fnce 30001 week 10 tsir revised(1)

FNCE 30001 InvestmentsSemester 2, 2011

12 & 14 October 2011Week 10: The Term Structure of Interest

RatesRatesProfessor Rob Brown

FNCE 30001 Investments 10.0

Lecture 9: The Term Structure of Interest Rates

Overview of Lecture1. Zero Rates, Forward Rates and Actual Rates,2. What Determines the Forward Rate?3. The Term Structure of Interest Rates (TSIR): Definition 3. T e Te St ct e o te est ates (TS ): e t o4. TSIR Explanations (1): The Expectations Hypothesis5. TSIR Explanations (2): The Liquidity Premium Hypothesis5. TSIR Explanations (2): The Liquidity Premium Hypothesis6. Interpreting the Slope of the TSIR

Reading: Bodie et al Chapter 15Reading: Bodie et al, Chapter 15.


1. Zero Rates, Forward Ratesd A l Rand Actual Rates


Zero Rates Forward Rates and Actual RatesZero Rates, Forward Rates and Actual Rates

Zero rate: the interest rate for the period from today (time 0) to a future date. Symbol: z.

30 1 2 3

z01z02

z03

The arrows go in both directions because the interest rate applies to both borrowing and lending.



F d h i d ( i 0) f hForward rate: the interest rate, set today (time 0), for the period from one future date to another future date. Symbol: f.

0 1 2 3

f12f23

f13

The arrows go in both directions because the interest rate applies to both borrowing and lending.



Example: f13 is the interest rate set today (time 0) and applying from time 1 until time 3.

Th i if i h l k i h i That is, if someone wishes to lock in the interest rate on a loan to begin at time 1 and end at time 3, they can go to the forward market and get a forward contract and thethe forward market and get a forward contract and the interest rate will be set at f13 .



Actual (future) rate: the market interest rate at a future time. ( )Symbol: r.

For example, r13 means the interest rate applying from time 1 to time 3. We wont know the value of r13 until we get to time 1.

By definition, r01= z01; r02= z02; r03= z03 etc

0 1 2 30 1 2 3

The arrows go in both directions because the interest rate applies to both borrowing and lending

r12 r23r01


applies to both borrowing and lending.


Discount factors

As we know, there is a discount factor associated with each zero rate.For example, d04 is defined as:

1

A d th r i for rd di t f t r i t d ith h 04 404

1 .1

dz

And there is a forward discount factor associated with each forward rate.For example d is defined as:For example, d14 is defined as:

14 3141 .

1d

f


141 f

2. What Determines the Forward R ?Rate?


What Determines the Forward Rate?What Determines the Forward Rate?

The forward rate can be determined from the current zero rates.Example: Forward rate f35Suppose I have $X that I want to invest for 5 years. Then:1. The obvious thing to do is to lend $X for a term of 5 years (ie buy

5 )a 5-year zero).But there are lots of other things I could do. For example, I could:

d b2. (a) Lend $X for a term of 3 years (ie buy a 3-year zero) AND(b) Buy a forward contract to buy a 2-year zero at time 3.



Example: Forward rate f35 (contd.)

0 2 3 41 5

f

(1) z05

(2) f35(2) z03

Effectively, (1) and (2) are the same thing.Therefore, the return on (1) and (2) should be the same.If the returns are not the same there is an arbitrage (a free lunch).



Example: Forward rate f35 (contd.)1. Lending $X for a term of 5 years (ie buying a 5-year zero).

At the end of the 5th year I will have $X (1+z05)5.2. (a) Lending $X for a term of 3 years (ie buying a 3-year

zero).At the end of the 3rd year I will have $X (1+z03)3.(b) Buying a forward contract to buy a 2-year zero at time 3.At the end of the 5th year I will have $X (1+z03)3 (1+f35)2.



Example: Forward rate f35 (contd.)

To prevent arbitrage we require that:

5 3 205 03 35$ 1 $ 1 1which implies that:X z X z f

1 2505

35 3

p

11

zf

3

031+z

The general formula (for ) when the current date is time 0:t T

1 ( )

0

0

1

1

T tTT

tT tt

zf

z1.


