tmv and its importance
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The Time Value
of Money
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The Time Value of Money
• Money grows over time when it earnsinterest.
• Therefore, money that is to be received atsome time in the future is worth less thanthe same dollar amount to be receivedtoday. Why?
• Similarly, a debt of a given amount to bepaid in the future is less burdensomethan that debt to be paid now. Why?
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Some Examples
• Bought Oakland house for $29,500 in 1969
$23,600 mortgage, $244 mo. pymt
I bought my house in Los Altos in 1979 for $135,000$70,000 30 yr mortgage, $500 mo
In 2009, would still paying $500 mo!
House sold for over $1.25 million in 2006
Current owner paying $5,500 per monthI now own $935,000 home, no mortgage!
Time value of money
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Indians – Manhattan Island
• In 1624, Indians got $24 for Manhattan island
• People think they were “taken”
• If invested at 8%, compounded annually, todaythey would have $206,635,300,000,000 (trillion)
• If compounded semiannually, $366 trillion
• If compounded quarterly, $493 trillion
• You could buy Manhattan today for around acouple $100 billion
• They could pay off the nat’l debt/buy back US!
• Time value of money!
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16 year old saves for retirement!
• Earns $2,000 per year for 6 years/stops
• Reinvests at 10% per year
• At 21 years old, she is worth $15,431
• At age 65, with no add’l investment, if she justlets it ride, she will be worth $1,022,535
• If she waits just one more year to get started, shewould be worth only $929,578
• She loses $92,957! (final years earnings)• So start saving now! You’ll never miss it.
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The Future Value of a Single Amount
• Suppose that you have $100 today andplan to put it in a bank account that earns
8% (k) per year.• How much will you have after 1 year?
• After one year:$100 + (.08 x $100) = $100 + $8 = $108
Or• If k = 8%, then 1 + k = 1 + .08 or 1.08 Then,
$100 x (1.08)1 = $108
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The Future Value of a SingleAmount• Suppose that you have $100 today and plan to put it
in a bank account that earns 8% per year.
• How much will you have after 1 year? 5? 15?
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After one year:$100 x (1.08)1 = $100 x 1.08 = $108
After five years:
$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
After fifteen years:
$100 x (1.08)15 = $100 x 3.1722 = $317.22
Equation:FV = PV (1 + k)n Table I, p. A-1
Appendix
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The Future Value of a Single Amount
Calculator solution:
N = 15
I/Y = 8PV = -$100
PMT = 0
Compute (CPT) FV = $317.22
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Present Value of a Single Amount
• Value today of an amount to be received or paidin the future.
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Example: Expect to receive $100 in one year. If caninvest at 10%, what is it worth today?
0 1 2
$100 PV = 100
(1.10)1=
PV = FVn
x 1
(1 + k)n
Table II, p. A-2, Appendix
$$100 x .9091 = $90.91
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Present Value of a Single Amount
• Value today of an amount to be received or paidin the future.
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Example: Expect to receive $100 in EIGHT years. Ifcan invest at 10%, what is it worth today?
=PV =100
(1+.10)8
0 1 2 3 4 5 6 7 8
$100
PV = FVn x 1
(1 + k)n
$100 x .4665 = $46.65
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Financial Calculator Solution - PV
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N I/YR PV PMT FV
- 46.65
PV =100
(1+.10)8
= 46.65Using Formula:
8 10 ? 100
Previous Example: Expect to receive $100 in EIGHTyears. If can invest at 10%, whatis it worth today?
Calculator Enter:N = 8I/YR = 10PMT = 0FV = 100
CPT PV = ?
0
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Financial Calculator Solution - FV
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N I/YR PV PMT FV
322.10
5 10 -200 0 ?
Calculator Enter:N = 5
I/YR = 10PV = -200PMT = 0CPT FV = ?
Using Formula: FV = $200 (1.10)5
= $322.10
Previous Example: You invest $200 at 10%. Howmuch is it worth after 5 years?
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Annuities
• An annuity is a series of equal cashflows spaced evenly over time.
• For example, you pay your landlord anannuity since your rent is the sameamount, paid on the same day of themonth for the entire year.
