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### calculating forward rates

Posted: **March 31st, 2008, 2:34 pm**

by **jimjohn**

caqn someone plz help me with this, its probably very basic for most of you guys. i have the following term structure: Term (in years) cost of funds rate: 0 2.556 0.0833 2.556 0.1667 2.57 0.25 2.588 0.5 2.68 0.75 2.725 1 2.77 then the annual compounding is computed for each term Term Annual Compounding .08333 2.586 .1667 2.598 .25 2.613 .5 2.698 .75 2.734 1 2.770 now we calculate the forward rates for each month in the future until the end of the year: (ie the forward rate from month 1 till end of the year, month 2 till the end of the year, etc.) fwd rate SA A 1 into 11 2.768 2.786817886 2 into 10 2.784 2.803132897 3 into 9 2.802 2.821286405 4 into 8 2.814 2.833481788 5 into 7 2.821 2.841226747 6 into 6 2.822 2.841965275 7 into 5 2.833 2.853183471 8 into 4 2.845 2.865118558 9 into 3 2.856 2.876415123 10 into 2 2.868 2.888505272 11 into 1 2.879 2.900225414 can someone plz explain first of all why do we do the annual compounding and then how do we get the formulas to calculate these forward rates. thx so much. im not sure if this is used in any of the formulas but the annual volatility rate is 22.5%. thx.

### calculating forward rates

Posted: **March 31st, 2008, 7:11 pm**

by **bilbo1408**

If I understand correctly, you want the 11 month forward rate that begins one month from now. The idea is to invest $100 today at the one month rate, then reinvest those earnings at the end of that month at the 11 month rate. This should be equal to investing that same $100 today in the 12 month rate.Ex: $100(1 + (.02586/12) ) = 100.2155 Reinvesting at the 11 month forward rate (annualized rate given below) gives 100.2155(1 + (.027868/12) )^11 = 102.806 This amount should equal to 100(1 + (.0277/12) )^12 = 102.805439Use the same principle for all maturities. So (7 month rate 5 months from now) = 100(1 + (.026696/12) )^5 = 101.1173 Then, I got the .026696 above by taking { [ (2.698 - 2.613) / 3 ] * 2} + 2.613

### calculating forward rates

Posted: **March 31st, 2008, 7:24 pm**

by **daveangel**

the first set of rates are simple rates (money market). These are used to calculate discount factors as follows:df = 1/(1 + R*T) where R is the rate and the T is the time in years.the annually compounded rates are obtained from the above discount factors as follows:df = (1+r)^-T where r is the annually compounded rate. The choice of annual compounding is arbitrary - other compounding can and is often used. if semi-annual compounding was used then df = (1+r/2)^(-2T) and so on.once you have a set of rates, you can then use these to calculate forward rates.the forward rate for period 1,2 is obtained as follows:df(0,1) * df(1,2) = df(0,2)hence df(1,2) = df(0,2)/df(0,1)for points along the curve where you do not have an explicit rate obtained from money market instruments, then you can just interpolate.for example, taking the 6 into 6 forward rate the forward discount factor is 0.973047/0.986777 which is 0.986085. This gives a forward annual compounding rate of 2.8421%.hth

### calculating forward rates

Posted: **April 1st, 2008, 12:55 pm**

by **jimjohn**

thanks guys, great explanations!

### calculating forward rates

Posted: **June 25th, 2010, 8:30 am**

by **hanh510**

Dear Daveangel,I infer from your answer that the formula of forward rate in case of simple rates (money market) is:Forward rate (1,2)= (1+R02*t02)/(1+R01*t01) - 1and in case of compound rates: Forward rate (1,2)= ((1+r02)^(t02))/((1+r01)^(t01))-1I do agree with U on both of cases. However, I have seen many documents in which the forward rate is calculated by this formula:Forward rate (1,2)= ((1+r02*t02)/(1+r01*t01) - 1) * (1/(t02 - t01))Could you please explain to me about this formula (when and why they use it but not the other formulas)?

### calculating forward rates

Posted: **June 25th, 2010, 11:56 am**

by **cemil**

QuoteI do agree with U on both of cases. However, I have seen many documents in which the forward rate is calculated by this formula:Forward rate (1,2)= ((1+r02*t02)/(1+r01*t01) - 1) * (1/(t02 - t01))Look at this formula: (1+r01*t01) * (1+r12*(t02-t1)) = (1+r02*t02)

### calculating forward rates

Posted: **June 27th, 2010, 4:55 am**

by **hanh510**

QuoteOriginally posted by: cemilQuoteI do agree with U on both of cases. However, I have seen many documents in which the forward rate is calculated by this formula:Forward rate (1,2)= ((1+r02*t02)/(1+r01*t01) - 1) * (1/(t02 - t01))Look at this formula: (1+r01*t01) * (1+r12*(t02-t1)) = (1+r02*t02)Thanks for your answer! This is the original formula of the calculating forward rate in question. However, my question is "why and when they use this formula?". What is the concept of this formula?I have the same opinion as Daveangel:" the first set of rates are simple rates (money market). These are used to calculate discount factors as follows:df = 1/(1 + R*T) where R is the rate and the T is the time in years.the annually compounded rates are obtained from the above discount factors as follows:df = (1+r)^-T where r is the annually compounded rate. The choice of annual compounding is arbitrary - other compounding can and is often used. if semi-annual compounding was used thendf = (1+r/2)^(-2T) and so on.once you have a set of rates, you can then use these to calculate forward rates.the forward rate for period 1,2 is obtained as follows:df(0,1) * df(1,2) = df(0,2)hencedf(1,2) = df(0,2)/df(0,1) "