Present Value The formula for present value depends on the interest rate that is available from a bank or other source. Example Suppose that the annual interest rate r is compounded at the end of each of m equal periods each year;and suppose that a cash payment of amount A will be received at the end of the kth period. Then the appropriate discount factor is 成=+) and thus the present value of a payment of A to be received k periods in the future is PV=dkA. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 16/174
Present Value The formula for present value depends on the interest rate that is available from a bank or other source. Example Suppose that the annual interest rate r is compounded at the end of each of m equal periods each year; and suppose that a cash payment of amount A will be received at the end of the kth period. Then the appropriate discount factor is dk = 1 + r m k −1 , and thus the present value of a payment of A to be received k periods in the future is PV = dkA. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 16 / 174
Present and Future Value of Streams Outline Principal and Interest Present Value 3 Present and Future Value of Streams Internal Rate of Return Evaluation Criteria Applications and Extensions 4口40+4三4至，至)只0 Xi CHEN (chenxi01090bfsu.edu.cn) Investment Science 17/174
Present and Future Value of Streams Outline 1 Principal and Interest 2 Present Value 3 Present and Future Value of Streams 4 Internal Rate of Return 5 Evaluation Criteria 6 Applications and Extensions Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 17 / 174
Present and Future Value of Streams The ldeal Bank An ideal bank applies the same rate of interest to both deposit,and loans,and it has no service charges or transactions fees. o The interest rate applies equally to any size of principal. o Separate transactions in an account are completely additive in their effect on future balances. Interest rates for all transactions may not be identical. Example OA 2-year CD might offer a higher rate than a 1-year CD. A 2-year CD must offer the same rate as a loan that is payable in 2 years. o If an ideal bank has an interest rate that is independent of the length of time for which it applies,and that interest is compounded according to normal rules,it is said to be a constant ideal bank. Q0￥ Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 18/174
Present and Future Value of Streams The Ideal Bank An ideal bank applies the same rate of interest to both deposit, and loans, and it has no service charges or transactions fees. The interest rate applies equally to any size of principal. Separate transactions in an account are completely additive in their effect on future balances. Interest rates for all transactions may not be identical. Example 1 A 2-year CD might offer a higher rate than a 1-year CD. 2 A 2-year CD must offer the same rate as a loan that is payable in 2 years. If an ideal bank has an interest rate that is independent of the length of time for which it applies, and that interest is compounded according to normal rules, it is said to be a constant ideal bank. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 18 / 174
Present and Future Value of Streams Future Value Theorem (Future value of a stream) Given a cash flow (xo,x1,...,xn),and interest rate r each period,the future value of a stream is n FV=∑x(1+r)n-=o(1+rP+x(1+r)n-1+.+x =0 Example Consider the cash flow stream(-2,1,1,1)when the periods are years and the interest rate is 10%.The future value is 0.648. 4口4+4三4至，至)只0 Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 19/174
Present and Future Value of Streams Future Value Theorem (Future value of a stream) Given a cash flow (x0, x1, . . . , xn), and interest rate r each period, the future value of a stream is FV = Xn i=0 xi(1 + r) n−i = x0(1 + r) n + x1(1 + r) n−1 + . . . + xn. Example Consider the cash flow stream (−2, 1, 1, 1) when the periods are years and the interest rate is 10%. The future value is 0.648. Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 19 / 174
Present and Future Value of Streams Present Value Theorem (Present value of a stream) Given a cash flow(xo,为，，xn),and interest rate r each period,the present value of a stream is =a中为+，+a++a X2 Xn Example Consider the cash flow stream(-2,1,1,1).Using an interest rate of 10%, the present value is 0.487. The relationship between PV and FV is FV PV= (1+r）n Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 20/174
Present and Future Value of Streams Present Value Theorem (Present value of a stream) Given a cash flow (x0, x1, . . . , xn), and interest rate r each period, the present value of a stream is PV = Xn i=0 xi (1 + r) i = x0 + x1 1 + r + x2 (1 + r) 2 + . . . + xn (1 + r) n . Example Consider the cash flow stream (−2, 1, 1, 1). Using an interest rate of 10%, the present value is 0.487. The relationship between PV and FV is PV = FV (1 + r) n . Xi CHEN (chenxi0109@bfsu.edu.cn) Investment Science 20 / 174