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Chapter 3-- Measuring Wealth:
Time Value of Money
Why
must future dollars be put on a common
basis before adding?
Cash is a limited and controlled resource.
Those controlling the resource can charge
for its use.
The longer the period of use the higher the
interest (or rental fee) for the use of the cash.
Therefore, you cannot add a 1 year dollar
with a two year dollar.
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Future Dollar Equivalent (Future
Value) of a Present Amount
These
can be solved using formulas, tables, a
financial calculator or a computer spreadsheet
package
solution FV = PV (1+i)n
Example -- How much will $1000 grow
to at 12% in 15 years?
Formula
Enter
1.12, yx, 15, times 1000 = $5473.56
2
Present Dollar Equivalent (Present
Value) of a Future Amount
These
can be solved using formulas, tables, a
financial calculator or a computer spreadsheet
package
Formula
solution PV = FV/ (1+i)n
Example
-- how much do you need today to
have $1,000,000 in 40 years if your money is
earning 12%?
Enter 1.12, yx, 40, =, 1/x, times 1000000 =
$10,747
3
Finding the Rate Between
Two Single Amounts
These
can be solved using formulas, tables, a financial
calculator or a computer spreadsheet package
Formula
solution -- i = (FV/PV)1/n -1
– you purchased your house for $76,900 in
1994. Your neighbor’s house of similar value sold for
$115,000 in 2004 ( 10 years later). What rate of return
are you earning on your house?
Example
Enter 115000 / 76900, yx, .1, –, 1, =, .0411 or 4.11%
4
Finding the Number of Periods
Needed Between Two Amounts
These
can be solved using formulas, tables, a financial
calculator or a computer spreadsheet package
Formula
solution -- n = LN(FV/PV)/LN(1+i)
– you inherit $120,000 from your great aunt
and invest it to earn 8% interest. How long will it take
for this to grow to $1,000,000?
Example
Enter – (1000000 / 120000) ,=, ln -- this gives you 2.1203
Enter – (1.08), ln -- this gives you .0770
Divide the two results to get 27.55 years
5
Different Types of Annuities.
Ordinary
annuities -- dollars are received or paid at the
end of the period and grow until the end of the period.
All annuity formulas to be discussed will work for
ordinary annuities with no adjustments.
Annuities due -- dollars are received or paid at the
beginning of the period and grow until the end of the
period.
All annuity formulas to be discussed will need
adjustment (for the extra year’s worth of interest).
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Future Value of an Ordinary Annuity
and an Annuity Due
Example
-- How much will you have at the end of 35
years if can earn 12% on your money and place
$10,000 per year in you 401k account at the beginning
of the year? (at the end of the year?)
Formula solution ordinary annuity –
FV = [((1+i)r –1) / r ] payment
Enter
– (1.12,yx, 35, =, –1, = ) / .12 times 10000
The answer is $4,316,635
solve for an annuity due just remove the –1 from the
formula above – the answer is then $4,834,631
To
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Present Value of an Ordinary
Annuity and an Annuity Due
Example
-- How much is a trust fund worth today that
promises to pay you $10,000 at the end (or beginning) of
each year for 35 years if can earn 12% on your money?
Formula solution ordinary annuity –
FV = [[1-(1/(1+i)r)] / r ] payment
Enter – 1.12,yx, 35, =, 1/x, –,1, +/-, = ) / .12 times 10000
this will give you the answer of $81,755
To
solve for an annuity due, change the 35 to 34 in the
formula above then add an additional 10000 payment to the
answer of $81,566 to get $91,566
8
Present Value of an Uneven Stream
of Year-end Cash Flows
– You can invest in an athletic endorsement
that will increase net cash flows to your firm by:
Example
$800,000 at the end of year 1
$600,000 at the end of year 2
$400,000 at the end of year 3
After that, you do not expect any additional benefit from her
endorsement. What is the present value of this endorsement if
the firm has a cost of funds of 8 percent?
Formula solution discount each future cash flow to present by
dividing by (1+i)n and then add up these results
Answer
-- $1,572,679
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Rate of Return on an Uneven Stream
of Year-end Cash Flows
– you can invest in an athletic endorsement
that will increase net cash flows to your firm by:
Example
$800,000 at the end of year 1
$600,000 at the end of year 2
$400,000 at the end of year 3
After that, you do not expect any additional benefit from her
endorsement. If this endorsement cost the firm $1,000,000
today, what is the rate of return of this endorsement?
Calculator solution – 2nd , CLR WRK, CF 1000000, +/-, enter,
800000, enter, 600000, enter, 400000, enter, IRR, CPT
The answer 42.06% appears
10
Adjusting for Compounding More
Than Once a Year
In
the formula, you divide the interest rate by the number of
compoundings and multiple the n by the number of
compoundings to account for monthly, quarterly or semi-annual
compounding
Excel Example -- What will $5,000 dollars invested today grow
to at the end of 10 years if your account promises a 10% APR
compounded monthly? You Enter -- for the monthly answer -=FV(.10/12,10*12,0,-5000,0)
You
Enter -- .10/12, =, +1, =, yx ,120 times 5000 = $13,535
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Adjusting for Compounding More
Than Once a Year
To
adjust an APR or nominal rate to an effective rate use the
following formula:
Effective rate = [(1+ nominal rate / # of comp.)n times # of comp]-1
12
Adjusting for When Cash Flows Are
Received Daily
A
close approximation for level daily cash
flows is the use of mid-year cash flows.
When using a computer package with both mid
year and year-end cash flows it is easiest to use
the PV function to discount each period’s cash
flow back to present individually.
When looking for the internal rate of return of
daily cash flows the problem must be worked
as a goal seek (solving for the interest rate).
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Valuing Perpetuities
Value perpetual no-grow cash flows
Formula
Present
value = cash flow / discount rate
Value perpetual growing cash flows
Formula
Present
value =
cash flow /(discount rate - growth rate)
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