chp05a forward rates and yield curve

8/13/2019 Chp05a Forward Rates and Yield Curve

1/33

K. Cuthbertson, D. Nitzsche 1

Lecture

Forward Rates and the Yield Curve

The material in these slides is taken from

Chapter 8Investments: Spot and Derivatives Markets(J. Wiley, 2001)K. Cuthbertson and D. Nitzsche

These slides provide the introductory materialrequired forChapters 5 and 6 of Financial Engineering

Version 1/9/2001


2/33


Uses of Forward Rates

Calculation of Forward Rates

Calculation of no arbitrage (equilibrium) forward rate

Forward Rate Agreement, FRA

Yield Curve- expectations hypothesis

- slope of the yield curve

- forecasting inflation

Topics


3/33



1)Today, you can lock in an interest rate which will applybetween two

periods in the future (e.g. between end of year-1 and end of year-2,denoted f12)

Analytically this would simply involve ringing up a bank TODAY and itwould quote you a forward rate f12= 11% , say for a deposit of $100.

You would then be obliged to give the bank $100 IN ONE YEARS TIMEand the bank would pay you $110 one year later (ie. at the end of year-2)

In practice, things are a little more complicated than this but the sameresult is achieved by using a Forward Rate Agreement, FRA.


4/33



2)Pricing :

Forward Agreements -strictly forward-forward aggmts(FFA)Forward Rate Agreements , FRAs (hedging- practice)

3) Forecasting future inflation

4) Also used in Pricing-Floating Rate Notes, FRNs

-Interest Rate Futures Contracts

-floating rate receipts, in an interest rate swap


5/33


Calculation of Forward Rates


6/33


The bank calculates the forward rate by using two spot rates which itobserves on its dealers screen.

, r02= interest rate on money lent today for 2-years (ie. 2-year spot rate) -often just denoted as r2 .

Consider a fixed 2-year investment horizon.

Choices

1) Invest $1 for 2-years at r021) Receipts at t=2 (with certainty) are $1 ( 1 + r02)

2

2) Invest $1 for 1-year at r01and today also enter into a forward agreementto invest between t=1 and t=2 at a quoted rate f12

2) Receipts at t=2 (with certainty) are $1( 1 + r01) (1 + f12)

These transactions are riskless hence the amounts received at t=2, must beequal.

HOW DOES THE BANK KNOW WHAT FORWARD RATE TO QUOTE?


7/33 K. Cuthbertson, D. Nitzsche 7

What is the relationship between the (no arbitrage)forward rate and current spot rates ?

Equating 1 and 2

$1( 1 + r2)2 = $1( 1 + r1) (1 + f12)

Therefore

( 1 + f12) = ( 1 + r2)2 / ( 1 + r1)

Or, approximately (Let r1 = 9% p.a. and r2= 10% pa )

f12 = 2 . r2 - r1= 2 (10) - 9 = 11%

1) The correct forward rate is derived from current spot rates (yield curve)2) f12is the rate a bank should quote if it does not want to be ripped off -

see later

3) Also it can be shown (later) that f12is the markets best forecast of whatthe the one-year rate in one-years time (denoted Er1t+1) will be



Algebra of General Calculation of Forward Rates

Calculate other forward rates from todays spot rates is

pretty intuitive since the superscripts and subscriptsadd up to the same amount on each side of theequals sign

( 1 + r03)

3

= ( 1 + r02)

2

. (1 + f23)

1

( 1 + r03 )3 = ( 1 + r01)

1. (1 + f13)2

In general (there is no need to memorise this!)

fm,n = [ n / (n -m) ] rn - [ m / (n -m) ] rm

e.g. f1,3 = [ 3 / 2 ] r3 - [ 1 / 2 ] r1



Spot Rates and Forward Rates

0 1 2 3

r1r2

r3

1 year forward rates

0 1 2 3

f12 f23

Next years spot curve (forecast at t = 0)

0 1 2 3

f12f13



Calculationof the

No-arbitrageor

(Equilibrium) Forward Rate



Market behaviour behind the determination of theno-arbitrage forward rate

Problem:

You will RECEIVE $100 in one years time ( t=1) from payment on defencecontract and wish to place the money on deposit for a further year.

You are worried that in 1-years time 1-year interest rates will have fallenCan you TODAY, lock in an interest rate on your future deposit ?

