Time Value of Money Analysis: Financial Analysis Techniques

  Time Value of Money Analysis: Financial Analysis Techniques Question 1 The future value of an initial $100 after three years, with an annual interest rate of 10%, can be calculated using the formula for compound interest: Future Value = Present Value x (1 + Interest Rate)^Time Period Future Value = $100 x (1 + 0.10)^3 Future Value = $100 x (1.10)^3 Future Value = $100 x 1.331 Future Value = $133.10 Therefore, the future value of the initial $100 after three years at a 10% annual interest rate is $133.10. Question 2 The present value of $100 to be received in three years, with an appropriate interest rate of 10% per year, can be calculated using the formula for present value: Present Value = Future Value / (1 + Interest Rate)^Time Period Present Value = $100 / (1 + 0.10)^3 Present Value = $100 / (1.10)^3 Present Value = $100 / 1.331 Present Value = $75.13 Therefore, the present value of $100 to be received in three years at a 10% annual interest rate is $75.13. Difference Between Ordinary Annuity and Annuity Due An ordinary annuity refers to a series of equal cash flows or payments that occur at the end of each period. For example, if you receive $100 at the end of each year for three years, it is an ordinary annuity. An annuity due, on the other hand, refers to a series of equal cash flows or payments that occur at the beginning of each period. To convert an ordinary annuity to an annuity due, you simply need to shift the cash flows or payments one period earlier. Question 3 The cash flow time line represents an ordinary annuity since the cash flows are occurring at the end of each period. To convert it to an annuity due, you would shift all the cash flows or payments one period earlier. In this case, you would move each cash flow one period to the left on the time line. Question 4 To calculate the future value of a 3-year ordinary annuity of $100 with an appropriate interest rate of 10%, you can use the formula for future value of an ordinary annuity: Future Value = Payment x [(1 + Interest Rate)^Time Period - 1] / Interest Rate Future Value = $100 x [(1 + 0.10)^3 - 1] / 0.10 Future Value = $100 x [1.331 - 1] / 0.10 Future Value = $100 x 0.331 / 0.10 Future Value = $100 x 3.31 Future Value = $331 Therefore, the future value of a 3-year ordinary annuity of $100 at a 10% interest rate is $331. To calculate the present value of the annuity, you would use the formula for present value of an ordinary annuity: Present Value = Payment x [1 - (1 + Interest Rate)^(-Time Period)] / Interest Rate Present Value = $100 x [1 - (1 + 0.10)^(-3)] / 0.10 Present Value = $100 x [1 - (1.10)^(-3)] / 0.10 Present Value = $100 x [1 - 0.751] / 0.10 Present Value = $100 x 0.249 / 0.10 Present Value = $100 x 2.49 Present Value = $249 Therefore, the present value of the annuity is $249. To calculate the future and present values if the annuity were an annuity due, you would use the same formulas as above but with one additional period. This is because in an annuity due, the cash flows occur at the beginning of each period rather than at the end. Question 5 The stated, or quoted, or simple rate (rSIMPLE) refers to the nominal interest rate expressed on an annual basis without considering compounding periods. The annual percentage rate (APR) represents the nominal interest rate expressed as a yearly rate that includes any fees or additional costs associated with borrowing. The periodic rate (rPER) is the interest rate applied to each compounding period when calculating future or present values. The effective annual rate (rEAR) represents the actual annual interest rate considering compounding effects on a loan or investment. Question 6 To calculate the effective annual rate (rEAR) for a simple rate of 10% compounded semiannually, we use the formula: rEAR = (1 + rSIMPLE/n)^n - 1 where rSIMPLE is the stated or simple rate, and n is the number of compounding periods per year. For semiannual compounding: rSIMPLE = 10% = 0.10 n = 2 (compounded semiannually) rEAR = (1 + 0.10/2)^2 - 1 rEAR = (1 + 0.05)^2 - 1 rEAR = (1.05)^2 - 1 rEAR = 1.1025 - 1 rEAR = 0.1025 or 10.25% Therefore, the effective annual rate for a simple rate of 10% compounded semiannually is 10.25%. To calculate the effective annual rate for compounded quarterly or daily, similar calculations would be performed by adjusting n accordingly. Question 7 To construct an amortization schedule for a $1,000 loan with a 10% annual interest rate repaid in three equal installments, we can use the following steps: Step 1: Calculate the payment amount: Payment Amount = Loan Amount / Number of Payments Payment Amount = $1,000 / 3 Payment Amount ≈ $333.33 Step 2: Calculate the interest for each payment: Interest Payment = Remaining Loan Balance x Interest Rate Interest Payment for Payment 1 = $1,000 x 10% Interest Payment for Payment 1 = $100 Interest Payment for Payment 2 = Remaining Loan Balance after Payment 1 x Interest Rate Interest Payment for Payment 3 = Remaining Loan Balance after Payment 2 x Interest Rate Step 3: Calculate the principal repayment for each payment: Principal Repayment = Payment Amount - Interest Payment Principal Repayment for Payment 1 ≈ $333.33 - $100 Principal Repayment for Payment 2 ≈ $333.33 - Interest Payment for Payment 2 Principal Repayment for Payment 3 ≈ Remaining Loan Balance after Payment 2 - Interest Payment for Payment 3 Step 4: Calculate the remaining loan balance after each payment: Remaining Loan Balance after Payment n = Remaining Loan Balance after Payment n-1 - Principal Repayment for Payment n Step 5: Repeat steps 2-4 until all payments are accounted for. Here is an example amortization schedule for a $1,000 loan repaid in three equal installments: Payment Payment Amount Interest Payment Principal Repayment Remaining Loan Balance 1 $333.33 $100 ≈$233.33 ≈$766.67 2 $333.33 ≈$76.67 ≈$256.67 ≈$510 3 $333.33 ≈$51 ≈$282.33 ≈$227.67 This amortization schedule demonstrates how a $1,000 loan with a 10% annual interest rate can be repaid in three equal installments over time until the remaining loan balance reaches zero at the end of the schedule.

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