Forward Rate

[hellohi]

hello @David Harper CFA FRM​

In financial markets and products topic practise question in official GARP book, they ask the following question:

below is a table of term structure of swap rates:

Maturity in Years / Swap Rate
1 / 2.5%
2 / 3.0%
3 / 3.5%
4 / 4.0%
5 / 4.5%


The two year forward swap rate starting in three years is closest to:
A. 3.5%
B. 4.5%
C. 5.51%
D. 6.02%

The answer was D

may you give a simple solving way to get this answer? or more than one solving way?

thanks

 

[Mkaim]

hello @David Harper CFA FRM​

In financial markets and products topic practise question in official GARP book, they ask the following question:

below is a table of term structure of swap rates:

Maturity in Years / Swap Rate
1 / 2.5%
2 / 3.0%
3 / 3.5%
4 / 4.0%
5 / 4.5%


The two year forward swap rate starting in three years is closest to:
A. 3.5%
B. 4.5%
C. 5.51%
D. 6.02%

The answer was D

may you give a simple solving way to get this answer? or more than one solving way?

thanks

May you tell us how you would go about solving it? And where you would have challenges we can assist if possible.

 

[hellohi]

May you tell us how you would go about solving it? And where you would have challenges we can assist if possible.

I solve it as I know and my result was exactily as answer C but in the book it solved like this:
[(1.045^5/(1.035^3))^(1/2)] = 6.02%
I did not get the concept behind this equation
thanks

 

[ShaktiRathore]

Hi,
You can think of nullifying arbitrage opportunity as is common in finance,
You can enter into a 3 year swap with receiving fixed rate of 3.5% at the same time enter into FRA 3 years into the future with pay on two years(yr 3-5) receiving the fixed swap rate at x%( two year forward swap rate ), you can at the same time enter into a 5 year swap with paying fixed rate of 4.5%.
Thus net arbitrage profit =((1.035)^3*(1+x%)^2-1.045^5)*NP
For no arbitrage, net arbitrage profit =0 =>((1.035)^3*(1+x%)^2-1.045^5)*NP=0
=> (1.035)^3*(1+x%)^2=1.045^5
=>(1+x%)^2=1.045^5/(1.035)^3
=> x%=(1.045^5/1.035^3)^0.5 - 1
=> x%=(1.246181/1.108717875)^0.5 - 1
=> x%=(1.12398386)^0.5 - 1=1.06018-1=.06018=6.02%

or you can convert the discrete rates to continuous as 3 / 3.5%=ln(1+0.035)=0.0344 and 5 / 4.5%=ln(1+0.045)=0.044
Thus the continuous two year forward swap rate =(0.044*5-0.0344*3)/2=0.0584 is converted to discrete as exp(0.0584/1)-1=6.02%
thanks

 

[hellohi]

Hi,
You can think of nullifying arbitrage opportunity as is common in finance,
You can enter into a 3 year swap with receiving fixed rate of 3.5% at the same time enter into FRA 3 years into the future with pay on two years(yr 3-5) receiving the fixed swap rate at x%( two year forward swap rate ), you can at the same time enter into a 5 year swap with paying fixed rate of 4.5%.
Thus net arbitrage profit =((1.035)^3*(1+x%)^2-1.045^5)*NP
For no arbitrage, net arbitrage profit =0 =>((1.035)^3*(1+x%)^2-1.045^5)*NP=0
=> (1.035)^3*(1+x%)^2=1.045^5
=>(1+x%)^2=1.045^5/(1.035)^3
=> x%=(1.045^5/1.035^3)^0.5 - 1
=> x%=(1.246181/1.108717875)^0.5 - 1
=> x%=(1.12398386)^0.5 - 1=1.06018-1=.06018=6.02%

or you can convert the discrete rates to continuous as 3 / 3.5%=ln(1+0.035)=0.0344 and 5 / 4.5%=ln(1+0.045)=0.044
Thus the continuous two year forward swap rate =(0.044*5-0.0344*3)/2=0.0584 is converted to discrete as exp(0.0584/1)-1=6.02%
thanks

thank you dear @ShaktiRathore for help

I just want to show you my solving way, I found it easy to do:

I calculate forward interest rates from a spot rates
using this equation Rf = R2*T2 - R1*T1/R2-R1
.045*5 - .035*3 / 5-3 = .06

and then convert it to discrete compounding
m(e^rc/m - 1)
1(e^.06/1 - 1) = .0618


what is your opinion?

thanks

 

[Deepak Chitnis]

Hi @hellohi , I was also liked the indirect way as your's. @ShaktiRathore way is more easy and direct. You can use both!

 

[hellohi]

Hi @hellohi , I was also liked the indirect way as your's. @ShaktiRathore way is more easy and direct. You can use both!

thanks a lot dear @Deepak Chitnis

 

[ShaktiRathore]

thank you dear @ShaktiRathore for help

I just want to show you my solving way, I found it easy to do:

I calculate forward interest rates from a spot rates
using this equation Rf = R2*T2 - R1*T1/R2-R1
.045*5 - .035*3 / 5-3 = .06

and then convert it to discrete compounding
m(e^rc/m - 1)
1(e^.06/1 - 1) = .0618


what is your opinion?

thanks

Hi,
These rates are given are discrete(in discrete compounding), you cannot use Rf = R2*T2 - R1*T1/R2-R1 when rates given are discrete rather you need to first convert the rate to continuous then use the continuous rates in the above formula. The formula is meant for continuous rates not discrete rates.You convert the rates from discrete as ,3.5%-->ln(1+0.035)=0.0344 and 4.5%-->ln(1+0.045)=0.044, these 0.0344 and 0.044 are continous rates so you can now use the formula to get Rf = R2*T2 - R1*T1/R2-R1=(0.044*5-0.0344*3)/2=0.0584 and then convert it to discrete compounding
(e^rc/m - 1)
(e^0.0584/1 - 1) = .0601

thanks

 

[hellohi]

Hi,
These rates are given are discrete(in discrete compounding), you cannot use Rf = R2*T2 - R1*T1/R2-R1 when rates given are discrete rather you need to first convert the rate to continuous then use the continuous rates in the above formula. The formula is meant for continuous rates not discrete rates.You convert the rates from discrete as ,3.5%-->ln(1+0.035)=0.0344 and 4.5%-->ln(1+0.045)=0.044, these 0.0344 and 0.044 are continous rates so you can now use the formula to get Rf = R2*T2 - R1*T1/R2-R1=(0.044*5-0.0344*3)/2=0.0584 and then convert it to discrete compounding
(e^rc/m - 1)
(e^0.0584/1 - 1) = .0601

thanks

dear @ShaktiRathore

thanks again for your great clarification

Nabil

 

[daniel_ma]

hi may i know how to distinguish when should we use dicrete or when should we use compounding rate?

 

[ShaktiRathore]

Hi,
normally we use continuous compounding for discounting purpose becuse its more simple to discount this way rather than discounting via discrete compounding. Hull uses continous compunding rather than discerte to simplify calculations.
Thanks