0tz


Numerical ExampleThe zero rate for t = 5 years (z05) is 7.25% pa.The zero rate for T = 8 ears (z ) is 8 755% paThe zero rate for T = 8 years (z08) is 8.755% pa.What is the forward rate (pa) for the period from t = 5 to T = 8? (ie what is f58?)Answer 1

0

0

11

1

T T tT

tT tt

zf

z

18 8 5

58 51.0875

11 0725

f

1/3

1.0725

1.37862967 111 297% pa


11.297% pa


We can rearrange:1

1

0

0

11

1

T T tT

tT tt

zf

z

to find that:

1 1 1T t T tf What is the 0 01 1 1 .T t tTz z f

1 1 1Hence:intuition here?

0 0

0 0

Hence: 1 1 1T t T tT t tT

T t tT

z z fie d d d

00

TtT

t

ddd


3. The Term Structure of Interest R D fi i iRates: Definition


The Term Structure of Interest Rates: Definition

Th f i (TSIR) f h diff The term structure of interest rates (TSIR) refers to the different interest rates that, at a given time, apply to different investment terms.

Various curves describe the relationship(s) between interest rate and term. The three most often used interest rate curves are:

z : zero rate curves;f : forward rate curves (for one-period investments); andy : yield curves.

These represent investors views about the interest rates on: b d current zero-coupon bonds; future zero-coupon bonds; and current coupon bonds


current coupon bonds.

The Term Structure of Interest Rates: Definition

Flat curve: f = z = y ;Rising curve: f > z > y ; andFalling curve: f < z < y .

We will use term structure of interest rates to mean todays f f h i iset of zero rates for the various terms to maturity.

We established these r l ti n hip in W k 8relationships in Week 8


4. TSIR Explanations (1):Th E i H h iThe Expectations Hypothesis


TSIR Explanations (1): The Expectations Hypothesis

Usually, long-term interest rates are higher than short-term interest rates why?B i l i l h hBut sometimes, long-term interest rates are lower than short-term interest rates why?

It is critical to know the term structure to price bonds It is critical to know the term structure to price bonds. Two major explanations:

Th E p t ti H p th i The Expectations Hypothesis The Liquidity Premium Hypothesis.



Assume that investors are risk neutral. By definition, risk neutral investors ignore risk. They care

l b donly about expected returns. Faced with a choice between two investments, a risk neutral

investor will always choose the one with the higher expectedinvestor will always choose the one with the higher expected return even if it is much riskier and the expected return is only a even if it is much riskier and the expected return is only a

little higher.



C id h f ll i h i $X f 3 Consider the following three ways to invest $X for 3 years:1. Buy a 3-year zero today;2 I iti ll shorter th th i tm t h riz2. Initially, go shorter than the investment horizon:

a. Buy a 2-year zero today andb At the end of the 2 years collect the par value andb. At the end of the 2 years, collect the par value, and

then buy a 1-year zero.3. Initially, go longer than the investment horizon:y, g g

a. Buy a 4-year zero today andb. Sell it at the end of 3 years.

The return from Strategy 1 is certain why? But the returns from Strategies 2 and 3 are risky why?



S 1Strategy 1 The sum that will be accumulated by Strategy 1 is:

S = $X (1+ )3S1 = $X (1+ z03)3.Strategy 2 The sum that will be accumulated by Strategy 2 is: The sum that will be accumulated by Strategy 2 is:

S2 = $X (1 + z02)2 (1 + r23). Today (time 0), r23 is unknown, but investors today expect itToday (time 0), r23 is unknown, but investors today expect it

to be E(r23). So, the sum expected to be accumulated by Strategy 2 is:

E(S2) = $X (1 + z02)2 [1+E(r23 )]



C i S i 1 d 2Comparing Strategies 1 and 2: If S1 > E(S2) then risk-neutral investors will choose Strategy 1.

