Time Value of Money
Objectives
 Calculate the future value of a dollar
amount that you save today
 Calculate the present value of a
dollar amount that will be received
in the future
 Calculate the future value of an annuity
 Calculate the present value of
an annuity
TVM importance
 Opportunity cost causes money to change value as it
moves over time.
 Can be applied to a single dollar amount—also
called a lump sum
 Can also be applied to an annuity
 Annuity: a series of equal cash flow payments that
occur at the end of each period
 An example would be a monthly deposit of $50
into your savings account
Time Lines – Picture Time
Future Value of a Single Dollar Amount
 Compounding – process by which money
accumulates interest.
 Future Value
 Amount of money deposited
 Interest rate per period
 Number of periods the money will grow
Picture Time
 Future Value of a lump sum
Nominal vs. Periodic
 Nominal Rate – Stated Annual Rate
 Periodic Rate – rate per period
Example: Nominal = 10%
Monthly Periodic Rate = ?
Future Value Interest Factor
 FVIF - what is this?
 How to calculate
Table Value
Formula
Business Calculator
Example:
 Suppose you want to know how much money you
will have in five years if you invest $5,000 now
and earn an annual return of 9 percent
 The present value of money (PV) is the amount
invested, or $5,000
 Find the interest rate of 9 percent and a time
period of five years on the table
Answer:
 Suppose you want to know how much money you
will have in five years if you invest $5,000 now
and earn an annual return of 9 percent
$5,000 x ___________ = _______________
Or by formula:
Another Problem 1
 If you deposit $1,000 at a nominal rate of 10%, compounded
annually for one year, how much will you have at the end of
the year? If it is compounded monthly?
 Table Factor:
 Formula:
 Answers: $1,100.00; $1,104.71
Another Problem 2
 If you deposit $1,000 at a nominal rate of 10%, compounded
annually for three years, how much will you have at the end
of the year? If it is compounded monthly?
Answers: $1,331.00; $1,348.18
Present Value
 Discounting – the process of obtaining
present values.
 Need:
$ amount to be received in the future
Periodic rate to be earned on the money
Number of periods invested
Picture Time
 Present Value of a lump sum
Present Value Interest Factor
 PVIF - what is this?
 How to calculate
Table Value
Formula
Business Calculator
Example Present Value
You would like to accumulate $50,000
in five years by making a single investment
today.You believe you can achieve a return
from your investment of 8 percent annually.
What is the dollar amount that you need to
invest today to achieve your goal?
continued
 You would like to accumulate $50,000 in five years by
making a single investment today.You believe you can achieve
a return from your investment of 8 percent annually.
What is the dollar amount that you need to invest today to
achieve your goal?
 50,000 x _________ = ________________
 Formula:
 Answer: $34,029.15
Example 2
 You would like to accumulate $50,000 in five years by
making a single investment today.You believe you can achieve
a return from your investment of 8 percent annually, but
compounded monthly. What is the dollar amount that
you need to invest today to achieve your goal?
 Answer: 33,560.52
Example 3
How much would you pay for an investment that is
going to earn you 6% over 3 years (annually) and at
the end of the 3 years you will receive $17,000?
 Present value of 17,000 at 6% for 3 years =
 Table:
 Formula:
 Answer: $14,273.53
Annuities – Picture Time
 Stream of equal amounts paid or received
 Ordinary Annuity –
 Annuity Due -
 Perpetuity -
Future Value of an Annuity (FVA)
 How to read the table
 I = periodic interest rate
 N = number of payments (payment frequency MUST match
the interest rate for that period)
 Answers: How much money will I have if I deposit $X each
period for N periods at I rate per period? (end of period
payments)
Example 1 - FVA
Suppose that you have won the lottery and will
receive $150,000 at the end of every year for the
next 20 years. As soon as you receive the
payments, you will invest them at your bank at an
interest rate of 7 percent annually. How much
will be in your account at the end of 20 years,
assuming you do not make any withdrawals?
Continued
 Suppose that you have won the lottery and will receive $150,000 at the end of
every year for the next 20 years. As soon as you receive the payments, you will
invest them at your bank at an interest rate of 7 percent annually. How much
will be in your account at the end of 20 years, assuming you do not make any
withdrawals?
 Picture:
 Table: $150,000 x __________ = _____________
 Calculator:
Answer: $6,149,323.85
Example 2
 You need to have $1,000,000 in 30 years when you retire.
