1.4A

Compound Interest Formula

  • When interest is compounded periodically across $n$ years at nominal annual percentage rate $r\%$, the future value $FV$ is: $$FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$$
  • Variables:
    • $PV$: Present Value (initial principal invested or borrowed).
    • $FV$: Future Value (accumulated total balance).
    • $r$: Nominal annual rate of interest (expressed as a percentage, e.g. $4.5$).
    • $k$: Number of compounding periods per year ($k=1$ annual, $k=2$ semi-annual, $k=4$ quarterly, $k=12$ monthly, $k=365$ daily).
    • $n$: Number of years.
  • Total Interest Earned: $I = FV - PV$.
1.4B

Reducing Balance Depreciation

  • Assets such as vehicles and electronics lose value at a constant annual depreciation percentage rate $r\%$.
  • The book value $V$ after $n$ years is given by: $$V = V_0 \times \left(1 - \frac{r}{100}\right)^n$$
  • This is a decreasing geometric sequence with common ratio $R = 1 - \frac{r}{100}$.
1.4C

GDC TVM Solver Workflow

  • For regular recurring payments, loan repayments, and retirement annuities, the GDC TVM (Time Value of Money) Solver can be used.
  • Parameter Meanings:
    • N: Total number of compounding periods ($n \times k$).
    • I%: Annual interest rate (entered directly as a percentage, e.g. $5.4$).
    • PV: Present value.
    • PMT: Regular payment amount per period.
    • FV: Future value ($0$ when a loan is fully repaid).
    • P/Y and C/Y: Payments per year and compounding periods per year (normally equal).
  • Cash flow sign convention:
    • Negative ($-$): Money leaving your pocket (deposits, investments, payments).
    • Positive ($+$): Money entering your pocket (loan proceeds received, withdrawals).

Payments, annuities and effective rates

  • Align the time units: with periodic decimal rate $i$ and N payments, END mode means payments occur after each period’s interest. BEGIN mode means before it. For unequal compounding and payment frequencies, the effective payment-period rate is $(1+j/k)^{k/m}-1$ for annual decimal nominal rate j, k compounds and m payments.
  • Ordinary savings annuity: deposits P at period ends give $FV=P((1+i)^N-1)/i$. BEGIN deposits give an extra factor $(1+i)$. At zero interest both formulas reduce to NP.
  • Loan repayments: $P=PV\,i/(1-(1+i)^{-N})$. After an END payment, $B_t=B_{t-1}(1+i)-P$; interest is $iB_{t-1}$ and principal repaid is P minus interest.
  • Effective annual rate: $(1+j/k)^k-1$. Compare offers on the same effective basis, including fees and realistic repayment terms.
  • Calculator workflow: clear old settings; enter N, annual I%, P/Y, C/Y, PV, PMT, FV and END/BEGIN. Use opposite cash-flow signs for receipts and payments. Substitute the output into a recurrence or formula to check it.
  • Keep unrounded values during calculation. Actual payments rounded to cents may require an adjusted final payment. A negative outstanding balance means an overpayment, not another full payment.

Worked amortization schedule

  • Loan 1000, interest 1% per month, end-of-month payments 400: month 1 interest10, principal390, balance610; month2 interest6.10, principal393.90, balance216.10.
  • Month3 balance before payment is $216.10(1.01)=218.261$. Final payment is 218.26 rounded to cents, rather than 400. Total interest is approximately24.26.
  • Check balances, total principal and payment timing. Taxes, changing rates and late fees are outside this fixed-rate model.

10 Practice Exercises

Attempt each problem before opening the solution.

Exercise 1 (Paper 1 — 3 Marks): Elena invests $\$8,000$ in a savings account paying a nominal annual interest rate of $3.6\%$, compounded monthly. Find the total value of her investment after 5 years, correct to the nearest dollar.
Formula: $FV = PV\left(1 + \frac{r}{100k}\right)^{kn}$ with $PV = 8000, r = 3.6, k = 12, n = 5$ [M1]
$FV = 8000\left(1 + \frac{3.6}{1200}\right)^{12 \times 5} = 8000(1 + 0.003)^{60} = 8000(1.003)^{60}$ [M1]
$FV \approx 8000(1.1968948029\ldots) = \$9575.1584\ldots \approx \$9575$ [A1]

