Mathematics of Finance for CA Foundation
Simple and compound interest, effective rates, present and future values, annuities and depreciation: the money maths of Paper 3.
Quantitative AptitudePaper 3, Business Mathematics13 MCQs2 lessons2 flashcards3 memory tricks
Practise Mathematics of Finance free
What you'll learn
- Simple and compound interest
Calculate SI, CI, effective rates and compounding for different periods. - Present value, annuities and depreciation
Use present value, annuity and depreciation formulas.
Key formulas
| Simple interest | SI = P × R × T / 100 |
|---|---|
| Compound amount | A = P(1 + r/100)ⁿ |
| Compounded m times a year | A = P(1 + r/100m)ᵐⁿ |
| Effective rate | E = (1 + i/m)ᵐ − 1 |
| CI − SI for 2 years | P(r/100)² |
| Present value | PV = A ÷ (1 + i)ⁿ |
| FV of ordinary annuity | A = P[(1 + i)ⁿ − 1] ÷ i |
| PV of ordinary annuity | V = P[1 − (1 + i)⁻ⁿ] ÷ i |
Concepts explained
Simple interest vs compound interest
Simple interest is earned only on the principal; compound interest is earned on the principal plus the interest already added.
What it means
- SI = PRT/100: the same interest every year.
- CI: A = P(1 + r/100)ⁿ; CI = A − P. Interest grows each year because it is earned on interest too.
- For 2 years, CI − SI = P(r/100)²; for 3 years, CI − SI = P(r/100)²(3 + r/100).
- At SI, money doubles when R × T = 100; at CI, by the rule of 72, in about 72 ÷ r years.
- P = ₹5,000, R = 8%, T = 3
- SI = 5,000 × 8 × 3 ÷ 100
Answer: ₹1,200
- P = ₹10,000, R = 10%, n = 2
- A = 10,000 × 1.21 = 12,100
Answer: CI = ₹2,100
- Same figures
- CI − SI = 10,000 × (0.1)²
Answer: ₹100
Related terms
- Amount: principal plus interest
- Rule of 72: years to double ≈ 72 ÷ rate
Exam tip: Read whether the question asks for the interest or the amount: both appear as options.
Nominal rate, compounding frequency and effective rate
The effective rate is the actual annual return after compounding: E = (1 + i/m)ᵐ − 1, where i is the nominal rate and m the number of compoundings a year.
What it means
- Compounding more often raises the effective rate above the nominal rate.
- For calculations: rate per period = i/m, number of periods = m × years.
- Continuous compounding gives the upper limit: E = eⁱ − 1.
- 10% compounded half-yearly
- E = (1.05)² − 1
Answer: 10.25%
- 12% compounded monthly
- E = (1.01)¹² − 1
Answer: ≈ 12.68%
- ₹1,000 at 12% quarterly for 1 year
- A = 1,000 × 1.03⁴
Answer: ≈ ₹1,125.51
Related terms
- Nominal rate: the stated annual rate before compounding
- Continuous compounding: A = Peⁱᵗ
Exam tip: Options often include the half-yearly answer when the question says quarterly: check m.
Present value, annuities and perpetuities
Present value discounts future money to today: PV = A ÷ (1 + i)ⁿ. An annuity is a series of equal payments at regular intervals.
What it means
- Ordinary annuity: payments at the end of each period. FV = P[(1 + i)ⁿ − 1] ÷ i; PV = P[1 − (1 + i)⁻ⁿ] ÷ i.
- Annuity due: payments at the start; multiply ordinary values by (1 + i).
- Perpetuity: payments forever; PV = P ÷ i.
- Sinking fund: periodic deposits to build a known future sum; reducing-balance depreciation: V = P(1 − r)ⁿ.
- ₹1,100 due in 1 year at 10%
- PV = 1,100 ÷ 1.1
Answer: ₹1,000
- ₹1,000 a year forever at 10%
Answer: PV = ₹10,000
- ₹10,000 a year for 3 years at 10%
- PV = 10,000 × 2.487
Answer: ≈ ₹24,869
Related terms
- EMI: the instalment of a loan treated as an annuity
- NPV: PV of inflows − PV of outflows
Exam tip: Discounting divides by (1 + i)ⁿ; subtracting a percentage gives the wrong answer.
Memory tricks
Rate per period = r/m; number of periods = m × n.
PV = FV ÷ (1 + i)ⁿ.
FV and PV of an annuity due are the ordinary annuity values × (1 + i).
Try 3 CA Foundation-style questions
Q1. ₹1,000 is invested at 12% per annum compounded quarterly for 1 year. The amount is about
- ₹1,120.00
- ₹1,123.60
- ₹1,480.24
- ₹1,125.51
Show answer
Answer: (4) ₹1,125.51
Quarterly rate 3%, 4 periods: A = 1,000 × 1.03⁴ ≈ ₹1,125.51.
Q2. Compared with an ordinary annuity, the future value of an annuity due of the same payments is
- lower by a factor of (1 + i)
- exactly the same
- higher by a factor of (1 + i)
- half as much
Show answer
Answer: (3) higher by a factor of (1 + i)
In an annuity due, each payment is made at the start of the period, earning one extra period of interest: FV(due) = FV(ordinary) × (1 + i).
Q3. A sinking fund is created mainly to
- pay today's expenses
- lend money to customers
- accumulate money by regular deposits to repay a liability or replace an asset at a future date
- avoid paying interest
Show answer
Answer: (3) accumulate money by regular deposits to repay a liability or replace an asset at a future date
Regular deposits earn compound interest and build up the amount needed on a known future date.
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