Financial mathematics is based on the principle that one person's money may be used by another for a specified period. In return, the borrower pays interest, usually in the form of periodic payments, to compensate the lender for the use of the funds. Interest is fundamental to banking, insurance, investments, mortgages, annuities, perpetuities, and sinking funds. Without periodic interest payments, the concept of lending and borrowing would not function as it does in financial systems. Therefore, Option C correctly identifies the financial return provided in exchange for the use of another person's money.

The logical functioning of a financial institution follows these steps: Step 1: People establish finance companies to provide financial services. Step 2: Individuals borrow or use money belonging to others. Step 3: Borrowers make periodic interest payments as compensation. Step 4: These payments accumulate and generate profits for the finance company. Thus, the correct sequence is II → I → III → IV Hence, Option D is correct.

Each financial concept has a distinct definition. Perpetuity is an annuity whose payments continue indefinitely. Sinking Fund consists of regular periodic deposits made to accumulate money for a future obligation. Interest is the return earned by the lender for allowing another party to use money. Therefore, the correct matching is 1 → a 2 → b 3 → c Hence, Option A is correct.

The present value formula depends on when the payments are made. For an end-period perpetuity (ordinary perpetuity), P = R/i For a beginning-period perpetuity (annuity due), P = R + R/i = R(1 + i)/i Statement (III), P = R(1 + i)ⁿ represents a compound amount expression rather than the present value of a perpetuity. Therefore, only Statements I and II are correct. Hence, Option B is correct.

A perpetuity consists of an infinite sequence of equal periodic payments. Although its present value is finite because future payments are discounted, P = R/i the total undiscounted payments continue forever and therefore are not finitely bounded. Thus, Statement C is incorrect because it contradicts the defining characteristic of a perpetuity. Hence, Option C is the correct answer.

A perpetuity pays an equal amount indefinitely. If we consider the undiscounted cumulative payments, the total after (n) payments is nR As n → ∞ the cumulative amount becomes ∞ It is important to distinguish this from the present value, P = R/i which is finite because future payments are discounted. Since the question refers to the undiscounted accumulation, the function is unbounded. Hence, Option D is correct.

The perpetuity formula P = R/i is derived under the assumption that the borrower always makes the required periodic payments. It is a deterministic model, meaning that all future cash flows are assumed to occur with certainty. Since default risk is not included in this model, the implied probability of default is 0%. Therefore, Option A is correct.

For a beginning-period perpetuity, P = R + R/(1 + i) + R/(1 + i)² + ⋯ Factoring out R, P = R[1 + (1 + i)⁻¹ + (1 + i)⁻² + ⋯] The infinite geometric series (1 + i)⁻¹ + (1 + i)⁻² + ⋯ = 1/i Hence, 1 + (1 + i)⁻¹ + (1 + i)⁻² + ⋯ = 1 + 1/i Therefore, Option B is correct.

The terms of the series are R/(1 + i), R/(1 + i)², R/(1 + i)³, … The ratio of consecutive terms is (R/(1 + i)²) ÷ (R/(1 + i)) = 1/(1 + i) = (1 + i)⁻¹ Since every consecutive pair has the same ratio, r = (1 + i)⁻¹ Therefore, Option C is correct.

The standard perpetuity formula is P = R/i where (P) = Present value, (R) = Periodic payment, (i) = Interest rate per period. The variable (i) does not represent the number of periods. A perpetuity continues indefinitely, so there is no finite value of (n) involved in the formula. Therefore, the Assertion is false. The Reason correctly states that (i) denotes the periodic interest rate, making it true. Hence, Option D is the correct answer.

For a nominal annual interest rate compounded (m) times per year, i = r/m where r = 0.04 m = 2 Therefore, i = 0.04/2 = 0.02 This periodic rate is substituted into the perpetuity formula P = R/i Hence, the correct value of i is 0.02 Therefore, Option A is correct.

Since the lectures are paid for at the beginning of each year, this is a beginning-period perpetuity (annuity due). The appropriate formula is P = R + R/i Substituting, R = 2500, i = 0.03 gives, P = 2500 + (2500/0.03) = 2500 + 83333.33 = 85833.33 Thus, the Assertion is true. However, the Reason is false because the formula P = R/i applies only to an end-period perpetuity, not a beginning-period perpetuity. Hence, Option B is correct.

The sinking fund receives equal quarterly deposits throughout the 10-year period. If every quarterly deposit is R, R, R, R, … then the simple moving average over any set of consecutive quarters is (R + R + ⋯ + R) / n = R where (n) is the number of quarters considered. Since all deposits are identical, the moving average is always equal to the deposit amount itself. Hence, Option C is correct.

The passage explicitly states that the sinking fund is created so that sufficient money is available to purchase a new machine when the existing machine reaches the end of its 12-year useful life. The fund accumulates through regular annual deposits while earning interest. At the end of the machine's life, the accumulated amount, together with the scrap value, helps finance the replacement. Therefore, the principal operational purpose of the sinking fund is replacement of the machine. Hence, Option D is correct.

The sinking fund needs to accumulate only the net replacement amount, since the machine's scrap value is recovered at the end of its useful life. Using the standard relationship, A = Replacement Cost − Scrap Value Substituting the given values, A = 100000 − 5000 = 95000 Therefore, the required accumulated amount is ₹95,000 Hence, Option A is correct.

The sinking fund formula is A = R(Sₙ|ᵢ) Rearranging, R = A / Sₙ|ᵢ Substituting the given values, R = 48000 / 13.8164480 ≈ 3474.12 Therefore, the annual deposit required is ₹3,474.12 Hence, Option B is correct.

A sinking fund is established for a specific financial obligation, such as repaying debt or replacing an asset. Equal periodic contributions are calculated using financial mathematics so that the required amount is available on the due date. Unlike a general savings account, which may be used for any purpose, a sinking fund enforces systematic and planned deposits toward a clearly defined objective. Therefore, Option C correctly explains why a sinking fund is more effective for meeting future debt obligations.

For a sinking fund with deposits made at the beginning of each period, the payments form an annuity due rather than an ordinary annuity. The accumulated amount is expressed as A = R(S̄₍ₙ₊₁₎|ᵢ − 1) which is equivalent to multiplying the ordinary annuity accumulation factor by an additional growth factor. Therefore, Option D correctly represents the annuity due form of the sinking fund formula.

Since deposits are made at the beginning of each year, the accumulation follows the annuity due formula. The accumulated amount is A = R(S̄₁₁|₀.₁₂ − 1) Rearranging, R = A / (S̄₁₁|₀.₁₂ − 1) Substituting the given values, R = 100000 / 19.65458 ≈ 5087.87 Therefore, the required annual deposit is ₹5,087.87 Hence, Option A is correct.

The amount that must be accumulated in a sinking fund is A = Replacement Cost − Salvage Value If the replacement cost remains unchanged while the salvage value increases, then A ↓ (i.e., the required accumulated amount decreases). Since the periodic deposit is calculated using R = A / S̄ₙ|ᵢ a smaller value of (A) results in a smaller value of (R). Therefore, the required periodic contribution decreases. Hence, Option B is correct.