YES : use a forward (FFA) agreement with the bank (- loosely speakingthis is equivalent to an FRA)



Fig 1: Actual Cash Flows -Real Forward Agreement

Will RECEIVE $100 at t=1 from payment on defence contract.

So:

Agree today, to lend (pay out) $100, in 1-years time and receive the

principle plus interest at end year-2.

Banks Quoted forward rateis, f12= 10.5% p.a.

(Note: No own funds are used at t=0)

f12 = 10.5 %

110.5

210

100 (pay to bank)



Should you take the banks offer or, can you

manufacture/engineer cash flows which exactly match the banksFRA, but at a lower interest cost ?

The home made FRA we call the synthetic forward

The synthetic is a simple example of financial engineeringIt provides a way of obtaining the fair or true (forward) rate

for the actual forward contract offered by the bank.

If you engineer your synthetic to have exactly the same pattern

of cash flows as the actual forward contract then actual forwardrate f12quoted by the bank must equal the interest cost ofcreating your synthetic contract - if not then you can rip off thebank. Great!

Calculation of the no-arbitrage forward rate



Creating a synthetic forward contract

What can we use to make our synthetic forward ?

We borrow and lend using the 1-year and 2-year moneymarket spot rates.

We will borrow and lend equal amounts at t=0 and henceuse no own funds - just as in the real FRA.

Let r1= 9% and r2= 10%



Figure 8.2 :Synthetic Forward Rate: sf12

.

r2=10%

r1= 9%sf12= ?? = 11%

1) Pay out $100

2) Borrow $91.74

at r1= 9%

3) Lend $91.74 at r2=10%

4) Receive $111

To reproduce the pattern of cash flows in the actual FRA, follow the

steps 1-4

Create cash flows with timing equivalent to actual FRA



Synthetic Forward

1)Borrow $91.743 at t=0, for 1-year at r1:

Owe $100 at t=1

(paid for from proceeds of the defence contract)

2)Use this $91.743 at t=0 to

3) Invest for 2-years at r2 : receive $111 at t=2

You have not used any of your own funds at t=0

The above cash flows are equivalent to

Lending $100 at t=1 with payment $111 at t=2

Hence

Synthetic forward rate sf12= 11 % (also note sf12 = 2 r2- r1)

Dont take the banks offer of f12= 10.5 % !!



Synthetic Forward,Algebra:(for afficionados only)

1)At t=0 borrow 100/(1 + r1) = $91.74 for 1-year (where r1= 0.09 )

This means you will have a cash outflow of 100 at t=1 (to match that in real

FRA)

2) Then lend the $91.74 you have just borrowed, for 2 years

This means you will receive in 2 years time:

$91.74 (1+ r2)2 = 111 (at t=2) (where r2= 10%)

3)The synthetic futures rate between t=1 and t=2 must therefore be

given by

(1 + sf12) = 111/100 = $91.74 (1+ r2)2 / $91.74 (1 + r1)

= (1+ r2)2 / (1 + r1)

Simplifying: sf12= 2 r2- r1 (we have set (r x f) or (r x r) to zero)



Can we Rip-Off the Bank ?

When sf = 11% > f = 10.5%, can we rip-off the bank?Can we also dispense with the defence contract cash ?YES WE CAN !!

1)Today, go to Bank-A and agree a real FRA at f12= 10.5% which involves

a) BORROW(ie. receive) $100 at t=1 and hence

b) PAY OUT $110.5 at t=2

2) Create your own synthetic forward contract as in Fig. 8.2 above, whichimplies no net cash flows at t=0 but you have to

a) PAY OUT $100 at t=1 (use 1a to pay this)

b)YOU WILL RECEIVE $111 at t=2

3). Net result is a RISKLESS PROFIT of

111 - 110.5 = $0.5 at t=2

and you have used non of your own money(capital) at t=0, 1 or 2 !



ARBITRAGE: Equalises f and sf

sf = 11% and f = 10.5%

Everyone is buying FRA (ie borrowing) from Bank-A, so as the demandfor its (forward) loans has increased, it raises its quoted rate f

Everyone is also using the synthetic forward route that is

Borrowing at r1and lending at r2 (see fig 2 above)

Hence: r1will rise , and r2 will fall

Fall in r2means less $s at t=2 from the investment in the synthetic

forward and hence a fall in sf ( = 2r2- r1) .

