Euribor rates, forward rates and swap rates
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Jukka Perttunen
University of Oulu - Department of Finance
Fall 2015
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
What next
In the following we consider
Euribor spot rate, Euribor forward rate and Euribor swap rate,
how periodic discount factors can be solved from the quoted interest rates,
how spot rates, forward rates and swap rates span each other.
Motivation
spot rates are not quoted for maturities longer than one year,
forward and swap rates must be used to determine periodic discount factors,
forward and swap rates are used in connection with interest rate derivatives.
We will find out that
spot rate is a single-period simple rate for an instant period,
forward rate is a single-period simple rate for a future period,
swap rate is a multi-period simple rate over several future periods,
the three rates always provide the same payoff in terms of present value.
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor rates
Euribor spot rate
offered simple spot rate of euro interbank term deposits,
quoted maturities: 1w, 2w, 1m, 2m, 3m, 6m, 9m, 12m,
begins +2 business days from the publishing,
ends at the corresponding calendar day (modified following),
quotation in terms of Actual/360 day count.
Euribor forward rate
implicit future period simple rate,
can be calculated for any future period on the basis of spot/swap rates.
Euribor swap rate
constant multi-period simple rate,
pays annually against floating 6m/3m Euribor rate,
quoted in the interdealer market,
maturities up to 30 years,
quotation in terms of 30/360 day count.
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor spot rate quotation
Interest period in Euribor spot rate quotation
length of the interest period varies from day to day,
actual number of days from the value date to the maturity date is used,
actual number of days is divided by 360.
Value date
the first day of the interest period,
the second business day (BD) following the quote date.
Maturity date
the last day of the interest period,
the target month date corresponding to the value date,
if not a BD, then replaced with the next BD in the target month,
if such is not available, then the last BD of the target month,
if value date is the last BD of a month, then the last BD of the target month.
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Interest rate quotations
Rate
Maturity
Euribor
6m
12m
6m forward
Quotation
Days
2.190
2.332
180
360
0.5y × 1.0y
2.447
180
1.0y × 1.5y
1.5y × 2.0y
2.0y × 2.5y
2.808
2.851
3.143
180
180
180
2.5y × 3.0y
3.0y × 3.5y
3.5y × 4.0y
3.193
3.395
3.456
180
180
180
4.0y × 4.5y
4.5y × 5.0y
3.620
3.688
180
180
2y/6m
3y/6m
4y/6m
2.587
2.783
2.943
2 × 360
3 × 360
4 × 360
5y/6m
...
30y/6m
3.082
5 × 360
4.050
30 × 360
...
Swap
For simplicity, each six-month period is assumed to be 180 days long.
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor rates and discount factors
2.190%
6m Euribor rate
d0.5 = 0.98917
2.332%
12m Euribor rate
d1.0 = 0.97721
2.513%
Implicit 18m Euribor rate
d1.5 = 0.96368
Implicit 24m Euribor rate
2.624%
d2.0 = 0.95014
Euribor-rates determine the corresponding market discount factors:
d0.5 =
d1.0 =
d1.5 =
d2.0 =
1
1 + 2.190% × 0.5
1
1 + 2.332% × 1.0
1
1 + 2.513% × 1.5
1
1 + 2.624% × 2.0
Jukka Perttunen
= 0.98917
= 0.97721
= 0.96368
= 0.95014
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor rates and discount factors – bond approach
2.190%
6m Euribor rate
d0.5 = 0.98917
12m Euribor rate
2.332%
d1.0 = 0.97721
2.513%
Implicit 18m Euribor rate
d1.5 = 0.96368
Implicit 24m Euribor rate
2.624%
d2.0 = 0.95014
An alternative approach is to consider a par-selling one-euro bond paying Euribor-rate:
1 = d0.5 × (1 + 2.190% × 0.5)
⇒
d0.5 = 0.98917
1 = d1.0 × (1 + 2.332% × 1.0)
⇒
d1.0 = 0.97721
1 = d1.5 × (1 + 2.513% × 1.5)
⇒
d1.5 = 0.96368
1 = d2.0 × (1 + 2.624% × 2.0)
⇒
d2.0 = 0.95014
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor forward rates
2.190%
6m Euribor rate
d0.5 = 0.98917
2.332%
12m Euribor rate
d1.0 = 0.97721
2.513%
Implicit 18m Euribor rate
