MGT 4068 FIXED INCOME SECURITIES

Week 3

Alex Hsu, Ph.D.

• The term structure of interest rates is the series of interest rates ordered by term-to-maturity at a given time

• The nature of interest rate determines the nature of the term structure

– The term structure of yields to maturity

– The term structure of zero coupon rates

– The term structure of forward rates

Types of Term Structures

The shape of Term Structures

Quasi-Flat

4.00%

4.50%

5.00%

5.50%

6.00%

6.50%

7.00%

0 5 10 15 20 25 30

par

yie

ld

maturity

US YIELD CURVE AS ON 11/03/99

Quasi-FlatQuasi-Flat

Increasing

IncreasingIncreasing

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

0 5 10 15 20 25 30

par

yie

ld

maturity

JAPAN YIELD CURVE AS ON 04/27/01

Decreasing

Decreasing (or inverted)Decreasing (or inverted)

4.50%

5.00%

5.50%

6.00%

0 5 10 15 20 25 30

pa

r yi

eld

maturity

UK YIELD CURVE AS ON 10/19/00

Humped (1)

Humped (decreasing then increasing)Humped (decreasing then increasing)

4.00%

4.50%

5.00%

5.50%

0 5 10 15 20 25 30

par

yie

ld

maturity

EURO YIELD CURVE AS ON 04/04/01

Humped (2)

Humped (increasing then decreasing)Humped (increasing then decreasing)

5.50%

6.00%

6.50%

7.00%

0 5 10 15 20 25 30

pa

r yi

eld

maturity

US YIELD CURVE AS ON 02/29/00

Dynamics of the Term Structure

• The term structure of interest rates changes in response to – Wide economic shocks – Market-specific events

• Example– On 10/31/01, Treasury announces that there will not be any

further issuance of 30 year bonds– Price of existing 30 year bonds is pushed up (buying pressure)– 30 year rate is pushed down

Stylized Facts (1) : Mean ReversionMean reversion: high (low) values tend to be followed by low (high) values

• Rates with different maturities are – Positively correlated one another– Not perfectly correlated though (more than one factor)– Correlation decreases with difference in maturity

• Example: France (1995-2000)

1M 3M 6M 1Y 2Y 3Y 4Y 5Y 7Y 10Y 1M 1 3M 0.999 1 6M 0.908 0.914 1 1Y 0.546 0.539 0.672 1 2Y 0.235 0.224 0.31 0.88 1 3Y 0.246 0.239 0.384 0.808 0.929 1 4Y 0.209 0.202 0.337 0.742 0.881 0.981 1 5Y 0.163 0.154 0.255 0.7 0.859 0.936 0.981 1 7Y 0.107 0.097 0.182 0.617 0.792 0.867 0.927 0.97 1 10Y 0.073 0.063 0.134 0.549 0.735 0.811 0.871 0.917 0.966 1

Stylized Facts (2) : Correlation

Stylized Facts (3)• The evolution of the interest rate curve can be split into three

standard movements– Shift movements (changes in level), which account for 70 to 80%

of observed movements on average– A twist movement (changes in slope), which accounts for 15 to

30% of observed movements on average– A butterfly movement (changes in curvature), which accounts

for 1 to 5% of observed movements on average

• In general, 3 factors account for more than 90% of the changes in the TS

Shift Movements

0

1

2

3

4

5

6

7

0 5 10 15 20 25 30

yiel

d (i

n %

)

maturity

Upward -Downward Shift Movements

Twist Movements

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25 30

yie

ld (in

%)

maturity

Flattening - Steepening Twist Movements

Butterfly Movements

2

2.5

3

3.5

4

4.5

5

5.5

6

0 5 10 15 20 25 30

yie

ld (

in %

)

maturity

Concave - Convex Butterfly Movements

• Studying the TS boils down to wondering about the preferences of participants' for curve maturities– Investors– Borrowers

• Indeed, if they were indifferent in terms of maturity– Interest rate curves would be invariably flat

• Market participants' preferences can be guided – By their expectations– By the nature of their liability or asset– By the level of the risk premiums they require

Theories of the Term Structure

• Term structure theories attempt to account for the relationship

between interest rates and their residual maturity

• They fall within the following categories1. Pure expectations2. Pure risk premium

• Liquidity premium • Preferred habitat

3. Market segmentation

Theories of the Term Structure

Pure expectation

• TS reflects market expectations of future short-term rates

– An increasing (decreasing) structure means that the market expects an increase (decrease) in future short-term rates

– A flat structure means that the market expects a stagnation in future short-term rates

“Future short-term rates equal Forward short-term rates”

Pure expectation : limitation

The pure expectations theory has an important limitation

• Investors behave in accordance with their expectations for the unique purpose of maximizing their investment return

• They do not take into account the fact that their expectations may be wrong

• The pure risk premium theory includes this contingency

Pure Risk Premium• Indeed, if forward rates were perfect predictors of future rates, future

bond prices would be known with certainty

• Unfortunately, it is not the case– Future interest rates are unknown (re-investment risk)– Future bond prices are unknown (market risk)

• What would you prefer?(a) Invest in a 3-year zero coupon bond and holding it until maturity(b) Invest in a 5-year zero coupon bond and selling it in 3 years(c) Invest in a 10-year zero coupon bond and selling it in 3 years

Risk and return tradeoff?• Rational asset pricing theory would suggest that:

If asset “A” is riskier than asset “B”, then “A” should be traded at a relatively lower price than “B”, or equivalently a promise for a higher return?

