Find the present value of an annuity, where the payments are $[tex]$1590[/tex] per month over 4 years, at an annual interest rate of [tex]$5 \%$[/tex], compounded monthly.
Answer (1)
Calculate the monthly interest rate: i = 12 0.05 .
Calculate the total number of payments: n = 4 × 12 = 48 .
Apply the present value of an ordinary annuity formula: P V = PMT × i 1 − ( 1 + i ) − n .
Substitute the values and calculate the present value: P V = 69042.50 .
Explanation
Problem Analysis We are asked to find the present value of an annuity with monthly payments of $1590 over 4 years at an annual interest rate of 5%, compounded monthly.
Calculations of Monthly Interest Rate and Number of Payments First, we need to determine the monthly interest rate. The annual interest rate is 5%, so the monthly interest rate is calculated as follows: i = 12 0.05 = 0.004166666... Next, we need to calculate the total number of payments. Since the payments are made monthly over 4 years, the total number of payments is: n = 4 × 12 = 48 Now, we can use the formula for the present value of an ordinary annuity: P V = PMT × i 1 − ( 1 + i ) − n where:
P V is the present value of the annuity
PMT is the payment amount per period ($1590)
i is the interest rate per period (0.05/12)
n is the number of periods (48)
Calculating the Present Value Substituting the given values into the formula, we get: P V = 1590 × 12 0.05 1 − ( 1 + 12 0.05 ) − 48 P V = 1590 × 0.004166666... 1 − ( 1.004166666... ) − 48 P V = 1590 × 0.004166666... 1 − 0.81930694 P V = 1590 × 0.004166666... 0.18069306 P V = 1590 × 43.366399966 P V = 69042.575946
Final Answer Rounding the present value to the nearest cent, we get: P V = 69042.50 Therefore, the present value of the annuity is $69042.50.
Examples
Understanding the present value of an annuity is crucial in financial planning. For instance, when considering a mortgage, knowing the present value helps determine the initial loan amount you can afford based on your monthly payment capacity. Similarly, in retirement planning, calculating the present value of expected future income streams allows you to estimate the lump sum needed today to fund your retirement. This concept is also applicable in investment decisions, where you can assess the current worth of future returns from an investment.
