Monthly Compounding: Effective Rates, Present Value, and Mortgage Payments

Summary

The document clarifies how to interpret a quoted annual percentage rate when interest compounds monthly. For a nominal APR and twelve compounding periods per year, the monthly rate is the APR divided by twelve. Compounding that monthly rate over a full year gives the effective annual rate; it is not the monthly rate itself. The response confirms a discounting expression for a single future amount over a specified number of monthly periods.

For a level-payment loan, the corrected payment formula uses the monthly rate both as the numerator rate applied to principal and in the annuity discount factor over the total number of monthly payments. This corrects the question’s expression, which mixes annual and monthly rates and omits the loan term in the exponent. The formulas are standard fixed-rate cash-flow relationships. The document does not discuss fees, changing rates, payment timing conventions, taxes, or other real-world mortgage terms, so those would need separate treatment in an applied valuation.

Key ideas

• Divide a nominal APR by the number of compounding periods per year to obtain the periodic rate.
• The annual effective rate reflects compounding of the periodic rate across a year.
• Discounting a future payment requires the periodic rate and the total number of periods.
• A level-payment loan formula uses the periodic rate in both the payment numerator and the annuity factor.
• The formulas assume fixed rates and regular payments, without addressing fees or other contract terms.

Tags

# Interest rates compounded monthly


# Interest rates compounded monthly












Suppose the quoted APR is $r_0 = x-1$ and interest is compounded monthly;

Am I correct in saying the formula for the monthly interest rate $r$ is:

$$r = (1+ (\frac{r_0}{m}))^m -1 $$

Is it also correct to say that the present value of monthly repayments each of $A$ at an APR of $r0$ compounded monthly is:

$$PV = \frac{A}{(1+r_0/m)^{mt}} $$

And finally that the monthly payments on a mortgage of $P$ over $t$ years at an APR of $r0$ is:

$$R = \frac{P \cdot r0}{[1-(1+r_0)^{-m}]}$$

## Answer by Alex C (score 0, accepted)

https://quant.stackexchange.com/a/49033

The monthly interest rate is $\frac{r_0}{m}$ where $m=12$. The formula you give $$r = \left(1+ \frac{r_0}{m}\right)^m -1 $$ is the Effective Annual Rate corresponding to $r_0$ compounded monthly.

The second formula is correct.

In the third formula there seem to be several typographical errors involving "m" and "t" (which is missing).

$$R=\frac{(r_0/m)P}{1-(1+r_0/m)^{-mt}}$$

A good reference for these basic formulas is Wikipedia.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.