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To find the present value of money, we must know:To find the present value of money, we must know: The Cash Flow Amount (cash inflow or outflow)The Cash Flow Amount (cash inflow or outflow)

Time (the “n” # of periods to the future point in time)Time (the “n” # of periods to the future point in time)

Discount Rate (“i” to Discount Rate (“i” to discountdiscount back to find the PV) per period back to find the PV) per period

Present Value of MoneyPresent Value of Money

The Present Value of a Single Lump The Present Value of a Single Lump Sum Amount...Sum Amount...

… … is described as the amount that is described as the amount that must be invested today (at time “0”) must be invested today (at time “0”)

to produce a known future valueto produce a known future valuePVFPVF

nn

ii

(1 + (1 + i)i)

PVPV = = Amount Amount xx ---------- ----------nn11

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$100$100 $100$100 $100$100 $100$100 $100$100

An investment that involves a series of An investment that involves a series of identical cash flows at the end of each identical cash flows at the end of each year is called an year is called an Ordinary AnnuityOrdinary Annuity..

The Present Value of an Ordinary Annuity equals ...The Present Value of an Ordinary Annuity equals ...

(AnnualAmnt) x [PVF-OA(n=#, r=%)](AnnualAmnt) x [PVF-OA(n=#, r=%)]

For an For an Annuity DueAnnuity Due, , the “rents” or payments occur at the the “rents” or payments occur at the

beginningbeginning of each period (pay in advance) of each period (pay in advance)

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$100$100 $100$100 $100$100 $100$100 $100$100

00

The Present Value of an Annuity Due equals ...The Present Value of an Annuity Due equals ...

(AnnualAmnt) x [PVF-AD(n=#, i=%)](AnnualAmnt) x [PVF-AD(n=#, i=%)]

Calculating the Present Value of Calculating the Present Value of an Annuity Duean Annuity Due

1) Find the present value of an ordinary annuity 1) Find the present value of an ordinary annuity factor (PVF-OA factor (PVF-OA @@ n=N-1, i=%) for n-1 n=N-1, i=%) for n-1 periodsperiods

2) Multiply that factor time (1+i), that is, 1 plus 2) Multiply that factor time (1+i), that is, 1 plus the interest rate -- to get the “PVF-AD” factorthe interest rate -- to get the “PVF-AD” factor

3) Multiply the periodic “amount” times this 3) Multiply the periodic “amount” times this PVF-AD to get the present value of paymentsPVF-AD to get the present value of payments

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$0$0 $0$0 $100$100 $100$100 $100$100

If the cash flows from an ordinary annuityIf the cash flows from an ordinary annuity begin begin after a period of time, we have a after a period of time, we have a

Deferred Ordinary AnnuityDeferred Ordinary Annuity..

Present Value of a Deferred Ordinary Annuity ...Present Value of a Deferred Ordinary Annuity ...

““deferred factordeferred factor” ” for payments extending through period “#” beginning after period “d”for payments extending through period “#” beginning after period “d”

= (PVF-OA N=#, r=%) - (PVF-OA N== (PVF-OA N=#, r=%) - (PVF-OA N=dd, r=%), r=%)

PV of DA = (AnnualAmnt) x [PV of DA = (AnnualAmnt) x [deferred factordeferred factor]]