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$500 $500 $500 $500 $500
Jan Feb Mar Dec
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Future Value of an Annuity
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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 3years if interest were earned at a rate of 8% annually?
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Future Value of an Annuity
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$100(1.08)2 $100(1.08)1
$108.00$116.64$324.64
You deposit $100 each year (end of year) into asavings 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
$0 $100 $100 $100
$100.00
$100(1.08)0
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Future Value of an Annuity
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How much would this account have in it at the end of 3 yearsif interest were earned at a rate of 8% annually?
= 100(3.2464) = $324.64FVA = PMTx( ) (1+k) - 1
k
n = 100 (1+.08)3 - 1.08
( )
0 1 2 3
$0 $100 $100 $100 $100(1.08)2 $100(1.08)1
$108.00$116.64$324.64
$100.00
$100(1.08)0
Table III . A-3 A endix
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Future Value of an AnnuityCalculator Solution
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N I/YR PV PMT FV
3 8 0 -100 ?
Enter:N = 3
I/YR = 8PV = 0PMT = -100CPT FV = ?
0 1 2 3
$0 $100 $100 $100
324.64
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Present Value of an Annuity
• How much would the following cash flowsbe worth to you today if you could earn 8%on your deposits?
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0 1 2 3
$0 $100 $100 $100
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Present Value of an Annuity
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$100 / (1.08)2
$92.60$85.73$79.38
$100/(1.08)1 $100 / (1.08)3
$257.71
How much would the following cash flows be worthto you today if you could earn 8% on yourdeposits?
0 1 2 3
$0 $100 $100 $100
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Present Value of an Annuity
• How much would the following cash flows be worth to youtoday if you could earn 8% on your deposits?
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= 100(2.5771) = $257.71PVA = PMTx( )
1(1+k)n1 -
k
$100 / (1.08)2
$92.60$85.73$79.38
$100/(1.08)1 $100 / (1.08)3
$257.71
0 1 2 3
$0 $100 $100 $100
.08= 100
1 - 1(1.08)3( )
Table IV, p. A-4, Appendix
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Present Value of an Annuity
Calculator Solution
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N I/YR PV PMT FV
3 8 ? 100 0
PV=?
Enter:N = 3I/YR = 8PMT = 100FV = 0CPT PV = ?
0 1 2 3
$0 $100 $100 $100
-257.71
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Future Value of an Annuity Due
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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 yearsif interest were earned at a rate of 8% annually?
0 1 2 3
$100 $100 $100 FVA=?
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Future Value of an Annuity Due
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$100(1.08)2 $100(1.08)1$100(1.08)3
$108$116.64
$125.97
$350.61
0 1 2 3
$100 $100 $100
You deposit $100 each year (beginning of year) into asavings account.
How much would this account have in it at the end of 3 yearsif interest were earned at a rate of 8% annually?
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Future Value of an Annuity Due
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$100(1.08)2 $100(1.08)1$100(1.08)3
$108
$116.64
$125.97
$350.61
0 1 2 3
$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)(1.08)=$350.61
FVA = PMTx( ) (1+k)(1+k)n - 1k
(1+.08)3 - 1.08
= 100 (1.08)( )
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Calculator solution to annuity due
• Same as regular annuity, except
• Multiply your answer by (1 + k) to account for theadditional year of compounding or discounting
• Future value of an annuity due:
n = 3, i/y = 8%, pmt = -100, PV = 0
CPT FV = 324.64 (1.08) = 350.61
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Present Value of an Annuity Due
• How much would the following cash flows be
worth to you today if you could earn 8% onyour deposits?
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PV=?
0 1 2 3
$100 $100 $100
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Present Value of an Annuity Due
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$100 / (1.08)2
$92.60$85.73
$100/(1.08)1
$278.33
0 1 2 3
$100 $100 $100
How much would the following cash flows be worthto you today if you could earn 8% on your deposits?
$100.00
$100/(1.08)0
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Present Value of an Annuity Due29
= 100(2.5771)(1.08) = 278.33PVA = PMTx( )
1(1+k)n1 -
k(1+k)
$100 / (1.08)2
$92.60$85.73
$100/(1.08)1
$278.33
0 1 2 3
$100 $100 $100
How much would the following cash flows be worth
to you today if you could earn 8% on your deposits?
