Compound Interest and Equivalent Rates
Prerequisite map
The Week 8 Quiz 2 syllabus covers annuities and sinking funds. This unit supplies their core machinery: a periodic effective rate and a number of periods. Compound interest was a Quiz 1 topic. If these conversions are wrong, every annuity formula built on them will also be wrong.
One sum on a timeline
For nominal annual rate j compounded m times per year, i = j/m and n = mt. Then F = P(1 + i)^n, P = F(1 + i)^(−n), and interest earned is F − P. This division applies because j is expressly a nominal rate. An effective annual rate is converted differently.
Example: PHP 90,000 at 8% nominal compounded quarterly for three years has i = .08/4 = .02 and n = 4 × 3 = 12. F = 90,000(1.02)^12 = PHP 114,141.76; compound interest = PHP 24,141.76. The exponent is 12 quarters, not three years. Simple interest would give PHP 111,600 and understates this compound accumulation.
Nominal versus effective annual rates
The effective annual rate is EAR = (1 + j/m)^m − 1. At 12% nominal compounded monthly, EAR = (1.01)^12 − 1 = 12.6825%. Quoted nominal rates should be compared through a common effective period.
If EAR = e, the equivalent monthly rate is i_month = (1 + e)^(1/12) − 1, not e/12. If e = 12%, the monthly rate is about .948879%. Dividing 12% by 12 produces a 1% monthly rate with a higher EAR of 12.6825%.
More generally, if j is compounded m times annually but payments occur p times annually, the effective payment-period rate is i_payment = (1 + j/m)^(m/p) − 1. Example: 12% nominal compounded quarterly and monthly payments gives i_month = (1.03)^(1/3) − 1 = .990163%. This is the rate to use in a monthly annuity when such a general-annuity case is explicitly specified.
Solve for the unknown rather than guessing
| Unknown | Equation | Units/check |
|---|---|---|
| Nominal annual j | m[(F/P)^(1/n) − 1] | n is total compounding periods |
| Period count n | ln(F/P)/ln(1 + i) | answer is in periods |
| Years t | ln(F/P)/[m ln(1 + j/m)] | divide period count by m |
| Minimum complete periods | ceiling of calculated n | verify the prior period falls short |
At 9% nominal compounded monthly, doubling takes ln(2)/ln(1.0075) = approximately 92.7658 months, or 7.7305 years. If interest is credited only at full month ends and the target must be reached then, 93 full months are needed. A fractional-time formula assumes fractional-period exponential accumulation; state the convention before using it.
Different dates require an equation of value
Choose one focal date and move every cash flow to it using the same compound rate. At 2% per quarter, PHP 20,000 due now plus PHP 30,000 due two quarters later is equivalent at quarter 4 to X = 20,000(1.02)^4 + 30,000(1.02)^2 = PHP 52,860.64. Compounding the sum PHP 50,000 for four quarters incorrectly gives the second cash flow two extra quarters of interest.
Alternatively discount everything to time 0. With consistent compound factors, moving the entire equation to another focal date gives the same economic answer. This time-consistency is why compound factors support annuity models cleanly.
Present value and a dated choice
An equipment price of PHP 150,000 payable in two years is not directly comparable with PHP 135,000 cash today. At an 8% effective annual opportunity rate, the deferred price has PV = 150,000/(1.08)^2 = PHP 128,600.82. On the stated rate, dates and no-fee assumption, the deferred payment has the lower PV cost. A lower sticker amount does not automatically mean a lower economic cost.
A down payment is already at time 0 and is added to the present value of financed installments. It is not discounted again and is not included in the principal on which the loan-payment formula operates.
Calculator discipline
- Label the focal date and every cash flow.
- Label the rate as nominal or effective and its compounding frequency.
- Convert to the period required by the cash flows.
- Compute periods independently of the rate; months and quarters must not be mixed.
- Evaluate powers before rounding; use parentheses around the full numerator or denominator.
- State whether the answer is F, interest F − P, present value, a percent rate, periods or years.
Diagnostic checks
At a positive rate F > P, and the PV of a future positive cash flow is below its amount. More frequent compounding raises EAR for the same positive nominal rate. A zero-rate limit gives F = P; the annuity factors in later units tend to n. A negative result from a positive deposit model usually means a missing negative exponent or reversed subtraction.
Source basis
Concepts follow the course compound-interest files, Time Value of Money basics and Slater Chapter 19. Cases, amounts and explanations here are original. Rate conversions with differing payment/compounding periods are labeled extension practice when used in later units.