Formula for Calculating Periodic Fixed Payments with Multiple Rate Changes

Question

I'm trying to figure out how to extend the formula provided by Chris Degnen in this previous question What is the formula for the monthly payment on an adjustable rate mortgage? to add in a third interest rate, e.g. for an extra 2 years after the 2nd term has run for 5 years. So the question would be:

Calculating fixed repayments for a £100,000 loan repaid by 7 annual payments. The first 2 years at 3%, the following 3 years at 4% and the final 2 years at 5%.

I would appreciate any help on this - thanks in advance!

Answer 1

Extending the example in the link.

d is the periodic payment
p is the loan amount
r1 is the periodic rate for the first m periods
r2 is the periodic rate for the next n periods
r3 is the periodic rate for the next o periods

p = 100000
r1 = 0.03
m = 2
r2 = 0.04
n = 3
r3 = 0.05
o = 2

Discounting each payment to net present value:

pv1 = d/(1 + r1)
pv2 = d/((1 + r1) (1 + r1))
pv3 = d/((1 + r1) (1 + r1) (1 + r2))
pv4 = d/((1 + r1) (1 + r1) (1 + r2) (1 + r2))
pv5 = d/((1 + r1) (1 + r1) (1 + r2) (1 + r2) (1 + r2))
pv6 = d/((1 + r1) (1 + r1) (1 + r2) (1 + r2) (1 + r2) (1 + r3))
pv7 = d/((1 + r1) (1 + r1) (1 + r2) (1 + r2) (1 + r2) (1 + r3) (1 + r3))

p = pv1 + pv2 + pv3 + pv4 + pv5 + pv6 + pv7

6.08738 d

So p = 6.08738 d therefore d = 100000/6.08738 = 16427.43

Expressing the above using summations and formulae:

p = (d - d (1 + r1)^-m)/r1 +

    1/(1 + r1)^m (d - d (1 + r2)^-n)/r2 +

    1/((1 + r1)^m (1 + r2)^n) (d - d (1 + r3)^-o)/r3

6.08738 d

There is a clear pattern to extend the formula for any number of changes. Here is the mathematical formula, but it does not simplify:

With

z[1] = r1, z[2] = r2, z[3] = r3, etc.
v[1] = m,  v[2] = n,  v[3] = o, etc.

f[3]

6.08738 d

∴ d = 100000/6.08738 = 16427.43

Returning to the three-rate formula and rearranging for d:

p = (d - d (1 + r1)^-m)/r1 +
    1/(1 + r1)^m (d - d (1 + r2)^-n)/r2 +
    1/((1 + r1)^m (1 + r2)^n) (d - d (1 + r3)^-o)/r3

∴ d = (p r1 (1 + r1)^m r2 (1 + r2)^n r3 (1 + r3)^o)/
     (-r1 r2 + (1 + r3)^o (r1 (r2 - r3) +
          (1 + r2)^n (r1 + (-1 + (1 + r1)^m) r2) r3))

    = 16427.43