%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.
interest value: \(3\% K = 0.03 K\).
increase : \(K + 3\% K = (1+3\%) K = 1.03 K\).
decrease : \(K - 3\% K = (1-3\%) K = 0.97 K\).
\(n\) 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)
\(i\) normally refers to a full year (annual interest rate, effective annual interest rate)
\(i_{12}\) is a monthly interest rate
\(i_4\) is a quarterly interest rate
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:
\(K_n = K_0 q^n\)
compound interests: multiply the starting capital (principal) with \(q^n\) to get the value \(n\) periods later.
discount interests: multiply the capital with \(q^{-n}\) to get the value \(n\) periods earlier.
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
\(i_{12} = i/12\)
\(i_4 = i/4\)
With compounding the effective annual interest rate is calculated with the conformal conversion: Effective \(i_{eff}\) distinguishes from nominal interest rate \(i\).
\(i_{eff} = (i_{12} + 1)^{12} - 1\)
\(i_{eff} = (i_4 + 1)^4 - 1\)
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
First one finds the future value \(R_n\).
This \(R_n\) needs to be compounded to the end of the other annuity.
Using the annuity formula one can calculate the requested quantity (\(n\), \(q\), \(r\)) of the new annuity.
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.