FACT: Longer term to maturity bond prices (not yields) are more sensitive to the interest rate fluctuation than a short-term bond

• If it is riskier then shouldn’t it command a lower price ? Therefore R(0,10) > R(0,1) according to the Pure Risk

Premium theory

Why the price of long-term bonds are more volatile ?

F + C

C C C F + C

Market Segmentation Theory• Shape of the curve determined

– By supply and demand on short and long-term bond markets– Insurance comp, pension funds are structural buyers of long-

term bonds

• Their demand for short-term bonds is influenced by business conditions– During growth periods, sell bonds to meet corporations' and

individuals' demand for loans => relative increase in short-term yields

– During slow-down periods, corporations and individuals pay back their loans, thus increasing bank funds; then banks invest in short-term bonds => relative decrease in short-term yields compared to long-term yields

What is the best theory?

1. Definitely a combination of these three

2. Pure Expectation and Pure Risk Premium are a bit too rigid, but probably work when viewed from a long-run

3. Market-Segmentation Hypothesis is too flexible, though it can explain pretty much every shape • However, critiques think it’s too behavioral • Lacks a robust rationale • Works well at explaining short-term phenomenon• “Theory fitting”

Forward Rates

• The forward rate is the future zero rate implied by today’s term structure of interest rates

• The rate that you agree to borrow/lend at in the future period

• We will talk about the difference between futures and forward rates next week

Forward Rates: notations

R(0, y) - Is the spot rate (%) quoted today for maturity at time “y”

F0(x, y) - Forward interest rate quoted today to be applied at the start in the future time “x”, and ending at a later time “y”

t=0 t=yt=x

R(0,y)

F0(x, y)

Forward rate” contract: another view point

t=0 t=yt=x

R(0,y)

F0(x,y)

A long position in a “forward rate” contract is:

An agreement at time t=0 to buy, at time t=x, a zero-coupon bond with maturity y-x and the YTM F0(x,y).

A short position in a “forward rate” contract is:

t=0 t=yt=x F0(x,y)

- You want to invest $1 without any risk from time 0 to time y- There are two ways to do this

Determining the forward rate : No arbitrage method

R(0,y)

R(0,x)

First way )0(,01 1$ yyReP

Second way xyyxFxxR eeP ,)0)(,0(2

01$

Since both methods for investment have no risk (i.e. you locked in your cash flow at time t=0), they must give equal pay out at time T.

No arbitrage price of the forward rate (1)

I assume that the spot rates are for continuous compounding.

21 PP

Forward rate formulae• For continuous compounding spot rates

or, after re-arranged:

• For a simple compounding:

)(

),0(),0(),(0 xy

xxRyyRyxF

1)),0(1(

)),0(1(),(

1

0

xy

x

y

xR

yRyxf

xy

xxRyRyRyxF

),0(),0(),0(),(0

Arbitrage trading strategy

Consider two portfolios/assets that provide you with identical cash flows

You can squeeze out arbitrage profits by:

1.If they both cost the same, invest in the one that gives you the highest yield, and finance your investment using the one that gives you the lowest yield.

2.If they both lead to the same future cash flow, buy (long) the cheaper one, and sell (short) the more expensive one

t=0 t=yt=x F0(x,y)

No arbitrage method: if they cost the same, invest in the asset that yields the highest return

R(0,y)

R(0,x)

Two ways to invest your money

$1

1.Invest in a zero-coupon bond with maturity at time “y”. Earning an interest rate of R(0,y)

2. Invest in a zero-coupon bond with maturity at time “x”. At the same time, enter into a forward contract to buy a zero-coupon bond at time “x” that matures at time “y”.

• Lock in your interest of R(0,x) from now to time “x” , as well as at the rate of F0(x,y) from time “x” to time “y”.

First way )0(,01 1$ yyReP

Second way xyyxFxxR eeP ,)0)(,0(2

01$

21 PP What if ?

No arbitrage method: invest in the asset that yields the highest return (2) No arbitrage method: if they cost the same, invest in the asset that yields the highest return

No arbitrage method: if they give the same cash flow, buy the cheapest asset that you can replicate

Identical cash flow of $1 at time “y”

1.Invest in a zero-coupon bond with the face value of $ 1, and maturity at time “y”. The cost of to replicate this cash flow is:

2.Enter into a forward contract to buy a zero-coupon bond with the face value of $1 at time “x” that matures at time “y”. The amount required to invest in this contract is:

Invest in a zero-coupon bond with the face value of $Q, and maturity at time

“x”. The cost to replicate this cash flow is:

