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FINM7405-finm7405代写
时间：2023-03-24
FINM7405
Lecture 5: Interest rate swaps and currency swaps
• Swaps
• Interest rate swaps
-Mechanism
-Why use interest rate swaps?
-Pricing
• Currency swaps
-Mechanism
-Why use currency swaps?
-Pricing
Outline
2
•Agreements to exchange a series of cash flows on periodic dates
•Swaps are similar to forwards:
- Swaps typically do not require payment by either party at initiation (except currency
swaps)
- Swaps are traded OTC and not standardized
- Swaps are regulated by International Swaps and Derivatives Association (ISDA) in
the global market, and by Australian Financial Markets Association (AFMA) in
Australia
- Although regulations exist, default risk is high for swap transactions
•Two most common swaps:
- Interest rate swap
- Currency swap
Swaps
3
• An agreement to exchange fixed rate for floating (variable) rate over the tenor (i.e., life) of the
swap
- Suppose I lend you $100 at 4% pa fixed rate and you lend me $100 at variable (i.e., floating)
rate for 3 years. Interest is paid semi-annually.
- Thus we have an interest rate swap with a tenor of 3 yrs
- Both loan amounts are equal, hence it is pointless to exchange $100.
- Also, it is pointless to exchange the full interest amount every 6 mths. Only the party with the
larger payment liability pays the difference. Example: If the floating rate is 6% pa, I pay you
$1. If the floating rate is 3% pa, you pay me $0.50
- In the above, you are the fixed rate payer (i.e., ‘payer’) and I am the fixed rate receiver (i.e.,
‘receiver’). As a fixed rate receiver, I pay floating rate.
- We don’t usually use the terms ‘floating rate payer’ or ‘floating rate receiver’.
Interest rate swap
4
•To transform a liability
-Suppose company A borrowed at LIBOR floating rate + a margin of 50 bps.
Company A already has many variable rate borrowings and wanted to convert
the above into fixed rate borrowing.
-Suppose company B has just issued an 8% coupon bond, in addition to other
fixed rate borrowings. Company B wanted to convert the above to variable
rate borrowing.
-Hence, companies A and B can do an interest rate swap. Let’s assume the
fixed swap rate is 6%
Why do we use interest rate swap?
5
6Company B
Effective
borrowing rate
for Company A
= 6.5%
Effective
borrowing rate
for Company B
= LIBOR + 2%
Why do we use interest rate swap?
Variable Bank
Loan
Company A
LIBOR +
50 bp
Fixed Coupon
Bond
8% fixed
6% LIBOR
• Suppose company F holds floating rate notes issued in Australia. Thus, it receives BBSW
minus 30 bps margin. BBSW (Bank Bill Swap Rate) is analogous to LIBOR; it is the reference
rate at which variable rates in Australia are set upon. Company F wanted to convert its cash
inflow into fixed amount.
• Suppose company V holds 5.8% fixed coupon bonds and wanted to convert its cash inflow
into variable amount
• Thus, companies F and V can do an interest rate swap. Let’s assume the fixed swap rate is
6%.
To transform an asset
7
-Usually, companies F and V won’t directly transact the interest rate swap.
They will do this through a financial intermediary (FI)
- In the example below, FI earns a 4 bps spread but each of F and V receives 2
bps less.
To transform an asset
8
- If you predict that the yield will increase (price will decrease) in the future, you
can try to speculate on your prediction by
Sell bond futures (sell high now, close out by buy low later), or
Enter into an interest rate swap as a fixed-rate payer (pay fixed, receive
floating). Reason:
- It can be shown (later) that if yield increases, the fixed coupon bond will trade
at discount. The effect is such that the swap has a positive value to fixed rate
payer
To speculate:
9
- Suppose you were a liability manager and you have issued/sold a fixed coupon bond. To hedge the risk
of a falling yield (bond price or value of your liability increases), you can
Buy back the bond, or
Enter into an interest rate swap as a fixed-rate receiver (receive fixed, pay floating). Reason:
• It can be shown (later) that if yield decreases, the fixed coupon bond will trade at premium.
The effect is such that the swap has a positive value to fixed rate receiver. This positive swap
value offsets the increase in your liability zero net effect
• It can be shown (later) that if yield increases, the fixed coupon bond will trade at discount.
The effect is such that the swap has a negative value to fixed rate receiver. This negative
swap value offsets the decrease in your liability zero net effect
- Note also that a receive fixed, pay floating swap has the same effect as decreasing duration. This is
because this swap implies that you issue short-term floating rate notes (thus pay floating rate) and use
the proceeds to buy back long-term bonds (thus receive fixed rate)
To hedge:
10
•Suppose we are the fixed rate payer (we pay fixed rate and receive floating rate).
