Loan Analysis & Period Calculations

Beyond the Formula

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Most Grade 12 Finance questions go beyond simply plugging numbers into the annuity formulas. They test whether you can analyse a financial scenario — calculate how long an investment takes, find the outstanding balance after several payments, or compare different options.

1. Calculating the Period ($n$)

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The Problem

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“How long will it take for an investment to reach a target amount?” or “How many payments are needed to pay off a loan?”

The Method: Use Logarithms

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For compound interest ($A = P(1 + i)^n$):

Example: Compound Interest Period

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How long will it take for R5 000 to grow to R20 000 at 8% p.a. compounded annually?

It takes approximately 18 years.

Example: Annuity Period

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You invest R2 000 per month at 9% p.a. compounded monthly. How many months until you have R500 000?

$i = \frac{0.09}{12} = 0.0075$

Using the Future Value formula:

Since you can’t make a fraction of a payment, round up to 142 months (about 11 years and 10 months).

Always round UP for period calculations — you need the full number of payments to reach the target.

2. Outstanding Balance on a Loan

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The Two Methods

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There are two ways to calculate the balance still owed after $k$ payments:

Method 1: “Retrospective” (Looking Back)

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Calculate the loan amount grown forward, minus all payments grown forward:

Where $P$ = original loan, $x$ = payment amount, $k$ = number of payments made.

Method 2: “Prospective” (Looking Forward)

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Calculate the present value of the remaining payments:

Where $n - k$ = number of payments remaining.

Example

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A loan of R800 000 is repaid over 20 years at 10.5% p.a. compounded monthly.

$i = \frac{0.105}{12} = 0.00875$, $n = 240$ months.

Step 1: Find the monthly payment:

Step 2: Find the balance after 5 years (60 payments):

Using Method 2 (prospective — 180 payments remaining):

After 5 years of payments, you still owe R733 841 on a R800 000 loan! This shows how most early payments go toward interest, not the principal.

3. The Final Payment

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In reality, the last payment is almost never exactly equal to the regular payment (because $n$ is usually not a perfect whole number).

The Method

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1. Calculate the balance after $n - 1$ payments (one payment before the end).
2. The final payment = Balance × $(1 + i)$ (the remaining balance plus one more period of interest).

Example

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If the balance after 239 months is R7 823.41:

This is slightly different from the regular R7 887.59 payment.

4. Deferred Payments (Payment Holiday)

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Some loans allow a “payment holiday” — you don’t pay for the first few months, but interest still accumulates.

The Strategy

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1. Grow the loan during the holiday period using compound interest: $A = P(1 + i)^k$ where $k$ is the number of deferred periods.
2. Use this new amount as the “loan” in the Present Value formula to calculate payments.

Example

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A loan of R500 000 at 12% p.a. compounded monthly. First payment is made 4 months after the loan is granted. The loan must be repaid in 180 equal monthly payments.

$i = 0.01$

Step 1: Grow the loan for 3 months (interest accrues but no payments):

Step 2: Calculate the monthly payment based on R515 150.50 over 180 payments:

The deferred period makes the loan more expensive because interest accumulated while you weren’t paying.

5. Comparing Financial Options

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Exam questions sometimes ask you to compare two investment or loan options.

Strategy

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Convert everything to the same basis:

• Same time period
• Same compounding frequency (use effective interest rate if needed)
• Compare the total amount paid or received

Example

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Option A: R10 000 invested at 8% p.a. compounded monthly for 5 years. Option B: R10 000 invested at 8.2% p.a. compounded annually for 5 years.

Option A: $A = 10\,000\left(1 + \frac{0.08}{12}\right)^{60} = 10\,000(1.00\overline{6})^{60} = \text{R}14\,898.46$

Option B: $A = 10\,000(1.082)^5 = \text{R}14\,840.87$

Option A is better by R57.59 — monthly compounding at 8% beats annual compounding at 8.2%.

6. Timelines: The Essential Tool

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For every finance question, draw a timeline:

|--------|--------|--------|--------|--------|
T0       T1       T2       T3       ...      Tn
Loan     Pay 1    Pay 2    Pay 3             Pay n
granted

• Mark when payments start and end.
• Mark any deferred periods or lump sum payments.
• All values must be at the same point in time before you can compare or combine them.

🚨 Common Mistakes

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1. Rounding $n$ down instead of up: If $n = 141.3$ months, you need 142 payments, not 141. The 142nd payment will be smaller than the rest.
2. Using the wrong $i$: If the rate is “12% p.a. compounded monthly”, then $i = \frac{0.12}{12} = 0.01$ per month. Students often use $i = 0.12$.
3. Deferred payment timing: If the first payment is in month 4, interest accumulates for 3 months (not 4). Count the gap between the loan date and the first payment.
4. Forgetting interest on the final payment: The last payment must cover the outstanding balance PLUS one period of interest.
5. Not drawing a timeline: This is the single biggest source of errors in finance. Always draw one.

💡 Pro Tip: The “Interest vs Principal” Insight

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In the early years of a home loan, nearly all your payment goes to interest and very little reduces the actual debt. This is why paying extra in the first few years has such a huge long-term impact — it reduces the principal that future interest is calculated on.

⏮️ Nominal vs Effective Interest | 🏠 Back to Finance