d1.5 = 0.96368
2.624%
Implicit 24m Euribor rate
d2.0 = 0.95014
2.447%
6m forward rate (0.5 × 1.0)
2.808%
6m forward rate (1.0 × 1.5)
2.851%
6m forward rate (1.5 × 2.0)
(1 + 2.190% × 0.5)(1 + 2.447% × 0.5) = 1 + 2.332% × 1.0
(1 + 2.332% × 1.0)(1 + 2.808% × 0.5) = 1 + 2.513% × 1.5
(1 + 2.513% × 1.5)(1 + 2.851% × 0.5) = 1 + 2.624% × 2.0
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor forward rates – bond approach
2.190%
6m Euribor rate
d0.5 = 0.98917
2.332%
12m Euribor rate
d1.0 = 0.97721
2.513%
Implicit 18m Euribor rate
d1.5 = 0.96368
2.624%
Implicit 24m Euribor rate
d2.0 = 0.95014
2.190%
6m Euribor/forward rate
2.447%
-
2.808%
-
2.851%
-
-
1 = 0.98917 × (1 + 2.190% × 0.5)
1 = 0.98917 × 2.190% × 0.5 + 0.97721 × (1 + 2.447% × 0.5)
1 = 0.98917 × 2.190% × 0.5 + 0.97721 × 2.447% × 0.5 + 0.96368 × (1 + 2.808% × 0.5)
1 = 0.98917 × 2.190% × 0.5 + 0.97721 × 2.447% × 0.5 + 0.96368 × 2.808% × 0.5 + 0.95014 × (1 + 2.851% × 0.5)
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor swap rate
2.190%
6m Euribor rate
d0.5 = 0.98917
2.332%
12m Euribor rate
d1.0 = 0.97721
2.513%
Implicit 18m Euribor rate
d1.5 = 0.96368
2.624%
Implicit 24m Euribor rate
d2.0 = 0.95014
2.190%
6m Euribor/forward rate
2.447%
-
2.808%
-
2.851%
-
2.587%
2y swap rate
2.587%
-
-
1 = 0.98917 × 2.190% × 0.5 + 0.97721 × 2.447% × 0.5 + 0.96368 × 2.808% × 0.5 + 0.95014 × (1 + 2.851% × 0.5)
1 = 0.97721 × 2.587% × 1.0 + 0.95014 × (1 + 2.587% × 1.0)
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
What next
In the following we consider
Euribor forward rate agreements and Euribor swaps,
the use of interest rate derivatives for hedging against interest rate risk,
Motivation
forward rate agreements and swaps are primary tools for risk management,
it is essential to understand the mutual insurance mechanism of the two rates.
We will find out that
forward rate agreements and swaps insure each other in the derivatives market,
forward rates can be considered risk-neutral expectations of future spot rates.
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor Forward Rate Agreements and Swaps
Euribor Forward Rate Agreement
non-standardized over-the-counter product,
single-period contract,
pays fixed forward rate against floating Euribor rate,
buyer pays a fixed forward rate and receives floating rate,
seller pays floating Euribor rate and receives a fixed forward rate,
cash settlement at the beginning of the interest period.
Euribor swap
over-the-counter product quoted in the interdealer market,
multi-period contract,
pays fixed swap rate against floating Euribor rate,
payer pays fixed swap rate annually and receives floating rate semiannually,
receiver pays floating rate semiannually and receives fixed swap rate annually,
cash settlement at the end of each interest period.
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor Forward Rate Agreement
Firm A
Bank B
−(r + 1.00%)
+(r + 2.00%)
e
6
er + 2.00%
Situation:
er + 1.00%
?
Bank A
+(r + 1.00%)
e
e
Firm A is to pay interest on 1 million
- over a six-month period one year from now
- 6m Euribor + 1% premium
- wants to hedge against a rise in 6mE
Bank B is to receive interest on 1 million
- over a six-month period one year from now
- 6m Euribor + 2% premium
- wants to hedge against a decline in 6mE
Jukka Perttunen
Fundamentals of Finance
Firm B
−(r + 2.00%)
e
Euribor rates, forward rates and swap rates
Euribor Forward Rate Agreement
Dealer
Firm A
−(r + 1.00%)
= −3.808%
−r + r
er
e e
=0
+(r + 2.00%)
e
+2.808%
−r
er
e
= 4.808%
6
er + 2.00%
Situation:
er + 1.00%
?
Bank A
+(r + 1.00%)
e
+2.808%−2.808%
e
2.808%
-
−2.808%
+r
Bank B
2.808%
e
Firm A is to pay interest on 1 million
- over a six-month period one year from now
- 6m Euribor + 1% premium
- wants to hedge against a rise in 6mE
Bank B is to receive interest on 1 million
- over a six-month period one year from now
- 6m Euribor + 2% premium
- wants to hedge against a decline in 6mE
Jukka Perttunen
Fundamentals of Finance
Firm B
−(r + 2.00%)
e
Euribor rates, forward rates and swap rates
Euribor swap
Firm A
Firm B
−(r + 1.00%)
−4.00%
e
Situation:
er + 1.00%
?