•We can replicate this effect using a clever combination of basic bonds:
- Issue fixed coupon bond
- Invest the proceeds in floating rate coupon bond with the same maturity and
payment dates as the above fixed coupon bond
- On each interest payment date, we pay fixed coupon and receive floating rate
- OR
- Issue fixed coupon bond
- Invest the proceeds in short-term floating rate notes (e.g., 6-mth floating rate note)
and roll-over successively until the maturity of the above fixed coupon bond
Replicating an interest rate swap
11
•Thus, we can value an interest rate swap by aggregating the value of each of the
components in the portfolio:
- Value a fixed coupon bond (discussed a few weeks ago)
- Value a floating rate coupon bond with the same maturity and payment dates as the
above fixed coupon bond
The value is equal to the par value on interest repayment dates
The value is greater than the par value on non-interest repayment dates
•Why is the floating rate coupon bond always repriced to par on interest repayment
dates?
- To understand this, remember first that the floating rate interest is always set in
advance, but paid at the next period
12
- The floating rate interest is always set in advance, but paid at the next period
- For a floating rate coupon bond, the coupon rate, which determines the coupon payment at
next periods, is set to the market yield.
- When coupon rate = yield, the bond is priced on par
- Example: Suppose the current YTM for the floating rate coupon bond = 4% pa. The coupon
rate of this bond will be set equal to this yield, and thus, today (t=0), the floating rate coupon
bond is priced at 100.
Why is the floating rate coupon bond always repriced to par on interest
repayment dates?
13
22 2 2 2
100+
2
remaining
periods
Coupon rate set
today (i.e., t=0)
but paid at
t=1,2,…
One period has passed. Suppose the current YTM for the floating rate coupon bond =
6% pa. The coupon rate of this bond will be reset (updated) equal to this yield, and
thus, today (t=0), the floating rate coupon bond is repriced to 100, again.
The above process is repeated on each repayment date i.e., coupon rate is reset (i.e.,
updated) to the yield of the floating rate bond on each repayment date, which means
that the floating rate coupon bond is always repriced to par
Why is the floating rate coupon bond always repriced to par on
interest repayment dates?
14
3 3 3 3
100 +
3
remaining
periods
Coupon rate set
today (i.e., t=0)
but paid at
t=1,2,…
0 1 2 3 4 5
o Note also we can value an interest rate swap by aggregating the value of
each of the components in the portfolio:
Value a fixed coupon bond
Value a short-term floating rate note (e.g., 6-mth floating rate note) and
roll-over successively until the maturity of the above fixed coupon bond
The value is equal to the par value on interest repayment dates
The value is greater than the par value on non-interest repayment dates
15
Example: Suppose today’s 6-mth LIBOR is 4% pa. LIBOR is a “simple rate” with
no compounding effect. Suppose you invest $100 in this 6-mth note, thus
getting an interest of $2 plus your $100 initial investment in the next period.
Why is the short-term floating rate note always repriced to par on
interest repayment dates?
16
100
+2
remaining
periods
LIBOR observed
today is 4%.
Invest $100
today
Get back 100 +
2
Example: One period has passed. Suppose today’s 6-mth LIBOR is 6% pa. Now, you
roll-over your $100 investment in this new 6-mth note, thus getting an interest of $3
plus $100 initial investment in the next period.
Why is the short-term floating rate note always repriced to par on
interest repayment dates?
17
100
+3
remaining
periods0 1 2 3 4 5
LIBOR observed
today is 6%.
Continue
investing $100
Get back 100 +
3
You keep on rolling-over to the next 6-mth floating rate note, and every time,
you invest only your initial $100 investment.
That means that the 6-mth short-term floating rate note is always repriced to
par on interest repayment dates.
Why is the short-term floating rate note always repriced to par on
interest repayment dates?
18
Value a fixed coupon bond (discussed a few weeks ago)
Value a floating rate coupon bond with the same maturity and payment dates of the above
fixed coupon bond OR
Value a short-term floating rate note (e.g., 6-mth floating rate note) and roll-over
successively until the maturity of the above fixed coupon bond
The value of the floating rate coupon bond or floating rate note is always equal to the par value on
interest repayment dates
o But first, we need to fix the swap rate at t=0
The swap rate is set so that the value of the fixed coupon bond is equal to the value of the
floating rate coupon bond or short-term floating rate note, which is always equal to par (see
above). That is, set the swap rate so that the value of the fixed coupon bond is equal to par
When a coupon bond is priced on par, its coupon rate is equal to the yield
Thus, the swap rate at t =0 is equal to the yield at initiation of the swap agreement.
