Equations and Business Applications
Mathematics
Equations and Business Applications
Syllabus tag: KASNEB CPA | Foundation Level | CA15 Quantitative Analysis | Topic 2 Equations and Business Applications
Lesson objectives
By the end of this topic, you will be able to:
- Compute simple and compound interest
- Compute present and future values
- Value an annuity and a perpetuity
- Compute a loan instalment and a sinking fund payment
- Distinguish nominal from effective rates
Why this matters
This is the arithmetic of money over time, and it is the foundation of the whole Financial Management syllabus. Every discounted cash flow in CA22 and CA33 uses the formulae below.
Simple and compound interest
Simple interest = P × r × n. The principal earns; the interest does not.
Compound interest: A = P(1 + r)ⁿ
KES 250,000 at 9% for 6 years: 250,000 × 1.09⁶ = KES 419,275.03
Simple interest would give 250,000 + (250,000 × 0.09 × 6) = 385,000. The KES 34,275 difference is interest earned on interest — and it grows sharply with the term, which is why the distinction matters more over long periods than short ones.
Present value
Discounting is compounding reversed:
PV = FV / (1 + r)ⁿ
KES 500,000 receivable in 5 years at 11%: 500,000 / 1.11⁵ = KES 296,725.66
That figure is what the future sum is worth today. The discount factor 1 / 1.11⁵ = 0.5935 is the same number expressed per shilling, and tables give it directly.
Annuities
An annuity is a constant sum each period for a fixed number of periods.
PV of an annuity = A × [1 − (1 + r)⁻ⁿ] / r
KES 60,000 a year for 8 years at 10%:
60,000 × [1 − 1.10⁻⁸] / 0.10 = 60,000 × 5.3349 = KES 320,095.57
The bracketed figure 5.3349 is the annuity factor, and tables give it directly. Note it is not 8 — receiving 60,000 for eight years is worth far less than eight times 60,000.
Ordinary annuity or annuity due. The formula above assumes payments at the end of each period. Where payments are made at the beginning, multiply by (1 + r). An examiner will state which, and applying the wrong one is a common error.
Perpetuity
A payment continuing for ever:
PV = A / r
KES 40,000 a year in perpetuity at 8% is worth 40,000 / 0.08 = KES 500,000
Growing perpetuity: PV = A / (r − g), which is the dividend growth model from CA22 seen in its algebraic form.
Loan repayments
Rearranging the annuity formula for the instalment:
A = PV × r / [1 − (1 + r)⁻ⁿ]
A loan of KES 1,500,000 over 5 years at 12%:
A = 1,500,000 × 0.12 / [1 − 1.12⁻⁵] = KES 416,114.60 a year
Total repaid = 5 × 416,114.60 = 2,080,573, so interest is 580,573 on a 1,500,000 loan.
An amortisation schedule shows each instalment split between interest and capital. Early instalments are mostly interest, because interest is charged on the larger outstanding balance; later ones are mostly capital. The instalment never changes, but its composition does.
:::checkpoint A borrower complains that after two years of a five-year loan, less than half the principal has been repaid. Explain why this is normal rather than an error. :::
Sinking funds
A sinking fund accumulates a sum by regular deposits — the reverse of a loan.
A = FV × r / [(1 + r)ⁿ − 1]
To accumulate KES 2,000,000 in 6 years at 8%:
A = 2,000,000 × 0.08 / [1.08⁶ − 1] = KES 272,630.77 a year
Total deposited = 1,635,785, so interest contributes the remaining 364,215. Sinking funds are used to provide for a known future obligation — replacing an asset, or redeeming a debenture on its maturity date.
Nominal and effective rates
A nominal rate is quoted annually but compounded more often. The effective rate is what is actually earned or paid.
Effective = (1 + i/m)^m − 1, where m is the number of compounding periods.
12% nominal compounded quarterly:
(1 + 0.03)⁴ − 1 = 12.55%
Compounded monthly it rises to 12.68%, and daily to 12.75%.
Always compare effective rates. Two loans quoted at 12% are not the same loan if one compounds monthly and the other annually, and the comparison is meaningless until both are expressed on the same basis.
:::checkpoint Lender A offers 11.8% compounded monthly and Lender B offers 12% compounded annually. Set out the calculation you would perform to choose between them, and say which figure decides it. :::