### Chapter 5 Time Value of Money

```Chapter 5 Time Value of Money
Time value
• What is the difference between
simple interest and compound
interest?
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Let’s Look at the worksheet
• In the worksheet, is there something special that needs to
be done for a present value amount?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
\$75.00
-\$50.00
5.00
Compounding Freq. (m) (P/Y)
Solve for PV
Solve for FV
FV (Continuous Compounding)
Solve for Interest Rate
\$50.00
8.45%
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
Effective Interest Rate
PVA
PMT for PVA
Interest for PVA (per period)
FVA
PMT for FVA
Interest for FVA
PV of Perpetuity
PV of Growing Annuity
PV of Growing Perpetuity
8.45%
\$
\$
(5.00)
0.08
#DIV/0!
\$
0.08
#DIV/0!
#DIV/0!
FV Example
• If you were to invest \$3,000
today, what would it be worth in
1 year if you can earn 10% on
Years? 5 years?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
-\$3,000.00
10.00%
1.00
Solve for FV
\$3,300.00
FV (Continuous Compounding)
\$3,315.51
Solve for Interest Rate
1
Solve for PV
-\$3,000.00
10.00%
2.00
Solve for FV
\$3,630.00
FV (Continuous Compounding)
\$3,664.21
Solve for Interest Rate
1
Solve for PV
-\$3,000.00
10.00%
5.00
1
Solve for FV
\$4,831.53
FV (Continuous Compounding)
\$4,946.16
Solve for Interest Rate
• What does the word
value or worth mean to
you?
PV Example
• When you were 10 years old
he will give you \$30,000.
Inflation is expected to be 3.5%.
What is this amount worth as a
10 year old?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
\$30,000.00
3.50%
12.00
1
Solve for PV
-\$19,853.50
Solve for FV
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
PV example
• When you retire at age 65, your
retirement fund promises to pay you
\$150,000 the first year of your
retirement. You are now 25. If inflation
is 4%, what is this worth to you in
today’s money?
• If the company can earn 12% on its
retirement investments, how much
must they put away today to get the
above payment?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
\$150,000.00
Compounding Freq. (m) (P/Y)
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
4.00%
40.00
-\$31,243.36
Solve for FV
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
\$150,000.00
12.00%
40.00
1
Solve for PV
-\$1,612.02
Solve for FV
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
Annuities
• What are some examples of cash
flows that are annuity cash flow
streams?
• What is the difference between
an annuity due and an ordinary
annuity?
Annuity Example
• If you started at age 19 to save
2,000 per year at the end of the
year and could average 11% per
year in earnings, how much
would you have at retirement at
age 65?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
11.00%
46.00
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
-\$2,000.00
Effective Interest Rate
11.00%
PVA
PMT for PVA
Interest for PVA (per period)
#NUM!
FVA
\$ 2,192,337.60
• Would it make a difference if you
started at the beginning of the
year instead of the end? How
much?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
11.00%
46.00
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
n
-\$2,000.00
Effective Interest Rate
11.00%
PVA
PMT for PVA
Interest for PVA (per period)
#NUM!
FVA
\$ 2,433,494.74
Building Wealth
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Annuity Example
• If you started saving \$1,500 per
year on a monthly basis for 18
education, how much would you
have if you invested and earned
8%? 12%?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
8.00%
18.00
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
12
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
y
-\$125.00
Effective Interest Rate
8.30%
PVA
PMT for PVA
Interest for PVA (per period)
#NUM!
FVA
\$
60,010.77
Solve for PV
12.00%
18.00
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
12
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
-\$125.00
Effective Interest Rate
12.68%
PVA
PMT for PVA
Interest for PVA (per period)
#NUM!
FVA
\$
94,732.58
Annuity Example
• You have analyzed your retirement
plan and have concluded that you
need 3,850,000 at age 62. You are
currently 25. If you can invest at
12% on average, how much must
you invest monthly to achieve the
financial goal?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
\$3,850,000.00
12.00%
37.00
Compounding Freq. (m) (P/Y)
Solve for PV
-\$46,427.02
Solve for FV
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
12
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
Effective Interest Rate
12.68%
PVA
PMT for PVA
\$
(469.94)
Interest for PVA (per period)
#NUM!
