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Safe
Money Math
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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. |
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 |
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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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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Annual
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Effective
(APY) Rate |
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Rate
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Quarterly |
Monthly |
Daily |
| 3.00% |
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3.03
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3.04 |
3.05 |
| 3.25% |
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3.29 |
3.30 |
3.30 |
| 3.50% |
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3.55 |
3.56 |
3.56 |
| 3.75% |
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3.80 |
3.81 |
3.82 |
| 4.00% |
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4.06 |
4.07 |
4.08 |
| 4.25% |
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4.32 |
4.33 |
4.34 |
| 4.50% |
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4.58 |
4.59 |
4.60 |
| 4.75% |
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4.83 |
4.85 |
4.86 |
| 5.00% |
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5.09 |
5.12 |
5.13 |
| 5.25% |
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5.35 |
5.38 |
5.39 |
| 5.50% |
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5.61 |
5.64 |
5.65 |
| 5.75% |
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5.88 |
5.90 |
5.92 |
| 6.00% |
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6.14 |
6.17 |
6.18 |
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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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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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Safe Money Places®