In summary, we can value an interest rate swap by aggregating the value of
each of the components in the portfolio:
19
Today, we reach an interest rate swap agreement in which we are the payer (we pay fixed and
receive floating)
The tenor of the swap is 3 years
Interest is repaid every 6 mths. The face value is $100
The current yield to maturity of a 3-yr fixed coupon bond is 6% p.a.
The current 6-mth LIBOR rate is 5% p.a.
Questions:
What is the swap rate? Note that this swap rate, once set today, will be fixed over the tenor of the
swap.
What is the current value of the swap?
Note:
The swap rate is set so that the value of the fixed coupon bond is equal to par
We don’t pay or receive anything at the initiation of the swap (recall that a swap is like a forward contract)
Example:
20
Swap timeline
21
remaining
periods
-3-3 -3-3 -3 -3
+2.5 ? ? ? ? ?
Only known at
t=1 (i.e., 6 mths
from today)
today
We pay a net
amt of $0.5 at
t=1
Fixed swap rate = 6% p.a.
The current 6-mth LIBOR rate is 5% p.a.
Example:
22
-3 -3
remaining
periods
-3 -3 -3
+4 ? ? ? ?
Only known at
t=1 (i.e., 6 mths
from today)
0 1 2 3 4 5
today
We receive a
net amt of $1 at
t=1
6 mths have passed.
The current yield to maturity of a 2.5-yr fixed
coupon bond is 7% p.a.
The current 6-mth LIBOR rate is 8% p.a.
Questions:
What is the current value of the swap?
Note:
The swap is valued at zero only if the market yield
is equal to the swap rate, which is not the case
over here.
The current yield to maturity of a 2.5-yr fixed coupon bond is 7% p.a.
The current 6-mth LIBOR rate is 8% p.a.
Step 1: Price the fixed side (i.e., price the “fixed coupon bond”)
Step 2: Price the floating side (i.e., price the “floating rate coupon bond”)
It is always repriced to par on interest repayment dates
Step 3: Get the net sum (note: We are the swap payer)
• -97.74 + 100 = $2.26
Example:
23
( )
( )
74.97
2
07.0
2
07.011
2
6
2
07.01
100
5.22
5.22 =
+−
+
+
=
×−
×P
If yield increases, the fixed coupon bond will trade at discount. The
effect is such that the swap has a positive value to fixed rate payer
Usually, the swap counterparties will complete the swap
Sometimes, we may want to terminate the swap. Two general ways:
1. Mutual termination: One party pays a fair price to another party and the swap is
terminated. In our example, since the swap is valued at +2.26 to us, the swap receiver
pays us $2.26
2. Offsetting contract: Enter into a new swap contract with another party to offset the
current (“old”) swap.
In offsetting our existing “old” contract (in which we are the payer)
We need to be the receiver of the new swap
The new swap’s tenor is 2.5 years
Recall that the swap rate is set so that the value of this new 2.5-yr fixed coupon
bond is equal to par
Since the current yield to maturity of the 2.5-yr fixed coupon bond is 7% p.a, the
swap rate of this new swap is 7% pa
Interpreting the swap value from another perspective
24
25
-3 -3
remaining
periods
-3 -3 -3
+4 ? ? ? ?
0 1 2 3 4 5
today
3.5 3.5
remaining
periods
-4 ? ? ? ?
0 1 2 3 4 5
3.5 3.5 3.5
Old swap
New swap
created to
offset the
old swap
• The floating interest that we pay
(under the new swap) always
offset the floating interest that we
receive (under the existing old
swap)
• Every 6 mths, the net effect of the
fixed interest is that we receive
$0.50. The present value of this
annuity is …
( )
?
2
07.0
2
07.011
50.0
5.22
=
+−
=
×−
PV
If you predicted (at the time when the old swap
was initiated 6 mths ago i.e., when the yield was
only 6%) that the yield would increase in the
future, you can speculate by being the payer of
this (old) swap
- Another 2 mths have passed
The existing swap still has 28 mths remaining
Assume that we didn’t terminate this swap 2 mths
ago
- The current yield to maturity of a 28-mth fixed
coupon bond is 9% p.a.
- Recall that, 2 mths ago, the 6-mth LIBOR rate was
8% p.a.
- Questions:
What is the current value of the swap?
