Let us assume our discounting curve is flat in 4% (continuous). We further assum
ID: 2764839 • Letter: L
Question
Let us assume our discounting curve is flat in 4% (continuous). We further assume that our Swap-Curve for 3-month Tenor is defined as:
Time Rate (continuous)
1 4.5%
2 4.7%
3 4.9%
4 5%
5 5.05%
We also assume that our Swap-Curve 6-month Tenor is defined as:
Time Rate (continuous)
1 4.75%
2 4.87%
3 5.02%
4 5.15%
5 5.25%
Let us assume we have two 5-years interest rate swaps. Where the fixed leg is paid annually (no day-correction and 30E/360 – day-convention). On one of the swaps the floating leg is paid quarterly and on the other swap it is paid semi-annually. Calculate the Swap-Rate on a 5-year Tenor-Basis Swap (where the Tenor Basis Swap is defined as two swaps with the same frequency fixed leg)
Explanation / Answer
the floating leg is paid quarterly,fc(t)=(rc(t)*t-rc(t-.25)*(t-.25))/.25 fixed leg is paid annually,obtain in between continous rates through interpolation Rate (continuous) Forward rate Rate (quarterly) floating pay 3m Time(t) rc(t) fc(t) (rq(t)=4*(exp(fc(t-1)/4)-1) fl1=rq(t)/4 0 4.0000% 4% 0.25 4.1250% 4.13% 4.02% 0.010050167 0.5 4.2500% 4.38% 4.15% 0.010365857 0.75 4.3750% 4.63% 4.40% 0.010997533 1 4.5000% 4.88% 4.65% 0.011629604 1.25 4.5500% 4.75% 4.90% 0.01226207 1.5 4.6000% 4.85% 4.78% 0.011945788 1.75 4.6500% 4.95% 4.88% 0.012198806 2 4.7000% 5.05% 4.98% 0.012451887 2.25 4.7500% 5.15% 5.08% 0.012705032 2.5 4.8000% 5.25% 5.18% 0.01295824 2.75 4.8500% 5.35% 5.28% 0.013211511 3 4.9000% 5.45% 5.39% 0.013464845 3.25 4.9250% 5.22% 5.49% 0.013718243 3.5 4.9500% 5.28% 5.26% 0.013148187 3.75 4.9750% 5.33% 5.31% 0.013274839 4 5.0000% 5.38% 5.36% 0.013401506 4.25 5.0125% 5.21% 5.41% 0.013528189 4.5 5.0250% 5.24% 5.25% 0.013116527 4.75 5.0375% 5.26% 5.27% 0.013179849 5 5.0500% 5.29% 5.30% 0.013243174 Total PV(fl3m) 0.250851853 the floating leg is paid semiannualy,fc(t)=(rc(t)*t-rc(t-.5)*(t-.5))/.5 fixed leg is paid annually,obtain in between continous rates through interpolation(mid) Rate (continuous) Forward rate Rate semiannualy floating pay 3m Time(t) rc(t) fc(t) (rs=2*(exp(fc(t-1)/2)-1) fl1=rs/2 0 4% 4% 0.5 4.38% 4.38% 4.04% 0.02020134 1 4.75% 5.13% 4.42% 0.022116012 1.5 4.81% 4.93% 5.19% 0.025956143 2 4.87% 5.05% 4.99% 0.024956323 2.5 4.95% 5.25% 5.11% 0.025571481 3 5.02% 5.40% 5.31% 0.026571901 3.5 5.09% 5.48% 5.47% 0.027342119 4 5% 5.61% 5.55% 0.027753138 4.5 5.20% 5.60% 5.68% 0.028421395 5 5.25% 5.70% 5.68% 0.028395684 Total PV(fl6m) 0.257285536 difference in the PVs of the floating legs(fl6m - fl3m) 0.006433683 PV of fixed swap rate pays=s*(exp(-.04)+exp(-2*.04)+exp(-3*.04)+exp(-4*.04)+exp(-5*.04) PV of fixed swap rate pays=s*(1-exp(-5*.04))/(1-exp(-.04)=PV= 0.006433683 PV of fixed swap rate pays=s*(1-exp(-5*.04))/(1-exp(-.04)= 0.006433683 A=(1-exp(-5*.04))/(1-exp(-.04)= 4.622970011 s=PV/A 0.1392% the Swap-Rate on a 5-year Tenor-Basis Swap is .1392% or 13.92 bps
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