Euribor rates, forward rates and swap rates University of Oulu - Department of Finance Fall 2015
What next Euribor rates, forward rates and swap rates 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.
Euribor rates Euribor rates, forward rates and swap 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.
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.
Interest rate quotations Rate Maturity Quotation Days Euribor 6m 2.190 180 12m 2.332 360 6m forward 0.5y 1.0y 2.447 180 1.0y 1.5y 2.808 180 1.5y 2.0y 2.851 180 2.0y 2.5y 3.143 180 2.5y 3.0y 3.193 180 3.0y 3.5y 3.395 180 3.5y 4.0y 3.456 180 4.0y 4.5y 3.620 180 4.5y 5.0y 3.688 180... Swap 2y/6m 2.587 2 360 3y/6m 2.783 3 360 4y/6m 2.943 4 360 5y/6m 3.082 5 360... 30y/6m 4.050 30 360 For simplicity, each six-month period is assumed to be 180 days long.
Euribor rates and discount factors 6m Euribor rate 12m Euribor rate Implicit 18m Euribor rate Implicit 24m Euribor rate 2.190% d 0.5 = 0.98917 2.332% d 1.0 = 0.97721 2.513% d 1.5 = 0.96368 2.624% d 2.0 = 0.95014 Euribor-rates determine the corresponding market discount factors: d 0.5 = d 1.0 = d 1.5 = d 2.0 = 1 1 + 2.190% 0.5 = 0.98917 1 1 + 2.332% 1.0 = 0.97721 1 1 + 2.513% 1.5 = 0.96368 1 1 + 2.624% 2.0 = 0.95014
Euribor rates and discount factors bond approach 6m Euribor rate 12m Euribor rate Implicit 18m Euribor rate Implicit 24m Euribor rate 2.190% d 0.5 = 0.98917 2.332% d 1.0 = 0.97721 2.513% d 1.5 = 0.96368 2.624% d 2.0 = 0.95014 An alternative approach is to consider a par-selling one-euro bond paying Euribor-rate: 1 = d 0.5 (1 + 2.190% 0.5) d 0.5 = 0.98917 1 = d 1.0 (1 + 2.332% 1.0) d 1.0 = 0.97721 1 = d 1.5 (1 + 2.513% 1.5) d 1.5 = 0.96368 1 = d 2.0 (1 + 2.624% 2.0) d 2.0 = 0.95014
Euribor forward rates 6m Euribor rate 12m Euribor rate Implicit 18m Euribor rate Implicit 24m Euribor rate 6m forward rate (0.5 1.0) 2.190% d 0.5 = 0.98917 2.332% d 1.0 = 0.97721 2.447% 2.513% d 1.5 = 0.96368 2.624% d 2.0 = 0.95014 6m forward rate (1.0 1.5) 2.808% 6m forward rate (1.5 2.0) 2.851% (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
Euribor forward rates bond approach 6m Euribor rate 12m Euribor rate Implicit 18m Euribor rate Implicit 24m Euribor rate 2.190% d 0.5 = 0.98917 2.332% d 1.0 = 0.97721 2.513% d 1.5 = 0.96368 2.624% d 2.0 = 0.95014 6m Euribor/forward rate 2.190% 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)
Euribor swap rate 6m Euribor rate 12m Euribor rate Implicit 18m Euribor rate Implicit 24m Euribor rate 2.190% d 0.5 = 0.98917 2.332% d 1.0 = 0.97721 2.513% d 1.5 = 0.96368 2.624% d 2.0 = 0.95014 6m Euribor/forward rate 2.190% 2.447% 2.808% 2.851% 2y swap rate 2.587% 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)
What next Euribor rates, forward rates and swap rates 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.
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.
Euribor Forward Rate Agreement Firm A ( r + 1.00%) Bank B +( r + 2.00%) r + 1.00% Situation: r + 2.00% Bank A +( r + 1.00%) 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 Firm B ( r + 2.00%)
Euribor Forward Rate Agreement Firm A ( r + 1.00%) 2.808% + r = 3.808% Dealer 2.808% 2.808% +2.808% 2.808% r + r r = 0 r Bank B +( r + 2.00%) +2.808% r = 4.808% r + 1.00% Situation: r + 2.00% Bank A +( r + 1.00%) 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 Firm B ( r + 2.00%)
Euribor swap Euribor rates, forward rates and swap rates Firm A ( r + 1.00%) Firm B 4.00% r + 1.00% Situation: 4.00% Bank +( r + 1.00%) 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 Bondholders +4.00%
Euribor swap Euribor rates, forward rates and swap rates Firm A ( r + 1.00%) 2.587% + r = 3.587% Dealer 2.587% 2.587% +2.587% 2.587% r + r r = 0 r Firm B 4.00% +2.587% r = ( r + 1.413%) r + 1.00% Situation: 4.00% Bank +( r + 1.00%) 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 Bondholders +4.00%
Neutralizing interest rate risk Investor A Investor A borrowing for two years with a reference to the six-month Euribor Euribor loan (2.190% + 1%) (Ẽ + 1%) (Ẽ + 1%) (Ẽ + 1%) 0.0 0.5 1.0 1.5 2.0 Investor A with an interest rate swap Euribor loan 2y/6m swap fixed leg 2y/6m swap floating leg (2.190% + 1%) (Ẽ + 1%) { 2.587% +2.190% +Ẽ (Ẽ + 1%) +Ẽ (Ẽ + 1%) 2.587% +Ẽ Investor A net interest position Net position 1% 2.587% 1% 1% 2.587% 1%