Example:
26
-3 -3
remaining
periods
-3 -3 -3
+4 ? ? ? ?
today
27
( ) ( ) 1012/09.014100 6/4 =+×+= −P
Step 1: Price the fixed side (i.e., value the dirty (full) price of the “fixed coupon bond”)
Step 2: Price the floating side (i.e., price the “floating rate coupon bond”)
It is always repriced to par on interest repayment dates
But today is not an interest repayment date!
Solution: Add the next known floating interest ($4, which was determined based on the last LIBOR rate) to
the par value and then get the PV of this total, discounted for the period btw today and the next interest
repayment
Step 3: Get the net sum (note: We are the swap payer)
-94.80 + 101 = $6.20
( )
( )
( ) 80.942/09.0142.93
42.93
2
09.0
2
09.011
2
6
2
09.01
100
6/2
5.22
5.22
=+×=
=
+−
+
+
=
×−
×
FV
P
Me You
Currency swap
28
AUD120
USD100
Me You
Currency swap
29
AUD120
USD100
Eurodollar
Investors
Issued
Eurodollar
bond
USD100
Example: I have just issued a Eurodollar
bond on par. To minimize currency risk, and
given that my company is an Australian
company, I want to convert the borrowings
to AUD.
I could do a currency swap with you (swap’s
counterparty)
At t = 0
Me You
Currency swap
30
USD Interest
AUD Interest
Eurodollar
Investors
USD
Interest
Every six mths, I repay my interest in AUD,
and you repay your interest in USD.
On interest repayment dates
Me You
Currency swap
31
USD100
AUD120
Eurodollar
Investors
Pay bond’s
face value
of USD100
On maturity, we “swap” back the notional
amount.
At maturity
Every six mths, I repay fixed AUD rate and you repay fixed USD rate.
Every six mths, I repay floating AUD rate and you repay fixed USD rate.
Every six mths, I repay fixed AUD rate and you repay floating USD rate.
Every six mths, I repay floating AUD rate and you repay floating USD rate.
There are 4 general types of currency swap:
32
Today, we reach a currency swap agreement in which I pay AUD-LIBOR rate and
receive USD-LIBOR rate.
I lend you (the swap’s counterparty) USD100.
The tenor of the swap is 3 years
Interest is repaid every 6 mths.
The current exchange rate is AUD1.2/USD.
Questions:
How much am I borrowing? And in what currency?
What is the current value of the swap to me (and to you)?
Solution:
I borrow AUD120.
The swap value is 0 at the start of the agreement. Reason: I lend USD100, which is equal to AUD120
(after currency conversion). At the same time, I borrow AUD120.
Example for ‘floating-for-floating’ currency swap:
33
One year has passed and the swap now has 2 years remaining.
The current exchange rate is AUD1.30/USD
The current 6-mth AUD-LIBOR is 4.2% pa
The current 6-mth USD-LIBOR is 3.2% pa
Questions:
What is the current value of the swap ?
Solution:
The floating rate coupon bond is always repriced to par on interest repayment dates.
The AUD floating rate coupon bond is repriced to AUD120
The USD floating rate coupon bond is repriced to USD100, which is equal to AUD130
Note that I pay AUD interest and receive USD interest:
Swap value to me (in AUD) = -AUD120 + AUD130 =AUD10
Example:
34
Another 1 mth has passed and the swap now has 23 mths remaining.
The current exchange rate is AUD1.25/USD
On the last LIBOR reset date (i.e., last month), the 6-mth AUD-LIBOR was 4.2% pa
On the last LIBOR reset date (i.e., last month), the 6-mth USD-LIBOR was 3.2% pa
The current AUD yield to maturity is 5.0% pa
The current USD yield to maturity is 3.5% pa
Note that I pay AUD interest and receive USD interest
Questions: What is the current value of the swap ?
Example:
35
remaining
periods
-2.1%
? ?
today
+1.6%
AUD
USD
- I pay AUD interest = AUD120*0.021=AUD2.52 five mths from today
- Add the next known floating interest (AUD2.52, which was determined based on the last AUD-LIBOR rate) to the par
value and then get the PV of this total, discounted for the period btw today and the next interest repayment
- I receive USD interest = USD100*0.016=USD1.60 five months from today. Then, do the same as above:
- Swap value to me (in AUD) = -120.02 + 100.14*1.25 = AUD5.15
Example:
36
( ) ( ) 02.1202/05.0152.2120 6/5 AUDP =+×+= −
( ) ( ) 14.1002/035.0160.1100 6/5 USDP =+×+= −
Lecture 5: Interest rate swaps and currency swaps
• Swaps
• Interest rate swaps
-Mechanism
-Why use interest rate swaps?
