Basic Finexec Terminology

Basic Finance Terminology

The following terminology is applied to the definitions for the Basic Finance category.

Terms

Effective discount rate
Effective interest rate (Compound interest rate)
Force of interest (Continuous compounding)
Frequency of payment within a time unit
Interest rate
Nominal interest rate (Compound interest)
Period
Simple interest rate (Flat interest rate)
Time Unit
Variable interest rates

The time unit is any period or interval of time that has been selected for the purposes of the financial calculation to be carried out. The selection of a suitable time unit is important as it can impact on the way a calculation is carried out. Common time units are:
   yearly
   half yearly
   quarterly
   monthly
   daily
Time units are generally of a constant period for a calculation.

Payment frequencies within a time unit are also commonly used to describe how the interest rate is to be applied for the calculation. For example the interest rate might be described as yearly with payments made monthly.

A period is measured as a number of time units. For example a period can be one time unit or a multiple of time units such as 12, 10.4 or 0.55.

A number of terms are used to describe an interest rate and how the rate is to be used in a calculation. Often terms in the specification of the interest rate are implicit or assumed from general usage eg: 6% pa. An interest rate requires a number of terms for its specification so that it can be interpreted and used for financial calculation in a non ambigious way. The four main terms used for the specification of an interest rate are:

The amount of interest can be described as 6.0% or 0.06 to be used in calculation. To aviod confusuion the amount of interest in the Finexec Project is expressed as a number. For example 0.06 would be used for 6.0%.

The time unit is the period that the amount of interest relates for describing the interest rate. For example if the amount of interest is 0.06 and we have time units of a year or half year then the of interest rate can be described as 0.06 yearly or 0.06 half yearly.

We consider four types of interest rates:
(a) Simple interest rate (Flat interest)
(b) Effective interest rate (Compound interest)
(c) Nominal interest rate (Compound interest)
(d) Force of Interest (Continuous compounding)
These are discussed in further detail below.

This is the time from a starting point of time = 0 from which the interest rate applies and is usually expressed in time units. For example the interest rate that applies from a time 2 years from a starting point.

The simple interest rate is also often called the flat interest rate. Under simple interest the amount of accumulated interest is not compounded at the end of the time unit applying to the simple interest rate. At a 10% yearly simple interest rate $100 accumulates to $110 after 1 year and $120 after two years.

   100 * (1 + 0.1*2)
120

At a 1% monthly simple interest rate $100 accumulates to $136 after 36 months or 3 years.

   100 * (1 + 0.01*36)
136

This is the most common type of interest rate used in financial calculations and is also referred to as the compound interest rate. The time unit is the compunding period for the accumulation of interest irrespecive of any frequency with which interest is paid during the time unit.

$100 invested at a 12% yearly effective interest rate accumulates to $112 at end of the year, irrespective of the frequency with which interest is paid during the year.

   100 * (1 + 0.12)
112

At the same rate $100 accumulates to $105.83 half way through the year.

   100 * (1 + 0.12)^0.5
105.83

$100 invested at a 6% half yearly effective interest rate accumulates to $126.25 at the end of year 2, irrespective of the frequency with which interest is paid during the period.

   100 * (1 + 0.06)^4 
126.248

A nominal interest rate is described in terms of a nominal period. A nominal interest rate then requires an additional item for its description. A payment frequency within the nominal period is required to describe at what intervals or time units interest is compounded.

For example if N is a nominal interest rate for a nominal period and Z and F is the payment frequency within the the nominal period then E = N % F is the effective interest rate for a time unit of Z/F.

For example a 0.12 (ie: 12%) yearly nominal interest rate payable monthly is is equivalent to a 0.01 (ie: 1%) monthly effective interest rate. In this case the nominal period is a year and the payment freqeuncy within the nominal period is monthly (12).

A 0.12 (ie: 12%) yearly nominal interest rate payable at periods of 1.5 years is equivalent to a 0.18 (ie: 18%) effective interest rate for the new time unit of 1.5 years. Then $100 invested for 1.5 years would accumulate to $118.

   100 * 1.18^1
118   

After 1 year the $100 would accumulate to 111.666

   100* 1.18^((1%1.5) % 1)
111.666   

A 0.12 (ie: 12%) yearly nominal interest rate payable at periods of 0.4 years is is equivalent to a 0.048 (ie: 4.8%) effective interest rate for a new time unit of 0.4 years.

$100 invested would accumulate to $112.435 after 1 year.

  100 * (1.048)^(2.5 % 1) 
112.435  

Under a force of interest, interest is compounded continuously over the time unit. The force of interest is usually assumed to be constant over the time unit but can be allowed to vary adding to the comlexity of the calculations.

For example at a 10% yearly force of interest, assuming it is constant over the year, $100 invested for 1 year accumulates to $110.52 at the end of the year.

      100*^0.1
110.517

Assuming a constant force of interest for 5 years $100 accumulates to $164.87

   100*^(0.1*5)
164.872

The relationship between an effective interest rate i and a constant force of interest c (instead of δ) for a time unit is as follows:
ci(i) = constant force of interest for effective interest rate i
$$ci\left(i\right)={\log }_e\left(1+i\right)$$ ic(c) =effective interest rate for constant force of interest c
$$ic\left(c\right)=e^c-1$$

   ci(0.05)
0.0487902
 
   ic(0.05)
0.0512711

   ic(ci 0.05)
0.05   

A number of the definitions and calculators allow for the interest rate for a time unit to vary over the period of the calculation. For example the variable interest rate is entered as a list for the time units with the last interest rate extended for the period of the calculation.

Examples of variable interest rates:
(1) 0.07
the same rate is applied for the total period
(2) 0.07 0.06 
the rate 0.07 applied to the first time unit and then 0.06 for the 
subsequent time units
(3) 0.07 0.07 0.075 0.06           
the rate 0.06 is applied to the fourth and subsequent time units.

   vzt(0.07;0 1 2 3 4 5) 
1 0.934579 0.873439 0.816298 0.762895 0.712986

   vzt(0.07 0.06;0 1 2 3 4 5) 
1 0.934579 0.881679 0.831772 0.784691 0.740274

  vzt(0.07 0.07 0.075 0.06;0 1 2 3 4 5) 
1 0.934579 0.873439 0.812501 0.766511 0.723123

If we consider one time unit and an effective interest rate i for the time unit then and amount of 1 invested at the start of the time unit would accumulated to (1+i) at the end of the time unit. With i being the accumulated interest at the end of the time unit. It we now consider an investment of (1-d) at the start of the time unit such that the accumulated amount at the effective interest rate i the end of the time unit is 1 then:

(1-d)(1+i) = 1 and therefore
d = i/(1+i)
d = (1-v) where: v = 1/(1+i)