.08= 100
1 - 1(1.08)3
(1.08)( )
$100.00
$100/(1.08)0
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Calculator solution to annuity due
• Same as regular annuity, except
• Multiply your answer by (1 + k) to account for theadditional year of compounding or discounting
• Present value of an annuity due:N = 3, i/y = 8%, PMT = 100, FV = 0,
CPT PV = -257.71 (1.08) = -278.33
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Amortized Loans
• A loan that is paid off in equal amounts thatinclude principal as well as interest.
• Solving for loan payments (PMT).
• Note: The amount of the loan is the present value(PV)
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Amortized Loans• You borrow $5,000 from your parents to purchase a used car.
You agree to make payments at the end of each year for thenext 5 years. If the interest rate on this loan is 6%, how much isyour annual payment?
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N I/YR PV PMT FV
0 1 2 3 4 5
$5,000 $? $? $? $? $?
–1,186.98
5 6 5,000 ? 0
ENTER:N = 5
I/YR = 6PV = 5,000FV = 0CPT PMT = ?
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Compounding more than once per Year
• If m = number of compounds, thenN = n x m and K = k / m
• Annual i.e. N = 4 K = 12%
• Semi-annual N = 4 x 2 = 8
• K = 12% / 2 = 6%
• Quarterly N = 4 x 4 = 16
• K = 12% / 4 = 3%• Monthly N = 4 x 12 = 48
• K = 12% / 12 = 1%
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Amortized Loans
• You borrow $20,000 from the bank to purchase a used car.You agree to make payments at the end of each month forthe next 4 years. If the annual interest rate on this loan is 9%,how much is your monthly payment?
$20,000 = PMT(40.184782)PVA = PMTx( )
1
(1+k)n1 -k
PMT = 497.70
= PMT.0075
1 -1
(1.0075)48
$20,000 ( )
Note: Tables no longer work
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Amortized Loans
• You borrow $20,000 from the bank to purchase a used car.You agree to make payments at the end of each month forthe next 4 years. If the annual interest rate on this loan is 9%,how much is your monthly payment?
ENTER:N = 48I/YR = .75PV = 20,000
FV = 0CPT PMT = ?
N I/YR PV PMT FV
– 497.70
48 .75 20,000 ? 0
Note:
N = 4 * 12 = 48
I/YR = 9/12 = .75
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Perpetuities
• A perpetuity is a series of equal paymentsat equal time intervals (an annuity) that willbe received into infinity.
PMTk
PVP =
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Perpetuities
If k = 8%: PVP = $5/.08 = $62.50
Proof: $62.50 x .08 = $5.00
PMTkPVP =
A perpetuity is a series of equal payments at equaltime intervals (an annuity) that will be received intoinfinity (i.e., retirement payments)
Example: A share of preferred stock pays a constantdividend of $5 per year. What is the presentvalue if k =8%?
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Solving for k
Example: A $200 investment has grown to $230 overtwo years. What is the ANNUAL return onthis investment?
0 1 2
$230 $200
FV = PV(1+ k)n
230 = 200(1+ k)2
1.15 = (1+ k)2
1.0724 = 1+ k
1.15 = (1+ k)2
k = .0724 = 7.24%
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Solving for k - Calculator Solution
N I/YR PV PMT FV
Enter known values:
N = 2I/YR = ?PV = -200PMT = 0FV = 230
Solve for:
I/YR = ?
2 -200 230?
Example: A $200 investment has grown to $230 overtwo years. What is the ANNUAL return onthis investment?
7.24
0
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Solving for N
Example: A $200 investment has grown to $230.If the ANNUAL return on this investment is 7.24%,how long did it take?
• Enter known values:
• N = ?
• I/YR = 7.24
• PV = -200
• PMT = 0• FV = 230
N = 1.9995, or 2 years
N I/YR PV PMT FV
? 7.24 -200 0 230
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Compounding more than Once per Year
• $500 invested at 9% annual interest for 2 years.Compute FV.
$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
Compounding
Frequency