-Pricing
• Currency swaps
-Mechanism
-Why use currency swaps?
-Pricing
Outline
2
•Agreements to exchange a series of cash flows on periodic dates
•Swaps are similar to forwards:
- Swaps typically do not require payment by either party at initiation (except currency
swaps)
- Swaps are traded OTC and not standardized
- Swaps are regulated by International Swaps and Derivatives Association (ISDA) in
the global market, and by Australian Financial Markets Association (AFMA) in
Australia
- Although regulations exist, default risk is high for swap transactions
•Two most common swaps:
- Interest rate swap
- Currency swap
Swaps
3
• An agreement to exchange fixed rate for floating (variable) rate over the tenor (i.e., life) of the
swap
- Suppose I lend you $100 at 4% pa fixed rate and you lend me $100 at variable (i.e., floating)
rate for 3 years. Interest is paid semi-annually.
- Thus we have an interest rate swap with a tenor of 3 yrs
- Both loan amounts are equal, hence it is pointless to exchange $100.
- Also, it is pointless to exchange the full interest amount every 6 mths. Only the party with the
larger payment liability pays the difference. Example: If the floating rate is 6% pa, I pay you
$1. If the floating rate is 3% pa, you pay me $0.50
- In the above, you are the fixed rate payer (i.e., ‘payer’) and I am the fixed rate receiver (i.e.,
‘receiver’). As a fixed rate receiver, I pay floating rate.
- We don’t usually use the terms ‘floating rate payer’ or ‘floating rate receiver’.
Interest rate swap
4
•To transform a liability
-Suppose company A borrowed at LIBOR floating rate + a margin of 50 bps.
Company A already has many variable rate borrowings and wanted to convert
the above into fixed rate borrowing.
-Suppose company B has just issued an 8% coupon bond, in addition to other
fixed rate borrowings. Company B wanted to convert the above to variable
rate borrowing.
-Hence, companies A and B can do an interest rate swap. Let’s assume the
fixed swap rate is 6%
Why do we use interest rate swap?
5
6Company B
Effective
borrowing rate
for Company A
= 6.5%
Effective
borrowing rate
for Company B
= LIBOR + 2%
Why do we use interest rate swap?
Variable Bank
Loan
Company A
LIBOR +
50 bp
Fixed Coupon
Bond
8% fixed
6% LIBOR
• Suppose company F holds floating rate notes issued in Australia. Thus, it receives BBSW
minus 30 bps margin. BBSW (Bank Bill Swap Rate) is analogous to LIBOR; it is the reference
rate at which variable rates in Australia are set upon. Company F wanted to convert its cash
inflow into fixed amount.
• Suppose company V holds 5.8% fixed coupon bonds and wanted to convert its cash inflow
into variable amount
• Thus, companies F and V can do an interest rate swap. Let’s assume the fixed swap rate is
6%.
To transform an asset
7
-Usually, companies F and V won’t directly transact the interest rate swap.
They will do this through a financial intermediary (FI)
- In the example below, FI earns a 4 bps spread but each of F and V receives 2
bps less.
To transform an asset
8
- If you predict that the yield will increase (price will decrease) in the future, you
can try to speculate on your prediction by
Sell bond futures (sell high now, close out by buy low later), or
Enter into an interest rate swap as a fixed-rate payer (pay fixed, receive
floating). Reason:
- It can be shown (later) that if yield increases, the fixed coupon bond will trade
at discount. The effect is such that the swap has a positive value to fixed rate
payer
To speculate:
9
- Suppose you were a liability manager and you have issued/sold a fixed coupon bond. To hedge the risk
of a falling yield (bond price or value of your liability increases), you can
Buy back the bond, or
Enter into an interest rate swap as a fixed-rate receiver (receive fixed, pay floating). Reason:
• It can be shown (later) that if yield decreases, the fixed coupon bond will trade at premium.
The effect is such that the swap has a positive value to fixed rate receiver. This positive swap
value offsets the increase in your liability zero net effect
• It can be shown (later) that if yield increases, the fixed coupon bond will trade at discount.
The effect is such that the swap has a negative value to fixed rate receiver. This negative
swap value offsets the decrease in your liability zero net effect
- Note also that a receive fixed, pay floating swap has the same effect as decreasing duration. This is
because this swap implies that you issue short-term floating rate notes (thus pay floating rate) and use
the proceeds to buy back long-term bonds (thus receive fixed rate)
To hedge:
10
•Suppose we are the fixed rate payer (we pay fixed rate and receive floating rate).