How much will you have to deposit at the end of each year if
the deposit earns 5% to have the $1,000,000?
 FVA = PMT x FVAF;
PMT = FV/FVAF
 1,000,000 / 66.4388 = 15,051.45 (table)
 Calculator: $15,051.44
Present Value of an Annuity (PVA)
 Discounting the individual cash flows of the annuity and
adding them up.
 Factor x annuity = PVA
 I = periodic interest rate
 N = number of periods in the annuity
 Rate per period must match the annuity frequency
Picture Time- PVA Time Line
Example 1
 Suppose you have just won the lottery. As a result of your
luck, you will receive $82,000 at the end of every year for
the next 25 years. Now, a financial firm offers you a lump
sum of $700,000 in return for these payments. If you can
invest your money at an annual interest rate of 9 percent,
should you accept the offer?
Continued
Suppose you have just won the lottery. As a result of your luck, you will receive
$82,000 at the end of every year for the next 25 years. Now, a financial firm offers
you a lump sum of $700,000 in return for these payments. If you can invest your
money at an annual interest rate of 9 percent, should you accept
the offer?
Present value of the Annuity = ?
82,000 x ________ = ______________
Should you accept or reject the offer?
Answer: $805,486
Example 2
You want to buy a house. Based on your cash flows you figure
that you can afford $750 each month on the mortgage
payment (not including taxes and insurance escrow).
Mortgage rates are 12% nominal (1% per month). How
much house can you afford to buy if you want to pay for it
over 50 months. (50/12 = 4 years + 2 months).
PVA = PVAF x PMT
PVA = 39.1961 x $750 = 29,397.07
Example 3
You want to buy a car over 5 years. The cost is 25,000. What
will be your annual payment be if the interest rate is 10%?
Annual
PMT = PVA/PVAF = 25,000/3.7908 = 6,594.91
Monthly
I = 10/12; 5 x 12 periods = 60; PVIF = 47.06537 (by
calculator)
PMT = $531.18
Amortization – the annual car loan
Year
1
2
3
4
5
Beg. Bal.
25,000.00
20,905.06
16,400.63
11,445.76
5,995.40
Payment
6,594.94
6,594.94
6,594.94
6,594.94
6,594.94
Interest
2,500.00
2,090.51
1,640.06
1,144.58
599.54
Prin. Pay
4,094.94
4,504.43
4,954.87
5,450.36
5,995.40
End Bal.
20,905.06
16,400.63
11,445.76
5,995.40
0.00
pmt = 6,594.937
n=5
I = .10
The ending balance is also the present value of the remaining
payments.
Perpetuity – Example
An investment promises you a payment of $724.00 at the end
of each year forever. How much would you pay for this
investment if your opportunity cost is 6.4%?
PV = PMT / Rate = 724.00/0.064 = $11,312.50
So: $11,312.50 x 0.064 = 724.00
Annuity Due - Example
You need to save $50,000 to help put your child through
college in 10 years. You want to make 10 payments starting
now to have the 50,000 in 10 full years. How much do you
have to deposit at the beginning of each year if you can earn
5% on the money?
Table answer: PMT = FV / FVAF (1+i)
50,000 / [FVAF x (1+i)] = 50,000 /12.5779 (1.05)
= 50,000 / 13.206795 = 3,785.93
Effective Rate – Simple example
 APR = Annual Percentage Rate: Required in Federal
Legislation for disclosure. (Rate/per) x periods adjusted for
costs.
 Effective Rate = the actual rate considering compounding.
Suppose you borrow 12,000 to buy a car at a nominal rate of
8.5% paid annually vs. another bank 8.3% paid monthly.
Which loan has the lowest effective rate? No loan costs.
Effective Rate Annual = 8.5%, APR = 8.5%
Effective rate Monthly = ([1+ .083/12]^12) – 1
= 1.08623 – 1 = 8.623%
APR = 8.3%
Effective Rate – Excel Spreadsheet
Motrgage Example
Change this:
Nominal Rate
Years
Per per year
Amt. Borrowed
Costs
0.07
30
12
100,000
2000
Nominal Rate
Periodic Rate
Yrs x per/yr
Payment
Used Funds
Payment
Effective Periodic
APR
Effective Annual
0.0700000
0.0058333
360
$665.30
98,000.00
$665.30
0.006001131
0.072013573
0.074438664