equivalent monthly cost with multiple interest rates

Question

Is equivalent monthly cost equal to equivalent annual cost divided by 12?

Also, using the equivalent annual cost HERE, how would I change the formula for differing rates over the period of time.

Example:

6 years, PV = $15,000,
Interest Rates:
Year 1 = 2% p.a.
Year 2 = 3% p.a.
Year 3 = 2% p.a.
Year 4 = 2% p.a.
Year 5 = 3.5% p.a.
Year 6 = 3% p.a.

Would I take the average interest rate and use that? So:
(2+3+2+2+3.5+3)/6 = 2.58%
Then divide 2.58%/12 = 0.215% for monthly



Thanks

Answer 1

Use the geometric mean for the average interest rate in a time series.

gm = (1.02*1.03*1.02*1.02*1.035*1.03)^(1/6) - 1 = 0.02581541058

Comparing the interest over six years.

1.02*1.03*1.02*1.02*1.035*1.03 - 1 = 16.5239812 %

                    (1 + gm)^6 - 1 = 16.5239812 %

The equivalent annual annuity calculation (shown in the derivation) becomes

pv = (c 1.02^5 + c 1.03^4 + c 1.02^3 + 
      c 1.02^2 + c 1.035^1 + c 1.03^0)/(1 + gm)^6

∴ c = (pv (1 + gm)^6)/
        (1.02^5 + 1.03^4 + 1.02^3 + 1.02^2 + 1.035 + 1)

pv = $15000

∴ c = $2745.53

where c is the annuity cash flow.

For a monthly calculation the monthly rate is (1 + r)^(1/12) - 1

where r is the annual effective rate or the nominal annual rate compounded annually.

(The equivalent annual annuity calculation should use the annual effective rate or the nominal annual rate compounded annual, which is the same. However, if you are calculating an equivalent monthly annuity the monthly rate can be taken as the nominal annual rate 'compounded monthly' divided by twelve.)

Derivation & Check

The equivalent annuity is based on the following summation, which shows the present value pv equal to the future value of the sum of the periodic cash flows (made at the beginning of each period) discounted to present value by division by (1 + r)^n.

By induction, the closed form is pv = (c - c (1 + r)^-n)/r

∴ c = (r pv)/(1 - (1 + r)^-n)

which matches the formula provided by the OP.

Adding in webpage example figures.

pv = 100000
n = 4
r = 0.08

∴ c = (r pv)/(1 - (1 + r)^-n) = 30192.08

Expressed as summation with geometric mean.

gm = (1.08*1.08*1.08*1.08)^(1/4) - 1 = 0.08

pv = (c 1.08^3 + c 1.08^2 + c 1.08^1 + c 1.08^0)/(1 + gm)^4 = 100000

So the summation expression checks out, although the example is a simplified case with a constant interest rate.

Answer 2

1.02 * 1.03 * 1.02 * 1.02 * 1.035 * 1.03 = 1.165239
1.0258^6 = 1.165134

It's not the exact same, because of compounding. It's really close, though.