A man borrows $50,000 towards the cost of a house. Compound interest is charge on the loan at 125 per annum. He agrees to pay back the loan in 25 equal installments, at yearly intervals, the first repayment being made exactly 1 year after the loan is taken out. Calculate the value of each installment
I assume you really mean 12.5% per annum.
R = Pi/[1 - (1+i)^(-n)]
where R = the periodic payment, n = the number of interest bearing periods and the number of periodic payments and i = the periodic interest in decimal form.
Thus, R = 50,000(.125)/[(1 - (1.125)^(-25)] = $5,921.10 per year.
To calculate the value of each installment, we need to take into account the compound interest charged on the loan and spread the repayment amount evenly over 25 installments.
The compound interest formula is given by:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = principal amount (initial loan amount)
r = annual interest rate (in decimal form)
n = number of times that interest is compounded per year
t = number of years
In this case, the principal amount (P) is $50,000, the annual interest rate (r) is 125% (or 1.25 as a decimal), the number of times interest is compounded per year (n) is 1 since the installments are made on a yearly basis, and the total repayment period (t) is 25 years.
Using these values, we can calculate the final repayment amount (A) after 25 years. Let's substitute the values into the formula:
A = 50000(1 + 1.25/1)^(1*25)
A = 50000(1 + 1.25)^(25)
A = 50000(2.25)^25
A ≈ 50000(2.8101...), rounded to the nearest dollar
A ≈ $140,505.23
Therefore, the value of each installment can be determined by dividing the final repayment amount by the number of installments:
Installment Value = Final Repayment Amount / Number of Installments
Installment Value = $140,505.23 / 25
Installment Value ≈ $5,620.21
So, the value of each installment is approximately $5,620.21.