2-1

FV Annuity Formula

The future value of an annuity with n periods and an interest rate of i can be found with the following formula:

.33110.

100

0.10

1)0(1

i

1i)(1PMT

3

n

2-2

Financial calculators solve this equation:

There are 5 variables. If 4 are known, the calculator will solve for the 5th.

.0i

1ni)(1PMTn

i1PVnFV

Financial Calculator Formula for Annuities

2-3

3 10 0 -100

331.00N I/YR PV PMT FV

Financial Calculator Solution

Have payments but no lump sum PV, so enter 0 for present value.

INPUTS

OUTPUT

2-4

Spreadsheet Solution

Use the FV function: see spreadsheet.

= FV(Rate, Nper, Pmt, Pv)

= FV(0.10, 3, -100, 0) = 331.00

2-5

What’s the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.64

75.13248.69 = PV

2-6

PV Annuity Formula

The present value of an annuity with n periods and an interest rate of i can be found with the following formula:

69.24810.

100

0.10)0(1

11-

ii)(1

11-

PMT

3

n

2-7

Have payments but no lump sum FV, so enter 0 for future value.

3 10 100 0N I/YR PV PMT FV

-248.69

INPUTS

OUTPUT

Financial Calculator Solution

2-8

Spreadsheet Solution

Use the PV function: see spreadsheet.

= PV(Rate, Nper, Pmt, Fv)

= PV(0.10, 3, 100, 0) = -248.69

2-9

Find the FV and PV if theannuity were an annuity due.

100 100

0 1 2 3

10%

100

2-10

PV and FV of Annuity Due vs. Ordinary Annuity

PV of annuity due:

= (PV of ordinary annuity) (1+i)

= (248.69) (1+ 0.10) = 273.56

FV of annuity due:

= (FV of ordinary annuity) (1+i)

= (331.00) (1+ 0.10) = 364.1

2-11

3 10 100 0

-273.55 N I/YR PV PMT FV

Switch from “End” to “Begin”.Then enter variables to find PVA3 = $273.55.

Then enter PV = 0 and press FV to findFV = $364.10.

INPUTS

OUTPUT

2-12

Excel Function for Annuities Due

Change the formula to:

=PV(10%,3,-100,0,1)

The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due:

=FV(10%,3,-100,0,1)

2-13

What is the PV of this uneven cashflow stream?

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39-34.15

530.08 = PV

2-14

Financial calculator: HP10BII

Clear all: Orange Shift key, then C All key (in orange).

Enter number, then hit the CFj key.

Repeat for all cash flows, in order.

To find NPV: Enter interest rate (I/YR). Then Orange Shift key, then NPV key (in orange).

2-15

Financial calculator: HP10BII (more)

To see current cash flow in list, hit RCL CFj CFj

To see previous CF, hit RCL CFj –

To see subseqent CF, hit RCL CFj +

To see CF 0-9, hit RCL CFj 1 (to see CF 1). To see CF 10-14, hit RCL CFj . (period) 1 (to see CF 11).

2-16

Input in “CFLO” register:

CF0 = 0

CF1 = 100

CF2 = 300

CF3 = 300

CF4 = -50

Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

2-17

Spreadsheet Solution

Excel Formula in cell A3:

=NPV(10%,B2:E2)

A B C D E

1 0 1 2 3 4

2 100 300 300 -50

3 530.09