Unit 7: Financial Mathematics
Perpetuity, Bonds, EMI, CAGR & Depreciation
1. Perpetuity and Sinking Funds
| Perpetuity (Present Value) | $$PV = \frac{R}{i}$$ (R = Payment, i = Interest rate) |
| Sinking Fund (Periodic Payment) | $$R = \frac{A \cdot i}{(1+i)^n – 1}$$ (A = Future Amount required) |
2. Valuation of Bonds
Value ($$V$$) = PV of Coupons + PV of Redemption Value
| Bond Valuation Formula | $$V = R \left[ \frac{1 – (1+i)^{-n}}{i} \right] + F(1+i)^{-n}$$ |
| Variables | $$R$$ = Periodic Coupon Payment $$F$$ = Face/Redemption Value $$i$$ = Yield to Maturity (Market Rate) $$n$$ = Periods to maturity |
3. EMI Calculation Methods
| Flat-Rate Method | $$\text{EMI} = \frac{P + I}{n} = \frac{P + (P \times r \times t)}{n}$$ |
| Reducing Balance Method | $$\text{EMI} = P \times \frac{i(1+i)^n}{(1+i)^n – 1}$$ (Here $$i$$ is monthly rate, $$n$$ is months) |
4. CAGR & Linear Depreciation
| CAGR Formula | $$\text{CAGR} = \left( \frac{\text{End Value}}{\text{Begin Value}} \right)^{\frac{1}{n}} – 1$$ |
| Linear Depreciation (Annual) | $$D = \frac{C – S}{n}$$ |
| Book Value (at year k) | $$BV_k = C – (D \times k)$$ |