Time Value of Money *Financial calculator functions: N I/Y PV PMT FV Enter four of the five keys, and solve the 5th one. *Excel functions: Future value FV FV(rate, nper, pmt, pv) Present value PV PV(rate, nper, pmt, fv) Number of periods nper nper(rate, pmt, pv, fv) Interest rate rate rate(nper, pmt, pv, fv) Payment pmt pmt(rate, nper, pv, fv) FV, PV, rate, nper, pmt Enter four of the five parameters, and solve the 5th one. Calculate Future Value (1) Compute the future of $1000 compounded annually for 10 years at 6 percent. 1. Use the mathematical formula To find the FV of a lump sum, we use: FV = PV(1 + r)t FV = $1,000(1.06)10 = $1,790.85 2. Use financial calculator Enter 10 6 โฆ

Time Value of Money

*Financial calculator functions:

N I/Y PV PMT FV

Enter four of the five keys, and solve the 5th one.

*Excel functions:

Future value FV FV(rate, nper, pmt, pv)

Present value PV PV(rate, nper, pmt, fv)

Number of periods nper nper(rate, pmt, pv, fv)

Interest rate rate rate(nper, pmt, pv, fv)

Payment pmt pmt(rate, nper, pv, fv)

FV, PV, rate, nper, pmt

Enter four of the five parameters, and solve the 5th one.

Calculate Future Value

(1) Compute the future of $1000 compounded annually for 10 years at 6 percent.

1. Use the mathematical formula

To find the FV of a lump sum, we use:

FV = PV(1 + r)t

FV = $1,000(1.06)10 = $1,790.85

2. Use financial calculator

Enter 10 6 -1,000 0

N I/Y PV PMT FV

Solve for 1790.85

3. Use Excel Spreadsheet

FV(rate, nper, pmt, pv)

FV(0.06, 10, 0, -1000)= 1790.85

(2) Compute the future of $1000 compounded semiannually for 10 years at 6 percent.

1. Use the mathematical formula

To find the FV of a lump sum, we use:

๐น๐ = ๐๐(1 + ๐

๐ )๐กร๐

m=2, r =6%, t=10, PV=1000

FV = $1,000(1.03)20 = $1,806.11

2. Use financial calculator

Enter 20 3 -1,000 0

N I/Y PV PMT FV

Solve for 1806.11

3. Use Excel Spreadsheet

FV(rate, nper, pmt, pv)

FV(0.03, 20, 0, -1000)= 1806.11

Calculate Present Value

What is the present value of $500,000 to be received ten years from today if the discount rate is

6% annually?

1. Use the mathematical formula

To find the PV of a lump sum, we use:

๐๐ = ๐น๐

(1+๐)๐ก

PV = 500000 / (1.06)10 = $279,197

2. Use financial calculator

Enter 10 6 0 500000

N I/Y PV PMT FV

Solve for -279,197

3. Use Excel Spreadsheet

PV(rate, nper, pmt, fv)

PV(0.06, 10, 0, 500000)= -279,197

Present value is $279,197.

Solve for the number of periods

How many years will it take for your initial investment of $7,752 to grow to $20,000 if it is

invested so that it earns 9% compounded annually?

1. Use the mathematical formula

FV = PV(1 + r)t

20,000 = 7,752(1.06)t = $1,790.85

FV=PV (1+i)n

N =ln (FV/PV) /ln (1+i)

N=ln (20,000/7,752) /ln (1.09)

N=11 years

2. Use financial calculator

Enter 9 -7,752 0 20,000

N I/Y PV PMT FV

Solve for 11

3. Use Excel Spreadsheet

nper(rate, pmt, pv, fv)

nper(0.09, 0, -7752, 20000)= 11

Solve for the interest rate, i

At what rate must your initial investment of $7,752 be compounded annually for it to grow to

$20,000 in 11 years?

1. Use the mathematical formula

FV = PV(1 + r)t

20,000 = 7,752(1+r)11

(1+r)11 = 2.58

1 + ๐ = โ2.58 11

= 1.09

๐ = 0.09

2. Use financial calculator

Enter 11 -7,752 0 20,000

N I/Y PV PMT FV

Solve for 9

3. Use Excel Spreadsheet

rate(nper, pmt, pv, fv)

rate(11, 0, -7752, 20000) = 9%

Answer: The interst rate is 9%.

Orninary Annuity

Future value of an ordinary annuity:

๐น๐๐ = ๐๐๐ [ (1 + ๐)๐ โ 1

๐ ]

Present value of an ordinary annuity:

๐๐๐ = ๐๐๐ [ 1 โ

1 (1 + ๐)๐

๐ ]

Annuity Due

Future value of an ordinary annuity:

๐น๐๐ = ๐๐๐ [ (1 + ๐)๐ โ 1

๐ ] (1 + ๐)

Present value of an ordinary annuity:

๐๐๐ = ๐๐๐ [ 1 โ

1 (1 + ๐)๐

๐ ] (1 + ๐)

Perpetuities

PV of level perpetuity

๐๐ = ๐๐๐

๐

PV of growing perpetuity

๐๐ = ๐๐๐

๐ โ ๐

Effective Annual Rate: EAR

APR: Annual Percentage Rate, Quoated Annual Rate

m: compounding periods per year

๐ธ๐ด๐
= (1 + ๐ด๐๐

๐ )

๐

โ 1

Ordinary Annuity Examples

(1) Youโve taken your first job and you plan to same $5000 each year for the next five years for

your grad school fund. How much money will you accumulate by the end of year five? The

rate of interest is 6% annually.

1. Use the mathematical formula

๐น๐๐ = ๐๐๐ [ (1 + ๐)๐ โ 1

๐ ]

๐น๐๐ = 5000 ร [ (1 + 0.06)5 โ 1

0.06 ] = 5000 ร 5.63709296 = 28,185.46

2. Use financial calculator

Enter 5 6 0 -5000

N I/Y PV PMT FV

Solve for 28,185.46

2. Use Excel Spreadsheet

FV(rate, nper, pmt, pv)

FV(0.06, 5, -5000, 0) = 28,185.46

Answer: The prevent value is $28,185.46.

Ordinary Annuity Examples

(2) How much of the annual payment must you deposit in a savings account earning 6% annual

interest in order to accumulate $5000 at then end of 5 years?

1. Use the mathematical formula

๐น๐๐ = ๐๐๐ [ (1 + ๐)๐ โ 1

๐ ]

5000 = ๐๐๐ ร [ (1 + 0.06)5 โ 1

0.06 ]

5000 = ๐๐๐ ร 5.63709296

๐๐๐ = 886.98 2. Use financial calculator

Enter 5 6 0 5000

N I/Y PV PMT FV

Solve for -886.98

3. Use Excel Spreadsheet

pmt(rate, nper, pv, fv)

pmt(0.06, 5, 0, 5000) = -$886.98

Answer: The annual payment is $886.98.