•We can replicate this effect using a clever combination of basic bonds:
- Issue fixed coupon bond
- Invest the proceeds in floating rate coupon bond with the same maturity and
payment dates as the above fixed coupon bond
- On each interest payment date, we pay fixed coupon and receive floating rate
- OR
- Issue fixed coupon bond
- Invest the proceeds in short-term floating rate notes (e.g., 6-mth floating rate note)
and roll-over successively until the maturity of the above fixed coupon bond
Replicating an interest rate swap
11
•Thus, we can value an interest rate swap by aggregating the value of each of the
components in the portfolio:
- Value a fixed coupon bond (discussed a few weeks ago)
- Value a floating rate coupon bond with the same maturity and payment dates as the
above fixed coupon bond
The value is equal to the par value on interest repayment dates
The value is greater than the par value on non-interest repayment dates
•Why is the floating rate coupon bond always repriced to par on interest repayment
dates?
- To understand this, remember first that the floating rate interest is always set in
advance, but paid at the next period
12
- The floating rate interest is always set in advance, but paid at the next period
- For a floating rate coupon bond, the coupon rate, which determines the coupon payment at
next periods, is set to the market yield.
- When coupon rate = yield, the bond is priced on par
- Example: Suppose the current YTM for the floating rate coupon bond = 4% pa. The coupon
rate of this bond will be set equal to this yield, and thus, today (t=0), the floating rate coupon
bond is priced at 100.
Why is the floating rate coupon bond always repriced to par on interest
repayment dates?
13
22 2 2 2
100+
2
remaining
periods
Coupon rate set
today (i.e., t=0)
but paid at
t=1,2,…
One period has passed. Suppose the current YTM for the floating rate coupon bond =
6% pa. The coupon rate of this bond will be reset (updated) equal to this yield, and
thus, today (t=0), the floating rate coupon bond is repriced to 100, again.
The above process is repeated on each repayment date i.e., coupon rate is reset (i.e.,
updated) to the yield of the floating rate bond on each repayment date, which means
that the floating rate coupon bond is always repriced to par
Why is the floating rate coupon bond always repriced to par on
interest repayment dates?
14
3 3 3 3
100 +
3
remaining
periods
Coupon rate set
today (i.e., t=0)
but paid at
t=1,2,…
0 1 2 3 4 5
o Note also we can value an interest rate swap by aggregating the value of
each of the components in the portfolio:
Value a fixed coupon bond
Value a short-term floating rate note (e.g., 6-mth floating rate note) and
roll-over successively until the maturity of the above fixed coupon bond
The value is equal to the par value on interest repayment dates
The value is greater than the par value on non-interest repayment dates
15
Example: Suppose today’s 6-mth LIBOR is 4% pa. LIBOR is a “simple rate” with
no compounding effect. Suppose you invest $100 in this 6-mth note, thus
getting an interest of $2 plus your $100 initial investment in the next period.
Why is the short-term floating rate note always repriced to par on
interest repayment dates?
16
100
+2
remaining
periods
LIBOR observed
today is 4%.
Invest $100
today
Get back 100 +
2
Example: One period has passed. Suppose today’s 6-mth LIBOR is 6% pa. Now, you
roll-over your $100 investment in this new 6-mth note, thus getting an interest of $3
plus $100 initial investment in the next period.
Why is the short-term floating rate note always repriced to par on
interest repayment dates?
17
100
+3
remaining
periods0 1 2 3 4 5
LIBOR observed
today is 6%.
Continue
investing $100
Get back 100 +
3
You keep on rolling-over to the next 6-mth floating rate note, and every time,
you invest only your initial $100 investment.
That means that the 6-mth short-term floating rate note is always repriced to
par on interest repayment dates.
Why is the short-term floating rate note always repriced to par on
interest repayment dates?
18
Value a fixed coupon bond (discussed a few weeks ago)
Value a floating rate coupon bond with the same maturity and payment dates of the above
fixed coupon bond OR
Value a short-term floating rate note (e.g., 6-mth floating rate note) and roll-over
successively until the maturity of the above fixed coupon bond
The value of the floating rate coupon bond or floating rate note is always equal to the par value on
interest repayment dates
o But first, we need to fix the swap rate at t=0
The swap rate is set so that the value of the fixed coupon bond is equal to the value of the
floating rate coupon bond or short-term floating rate note, which is always equal to par (see
above). That is, set the swap rate so that the value of the fixed coupon bond is equal to par
When a coupon bond is priced on par, its coupon rate is equal to the yield
Thus, the swap rate at t =0 is equal to the yield at initiation of the swap agreement.
