Recursion: Boot Strapping Forward Rates Extended

In the CFA curriculum, one topic which is covered is a method of boot strapping in order to find forward yield rates based on Treasury (risk free) securities. While the CFA level I text looks at how to boot strap between years to determine basic 1 year forward rates between different years (based on posted rates), let's have a look at what else we can do with the same concept but more applied math. First the basics:

Assume that the following stated yields and horizons:

[y]f[r] --> Notation
I can't remember if this is the official CFA notation, but I'll use this:
y - year (0 is current year)
r - rate for r years into the future

0f1: Current 1 year bond rate: 3.5%
0f2: Current 2 year bond rate: 4.6%
0f3: Current 3 year bond rate: 3.8%
0f4: Current 4 year bond rate: 4.2%

(I've deliberately put rates all over the place to illustrate some points. Plus these rates are rather high given the current economic conditions, but for the purpose of illustrating the math that won't matter). The CFA points out that it is possible to find out the forward rates between any two years by doing "boot strapping":

For instance, at the two year rate, the gain is compounded over two years (4.6% compounded twice over the two years or (104.6%)^2 = 9.4% gain).

At the three year rate, the gain is compounded over three years (3.8% compounded three times over three years or (103.8%) ^3 = 11.8% gain)

To determine (by approximating an arbitrage free value that shows expected interest yield increases) what the forward one year rate would be between the second and third year, you would take

(1+0f3)^3 / (1+0f2)^2 = (1+2f1)

Now let's extend our notation and math. Notice that there is a recursive element involved here:

For any given year and rate:

(1+yf1) = (1+0f[y+1])^(y+1) / (1+0fy)^y

That is to say that the 1 year forward rate for any given year, y, can be approximated by the total return from the current year to year [y+1] compounded (y+1) times divided by the total return from the current year to year [y] compounded y times. Notice in this case, we have deliberately selected a difference of 1 year (a 1 year forward rate) to keep the math simple.

Extending the formula to solve to generalize for yfr (any case):

(1+yfr)^r = (1+0f[y+r])^(y+r) / (1+0fy)^y

Now we have a formula which can approximate for any given year and any given period of time (assuming that you have all the current rates available). This formula says that the total gains achieved using the rth year forward rate for any given year, y, can be approximated by the total return from the current year to year [y+r] compounded (y+r) times divided by the total return from the current year to year [y] compounded y times. This is the ultimate case, flexible to encompass any starting point and any duration.

Let's try this for 2f2 (this should be the two year forward rate in two years):

(1+2f2)^2 = (1+0f[4])^(4) / (1+0f2)^2
= (1.042)^4 / (1.046)^2
= 1.179 / 1.094
= 1.078

The total gain over the two years between years 2 and 4 is expected to be 7.8% (or 3.8% compounded each year). Note that this seems to be in the ball park as the rate is lower and that the further into the future the larger the effect of the compounding is reduced for the 4 year rate versus the higher rate at 2 years.