Renty

Ogólne wzory na renty

$P = \sum_{j = 1}^{n}{R_{j}\left( 1 + i \right)^{- j}}\ $

$F = \sum_{j = 1}^{n}{R_{j}{(1 + i)}^{n - j}}$

P(1 + i)ⁿ = F

Stałe raty płatne z dołu

$P = R\sum_{j = 1}^{n}\left( 1 + i \right)^{- j} = R\frac{1 - \left( 1 + i \right)^{- n}}{i} = Ra_{\overset{\overline{}}{n}|i} = R*PVIFA\left( n,i \right)$

$F = \sum_{j = 1}^{n}{R_{j}\left( 1 + i \right)^{n - j}} = R\frac{{(1 + i)}^{n} - 1}{i} = Rs_{\overset{\overline{}}{n}|i} = R*FVIFA(n,i)\ $

$n = - \frac{\ln\left( 1 - \frac{\text{iP}}{R} \right)}{\ln\left( 1 + i \right)} = \frac{\ln\left( 1 + \frac{\text{iF}}{R} \right)}{\ln\left( 1 + i \right)}$

$\frac{P}{R} - \frac{1 - \left( 1 + i \right)^{- n}}{i} = 0 = \frac{F}{R} - \frac{\left( 1 + i \right)^{- n} - 1}{i}\ $

Stałe raty płatne z góry

$P^{( + 1)} = R\sum_{j = 0}^{n - 1}\left( 1 + i \right)^{- j} = R\left( 1 + i \right)\frac{1 - \left( 1 + i \right)^{- n}}{i} = R{\ddot{a}}_{\overset{\overline{}}{n}|i} = R\left( 1 + i \right)a_{\overset{\overline{}}{n}|i} = R\left( 1 + a_{\overset{\overline{}}{n - 1}|i} \right)$

$F^{( + 1)} = R{(1 + i)}^{n}\sum_{j = 0}^{n - 1}\left( 1 + i \right)^{- j} = R\left( 1 + i \right)\frac{\left( 1 + i \right)^{- n} - 1}{i} = R{\ddot{s}}_{\overset{\overline{}}{n}|i} = R\left( 1 + i \right)s_{\overset{\overline{}}{n}|i} = R(s_{\overset{\overline{}}{n - 1}|i} - 1)$

Renta odroczona stałe raty płatne z dołu, pierwsza płatność za H+1 mcy/lat

$P^{( - H)} = R\sum_{j = 1}^{n}\left( 1 + i \right)^{- H - j} = R\left( 1 + i \right)^{- H}a_{\overset{\overline{}}{n}|i}$

$F^{( - H)} = R\sum_{j = 1}^{n}\left( 1 + i \right)^{n - j} = Rs_{\overset{\overline{}}{n}|i}$

F^( − H) = (1 + i)^n + HPV^( − H)

Renta wieczysta, płatna z dołu

$P_{\infty} = R\operatorname{}\frac{1 - \left( 1 + i \right)^{- n}}{i} = \frac{R}{i}$

$\frac{P_{\infty} - P}{P} = \frac{\frac{R}{i} - Ra_{\overset{\overline{}}{n}|i}}{Ra_{\overset{\overline{}}{n}|i}} = \frac{1}{ia_{\overset{\overline{}}{n}|i}} - 1 = \frac{1}{\left( 1 + i \right)^{- n} - 1}$

Renta o ratach seriami stałych, płatnych z dołu

$P = \sum_{l = 1}^{L}{R_{l}\left( 1 + i \right)^{- \left( t_{l} - 1 \right)}}a_{\overset{\overline{}}{n_{l}}|i}$

$F = {(1 + i)}^{n}\sum_{l = 1}^{L}{R_{l}\left( 1 + i \right)^{- \left( n_{l} + t_{l} - 1 \right)}}s_{\overset{\overline{}}{n_{l}}|i}$

Renta o ratach tworzących ciąg arytmetyczny płatnych z dołu

$P_{a} = R_{1}a_{\overset{\overline{}}{n}|i} + \frac{d}{i}(a_{\overset{\overline{}}{n}|i} - n{(1 + i)}^{- n})$

$F_{a} = R_{1}s_{\overset{\overline{}}{n}|i} + d\sum_{j = 1}^{n - 1}s_{\overset{\overline{}}{j}|i} = R_{1}s_{\overset{\overline{}}{n}|i} + \frac{d}{i}(s_{\overset{\overline{}}{n}|i} - n)$

