Mathematical Finance

Formulas

The following variables are used in the points given below:

−  $C_0$ : initial value or actual value
−  $C_n$ : final value after n periods, or future value
−  $i_N$ : nominal or annual interest rate
−  $i$ : periodic interest rate expressed in decimal notation
−  $n$ : number of periods being considered
−  $Pmt$ : value of the annuity (reimbursement or investment)
−  $S_n$ : sum of the capitalized values after $n$ payments
−  $S_0$ : initial value to invest to obtain an annuity of $Pmt$

Specific terminology is written in italics.

• Capitalization after 1 year: $C_n = C_0 + i × C_0$
• Annual simple interest: $I_A = C_0 × i$
• Profit :  $I_{total} = C_n − C_0$
• Final value (simple interest): $C_n = C_0 + (C_0 · i · n)$
• Return rate (expressed as a percentage): $Return = \dfrac{Profit}{C_0} · 100$
• Periodic interest rate : $i = \dfrac{i_N}{n}$

Formulas for an initial fixed amount and compound interest

• Initial value: $C_0 = \dfrac{C_n}{(1 + i)^n}$
• Final value: $C_n = C_0 · (1 + i)^n$
• Number of periods: $n = \dfrac{\textrm{ln} (\frac{C_n}{C_0})}{\textrm{ln}(1 + i)}$
• Interest rate: $i = \sqrt[n]{\dfrac{C_n}{C_0}} \space – 1$

Formulas for a series of regular payments (Pmt) having compound interest that is reinvested at the end of each period.

• Initial value: $S_0 = Pmt · \left(   \dfrac{1 \space – (1+i)^{-n}}{i} \right)$
• Final value:  $S_n = Pmt · \left( \dfrac{(1 + i)^n \space – 1}{i} \right)$
• Number of periods when the present value is known:: $n = \dfrac{- \textrm{ln} \left( 1 \space – \frac{S_0}{Pmt} · i \right)}{\textrm{ln} (1 + i)}$
• Number of periods when the final value is known: $n = \dfrac{ \textrm{ln} \left( 1 \space + \frac{S_0}{Pmt} · i \right)}{\textrm{ln} (1 + i)}$
• Amount of the payments when the present value is known formula used to calculate regular repayments $Pmt$ of a debt $C_0$ :
  $Pmt = S_0 · \left(  \dfrac{i}{1 \space – (1 + i)^{-n}}    \right)$
• Amount of the payments when the final value is known (formula used to calculate the regular deposits $Pmt$ that allow funds to be generated for a given term $C_n$ :
  $Pmt = S_n · \left( \dfrac{i}{(1 + i)^n \space -1}  \right)$