If E(S ) > S th ri k tr l i t r ill h Str t 2If E(S2) > S1 then risk-neutral investors will choose Strategy 2. Therefore, in equilibrium, S1 = E(S2). That is $ X (1+ z )3 = $X (1 + z )2 [1+E(r )] That is, $ X (1+ z03) = $X (1 + z02) [1+E(r23 )]. Hence (1+ z03)3 = (1 + z02)2 [1+E(r23 )]. Generalising from this example, for two future dates t and TGeneralising from this example, for two future dates t and T

(where t < T ):(1+ z0T)T = (1 + z0t)t [1+E(rt T )]T t.



Strategy 3

The sum that will be accumulated by Strategy 3 is:

4

043

34

$ 1.

1X z

Sr

Today (time 0), r34 is unknown, but investors today expect it to be E(r34).

b d b So, the sum expected to be accumulated by Strategy 3 is:

4

04$ 1EX z

S 3 34E 1 ES r



Comparing Strategies 1 and 3:

If S1 > E(S3) then risk-neutral investors will choose Strategy 1.1 ( 3) gyIf E(S3) > S1 then risk-neutral investors will choose Strategy 3.

Therefore, in equilibrium, S1 = E(S3)., q , 1 ( 3) That is,

4$ 1X z

3 04

03344

$ 1$ 1

1 E

1

X zX z

r

43 04

0334

1Hence 1

1 Ez

zr



Comparing Strategies 1 and 3 (contd.):

Generalising from this example, for two future dates t and T( h t < T )(where t < T ):

0

01

1T

t Tt T t

zz

Rearranging this equation, we find: 1 E T ttTr

0 01 1 1 Ehi h i h l f d h

T tT tT t tTz z r

which is the same result we found when we compared Strategies 1 and 2.


TSIR Explanations (1):TSIR Explanations (1): The Expectations Hypothesis

Interpretation:

If the expected returns are the same, risk-neutral investors do h h h f h i inot care whether the term of their investment:

Matches their investment horizon (Strategy 1)I h h h i i h i (S 2) Is shorter than their investment horizon (Strategy 2)

Is longer than their investment horizon (Strategy 3).

This is because, by definition, they do not care about risk.If the expected returns are the same, they place no value on the fact that Strategy 1 is less risky than Strategies 2 and 3.



Compare this equation:p q(1+ z0T)T = (1 + z0t)t [1+E(rt T )]T t

with this one we covered earlier:Assumes risk neutrality

(1 + z0T )T = (1 + z0t )t (1 + ft T )T t Assumes there are no arbitrage opportunities

Thus, according to the expectations hypothesis:E(rt T ) = ft T

This is the heart of the Expectations Hypothesis: Th k d f h hThe market sets todays term structure of zero rates such that the resulting forward rates are equal to the markets expectations of future interest rates.


f


Thousands of thingsg

Expected future interest rates

Todays zero rates (ie todays term structure)

Todays forward rates

Trade bonds

Do forwards = expected?

Trade bonds

Equilibrium TSIR

YES NO


Equilibrium TSIR


ll h d h h hRecall that according to the expectations hypothesis:

0 01 1 1 E T tT tT t tTz z r Suppose t = 2 and T = 3. Then:

3 21 1 1 Ez z r

But the expectations hypothesis also maintains that when 1 d T 2 h

03 02 231 1 1 Ez z r

t = 1 and T = 2 then:

202 01 121 1 1 Ez z r 303 01 12 23So 1 1 1 E 1 Ez z r r



l f h f h Generalising from this gives us a more convenient statement of the expectations hypothesis.

For any given term to maturity of T, the hypothesis states that the current y g y , ypzero rate for that term (z0T) is set by the market such that:

0 01 12 23 11 1 1 E 1 E ... 1 ETT T Tz z r r r

0 01 12 23 1,

1

0 01 12 23 1

...

Therefore:

1 1 E 1 E ... 1 E 1

T T T

T

T T T

z z

z z r r r

This is a geometric average of the zero and the expected rates.