This completes the argument of why it must be the case that at all times:

( 1 + r2)2 = ( 1 + r1) (1 + f12)




Analytically this is the same as a forward(-forward) agreement at arate f = 11% agreed today ( t = 0)

However in practice only differences are exchanged (and there aresome other nuances, which need not concern us)




Example:

You will receive $100 in one years time and want to then place it ondeposit for a further year. You fear a fall in interest rates over thecoming year. You therefore take out an FRA at f = 11%.

At the end of the year suppose interest rates have fallen to say 8%.The FRA pays out:

f - 8% = 3% on $100 = $3

You only get 8% from the bank deposit but the 3% from the FRAmeans you have locked in 11% .

(Actually you get the PV of $3, but this need not concern you)


23/33 K. Cuthbertson, D. Nitzsche

Figure 8.4 : Buy 3m x 6m FRA

f3,6

Recieve$1m plus

interestat f3,6

Notional Cash Flows

3m 6m0FRA negotiated

Lend/Depositof $1m

90 days

S



Summary : Forward Rates

Forward rates apply between two periods in the future but the rateis agreed today

Locks in an interest rate on a loan or deposit between two future dates

It can be shown that f12is the markets best forecast of what the the one-year interest rate in one-years time will be.

This can then be used to give an inflation forecast. For example if f12= 11%and the real interest rate is 3% (= growth rate of the economy) then theforecast for annual inflation in one years time would be 8% (=11 - 3).

Forward rates are used to price FRAs, ( also FRNs, interest ratefutures and swaps)



The Yield Curve

and theExpectations Hypothesis

THE YIELD CURVE



THE YIELD CURVE

Why are long rates of interest often higher than shortrates of interest ?

- can long rates be lower than short rates ? Yes !

= Expectations Hypothesis

If we know the shape of the yield curve (ie. All the spotrates) then we can calculate forward rates for all

maturities


27/33


Figure 5 :YIELD CURVE

We can show that if the yield curve is upward sloping then(one-period) short rates in the futureare expected to rise.

Yield

6

4

1 2Time to

maturity

A

A

B

B

3

7

Expectations Hypothesis (EH): Term Structure


28/33


Expectations Hypothesis (EH): Term Structure

Risky Arbitrage but assuming risk neutrality

$1.( 1+ r2 ) 2 = $1. (1+r1) . [ 1 + Er12 ]

Approx.r2 = ( 1 / 2 ) . [ r1 + Er12 ]

EH implies

1.Long-rate r2is weighted average of current (r1) and

expectedfuture (one-period) short rates Er12

Upward Sloping Yield Curve


29/33


Upward Sloping Yield Curve

Three Period Rate : r3t

r03 = ( 1 /3 ) . [ r01 + Er12+ Er23 ]

Suppose r1 = 4 and future (one-period) rates areexpectedto rise, Er12 = 8, Er23 = 9

then

r01= 4

r02 = ( 1 /2 ) . [ r01 + Er12 ] = 6r03 = ( 1 /3 ) . [ r01 + Er12+ Er23 ] = 7

Hence, yield curve will be upward sloping (4


30/33


Inflation Prediction from the Yield Curve

Rising yield curve implies that short rates are expected to be higherin the future and this is probably because inflation is expected torise in future years

Observe the current yield curve

r2= 10%, r1= 5%, then f12= 15.2%

If real rate = 3%, then ( from Fisher effect)

Expected inflation in 1-years time

= 15.2 - 3 = 12.2%

= Bank of England inflation forecast ?

Li idit P f


31/33


Liquidity Preference

Lenders like to lend at short horizon, borrowers like to

borrow for long horizon, so long rates contain positiveliquidity premia

ExpectationsHypothesis

liquidity premium

L.Preference

Maturity

Yield


32/33


Expectations Hypothesis and Forward Rates

EH mathematically is:

1) r02 = ( 1 /2 ) . [ r01 + Er12 ]

Definition/ Calculation of no arbitrage forward rate:

2) r02 = ( 1 /2 ) . [ r01 + f12 ]

Hence the EH implies that the current forward rate is the markets bestpredictor of the future spot rate (not 100% accurate though!) - hencefrom 1 and 2, above

f12= Er12

So why not disband the MPC on whom we spend a lot of money so they cantry and forecast future inflation, better than an average of all marketparticipants, (who at least put their money where there mouth is whenplaying the yield curve - ie. the EH).


33/33

chp05a forward rates and yield curve

Documents