Formulario di matematica finanziaria

It appears you have a set of notes or formulas related to financial mathematics, particularly focusing on annuities and their values under different conventions. Here's an attempt to interpret and present the information in a more readable format: ### 1. Non-Standard Time Periods For time periods that are not multiples of the capitalization period: - **Exponential Convention**: \[ (1 + i)^n = (1 + i)^h \] - **Mixed Linear Convention**: \[ (1 + i)^n = (1 + i)^h(1 + i^{(2)}_h) \] - **Nominal Rate Convertible Convention**: \[ (1 + i)^n = (1 + i)^h\left(1 + \frac{i^{(2)}}{h}\right) \] ### 2. Rendite (Annuities) #### General Formulas - **Value at a Generic Time \( t \)**: \[ A_t = \sum_{k=1}^n a_k \cdot v(t, k) + A_n \] - **Present Value of the Annuity** (\( t = 0 \)): \[ V_0 = \sum_{k=1}^n a_k \cdot v(k, 0) \] - **Future Value of the Annuity** (\( t = n \)): \[ A_n = \sum_{k=1}^n a_k \cdot v(k, k) \] #### Scindible Regime: \[ V_0 = V_0 \cdot v(n, n) \rightarrow V_0 = V_0 \cdot a(0, n) \cdot v(n, 0) \] \[ a(n, n) = a(0, n) \cdot v(0, n) \] ### 3. Periodic Rendite (Annuities) #### Montants - **Future Value of an Annuity**: \[ A_n = \sum_{k=1}^n a_k \cdot v(k, k) \] - **Present Value of an Annuity** (\( t = 0 \)): \[ V_0 = \sum_{k=1}^n a_k \cdot v(k - 1, 0) \] #### Scindible Regime: \[ V_0 = V_0 \cdot v(n, n) + A_n \] \[ A_n = V_0 \cdot a(0, n) \cdot v(n, 0) \] ### 4. Simple Rendite - **Value of the Annuity Using Rational Discount**: \[ /A_{n|y} (y) = \sum_{k=1}^{n+1} a_k [1 + d(y - y_k)] + \sum_{k=1}^n \frac{1}{1 + d(y - y_k)} \] - **Value of the Annuity Using Commercial Discount**: \[ /A_{n|y} (y) = \sum_{k=1}^{n+1} a_k (1 + d(y - y_k)) \] #### Future Value \[ A_n = \sum_{k=1}^n a_k [1 + d(n - k)] \] - **Present Value Rational**: \[ /V_{n|y} (y) = \sum_{k=1}^{n+1} \frac{a_k}{1 + d(y - y_k)} \] - **Present Value Commercial**: \[ /V_{n|y} (y) = \sum_{k=1}^{n+1} a_k (1 - d(y - y_k)) \] ### 5. Unitary Periodic Rendite - **Deferred Annuity (Present Value Commercial)**: \[ V_0 = \left(\frac{1}{v}\right) - v [a(4) + a\left(\