Essential Financial Mathematics & Index Numbers

• Arithmetic & Geometric Progressions

• AP: fixed addition d

  • General term T_n = a + (n-1)d

  • Sum S_n = \frac{n}{2}\,[2a+(n-1)d]

• GP: fixed multiplier r

  • General term T_n = ar^{n-1}

  • Sum S_n = a\,\frac{1-r^{n}}{1-r}\;(r\neq1)

  • Sum to infinity (|r|<1) S_{\infty}=\frac{a}{1-r}

Simple Interest

• Interest on principal only: I = P i n

• Accrued amount: A_n = P(1+i n)

Compound Interest

• Re-invest interest each period

  • Accrued amount: A_n = P(1+i)^n

  • Present value: P = \frac{A_n}{(1+i)^n}

• Finding rate: i = \sqrt[n]{\tfrac{A}{P}}-1 (use logs if needed)

Nominal vs Effective (APR)

• Nominal rate i_{nom} quoted p.a.; compounded n times per year

• Periodic rate ip = \tfrac{i{nom}}{n}

• APR (effective): APR = (1+i_p)^n-1

Multi-Period Compounding

• Future value with m compounding periods/yr: A = P\,[1+\tfrac{r}{m}]^{n m}

Future Value of a Lump Sum

• FV = P(1+r)^n

Future Value of Annuities

• Ordinary (end-period): FV = A\,\frac{(1+r)^n-1}{r}

• Annuity-due (start-period): FV_{ad}= FV\,(1+r)

• Sinking-fund deposit: A = FV\,\frac{r}{(1+r)^n-1}

Present Value

• Lump sum: PV = FV\,(1+r)^{-n}

• Ordinary annuity: PV = A\,\frac{1-(1+r)^{-n}}{r}

• Uneven cash flows: discount each flow individually and sum.

Loan Amortisation (Capital Recovery)

• Installment for loan P, rate r, term n: A = P\,\frac{r(1+r)^n}{(1+r)^n-1}

• Amortisation schedule splits each payment into interest P_{bal}\,r and principal.

Depreciation

• Straight-line: annual dep. =\frac{Cost-\text{Residual}}{Life}

• Reducing balance: D_n = B(1-i)^n; rate i = 1-\sqrt[n]{\tfrac{D}{B}}

Index Numbers

• Simple price index: \tfrac{pn}{p0}\times100

• Simple quantity index: \tfrac{Qn}{Q0}\times100

• Aggregate (Laspeyres): \tfrac{\sum pn q0}{\sum p0 q0}\times100

• Aggregate (Paasche): \tfrac{\sum pn qn}{\sum p0 qn}\times100

• Fisher (ideal): geometric mean of Laspeyres & Paasche.

• Chain index: each year’s index uses preceding year as base.

• Retail Price Index (RPI): weighted Laspeyres using household weights.

• Deflating series: Real value = \tfrac{Nominal}{Index}\times100

Key Formula Recap

• Simple: I=Pin | Compound: A=P(1+i)^n

• APR: (1+\tfrac{i_{nom}}{n})^{n}-1 | PV annuity: \frac{1-(1+r)^{-n}}{r}

• FV annuity: \frac{(1+r)^n-1}{r} | Loan payment: P\,\frac{r(1+r)^n}{(1+r)^n-1}

• Reducing balance dep.: B(1-i)^n

• Laspeyres price: \frac{\sum pn q0}{\sum p0 q0}\times100