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 %

Answer 2

The easiest way in this age of computers and spreadsheets would be to set up a spreadsheet with one row for each month, enter the appropriate growth rate on each month, and get the final total at the bottom. Then calculate the annual growth rate based on that total and your initial investment.

I'm assuming that when you say "10% for the first 24 months" that you mean an annual rate of 10%, and not 10% per month or 10% for the entire 24 months. Also I'm assuming your numbers are simple interest, before compounding.

I'm also assuming the money remains in the account, growing and compounding, with no withdrawals. That assumption may well not be true and if not you'd have to adjust the spreadsheet.

With those assumptions, I find that an initial investment of $100 would grow to $110 after 12 months (compounding at that point is less than 50 cents), to $122 after 24 months, to $132 after 36 months, and to $138 after 48 months. So growth from $100 to $138 in 48 months would require an average (geometric mean) growth of 0.67% per month or 8.0% per year.

Just for amusement, note that if you computed a simple arithmetic mean, (10% x 24 + 8% x 12 + 4% x 12) ÷ 48, you'd also get 8.0%. The exact number I got the long way was 7.9975%, so the difference comes to only .0025%.