$1

t=0 t=yt=x F0(x,y)

R(0,y)

R(0,x)

)(),(exp1$ 0 xyyxFQ

First way )0(,01 1$ yyReC

Second way xyyxFxxR eeC ,)0)(,0(2

01$

12 CC What if ?

No arbitrage method: if they give the same cash flow, buy the cheapest asset that you can replicate

Forward and Futures

1. What is a future contract

2. Difference between forwards and futures

3. Pricing a forward contract on a bond

• Available on a wide range of underlying

• Exchange traded

• Specifications need to be defined:What can be delivered,Where it can be delivered, & When it can be delivered

• Settled daily – Marking to market

Futures Contracts

Forward Contracts vs Futures Contracts

Private contract between 2 parties Exchange traded

Non-standard contract Standard contract

Usually 1 specified delivery date Range of delivery dates

Settled at end of contract Settled daily

Delivery or final cashsettlement usually occurs

Contract usually closed outprior to maturity

FORWARDS FUTURES

Some credit risk Virtually no credit risk

Margins : futures trading

• A margin is cash or marketable securities deposited by an

investor with his or her broker

• The balance in the margin account is adjusted to reflect daily

settlement

• Margins minimize the possibility of a loss through a default on

a contract

Margins: investors/broker/exchange

Investors BROKER

The exchange clearing house

Margin

-Investors maintain margin account with a broker

-Brokers maintain margin account with the exchange clearing house

Margin

Closing Out PositionsThis is a trading terminology for clearing your positions in the portfolio free such derivative contract

The vast majority of futures/(some forwards) do not lead to delivery of the underlying asset

• Settle in cash• Settle using the cheapest to deliver product that is closest to the underlying asset

If delivery is expensive – maybe it is cheaper to close your positions before it’s due

Closing Out Positions : convergence of future and spot prices

Time (t)

Maturity of the contract

“Delivery date”

T - maturity

F(t,T) - Price of a future contract at time “t” for buying/selling the underlying asset “S” at maturity “T”

S(t) – Price of the underlying asset “S” at time “t”

Orange Producers is FloridaWant insurance against low Orange price

Beverage Factory in CA

Want insurance against high Orange price

Trade here

Why delivery for futures is not so common?

Why delivery for futures is not so common?

- Both enter into futures contract in an exchange in Chicago. They do not necessarily know of each other

- Both Seek insurance against price fluctuation.

- Thus producer enters to sell at fixed price and beverage producer enters to buy at fixed price.

Time

FuturesPrice

Spot Price

As maturity approaches future prices will approach the spot price. This is intuitive!

- At this point, the futures price is very close to the spot price!! Well, the asset is soon to be delivered.

- If they wait until contract expires, the orange producer has to deliver oranges across the continent to the producer in California. Costly! - In order to avoid the delivery cost, both beverage factory and orange producer will close out their position.

- Let’s say that the market price of orange rises!!!

- If you were the orange producer, would you be happy about this ?

What will happen when delivery date gets close ?

What happen when delivery date gets close (continue..)

- Orange producers endures loss in futures trade, but this loss is offset by selling orange in the local market (Florida) at the higher price

- Beverage producers makes profit in futures trade, but this profit is offset by having to buy orange in the local market (California) at the higher price

What happens if the orange price rises?

- Your optimal choice of maturities/cash-flows might not exist

- Not all government bonds can be easily purchased in the off-the-run market, which by nature is less liquid

- Settle for it in cash

- Settle with a similar bond/cheapest to deliver, but this will require a price adjustment factor

What if the underlying asset is a bond ?

WHAT about OTC Market ?

• Remember they are not traded on the exchange!! So are exposed to credit risk

• It is becoming increasingly common for contracts to be collateralized in OTC markets

• Failure to find collateral when entering a contract

can lead to serious mess

Pricing forwards and futures

How do you determine whether it is too cheap or too expensive?

Forward and Future price?

St: Spot price today

F(t,T): Futures or forward price quoted today with time until delivery – T . This is F0 in my excel sheet. It is often assumed that t=0.

T: Time until delivery date

r: Risk-free interest rate (spot rates) for maturity T, i.e. R(0,T)

Notation for Valuing Futures and Forwards

Assumptions1) No transaction cost!! << Trading occurs on exchange – transaction cost is

unavoidable >>

2) Everyone pays the same tax rate! << Not in the US, but in Eastern European countries like Georgia >>

3) You can borrow and lend money at the same rate. << not a chance in reality >>

4) If there is an arbitrage opportunity, everyone jumps at it. <<Very TRUE>>

The Forward/Future Price If the spot (current) price of the underlying asset is B0

then its forward/future price for delivery at time T is

If interest rate is cont-compounded

F(0,T) = B0 exp(rT )

If interest rate is simple-compounded

F(0,T) = B0 (1+r)T

F(0,T) = B0exp(rT )

IF F > B0 exp(rT )

IF F < B0exp(rT )

Too Expensive. Short the contract

Too Cheap. Long the contract

Show arbitrage opportunity if

( I expect you to be able to construct the payoff )