Math Problem Statement

Suppose the interest rate is

6.9 %6.9%

APR with monthly compounding. What is the present value of an annuity that pays

$ 90$90

every

sixsix

months for

sevenseven

​years?

​(Note​:

Be careful not to round any intermediate steps less than six decimal​ places.)

Solution

To find the present value of an annuity, we use the formula for the present value of an ordinary annuity:

PV=P×(1(1+r)n)÷rPV = P \times \left(1 - (1 + r)^{-n}\right) \div r

Where:

  • PVPV is the present value of the annuity.
  • PP is the payment per period.
  • rr is the interest rate per period.
  • nn is the total number of periods.

Step 1: Determine the payment per period.

Here, the payment P=90P = 90 dollars.

Step 2: Adjust the interest rate for semiannual payments.

Since the interest rate is 6.9% APR (Annual Percentage Rate) compounded monthly, we first need to adjust it to fit the semiannual (every six months) payment structure. The monthly interest rate is:

Monthly interest rate=6.9%12=0.00575\text{Monthly interest rate} = \frac{6.9\%}{12} = 0.00575

Now, to get the semiannual interest rate, we compound the monthly interest rate over 6 months:

Semiannual interest rate=(1+0.00575)61\text{Semiannual interest rate} = (1 + 0.00575)^6 - 1

Step 3: Determine the total number of periods.

Since the annuity is paid every six months for seven years, the total number of periods is:

n=7 years0.5 years per period=14 periodsn = \frac{7 \text{ years}}{0.5 \text{ years per period}} = 14 \text{ periods}

Step 4: Calculate the present value.

Once we have rr (semiannual interest rate) and n=14n = 14, we plug these into the formula to get the present value.

Let me calculate the result.The semiannual interest rate is approximately 3.49998%, and the present value of the annuity is $982.85 (rounded to two decimal places).

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

  1. How would the present value change if the payments were made monthly instead of semiannually?
  2. What is the future value of this annuity after 7 years?
  3. How does changing the interest rate to 5% affect the present value?
  4. What if the annuity lasts for 10 years instead of 7?
  5. How do different compounding frequencies (e.g., annual vs. quarterly) impact the annuity’s present value?

Tip: When solving annuity problems, always adjust the interest rate to match the payment period (monthly, quarterly, semiannual, etc.).

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Math Problem Analysis

Mathematical Concepts

Annuities
Present Value
Interest Rates
Compounding

Formulas

Present value of an ordinary annuity: PV = P × (1 - (1 + r)^-n) ÷ r
Monthly interest rate: r_monthly = APR ÷ 12
Semiannual interest rate: r_semiannual = (1 + r_monthly)^6 - 1
Total number of periods: n = 7 years ÷ 0.5 years per period

Theorems

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Suitable Grade Level

Grades 11-12, College Introductory