riaology
Learning path 3 · Topic 15 of 30

Match interest to payment periods

Cash-flow periods and compounding periods can differ. Convert to the effective rate over one payment interval, then count payment intervals.

Change a scenario. Watch the consequence.
0123456789101112Receipts (+)Expenses (−)

Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

present worth of $1,000 at each payment-period end · $ip=(1+r/m)m/p−1=3.0301%,N=12,P=1000(P/A,ip,N)=9936.142384i_p=(1+r/m)^{m/p}-1=3.0301\%,\quad N=12,\quad P=1000(P/A,i_p,N)=9936.142384

$1,000 arrives at each payment-period end. If deposits occur between posting dates, this assumes fractional-period compounding; follow any stated bank convention.

Follow the reasoning

Convert nominal annual r into effective rate over one PAYMENT period.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

01

What the model means

If r is nominal per year, m compoundings per year and p payments per year, the equivalent payment-period rate is (1+r/m)ᵐ/ᵖ−1. With quarterly deposits and monthly compounding, three monthly growth factors form one quarterly factor.

02

Build the calculation

If payments occur more frequently than compounding postings, fractional-period compounding is an assumption, not an automatic bank rule. Some contracts credit no interest between posting dates or use simple prorating. Read the stated convention before using an equivalent-rate formula.

ip=(1+r/m)m/p−1,Np=pTi_p=(1+r/m)^{m/p}-1,\quad N_p=pT
Practice with feedback

Set up. Calculate. Explain.

Money: enter dollars

Rebuild the starting scenario

Nominal annual rate: 12 % per year; Compoundings per year: 12 periods/year; Payments per year: 4 payments/year; Number of years: 3 years. Calculate the present worth of $1,000 at each payment-period end.

Setup hint

Convert nominal annual r into effective rate over one PAYMENT period.

Open the visual solutionip=(1+r/m)m/p−1=3.0301%,N=12,P=1000(P/A,ip,N)=9936.142384i_p=(1+r/m)^{m/p}-1=3.0301\%,\quad N=12,\quad P=1000(P/A,i_p,N)=9936.142384
  1. Convert nominal annual r into effective rate over one PAYMENT period.
  2. Number of payments N = years × payments/year = 12.
  3. Use i_p and N together. Never insert an annual rate beside a monthly N.
0123456789101112Receipts (+)Expenses (−)

Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

present worth of $1,000 at each payment-period end · $ip=(1+r/m)m/p−1=3.0301%,N=12,P=1000(P/A,ip,N)=9936.142384i_p=(1+r/m)^{m/p}-1=3.0301\%,\quad N=12,\quad P=1000(P/A,i_p,N)=9936.142384

$1,000 arrives at each payment-period end. If deposits occur between posting dates, this assumes fractional-period compounding; follow any stated bank convention.

Visual solution

Convert nominal annual r into effective rate over one PAYMENT period.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.

Money: enter dollars

Transfer the idea to a new scenario

A 12% nominal annual rate compounds monthly. Receive $1,000 at each quarter-end for four years. Find PW.

Setup hint

Convert nominal annual r into effective rate over one PAYMENT period.

Open the visual solutionip=(1+r/m)m/p−1=3.0301%,N=16,P=1000(P/A,ip,N)=12532.246293i_p=(1+r/m)^{m/p}-1=3.0301\%,\quad N=16,\quad P=1000(P/A,i_p,N)=12532.246293
  1. Convert nominal annual r into effective rate over one PAYMENT period.
  2. Number of payments N = years × payments/year = 16.
  3. Use i_p and N together. Never insert an annual rate beside a monthly N.
0246810121416Receipts (+)Expenses (−)

Period 0: +$0.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.

present worth of $1,000 at each payment-period end · $ip=(1+r/m)m/p−1=3.0301%,N=16,P=1000(P/A,ip,N)=12532.246293i_p=(1+r/m)^{m/p}-1=3.0301\%,\quad N=16,\quad P=1000(P/A,i_p,N)=12532.246293

$1,000 arrives at each payment-period end. If deposits occur between posting dates, this assumes fractional-period compounding; follow any stated bank convention.

Visual solution

Convert nominal annual r into effective rate over one PAYMENT period.

Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.

Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.

Explain the distinction

What should enter P/A for quarterly deposits with monthly compounding?

Can you defend your setup?

Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?

← PreviousNext →

Completion is saved on this browser.