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Compounding, Discounting and Effective Annual Rate


Author: Awais Ahmad (comsian027@gmail.com)

Compounding:

The process of going from today’s value (Present Value; denoted by PV) to Future Value (denoted by FV) is called Compounding. If i is the Interest Rate, then Interest Amount (INT) can be calculated as:

INT ($) = PV x i

The Future Value will be the Present Value plus the amount of Interest, so:

FV = PV + INT

FV = PV + PV x i

FV = PV (1 + i)

If the amount is deposited or invested for n periods, the same formula can be written as:

FVn = PVn (1 + i) n

The term (1 + i)n is known as Future Value Interest Factor and is denoted by FVIFi,n, so:

FVn = PVn (1 + i)n = PV (FVIFi,n)

In some cases, Interest is paid semiannually, which means Interest is paid twice a year. Similarly Interest payment 4 times a year means Interest is paid quarterly. For such cases, the above formula can be more generalized:

Where m is the number of times Interest Payment is made in a year. However, in such case, i is taken as Nominal Rate of Interest.

Discounting:

The process of calculating Present Value (PV) from Future Value (FV) is called Discounting. As we know:

FVn = PVn (1 + i) n

Solving for PV, we have:

Or we can write it as under:

PVn = FVn (1 + i)-n

The term (1 + i)-n is called Present Value Interest Factor, and is denoted by PVIFi,n; therefore:

PVn = FVn (1 + i)-n = FV (PVIFi,n)

Same as previous case, if the Interest is paid semiannually or quarterly, a more general formula is applicable:

Where i is taken as Nominal Rate of Interest.

Effective Annual Rate (EAR):

Effective Annual Rate is defined as the rate which would produce the same Future Value, if annual Compounding had been used. It is also called Equivalent Annual Rate, and can be calculated as:

As we have taken Annual Compounding, therefore n is not shown in the formula, and i will be taken as Nominal Rate of Interest.

References:

Financial Management – Theory & Practice by Eugene F. Brigham & Michael C. Ehrhardt

Investment Analysis & Portfolio Management – Lectures

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