Question

A student gets a mortgage of \(£ 100000\) (or \(\$ 100000\) ). This has to be paid back in 25 years with an interest rate of \(10 \%\). Calculate the annual payment on basis of annuity

Short answer

The annual payment is approximately £11,045.98.

Step-by-step solution

Determine the Variables

To solve an annuity mortgage problem, it's essential first to identify the variables in the problem. The principal amount \( P \) is £100,000, the annual interest rate \( r \) is 0.10, and the total number of payments \( n \) over 25 years is 25.

Apply the Annuity Formula

The formula for calculating the annual payment \( A \) of a mortgage based on annuity is \( A = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \). This formula helps us find the constant annual payment over the term of the mortgage.

Substitute the Values

Substitute the identified values into the annuity formula. Using \( P = 100000 \), \( r = 0.10 \), and \( n = 25 \), substitute into the formula: \[ A = 100000 \frac{0.10(1 + 0.10)^{25}}{(1 + 0.10)^{25} - 1} \].

Compute the Power and Product

First, compute \( (1 + 0.10)^{25} \). This gives approximately 10.8353. Then compute \( 0.10 \times 10.8353 \), which equals approximately 1.08353.

Calculate the Denominator

Calculate \( (1 + 0.10)^{25} - 1 \), which is \( 10.8353 - 1 = 9.8353 \).

Solve the Annuity Formula

Substitute these computed values back into the formula: \[ A = 100000 \frac{1.08353}{9.8353} \].

Calculate the Final Result

After dividing 1.08353 by 9.8353 and multiplying by 100000, the result for the annual payment \( A \) is approximately £11,045.98.

Key concepts

Principal Amount

The principal amount, often referred to simply as the "principal," is a key element in any loan calculation, including mortgage annuities. In this context, the principal is the original sum of money borrowed, upon which interest is calculated. It is the foundation of your mortgage loan.
For example, when a student takes out a mortgage of £100,000, this £100,000 is considered the principal amount.

• It is the initial debt that the borrower agrees to repay, typically over a set number of years.
• In our example, this principal forms the basis on which the annual interest will accrue.

Understanding the principal amount is crucial as it influences the longevity of repayment and the total interest paid over time. A higher principal generally means higher interest costs unless there are adjustments in interest rates or loan term.

Interest Rate

The interest rate is the cost of borrowing money, typically expressed as a percentage of the principal amount. In an annuity mortgage, it represents the annual cost that the borrower must pay to the lender in addition to repaying the principal.

Given an interest rate of 10%, it means that 10% of the outstanding mortgage principal is added to the loan balance each year.

• For example, with £100,000 borrowed, the interest for the first year would be £10,000.
• This interest applies each year, recalculated on the decreasing principal following each payment.
• An annuity mortgage spreads this interest evenly over the repayment period, shaping the amount paid annually.

Interest rates can significantly affect the total cost of a loan. Larger rates translate to higher total payments over the life of the mortgage.

Annual Payment Formula

The annual payment formula for an annuity mortgage is a mathematical expression used to calculate the amount the borrower needs to pay each year.
The formula is:\[A = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]where:

• \(A\) is the annual payment,
• \(P\) is the principal amount,
• \(r\) is the annual interest rate as a decimal, and
• \(n\) is the number of payments.

This formula ensures that each payment not only covers the interest costs but also reduces the principal over time so that the loan is fully repaid by the end of its term.
The regular, fixed payments provided by this formula make budget planning easier for borrowers.

Loan Repayment Schedule

A loan repayment schedule outlines how the loan will be paid off over time, showing each payment and its allocation between principal and interest.
In an annuity mortgage, each annual payment is the same amount.

However, within each payment, the proportion going towards interest versus principal shifts over the lifetime of the loan.

• Initially, a larger portion of the payment covers interest.
• As the principal decreases through regular repayments, the interest cost in subsequent payments also decreases.
• Thus, later payments contribute more to reducing the principal.

Understanding this schedule is vital for borrowers to grasp how their debt is shrinking and to predict future financial commitments. It also allows them to see how extra payments can shorten the loan term and reduce interest costs.