Renta o ratach tworzących ciąg geometryczny płatnych z dołu

$P_{g} = R_{1}\sum_{j = 1}^{n}{q^{j - 1}{(1 + i)}^{- j}} = \frac{R_{1}}{q}\sum_{j = 1}^{n}{(\frac{1 + i}{q})}^{- j} = \frac{R_{1}}{q}a_{\overset{\overline{}}{n}|p} = \frac{R_{1}}{q}\ \frac{1 - \left( 1 + p \right)^{- n}}{p}\ \ \ \ \ \ \ \ \ \ p = \frac{1 + i}{q} - 1$

$F_{g} = R_{1}\sum_{j = 1}^{n}{q^{j - 1}{(1 + i)}^{n - j}} = R_{1}q^{n - 1}\sum_{j = 1}^{n}{(\frac{1 + i}{q})}^{n - j} = R_{1}q^{n - 1}s_{\overset{\overline{}}{n}|p} = R_{1}q^{n - 1}\frac{{(1 + p)}^{n} - 1}{p}\ \ \ \ \ \ \ \ \ \ \ p = \frac{1 + i}{q} - 1$

Renta uogólniona

TI   R = R_O      n = n_O      i = (i + p)^m_K − 1

$TII\ R = R_{O}\ \ \ \ \ \ n = n_{O}\ \ \ \ \ \ i = {(i + p)}^{\frac{1}{m_{B}}} - 1$

$TI\ \ \ R = \frac{R_{O}}{s_{\overset{\overline{}}{m_{K}}|i_{O}}}\ \ \ \ \ \ n = n_{O}m_{K}\ \ \ \ \ \ i = p$

$TII\ R = R_{O}\ s_{\overset{\overline{}}{m_{B}}|i_{O}}\ \ \ \ \ n = \frac{n_{O}}{m_{B}}\ \ \ \ \ \ i = p$

Skumulowana rata końcowa

$R = R_{O}\sum_{j = 0}^{m_{B} - 1}{\left( 1 + \frac{j}{m_{B}} \right)i} = R_{O}\left( m_{B} + \frac{m_{B} - 1}{2}i \right)\text{\ \ \ \ \ \ \ \ \ \ \ }$ n = n_O      i = p

Skumulowana rata początkowa

$R = R_{O}\sum_{j = 1}^{m_{B}}\left( 1 + \frac{j}{m_{B}} \right)^{- 1}$ $n = \frac{n_{O}}{m_{B}}\ \ \ \ \ \ \ \ \ i = p\ \text{\ \ \ }$

Dług

$K_{0} = \sum_{j = 1}^{n}{R_{j}\left( 1 + i \right)^{- j}}\ $

$K_{0}\left( 1 + i \right)^{n} = \sum_{j = 1}^{n}{R_{j}\left( 1 + i \right)^{n - j}}\ $

$K_{0}\left( 1 + i \right)^{j} = \sum_{m = 1}^{j}{R_{j}\left( 1 + i \right)^{j - m}} + \sum_{m = j + 1}^{n}{R_{j}\left( 1 + i \right)^{j - m}}\ $

$K_{j} = \sum_{m = j + 1}^{n}{R_{j}\left( 1 + i \right)^{j - m}} = K_{j - 1}\left( 1 + i \right) - R_{j}$

$K_{O} = \sum_{j = 1}^{n}\left( K_{j - 1} - K_{j} \right)$

I_j = K_j − 1i

R_j = I_j + U_j

U_j = K_j − 1 − K_j

$K_{O} = \sum_{j = 1}^{n}U_{j}$

$K_{j} = K_{O} - \sum_{m = 1}^{j}U_{m} = \sum_{m = j + 1}^{n}U_{m}$

$I_{(0)} = \sum_{j = 1}^{n}{I_{j}{(1 + i)}^{- j}}$

Rata annuitetowa

$K_{0} = R\sum_{j = 1}^{n}\left( 1 + i \right)^{- j} = Ra_{\overset{\overline{}}{n}|i}$

$K_{j} = K_{0}\left( 1 + i \right)^{j}\left( 1 - \frac{a_{\overset{\overline{}}{j}|i}}{a_{\overset{\overline{}}{n}|i}} \right) = K_{0}\left( 1 + i \right)^{j} - Rs_{\overset{\overline{}}{j}|i} = K_{0}\frac{a_{\overset{\overline{}}{n - j}|i}}{a_{\overset{\overline{}}{n}|i}}$

U_j = (R−K₀i)(1+i)^j − 1 = U₁(1+i)^j − 1 = U_j − 1(1+i)

