# Managerial Economics/Interest Calculations

Interest calculations are the relationship between time and money. For example, what's the difference between having $1,000,000 (1 million) now, versus having it a year later? If the money could have been received earlier and invested at 10%, then you would have had an additional 100,000 dollars the next year. In a loan situation, you are taking money from the bank. This is money that you can spend now, so you are in effect, paying (paying the interest) to have this money on hand immediately. Conversely, when you invest, you are granting immediate money to another party, and they are paying you for this money, through interest. In this section, we will be covering interest in depth and the different ways that interest can compound, and how it's calculated. In later sections we will cover cash flows such as payments in geometric and linear patterns. ## Terminology Principal The initial money amount received in a loan or spent in an investment. Interest Rate The interest rate is the "cost" of the money, it is a percentage per period of time. Interest Period A measurement of the frequency that interest rate is applied to the current loan/investment. Typically Annually, Semi Annually, Quarterly or Monthly. Annually Interest is calculated once a year; Interest is usually calculated at the end of the year. Semi Annually The interest is calculated twice a year; Typically the interest will be calculated half-way through the year, and at the end of the year. Quarterly The interest is calculated 4 times a year, or every three months. Monthly Interest is applied at the end of every month. ## Converting Interest Rates to different Interest Periods Typically loans are given with a certain Annual interest rate, regardless of whether or not interest is calculated annually or not. To convert from an interest period where interest is applied once, to a semi annually (were it's applied twice), the annual interest rate is converted to a semi annual interest rate. This is best explained with an example. Suppose the annual interest rate is 10% and is to be applied semi-annually on a loan amount of$1000, then the interest is calculated as below:

The annual interest rate: 10% Number of semi-annual time periods in an year = 2 Therefore, semi-annual interest rate = 10/2 = 5%

Amount due at the end of the first half year = 1000*(1.05) = 1050

Amount due at the end of the second half year = 1050*(1.05) = 1102.50

Effectively you are halving the interest rate but multiplying it to to the amount twice.

Similarly, if the interest was to be applied on a quarterly basis then you would divide the annual interest rate by 4, but also multiply it to the amount 4 times.

In other words, if R is the annual interest rate, n is the number of time periods at which interest is applied then the effective rate E is given by

$E=\left(1+{\frac {R}{n}}\right)^{n}$ Bond equivalent yields (also called APR on most consumer loan documents) add these sub period rates to get an annual rate. The actual or effective rate compounds the interest according to the periods specified in the loan documents (normally continuous, daily, monthly, quarterly, semi-annually, or annually). To calc the effective rate you would add one to each of the period rates and multiply them together, then subtract one. If all your period rates are the same length and carry the same rate you can simplify the function to the form $I=P*\left(\left(1+{\frac {r}{n}}\right)^{n}-1\right)$ As an example for a quarterly 12% APR you have four compounding periods (each with 3% interest per period). Your effective rate is (1+0.03)^4-1 or 12.55% (rounded to two decimal places). As your n approaches infinity the formula changes to P*e^r where e is a constant that is approximately 2.71828 (it is irrational).

By convention different security types use different month and year assumptions. A few common assumptions are actual number of months/actual days in the year, 30 day months/360 day year, 30 day months/365 day year.

## Simple Interest

Simple interest is where interest is calculated only of the principal. This will mean that the added amount of interest added during the interest period is constant throughout the duration of the transaction.

For example if I were to take a loan of 1000 dollars at 10% interest annually. At the end of each year I would owe an addition $100 dollars, regardless of however much the loan has shrunk. Under simple interest the final amount due can be easily calculated $I=(iP)N$ Where I is the total interest due. P is the Principal. i is the interest rate. N is the number of times interest is applied. ie Number of Interest periods per year times number of years. The Final value of the transaction can be easily calculated under simple interest. Since the final value is the Principal plus the interest you get the following formula. $F=P+I=P+iPN=P(1+iN)$ Where F is the final value. I is the total interest due. P is the Principal. i is the interest rate. N is the number of times interest is applied. ie Number of Interest periods per year times number of years. P =$1000
i = 10%
N = 1 year

Calculation Simple Interest Formula
$F=P(1+iN)$ $F=1000*(1+(10/100)*1)$ $F=1000*1.1$ $F=\1100$ Simple interest is less common than compound interest and is occasionally found in add-on loans or bonds.

## Compound Interest

test Compound interest is denoted by the formula $A=P(1+i)^{n}$ .8511

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