# Explain how much it will cost in interest to pay a loan off in 5 years if compounded monthly.

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The catalog of payments over a time period describing payments on principle and interest is called an amortization schedule. The amount of interest paid over the life of the loan depends on the payment made each term. It is common to calculate the minimum payment amount using the formula:

A = P * r / (1 - (1 + r)^-n)

where A is the payment, P is the loan principle, and r is the interest rate per period, and n is the total number of periods, i.e. 5*12 months in your question. Once this payment amount is known, then an amortization schedule may be constructed.

Say your loan is for $1000, and the payment is $20 per month, and the interest rate is 1%/month. Then the interest charge is $1000*.01 = $10. So, in the first payment, the bank gets $10 in interest; the remaining $10 goes to pay off your loan, so now you owe $990. Next month, you pay $100, and the bank gets $9.90, so now you owe $979.90. In this scenario, paying off the $1000 takes about 6 years, and the total interest paid is about $400.

We assume the nominal rate of the interest to be** r **percent per annum, the principal loan amount burrowed to be **P **.

Then the monthly interest works out (r/12)% =** r/1200** per rupee or per dollar or per unit of money. Period of compounding is month.

The loan amount growing by compound interest: The loan amount of P grows in one month=**P(1+r/1200)** along with interest.

The loan amount P grows to P(1+r/1200)^ 2 in 2 months, along with interest.

The loan amount grows to become P(1+r/1200)^12 in 12 months or one year period together with interest.

The loan amount with interest,therefore, grows to become P(1+r/1200)^60 in 60 months or 5 years, period. Take off the principal loan amount from the grown amount with interest in 5 years, and then you get what costed you the compound interest over your loan of P .

Therefore , **the cost of compound interest **for the Principal loan amount ,P for 5 years, compounding being monthly = the amount loan grown in 5 years ** minus **principal amount of loan=

=P(1+r/1200)^60 -P= P{(1+r/1200)^60 - 1}

Therefore, the cost of compound interest for 5 years on 1 unit of money =** **P(1+r/1200)^60 -1}/P = **(1+r/1200)^60 -1**. From this we can construct a table for different rates of interest for different principal loan amount and get the cost of compound interest for 5 years ,compounding monthly, as below:

Loan Vs Interest rate : cost of compound interest for 5 years at rates:

Amoun(principal)Loan : 1% 2% 3% 4% 5%

1 0.0512 0.1051 0.1616 0.2210 0.2834

100 5.12 10.50 16.16 22.10 28.34

10000 512.49 1050.08 1616.16 2209.97 2833.59

The construction of the table can be extended simimarly for different rate of nominal interests, and for different principles and we can use it to get any intermediate values , by interpolation if requred.

The total interest will of course depend on the interest rate and the principle amount, which are not given in the question. So what we can do is to workout the formula which will enable us to calculate the interest.

Let us say that the principle amount is just 1 dollar, and the rate of interest is 'i' percent per month. It is customary to specify interest rate in percent per month, but in this case we have taken interest per month because compounding is done on monthly. If yearly interest rate is specified, we can convert it in monthly interest by dividing it by 12.

Now a loan of 1 dollar will attract an interest of i/100 for the first month. So the total amount of principal plus interest at the end of one month will be 1+(i/100).

The interest in the second month will be applicable on 1+(i/100). This works out t:

[1+(i/100)]*(i/100) = (i/100)+[(1/100)^2]

Therefore Principal plus interest at the end of second month:

= [1+(i/100)]+{(i/100)+[(1/100)^2]}

= 1+[2*(i/100]+[(i/100)^2]

= [1+(i/100]^2

Thus we can show that amount of principal plus interest at the end of nth. month is:

= [1+(i/100]^n

and total interest at the for the complete period of n months will be:

= {[1+(i/100]^n} -1

Thus interest for five years or 60 months will be:

= {[1+(i/100]^60} -1

Manual calculation of quantities such as [1+(i/100]^60 can be quite tedious therefore ready-made tables are available that give information on interest for different rates of interest, compounding period, and total loan duration.

Please note that the previous answer to this question assumes that the amount is repaid monthly in equal installments. As this is not the assumption stated in question I have given answer on the basis that no repayment is made for the five year period.

Also please note that the previous answer does not even stick to the 5 year repayment period.