TIME VALUE OF MONEY
Interest is the price of the money. As it is in other commodities which have an exchanging price in the market, in payment for the use of someone else’s money, the payment or the fee that is paid is called as interest and it is expressed as a percentage of the principal or as a total amount.
The interest system is a system that consists of money inflows (+) and outflows (-) along the time line in a given period and can be expressed visually as follows:
This visual aspect of the interest system called as Cash Flow Diagram. As viewing the interest action as a system, it is possible to see its subsystems. The subsystems of the interest system are future value (FV) or end value, present value(PV) or principal, number of periods(t), equal payments or annuities(A) or equal installments and interest rate(r).
These subsystems are expressed with the following symbols. FV= future value; PV= present value; A= annuity; r= interest rate; t= number of periods
As it is in any kind of system; there are mutual relations and interactions among subsystems and these relations can be expressed mathematically. And depending to the types of these mathematical relations; basically there are three types of interest. (1) Simple Interest; (2) Discrete Compounding and (3) Continuous Compounding.
Simple Interest
The simple interest is the case that the interest payment that is earned at the end of the period is not added to the principal and can be calculated with the following relations formula. In other words there are direct mathematical relations in the simple interest system. Future Value is the sum of principal and interest payment: (FV = PV + I) and Interest payment : I = (PV * t * I); so by combining these equations we can write;
FV = PV ( 1 + t * r )
In the formula; ( P ) represents the principal or present value and ( t ) is expressed as a year, but for the shorter periods than a year it is calculated as a fraction of a year such as 4 months 12 days is equal (4 *30 + 12) / 365 0,362 and of course ( r ) is expressed as a percentage.
EXAMPLE:
What is the future amount that will be available in four years if 8,000 YTL is invested at 12% per year simple interest now? Ans.11,840,-YTL
When we directly install the given data in the following formula; the result is:
FV = PV ( 1 + n * i )
FV = 8,000, - * ( 1 + 4 * 0,12 )
FV = 11,840, - YTL.
EXAMPLE:
What is the annual rate of simple interest if 465, - YTL is earned in five months on an investment of 18,600, - TL? Ans. 6%
FV = PV + I FV = 18,600 + 465
FV = PV ( 1 + n * i )
19,065= 18,600 ( 1 + 5/12 * i )
i = 6%
EXAMPLE:
Determine the principal that would have to be invested to provide 250, - YTL of simple interest at the end of two and half years if the interest rate is 10%? Ans. 1,000,-TL
I = PV * t * r
250 = PV * 2,5 * 10%
PV = 1,000, -YTL
Discrete Compounding
The discrete compounding interest is the case that the interest payment that is earned at the end of each period is added to the principal and this operation continues consecutively for the given period.
And it can be calculated with the following relation formula. In other words there are exponential mathematical relations in the discrete compounding interest system.
FV = PV ( 1 + r )t
or the annuities
FV = A [( 1 + r) t - 1 ) / r ]
These exponential mathematical relations bring out the importance of economic equivalency. Economic equivalency means any amount of money at any time point is equivalent to another amount of money at another time point under the given interest rate. This concept can be expressed on the cash flow diagrams as it is in the following examples.
Model 1.
Today’s 100, -YTL is equivalent to one-year later 110, -YTL under a 10% interest rate.
Model 2.
The 100, -YTL on the time points of 0, 1, 2, 3 are equivalent 453,-YTL in the time point 4 under a 5% interest rate.
Model 3.
The 100, - ; 150,- ; 200,- and 250,- TL s on the time points of 0, 1, 2, 3 are equivalent on the time points of 4 and 5 the 500,- and 292,- TL s under a 5% interest rate.
Of course; the economic equivalency of the amount that place in these models are true for the reverse of the process. That means the cash inflows can be replaced with the cash outflows or vice versa.
Interest is the price of the money. As it is in other commodities which have an exchanging price in the market, in payment for the use of someone else’s money, the payment or the fee that is paid is called as interest and it is expressed as a percentage of the principal or as a total amount.
The interest system is a system that consists of money inflows (+) and outflows (-) along the time line in a given period and can be expressed visually as follows:
This visual aspect of the interest system called as Cash Flow Diagram. As viewing the interest action as a system, it is possible to see its subsystems. The subsystems of the interest system are future value (FV) or end value, present value(PV) or principal, number of periods(t), equal payments or annuities(A) or equal installments and interest rate(r).
These subsystems are expressed with the following symbols. FV= future value; PV= present value; A= annuity; r= interest rate; t= number of periods
As it is in any kind of system; there are mutual relations and interactions among subsystems and these relations can be expressed mathematically. And depending to the types of these mathematical relations; basically there are three types of interest. (1) Simple Interest; (2) Discrete Compounding and (3) Continuous Compounding.
Simple Interest
The simple interest is the case that the interest payment that is earned at the end of the period is not added to the principal and can be calculated with the following relations formula. In other words there are direct mathematical relations in the simple interest system. Future Value is the sum of principal and interest payment: (FV = PV + I) and Interest payment : I = (PV * t * I); so by combining these equations we can write;
FV = PV ( 1 + t * r )
In the formula; ( P ) represents the principal or present value and ( t ) is expressed as a year, but for the shorter periods than a year it is calculated as a fraction of a year such as 4 months 12 days is equal (4 *30 + 12) / 365 0,362 and of course ( r ) is expressed as a percentage.
EXAMPLE:
What is the future amount that will be available in four years if 8,000 YTL is invested at 12% per year simple interest now? Ans.11,840,-YTL
When we directly install the given data in the following formula; the result is:
FV = PV ( 1 + n * i )
FV = 8,000, - * ( 1 + 4 * 0,12 )
FV = 11,840, - YTL.
EXAMPLE:
What is the annual rate of simple interest if 465, - YTL is earned in five months on an investment of 18,600, - TL? Ans. 6%
FV = PV + I FV = 18,600 + 465
FV = PV ( 1 + n * i )
19,065= 18,600 ( 1 + 5/12 * i )
i = 6%
EXAMPLE:
Determine the principal that would have to be invested to provide 250, - YTL of simple interest at the end of two and half years if the interest rate is 10%? Ans. 1,000,-TL
I = PV * t * r
250 = PV * 2,5 * 10%
PV = 1,000, -YTL
Discrete Compounding
The discrete compounding interest is the case that the interest payment that is earned at the end of each period is added to the principal and this operation continues consecutively for the given period.
And it can be calculated with the following relation formula. In other words there are exponential mathematical relations in the discrete compounding interest system.
FV = PV ( 1 + r )t
or the annuities
FV = A [( 1 + r) t - 1 ) / r ]
These exponential mathematical relations bring out the importance of economic equivalency. Economic equivalency means any amount of money at any time point is equivalent to another amount of money at another time point under the given interest rate. This concept can be expressed on the cash flow diagrams as it is in the following examples.
Model 1.
Today’s 100, -YTL is equivalent to one-year later 110, -YTL under a 10% interest rate.
Model 2.
The 100, -YTL on the time points of 0, 1, 2, 3 are equivalent 453,-YTL in the time point 4 under a 5% interest rate.
Model 3.
The 100, - ; 150,- ; 200,- and 250,- TL s on the time points of 0, 1, 2, 3 are equivalent on the time points of 4 and 5 the 500,- and 292,- TL s under a 5% interest rate.
Of course; the economic equivalency of the amount that place in these models are true for the reverse of the process. That means the cash inflows can be replaced with the cash outflows or vice versa.