Formula for Calculating Periodic Fixed Payments with Multiple Rate Changes

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!

One thought on “Formula for Calculating Periodic Fixed Payments with Multiple Rate Changes

  1. Chris Degnen

    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:

    enter image description here

    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.
    

    enter image description here

    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
    

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