fnce 30001 week 11 floaters and swaps

8/22/2019 FNCE 30001 Week 11 Floaters and Swaps

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FNCE 30001 Investments 11.0

FNCE 30001 Investments

Semester 2, 2011

19 & 21 October 2011

Week 11: Floaters and Swaps

Professor Rob Brown


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Week 11: Floaters and Swaps

Overview of Lecture

1. Floaters1.1 What is a Floater?

1.2 How Much is a Floater Worth?

1.3 The Duration of a Floater

2. Interest Rate Swaps2.1 What is an Interest Rate Swap?

2.2 Why do Interest Rate Swaps Exist?

2.3 How Much is an Interest Rate Swap Worth?

2.4 Interest Rate Swaps in Practice

Reading: Bodie et al, Chapter 16.


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1. Floaters1.1 What is a Floater?


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What is a Floater?

Floater is short for floating rate bond or floating rate note.

Also known as FRNs or frens.

A floater is the same as a fixed coupon bond except that the coupon ratechanges over time in line with an indicator rate.

The indicator rate is usually a short-term rate

eg6-month USD LIBOR

There is usually a fixed margin (spread) above the indicator rate

eg6-month USD LIBOR plus 100 basis points

The coupon rate is reset on every coupon payment date.

Otherwise, floaters are like fixed coupon bonds:

Fixed schedule of coupon payment dates

Fixed par value

Fixed maturity date

although some floaters are perpetuities


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What is a Floater?

The greater the credit risk of the borrower, the greater the

spread over the indicator rate we assume that credit risk doesnt change over time.

After the payment of a coupon, the coupon rate is set for theperiod just starting.

Therefore, the current periods coupon rate is knowntoday (but none thereafter).


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What is a Floater?

Is a floater a long-term or short-term instrument?

Good question! Floaters may last a long time some are even perpetuities

known as perpetual FRNs.

This sounds like a long-term security. But because the coupon rate is reset frequently, a floaters

interest rate risk is equal to that of a short-term security.

The interest cost (revenue) is the same as a series of short-term debt securities.


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1.2 How Much is a Floater Worth?


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How Much is a Floater Worth?

You are an investor and you are considering whether to buy afloater with these features:

Coupons are paid every 180 days.

The indicator rate is the 180-day bank bill rate.

The remaining term to maturity is 360 days.

The credit risk of the issuer is extremely small

equal to that of a big Australian bank.

A coupon has just been paid.

Note that in practice floaters pay coupons on a calendar-month basis like 15 March and 15 September each year.

Im doing it this way to keep the comparison watertight.


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How much would you pay for this floater?

Valuation by replication: an investment in this floater is verysimilar to an investment in 180-day bank bills:

Same credit risk

Same interest rate The next cash flow will occur on the same date.

So, lets do a detailed comparison and see what differencesthere are (if any) between an investment in this floater and aninvestment in 180-day bank bills.


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Day 180 Day 360Now

180-day

bill rate on

Day 180

180-day

bill rate on

Day 0

The Floater:

180-day

bill rate on

Day 0

A 180-day Bill:

Obvious difference:

The floaterprovides an investment from Day 180 to

Day 360. A180-day bill doesnt.

?


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Day 180 Day 360Now

180-day

bill rate on

Day 180

180-day

bill rate on

Day 0

The Floater:

180-day

bill rate on

Day 0

Consecutive

180-day Bills:

180-day

bill rate on

Day 180

Only difference:

The floater automatically provides a second 180-day

bill. A180-day bill doesnt.


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How much is this difference worth?

What does the floater provide that a bank bill doesnt?1. An automatic reinvestment but:

This is worth nothing because there is no shortage of 180-day billsto buy.

2. The reinvestment incurs no transaction cost but: This is worth nothing because transaction costs in the bill market

are very low (especiallymarginal transaction costs).

What does a bank bill provide that a floater doesnt?

1. Automatic payout of par value at Day 180 but: This is worth nothing because if the investor in the floater wants to

receive the par value on Day 180, he simply sells the floater.


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How much is this difference worth? (contd.)

Conclusion: there is no difference. Hence, they should be priced the same.

But be careful not to get terminology mixed up.

In a bank bill, the payment at maturity is the par value(usually assumed to be $100).

In a floater, the payment at maturity is the par value plusone coupon payment.

This is a difference in terminology, not in economic substance.