Answer: $\$9575$

Exercise 2 (Paper 1 — 4 Marks): A delivery van is purchased for $\$35,000$ and depreciates at a rate of $15\%$ per year.
  1. Find the value of the van after 4 years.
  2. Determine the number of complete years before the van's value drops below $\$10,000$.
(a) $V = 35000(1 - 0.15)^4 = 35000(0.85)^4 \approx \$18,270.22 \approx \$18,300$ [M1][A1]
(b) $35000(0.85)^n < 10000 \implies (0.85)^n < \frac{10000}{35000} \approx 0.28571$ [M1]
$n \ln(0.85) < \ln(0.28571) \implies n > \frac{\ln(0.28571)}{\ln(0.85)} \approx \frac{-1.25276}{-0.16252} \approx 7.71$
Number of complete years = 8 years [A1]

Answer: (a) $\$18,270$ (or $\$18,300$), (b) 8 years

Exercise 3 (Paper 2 — 5 Marks): Marcus takes out a car loan of $\$24,000$ at an annual interest rate of $6.5\%$, compounded monthly. The loan is to be fully repaid over 4 years with equal monthly installments.
  1. Calculate the required monthly repayment amount.
  2. Calculate the total amount paid over the 4 years and hence determine the total interest charged.
(a) TVM Solver values: $N = 4 \times 12 = 48, I\% = 6.5, PV = 24000, FV = 0, P/Y = 12, C/Y = 12$, with payments at the end of each month (END mode) [M1]
Solve for $PMT = -\$569.16$ [A1]
Monthly payment = $\$569.16$ [A1]
(b) Using the monthly payment rounded to cents, total amount paid = $48 \times 569.16 = \$27,319.68$ [M1]
Total interest = $\$27,319.68 - \$24,000 = \$3319.68$ [A1]

Answer: (a) $\$569.16$ per month, (b) Total paid = $\$27,319.68$, Total interest = $\$3319.68$

Exercise 4

Deposit 200 at each year end for 3 years at 5% annual interest. Find the balance and explain the timeline.

$200(1.05^2+1.05+1)=630.50$. The first deposit earns two years’ interest, the last none.

Exercise 5

Repeat with deposits at the beginning of each year.

Each deposit earns one more year: $630.50(1.05)=662.025$, or662.03. BEGIN and END are different models.

Exercise 6

Compare 6% compounded monthly with 6.1% compounded annually, ignoring fees.

Monthly effective rate $(1.005)^{12}-1\approx6.16778\%$, so it produces the greater one-year growth. Compare effective rates, not nominal percentages.

Exercise 7

A loan of1000 at1% monthly has payments400 at month ends. Construct its repayment schedule.

Month1: interest10, balance610. Month2: interest6.10, balance216.10. Month3: interest2.161, final payment218.261 before rounding. Total interest24.261.

Exercise 8

How many complete years are needed for1000 to exceed1500 at5% annual interest?

$n>\ln1.5/\ln1.05\approx8.3104$. Minimum9: year8 about1477.46, year9 about1551.33.

Exercise 9

A balance5000 accrues1% monthly interest. A customer pays40 at each month end. Describe the first month and whether the debt shrinks.

Interest50 exceeds payment40; new balance5010. Under this fixed rate and payment, the balance grows. Paying regularly does not guarantee amortization.

Exercise 10

Find the year-end deposit needed to accumulate5000 in4years at4% annual interest.

$P=5000(0.04)/(1.04^4-1)\approx1177.45023$, or1177.45. N4, END, PV0, FV5000; PMT has the opposite sign to FV.

Formulae

Financial Mathematics Formulas

Compound Interest:

$$FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$$

Depreciation:

$$V = V_0 \left(1 - \frac{r}{100}\right)^n$$
Examiner Pitfalls

Common Exam Traps in Financial Math

  • Mixing up $P/Y$ and Compounding Frequency: Always verify that $k$ aligns with the compounding frequency stated in the problem ($k=12$ monthly, $k=4$ quarterly, $k=1$ annually).
  • Sign Errors in TVM Solver: For a loan, PV and PMT have opposite signs. For savings, the initial investment and recurring deposits can have the same sign; the eventual withdrawal FV has the opposite sign. Pay close attention to direction of cash flow.
Topic 1.4 Check

Exam Mini Test (5 Questions)

Score: 0 / 5

1. If $\$5,000$ is invested at $4\%$ compounded quarterly, what is the value of $k$ in the formula?

2. An asset depreciates by $10\%$ per year. What is the multiplier for each passing year?

3. When solving for monthly mortgage repayments in the GDC TVM Solver, what is the value of $FV$ when the loan is fully paid off?

4. A principal of $\$10,000$ grows to $\$12,500$ after 3 years. How much total interest was earned?

5. Which compounding frequency yields the highest future value for a fixed nominal rate of $5\%$?