【正文】 Benjamin Franklin Definition: Compound interest is puted on a principal amount and any accumulated interest. A bank that pays pound interest on a savings account putes interest periodically (., daily or quarterly) and adds this interest to the original principal. The interest for the following period is puted by using the new principal (., the original principal plus interest). The formula for the amount, A, you will receive at the end of period n is: A = p (1 + )nt where, p = the principal r = the annual interest rate n = the number of times pounding is done in a year t = the number of years r n Notes: As the number of times pounding is done per year approaches infinity (as in continuous pounding), the amount, A, you will receive at the end of period n is calculated using the formula: A = pert The effective annual interest rate (or yield) is the simple interest rate that would generate the same amount of interest as would the pound rate 26 CU7112997ECA Bain Math Compound Interest Application $1, $ $1, $ $1, $ $1, $ $1, $0 $250 $500 $750 $1,000 $1,250 Dollars i1 i2 i3 i4 A1 A2 A3 A4 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter Given: What amount will you receive at the end of one year if you invest $1,000 at an annual rate of 12% pounded quarterly? Answer: A = p (1+ ) nt = $1,000 (1 + ) 4 = $1, r n 4 Detailed Answer: At the end of each quarter, interest is puted, and then added to the principal. This bees the new principal on which the next period’s interest is calculated. Interest earned (i = prt): i1 = $1, i2 = $1, i3 = $1, 14 = $1, = $ = $ = $ = $ New principle A1 = $1,000+$30 A2 = $1,030+ A3 = $1,+ A4 = $1,+ = $1,030 = $1, = $1, = $1, 27 CU7112997ECA Bain Math Present Value Definitions (1) Time Value of Money: At different points in time, a given dollar amount of money has different values. One dollar received today is worth more than one dollar received tomorrow, because money can be invested with some return. Present Value: Present value allows you to determine how much money that will be received in the future is worth today The formula for present value is: PV = Where, C = the amount of money received in the future r = the annual rate of return n = the number of years is called the discount factor The present value PV of a stream of cash is then: PV = C0+ + + Where C0 is the cash expected today, C1 is the cash expected in one year, etc. 1 (1+r)n C (1+r)n C1 1+r C2 (1+r)2 Cn (1+r)n 28 CU7112997ECA Bain Math Present Value Definitions (2) The present value of a perpetuity (., an infinite cash stream) of is: PV = A perpetuity growing at rate of g has present value of: PV = The present value PV of an annuity, an investment which pays a fixed sum, each year for a specific number of years from year 1 to year n is: Perpetuity: Growing perpetuity: Annuity: C r C rg PV = C r 1 (1+ r)n C r 29 CU7112997ECA Bain Math Present Value Exercise (1) 1) $ today 2) $ five years from today 3) A perpetuity of $ 4) A perpetuity of $, growing at 5% 5) A six year annuity of $ Assume you can invest at 16% per year Which of the following would you prefer to receive? 30 CU7112997ECA Bain Math Present Value Exercise (2) *The present value is negative because this is the cash outflow required today receive a cash inflow at a later time 1) $ today, PV = $ 2) $ five years from today, For HP12C: 5 16 3) A perpetuity of $, PV = = $ 4) A perpetuity of $, growing at 5%, PV = = $ 5) A six year annuity of $, PV = =$ $ $ The option with the highest present value is 1, receiving $ today $ 1 (1+ )5 $ FV i PV N =()* 20( ) ( ) PV = = $ $ (1+)5 Answer: 31 CU7112997ECA Bain Math Risk and Return ?Not all investments have the same risk –investing in the . stock market is more risky than investing in a . government treasury bill, but less risky than investing in the stock market of a developing country ?Most investors are risk averse they avoid risk when they can do so without sacrificing return ?Risk averse investors demand a higher return on higher risk investments A safe dollar is worth more than a risky one. 32 CU7112997ECA Bain Math Net Present Value ?Net present value (NPV) is the method used in evaluating investments whereby the present value of all case outflows required for the investment are added to the present value of all cash inflows generated by the investment ?Cash outflows have negative present values。 cash inflows have positive present values ?The rate used to calculate the present values is the discount rate. The discount rate is the required rate of return, or the opportunity cost of capital (., the return you are giving up to pursue this project) ?An investment is acceptable if the NPV is positive ?In capital budgeting, the discount rate used is called the hurdle rate Definition: 33 CU7112997ECA Bain Math Internal Rate of Return ?The internal rate of return (IRR) is the discount rate for which the present value is zero (., the cost of the investment equals the future cash flows generated by the investment) ?The investment is acceptable when the IRR is greater than the required rate of return, or hurdle rate ?Unfortunately, paring IRRs and c