the course, students should learn about

No arbitrage pricing, with some applications to bond and forward

markets

Basics of derivative pricing.

Risk premia and risk neutrality

Assessment

A short multiple choice test taken at the same time as the qualifying

examination in mathematics.

3

Teaching Arrangements

The course is taught over 3 weeks: as follows

Lecture 1 Wednesday 5 September

Lecture 2 Wednesday 12 September

Lecture 3 Thursday 20 September

NB: note change of timing for Lecture 3 from original timetable. Slots

for Statistics and this course will be swapped this week.

4

Textbooks

There is no set text, and no required reading. Handouts will be provided

that cover key ideas. Those wishing for additional background on some

topics may also find the following texts useful. You are not required to

buy them at this stage, although they are likely also to come in handy

later in one or both programmes (as indicated below).

Hull J, Options, Futures and Other Derivative Securities, Prentice-Hall

(Finance and Financial Engineering, MSc level)

Cochrane, J, Asset Pricing (Finance, MSc level)

Bodie, S and Merton R, Finance (Finance, Introductory level)

Copeland, Thomas E & JF Weston, Financial Theory and Corporate

Policy, Addison Wesley (Finance, technical MBA level)

5

Introduction to Finance

Lecture 1, 2012 Some Key Definitions

No Arbitrage Conditions and the Law of One Price

A simple bond market

Application: the spot and forward price of oil

Bond pricing, yields and forward rates.

6

Financial assets and returns to investors Prices, Income, Payoffs and Returns

Suppose in period t you buy an asset j with price Pjt. In exchange you

(usually) get some income in the future, often over multiple periods

1 2jt jt jt kD D D

or at some earlier time t+s you may sell the asset for price Pjt+s

We can treat any asset, even one that will pay income for multiple

periods, as a one-period investment, by defining the next period’s

payoff

1 1 1jt jt jtY P D

What are we assuming about the liquidity of the asset?

7

In general payoffs are uncertain, hence most assets are risky assets.

Even assets with completely risk-free income, Djt+1, are usually risky

assets since the price they can be sold for in the next period will be

set by markets, and hence is uncertain.

The return in period t+1 is defined by

111 jt

jtjt

YR

PUsing the definition of the payoff we can also write the return as

1 1 1 111 jt jt jt jt

jtjt jt jt

P D P DR

P P P

Which element is usually taken to dominate, eg, stock returns?

8

Investment Horizons & Rational Investors

An investor at time t need only be directly concerned about the

return in t+1 since the asset could be sold in t+1.

This might be enough to satisfy a short-sighted investor.

But a rational investor knows that in order to realise the return on

the asset they must sell (presumably to another rational investor),

who will then hold it till t+2, etc, etc.

Hence the multiperiod nature of markets and prices cannot be

entirely assumed away. More on this in Week 3.

9

Nominal vs Real Returns

Rational investors should be interested in real returns

If returns are defined as above are in nominal terms, then if t is the

inflation rate, the real return t is defined by

11

1 1jt jt t

jt jt jt tt t

R RR

(The latter approximation is commonly used but does not work well

when inflation is high.)

If you compare returns between periods with different inflation rates,

inflation can be crucial (eg Japan in 90s vs UK today).

10

Broad Asset Classes

.... can be related fairly closely to the above definitions.

The risk-free asset has a payoff in period t+1 that is completely

predictable at time t. This can only be the case if:

a) it matures in period t+1, hence Yjt+1 = Djt+1.

b) Djt+1 has no default risk or inflation risk

Proxies in practical applications?

What makes an asset truly risk-free? A crucial issue at present!

11

Broad Asset Classes, continued: Bonds

A bond usually pays a fixed income (coupons) at regular intervals,

but at some (usually) fixed date in the future, at maturity, it pays

back a fixed amount (the principal, or “par value”)

Even if cash payments are entirely risk-free, payoff in t+1 is still

risky because future prices may change.

An index-linked bond guarantees both principal and coupons in real

terms, by linking to a price index (the RPI in the UK).

A corporate bond also pays fixed coupons and principal, but with

some risk of default, captured by credit ratings.

Some sovereign debt is also by no means risk-free! What

characteristics affect the risk of default?

12

Broad Asset Classes, continued: Equities and Real Estate

Equities (stocks, shares) are a claim to the ownership of a limited

liability corporation.

Hence give voting rights, potential control, etc. Irrelevant for most

shareholders; but a reminder of the function of equity markets.

Viewed as a financial asset, both future prices and income

(dividends) are uncertain.

Two sources of uncertainty usually implies higher volatility.

Real Estate can be viewed analogously. Income is rent, whether

actual or imputed.

