The Logic of Recurrence#
In Grade 11, you dealt with single amounts of money. In Grade 12, we deal with Annuities — regular, fixed payments made over time (like a monthly savings plan or a car loan).
The “Time Travel” Analogy#
Think of finance as moving money through time:
- Future Value ($F$): You are standing in the Future and looking back at all the small payments you saved. You want to know the total “mountain” of money at the end.
- Present Value ($P$): You are standing in the Present and looking forward. You get a “mountain” of money now (a loan), and you want to know how to break it into small payments to pay it back.
1. ALWAYS Start with a Timeline#
Before touching any formula, draw a timeline. This is the single most important tool in finance.
Future Value (Saving):
|--------|--------|--------|--------|--------|
T0 T1 T2 T3 ... Tn
Pay 1 Pay 2 Pay 3 Pay n
↑ F is HEREPresent Value (Loan):
|--------|--------|--------|--------|--------|
T0 T1 T2 T3 ... Tn
↑ P is Pay 1 Pay 2 Pay 3 Pay n
HERERules for timelines:
- Mark every payment, withdrawal, or lump sum on the line.
- Mark where $F$ or $P$ sits (the “big pile” of money).
- All values you compare must be at the same point in time.
2. Where Do the Formulas Come From? (The Geometric Series Link)#
The annuity formulas are NOT random — they come directly from the Geometric Series sum formula you learned in Sequences & Series. Understanding this connection helps you remember them and handle unusual problems.
Future Value Derivation#
Suppose you deposit $x$ rands at the end of each month for $n$ months at interest rate $i$ per month.
Each payment grows at compound interest for a different number of months:
- Payment 1 (at $T_1$) earns interest for $n-1$ months → worth $x(1+i)^{n-1}$ at $T_n$
- Payment 2 (at $T_2$) earns interest for $n-2$ months → worth $x(1+i)^{n-2}$ at $T_n$
- …
- Payment $n$ (at $T_n$) earns no interest → worth $x$ at $T_n$
The total future value is:
$$ F = x(1+i)^{n-1} + x(1+i)^{n-2} + \ldots + x(1+i) + x $$Written in reverse, this is a geometric series with:
- First term $a = x$
- Common ratio $r = (1+i)$
- Number of terms $= n$
Apply the geometric series sum formula $S_n = \frac{a(r^n - 1)}{r - 1}$:
$$ F = \frac{x[(1+i)^n - 1]}{(1+i) - 1} = \frac{x[(1+i)^n - 1]}{i} $$This is the Future Value formula — it’s just the sum of a geometric series!
Present Value Derivation#
For a loan of $P$ rands repaid with $n$ equal payments of $x$:
Each payment is “discounted” back to $T_0$:
- Payment 1 is worth $\frac{x}{(1+i)^1}$ today
- Payment 2 is worth $\frac{x}{(1+i)^2}$ today
- …
- Payment $n$ is worth $\frac{x}{(1+i)^n}$ today
This is a geometric series with $a = \frac{x}{(1+i)}$ and $r = \frac{1}{(1+i)}$:
$$ P = \frac{x}{(1+i)} \cdot \frac{1 - \left(\frac{1}{1+i}\right)^n}{1 - \frac{1}{1+i}} = \frac{x[1 - (1+i)^{-n}]}{i} $$This is the Present Value formula — again, a geometric series sum!
3. The Formulas (Summary)#
Future Value ($F$) — Saving / Sinking Funds#
$$ F = \frac{x[(1+i)^n - 1]}{i} $$- $x$: The regular payment.
- $n$: The number of payments.
- $i$: Interest rate per period.
- When: The “big pile” of money is at the END.
Present Value ($P$) — Loans / Pensions#
$$ P = \frac{x[1 - (1+i)^{-n}]}{i} $$- $n$: The number of payments remaining.
- When: The “big pile” of money is at the START.
4. The “Total Value Then Subtract” Strategy#
When a problem involves extra deposits, withdrawals, or missed payments, do NOT try to build one complicated formula. Instead:
- Calculate the total as if no changes happened (use the standard formula).
- Calculate the effect of the change separately (grow/discount the extra amount to the same point in time).
- Add or subtract as needed.
Example: Extra Deposit#
You save R1 000/month for 5 years at 8% p.a. compounded monthly. After 2 years, you make an additional once-off deposit of R10 000. What is the total at the end?
Step 1: Calculate the annuity as if the R10 000 didn’t exist:
$$ F_{\text{annuity}} = \frac{1000[(1.00\overline{6})^{60} - 1]}{0.00\overline{6}} $$Step 2: Grow the R10 000 separately for the remaining 3 years (36 months):
$$ F_{\text{extra}} = 10\,000(1.00\overline{6})^{36} $$Step 3: Total = $F_{\text{annuity}} + F_{\text{extra}}$
Example: Withdrawal#
If instead you withdrew R5 000 after 2 years, you would subtract:
$$ F_{\text{total}} = F_{\text{annuity}} - 5000(1.00\overline{6})^{36} $$Key insight: Always grow extra amounts to the same point in time as the main calculation before adding or subtracting.
🚨 Common Mistakes#
- Mixing up $F$ and $P$: Ask yourself: “Did I get the big pile of money at the START or at the END?”
- Start = Present Value ($P$) (Loans).
- End = Future Value ($F$) (Savings).
- The “$n$” calculation: $n$ is the number of payments, not just years. If you pay monthly for 5 years, $n = 5 \times 12 = 60$.
- The “One-Month Gap” Rule:
- For $F$: The formula assumes the first payment is at the end of the first period. If you pay “immediately”, multiply the result by $(1+i)$.
- For $P$: The formula assumes the first repayment happens one period after the loan is granted. If there is a delay, accumulate interest first.
- Not drawing a timeline: This is the single biggest source of errors. You cannot reliably solve finance problems without one.
- Trying one big formula for complex problems: Break it into pieces. Calculate the standard case, then handle extras separately.
💡 Pro Tip: The “Effective Rate” Check#
If your bank offers $12\%$ per annum compounded monthly, your money actually grows by slightly more than $12\%$ because of the “interest on interest” each month. This is the Effective Rate. Always ensure your $i$ value in the formula matches your compounding period ($i = \frac{0.12}{12} = 0.01$).