Three different jobs
- CpY: how often interest is added.
- PpY: how often money moves.
- N: how many payments. N = years × PpY.
12.2.3
When PpY ≠ CpY
12.2.3
When PpY ≠ CpY
By the end of this section
- PpY ≠ CpY: annual repayments with monthly or quarterly compounding
CpY: how often interest is added. PpY: how often money moves. N: how many payments.
Key points
When they differ
- Payments and compounding do not have to match. Read the question for each one separately.
- CpY never changes N.
Explore: the same loan, two rhythms
Interest is added monthly (CpY = 12). The loan is repaid monthly (PpY = 12) or once a year (PpY = 1). Find Pmt for each column with your Finance Solver, then press Show Pmt. Drag the orange point along the years to read what is still owed, month by month.
- Owed, paying monthly
- Owed, paying yearly
- Interest so far, monthly
- Interest so far, yearly
| Box | Monthly | Once a year |
|---|
Move a slider to compare the two.
- Which boxes are the same in the two columns, and which are different? Why does CpY stay 12 in both?
- Before you press Show Pmt: is the yearly payment more or less than 12 monthly payments? Why?
- Drag the point through one year, a month at a time. When does the yearly payer owe more than the monthly payer? When do they owe the same? Why does the yearly payer end up paying more interest?
- Set the rate to 0.5%. What happens to the extra cost of paying once a year? Why?
Worked examples
Problems like the ones in class. Pick one, try it, then open the working.
AI SL Finance and approximation slides · L04
Same loan, two rhythms
$150 000 is borrowed over 20 years at 3% per year, compounded monthly. Find the repayment if the loan is repaid monthly, and if it is repaid annually. How much more does paying annually cost in total?
Working
Have a go first. Then open the working one line at a time.
-
Repaid monthly
N = 240, I(%) = 3, PV = 150 000, FV = 0, PpY = 12, CpY = 12 → Pmt = −831.90
Total repaid: 240 payments, $199 655.
-
Repaid annually
N = 20, I(%) = 3, PV = 150 000, FV = 0, PpY = 1, CpY = 12 → Pmt = −10 121.17
Total repaid: 20 payments, $202 423.
-
Compare
$202 423 − $199 655 = $2768
Paying annually costs $2768 more. Only PpY and N changed.
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Why is annual dearer?
12 × $831.90 = $9983, but the annual payment is $10 121
Interest is charged on what you still owe. Monthly payments cut the balance eleven times before the annual payer pays anything. Pay more often, owe less for longer.
AI SL Finance and approximation slides · L04
Two rhythms: annual repayments, monthly interest
A business borrows €60 000 at 3.6% per year, compounded monthly. It repays the loan in equal instalments at the end of each year for 10 years. Find the annual repayment.
Working
Have a go first. Then open the working one line at a time.
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N
N = 10
N counts the repayments, so 10, not 120.
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PpY and CpY
PpY = 1, CpY = 12
One payment a year. CpY = 12 because the bank still adds interest monthly.
-
Solve for Pmt
I(%) = 3.6, PV = 60 000, FV = 0 → Pmt = −7272.78
€7272.78 a year. N counts payments, so N = years × PpY. CpY never touches N.
AI SL Finance and approximation slides · L04
Two rhythms: monthly withdrawals, quarterly interest
€150 000 is in an account paying 2.8% per year, compounded quarterly. Equal withdrawals are made at the end of each month for 20 years, emptying the account. Find the monthly withdrawal.
Working
Have a go first. Then open the working one line at a time.
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N
N = 20 × 12 = 240
N counts withdrawals.
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PpY and CpY
PpY = 12, CpY = 4
Now the payments are the more frequent of the two.
-
Solve for Pmt
I(%) = 2.8, PV = −150 000, FV = 0 → Pmt = 816.48
The money is paid in, so PV is negative. FV is 0 because the account is emptied. €816.48 a month.
AI SL Finance and approximation slides · L04
Annual instalments on a monthly-compounded loan
$45 000 is borrowed at 4.6% per year, compounded monthly, and repaid in equal annual instalments over 8 years. (a) Find the annual repayment. (b) Find the total amount repaid. (c) Suggest why repaying monthly would cost less in total.
Working
Have a go first. Then open the working one line at a time.
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(a)
N = 8, I(%) = 4.6, PV = 45 000, FV = 0, PpY = 1, CpY = 12 → Pmt = −6877.80
$6877.80 a year.
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(b)
8 × 6877.795… = $55 022.37
Eight repayments, using the unrounded payment.
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(c)
Monthly payments reduce the balance sooner, so less interest is charged on what is still owed.
Monthly deposits, quarterly interest
Sofia pays €180 into an account at the end of each month. The account pays 3.2% per year, compounded quarterly. Find the amount in the account after 7 years.
Working
Have a go first. Then open the working one line at a time.
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N
N = 7 × 12 = 84
N counts deposits.
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PpY and CpY
PpY = 12, CpY = 4
Money moves every month. Interest is added every quarter.
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PV and Pmt
PV = 0, Pmt = −180
Nothing is in the account at the start. She pays in, so Pmt is negative.
-
Solve for FV
I(%) = 3.2 → FV = 16 916.98
€16 916.98. She paid in €15 120, so €1796.98 is interest.
Fortnightly payments, repaid early
A loan of $9500 charges 7.8% per year, compounded monthly. It is repaid with equal payments at the end of every fortnight for 3 years. Take a year as 26 fortnights. (a) Find the fortnightly payment. (b) After 2 years the borrower pays off everything still owed. Find that amount.
Working
Have a go first. Then open the working one line at a time.
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N
N = 3 × 26 = 78
N counts payments, and there are 26 a year.
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PpY and CpY
PpY = 26, CpY = 12
Money moves every fortnight. Interest is added every month.
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(a)
I(%) = 7.8, PV = 9500, FV = 0 → Pmt = −136.75
$136.75 a fortnight.
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(b)
N = 2 × 26 = 52, Pmt = −136.754… → FV = −3415.81
Leave Pmt in its box, change N to the 52 payments made, and solve for FV. $3415.81 is still owed.