But real estate is also typically illiquid (cf equivalence of Pjt+1 and

Djt+1 for financial assets)

13

Broad Asset Classes, continued: Commodities

What are the payoffs on commodities such as gold, oil, or pork

bellies?

From an investor’s point of view virtually all that matters is future

prices, hence effectively these are purely speculative investments,

but at the same time with a clear link to the real economy.

For many commodities underlying supply and demand factors limit

price movements in speculative markets.

If an asset had no intrinsic value, it could still yield returns through

sustained capital appreciation, but this would be a “bubble”. More

on this later in the year for MSc Finance students.

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Annual Real Returns On US Stocks, Bonds and Bills: 1802-2010*

-60%

-40%

-20%

0%

20%

40%

60%

80%

1802 1822 1842 1862 1882 1902 1922 1942 1962 1982 2002

Stocks Bonds Bills

*Sources: Jeremy Siegel, Stocks for the Long Run (1802-1899); Dimson, Marsh & Staunton, Triumph of the Optimists (1900-2010)

15

Excess Returns and Risk Premia

.... are defined, for any asset j, relative to the risk-free rate, Rt , by

1 11 1 1

11

1 1jt jt t

jt jt jt tt t

R R RXR XR R R

R R

For any risky asset realised excess returns can in principle (and often

in practice) be positive or negative (asset prices can go down as well as up)

But historic average excess returns for risky assets (often loosely

referred to as “risk premia”) are usually positive.

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Excess Returns On US Stocks and Bonds: 1802-2010

-60%

-40%

-20%

0%

20%

40%

60%

80%

1802 1822 1842 1862 1882 1902 1922 1942 1962 1982 2002

Stocks Bonds

NB: Straight lines show averages: 5.2% for stocks, 0.9% for bonds

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A stricter definition of a risk premium is

1jt t jtRP E XR

The Et is not innocuous!

Rationale for use of average actual returns as proxy for risk premia?

1

1 1

1 1 1 1

1 1

By adding and subtracting we can write

where

Hence over sufficiently long samples we would hope to find

0

(Where a bar indic

jt

jt jt jt

jt jt jt jt t jt

jt jt jt jt

XRXR RP

XR RP XR E XR

XR RP RPates a sample average)

But these samples may have to be long!very

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Arbitrage and No Arbitrage Pricing Market Assumptions

1. Many buyers, many sellers2. All sell exactly same product (“homogenous product”), which can

be sold in any quantity (including fractions of a unit).3. One unit must sell at same price, P (“Law of One Price”) 4. N units of the good must sell for N.P (N can be any number,

including fractions)

That’s it! For now we need no more assumptions about behaviour

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(No) Arbitrage

Assumption 4 seems trivial but is very important. Together with law of one price it rules out profitable arbitrage opportunities

Eg, suppose I sell N units to a consumer who pays me £R forthem, and supply him by simultaneously buying them from someone else at cost £C I need no capital to engage in this trade, and my profits are R-C and are riskless. This is arbitrage.

But in this market I would not make any profits! C=R=N.PAbsence of arbitrage opportunities and the Law of One Price are

therefore essentially the same phenomenon, and arise out of the same (minimal) assumptions about behaviour.

The Law of One Price is not imposed by order of the state; nor by construction. But if it does not hold, there will be opportunities for arbitrage, which enforce the law. In some markets, therefore, it is treated as if were a true law.

But using No Arbitrage Pricing does not mean we believe arbitrage never takes place: indeed quite the contrary ( Lucas’s $100 bill)

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A Simple Risk-Free Bond Market

Now assume that the “good” is the simplest possible bond, characterised by maturity, n

Each bond pays a guaranteed £1 in n periods and nothing before.

Periods could be days, weeks, years. Law of one price/No Arbitrage implies all such bonds with same

n must trade at same price, Pn,t, and N bonds must trade at N.Pn,t

The same applies to anything that replicates the payoff from 1 or N bonds.

Typically we assume Pn,t <1 for all n. (for reasons that will become evident)

.... so such bonds are often called (pure) discount bonds.Contrast with coupon bonds that make regular payments

between period t and period nSo discount bonds also often referred to as zero coupon bonds.For now we set n=1 and suppress time subscripts.

21

One-Period Zero Coupon Bond Prices and Interest Rates

Assume there is a (government guaranteed) bank next to the bond market.

If I deposit £1 with the bank today it will pay me a risk-free interest rate of 100.R%

So it will repay £(1+R) in the next period.Our assumptions about the nature of the bond market require

11

1P

R

This is the simplest possible no arbitrage condition (think about what would happen if it did not hold)

It does not tell us what R and P1 will be, but it does tell us that if we know P1 we know R, or vice versa.

