The Time Value of Money. Chapter 8. October 3, 2012. Learning Objectives. The “time value of money” and its importance to you and business decisions The future value and present value of a single amount. The future value and present value of an annuity.
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The Time Value
of Money
Chapter 8
October 3, 2012
$23,600 mortgage, $175 mo. pymt
I bought my house in Los Altos in 1979 for $135,000
$40,000 30 yr mortgage, $300 mo
In 2009, would still paying $300 mo!
House sold for over $1.25 million in 2006
Current owner paying $5,500 per month
I now own $935,000 home, no mortgage!
Time value of money
FV = PV (1 + k)n
$100 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 = $146.93
$100 x (1.08)5 = $100 x 1.4693 = $146.93
$100 x (1.08)15 = $100 x 3.1722* = $317.22
*Table I, p. A-1
Appendix
Calculator solution:
N = 15
I/Y = 8
PV = -$100
PMT = 0
Compute (CPT) FV = $317.22
1
(1 + k)n
PV = FVn x
0 1 2
100
(1.10)1
PV =
=
*Table II, p. A-2, Appendix
Example: Expect to receive $100 in one year. If can invest at 10%, what is it worth today?
$100
$100 x .9091* = $90.91
$
1
(1 + k)n
PV = FVn x
0 1 2 3 4 5 6 7 8
100
(1+.10)8
=
PV =
Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today?
$100
$100 x .4665* = $46.65
*Table II, p. A-2, Appendix
100
(1+.10)8
= 46.65
PV =
Using Formula:
N
I/YR
PV
PMT
FV
100
8 10 ?
Previous Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today?
Calculator Enter:
N = 8
I/YR= 10
PMT = 0
FV= 100
CPT PV = ?
- 46.65
0
Jan Feb Mar Dec
$500
$500
$500
$500
$500
0 1 2 3
$0
$100
$100
$100
You deposit $100 each year (end of year) into a savings account (saving up for an IPad).
How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?
0 1 2 3
$0
$100
$100
$100
$100(1.08)2
$100(1.08)1
$100(1.08)0
$100.00
$108.00
$116.64
$324.64
You deposit $100 each year (end of year) into a savings account.
How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?
$100(1.08)2
$100(1.08)1
$100(1.08)0
$100.00
$108.00
$116.64
$324.64
)
(1+.08)3 - 1
.08
(
= 100
n
(1+k) - 1
k
FVA = PMTx( )
0 1 2 3
$0
$100
$100
$100
How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?
= 100(3.2464*) = $324.64
*Table III, p. A-3, Appendix
0 1 2 3
$0
$100
$100
$100
N
I/YR
PV
PMT
FV
Enter:
N = 3
I/YR = 8
PV= 0
PMT = -100
CPT FV = ?
324.64
3 8 0 -100 ?
0 1 2 3
$0
$100
$100
$100
0 1 2 3
$0
$100
$100
$100
$100/(1.08)1
$100 / (1.08)2
$100 / (1.08)3
$92.60
$85.73
$79.38
$257.71
0 1 2 3
$100/(1.08)1
$100 / (1.08)2
$100 / (1.08)3
$92.60
$0
$100
$100
$100
$85.73
$79.38
1
(1.08)3
$257.71
1 -
(
)
= 100
1
(1+k)n
1 -
.08
PVA = PMTx( )
k
= 100(2.5771*) = $257.71
*Table IV, p. A-4, Appendix
0 1 2 3
$0
$100
$100
$100
N
I/YR
PV
PMT
FV
PV=?
Enter:
N = 3
I/YR = 8
PMT = 100
FV= 0
CPT PV = ?
-257.71
3 8 ? 100 0
0 1 2 3
$100
$100
$100
FVA=?
You deposit $100 each year (beginning of year) into a savings account.
How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?
0 1 2 3
$100
$100
$100
$100(1.08)3
$100(1.08)2
$100(1.08)1
$108
$116.64
$125.97
$350.61
You deposit $100 each year (beginning of year) into a savings account.
How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?
0 1 2 3
(1+k)n - 1
k
FVA= PMTx( )
$108
$100(1.08)3
$100(1.08)2
(1+k)
$100(1.08)1
$100
$100
$100
$116.64
$125.97
$350.61
)
(
(1+.08)3 - 1
.08
= 100
(1.08)
How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?
=100(3.2464)(1.08)=$350.61
n = 3, i/y = 8%, pmt = -100, PV = 0
CPT FV = 324.64 (1.08) = 350.61
0 1 2 3
$100
$100
$100
PV=?
0 1 2 3
$100
$100
$100
$100/(1.08)1
$100 / (1.08)2
$100/(1.08)0
$100.00
$92.60
$85.73
$278.33
0 1 2 3
$100
$100
$100
1
(1.08)3
1 -
(
)
(1.08)
= 100
1
(1+k)n
1 -
.08
(1+k)
PVA = PMTx()
k
$100/(1.08)1
$100 / (1.08)2
$100/(1.08)0
$100.00
$92.60
$85.73
$278.33
= 100(2.5771)(1.08) = 278.33
N = 3, i/y = 8%, PMT = 100, FV = 0,
CPT PV = -257.71 (1.08) = -278.33
0 1 2 3 4 5
$5,000
$?
$?
$?
$?
$?
N
I/YR
PV
PMT
FV
ENTER:
N = 5
I/YR = 6
PV= 5,000
FV= 0
CPT PMT = ?
–1,186.98
5 6 5,000 ? 0
N = n x m and K = k / m
1 -
)
(
= PMT
$20,000
1
(1.0075)48
.0075
1
(1+k)n
1 -
PVA = PMTx( )
k
$20,000 = PMT(40.184782)
PMT = 497.70
Note: Tables no longer work
N
I/YR
PV
PMT
FV
ENTER:
N = 48
I/YR = .75
PV= 20,000
FV= 0
CPT PMT = ?
– 497.70
Note:
N = 4 * 12 = 48
I/YR = 9/12 = .75
48 .75 20,000 ? 0
PMT
k
PVP =
PMT
k
PVP =
If k = 8%: PVP = $5/.08 = $62.50
Proof: $62.50 x .08 = $5.00
Example:A share of preferred stock pays a constant dividend of $5 per year. What is the present value if k =8%?
0 1 2
$200
$230
FV= PV(1+ k)n
1.15 = (1+ k)2
Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment?
230 = 200(1+ k)2
1.15 = (1+ k)2
1.0724 = 1+ k
k = .0724 = 7.24%
N
I/YR
PV
PMT
FV
2
?
-200
230
Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment?
Enter known values:
N=2
I/YR=?
PV= -200
PMT = 0
FV= 230
Solve for:
I/YR= ?
7.24
0
N = 1.9995, or 2 years
N
I/YR
PV
PMT
FV
Example: A $200 investment has grown to $230. If the ANNUAL return on this investment is 7.24%, how long would it take?
? 7.24 -200 0 230
Compounding Frequency
$500(1.09)2= $594.05 Annual
$500(1.045)4= $596.26 Semi-annual
$500(1.0225)8= $597.42 Quarterly
$500(1.0075)24= $598.21 Monthly
$500(1.000246575)730= $598.60 Daily