Discussion 2
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:
𝐹𝑉 = 𝑃𝑉(1 + 𝑟
𝑚 )𝑡×𝑚
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:
𝑃𝑉 = 𝐹𝑉
(1+𝑟)𝑡
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 + 𝑟 = √2.58 11
= 1.09
𝑟 = 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:
𝐹𝑉𝑛 = 𝑃𝑀𝑇 [ (1 + 𝑟)𝑛 − 1
𝑟 ]
Present value of an ordinary annuity:
𝑃𝑉𝑛 = 𝑃𝑀𝑇 [ 1 −
1 (1 + 𝑟)𝑛
𝑟 ]
Annuity Due
Future value of an ordinary annuity:
𝐹𝑉𝑛 = 𝑃𝑀𝑇 [ (1 + 𝑟)𝑛 − 1
𝑟 ] (1 + 𝑟)
Present value of an ordinary annuity:
𝑃𝑉𝑛 = 𝑃𝑀𝑇 [ 1 −
1 (1 + 𝑟)𝑛
𝑟 ] (1 + 𝑟)
Perpetuities
PV of level perpetuity
𝑃𝑉 = 𝑃𝑀𝑇
𝑖
PV of growing perpetuity
𝑃𝑉 = 𝑃𝑀𝑇
𝑖 − 𝑔
Effective Annual Rate: EAR
APR: Annual Percentage Rate, Quoated Annual Rate
m: compounding periods per year
𝐸𝐴𝑅 = (1 + 𝐴𝑃𝑅
𝑚 )
𝑚
− 1
Ordinary Annuity Examples
(1) You’ve 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
𝐹𝑉𝑛 = 𝑃𝑀𝑇 [ (1 + 𝑟)𝑛 − 1
𝑟 ]
𝐹𝑉𝑛 = 5000 × [ (1 + 0.06)5 − 1
0.06 ] = 5000 × 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
𝐹𝑉𝑛 = 𝑃𝑀𝑇 [ (1 + 𝑟)𝑛 − 1
𝑟 ]
5000 = 𝑃𝑀𝑇 × [ (1 + 0.06)5 − 1
0.06 ]
5000 = 𝑃𝑀𝑇 × 5.63709296
𝑃𝑀𝑇 = 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.