0 01 12 23 1,1 1 E 1 E ... 1 E 1T T Tz z r r r If the expectations hypothesis is true, then the expected rates are equal to the

forward rates.



Th t i th p t ti h p th i b t t d thi That is, the expectations hypothesis can be stated this way:The current zero rate for a term of T years is a function of the current zero rate for Year 1 and the expected one yearthe current zero rate for Year 1 and the expected one-year rates for Year 2, Year 3, Year 4, etc up to T years.



Example 1

Suppose the expectations hypothesis is true. Todays one-year zero rate (z01) is 10.0%.The expected one-year rates are:

Year 1: E(r12) = 12.0%Year 2: E(r23) = 13.0%Year 3: E(r34) = 13.4%Year 4: E(r45) = 13.5%

What is todays term structure of interest rates?



Answer to Example 1According to the expectations hypothesis, the market will set todays zero rates such that todays forward rates equal expected future ratesrates such that today s forward rates equal expected future rates.Therefore f12 = E(r12) = 12.0%. 2021B d fi iti 1zf

0212

012

02

By definition 11

1Th f 0 12 1

zf

z

z

02

02

Therefore 0.12 11.10

which solves to give 0.10995 10.995%

z

z

By similar reasoning we can find z03, z04 and z05.



Or, just applying the formula:

10 01 12 23 1,1 1 E 1 E ... 1 E 1TT T Tz z r r r 0 0 3 ,1 2

we get

T T Tz z

1 202

1 303

1.10 1.12 1 10.995% pa

1.10 1.12 1.13 1 11.660% pa

z

z

031 4

04

1.10 1.12 1.13 1 11.660% pa

1.10 1.12 1.13 1.134 1 12.092% pa

z

z

1 505 1.10 1.12 1.13 1.134 1.135 1 12.372z % pa



Example 2

Given these expectations, an investor wonders what next ill l k likyears term structure will look like.

Assuming that the expectations hypothesis is true, and expectations are fulfilled (that is at the start of next year theexpectations are fulfilled (that is, at the start of next year, the one-year rate is indeed 12% pa, as expected), what should be the term structure next year?y



Answer to Example 2We can summarise the results so far in the following table:

Term Now (ie at t = 0)Start of next year

(ie at t = 1) At t = 2 At t = 3 At t = 4

g

1 year z01=10.000% z12 = 12.000% 13.000% 13.400% 13.500%2 years z02=10.995% E(z13)3 years z03=11.660% E(z14)4 years z04=12.092% E(z15)5 12 372%5 years z05=12.372%

We want to calculate E(z13), E(z14) and E(z15).



Answer to Example 2 (contd.)

Time has in effect moved on 1 year from 0 to 1t t

1

Time has, in effect, moved on 1 year from 0 to 1.So the relevant formula now is:

T

t t

1 12 23 34 1,E 1 1 E 1 E ... 1 E 1T T Tz z r r rHence:

1/2131/3

Hence:

E 1.12 1.13 1 12.4989% paz

1/314

1/415

E 1.12 1.13 1.134 1 12.7985% pa

E 1.12 1.13 1.134 1.135 1 12.9734% pa

z

z


5


Answer to Example 2Adding these results to the table:g

Term Now (ie at t = 0)Start of next year

(ie at t = 1) At t = 2 At t = 3 At t = 4(ie at t = 0) (ie at t = 1)1 year z01=10.000% z12 = 12.000% 13.000% 13.400% 13.500%2 years z02=10.995% E(z13) = 12.4989%2 years z02 10.995% E(z13) 12.4989%3 years z03=11.660% E(z14) = 12.7985%4 years z04=12.092% E(z15) = 12.9734%y z04 (z15)5 years z05=12.372%



This means the TSIR is forecast to rise and flatten next year:14.000

10.000

12.000

6.000

8.000

R

A

T

E

(

%

p

a

)

TSIR this yearTSIR next year

2 000

4.000

0.000

2.000

0 1 2 3 4 5 6

TERM (YEARS)


