# Effective compounded interest rate with multiple interest rates

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!

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 %
``````
• Thanks Chris! And is there a formula to essentially average out the interest rates over the lifetime of the investment? If my understanding is correct, would this be the geometric mean? May 2, 2020 at 10:54
• Hi, I take it you mean reversing the calculation to find out what the average rate would be over the 4 years. With known `FV`, `d` and `n` the formula `FV = (d (1 + r) ((1 + r)^n - 1))/r` can't be solved algebraically for `r`, but you can solve it numerically by graph or solver, e.g. money.stackexchange.com/a/120192/11768 May 2, 2020 at 11:07
• Yes that is what I meant. Ok great, thanks for the help! May 2, 2020 at 12:37
• Using a solver, with `d = 0.0181527 FV` and `n = 48`, `r = 0.00551856` ∴ effective annual rate is `(1 + r)^12 - 1 = 6.82701 %`. May 2, 2020 at 13:07
• Mathematica makes computational algebra a piece of cake. May 4, 2020 at 16:17