In depth: Time Value of Money

1
In depth Time Value of Money
17
Interest makes a dollar to be received tomorrow
less valuable than a dollar received today
2
Objective 17.1 Explain the effect of interest on
payment streams

• A payment stream can be
• a single amount,
• a series of payments
• any combination of both

O17.1
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Interest
Compensation or rent paid to the owner of cash
for its use by others over time
O17.1
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Future value
The value in the future of a payment stream with
the effect of interest included and expressed as
a single value
O17.1
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Future value (FV)
Computing future value assumes some rate of
interest
What will this dollar be worth one year from
today?
At 8 per year FV 1 x 1.08 FV 1.08
Interest
Principal
O17.1
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Future value (FV)
Any amount times 1 interest rate equals the FV
after one time period
EXAMPLES
500 x 1.08 540 or 12 x 1.08
12.96 or 15,560 x 1.08 16,805
O17.1
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Future value (FV)
Repeat the process for each interest period
What will this dollar be worth two years from
today?
FV 1 x 1.08
x 1.08
1.17
O17.1
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Future Value Single Amount
PresentValue Single Amount
Interest rate is 8 per year
Interest earned
Principal
Year 5
Year 4
Year 3
Year 2
Year 1
Year 0
X 1.08
X 1.08
X 1.08
X 1.08
X 1.08
O17.1
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Present value
The value today of a payment stream with the
effect of interest removed and expressed as a
single value
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Rule 1
A dollar received today is always more valuable
than a dollar received in the future and A
dollar received in the future is always less
valuable than a dollar received today
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Present value (PV)
If this dollar is received in one year, what is
it worth today?
To compute PV DIVIDE by 1 the interest rate
At 8 per year PV 1 / 1.08 PV .93
O17.1
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Present value (PV)
Any amount DIVIDED by 1 interest rate equals
the PV for one time period
EXAMPLES
500 / 1.08 463 or 12 / 1.08
11 or 15,560/ 1.08 14,407
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Present value
Present Value
FutureValue
4
1
2
3
0
DISCOUNTING

SINGLE AMOUNT
Reducing payment streams to their present value
is called discounting
O17.1
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Rule 2
The more time periods, the higher the future
value and the more time periods, the lower the
present value
EXAMPLE Future Value
Value after 1 year 50 x 1.08 54 Value after
2 years 50 x 1.08 x 1.08 58
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Rule 2
The more time periods, the higher the future
value and the more time periods, the lower the
present value
EXAMPLE Present value
Value today if received in 1 year 50 / 1.08
46 Value today if received in 2 years 50 /
1.08 /1.08 43
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Rule 3
The value in the future is always higher if the
interest rate is higher and the value today is
always lower if the interest rate is higher
EXAMPLE Present value
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Present Value
Future Value
Interest rate is 8 per year
Fifth Year Principal
Discounted Principal
Year 5
Year 4
Year 3
Year 2
Year 1
Year 0
1.08
1.08
1.08
1.08
1.08
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Present Value
Future Value
Interest rate is 25 per year
Fifth Year Principal
Discounted Principal
Year 5
Year 4
Year 3
Year 2
Year 1
Year 0
1.25
Much smaller than at 8
1.25
1.25
1.25
1.25
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Compounding
During a time period (year), the computation of
interest and the addition to or subtraction from
principle.
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Rule 4
The more compounding periods the higher the
future value and The more compounding periods
the lower the present value.
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Compounding
EXAMPLE Future value
A 5,000 savings deposit pays 4 per year
compounded every six months. 4/2 2 per six
months 5,000 x 1.02 x 1.02 5,202.00 A
5,000 savings deposit pays 4 per year
compounded every quarter. 4/4 1 per quarter
5,000 x 1.01 x 1.01 x 1.01 x 1.01 5,203.02
More compounding periods higher future value
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Objective 17.2 Compute the present and future
value of single amounts
Financial calculators also can be used
Time value of money tables provide a short cut to
the computation of present and future values.
O17.2
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Future value tables
The future value of 1 table below gives the
future value of 1 at various interest rates and
time periods.
At 3, the value of 1 in 4 years is 1.1255
Payment Stream is a Single Amount
O17.2
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Compute future value of single amount
A 10,000 savings deposit pays 6 per year ( no
compounding) What will the value of the deposit
be in 5 years?
EXAMPLE
From the table at 6, 1 would be worth 1.3382,
therefore, 10,000 x 1.3382 13,382
O17.2
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Present value tables
The present value of 1 table below gives the
present value of 1 at various interest rates and
time periods.
At 2, the value today of 1 received in 5 years
is .9057
Payment Stream is a Single Amount
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Compute present value of single amount
At 5, what would a promise to receive 10,000 in
four years ( no compounding) be worth today?
EXAMPLE
From the table at 5, 1 would be worth .8227,
therefore, 10,000 x .8227 8,227
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Objective 17.3 Compute the present and future
value of annuities
An annuity is series of equal payments, paid or
received, each time period.
A common example of an annuity is an installment
loan payment such as an auto loan with identical
payments every month.
O17.3
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Annuity vs Single amount
O17.3
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Present Value (annuity)
Future Value (annuity)
Interest rate is 8 per year
Annuity payment
Discounted Principal
Year 5
Year 4
Year 3
Year 2
Year 1
1.08
Year 0
1.08
1.08
1.08
1.08
O17.3
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Future value tables
At 5, the value of a 1 annuity for 5 years is
5.5256
The future value of an annuity of 1 table below
gives future values at various interest rates and
time periods.
Payment Stream is an annuity
O17.3
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Compute future value of an annuity
Tony will deposit 1,000 per year in a savings
account that pays 4 annually. What will the
value of the deposit be in 5 years?
EXAMPLE
From the table at 4, a 1 annuity would be worth
5.4163 in 5 years, therefore, 1,000 x 5.4163
5,416.30
O17.3
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Present value tables
At 4, the value today of 1 annuity for 4 years
is 3.6299
The present value of an annuity of 1 table below
gives present values at various interest rates
and time periods.
Payment Stream is an annuity
O17.3
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Compute present value of an annuity
A wealthy friend agrees to offer a loan to you at
6 for 5 years. You can promise to repay 1,000
per year. How much can you borrow?
EXAMPLE
From the table at 6, a 1 annuity would be worth
4.2124 today, therefore, 1,000 x 4.2124
4,212.40
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Reference Time value of money tables
For quick reference, the following slides are
time value of money tables
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End Unit 17