Factor Markets

Factor Markets

🟢 Lite — Quick Review (1h–1d)

Factor Markets in CA Foundation QA covers financial mathematics used to value capital and cash flows over time. The foundational principle is that money today is worth more than the same amount in the future due to earning capacity (time value of money).

Core formulas to memorise:

• Simple Interest: SI = P × r × n
• Compound Interest: FV = P(1 + r)^n
• Present Value: PV = FV ÷ (1 + r)^n
• PV of Ordinary Annuity: PMT × [1 − (1 + r)^−n] ÷ r
• PV of Annuity Due: multiply ordinary annuity formula by (1 + r)
• Capital Recovery Factor: CRF = r(1 + r)^n ÷ [(1 + r)^n − 1]

Exam pointers (CA Foundation Paper 3):

• Numerical questions on PV/FV of annuity carry 3–5 marks each
• Capital recovery and sinking fund questions appear in 2-mark and 4-mark variants
• Always check whether payments are at beginning (annuity due) or end (ordinary annuity) before applying a formula
• Rounding errors compound — carry 4–5 decimal places during calculation

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🟡 Standard — Regular Study (2d–2mo)

Time Value of Money

The time value of money holds that ₹100 available today is worth more than ₹100 receivable in Year 3 because today’s amount can be invested to earn interest. This principle underpins all factor market calculations — present value (PV) discounts future cash flows back to today’s terms, while future value (FV) projects current cash flows forward.

Simple vs Compound Interest

Simple Interest (SI) accrues only on the original principal:

• SI = P × r × n
• Total amount = P(1 + rn)

Compound Interest (CI) accrues on accumulated principal:

• FV = P(1 + r)^n
• Here r must be the rate per compounding period, and n must be the total number of compounding periods.

If compounding is semi-annual, divide r by 2 and multiply n by 2 before substituting.

PV and FV Relationships

$$PV = \frac{FV}{(1+r)^n}$$

Given a future obligation of ₹50,000 due in 5 years at 10% p.a. compounded annually: $$PV = \frac{50000}{(1.10)^5} = ₹31,046.05$$

Annuities

An annuity is a sequence of equal payments at uniform intervals. Two variants matter:

Type	Timing	Formula adjustment
Ordinary Annuity	Payment at period-end	None
Annuity Due	Payment at period-beginning	Multiply PV result by (1+r)

For a 4-year annuity of ₹10,000 at 8% p.a.:

• Ordinary annuity PV: 10,000 × [1 − (1.08)^−4] ÷ 0.08 = ₹33,122.78
• Annuity due PV: 33,122.78 × 1.08 = ₹35,772.60

Capital Recovery Factor

This finds the annual payment required to fully repay a loan (principal + interest) over n periods at rate r: $$CRF = \frac{r(1+r)^n}{(1+r)^n - 1}$$

For a ₹200,000 loan at 12% over 5 years, annual payment = 200,000 × 0.27741 = ₹55,482 per year.

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🔴 Extended — Deep Study (3mo+)

Effective vs Nominal Rate

When compounding frequency exceeds annual (e.g., monthly), the effective annual rate (EAR) must be used: $$EAR = \left(1 + \frac{r_{nominal}}{m}\right)^m - 1$$

A nominal 12% rate compounded monthly gives: $$EAR = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 1.2682 - 1 = 12.68%$$

Failing to convert nominal to effective rate inflates the true cost of borrowing or understates investment returns.

Sinking Fund Factor

Whereas CRF answers “what annual payment repays a loan?”, the Sinking Fund Factor (SFF) answers “what annual deposit accumulates to a target future sum?”:

$$SFF = \frac{r}{(1+r)^n - 1}$$

To accumulate ₹500,000 in 10 years at 9%: Annual deposit = 500,000 × SFF = 500,000 × 0.06565 = ₹32,825 per year.

Note that SFF is simply CRF minus r: CRF = SFF + r.

Common Mistakes

1. Omitting the (1+r) multiplier for annuity due — this is the most frequent error in CA Foundation exams, costing 2–4 marks.
2. Mismatched time units — if r is annual but compounding is quarterly, set r quarterly = r/4 and n = 4×years.
3. Treating non-uniform cash flows as annuity — use individual PV calculations for irregular streams.
4. Confusing SI and CI — SI is linear in n; CI is exponential. For periods >1 year at identical r, CI always produces a larger FV.

Practice Prompts

1. A finance lease requires payment of ₹15,000 at the beginning of each quarter for 3 years at 16% p.a. compounded quarterly. Find the PV of the lease obligation.

2. A company plans to retire a ₹800,000 bond issue in 6 years by accumulating a sinking fund earning 10% p.a. The annual sinking fund deposit is ₹97,695. Verify whether the deposit amount is correct using SFF.

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