%path = "maths/finance/interest" %kind = kinda["texts"] %level = 10

\(K\) Capital

An amount of money.

\(i\) Interest rate

The increase or decrease of capital \(K\) is notated in percent %=1/100.

\(n\) Period (Year/Quarter/Month/Day)

The interest rate \(i\) always refers to a time period, in which the increase or decrease takes place (is compounded)

After this time period \(K\) has grown by \(iK\), i.e. \(K_{n=1} = K_0 (1+i) = K_0 q\) (q = 1+i).

Compound interest

After one period the capital becomes \(K_{n=1} = K_0 (1+i) = K_0 q\), after n=2 periods \(K_0 q^2\), after n=3 periods \(K_0 q^3\)

After n periods:

Annuity

An annuity is a payment \(r\) in regular time periods. The number of periods for the annuity depends on the payment. The accrued payments make up the lump-sum. This is the pension or annuity formula:

\(R_n = \sum_{m=0}^{n-1} r_m = \sum_{m=0}^{n-1} r q^m = r \frac{q^n - 1}{q-1}\)

The formula can be used to calculate the future value (FV)`R_n` when the interests are compounded at the end of the periods.

Annuity due is when compounded at the beginning: \(R_n^v = q R_n\)

The present value (PV) of an annuity is obtained by discounting from the FV: \(B_n = R_n q^{-n}\).

Compounding periods smaller than a year

To compare the effective annual rate of interest with the rate for the period one converts the rates.

In a linear conversion we use when there is no compounding taking place

With compounding the effective annual interest rate is calculated with the conformal conversion: Effective \(i_{eff}\) distinguishes from nominal interest rate \(i\).

Normally the annual interest rate is given. For a monthly or quarterly compounding this first needs to be converted.

Annuity rest

To calculate the remaining value of the annuity at a certain time one subtracts the future value of the annuity for that time from the capital value for that time.

Convert one annuity to another

Comparison of capitals or offers

To compare values one must first compound their values to the same time (time-value, e.g. present value) using the compounding or annuity formulas.