Engineering and finance formulas in PDF

# Engineering and finance formulas

## Signals and control

A first-order low-pass filter, its cut-off frequency and the sampling theorem:

$ H(s) = \frac{1}{1 + sRC} \qquad f_c = \frac{1}{2\pi RC} \qquad f_s > 2 f_{\max} $

$ |H(j\omega)| = \frac{1}{\sqrt{1 + (\omega R C)^2}} \qquad \text{gain in dB} = 20 \log_{10} |H(j\omega)| \qquad y[n] = \sum_{k=0}^{M} b_k\, x[n-k] - \sum_{k=1}^{N} a_k\, y[n-k] $

The PID controller and the discrete Fourier transform:

$ u(t) = K_p\, e(t) + K_i \int_0^t e(\tau)\,\mathrm{d}\tau + K_d\, \frac{\mathrm{d}e(t)}{\mathrm{d}t} \qquad X_k = \sum_{n=0}^{N-1} x_n\, e^{-2\pi i k n / N} $

## Electrical

$ V = IR \qquad P = VI = I^2 R = \frac{V^2}{R} \qquad \sum_{k} I_k = 0 \qquad Z = R + j\left(\omega L - \frac{1}{\omega C}\right) \qquad \omega_0 = \frac{1}{\sqrt{LC}} $

## Mechanical and civil

Deflection of a simply supported beam under a central load, stress and strain, and the Reynolds number:

$ \delta_{\max} = \frac{F L^3}{48\, E I} \qquad \sigma = \frac{F}{A} = E\,\varepsilon \qquad I_{\text{rect}} = \frac{b h^3}{12} \qquad \mathrm{Re} = \frac{\rho v L}{\mu} $

$ Q = A v \qquad p_1 + \tfrac{1}{2}\rho v_1^2 + \rho g h_1 = p_2 + \tfrac{1}{2}\rho v_2^2 + \rho g h_2 \qquad \Delta p = f\, \frac{L}{D}\, \frac{\rho v^2}{2} $

## Interest and loans

Compound interest, continuous compounding, and the payment on an annuity loan of principal $P$ at periodic rate $r$ over $n$ periods:

$ A = P \left( 1 + \frac{r}{m} \right)^{mt} \qquad A = P e^{rt} \qquad \text{payment} = P\, \frac{r (1 + r)^n}{(1 + r)^n - 1} $

A loan of $P = 250{,}000$ at $0.35\,\%$ a month over $n = 300$ months costs $250{,}000 \times \dfrac{0.0035 \cdot 1.0035^{300}}{1.0035^{300} - 1} \approx 1{,}518$ a month.

## Investment appraisal

$ \mathrm{NPV} = \sum_{t=0}^{T} \frac{C_t}{(1 + r)^t} \qquad \mathrm{IRR}: \sum_{t=0}^{T} \frac{C_t}{(1 + \mathrm{IRR})^t} = 0 \qquad \mathrm{WACC} = \frac{E}{V} r_E + \frac{D}{V} r_D (1 - \tau) $

The capital asset pricing model and the Sharpe ratio:

$ \mathbb{E}[R_i] = R_f + \beta_i \left( \mathbb{E}[R_m] - R_f \right) \qquad \beta_i = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(R_m)} \qquad S = \frac{\mathbb{E}[R] - R_f}{\sigma} $

## Option pricing

The Black-Scholes partial differential equation and the price of a European call:

$ \frac{\partial V}{\partial t} + \tfrac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - rV = 0 $

$ C = S_0\, N(d_1) - K e^{-rT} N(d_2), \qquad d_1 = \frac{\ln(S_0 / K) + \left( r + \tfrac{1}{2}\sigma^2 \right) T}{\sigma \sqrt{T}}, \qquad d_2 = d_1 - \sigma \sqrt{T} $

| Greek | Definition | Call |
|:------|:----------:|:-----|
| Delta | $\dfrac{\partial C}{\partial S}$ | $N(d_1)$ |
| Gamma | $\dfrac{\partial^2 C}{\partial S^2}$ | $\dfrac{N'(d_1)}{S_0 \sigma \sqrt{T}}$ |
| Theta | $\dfrac{\partial C}{\partial t}$ | $-\dfrac{S_0 N'(d_1) \sigma}{2\sqrt{T}} - r K e^{-rT} N(d_2)$ |
| Vega | $\dfrac{\partial C}{\partial \sigma}$ | $S_0 \sqrt{T}\, N'(d_1)$ |