# equivalent monthly cost with multiple interest rates

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

• You mention cost. Are these loans which you have taken out, which you have taken out that have rates varying by year? – RonJohn Oct 7 '17 at 5:32
• Its a similar style to the example on the page I linked, under the heading "Example of the Equivalent Annual Annuity Formula". However, the interest rate changes, my issue is how do i adjust the formula to account for changing interest rates. – AllanP Oct 7 '17 at 5:45
• OK, I see the "annuity" tag. – RonJohn Oct 7 '17 at 6:03

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.

``````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.

``````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.