In summary, we can value an interest rate swap by aggregating the value of
each of the components in the portfolio:
19
Today, we reach an interest rate swap agreement in which we are the payer (we pay fixed and
receive floating)
The tenor of the swap is 3 years
Interest is repaid every 6 mths. The face value is $100
The current yield to maturity of a 3-yr fixed coupon bond is 6% p.a.
The current 6-mth LIBOR rate is 5% p.a.
Questions:
What is the swap rate? Note that this swap rate, once set today, will be fixed over the tenor of the
swap.
What is the current value of the swap?
Note:
The swap rate is set so that the value of the fixed coupon bond is equal to par
We don’t pay or receive anything at the initiation of the swap (recall that a swap is like a forward contract)
Example:
20
Swap timeline
21
remaining
periods
-3-3 -3-3 -3 -3
+2.5 ? ? ? ? ?
Only known at
t=1 (i.e., 6 mths
from today)
today
We pay a net
amt of $0.5 at
t=1
Fixed swap rate = 6% p.a.
The current 6-mth LIBOR rate is 5% p.a.
Example:
22
-3 -3
remaining
periods
-3 -3 -3
+4 ? ? ? ?
Only known at
t=1 (i.e., 6 mths
from today)
0 1 2 3 4 5
today
We receive a
net amt of $1 at
t=1
6 mths have passed.
The current yield to maturity of a 2.5-yr fixed
coupon bond is 7% p.a.
The current 6-mth LIBOR rate is 8% p.a.
Questions:
What is the current value of the swap?
Note:
The swap is valued at zero only if the market yield
is equal to the swap rate, which is not the case
over here.
The current yield to maturity of a 2.5-yr fixed coupon bond is 7% p.a.
The current 6-mth LIBOR rate is 8% p.a.
Step 1: Price the fixed side (i.e., price the “fixed coupon bond”)
Step 2: Price the floating side (i.e., price the “floating rate coupon bond”)
It is always repriced to par on interest repayment dates
Step 3: Get the net sum (note: We are the swap payer)
• -97.74 + 100 = $2.26
Example:
23
( )
( )
74.97
2
07.0
2
07.011
2
6
2
07.01
100
5.22
5.22 =
+−
+
+
=
×−
×P
If yield increases, the fixed coupon bond will trade at discount. The
effect is such that the swap has a positive value to fixed rate payer
Usually, the swap counterparties will complete the swap
Sometimes, we may want to terminate the swap. Two general ways:
1. Mutual termination: One party pays a fair price to another party and the swap is
terminated. In our example, since the swap is valued at +2.26 to us, the swap receiver
pays us $2.26
2. Offsetting contract: Enter into a new swap contract with another party to offset the
current (“old”) swap.
In offsetting our existing “old” contract (in which we are the payer)
We need to be the receiver of the new swap
The new swap’s tenor is 2.5 years
Recall that the swap rate is set so that the value of this new 2.5-yr fixed coupon
bond is equal to par
Since the current yield to maturity of the 2.5-yr fixed coupon bond is 7% p.a, the
swap rate of this new swap is 7% pa
Interpreting the swap value from another perspective
24
25
-3 -3
remaining
periods
-3 -3 -3
+4 ? ? ? ?
0 1 2 3 4 5
today
3.5 3.5
remaining
periods
-4 ? ? ? ?
0 1 2 3 4 5
3.5 3.5 3.5
Old swap
New swap
created to
offset the
old swap
• The floating interest that we pay
(under the new swap) always
offset the floating interest that we
receive (under the existing old
swap)
• Every 6 mths, the net effect of the
fixed interest is that we receive
$0.50. The present value of this
annuity is …
( )
?
2
07.0
2
07.011
50.0
5.22
=
+−
=
×−
PV
If you predicted (at the time when the old swap
was initiated 6 mths ago i.e., when the yield was
only 6%) that the yield would increase in the
future, you can speculate by being the payer of
this (old) swap
- Another 2 mths have passed
The existing swap still has 28 mths remaining
Assume that we didn’t terminate this swap 2 mths
ago
- The current yield to maturity of a 28-mth fixed
coupon bond is 9% p.a.
- Recall that, 2 mths ago, the 6-mth LIBOR rate was
8% p.a.
- Questions:
What is the current value of the swap?
