Let’s prove that the calculation of arbitrage yield for bond issues with out of safe harbor premium binds is equivalent to the following simpler, finite, rational, absolutely NO - not even one - permutations calculation.
For each out of safe harbor bond select the redemption date with the lowest price when using the arb yield instead of the market yield in the bond price formula, and these redemption dates are the ones we need to use to compute the arb yield. It's iterative, but it's finite and doable, and doesn't involve any crazy bazillion, bazillion, bazillion ridiculous permutations.
This gives us a simple, quick and efficient method for computing the arbitrage yield.
1) Start with an initial guess for the arbitrage yield
2) PV the issue’s cashflows to maturity (the standard PV calculation)
3) For each out of safe harbor bond use the price function in Excel, or DBC or whatever you’re using to price each bond to both maturity and call using the arb yield guess instead of the market yield.
4) Select the minimum price in step 3) and subtract it from the price to maturity in step 3).
5) Reduce the PV in step 2) by the sum of the differences in step 4)
6) Change the guess rate until the modified cashflow PV in step 5) equals the arb yield target.
Proof For each out of safe harbor bond use the arb yld instead of the market yield to compute each redemption date producing the lowest ‘price’ (so this isn’t the sale price or production price which uses the market yield, but this is the same technique). If these redemption dates aren’t the ones we used to produce our assumed arb yield then changing the redemption dates we used to produce our assumed arb yield to these new redemption dates will give us a lower PV than the target PV number that we used to compute our assume arb yield. And that means we can resolve and find a lower yield than the arb yield, which was supposedly lowest.
Next, the only redemption dates we care about are the maturity date and call dates where the premium changes.
Proof. Suppose we pick a redemption date n + k/180 between interest periods n and n + 1, where k is a regular integer counting number like 0, 1, 2, 3 that's between 0 and180. Given this, the last term of our PV formula is going to look like ((1 + c*k/360)/(1 + y/2)(k/180))/(1 + y/2)n. We want to show k is = to either 0 or 180 if the PV is minimized. First, since y, c and n are constants, then so is 1/(1 + y/2)n, so let’s dispense with it and deal with (1 + ck/360)/(1 + y/2)(k/180).
Let’s show if
(I) (1 + ck/360)/(1 + y/2)(k/180) < (1 + c(k + 1)/360)/(1 + y/2)((k + 1)/180)
then
(II) (1 + c(k - 1)/360)/(1 + y/2)((k - 1)/180) (1 + ck/360)/(1 + y/2)(k/180)
must also be true.
Can you see why (I) implies (II)?
Well, it’s enough to show (1 + c(k - 1)/360)/(1 + y/2)(k/180) < (1 + ck/360)/(1 + y/2)((k + 1)/180), and that's because the only difference between this inequality and (II) above is multiplication by a constant = 1/(1 + y/2)(1/180). And the rest is true because the only difference between (I) and the inequality (1 + c(k - 1)/360)/(1 + y/2)(k/180) < (1 + ck/360)/(1 + y/2)((k + 1)/180) is the obvious inequality (- c/360)/(1 + y/2)((k - 1)/180) < (- c/360)/(1 + y/2)(k/180).
By the way, although we began with the < sign, we could have just as easily started with the > sign and the same argument would have worked.
Hence, we’ve shown I implies II, and that means that when c and y are fixed and only k is allowed to vary, the maxs and mins are found only at interest point dates. QED, (and no calculus).
Note, if you don't like 'abstract' math or reasoning, you can check this on any spreasheet by picking two numbers for the arb yield and coupon, say, 4% and 6% and graphing (1 + .03*x)/(1.02)x with x between 0 and 1.