Question

What is the present value of an annuity consisting of monthly payments of an amount \(C\) continuing for \(n\) years? Express the answer in terms of the effective rate \(r_{\mathrm{e}}\).

Short answer

PV = \( C \left(1 - (1 + r_{\text{e}})^{1/12 - 12n} \right) / ((1 + r_{\text{e}})^{1/12} - 1) \.

Step-by-step solution

Identify Variables

Determine the key variables given in the problem. Here, the monthly payment amount is denoted by \( C \), the number of years by \( n \), and the annual effective rate by \( r_{\text{e}} \).

Convert Annual Rate to Monthly Rate

The monthly effective rate \( r_{\text{m}} \) can be found using the formula \ r_{\text{m}} = (1 + r_{\text{e}})^{1/12} - 1 \.

Determine the Number of Payments

Calculate the total number of monthly payments, which is \( 12n \), since there are 12 months in a year.

Present Value of Annuity Formula

The present value \( PV \) of an annuity can be computed using the formula: \ PV = C \left(1 - (1 + r_{\text{m}})^{-12n} \right) / r_{\text{m}} \.

Combine all Expressions

Substitute \( r_{\text{m}} \) back into the annuity formula and simplify: \ PV = C \left(1 - (1 + (1 + r_{\text{e}})^{1/12} - 1)^{-12n} \right) / ((1 + r_{\text{e}})^{1/12} - 1) \.

Key concepts

Effective Rate

Understanding the effective rate is essential when calculating present value of annuities. The effective annual rate, denoted as \( r_{\text{e}} \), measures the annual interest that's actually earned or paid due to compounding over a year. It differs from the nominal rate, which may not consider compounding.
This rate is crucial because it allows you to convert annual payments to monthly terms, ensuring accurate calculations. Essentially, it bridges the gap between annual and monthly rates, facilitating precise financial planning.

Monthly Payments

Monthly payments, marked as \( C \), are the consistent amounts paid or received each month in an annuity. They can represent loan repayments, rents, or any recurring monthly income or expense.
To determine the present value of these payments, it's important to recognize the total payments made over the entire period.
For instance, if payments are made over \( n \) years, you'll have \( 12n \) payments.
This makes it easier to break down the problem and apply formulas accurately.

Present Value Formula

The present value formula allows us to determine the current value of a series of future payments. It's essential in evaluating annuities, letting you understand how much a series of future payments is worth right now. For an annuity with monthly payments, the present value \( PV \) can be calculated by:\[ PV = C \left(1 - (1 + r_{\text{m}})^{-12n} \right) / r_{\text{m}} \]Here, \( r_{\text{m}} \) is the monthly rate derived from the annual effective rate.
Substituting \( r_{\text{m}} \) back into the formula integrates all our variables for a precise present value.

Annuity Calculation

Calculating the present value of an annuity involves multiple steps, combining effective rates, monthly payments, and the proper formulas.
First, convert the annual effective rate into a monthly rate using:
\[ r_{\text{m}} = (1 + r_{\text{e}})^{1/12} - 1 \]Second, determine the total number of payments: \( 12n \). Finally, apply these to the present value formula:\[ PV = C \left(1 - (1 + r_{\text{m}})^{-12n} \right) / r_{\text{m}} \]By systematically combining these elements, we obtain the present value of the annuity, offering a clear understanding of its current worth compared to its future payments.