### watch the thematic video

Bank Loan Interest and Payment Calculation in Excel in Tamil - Home loan calculation in excel - EMI### Learning Outcomes

- Calculate the balance on an annuity after a specific amount of time
- Discern between compound interest, annuity, and payout annuity given a finance scenario
- Use the loan formula to calculate loan payments, loan balance, or interest accrued on a loan
- Determine which equation to use for a given scenario
- Solve a financial application for time

### Conventional Loans

In the last section, you learned about payout annuities. In this section, you will learn about conventional loans (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.

One great thing about loans is that they use exactly the same formula as a payout annuity. To see why, imagine that you had $10,000 invested at a bank, and started taking out payments while earning mortgage loan payment formula as part of a payout annuity, and after 5 years your balance was zero. Flip that around, and imagine that you are acting as **mortgage loan payment formula** bank, and a car lender is acting as you. The car lender invests $10,000 in you. Since you’re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.

### Loans Formula

[latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex]

*P*is the balance_{0}**mortgage loan payment formula**the account at the beginning (the principal, or amount of the loan).*d*is your loan payment (your monthly payment, annual payment, etc)*r*is the annual interest rate in decimal form.*k*is the number of compounding periods in one year.*N*is the length of the loan, in years.

Like before, the compounding frequency is not always explicitly given, but is determined by how often you make payments.

### When do you use this?

The loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying interest on the loan.

- Compound interest: One deposit
- Annuity: Many deposits
- Payout Annuity: Many withdrawals
- Loans: Many payments

### Example

You can afford $200 per month as a car payment. If you can get an auto loan at 3% interest for 60 months (5 years), how expensive of a car can you afford? In other words, what amount loan can you pay off with $200 per month?

In this example,

d = $200 | the monthly loan payment |

r = 0.03 | 3% annual rate |

k = 12 | since we’re doing monthly payments, we’ll compound monthly |

N = 5 | since we’re making monthly payments for 5 years |

We’re looking for *P _{0}*, the starting amount of the loan.

[latex]\begin{align}&{{P}_{0}}=\frac{200\left(1-{{\left(1+\frac{0.03}{12}\right)}^{-5(12)}}\right)}{\left(\frac{0.03}{12}\right)}\\&{{P}_{0}}=\frac{200\left(1-{{\left(1.0025\right)}^{-60}}\right)}{\left(0.0025\right)}\\&{{P}_{0}}=\frac{200\left(1-0.861\right)}{\left(0.0025\right)}=\$11,120 \\\end{align}[/latex]

You can afford a $11,120 loan.

You will pay a total of $12,000 ($200 per month for 60 months) to the loan company. The difference between the amount you pay and the amount of the loan is the interest paid. In this case, you’re paying $12,000-$11,120 = $880 interest total.

Details of this example are examined in this video.

### Try It

### Example

You want to take out a $140,000 mortgage (home loan). The interest rate on the loan is 6%, and the loan is for 30 years. How much will your monthly payments be?

In this example, we’re looking for *d*.

r = 0.06 | 6% annual rate |

k = 12 | since we’re paying monthly |

N = 30 | 30 years |

P = $140,000_{0} | the starting loan amount |

In this case, we’re going to have to set up the equation, and solve for *d*.

[latex]\begin{align}&140,000=\frac{d\left(1-{{\left(1+\frac{0.06}{12}\right)}^{-30(12)}}\right)}{\left(\frac{0.06}{12}\right)}\\&140,000=\frac{d\left(1-{{\left(1.005\right)}^{-360}}\right)}{\left(0.005\right)}\\&140,000=d(166.792)\\&d=\frac{140,000}{166.792}=\$839.37 \\\end{align}[/latex]

You will make payments of $839.37 per month for 30 years.

You’re paying a total of $302,173.20 to the loan company: $839.37 per month for 360 months. You are paying a total of $302,173.20 – $140,000 = $162,173.20 in interest over the life of the loan.

View more about this example here.

### Try It

### Try It

Janine bought $3,000 of new furniture on credit. Because her credit score isn’t very good, the store is charging her a fairly high interest rate on the loan: 16%. If she agreed to pay off the furniture over 2 years, how much will she have to pay each month?

*d* = unknown

*r* = 0.16 16% annual rate

*k* = 12 since we’re making monthly payments

*N* = 2 2 years to repay

*P0* = 3,000 we’re starting with a $3,000 loan

[latex]\begin{array}{c}3000=\frac{{d}\left(1-\left(1+\frac{0.06}{12}\right)^{-2*12}\right)}{\frac{0.16}{12}}\\\\3000=20.42d\end{array}[/latex]

Solve for d to get monthly payments of $146.89

Two years to repay means $146.89(24) = $3525.36 in total payments. This means Janine will pay $3525.36 – $3000 = $525.36 in interest.

### Calculating the Balance

With loans, it is often desirable to determine what the remaining loan balance will be after some number of years. For example, if you purchase a home and plan to sell it in five years, you might want to know how much of the loan balance you will have paid off and how much you have to pay from the mortgage loan payment formula src="https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/1141/2016/12/07163930/5445898743_bef11a7c87_z.jpg" alt="Pair of glasses resting on a Mortgage Loan Statement" width="640" height="478">

To determine the remaining loan balance after some number of years, we first need to know the loan payments, if we don’t already know them. Remember that only a portion of your loan payments go towards the loan balance; a portion is going to go towards interest. For example, if your payments were $1,000 a month, after a year you will *not* have paid off $12,000 of the loan balance.