TERM (YEARS)


S o , if w e h av e a fo re cast o f n ex t y e a rs (t im e 1 's) te rm stru c tu re y ( )w e c an d ed u ce n ex t y e a r 's fo rw ard cu rv e .T h e fo rm u la w h en th e cu rren t d a te is t im e 1 is :

1 ( )11

11

11, w h ich im p

1

T tTT

tT tt

zf

zl ie s : 1 tz

2

2 3 2 31 .1 2 4 9 8 9

1 1 3 .0 0 0 % E1 .1 2 0 0 0

f r

3

3 4 3 42

1 .1 2 0 0 01 .1 2 7 9 8 5

1 1 3 .4 0 0 % E1 .1 2 4 9 8

f r

4

4 5 4 531 .1 2 9 7 3 4

1 1 3 .5 0 0 % E1 1 2 7 9 8 5

f r


1 .1 2 7 9 8 5

4. TSIR Explanations (2):Th Li idi P i H h iThe Liquidity Premium Hypothesis


TSIR Explanations (2): The Liquidity Premium Hypothesis

The Main Weakness of the Expectations HypothesisThe Main Weakness of the Expectations Hypothesis The expectations hypothesis assumes that investors are risk neutral.

Ri k li i h d d i i Fi Risk neutrality is not the standard assumption in Finance. If all investors were risk neutral, then (for example):

i h l h di ifi d f li f h in the long term, the return on a diversified portfolio of shares would be the same as the interest rate on bank deposits;no homeo ner o ld ins re their home g inst fire no homeowner would insure their home against fire.

This is a pretty strange world!



The standard assumption in finance is risk aversion. That is, investors regard risk as undesirable.

Thus, bearing risk must be compensated by a higher expected return.

If ll i i k h (f l ) If all investors were risk averse, then (for example): in the long run, the return on a diversified portfolio of

h r ill d th i t r t r t b k d p itshares will exceed the interest rate on bank deposits; at least some homeowners will insure against fire.

Thi i h lik h ld li i ! This is much more like the world we live in! The liquidity premium hypothesis assumes that investors

are risk averseFNCE 30001 Investments 10.44

are risk averse.


Consider again the following three ways to invest $X for 3 years:1 B 3 d1. Buy a 3-year zero today;2. Initially, go shorter than the investment horizon:

B 2 d da. Buy a 2-year zero today andb. At the end of the 2 years, collect the par value, and then

b 1 r z rbuy a 1-year zero.3. Initially, go longer than the investment horizon:

B 4 d da. Buy a 4-year zero today andb. Sell it at the end of 3 years.



With risk-neutral investors (ie the expectations hypothesis), the market will set interest rates such that the returns from Strategies 1 2 and 3 are equal:Strategies 1, 2 and 3 are equal:

43 2 04

03 02 2334

11 1 1 E

1 Ez

z z rr

Suppose instead that investors are risk averse. S pp l th t e er i t r i thi m rk t h 3 r

341 E r

Suppose also that every investor in this market has a 3-year investment horizon.



Th Then:

1 23

3 2 0311 1 1 1 dz

3 2 0303 02 23 02

23

4 1 43 304

1 1 1 E 1 and1 E

11 1 1 E 1

zz z r z

r

z

Thus, because of risk aversion, both the 2-year rate (z02) and

3 30403 04 03 34341 1 1 E 11 Ezz z z rrThus, because of risk aversion, both the 2 year rate (z02) andthe 4-year rate (z04) will be higher than in a risk-neutral world.

Investors have to be offered higher interest rates to induce them to go shorter (2 years) and longer (4 years) than their horizon.


TSIR Explanations (2):

B i i lik l h ll i (l d ) ld h

The Liquidity Premium Hypothesis

But it is most unlikely that all investors (lenders) would have the same investment horizon (3 years).

Some investors (lenders) would have a 2-year horizon some aSome investors (lenders) would have a 2-year horizon, some a 3-year horizon, some a 4-year horizon etc.

