Paper ZeroSign in

Compound interest and depreciation

  • In 8 of 20 recent exams
  • 1% of the marks
  • Paper 1 or 2
  • 4 to 8 marks
How it is set in the exam

Paper 2 (calculator), 4–8 marks. Rarely on Paper 1 (no calculator), and then the numbers let you work the power out by hand.

Spot it in the exam
Exam phrases to spot (6)

  • “an annual interest rate of 3.6%, compounded monthly”

  • “a nominal annual rate of 3.6%, compounded monthly”

  • “depreciates by 15% per year”

  • “find the value after 6 years”

  • “find the interest rate”

  • “how many complete years until…”

In disguise (4)
  • "loses a fifth of its value every year" (depreciation, r=20%r = 20\%)
  • "how long until the investment doubles" (time unknown)
  • "find the annual rate" given a start and end value (rate unknown)
  • "the real value, allowing for inflation of 2%"
Looks like it, but isn't (2)
  • simple interest (the same amount each year) is arithmetic: Arithmetic or geometric in context
  • a growth or decay formula in tt given by the question (N0ektN_0e^{kt}, N0btN_0b^t), used for a value, a constant or a time: Exponential and log models
Worked example

Final value, compounded monthly

Paper 2 (calculator)
Question

Maya invests €5000 in an account paying an annual interest rate of 3.6%, compounded monthly.
(a) Find the value of her investment after 6 years, to the nearest euro. [3]
(b) Find the interest she has earned, to the nearest euro. [1]

Try it yourself first, then compare.

Now try one like it

Write your working on paper or on screen. Paper Zero marks it step by step, like the worked example, and shows where each mark went. Free, with no account.

A handwritten page as Paper Zero marks it: ticks at the two lines that earned a mark, a cross on line 3 where +7 was copied as −7, and a short note beside that line.123456f′(x) = 4x − 124x − 12 = 0, so x = 3f(3) = 2(3)² − 12(3) − 7= 18 − 36 − 7 = −25minimum at (3, −25)M1 ✓A1 ✓A1 ✗11The +7 became −7: 18 − 36 + 7 = −11.
Which method?
  • Final value FVFV
    Use
    The formula directly
    Watch out
    kk and knkn: monthly means k=12k = 12
  • Final value on (no calculator)
    Use
    The formula, with the power worked out by hand: repeated multiplication, or a binomial expansion (worked line in Compound interest and depreciation)
    Watch out
    Keep every decimal until the last line, then round the money
  • Time
    Use
    Find n: an inequality in the number of periods, with multiplier 1+r100k1 + \frac{r}{100k} per period. GDC: table or the financial solver.
    Watch out
    Convert periods into the units asked, usually complete years: check the values at the two whole years either side
  • Rate rr
    Use
    GDC: financial solver, or rearrange with an nnth root
    Watch out
    Depreciation rates come out as a decrease
  • Present value PVPV
    Use
    The formula rearranged, or the GDC: financial solver
    Watch out
  • Real value
    Use
    Real rate ≈\approx interest rate −- inflation rate, then the formula
    Watch out
    Use the approximation when the question says so
Steps

  1. Identify PVPV, the rate per period (negative for depreciation) and the number of periods knkn.

  2. Write the formula, marking the unknown. With the financial solver, write down the values you enter instead. Write the values with PVPV and FVFV of opposite signs (money paid in is negative). Usually at least two correct values earn the method mark, and all of them the next mark. Some markschemes accept either sign, but N23 withheld the "all entries correct" mark unless PVPV and FVFV had opposite signs, and M26 required a positive final value.

  3. Solve by the row of the table.

  4. Convert to the question's units (complete years, months). Interest =FV−PV= FV - PV.

  5. Money

    goes to the accuracy the question states (2 d.p., or the nearest euro or dollar). The first money answer that isn't loses an .

Common mistakes

  • Wrong kk or knkn

    monthly for 6 years is 72 periods at r12%\frac{r}{12}\%, not 6 periods at r%r\%.

  • Money to the wrong accuracy

    if Example 8.1(a) had said "to 2 d.p.", the answer would be €6203.51, not €6203.5 or €6204. The first money answer at the wrong accuracy loses an (markschemes usually take it only once).

  • Solver values not written down

    a correct answer from the financial solver earns the method marks it implies, even with no values shown. The risk is a wrong or mis-rounded answer: then the written values are the only evidence of method, and without them the method mark is lost too. Write the number of periods, the rate, PVPV and FVFV (with opposite signs).

More mistakes (5)

  • Depreciation with a plus sign

    losing 15% per year means multiplying by 0.850.85.

  • "Complete years" rounded wrongly

    check the value at the ends of the two whole years either side.

  • Interest earned vs final value

    interest is FV−PVFV - PV.

  • Rounding the rate first

    using 12.4%12.4\% in Example 8.3(b) gives €8520 instead of €8550 and loses the . Keep the full multiplier (keep full-accuracy values).

  • Comparing a real value with a future price

    in Example 8.4(c), compare €3553 (money in 10 years) with the inflated price, not €3072 (today's money).

Keep going

Easy to confuse with

Next typeArithmetic or geometric in context

More in Sequences and series

Next: Arithmetic or geometric in context
Try one, marked free