Ba 332 Name Homework Feb 26 2013 If Four Years Of Coll

Ba 332name Homework Feb 26 20131 If Four Years Of Col

Suppose four years of college tuition is expected to cost $150,000 in 18 years. To determine how much must be deposited now into an account that will earn a specified interest rate to reach this future value, we use present value calculations based on compound interest. The question asks for the initial deposit needed at 8% annual interest and then how the required deposit changes if the annual interest rate increases to 11%. Additionally, the problem involves valuing a series of payments and understanding amortization of loans and mortgages, including calculating monthly payments and present values of annuities.

Paper For Above instruction

Financial planning and investment valuation involve calculating present values (PV) and future values (FV) based on specified interest rates and time horizons. When considering the future cost of college tuition, it is essential to determine how much money must be invested today to meet the projected expense in the future. This process relies on the fundamental principles of compound interest, which serve as the backbone for various financial calculations such as valuation of annuities, loan amortization, and mortgage payments.

To calculate the present value of a future sum, the formula used is: PV = FV / (1 + r)^n, where FV represents the future value, r is the annual interest rate, and n is the number of years. Applying this framework, if the future cost is $150,000 in 18 years, and the interest rate is 8%, the present deposit needed can be computed as PV = 150,000 / (1 + 0.08)^18. This yields approximately $30,797. If the interest rate increases to 11%, the present value becomes PV = 150,000 / (1 + 0.11)^18, which reduces the initial deposit required to approximately $22,661. This demonstrates how higher interest rates decrease the amount needed initially to reach a specific future sum because of the increased growth rate of invested funds over time.

Valuing an annuity, such as receiving equal payments over a period, involves calculating the present value of a series of cash flows. For example, suppose a prize of $5,000,000 is to be received in equal payments over 20 years, with the first payment starting today. At an interest rate of 7%, the present value can be calculated using the formula for an annuity with payments starting immediately, often called an annuity due:

PV = P × [(1 - (1 + r)^-n) / r] × (1 + r),

where P is the payment amount, r is the interest rate per period, and n is the number of periods. Solving for the payment P, given the total present value of $5,000,000, one can determine the individual payment amount. This calculation reflects the time value of money and adjust for the fact that payments begin immediately.

In the context of loans and savings accounts, the problem involving a $1,000 balance earning 9% interest illustrates the equivalence of savings and repayment schedules. Specifically, a savings account with a balance of $1,000 and an annual interest rate of 9% is exactly sufficient to make annual payments on a $1,000 loan over three years at the same interest rate. This involves calculating the annuity payment needed to amortize the loan, which can be derived from the present value of an annuity formula or directly through amortization schedules. The consistency of the account balance and the payment schedule ensures the loan's principal is fully repaid over the term, with interest paid annually.

Mortgages are a common application of amortization. To determine the monthly payment on a mortgage of $75,000 at an annual interest rate of 12% over 30 years, standard mortgage formulas are used. The monthly interest rate is r = 0.12 / 12 = 0.01 (1%), and the total number of payments is n = 30 × 12 = 360. The payment (PMT) can be calculated via:

PMT = P × [r(1 + r)^n] / [(1 + r)^n - 1],

where P is the loan amount. Plugging in the values yields the monthly payment, which can be used to generate an amortization schedule, illustrating how each payment allocates between interest and principal over the loan term. This schedule is critical for understanding equity buildup and interest costs over time.

Finally, considering home affordability, if homes in a certain price range require monthly payments of approximately $1,200 over 30 years at 9% interest, we can reverse-engineer to find the maximum mortgage size. Using the same mortgage formula and solving for the principal P, based on known monthly payments, interest rate, and term, provides the approximate mortgage amount one can afford under these conditions. This calculation is valuable for both buyers and financial planners assessing debt capacity and planning budgets.

Overall, mastering these financial calculations enhances understanding of how interest rates, time horizons, and periodic cash flows influence investment decisions, loan repayment plans, and personal financial planning. Accurate valuation techniques enable consumers and professionals to make informed choices aligned with their long-term financial goals, considering inflation, interest rate fluctuations, and individual circumstances. By employing precise formulas and generating amortization schedules, individuals can visualize the impact of different scenarios, optimize savings strategies, and negotiate better loan terms.

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