Chapter 9 Problems 6: Determine The Present Value Of $5000

Chapter 9problems6determine The Present Values If 5000 Is Received I

Determine the present values if $5,000 is received in the future (i.e., at the end of each indicated time period) in each of the following situations: a. 5 percent for ten years b. 7 percent for seven years c. 9 percent for four years.

Assume you are planning to invest $5,000 each year for six years and will earn 10 percent per year. Determine the future value of this annuity if your first $5,000 is invested at the end of the first year.

Determine the present value now of an investment of $3,000 made one year from now and an additional $3,000 made two years from now if the annual discount rate is 4 percent.

What is the present value of a loan that calls for the payment of $500 per year for six years if the discount rate is 10 percent and the first payment will be made one year from now? How would your answer change if the $500 per year occurred for ten years?

Determine the annual payment on a $500,000, 12 percent business loan from a commercial bank that is to be amortized over a five-year period.

Determine the annual payment on a $15,000 loan that is to be amortized over a four-year period and carries a 10 percent interest rate. Also, prepare a loan amortization schedule for this loan.

Assume a bank loan requires an interest payment of $85 per year and a principal payment of $1,000 at the end of the loan's eight-year life.

• a. At what amount could this loan be sold for to another bank if loans of similar quality carried an 8.5 percent interest rate? That is, what would be the present value of this loan?
• b. Now, if interest rates on other similar-quality loans are 10 percent, what would be the present value of this loan?
• c. What would be the present value of the loan if the interest rate is 8 percent on similar-quality loans?

Paper For Above instruction

Financial decision-making often hinges upon understanding the present value (PV) concept, which allows investors and borrowers to evaluate the worth of future cash flows in today’s dollars. This paper explores the fundamental principles of present value calculations through various practical scenarios, emphasizing its critical role in investment analysis, loan valuation, and financial planning. By examining different interest rates, time horizons, and cash flow structures, the discussion elucidates how present value influences financial decisions across diverse contexts.

Introduction to Present Value

The present value is the current worth of a future sum of money or a stream of cash flows given a specified rate of return or discount rate. It encapsulates the time value of money, asserting that a sum of money today is more valuable than the same sum in the future due to its potential earning capacity. The PV calculation depends on the discount rate, which reflects the opportunity cost of capital, inflation expectations, and risk factors associated with the cash flows.

Present Value of a Lump Sum in the Future

One of the fundamental calculations involves determining the present value of a future lump sum. For example, analyzing the PV of $5,000 to be received in the future at different interest rates showcases how the discount rate significantly impacts today’s valuation. Using the present value formula PV = FV / (1 + r)^n, where FV is the future value, r is the discount rate, and n is the number of periods, enables precise valuation. For instance, a future receipt of $5,000 in ten years at 5% yields PV = 5000 / (1.05)^10 ≈ $3,063.45.

Similar calculations at 7% for seven years (PV ≈ $3,312.63) and 9% for four years (PV ≈ $3,392.94) demonstrate how higher rates and longer horizons diminish present value.

Present Value of Annuities

Annuities involve equal payments made periodically over a specified period. Planning to invest $5,000 annually for six years with a 10% return results in a future value (FV) of the annuity. The FV of an ordinary annuity can be calculated using the formula FV = P * [( (1 + r)^n - 1) / r], where P is the payment. Applying this formula, the future value after six years can be computed, considering the timing of payments and the interest accrued.

Similarly, the present value of receiving $3,000 in one and two years at a 4% discount rate involves calculating PV of each cash flow separately and summing them: PV = 3000 / (1.04)^1 + 3000 / (1.04)^2 ≈ $2,884.62 + $2,774.67 ≈ $5,659.29.

Valuation of Annuity and Perpetuity

The valuation of loans with fixed payments involves discounting expected payments. For instance, a loan paying $500 annually for six years at 10% has a present value computed using the present value of an annuity formula: PV = P x [1 - (1 + r)^-n] / r. If extended to ten years, the PV increases due to the longer payment period, illustrating the importance of duration in loan valuation.

For example, PV = 500 * [1 - (1 + 0.10)^-6] / 0.10 ≈ $2,715.57 for six years. Extending to ten years yields a PV of approximately $3,790.78, reflecting the impact of a longer amortization period on the loan’s valuation.

Loan Amortization and Payments

Determining loan payments involves calculating fixed periodic payments that amortize the principal and interest over the term. Calculations for a $500,000 loan at 12% over five years yield an annual payment, which can be derived via amortization formulas. Similarly, a $15,000 loan at 10% over four years requires detailed amortization schedules, showing how each payment reduces the principal while covering interest accrued.

Creating these schedules involves systematic calculations of interest applied on the remaining principal and the reduction of the principal after each payment, providing borrowers and lenders with clear repayment pathways.

Valuing Loan Sales and Market Rate Impact

The valuation of an existing loan depends on current market interest rates. If a bank considers selling a loan with an $85 annual interest payment and a $1,000 principal due at the end, the present value of this loan is calculated by discounting the interest payments and the principal repayment at prevailing market rates.

At an 8.5% rate, the present value increases compared to a 10% rate, reflecting the inverse relationship between market interest rates and loan value. Conversely, a lower rate like 8% increases the loan's market value, illustrating the sensitivity of loan valuation to market rate fluctuations.

Conclusion

Understanding the principles of present value is vital for sound financial decision-making. Whether evaluating future cash inflows, loan payments, or investment opportunities, PV calculations provide a quantitative foundation to assess value over time. The impact of discount rates, payment structures, and market conditions emphasizes the need for accurate computations to make informed financial choices. As demonstrated through various examples in this paper, mastery of present value concepts enhances strategic financial planning and valuation accuracy, essential skills for managers, investors, and financial analysts alike.

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