Problem 11 - Precalculus Mathematics for Calculus

Question

What is the present value of an annuity that consists of 20 semiannual payments of \(\$ 1000\) at an interest rate of \(9 \%\) per year, compounded semiannully?

Step-by-Step Solution

Answer

The present value of the annuity is approximately $12,926.70.

Step 1: Understanding the Problem

We need to find the present value of an annuity, which is a series of equal payments made at regular intervals. In this problem, there are 20 semiannual payments of $1000, and the interest rate is 9% per annum, compounded semiannually.

Step 2: Converting to Semiannual Interest Rate

The annual interest rate is 9%. Because the payments are semiannual, we need the semiannual interest rate. We convert the annual rate by dividing by 2: \( i = \frac{9\%}{2} = 4.5\% = 0.045 \).

Step 3: Identifying Total Number of Periods

There are 20 semiannual payments, meaning the total number of periods \( n \) is 20.

Step 4: Using Present Value of Annuity Formula

The present value of an annuity formula is given by: \( PV = P \times \frac{1-(1+i)^{-n}}{i} \), where \( P \) is the annuity payment, \( i \) is the interest rate per period, and \( n \) is the number of periods.

Step 5: Substituting Values

Substitute the known values: \( P = 1000 \), \( i = 0.045 \), \( n = 20 \). The formula becomes: \[ PV = 1000 \times \frac{1-(1+0.045)^{-20}}{0.045} \]

Step 6: Calculating the Present Value

First calculate \((1+0.045)^{-20}\) which gives approximately 0.4183. Substitute back into the formula: \[ PV = 1000 \times \frac{1-0.4183}{0.045} = 1000 \times \frac{0.5817}{0.045} \]. Solve the fraction: \( \frac{0.5817}{0.045} \approx 12.9267 \). Thus, \( PV \approx 1000 \times 12.9267 = 12,926.70 \).

Key Concepts

Annuity Payments

An annuity involves making a series of equal payments at regular intervals. This might occur for a loan repayment, insurance policy payout, or other financial arrangements. Think of each payment as a regular "give-and-take" exchange happening over time. In our context, we're looking at **semiannual payments** — payments made every six months.

• Each payment, in this example, amounts to $1,000.
• There are 20 such payments over the life of this annuity, covering a ten-year span since they occur twice per year.

Understanding these payments helps lay the groundwork: knowing how much is paid and how often gives us the context to calculate everybody's main question — its value today.

Interest Rate Conversion

Interest rates can often be tricky because they need to match the payment frequency in an annuity. This means changing how we view a yearly interest rate to reflect payments that happen more often, like every six months or every month.

• Start with the annual interest rate, which is stated at 9%.
• Because payments in this example are semiannual, we divide this rate by 2, reflecting how often payments occur per year.
• So, our converted semiannual rate is \(4.5\% \) or \(0.045\) in decimal form.

By aligning how interest accrues with payment occurrences, you ensure your calculations mirror the actual flow of money, essential for accurate financial assessments.

Present Value Formula

The formula for calculating the Present Value (PV) of an annuity allows one to find the worth of a series of future payments in today's terms. It's like figuring out how much all those future \(1,000 payments would be worth right now given a certain interest environment.
To utilize this formula, remember:

• The formula itself is: \[ PV = P \times \frac{1-(1+i)^{-n}}{i} \]
• Here, \(P\) stands for each individual payment, \(i\) is the per-period interest rate, and \(n\) is the total number of periods.

By substituting into our specific example:

• \(P = 1000\), \(i = 0.045\), and \(n = 20\)
• We derive: \[ PV = 1000 \times \frac{1-(1+0.045)^{-20}}{0.045} \]
• Calculated fully, this results in a present value of approximately \(\\)12,926.70\)

In essence, it's a snapshot of future assets valued right now, a crucial concept when planning financially.