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Example

Suppose:

The par value of the 360-day floater is $100,000.

The floaters coupon rate equals the 180-day bank bill rate.

Todays 180-day bill rate is 7.75% pa.What is the price of the floater today?


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Answers

The 180-day rate today is by definition:

Hence, on the next coupon date (which is in 180 days time),the floater will pay a coupon of:

3.82192% x $100,000 = $3821.92

Therefore, the price of the floater today is:

180

7.75% 3.82192%365

$103,821.92

$100,0001.0382192

P


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Answers (contd.)

As weve just seen, the price of a floater immediately after acoupon payment (iethe one just paid today) is simply the parvalue.

The equivalent bank bill is one with a par value of$103,821.92.

In practice, of course, no bank bill has such an odd par valuebut this is unimportant.


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Hang on, is there a problem here?

On Day 180, the cash flow from the floater is $3,821.92 butthe cash flow from the bank bill is $103,821.92.

Yes, but the investor with the floater retains the floater.

And, having just paid a coupon, the floater is now worth its

par value, which is $100,000.

So if the investor wants the further $100,000, he can simplysell the floater.


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Another way to think of this

The present value of the Day 360 cash flow, as at Day 180, will be:

0 180 360

$Par(1 + r180,360)r0,180 x $Par

180,360

180

180,360

$ 1

1

Par r V Par

r

Floater cash flows


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So, the cum-interest price of the floater on Day 180 will be:

180 0,180 180

0,180

0,180

0,180

0

0,180

1

which, valued as at Day 0, gives the price today:

1

1

P r Par V

r Par Par

Par r

Par r P

rPar


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The price of a floater between coupon dates

After holding this floater for 120 days you decide to sell it.On that day, bank bill yields for various terms to maturity are:

30 days: 7.35% pa

60 days: 7.45% pa90 days: 7.55% pa

120 days: 7.65% pa

180 days: 7.70% paWhat is the price of the floater?

This is the relevant

rate to discount thefuture payment.


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On Day 120, the floater is equivalent to a 60-day bank bill

with a face value of $103,821.92.(Why?)

Therefore, the price of the floater is:

$103,821.92 $102,565.8460

1 0.0745365

P


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Practical Example

A floater has these features:Term to maturity is 7 years.

Coupon rate (pa) is 6-month LIBOR (pa) + 50 bp (pa).

Coupons are paid twice each year.Par value is $1 million.

It is three months since the last coupon payment.


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Practical Example (contd.)

Information on LIBOR is shown in the table below:

3-monthLIBOR

6-monthLIBOR

Today 5.65% pa 5.85% pa

3 months ago 5.55% pa 5.80% pa


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The current coupon rate (pa) was set 3 months ago at 6-month LIBOR (pa)plus 0.5% (pa)

= 5.80% pa + 0.5% pa = 6.30% pa = 3.15% (for 6 months).

Thus the next coupon is:

0.0315 x $1,000,000 = $31,500

The relevant discount rate today (pa) is todays 3-month LIBOR (pa)plus 0.5% (pa)

= 5.65% pa + 0.5% pa = 6.15% pa = 1.5375% (for 3 months).

Thus the price is:

$1,031,500

$1,015,8811.015375

P


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1.3 The Duration of a Floater


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The Duration of a Floater

Suppose you own a floater and it is now just an instant after acoupon has been paid and the coupon rate has been reset.

Your floater is now, in effect, a zero-coupon bond.

The par value of this zero is equal to the par value of thefloater plus the next coupon payment.

Recall that the duration of a zero-coupon bond is just theterm to maturity.

So, the duration of the floater is the period of time to the next

coupon payment.


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The Duration of a Floater

Question: But wont the floater continue to exist after (maybeyears after) the next coupon (rate set) date?

Answer: Yes, but thats not relevant to duration. (Why?)

So, what is the duration of the floater immediately before thenext coupon (rate set) date?

Answer: Zero


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2. Interest Rate Swaps

2.1 What is an Interest Rate Swap?


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What is an Interest Rate Swap?

Consider this game

Todays 1-year interbank interest rate (b0) is 5% pa.I put the following proposition to you:

In 1 years time, I will pay you 5% of $1m= $50,000.

Thereafter, each year for 2 more years, I will pay you

according to future (currently unknown) 1-year interbankinterest rates.

But each yearyou must pay me s% x $1m(sis fixed andhence known today).