$I_{(0)} = K_{0}\frac{n}{1 + i}\left( R - K_{0}i \right)$

Renta o stałej części kapitałowej

$U = \frac{K_{0}}{n}$

K_j = K₀ − jU

I_j = I_j − 1 − Ui = (n−j+1)Ui

R_j = [1+(n−j+1)i]U

$I_{(0)} = \left( n - a_{\overset{\overline{}}{n}|i} \right)U$

Spłata odsetek w jednej racie i stałe raty kapitałowe

$U = \frac{K_{0}}{n}$

$I_{m} = \left( n - a_{\overset{\overline{}}{n}|i} \right)\left( 1 + i \right)^{m}U$

$R_{j} = \left\{ \begin{matrix} I_{m} + U,j = m \\ U\ \ \ ,j \neq m \\ \end{matrix} \right.\ $
Bieżąca spłata odsetek i zwrot kapitału w ostatniej racie

I_j = K₀i

$R_{j} = \left\{ \begin{matrix} K_{0}i,j < n \\ K_{0}i + K_{0}\ \ \ ,j = n \\ \end{matrix} \right.\ $

I₍₀₎ = K₀[1−(1+i)⁻ⁿ]

Spłata długu przez fundusz umorzeniowy

$F = Zs_{\overset{\overline{}}{m}|i_{\text{FU}}} = K_{0}$

Spłata długu przy oprocentowaniu prostym
$K_{0} = \ \sum_{j = 1}^{n}{R_{j}\left( 1 + ij \right)^{- 1}}$

$K_{0}(1 + ni) = \ \sum_{j = 1}^{n}{R_{j}(1 + ij)}$

$\sum_{\alpha = 1}^{a}{A_{\alpha}\left( 1 + r \right)^{- t_{\alpha}}} = \sum_{\beta = 1}^{b}{B_{\beta}\left( 1 + r \right)^{- t_{\beta}}}$

Inwestycje

$NPV = \sum_{j = 0}^{n}{C_{j}\left( 1 + r \right)^{- t_{j}}}$

$\text{NPV}_{\text{Ob}} = - C_{0} + C\sum_{j = 1}^{n}{\left( 1 + r \right)^{- t_{j}} + \left( 1 + r \right)^{- t_{n}}}$

$\sum_{j = 0}^{n}{C_{j}\left( 1 + IRR \right)^{- t_{j}}} = 0$

$\text{IRR}_{\text{Ob}} = \left( 1 + \frac{C}{k} \right)^{k} - 1\ ,C_{0} = N$

Średni czas trwania (duration)

$C_{0} = P_{0} = \sum_{j = 1}^{n}{C_{j}\left( 1 + r^{*} \right)^{- t_{j}}}$

D=$\sum_{j = 1}^{n}{t_{j}\frac{C_{j}\left( 1 + r^{*} \right)^{- t_{j}}}{P_{0}}} = \frac{{P_{0}}^{'}\left( r \right)}{P_{0}\left( r \right)}\left( 1 + r \right) = \frac{\sum_{j = 1}^{n}{t_{j}C_{j}\left( 1 + r^{*} \right)^{- t_{j} - 1}}}{\sum_{j = 1}^{n}{C_{j}\left( 1 + r^{*} \right)^{- t_{j}}}}\left( 1 + r \right) = - \frac{\frac{dP_{0}}{d(1 + r)}}{\frac{P_{0}}{1 + r}}$

$\frac{P_{0}}{P_{0}} \approx - Dr$

Okres zwrotu inwestycji

$\sum_{j = 0}^{k}{C_{j}\left( 1 + r \right)^{- t_{j}}} < 0\ \ \ i\ \sum_{j = 0}^{k + 1}{C_{j}\left( 1 + r \right)^{- t_{j}}} = 0\ \ \ = > \ \ T = t_{k + 1}$

$\sum_{j = 0}^{k}{C_{j}\left( 1 + r \right)^{- t_{j}}} < 0\ \ \ i\ \sum_{j = 0}^{k + 1}{C_{j}\left( 1 + r \right)^{- t_{j}}} > 0\ \ = > \ \ \text{Tϵ}\left( t_{k};t_{k + 1} \right\rbrack\ \ i\ T = t_{k} + t^{*}\ ,\ \sum_{j = 0}^{k}{C_{j}\left( 1 + r \right)^{- t_{j}}} + C_{k + 1}\frac{t^{*}}{t_{k + 1} - t_{k}}{(1 + r)}^{- (t_{k} + t^{*})} = 0$