Neutralizing interest rate risk Investor B Investor B lending for a six-month future period, with a reference to the six-month Euribor Euribor loan +(Ẽ + 1%) 0.5 1.0 Investor B with a forward rate agreement (FRA) Euribor loan 0.5 1.0 FRA fixed payment 0.5 1.0 FRA floating payment { +(Ẽ + 1%) +2.447% Ẽ Investor B net interest position Net position +2.447% +1%
Neutralizing interest rate risk Investor C Investor C lending for a six-month future period, with a reference to the six-month Euribor Euribor loan +(Ẽ + 1%) 1.0 1.5 Investor C with a forward rate agreement (FRA) Euribor loan 1.0 1.5 FRA fixed payment 1.0 1.5 FRA floating payment { +(Ẽ + 1%) +2.808% Ẽ Investor C net interest position Net position +2.808% +1%
Neutralizing interest rate risk Investor D Investor D lending for a six-month future period, with a reference to the six-month Euribor Euribor loan +(Ẽ + 1%) 1.5 2.0 Investor D with a forward rate agreement (FRA) Euribor loan 1.5 2.0 FRA fixed payment 1.5 2.0 FRA floating payment { +(Ẽ + 1%) +2.851% Ẽ Investor D net interest position Net position +2.851% +1%
Neutralizing interest rate risk Market level Market participants interest rate positions in derivatives market 2y/6m swap fixed leg 2y/6m swap floating leg { 2.587% +2.190% +Ẽ +Ẽ 2.587% +Ẽ 0.5 1.0 forward rate agreement 1.0 1.5 forward rate agreement 1.5 2.0 forward rate agreement { Ẽ +2.447% { Ẽ +2.808% { Ẽ +2.851% Net interest rate positions in derivatives market 2y swap rate 6m Euribor/forward rate +2.190% 2.587% +2.447% +2.808% 2.587% +2.851%
Risk-neutral interest rate positions Net interest rate positions in the derivatives market 2y swap rate 6m Euribor/forward rate +2.190% 2.587% +2.447% +2.808% 2.587% +2.851% } {{ } 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 6m Euribor/forward rate Future 6m Euribor rate +2.190% +2.447% Ẽ +2.808% Ẽ +2.851% Ẽ } {{ } Forward rates can be considered as risk-neutral expectations of future periods Euribor spot rates.
What next Euribor rates, forward rates and swap rates 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.
Euribor rates and discount factors Maturity Euribor T /6m swap 6m forward Discount factor T r s s s f s d T 0.5 2.190 0.98917 1.0 2.332 2.447 0.97721 1.5 2.808 0.96368 2.0 2.587 2.851 0.95014 2.5 3.143 0.93544 3.0 2.783 3.193 0.92074 3.5 3.395 0.90537 4.0 2.943 3.456 0.88999 4.5 3.620 0.87417 5.0 3.082 3.688 0.85834...
Calculations of continuously compounded rates Continuously compounded discount rate discount rate can be solved from the corresponding discount factor. d T = e rt r = 1 T ln d T Continuously compounded forward rate of a period of t spot rate r T t together with the forward rate f equals to the spot rate of r T. e r T t (T t) e f t = e r T T r T T r T t (T t) f = t Forward rate change in a case of a parallel shift in spot rates both spot rates, r T t and r T, rise by r percentage units. f (r T + r)t (r T t + r)(t t) = = f + r t
Continuously compounded rates Maturity Euribor T /6m swap 6m forward Discount factor Discount rate Forward rate T r s s s f s d T r c f c 0.5 2.190 0.98917 2.1778 1.0 2.332 2.447 0.97721 2.3054 2.4330 1.5 2.808 0.96368 2.4664 2.7884 2.0 2.587 2.851 0.95014 2.5573 2.8300 2.5 3.143 0.93544 2.6695 3.1183 3.0 2.783 3.193 0.92074 2.7526 3.1681 3.5 3.395 0.90537 2.8403 3.3665 4.0 2.943 3.456 0.88999 2.9136 3.4267 4.5 3.620 0.87417 2.9885 3.5877 5.0 3.082 3.688 0.85834 3.0551 3.6545... r c = 1 r ct T r ct T ln d 0.5 (T 0.5) T f c = 0.5 Continuously compounded discount rate r c corresponds to the Euribor spot rate r s within numercial precision! Continuously compounded forward rate f c corresponds to the 6m Euribor forward rate f s within numercial precision!
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 r c f c f s rc fc fs 0.5 2.1778 2.1778 2.190 3.1778 3.1778 3.203 1.0 2.3054 2.4330 2.447 3.3054 3.4330 3.463 1.5 2.4664 2.7884 2.808 3.4664 3.7884 3.825 2.0 2.5573 2.8300 2.851 3.5573 3.8300 3.867 2.5 2.6695 3.1183 2.143 3.6695 4.1183 4.161 3.0 2.7526 3.1681 3.193 3.7526 4.1681 4.212 3.5 2.8403 3.3665 3.395 3.8403 4.3665 4.415 4.0 2.9136 3.4267 3.456 3.9136 4.4267 4.476 4.5 2.9885 3.5877 3.620 3.9885 4.5877 4.641 5.0 3.0551 3.6545 3.688 4.0551 4.6545 4.709... r c = rc + r = rc + 1% fc = f c + r = f c + 1% f s = 1 0.5 (ef c T 1)