Effective compounded interest rate with multiple interest rates

Question

I'm looking for the effective annual compounded interest rate of an investment which requires me to save a fixed monthly amount, but the interest rate decreases as the investment gets closer to maturity. For example:

The investment starts out in a high growth phase of 10% for 24 months, which is followed by a medium growth of 8% for 12 months and the finally a low growth phase of 4% for 12 months. With each month requiring the same fixed payment.

I've been trying like crazy to find a formula, but most annuity type formulas work on changing payments over time rather than changing interest rates.

Any help would be greatly appreciated!

Answer 1

This is the formula for an annuity with initial amount

where

a is the initial amount
d is the periodic deposit
r is the periodic interest rate
d is the periodic deposit
n is the number of periods

Chaining together three calculations

a1 = 0
r1 = (1 + 0.1)^(1/12) - 1
n1 = 24

a2 = (d (1 + r1) ((1 + r1)^n1 - 1))/r1 + a1 (1 + r1)^n1 = 26.5451 d

r2 = (1 + 0.08)^(1/12) - 1
n2 = 12

a3 = (d (1 + r2) ((1 + r2)^n2 - 1))/r2 + a2 (1 + r2)^n2 = 41.1826 d

r3 = (1 + 0.04)^(1/12) - 1
n3 = 12

FV = (d (1 + r3) ((1 + r3)^n3 - 1))/r3 + a3 (1 + r3)^n3 = 55.0884 d

∴ d = 0.0181527 FV

or without values, the general case for three rates is

FV = ((-d r1 (1 + r2) + (1 + r2)^n2 (d (r1 - r2) +
      (1 + r1)^n1 (d + (a1 + d) r1) r2)) (1 + r3)^n3)/
      (r1 r2) + (d (1 + r3) (-1 + (1 + r3)^n3))/r3

∴ d = (r1 r2 r3 (FV - a1 (1 + r1)^n1 (1 + r2)^n2 (1 + r3)^n3))/
       (-r1 r2 (1 + r3) + (1 + r3)^n3 (r1 (r2 - r3) +
       (1 + r2)^n2 (r1 - r2 + (1 + r1)^(1 + n1) r2) r3))

And if the initial amount a1 is zero

d = (r1 r2 r3 FV)/
     (-r1 r2 (1 + r3) + (1 + r3)^n3 (r1 (r2 - r3) +
     (1 + r2)^n2 (r1 - r2 + (1 + r1)^(1 + n1) r2) r3))

Obtaining the equivalent fixed rate

Solving for r where

FV = (d (1 + r) ((1 + r)^n - 1))/r
d  = 0.0181527 FV
n  = 48

r = 0.00551856

Effective annual rate = (1 + r)^12 - 1 = 6.82701 %