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Safe Money
Math
A brief primer on
safe math concepts
The math formulas of
saving may seem intimidating, but
nothing that follows is more difficult
than some of the basic algebra concepts
you learned in 9th grade. Trust me,
these concepts will be easier to
understand today (you had a 14 year old
brain then and you are smarter now).
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How Fast Money Doubles - Rule Of 72
A quick and easy way
to see how many years it takes money to
double is divide the interest rate
received into 72.
A rate of 6% divided
into 72 produces an answer of 12 years,
8% takes 9 years to double and so.
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72 / 8% = 9
years
72/ 6% = 12 years
72/ 4% = 18 years
Or if you
have a timeframe you can determine what
return is need to double your money
72/ 8 years
= 9% rate needed to double money
72/ 6 years = 12% rate needed to double
money
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Annual Percentage Yield (APY) versus
Annual Interest Rate
APY reflects the true
annual compounded yield earned on an
interest earning account. As examples, a
4.0% interest rate would also have an
APY of 4.0% if interest was compounded
annually, but the APY would be 4.06% if
interest were compounded quarterly. Why?
Compounding divides
up the interest rate into smaller bits
representing the way you are
compounding. If you were compounding
quarterly the annual interest rate is
divided into quarters or fourths, so the
annual rate of 4.0% is divided by 4
meaning you earn 1% for that quarter.
At the end of the
quarter you now have 101% of what you
started with representing the 100%
principal and the 1% interest earned.
When you begin the second quarter you
will again earn 1%, but you earn on both
the original principal and the interest
already earned. 101% times a 1% rate
means you earn 1.01% for the second
quarter. This is added to the 101%
giving you 102.01%. This process of
earning interest on both principal and
interest repeats in the third and fourth
quarter, so at the end of the year you
have 104.06% and that .06% reflects
extra interest earned
The formula to
convert a nominal or annual rate into an
annual percentage rate is
ER = [1 + (i/n)]n
-1
Say that you are
comparing a fixed rate annuity to a
certificate of deposit. The fixed rate
annuity has an annual rate of 6.1%
compounded annually, and the certificate
of deposit has a nominal rate of 6%,
compounded monthly. Because the fixed
annuity has only one compounding period
the fixed annuity’s nominal and
effective rate are the same — 6.1%.
However, because the CD compounds
monthly there are twelve compounding
periods. The CD’s effective rate is
6.17% computed as follows
[1+(.06/12) ]12
– 1 = 1.00512 – 1 = .0617.
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Compound vs. Simple
Interest
Suppose you had $1 and someone gave you a
choice between earning 10% calculated as simple
interest or 9% calculated as compound interest.
Which one do you choose? It depends on how long
you are going to leave your money at work.
At the end of one-year the simple interest
method would add a dime to your initial dollar
and you’d have $1.10. The compound interest
method would add nine cents and you’d
wind up with $1.09. If you’re only looking at
one period there’s no difference between
simple and compound interest.
If you went out two-years,
the simple method would add another dime to your
$1.10
giving you a balance of $1.20. Simple interest
means that interest is only earned
on the original principal.
Compound interest works a little differently.
Compound interest multiples the previous balance
- in this case $1.09, by the interest rate (9%),
and adds the numbers together to determine a new
value.
So, $1.09 multiplied by 9% produces not 9 cents
but 9.81 cents. This is added to the previous
balance (1.09 + .0981) to produce a second year
total of $1.1881. A short hand way to figure the
results of compounding is to put a one in front
of the interest rate ($1.0 x 1.09 = $1.09).
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Annual
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Effective (APY) Rate |
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Rate
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Quarterly |
Monthly |
Daily |
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3.00% |
3.03
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3.04 |
3.05 |
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3.25% |
3.29 |
3.30 |
3.30 |
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3.50% |
3.55 |
3.56 |
3.56 |
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3.75% |
3.80 |
3.81 |
3.82 |
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4.00% |
4.06 |
4.07 |
4.08 |
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4.25% |
4.32 |
4.33 |
4.34 |
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4.50% |
4.58 |
4.59 |
4.60 |
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4.75% |
4.83 |
4.85 |
4.86 |
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5.00% |
5.09 |
5.12 |
5.13 |
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5.25% |
5.35 |
5.38 |
5.39 |
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5.50% |
5.61 |
5.64 |
5.65 |
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5.75% |
5.88 |
5.90 |
5.92 |
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6.00% |
6.14 |
6.17 |
6.18 |
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Annual
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Effective (APY) Rate
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Rate
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Quarterly
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Monthly
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Daily
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3.00%
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3.03
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3.04
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3.05
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3.25%
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3.29
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3.30
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3.30
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3.50%
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3.55
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3.56
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3.56
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3.75%
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3.80
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3.81
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3.82
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4.00%
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4.06
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4.07
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4.08
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4.25%
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4.32
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4.33
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4.34
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4.50%
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4.58
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4.59
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4.60
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4.75%
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4.83
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4.85
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4.86
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5.00%
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5.09
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5.12
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5.13
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5.25%
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5.35
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5.38
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5.39
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5.50%
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5.61
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5.64
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5.65
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5.75%
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5.88
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5.90
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5.92
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6.00%
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6.14
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6.17
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6.18
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At the end of five
years the Simple Method would have
produced a total of $1.50
($1+ 0.1+ 0.1+ 0.1+
0.1+ 0.1)
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But, the Compound
Method generated $1.54
($1 x 1.09 x 1.09 x
1.09 x 1.09 x 1.09)
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If you are only looking at a few periods there
is not a lot of difference between the effects
of simple and compound interest. However, the
benefits of compound interest become
proportionately greater with each passing
period.
Rate
of Return
In an attempt to compare the annual
return of different investments some people will
divide the total return by the number of years
involved. However, unless the investments are of
equal duration, the results may lead to the
wrong conclusion.
Say that Investment A
returned 30% in three-years, Investment B
returned 60% in six-years, and Investment C
returned 90% in nine-years. If you divide each
investment’s total return by the number of years
in the period, the answer for each one is 10%.
But, the actual annual rate of return for
Investment A is 9.13%, the rate of return for
Investment B is 8.15%, and the rate of return
for Investment C is 7.18%.
The formula to calculate the
honest rate of return for an investment without
irregular cash flows is this:
i = (FV/PV)(1/n)
- 1
If the Present Value is
$10,000, the Future Value is $15,000, and the
number of periods is 5. The equation would be
($15,000/$10,000)(1/5) - 1= 1.500.2
- 1 = 8.45%.
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