Eg, if R=0.05 (5%),

11 0.952 1

1.05P R

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Note also that we could have started by assuming that the banking market had features 1 to 4, and used this to derive bond prices.If we started with R rather than P1 we can think of this condition as a “present value” calculation. Note that it implies a negative relationship between interest rates and bond prices. It also allows us to think of interest rates and prices interchangeably, depending on context.

23

Generalising (a bit)

Assume asset j, with price Pj in period t has guaranteed payoff Yj in period t+1

Equivalent to payoff from Yj zero coupon bonds.By assumption 4 it must cost the same. So for any such asset we

have the no arbitrage pricing condition

1 1j

j j

YP Y P

RThe price of any such asset can be viewed as the present value

of its payoffs.Eg, if asset j has guaranteed payoff of 20, and R=0.05

1120 19.05

1.05j jP Y P

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The return on any such asset is defined by

1 jj

j

YR

P

but our no arbitrage pricing condition reduces this to

1 1

11 1 for all j jj

j j

Y YR R j

P PY P

All risk-free investments of a given maturity must earn the same

return.

Eg, for numerical example:

1 1

20 20 11 1.0519.05 20jR

P P

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Constant R; multiple periods

Assume (just for now) that R, the return on a one period risk-free deposit, is constant for all time

If I invest £1 today and leave it in the bank for n periods I will end up with £(1+R)n (“compound interest”)

If I invest £1/(1+R)n today, in n periods I will have £(1+R)n/(1+R)n =£1

So by the replication argument, No Arbitrage implies that a zero coupon bond that pays £1 in n periods time must have price

1(1 )n nP

R

Link between price and maturity?

26

Pricing Forward Contracts by No Arbitrage

Assume we make a contract to supply one unit of some commodity

at some fixed point T periods in the future.

The forward price F is set today, and we assume no possibility of

default, so the contract implies a risk free payoff of F at time T.

Suppose we buy one unit of the commodity now, at the current spot

market price S, funded via borrowing at the risk-free rate (assumed

constant) and storage costs are zero.

At time T we supply the commodity, and have to repay

1 TS R

NB: Length of time period could be just one day, so R could be very

small indeed.

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This strategy requires no capital, and is entirely risk-free. The

profit/loss on the strategy in period T will be

1 TF S R

No arbitrage must imply that this strategy yields zero profits.

Hence we must have

1 TF S R

This does not say that S determines F or vice versa, it just says that

if this relationship does not hold someone can make risk-free

(arbitrage) profits.

28

Spot vs Forward Oil Price

40

60

80

100

120

140

160

07/07/2006 19/9/2006 30/11/2006 02/12/2007 25/4/2007 07/06/2007 18/9/2007 29/11/2007 02/11/2008 23/4/2008 07/04/2008

Crude Oil-Brent Cur. Month FOB U$/BBL

Crude Oil-Brent 3Mth Fwd FOB U$/BBL

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Ratio of Forward to Spot Oil Prices

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

07/07/2006 19/9/2006 30/11/2006 02/12/2007 25/4/2007 07/06/2007 18/9/2007 29/11/2007 02/11/2008 23/4/2008 07/04/2008

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Continuous vs Discrete Compounding

In textbooks (eg Hull), the forward price is usually written asrTF Se

where er=1+R

For reasonably small R, r R. Eg, if R=0.05,0.05

. .

1.0513 1.05 111 ;

1

R

T R T R TT

e e R

R e eR

For any R we can always find an r=ln(1+R) that makes this

expression exact. Conceptually, r is the “continuously

compounded interest rate”

You should get used to the equivalence of the two expressions

31

Bond Pricing and the Yield Curve Bond Prices and Yields

Generalise: R (one period safe return) is not constant. Market in zero coupon bonds with different maturity, 1 2P PEach pays off £1 (with no risk) when it matures, nothing before. Suppose we held the bond to maturity. Since we know the current price and the payoff in n periods, this would generate a guaranteed compound average risk-free return, Rn defined by

11 nn

n

RP

(But remember that the return over any horizon less than n is notguaranteed)We define this notional risk-free n-period return as the yield on the bond.

32

Equivalently we can think of the price of the bond as a present value calculation, using the yield as a discount rate (or “internal rate of return”), ie,

11n n

nP

RIf we have Pn we can automatically calculate Rn. Yield is just an indirect way of measuring price. But yields at different maturities are much easier to compare than prices. If R (the one period return) were constant, Rn =RGiven the equivalence of yields and prices they are used interchangeably, depending on context. Yields on zero coupon bonds are also known (eg in Hull) as “zerocoupon rates”

33

The “Yield Curve”

If we plot Rn against n this is the (zero-coupon) “yield curve”(see 1st chart)

In the postwar era this has on average (but by no means always, see 2nd chart) been upward sloping: ie, “term premia” have typically been positive.