Bank
+(r + 1.00%)
e
4.00%
Firm A is paying interest on 1 million
- over several six-month periods
- 6M Euribor + 1% premium
- prefers paying a fixed rate
Firm B is paying interest on 1 million
- over several six-month periods
- bond with a 4% semiannual coupon
- prefers paying a floating rate
Jukka Perttunen
Fundamentals of Finance
?
Bondholders
+4.00%
Euribor rates, forward rates and swap rates
Euribor swap
Dealer
Firm A
−(r + 1.00%)
= −3.587%
−r + r
er
e e
+2.587%
−r
er
=0
e
= −(r + 1.413%)
e
Situation:
er + 1.00%
?
Bank
+(r + 1.00%)
e
−4.00%
+2.587%−2.587%
e
2.587%
-
−2.587%
+r
Firm B
2.587%
e
4.00%
Firm A is paying interest on 1 million
- over several six-month periods
- 6M Euribor + 1% premium
- prefers paying a fixed rate
Firm B is paying interest on 1 million
- over several six-month periods
- bond with a 4% semiannual coupon
- prefers paying a floating rate
Jukka Perttunen
Fundamentals of Finance
?
Bondholders
+4.00%
Euribor rates, forward rates and swap rates
Neutralizing interest rate risk – Investor A
Investor A borrowing for two years with a reference to the six-month Euribor
−(Ẽ + 1%)
−(2.190% + 1%)
Euribor loan
0.0
−(Ẽ + 1%)
-
0.5
−(Ẽ + 1%)
-
1.0
-
1.5
2.0
Investor A with an interest rate swap
−(2.190% + 1%)
Euribor loan
−(Ẽ + 1%)
-
−(Ẽ + 1%)
−2.587%
2y/6m swap fixed leg
−2.587%
-
n
+2.190%
2y/6m swap floating leg
−(Ẽ + 1%)
-
-
+Ẽ
-
+Ẽ
-
+Ẽ
-
-
Investor A net interest position
−2.587%
Net position
−2.587%
−1%
-
Jukka Perttunen
−1%
−1%
-
Fundamentals of Finance
-
−1%
-
Euribor rates, forward rates and swap rates
Neutralizing interest rate risk – Investor B
Investor B lending for a six-month future period, with a reference to the six-month Euribor
+(Ẽ + 1%)
Euribor loan
0.5
1.0
Investor B with a forward rate agreement (FRA)
+(Ẽ + 1%)
Euribor loan
+2.447%
0.5 × 1.0 FRA fixed payment
-
n
−Ẽ
0.5 × 1.0 FRA floating payment
-
Investor B net interest position
+2.447%
-
Net position
+1%
-
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Neutralizing interest rate risk – Investor C
Investor C lending for a six-month future period, with a reference to the six-month Euribor
+(Ẽ + 1%)
Euribor loan
1.0
1.5
Investor C with a forward rate agreement (FRA)
+(Ẽ + 1%)
Euribor loan
+2.808%
1.0 × 1.5 FRA fixed payment
-
n
−Ẽ
1.0 × 1.5 FRA floating payment
-
Investor C net interest position
+2.808%
-
Net position
+1%
-
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Neutralizing interest rate risk – Investor D
Investor D lending for a six-month future period, with a reference to the six-month Euribor
+(Ẽ + 1%)
Euribor loan
1.5
2.0
Investor D with a forward rate agreement (FRA)
+(Ẽ + 1%)
Euribor loan
+2.851%
1.5 × 2.0 FRA fixed payment
-
n
−Ẽ
1.5 × 2.0 FRA floating payment
-
Investor D net interest position
+2.851%
-
Net position
+1%
-
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Neutralizing interest rate risk – Market level
Market participants’ interest rate positions in derivatives market
−2.587%
2y/6m swap fixed leg
−2.587%
-
n
+2.190%
2y/6m swap floating leg
-
+Ẽ
-
+Ẽ
-
+Ẽ
-
-
−Ẽ
-
n
0.5 × 1.0 forward rate agreement
+2.447%
n
1.0 × 1.5 forward rate agreement
−Ẽ
+2.808%
n
1.5 × 2.0 forward rate agreement
−Ẽ
+2.851%
Net interest rate positions in derivatives market
−2.587%
2y swap rate
6m Euribor/forward rate
−2.587%
+2.190%
+2.447%
-
Jukka Perttunen
+2.808%
-
Fundamentals of Finance
+2.851%
-
-
Euribor rates, forward rates and swap rates
Risk-neutral interest rate positions
Net interest rate positions in the derivatives market
−2.587%
2y swap rate
−2.587%
+2.190%
6m Euribor/forward rate
+2.447%
-
+2.808%
-
|
+2.851%
-
-
{z
}
Derivatives market determines the levels of forward and swap rates
so that the rates insure each other by the different derivative contracts.