FVA
PMT for FVA
\$
(469.94)
Annuity example
• Let’s assume that at age 62 you
have saved the amount in the
above slide. You think you will live
to be 88 years old. During
retirement, you plan to earn 8% on
you withdraw every month for the
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Solve for PV
-\$3,850,000.00
8.00%
26.00
Compounding Freq. (m) (P/Y)
Solve for FV
\$30,605,216.75
FV (Continuous Compounding)
\$30,817,205.32
Solve for Interest Rate
12
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
Effective Interest Rate
PVA
PMT for PVA
8.30%
\$
29,360.03
With respect to money what do the
• College student 19 years old.
• Young married person 31 years old with
a child age 5.
• A stable working married adult age 50,
children are out of the house and
college.
• A senior citizen age 64 recently retired.
Annuity Example
• You win a prestigious sweepstake award. The
company offers you \$50,000 per year for the next
30 years or a lump sum. Answer the following
questions.
• If the company can invest at 7%, how much would
they need to invest to pay the promised cash flow
stream? This is the amount of the lump sum offer?
• If you could invest at 9%, how much could you
withdraw for the 30 years, if you invest the lump
sum?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Solve for PV
7.00%
30.00
Compounding Freq. (m) (P/Y)
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
y
\$50,000.00
Effective Interest Rate
7.00%
PVA
\$ (620,452.06)
PMT for PVA
Interest for PVA (per period)
#NUM!
FVA
Solve for PV
-\$620,452.00
9.00%
30.00
Compounding Freq. (m) (P/Y)
Solve for FV
\$8,231,957.64
FV (Continuous Compounding)
\$9,232,159.31
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
Effective Interest Rate
PVA
PMT for PVA
9.00%
\$
60,392.53
Annuity Example
• At retirement you want to receive
\$60,000 per year for 25 years and can
earn 13%. How much must you invest to
achieve this goal?
• If you are concerned about the 3.75%
inflation rate, how much must you
invest?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
13.00%
25.00
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
y
\$60,000.00
Effective Interest Rate
PVA
Solve for PV
13.00%
25.00
13.00%
\$ (439,799.10)
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
\$60,000.00
3.75%
Effective Interest Rate
13.00%
PVA
\$ (439,799.10)
PMT for PVA
Interest for PVA (per period)
#NUM!
FVA
PMT for FVA
Interest for FVA
PV of Perpetuity
\$ (461,538.46)
PV of Growing Annuity \$ (571,956.47)
Annuity
• If you wanted to receive \$10,000
per year for ever, how much
would you need to invest at
12%?
• What kind of cash flow stream is
this?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
12.00%
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
\$0.00
y
\$10,000.00
Effective Interest Rate
#DIV/0!
12.00%
PVA
\$
PMT for PVA
Interest for PVA (per period)
#NUM!
FVA
PMT for FVA
Interest for FVA
PV of Perpetuity
\$ (83,333.33)
Change the interest rate to 10%. How much must you invest?
How does the perpetuity work?
Mixed Cash Flows
• A company is planning a project that will
provide the following cash flow stream. If
they can earn 14% on average, what is the
value of this project?
1
2
3
4
5
6
7
80,000
70,000
70,000
50,000
50,000
60,000
75,000
• If the company could reinvest the above
cash flow stream at 12%, what would they
have at the end of the 7th year?
Pds
Cash Flow
0
1
2
3
4
5
6
7
8
9
10
11
Pds
\$80,000
\$70,000
\$70,000
\$50,000
\$50,000
\$60,000
\$75,000
Discount Rate
14.00%
Number of Periods
PV of Future Cash Flows
Net Present Value
IRR
FV of Cash Flows
7
\$284,166.61
\$284,167
#NUM!
\$711,061.24
Cash Flow
0
1
2
3
4
5
6
7
8
9
10
11
\$80,000
\$70,000
\$70,000
\$50,000
\$50,000
\$60,000
\$75,000
Discount Rate
12.00%
Number of Periods
PV of Future Cash Flows
Net Present Value
IRR
FV of Cash Flows
7
\$301,528.07
\$301,528
#NUM!
\$666,582.49
Compounding frequency
• How do you change the
compounding frequency in a
time value problem?
• What is the Effective interest
rate for a 12%, monthly
compounded investment?