However, according to the liquidity premium hypothesis, g q y p ypthere is a pattern.

This pattern is: There is a systematic tendency for borrowers to want to b f l n r p i d th l d t t l dborrow for longer periods than lenders want to lend.

That is, investors (lenders) display a preference for liquidity (making shorter-term loans).(making shorter term loans).

Hence, investors (lenders) require extra inducement if they are to make long-term loans.



According to the liquidity premium hypothesis, the relevant case to look at is where an investor is induced to lend for a term that exceeds the investment horizon:term that exceeds the investment horizon: As in the case where investors have a 3-year horizon but

are induced to buy a 4-year zero:are induced to buy a 4 year zero:

43 04

031

1z

z

0334

43 04

1 E

1Th f 1

zr

z 04

0334 34

34

Therefore: 11 E

where is an extra return (a liquidity premium).

zz

r LL


34where is an extra return (a liquidity premium).L


404

34 34303

1Therefore: 1 E

1

zL r

z 34 34Ef r

34 34 34which we can rewrite as: Ef r L The liquidity premium hypothesis says that the zero rates

(term structure) set by market participants produce forward

34 34 34f

rates that are equal to the relevant expected rate plus a little bit this little bit being the liquidity premium

Th li idi i b d i i h The liquidity premium may be assumed to increase with term.



Thousands of thingsg

Expected future interest rates

Todays zero rates (ie todays term structure)

Todays forward rates

Trade bonds

Do forwards = expected + L ?

Trade bonds

Equilibrium TSIR

YES NO


Equilibrium TSIR


The general effect of having a liquidity premium is that there is a bias towards longer-term interest rates being higher than would be justified purely on the basis of expectations of future interestbe justified purely on the basis of expectations of future interest rates.

Note that the liquidity premium hypothesis builds on theNote that the liquidity premium hypothesis builds on the foundation of the expectations hypothesis; both hypotheses maintain that expectations are important.yp p p

To avoid confusion, they are sometimes renamed:expectations hypothesis pure expectations hypothesisexpectations hypothesis pure expectations hypothesisliquidity premium hypothesis biased expectations hypothesis



Suppose a time came when lenders had a longer-terminvestment horizon than borrowers.

Wh i h hi h ? When might this happen? Then lenders would need to be induced to lend for short

periodsperiods. The liquidity premium would be negative.

Th t i th r ld b p lt ppli d t l d r h That is, there would be a penalty applied to lenders who wanted to lend for long periods.


5. Interpreting the Slope of the TSIRTSIR


Interpreting the Slope of the TSIRInterpreting the Slope of the TSIR

d b l Suppose, today, we observe a particular term structure. Given belief in a theory of the term structure we may be able to infer what

investors expectations are. p For example,

if todays term structure is flat (at 10% pa), then

if we believe in the pure expectations hypothesis, investors must expect interest rates will not change in the future;investors must expect interest rates will not change in the future; but

if we believe in the liquidity premium hypothesis, and also believe that the liquidity premium rises with term, then investors must expect interest rates to fall in the future.



Todays TSIRP i

Interest Pure expectations:

All forward rates = 10% paAll expected rates = 10% pa

rate (pa)

10%

LLt,t+1

Rising liquidity premium:All forward rates = 10% paAll expected rates = 10% Lt, t+1

Term (years ahead)

FNCE 30001 Investments 10.56INTERPRETING THE TSIR WITH A RISING LIQUIDITY PREMIUM


Excluding the possibility of a negative liquidity premium, the slope of the term structure is interpreted as shown in the table:

If todays term structure of

Then market expectations of future interest rates are:

According to the (pure) According to the liquidity interest rates is:

g (p )expectations hypothesis

g q ypremium hypothesis

Increasing Increasing Increasing, Flat or Decreasing

Flat Flat Decreasing

Decreasing Decreasing Decreasing


fnce 30001 week 10 tsir revised(1)

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forward rate f35suppose

actual rateszero rates

actual rates example

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actual ratesactual future

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