Example:
26
-3 -3
remaining
periods
-3 -3 -3
+4 ? ? ? ?
today
27
( ) ( ) 1012/09.014100 6/4 =+×+= −P
Step 1: Price the fixed side (i.e., value the dirty (full) price of the “fixed coupon bond”)
Step 2: Price the floating side (i.e., price the “floating rate coupon bond”)
It is always repriced to par on interest repayment dates
But today is not an interest repayment date!
Solution: Add the next known floating interest ($4, which was determined based on the last LIBOR rate) to
the par value and then get the PV of this total, discounted for the period btw today and the next interest
repayment
Step 3: Get the net sum (note: We are the swap payer)
-94.80 + 101 = $6.20
( )
( )
( ) 80.942/09.0142.93
42.93
2
09.0
2
09.011
2
6
2
09.01
100
6/2
5.22
5.22
=+×=
=
+−
+
+
=
×−
×
FV
P
Me You
Currency swap
28
AUD120
USD100
Me You
Currency swap
29
AUD120
USD100
Eurodollar
Investors
Issued
Eurodollar
bond
USD100
Example: I have just issued a Eurodollar
bond on par. To minimize currency risk, and
given that my company is an Australian
company, I want to convert the borrowings
to AUD.
I could do a currency swap with you (swap’s
counterparty)
At t = 0
Me You
Currency swap
30
USD Interest
AUD Interest
Eurodollar
Investors
USD
Interest
Every six mths, I repay my interest in AUD,
and you repay your interest in USD.
On interest repayment dates
Me You
Currency swap
31
USD100
AUD120
Eurodollar
Investors
Pay bond’s
face value
of USD100
On maturity, we “swap” back the notional
amount.
At maturity
Every six mths, I repay fixed AUD rate and you repay fixed USD rate.
Every six mths, I repay floating AUD rate and you repay fixed USD rate.
Every six mths, I repay fixed AUD rate and you repay floating USD rate.
Every six mths, I repay floating AUD rate and you repay floating USD rate.
There are 4 general types of currency swap:
32
Today, we reach a currency swap agreement in which I pay AUD-LIBOR rate and
receive USD-LIBOR rate.
I lend you (the swap’s counterparty) USD100.
The tenor of the swap is 3 years
Interest is repaid every 6 mths.
The current exchange rate is AUD1.2/USD.
Questions:
How much am I borrowing? And in what currency?
What is the current value of the swap to me (and to you)?
Solution:
I borrow AUD120.
The swap value is 0 at the start of the agreement. Reason: I lend USD100, which is equal to AUD120
(after currency conversion). At the same time, I borrow AUD120.
Example for ‘floating-for-floating’ currency swap:
33
One year has passed and the swap now has 2 years remaining.
The current exchange rate is AUD1.30/USD
The current 6-mth AUD-LIBOR is 4.2% pa
The current 6-mth USD-LIBOR is 3.2% pa
Questions:
What is the current value of the swap ?
Solution:
The floating rate coupon bond is always repriced to par on interest repayment dates.
The AUD floating rate coupon bond is repriced to AUD120
The USD floating rate coupon bond is repriced to USD100, which is equal to AUD130
Note that I pay AUD interest and receive USD interest:
Swap value to me (in AUD) = -AUD120 + AUD130 =AUD10
Example:
34
Another 1 mth has passed and the swap now has 23 mths remaining.
The current exchange rate is AUD1.25/USD
On the last LIBOR reset date (i.e., last month), the 6-mth AUD-LIBOR was 4.2% pa
On the last LIBOR reset date (i.e., last month), the 6-mth USD-LIBOR was 3.2% pa
The current AUD yield to maturity is 5.0% pa
The current USD yield to maturity is 3.5% pa
Note that I pay AUD interest and receive USD interest
Questions: What is the current value of the swap ?
Example:
35
remaining
periods
-2.1%
? ?
today
+1.6%
AUD
USD
- I pay AUD interest = AUD120*0.021=AUD2.52 five mths from today
- Add the next known floating interest (AUD2.52, which was determined based on the last AUD-LIBOR rate) to the par
value and then get the PV of this total, discounted for the period btw today and the next interest repayment
- I receive USD interest = USD100*0.016=USD1.60 five months from today. Then, do the same as above:
- Swap value to me (in AUD) = -120.02 + 100.14*1.25 = AUD5.15
Example:
36
( ) ( ) 02.1202/05.0152.2120 6/5 AUDP =+×+= −
( ) ( ) 14.1002/035.0160.1100 6/5 USDP =+×+= −