To determine the remaining loan balance, we can think “how much loan will these loan payments be able to pay off in the remaining time on the loan?”

### Example

If a mortgage at a 6% interest rate has payments of $1,000 a month, how much will the loan balance be 10 years from the end the loan?

To determine this, we are looking for the amount of the loan that can be paid off by $1,000 a month payments in 10 years. In other words, we’re looking for P_{0} when

d = $1,000 | the monthly loan payment |

r = 0.06 | 6% annual rate |

k = 12 | since we’re doing monthly payments, we’ll compound monthly |

N = 10 | since we’re making monthly payments for 10 more years |

[latex]\begin{align}&{{P}_{0}}=\frac{1000\left(1-{{\left(1+\frac{0.06}{12}\right)}^{-10(12)}}\right)}{\left(\frac{0.06}{12}\right)}\\&{{P}_{0}}=\frac{1000\left(1-{{\left(1.005\right)}^{-120}}\right)}{\left(0.005\right)}\\&{{P}_{0}}=\frac{1000\left(1-0.5496\right)}{\left(0.005\right)}=\$90,073.45 \\\end{align}[/latex]

The loan balance with 10 years remaining on the loan will be $90,073.45.

This example is explained in the following video:

Oftentimes answering remaining balance questions requires two steps:

- Calculating the monthly payments on the loan
- Calculating the remaining ope sas andalucia balance based on the
*remaining time*on the loan

### Example

A couple purchases a home with a $180,000 mortgage at 4% for 30 years with monthly payments. What will the remaining balance on their mortgage be after 5 years?

First we will calculate their monthly payments.

We’re looking for *d*.

r = 0.04 | 4% annual rate |

k = 12 | since they’re paying monthly |

N = 30 | 30 years |

P = $180,000_{0} | the starting loan amount |

We set up the equation and solve for *d*.

[latex]\begin{align}&180,000=\frac{d\left(1-{{\left(1+\frac{0.04}{12}\right)}^{-30(12)}}\right)}{\left(\frac{0.04}{12}\right)}\\&180,000=\frac{d\left(1-{{\left(1.00333\right)}^{-360}}\right)}{\left(0.00333\right)}\\&180,000=d(209.562)\\&d=\frac{180,000}{209.562}=\$858.93 \\\end{align}[/latex]

Now that we know the monthly payments, we can determine the remaining balance. We want the remaining balance after 5 years, when 25 years will be remaining on the loan, so we calculate the loan balance that will be paid off with the monthly payments over those 25 years.

d = $858.93 | the monthly loan payment we calculated above |

r = 0.04 | 4% annual rate |

k = 12 | since they’re doing monthly payments |

N = 25 | since they’d be making monthly payments for 25 more years |

[latex]\begin{align}&{{P}_{0}}=\frac{858.93\left(1-{{\left(1+\frac{0.04}{12}\right)}^{-25(12)}}\right)}{\left(\frac{0.04}{12}\right)}\\&{{P}_{0}}=\frac{858.93\left(1-{{\left(1.00333\right)}^{-300}}\right)}{\left(0.00333\right)}\\&{{P}_{0}}=\frac{858.93\left(1-0.369\right)}{\left(0.00333\right)}=\$155,793.91 \\\end{align}[/latex]

The loan balance after 5 years, with 25 years remaining on the loan, will be $155,793.91.

Over that 5 years, the couple has paid off $180,000 – $155,793.91 = $24,206.09 of the loan balance. They have paid a total of $858.93 a month for 5 years (60 months), for a total of $51,535.80, so $51,535.80 – $24,206.09 = $27,329.71 of what they have paid so far has been interest.

More explanation of this example is available here:

### Solving for Time

Recall that we have used logarithms to solve for time, since it is an exponent in interest calculations. We can apply the same idea to finding how long it will take to pay off a loan.

### Try It

Joel is considering putting a $1,000 laptop purchase on his credit card, which has an interest rate of 12% compounded monthly. How long will it take him to pay off the purchase if he makes payments of $30 a month?

*d* = $30 The monthly payments

*r* = 0.12 12% annual rate

*k* = 12 since we’re making monthly payments

*P0* = 1,000 we’re starting with a $1,000 loan

We are solving for *N*, the time to pay off the loan

[latex]1000=\frac{30\left(1-\left(1+\frac{0.12}{12}\right)^{-N*12}\right)}{\frac{0.12}{12}}[/latex]

Solving for *N* gives 3.396. It will take about 3.4 years to pay off the purchase.

### FYI

Home loans are typically paid off through an amortization process, **amortization** refers to paying off a debt (often from a loan or mortgage) over time through regular payments. An **amortization schedule** is a **table** detailing each periodic payment on an **amortizing** loan as generated by an **amortization calculator**.

If you want to know more, click on the link below to view the website “How is an Amortization Schedule Calculated?” by MyAmortizationChart.com. This website provides a brief overlook of Amortization Schedules.

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