My question to you: what value ofswould induce you to playthis game?


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0 31 2

5% b1 b2

$50,000 b1 x $1m b2 x $1m

sx $1m sx $1m sx $1m

I will pay you:

You will pay me:

The critical question: what is the value ofs?


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If we can agree on a value for s, then we have just enteredinto a 3-year fixed-for-floating interest rate swap with annualswap payments.

sis known as the swap rate.

You are called the pay-fixed party

I am called the receive-fixed party.

The $1 million is called the notional amount

Note that in a plain vanilla interest rate swap the

notional amount is never paid.


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In practice, only the net swap payment is made

if I owe you $50,000 but you owe me $49,000 then I justpay you $1000.

Say we agree on a swap rate (s) of 5.9% pa.

I will agree to accept a swap payment of 5.9% only if I thinkthe swap value is zero or positive for me.

You will agree to pay a swap payment of 5.9% only if youthink the swap value is zero or positive for you.

In equilibrium, therefore, we expect the value of a newswapto be zero.


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The value of an old (or decayed) swap ieone made inthe past but still hasnt matured will usuallynot be zero:

time will have passed and interest rates may have changed.

Hence the price of a decayed swap is nearly always positivefor one party and therefore negative for the other.

But if a newinterest rate swap has a value of zero, why wouldanyone bother entering into one?

That is, why do interest rate swaps exist?


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2.2 Why do Interest Rate Swaps

Exist?


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Why do Interest Rate Swaps Exist?

Same answer as for futures contracts:

Futures contracts are also worth zero on initiation but arenevertheless useful for hedgers and speculators.

A swap converts an existing fixed-rate borrower into a floatingrate borrower and vice-versa.

Clearly, such a switch may serve hedging and speculativeends.


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A swap may even reduce borrowing costs for both parties.

Suppose our pre-swap borrowing opportunities are:

Initially, I borrow fixed-rate and you borrow floating rate

then we enter into the swap.

The swap will reduce borrowing costs for both of us.

Fixed-rate Loan Floating-rate Loan

Me 6.0% Interbank rate + 0.3%

You 6.5% Interbank rate + 0.4%


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ME YOU

Floating

rate lenders

Fixed

rate lenders

6.0% interbank rate + 0.4%

BEFORE THE SWAP


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ME YOU

Floating

rate lenders

Fixed

rate lenders

6.0%

5.9%

interbank rate

interbank rate + 0.4%

Net cost for me is interbank rate + 0.1% (saving = 0.2%).

Net cost for you is 6.3% (saving = 0.2%).

AFTER THE SWAP


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The effect of the interest rate swap has been:

To convert me from being a fixed rate borrower to afloating rate borrower

To convert you from being a floating rate borrower to afixed rate borrower.

Banks and other financial institutions frequently enter intoswaps for exactly this reason.

In December 2010, the total notional amount outstanding oninterest rate swaps worldwide was $US 364,378 billion (Source:

Bank for International Settlements www.bis.org). The gross market value of these swaps was $US 13,001 billion

(Source: BIS).


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2.3 How Much is an Interest Rate

Swap Worth?


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How Much is an Interest Rate Swap Worth?

At initiation: zero to each counterparty.

At any other time, a swap will (usually) have a positive value toone counterparty and a corresponding negative value to theother counterparty.

Hence, the totalvalue of the two positions is always zero.

This ignores default risk.


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How significant is default risk?

Only relates to the net swap payments:much

less than thenotional amount.

Typically, parties have high credit rating.

Some have a margin call feature; that is, you have to pay

money early if youve lost a lot already.

There have been few defaults in practice.


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Consider the pay-fixed party in a 3-year swap with annualswap payments.

Notation

The floating rate at time tis btand results in a swappayment at time t+ 1.

The fixed rate in the swap is s.

The notional amount of the swap isA dollars.

The year-by-year cash flows are shown on the next slide.


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0 31 2

b0 b1 b2

b0 xAReceive-floating

payments

Pay-fixed

payments sxA sxA sxA

b1 xA b2 xA

Net cash flow (b0s) xA (b1s) xA (b2s) xA

CASH FLOWS FOR THE PAY-FIXED PARTY


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Proposition: The future cash flows for the pay-fixed partyare the same as those from buying an appropriately

specified floater and selling an appropriately specifiedfixed-coupon bond.

