To find the present value of money, we must know: The Cash Flow Amount (cash inflow or outflow) Time (the “n” # of periods to the future point in time) Discount Rate (“i” to discount back to find the PV) per period Present Value of Money

The Present Value of a Single Lump Sum Amount... … is described as the amount that must be invested today (at time “0”) to produce a known future value PVF n i (1 + i) (1 + i) PV = Amount x PV = Amount x n 1

12345 $100$100$100$100$100 An investment that involves a series of identical cash flows at the end of each year is called an Ordinary Annuity. The Present Value of an Ordinary Annuity equals... (AnnualAmnt) x [PVF-OA(n=#, r=%)]

For an Annuity Due, the “rents” or payments occur at the beginning of each period (pay in advance) 12345$100$100$100$100$1000 The Present Value of an Annuity Due equals... (AnnualAmnt) x [PVF-AD(n=#, i=%)]

Calculating the Present Value of an Annuity Due 1) Find the present value of an ordinary annuity factor n=N-1, i=%) for n-1 periods 2) Multiply that factor time (1+i), that is, 1 plus the interest rate -- to get the “PVF-AD” factor 3) Multiply the periodic “amount” times this PVF-AD to get the present value of payments

12345$0$0$100$100$100 If the cash flows from an ordinary annuity begin after a period of time, we have a Deferred Ordinary Annuity. Present Value of a Deferred Ordinary Annuity... “deferred factor” for payments extending through period “#” beginning after period “d” “deferred factor” for payments extending through period “#” beginning after period “d” = (PVF-OA N=#, r=%) - (PVF-OA N=d, r=%) = (PVF-OA N=#, r=%) - (PVF-OA N=d, r=%) PV of DA = (AnnualAmnt) x [deferred factor]