Forward rates and future spot rates
+2.190%
6m Euribor/forward rate
+2.447%
-
+2.808%
−Ẽ
Future 6m Euribor rate
−Ẽ
|
+2.851%
{z
Forward rates can be considered as risk-neutral
expectations of future periods’ Euribor spot rates.
Jukka Perttunen
Fundamentals of Finance
−Ẽ
}
Euribor rates, forward rates and swap rates
What next
In the following we consider
calculation of continuously compounded discount and forward rates,
the effect of a parallel change in continuously compounded discount rates.
Motivation
continuously compounded rates are easier to analyze than simple rates,
interest rate risk is typically analyzed in terms of a parallel interest rate change.
We will find out that
forward rates change by the same amount as the discount rates.
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Euribor rates and discount factors
Maturity
Euribor
T /6m swap
6m forward
Discount factor
T
rs
ss
fs
dT
0.5
1.0
1.5
2.190
2.332
2.447
2.808
0.98917
0.97721
0.96368
2.783
2.851
3.143
3.193
0.95014
0.93544
0.92074
2.943
3.395
3.456
0.90537
0.88999
3.082
3.620
3.688
0.87417
0.85834
2.0
2.5
3.0
3.5
4.0
4.5
5.0
...
2.587
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Calculations of continuously compounded rates
Continuously compounded discount rate
discount rate can be solved from the corresponding discount factor.
dT = e
−rT
⇒
r =−
1
T
ln dT
Continuously compounded forward rate of a period of ∆t
spot rate rT −∆t together with the forward rate f equals to the spot rate of rT .
r
(T −∆t) f ∆t
r T
e T −∆t
e
=eT
⇒
f =
rT T − rT −∆t (T − ∆t)
∆t
Forward rate change in a case of a parallel shift in spot rates
both spot rates, rT −∆t and rT , rise by ∆r percentage units.
f
∗
=
(rT + ∆r )T − (rT −∆t + ∆r )(T − ∆t)
∆t
Jukka Perttunen
= f + ∆r
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Continuously compounded rates
Maturity
T
Euribor
rs
0.5
2.190
1.0
1.5
2.332
2.0
2.5
3.0
3.5
4.0
4.5
5.0
...
T /6m swap
ss
6m forward
fs
Discount factor
dT
0.98917
2.1778
2.447
2.808
0.97721
0.96368
2.3054
2.4664
2.4330
2.7884
2.851
3.143
3.193
0.95014
0.93544
0.92074
2.5573
2.6695
2.7526
2.8300
3.1183
3.1681
2.943
3.395
3.456
3.620
0.90537
0.88999
0.87417
2.8403
2.9136
2.9885
3.3665
3.4267
3.5877
3.082
3.688
0.85834
3.0551
3.6545
2.587
2.783
rc = −
1
T
ln dT
fc =
Discount rate
rc
Forward rate
fc
rcT T − rcT −0.5 (T − 0.5)
0.5
Continuously compounded discount rate rc corresponds to the Euribor spot rate rs within numercial precision!
Continuously compounded forward rate fc corresponds to the 6m Euribor forward rate fs within numercial precision!
Jukka Perttunen
Fundamentals of Finance
Euribor rates, forward rates and swap rates
Change in continuously compounded discount rate
Initial situation
Scenario with one percentage unit rise
Maturity
Discount rate
Forward rate
6m forward
Discount rate
Forward rate
6m forward
T
rc
fc
fs
rc∗
fc∗
fs∗
0.5
1.0
1.5
2.1778
2.3054
2.4664
2.1778
2.4330
2.7884
2.190
2.447
2.808
3.1778
3.3054
3.4664
3.1778
3.4330
3.7884
3.203
3.463
3.825
2.0
2.5
3.0
2.5573
2.6695
2.7526
2.8300
3.1183
3.1681
2.851
2.143
3.193
3.5573
3.6695
3.7526
3.8300
4.1183
4.1681
3.867
4.161
4.212
3.5
4.0
2.8403
2.9136
3.3665
3.4267
3.395
3.456
3.8403
3.9136
4.3665
4.4267
4.415
4.476
4.5
5.0
...
2.9885
3.0551
3.5877
3.6545
3.620
3.688
3.9885
4.0551
4.5877
4.6545
4.641
4.709
∗
rc = rc + ∆r = rc + 1%
∗
fc = fc + ∆r = fc + 1%
Jukka Perttunen
∗
fs =
Fundamentals of Finance
1
0.5
f ∗T
(e c
− 1)