Notation

The coupon rate in the floater at time tis equal to bt ,which is the floating rate in the swap.

The coupon rate in the fixed-coupon bond is equal to s,

which is the fixed rate in the swap. The par value of both bonds is Parand is equal to the

notional amount (A) of the swap.


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0 31 2

b0 b1 b2

b0 xABought Floater

(Par = A)

Sold Fixed Bond

(Par = A) sxA sxA sxAA

b1 xA b2 xA+A

(b0s) xA (b1s) xA (b2s) xANet cash flow

CASH FLOWS FOR BOUGHT FLOATER AND SOLD

FIXEDCOUPON BOND


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The value of the swap to the pay-fixed party is therefore:

At initiation of the swap, both the floater and the coupon bondare newly issued.

Therefore, both are priced at par. Hence:

In the future (t> 0), the prices of both the floater and thecoupon bond will change, resulting in a non-zero swap value.

t t t

swap floater fixed bond

V V V

0 0 00

swap floater fixed bond V V V Par Par


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Further implications:

Ignoring credit risk, the current swap rate (s) is thereforeequivalent to the yield on a coupon bond selling at par (ieitis a par yield).

We can also turn this around:

if we can observe swap rates (which we usually can),then that is equivalent to observing par yields (which inAustralia are often not directly observable).

Then from par yields we can estimate zero rates (whichoften are not directly observable).

And using zero rates we can price coupon bonds.


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Further implications(contd.)

If interest rates subsequently increase, then the prices ofboth the floater and the fixed-coupon bond will fall.

But the price of the fixed-coupon bond will fall furtherthan the price of the floater. (Why?)

Therefore, floater-price-minus-fixed-price increases.

That is, the value of the swap to the pay-fixedcounterparty increases.

This makes sense if I am paying fixed, then I amreceiving floating, and I am happy to receive higherswap payments.


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Further implications(contd.)

Therefore, if interest rates increase, then the swap has alower value for the pay-floating (= receive-fixed) party.

In a simple table:

If interestrates: Then the value of an existing swap to:the pay-fixed party the receive-fixed party

increase increases decreases

decrease decreases increases


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Example 1

A 6-year interest rate swap with a notional amount of$100mwas entered into 2 years ago. The swap rate is8.5% pa and a swap payment has just been made.

The floating rate is the 1-year interbank rate

(currently 7.7% pa). The swap has annual rate sets.Current zero rates are:

2 years: 8.0% pa

3 years: 8.2% pa

4 years: 8.3% pa


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Example 1 (contd.)

Ignoring credit risk, how much is the swap worthtoday:

(a) to the pay-fixed counterparty?

(b) to the receive-fixed counterparty?


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Answer to Example 1(a) Value to the pay-fixed counterparty

The fixed swap payment is 8.5% x $100m= $8.5m.

The value of the floater is $100m. (Why?)

The value of the equivalent 4-year bond is:

The value of the swap to the pay-fixed counterparty is therefore:$100m $100,760,630 = $760,630.

(b) Value of the swap to the receive-fixed counterpartyis +$760,630.

2 3 4

$8.5 $8.5 $8.5 $108.5$100,760,639

1.077 1.080 1.082 1.083

fixed bond m m m m V


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Example 2

Three months later the 9-month interbank rate(simple interest) is 8.4% pa and the relevant zero

rates (compound interest) are:

1.75 years: 8.65% pa2.75 years: 8.85% pa

3.75 years: 8.95% pa

Ignoring credit risk, how much is the swap worth

now to the pay-fixed counterparty?


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Answer to Example 2The next floating swap payment will be 7.7% x $100m= $7.7m.

The value of the floater is therefore:

The value of the equivalent 3.75-year bond is:

The value of the swap to the pay-fixed counterparty is therefore:$101,317,027 $100,752,875 = +$564,152.

1.75 2.75 3.75

$8.5 $8.5 $8.5 $108.5$100,752,875

1.063 1.0865 1.0885 1.0895

fixed bond m m m m V

$107.7 $107.7$101,317,027

1 0.75 0.084 1.063

floater m mV


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Compare Example 1 and Example 2:

The value of the swap to the pay-fixed party has increased from $760,630 to +$564,152. Why?

Because in the 3 months that elapses between Example 1 andExample 2 interest rates rise by 65-70 basis points.

Clearly, this change benefits the pay-fixed counterparty.

