Use The Appropriate Formula To Find How Much She Should Depo

Question

Use the appropriate formula to find how much she should deposit (in $) now at 6% interest, compounded semiannually, to yield this payment for 12 years. Kyla wants to receive an annuity payment of $800 at the end of each six months for 12 years. To determine the present value (the amount she should deposit now), we use the Present Value of an Ordinary Annuity formula, adapted for semiannual periods. The formula for the present value (PV) of an annuity is: PV = P × [1 - (1 + r)^-n] / r where P is the payment per period ($800), r is the interest rate per period (annual rate divided by 2 for semiannual compounding), and n is the total number of periods (number of years times 2). Given an annual interest rate of 6%, compounded semiannually, the rate per period is 0.06 / 2 = 0.03. Number of periods n = 12 years × 2 = 24 periods. Applying values: PV = 800 × [1 - (1 + 0.03)^-24] / 0.03 Calculating (1 + 0.03)^-24: (1.03)^-24 ≈ 0.491 Calculating numerator: 1 - 0.491 = 0.509 Dividing: 0.509 / 0.03 ≈ 16.967 Finally, PV ≈ 800 × 16.967 ≈ $13,573.60 Thus, Kyla should deposit approximately $13,573.60 now to receive $800 at the end of each six months for 12 years.

Dr. Jack HW Helper · Accepted Answer

Kyla's goal to secure a steady stream of payments of $800 every six months over a 12-year period requires a thorough understanding of present value calculations for annuities. This financial planning task involves estimating how much she must invest today, assuming a specific interest rate, to generate the desired payouts. In this context, it is crucial to select an appropriate formula that accounts for semiannual compounding. The key concept in solving this problem is the present value of an ordinary annuity, which represents the current worth of a series of future payments, discounted at a particular interest rate. For semiannual payments, the periodic interest rate is half of the annual nominal rate, reflecting the compounding frequency. Here, with a 6% interest rate compounded semiannually, the effective interest rate per period is 3%. Applying the formula PV = P × [1 - (1 + r)^-n] / r, where P is the payment amount per period ($800), r is 0.03, and n is 24 periods (12 years times 2 for semiannual periods), results in a present value of approximately $13,573.60. This is the amount Kyla needs to deposit now to meet her future obligation. Understanding the relationship between interest rates, compounding frequency, and payment periods is essential in such financial calculations. The model underscores how frequent compounding accelerates growth and impacts the initial investment necessary to produce a specified series of future payments. Sound financial planning demands accuracy in selecting formulas and interpreting parameters, such as interest rate and periods, to produce reliable estimates for investment and funding purposes. References Brigham, E. F., & Ehrhardt, M. C. (2016). Financial Management: Theory & Practice. Cengage Learning. Damodaran, A. (2012). Investment valuation: Tools and techniques for determining the value of any asset. John Wiley & Sons. Ross, S. A., Westerfield, R. W., & Jaffe, J. (2013). Corporate Finance. McGraw-Hill Educat

Use the appropriate formula to find how much she should deposit (in $) now at 6% interest, compounded semiannually, to yield this payment for 12 years.

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