Compounding, Discounting, Annuities & Loan Amortization

Explanation

The time value of money provides the quantitative framework for all asset valuation, bond pricing, and capital budgeting. By discounting future nominal cash flows back to the present epoch using an appropriate opportunity cost of capital i, financial analysts can equitably evaluate competing multi-year investment alternatives.

Key points

• Future Value of a Single Sum: FV_n = PV · (1 + i)^n.
• Present Value of a Single Sum: PV = FV_n / (1 + i)^n = FV_n · PVIF(i, n).
• Ordinary Annuity (payments at END of each period): PVIFA = [1 - (1 + i)^{-n}] / i; Annuity Due (payments at BEGINNING): PV_{due} = PV_{ordinary} · (1 + i).
• Perpetuity: Constant stream of cash flows forever: PV = PMT / i.
• Loan Amortization: Calculating equal annual installments (PMT = Principal / PVIFA) and dividing each payment into interest and principal reduction.

Example

Code examples

// Python Calculation: Loan Amortization Schedule
loan = 100000.0  # Loan amount
rate = 0.10      # 10% annual interest
n = 3            # 3 years
# PMT = Loan / PVIFA(10%, 3) = 100000 / 2.48685 = 40211.48
pmt = loan * (rate / (1 - (1 + rate)**(-n)))
print(f"Annual Installment: NPR {pmt:.2f}")

balance = loan
for year in range(1, n + 1):
    interest = balance * rate
    principal = pmt - interest
    balance -= principal
    print(f"Year {year}: Payment={pmt:.2f}, Interest={interest:.2f}, Principal={principal:.2f}, Balance={balance:.2f}")

Exam questions

• [8 Marks] A company borrows NPR 500,000 at 12% annual interest to be repaid in 5 equal annual installments. Construct a complete loan amortization schedule.
• [7 Marks] Differentiate between Ordinary Annuity and Annuity Due. Find the present value of an annuity of NPR 20,000 per year for 6 years at an 8% discount rate.