In Example 1, the rates were below the fixed rate of 8.5% pa.

In Example 2, most rates were above the fixed rate of 8.5% pa.

So the value to the pay-fixed counterparty changes fromnegative to positive.



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2.4 Interest Rate Swaps in Practice

I R S i P i


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Interest Rate Swaps in Practice

Specification of the floating rate Most common world-wide is 6-month USD LIBOR.

In Australia, the standard is BBSW, which is an average of bid and askyields for 90-day bank bills as at 10am each trading day.

Approach to pricing The approach we used (value of pay-fixed swap = + floater coupon

bond) isnt the only one. A swap can also be shown to be equivalent to a series of forward interest

rate agreements (FRAs).

Hence, a swap is almost equivalent to a series of interest rate futures

contracts. So in practice, swaps are also priced using FRAs and/or interest ratefutures.

I R S i P i


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Swap dealing I have told the story as if the end-users in a swap are in

contact with each other. In very early swaps (early 1980s) this was true but it isnt anymore.

Most swaps now are set up with at least one of the

counterparties being a bank. There is a parallel history in lending.

Centuries ago all borrowers and lenders would directlynegotiate.

Then banks evolved to borrow from the lenders and lendto the borrowers.

Recall the example in which you and I agreed to a swap

between us:

Wh d I R S E i ?


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ME YOU

Floating

rate lenders

Fixed

rate lenders

6.0%

5.9%

interbank rate

interbank rate + 0.4%


Net cost for you is 6.3% (saving = 0.2%).

AFTER THE SWAP

I t t R t S i P ti


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Today, banks will enter a swap in their own name and thenlook for the matching party, intending to take a profitable

position in the process.

There is avery active over-the-counter market in swaps.

So, today, a more realistic picture would see the creation of

two interest rate swaps, as shown on the next slide:

Int r t R t S p in Pr ti


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ME YOU

Floating

rate lenders

Fixed rate

lenders

6.0%

6.0%

i/b rate + 0.1%

i/b rate + 0.4%


Net cost of funds for you is 6.4% (saving = 0.1%).

The Bank makes 0.2%.

BANK

i/b rate

5.9%



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Frequency of swap payments

Most swaps require payments more frequently than annual.

In Australia, typical arrangements are:

For terms of 3 years or less: quarterly

For terms longer than 3 years: half-yearly.

This requires the exact number of days between payments tobe counted.



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ExampleIt is now October 2013 and you have been asked to reconstruct the cash

flows that should have occurred in a 1-year swap entered into on 16 July2012. The swap was BBSW for the swap rate and the notional amount was$10m. You have been given the following historical data:

Date BBSW Swap Rate16 July 2012 6.40% pa 6.56.% pa

16 October 2012 6.35% pa 6.49% pa

16 January 2013 6.50% pa 6.55% pa

16 April 2013 6.80% pa 6.75% pa

16 July 2013 7.00% pa 7.06% pa



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Answer

1. Identify the data we need (shown in red italics).

2. Calculate the number of days between swap payments.

Date BBSW Swap Rate Number of Days

16 July 2012 6.40% pa 6.56% pa

16 October 2012 6.35% pa 6.49% pa 15 + 31 + 30 + 16 = 92

16 January 2013 6.50% pa 6.55% pa 15 + 30 + 31 + 16 = 92

16 April 2013 6.80% pa 6.75% pa 15 + 28 + 31 + 16 = 90

16 July 2013 7.00% pa 7.06% pa 14 + 31 + 30 + 16 = 91

Note: I am ignoring the weekend problem. There are rules for dealing with

weekends.



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Answer (contd.)

We now calculate the swap payments. The relevant swap rate is 6.56% pa.

Notional amount is $10m.

Date BBSW Days Amount paid on indicated date

16 Jul 12 6.40%

16 Oct 12 6.35% 92 (6.40%6.56%) x 92/365 x $10m=$4032.8816 Jan 13 6.50% 92 (6.35%6.56%) x 92/365 x $10m=$5293.15

16 Apr 13 6.80% 90 (6.50%6.56%) x 90/365 x $10m=$1479.45

16 Jul 13 91 (6.80%6.56%) x 91/365 x $10m= + $5983.56Amounts shown as negative are paid by the pay-fixed party.Amounts shown as positive are paid by the receive-fixed party.

fnce 30001 week 11 floaters and swaps

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