Present Value of Future Cash Flows Whenever a contract establishes a relationship between an initial amount borrowed or loaned and one or more future cash flows, the initial amount borrowed or loaned is the present value of those future cash flows. The present value can be interpreted in two ways: From the borrower's viewpoint, it is the liability that will be exactly From the lender's viewpoint, it is the receivable balance that will be exactly satisfied by the future receipts. In understanding cash flows, cash flow diagrams that display both the amounts and the times of the cash flows specified by a contract can be quite helpful. In these diagrams, a time line runs from left to right. Inflows are represented as arrows pointing upward and outflows as arrows pointing downward. For example, suppose that the Hilliard Corporation borrows $100,000 from Citizens Bank of New Hope on January 1, 2021. The note requires three $38,803.35 payments, one each at the end of 2021, 2022, and 2023, and includes interest at 8% per year. The cash flows for Hilliard are shown in Exhibit A3.1. Exhibit A3.1 Cash Flow Diagram $100,000 $38,803.35 12/31/21 $38,803.35 12/31/22 $38,803.35 12/31/23 1/1/21 The calculation that follows shows, from the borrower's perspective, the relationship between the amount borrowed (the present value and the future payments (future cash flows) required by Hilliard's note. Amount borrowed, 1/1/21 Add: 2021 interest ($100,000.00 X 0.08) Subtract payment on 12/31/21 Liability at 12/31/21 Add: 2022 interest ($69,196.65 X 0.08) Subtract payment on 12/31/22 Liability at 12/31/22 Add: 2023 interest ($35,929.03 X 0.08) Subtract payment on 12/31/23 Liability at 12/31/23 $100,000.00 8,000.00 (38,803.35) $ 69,196.65 5,535.73 (38,803.35) $ 35,929.03 2,874.32 (38,803.35) $ 0.00 Present value calculations like this one are future value calculations in reverse. Here, the three payments of $38,803.35 exactly pay off the liability created by the note. Because the reversal of future value calculations can present a burdensome and sometimes difficult algebraic problem, shortcut methods using tables have been developed (see Exhibits A3.7, A3.8, A3.9, and A3.10, discussed later in this appendix). Exercise A3-17 (Algorithmic) Present Values Use Present Value Tables or your calculator to complete the requirements below. Required: a. Determine the present value of a single $13,700 cash flow in 7 years if the interest (discount) rate is 8% per year. Round your answer to the nearest cent, if rounding is required. b. Determine the number of periods for which $5,820 must be invested at an annual interest (discount) rate of 7% to produce an investment balance of $10,000. Round your answer to the nearest whole number of periods, if rounding is required. periods c. Determine the size of the annual cash flow for a 25-year annuity with a present value of $49,113 and an annual interest rate of 9%. One payment is made at the end of each year. Round your answer to the nearest cent, if rounding is required. ) d. Determine the annual interest rate at which an investment of $2,542 will provide for a single $4,000 cash flow in 4 years. Round your answer to the nearest whole percentage rate (for example, 10.8% rounds to 11%). e. Determine the annual interest rate earned by an annuity that costs $17,119 and provides 15 payments of $2,000 each, one at the end of each of the next 15 years. Round your answer to the nearest whole percentage rate (for example, 10.8% rounds to 11%). Present Value of Future Cash Flows Whenever a contract establishes a relationship between an initial amount borrowed or loaned and one or more future cash flows, the initial amount borrowed or loaned is the present value of those future cash flows. The present value can be interpreted in two ways: From the borrower's viewpoint, it is the liability that will be exactly From the lender's viewpoint, it is the receivable balance that will be exactly satisfied by the future receipts. In understanding cash flows, cash flow diagrams that display both the amounts and the times of the cash flows specified by a contract can be quite helpful. In these diagrams, a time line runs from left to right. Inflows are represented as arrows pointing upward and outflows as arrows pointing downward. For example, suppose that the Hilliard Corporation borrows $100,000 from Citizens Bank of New Hope on January 1, 2021. The note requires three $38,803.35 payments, one each at the end of 2021, 2022, and 2023, and includes interest at 8% per year. The cash flows for Hilliard are shown in Exhibit A3.1. Exhibit A3.1 Cash Flow Diagram $100,000 $38,803.35 12/31/21 $38,803.35 12/31/22 $38,803.35 12/31/23 1/1/21 The calculation that follows shows, from the borrower's perspective, the relationship between the amount borrowed (the present value and the future payments (future cash flows) required by Hilliard's note. Amount borrowed, 1/1/21 Add: 2021 interest ($100,000.00 X 0.08) Subtract payment on 12/31/21 Liability at 12/31/21 Add: 2022 interest ($69,196.65 X 0.08) Subtract payment on 12/31/22 Liability at 12/31/22 Add: 2023 interest ($35,929.03 X 0.08) Subtract payment on 12/31/23 Liability at 12/31/23 $100,000.00 8,000.00 (38,803.35) $ 69,196.65 5,535.73 (38,803.35) $ 35,929.03 2,874.32 (38,803.35) $ 0.00 Present value calculations like this one are future value calculations in reverse. Here, the three payments of $38,803.35 exactly pay off the liability created by the note. Because the reversal of future value calculations can present a burdensome and sometimes difficult algebraic problem, shortcut methods using tables have been developed (see Exhibits A3.7, A3.8, A3.9, and A3.10, discussed later in this appendix). Exercise A3-17 (Algorithmic) Present Values Use Present Value Tables or your calculator to complete the requirements below. Required: a. Determine the present value of a single $13,700 cash flow in 7 years if the interest (discount) rate is 8% per year. Round your answer to the nearest cent, if rounding is required. b. Determine the number of periods for which $5,820 must be invested at an annual interest (discount) rate of 7% to produce an investment balance of $10,000. Round your answer to the nearest whole number of periods, if rounding is required. periods c. Determine the size of the annual cash flow for a 25-year annuity with a present value of $49,113 and an annual interest rate of 9%. One payment is made at the end of each year. Round your answer to the nearest cent, if rounding is required. ) d. Determine the annual interest rate at which an investment of $2,542 will provide for a single $4,000 cash flow in 4 years. Round your answer to the nearest whole percentage rate (for example, 10.8% rounds to 11%). e. Determine the annual interest rate earned by an annuity that costs $17,119 and provides 15 payments of $2,000 each, one at the end of each of the next 15 years. Round your answer to the nearest whole percentage rate (for example, 10.8% rounds to 11%)