Quarterly?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Solve for PV
-\$1,000.00
12.00%
1.00
Compounding Freq. (m) (P/Y)
Solve for FV
\$1,120.00
FV (Continuous Compounding)
\$1,127.50
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
y
Effective Interest Rate
12.00%
Solve for PV
-\$1,000.00
12.00%
1.00
Compounding Freq. (m) (P/Y)
Solve for FV
\$1,125.51
FV (Continuous Compounding)
\$1,127.50
Solve for Interest Rate
4
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
y
Effective Interest Rate
12.55%
Solve for PV
-\$1,000.00
12.00%
1.00
Compounding Freq. (m) (P/Y)
Solve for FV
\$1,127.47
FV (Continuous Compounding)
\$1,127.50
Solve for Interest Rate
365
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
y
Effective Interest Rate
12.75%
Example
• A newly married couple is considering buying a new
home. The house of their dreams costs \$325,000.
They have 10% to put down on the home and can
borrow at 3.95%.
• How much must they borrow?
• What are the monthly payments on a 30 year
mortgage?
• What is the total cost of the loan for the 360
months?
• Loan Analysis worksheet
Loan Amount
\$292,500.00
Pmt per Period
\$1,388.02
Loan Maturity (yrs)
30
Total AMT Paid
\$499,687.71
PMT per Year (P/Y) m
12
Total Financing Costs
\$207,187.71
Annual Interest Rate
3.95%
What is a loan amortization table?
• How much interest is paid out of the first month’s
payment?
• When the couple pays their 180th payment, what is the
balance?
• If they paid an extra 50 payment per month, how much
interest would they save?
• If they paid biweekly payments, how much would they
save? (divide monthly payments by 12)
Period
PMT
0
1
2
3
180
181
360
Interest PMT
\$1,388.02
\$1,388.02
\$1,388.02
\$1,388.02
\$1,388.02
\$1,388.02
Loan Amount
\$962.81
\$961.41
\$960.01
\$622.30
\$619.78
\$4.55
Principal
Reduction
\$425.21
\$426.61
\$428.01
\$765.72
\$768.24
\$1,383.47
Remaining
Balance
\$292,500.00
\$292,074.79
\$291,648.18
\$291,220.17
\$188,286.76
\$187,518.52
\$292,500.00
Pmt per Period
\$1,388.02
Loan Maturity (yrs)
30
Total AMT Paid
\$499,687.71
PMT per Year (P/Y) m
12
Total Financing Costs
\$207,187.71
Impact of Accelerated PMTS
Years of Loan
Total AMT Paid
Interest Saved
28.08
\$484,525.24
\$15,162.47
Annual Interest Rate
Extra Periodic PMT
Biweekly impact
=PMT/12
Loan Amount
3.95%
\$50.00
\$292,500.00
Pmt per Period
\$1,388.02
Loan Maturity (yrs)
30
Total AMT Paid
\$499,687.71
PMT per Year (P/Y) m
12
Total Financing Costs
\$207,187.71
Impact of Accelerated PMTS
Years of Loan
Total AMT Paid
Interest Saved
25.93
\$467,856.90
\$31,830.81
Annual Interest Rate
Extra Periodic PMT
Biweekly impact
=PMT/12
3.95%
\$115.67
Example of returns
• When I was born my Grandfather purchased
a stock for \$25. When I was 25 the stock was
worth \$75. What did I earn on the
investment?
• What would I earn at age 65 if the stock was
then worth \$250?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
\$75.00
-\$25.00
25.00
Solve for PV
Solve for FV
FV (Continuous Compounding) \$25.00
Solve for Interest Rate
4.49%
1
\$250.00
-\$25.00
65.00
1
Solve for PV
Solve for FV
FV (Continuous Compounding) \$25.00
Solve for Interest Rate
3.61%
The Rule of 72.
What is this rule?
0
6
12
18
24
30
3%
6%
12%
2000
2000
2000
4000
4000
8000
8000
16000
32000
16000
64000
128000
4000
36
42
48
256000
8000
32000
512000
Time
feature=player_detailpage&v=_zp
GZfFbW4M
Retirement example
•
You currently earn 50,000 per year and have been able to save \$15,000 in a
retirement account. You will retire in 35 years at age 60 and inflation is 4%. What
will your income need to be in year 1 of retirement to maintain your current
lifestyle?
PV = 50K FV = ? N=35 I/Y=4% m=1 PMT = 0
•
If you live to 90, how much do you need in your pension fund at age 60 with 8%
return.
PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304
•
If you wanted your retirement income to keep up with an expected inflation rate
of 4.5%, how much would you need?
PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304
•
Growth of annuity = 4.5%
How much must you invest each month in your retirement plans to get your
desired growing retirement income if you can earn a 12% return?
PV = -15000 FV = 3539071 N=35 I/Y=12% m=12
PMT for FVA = ?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
-\$50,000.00
4.00%
35.00
1
Solve for FV
\$197,304.45
FV (Continuous Compounding)
\$202,760.00
Solve for Interest Rate
Retirement example
•
You currently earn 50,000 per year and have been able to save \$15,000 in a
retirement account. You will retire in 35 years at age 60 and inflation is 4%. What
will your income need to be in year 1 of retirement to maintain your current
lifestyle?
PV = 50K FV = ? N=35 I/Y=4% m=1 PMT = 0
•
If you live to 90, how much do you need in your pension fund at age 60 with 8%
return.
PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304
•
If you wanted your retirement income to keep up with an expected inflation rate
of 4.5%, how much would you need?
PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304
•
Growth of annuity = 4.5%
How much must you invest each month in your retirement plans to get your
desired growing retirement income if you can earn a 12% return?
PV = -15000 FV = 3539071 N=35 I/Y=12% m=12
PMT for FVA = ?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
8.00%
30.00
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
\$197,304.00
Effective Interest Rate
PVA
8.00%
\$ (2,221,205.68)
Retirement example
•
You currently earn 50,000 per year and have been able to save \$15,000 in a
retirement account. You will retire in 35 years at age 60 and inflation is 4%. What
will your income need to be in year 1 of retirement to maintain your current
lifestyle?
PV = 50K FV = ? N=35 I/Y=4% m=1 PMT = 0
•
If you live to 90, how much do you need in your pension fund at age 60 with 8%
return.
PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304
•
If you wanted your retirement income to keep up with an expected inflation rate
of 4.5%, how much would you need?
PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304
•
Growth of annuity = 4.5%
How much must you invest each month in your retirement plans to get your
desired growing retirement income if you can earn a 12% return?
PV = -15000 FV = 3539071 N=35 I/Y=12% m=12
PMT for FVA = ?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
Compounding Freq. (m) (P/Y)
Solve for PV
8.00%
30.00
\$0.00
Solve for FV
\$0.00
FV (Continuous Compounding) \$0.00
Solve for Interest Rate
1
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
\$197,304.00
4.50%
Effective Interest Rate
8.00%
PVA
\$ (2,221,205.68)
PMT for PVA
Interest for PVA (per period)
#NUM!
FVA
PMT for FVA
Interest for FVA
PV of Perpetuity
\$ (2,466,300.00)
PV of Growing Annuity \$ (3,539,071.58)
Retirement example
•
You currently earn 50,000 per year and have been able to save \$15,000 in a
retirement account. You will retire in 35 years at age 60 and inflation is 4%. What
will your income need to be in year 1 of retirement to maintain your current
lifestyle?
PV = 50K FV = ? N=35 I/Y=4% m=1 PMT = 0
•
If you live to 90, how much do you need in your pension fund at age 60 with 8%
return.
PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304
•
If you wanted your retirement income to keep up with an expected inflation rate
of 4.5%, how much would you need?
PVA = ? FV = 0 N=30 I/Y=8% m=1 PMT = 197304
•
Growth of annuity = 4.5%
How much must you invest each month in your retirement plans to get your
desired growing retirement income if you can earn a 12% return?
PV = -15000 FV = 3539071 N=35 I/Y=12% m=12
PMT for FVA = ?
Future Value (FV)
Present Value (PV)
Annual Interest Rate (I/Y)
Time in Years (N)
\$3,629,071.00
-\$15,000.00
12.00%
35.00
Compounding Freq. (m) (P/Y)
Solve for PV
Solve for FV
FV (Continuous Compounding)
\$1,000,294.97
Solve for Interest Rate
12
Solve for Time
Is this an Ordinary Annuity (y/n)
Payment (PMT) (A)
Growth of an Annuity
Growth of a Perpetuity
y
Effective Interest Rate
PVA
PMT for PVA
\$
Interest for PVA (per period)
\$
FVA
PMT for FVA
\$
12.68%
(411.98)
0.01
(411.98)
Retirement example
• You currently earn \$XXXX per year and have been able to save \$XXX in a
retirement account. You will retire in X years at age 65 and inflation is
3.1%. What will your income need to be in year 1 of retirement to