Calculating Forward Rates (from Spot Rates)

CFA Level I Fixed Income
#1

Hi,

i know this might sound very fundamental, but i seem to get more confused the more i read sad

I read S2000magician’s blog and i still don’t quite understand 2 things:

  1. which spot rates to use in calculating the BEY (why is S1 and S4 used in its computation below) and

  2. which spot rates to use in the use of its approximation (why 4 X 3.7% below).

Thanks in advance!

For example, suppose that you’re given these spot rates:

  • 6-months, 2.00%
  • 1-year, 3.00%
  • 18-months, 3.50%
  • 2-years, 3.70%

and you’re asked to calculate the 18-month forward rate starting 6 months from today. The calculation is based on semiannual rates:

S1 = 2%/2 = 1.00%

S4 = 3.7%/2 = 1.85%

3Y forward rate in 1year’s time = 2.1349% (after doing the appropriate calculations)

BEY = 2 x 2.1349% = 4.2698%

The approximation would be: (4 x 3.70% - 2%) / 3 = 4.2667%

#2

Draw out a timeline.

You require an 18 months forward rate, 6 months from today. So the end of the horizon is 24 months.

Therefore a 24 month investmement is equivalent to what? A 6 month investment starting today and an 18 month investment starting 6 months from today. Note that both investments start today and end in 24 months.

It should look something like this: r(0,4) = 24 month rate starting today (ive used 4 as the end becuase it is 4 6-months periods) r(0,1) = 6 month rate startinig today f(1,4) = 18 months rate starting in 6 months and ending in 24 months from today

(1+r(0,4)/2)^4 = (1+r(0,1)/2)*(1+f(1,4)/2)^3

#3

If you’re calculating the 18-month forward rate starting 6 months from now, then the start of that forward rate is one period (6 months) from now, and the end is four periods (2 years) from now; you can calculate the forward rate if you have the spot rate for the start of the forward rate and the spot rate for the end of the forward rate.

The approximation uses the same rates as the full calculation; here, the 1-period (6-month) and 4-period (2-year) spot rates.

#4

i spent so much time reading and re-reading about this…thanks for your very prompt response both dwheats and s2000magician!

#5

My pleasure.