Calculate the duration of a one-year fixed payments loan with monthly payments of $150 and yield to maturity of 12%. Use this number to determine the % change in the price of this loan if interest rates increase to 14%.
You have already specified that the duration is one year. Do you really want to know the principal that can be paid off in that time at that specified monthly payment rate?
When refering to loans, one does not speak of the "yield to maturity"
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I wrote out exactly what the problem says
To calculate the duration of a one-year fixed payments loan with monthly payments, we need to know the present value of the loan's cash flows. We are given that the monthly payment is $150, and the yield to maturity is 12%.
First, let's calculate the present value of the loan's cash flows. Since the loan has monthly payments, we need to discount each monthly payment to its present value using the yield to maturity.
PV = (Payment / (1 + Yield)^n) + (Payment / (1 + Yield)^(n-1)) + ... + (Payment / (1 + Yield)^1)
Where:
PV = Present value of loan's cash flows
Payment = Monthly payment
Yield = Yield to maturity
n = Number of payments in one year (12 in this case)
Using the given values, we can substitute them into the formula to find the present value of the loan's cash flows:
PV = ($150 / (1 + 0.12/12)^12) + ($150 / (1 + 0.12/12)^11) + ... + ($150 / (1 + 0.12/12)^1)
Now, let's calculate the duration of the loan. Duration is a measure of the loan's sensitivity to changes in interest rates. The formula to calculate the duration is:
Duration = (∑(PVt * t)) / ∑PVt
Where:
Duration = Duration of the loan
PVt = Present value of the cash flow at time t
t = Time of the cash flow
Calculate the present value of each cash flow at its respective time (t) and sum them up to get the numerator. Also, sum up the present values of all cash flows to get the denominator. Divide the numerator by the denominator to get the duration.
Now that we have the duration of the loan, we can calculate the percentage change in price if the interest rate increases to 14%. The formula to calculate the percentage change in price is:
Percentage Change in Price = - Duration * Change in Yield
Where:
Percentage Change in Price = Percentage change in the price of the loan
Duration = Duration of the loan (calculated earlier)
Change in Yield = New yield - Current yield
By substituting the values into the formula, we can calculate the percentage change in the price of the loan if the interest rate increases to 14%.
Note: To simplify the calculations, you can use a financial calculator or spreadsheet software that has built-in functions to calculate present value and duration.