It is often assumed that this is the normal state of affairs, hence when the yield curve slopes downwards it is referred to as “inverted”.

It is also often assumed that the slope of the yield curve tells us something about market expectations.

We shall discuss the basis for both of these assumptions. We shall also see how “forward rates” can be related to the yield

curve.

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The UK Yield Curve, 1 May, 2012

0.00

0.50

1.00

1.50

2.00

2.50

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

35

36

Coupon Bonds

In actual bond markets we mainly observe “coupon bonds”, that make regular payments (coupons) usually at a fixed rate, before they mature.Eg, a ten year bond with face value of £100, and coupon rate of 6.25%, will pay £6.25 every year (usually in two half-yearly instalments of £3.125), and will then pay £103.125 when it matures (principal plus the last coupon).There are far more coupon bonds traded than zero coupon bonds (especially at longer maturities), since for many investors the regular income flow from coupon bonds is convenient (hence the term “fixed income securities”)

37

We can price any coupon bond as a portfolio of zero coupons. By no arbitrage, the price of the coupon bond must be the same as the price of the equivalent portfolio of zero coupon bonds.Eg, our ten year 6% annual coupon bond is equivalent to holding the following portfolio of zero coupon bonds….

6 1-year zero coupon bonds+ 6 2-year zero coupon bonds + 6 3- zero coupon bonds+….. +106 10-year zero coupon bonds

… and hence the bond must have the same price as the portfolio. Most analysis of bond pricing ignores coupon bonds But beware: a lot of published data relate to coupon bonds!(And even published zero coupon yields, eg, Bank of England data shown in charts, are actually derived from data on coupon bonds, especially at longer maturities)

38

Forward Rates: Another No Arbitrage Argument

We’ve seen already that a forward contract is an agreement to trade some commodity at some point in the future, at a price agreed today.Generalising our earlier formula, for any commodity, the forward price with settlement date in T periods must be given by

F=S(1+RT)T

where RT is the yield on a T-period zero coupon bond. Now consider the case where the commodity being traded is itself abond: for simplicity a 1-period zero coupon bond, that will mature in n=T+1 periods, and pay £1.What is the equivalent spot price, S?(Hint, we are looking for the price of something traded today that will give us the same cash payment, on the same date)

39

To simplify, assume n=2, and let the forward rate on a one period zero coupon bond, maturing 2 periods in the future (hence to be traded in one year) be 1 2FR , ie the rate that satisfies

11

121 2

FFF P

RThe forward rate is fixed today, whereas if you wait to buy a 1 period bond in a year’s time, you don’t know what its price will be. From our general formula we must have

11

12 1 2

21

11 11

1 2 1F

F S RRP R

R R

Implying2

1 1 21 1 2 1FR R R

Equivalently, we have two different ways of getting exactly the same payoff in two years’ time: these must on average yield the same return.

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It is straightforward to show that this generalises, for any n-periodbond, to give

1 1 11 1 1 2 ... 1 1nF F F

nR R R n R

or equivalently, to a very good approximation, 1 1 11 2 ....F F F

n

R R R nR

n

So for any observed yield curve there is an associated “forward

curve”: these are two different ways of showing exactly the same

information.

41

Yield and Forward Curves, 1 May 2012

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

2.0 4.0 6.0 8.0 10.0

forward curve

yield curve

42

Forward Rates and Expectations

Under strong assumptions (if investors all investors are risk-neutral, and markets are informationally efficient), then forward rates are unbiased and minimum variance forecasts of actual one year rates, and we get the “Pure Expectations Theory of the Term Structure”:

1, 1, 1 1, 1...t t t nn t

R R RR E

nNB: This is much more than a simple no arbitrage condition. The bond market as a forecasting mechanism. If, eg, yields are higher than short-term rates what does this imply? What does this say about the patterns of yields we saw in the charts?

43

Yields vs Short-Term Returns on (Zero Coupon) Bonds

Unless held to maturity, yields on bonds are not the same as returns

on bonds.

Recall the relation between price and yield for a zero coupon bond :

11n n

nP

R

Remember that this bond generates no income, so short-term

returns (“holding returns”) are entirely due to capital

gains/losses.

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The following relationship holds to a reasonable approximation:

% holding return - % percentage point rise in nn R

Reflects inverse relationship between yield changes and price

changes (hence holding returns).

So longer-dated bonds tend to have much more volatile returns than

do short-dated bonds.

This may help to explain why the yield curve has tended to slope

upwards on average

But the steep upward slope in the yield curve in the recent data

mainly reflects market expectations that short-term rates will rise in

coming years.

45

Next Week

Derivatives, with particular focus on options

Risk premia, risk neutrality and risk-neutral

pricing.