The Rule of 72.

In our last post, we used our equations to calculate yield for a specific interest rate over a specific time.  Our last problem, we looked at a $10,000 principle at 8% for 9 years, and at 9% for 8 years.  They were pretty close in amounts – $20,495 and $20,489 respectively.  But they shared one special characteristic…they both doubled the principle at those rates over that specific time. 

Today we will look at a very simple rule that helps you approximate how long it will take to double your principle at a given rate.  It is called the Rule of 72.  The Rule of 72 states that if you divide 72 by an interest rate percentage, you will have a good approximation how many years it will take to double your initial principle.  This also works for if you want to find out what interest rate you need to invest at in order to double your principle over a determined time period. 

Let’s make this equation, its rather simple.  First let’s arrange our variables:

I = Interest Rate Percentage

T= Time (in years)  

Ok, there is only one tiny thing about this equation.  Lets say we are talking about percentages – 5% for instance.  Normally for equations we make 5% = .05, but for the purposes of this equation it must remain the whole number percentage (or else we would call it the rule of .72!) 

Our equation will look like this if we want to find out how long it will take to double the principle at a given interest rate: 

T = 72 / I  

Our equation will look like this if we want to find out what rate you would need to double the principle over a given time: 

I = 72 / T 

Let’s take this equation for a test drive.

How long will it take to double the principle at 3%, 6%, and 12%?

What interest rate would you need to double your investment in 5 years, 10 years, and 20 years? 

Ok, we will start with 3%. 

T = 72 / 3    Divide.

T = 24 years.   

Next, 6%.

T = 72 / 6    Divide.

T = 12 years. 

Finally, 12%

T = 72 / 12    Divide.

T = 6 years.  

For the next set of problems, we need to use the second form of the equation. 

Ok, we will start with 5 years.

I = 72 / 5    Divide.

I = 14.4%. 

Next, 10 years.

I = 72 / 10    Divide.

I = 7.2%.

Finally,  20 years.

I = 72 / 20    Divide.

I = 3.6%. 

The Rule of 72 is a pretty simple way to approximate these two things.  It is important to note that for higher interest rates, the equation has to be adjusted.  Wikipedia has a good explanation why this is and how to adjust: http://en.wikipedia.org/wiki/Rule_of_72 

They also go into other conclusions that can be derived from this formula.  It’s a good read.  CD

Interest Rate vs Annual Percentage Yield – Part 2 – Applications

Yesterday we came up with a formula that helps us find the Annual Percentage Yield based on the interest rate a bank offers. Today we are going to check this formula and see if we can test how this applies to real, everyday life.

First,we will check the current interest rate for a popular online savings institution – INGDirect (www.ingdirect.com) – to see if our formula works. They show the interest rate at 3.590% and the APY at 3.650%.

Second, we can check to see how much $10,000 will grow to at a 5% interest rate over 40 years compounding each month.

Finally, we will check to see which earns more money: $10,000 at 8% for 9 years, or $10,000 at 9% for 8 years?

For the first problem, lets organize our variables:

I = 3.590% (or .0359) This is our INTEREST RATE. (Note: This comes from the information in the problem.)

P = 1 month This is our PERIOD. (Note: The interest compounds monthly, so the period has to be one month.)

N = 12 This is the NUMBER OF PERIODS. (Note: The number of periods is 12 because there are 12 months/periods in a year and we and to check the Annual Percentage Yield.)

Ok, let’s take our equation from yesterday (1+(I/P))^N and plug the variables in:

(1+(.0359/12))^12 Simplify – Division.

(1+(.002992))^12 Simplify – Addition.

(1.002992)^12 Now we will need a calculator. After plugging it in our solution is:

1.036500769 Rounding gives us 1.03650 or 103.650% of our original investment after 1 year of compounding interest at 3.59%

That makes the Annual Percentage Yield 3.650%! The formula checks out!

For the second problem, lets start again by organizing our variables:

I = 5% (or .05)

P = 1 month

N = 480 (Note: We want to find the yield for 40 years. 40 years * 12 months (or periods) = 480 months (or periods). It is just like finding the APY for 1 year, but we are finding a similar percentage for 40 years and we will multiply this percentage by the Initial Principle.

$ = 10,000 (This is our initial principle.)

So we will use the formula from the problem above again, but since we want to find the dollar amount after 40 years, we need to multiply by the initial principle after we find the yield for 40 years. It will look like this: $ * (1 + (I/P)^N)) Let’s plug in the variables:

10000 * (1 + (.050/12)) ^ 480 Simplify – Division.

10000 * (1 + (.004166666)) ^ 480 Simplify – Addition.

10000 * (1.004166666) ^ 480 Use a calculator to figure out 1.0041666 ^ 480.

10000 * 7.358417 Multiply.

$73,584.17 – You will have 73,584.17 after 40 years.

For the last problem, lets organize the variables for the first part:

I = 8% (or .08)

P = 1 month

N = 108 (9 years * 12 months = 108 months)

$ = 10,000

Lets plug these variables into our formula:

10000 * (1 + (.08/12)) ^ 108 Simplify – Division.

10000 * (1 + (.00666666)) ^ 108 Simplify – Addition.

10000 * (1.00666666)) ^ 108 Use a calculator to figure out 1.00666666 ^ 108.

10000 * 2.04953026 Multiply.

$20,495.30 for the first part.

Now, lets organize the variables for the second part:

I = 9% (or .09)

P = 1 month

N = 96 (8 years * 12 months = 96 months)

$ = 10,000

Lets plug these variables into our formula:

10000 * (1 + (.09/12)) ^ 96 Simplify – Division.

10000 * (1 + (.0075)) ^ 96 Simplify – Addition.

10000 * (1.0075)) ^ 96 Use a calculator to figure out 1.0075 ^ 96.

10000 * 2.048922128 Multiply.

$20,489.22 for the second part.

So, the two totals are relatively close. The 8% for 9 years is $6.08 higher. There are two things to note based on this last problem. Time is the biggest factor in compounding interest. The longer that money is invested, the higher the return will be. For all of you people putting off investments, this equation proves that the earlier you get your money in, the better. It also shows the importance of shopping around (or choosing your investment wisely) in order to ensure the best rate possible. For example, you can make up for lost time by choosing an investment vehicle that has a high return. But there are many, many, many more factors that are involved in managing a portfolio – risk, target dates, goals, etc. This sort of math would just help compare one single facet of investing (time vs. rate) – and thats only practical for investments like savings accounts or Certificates of Deposit, where rates fluctuate much less. I am far from an expert on investing, but I just thought I would point out these trends.

The second thing to note is that we started with $10,000 and ended with more than $20,000. Both investments more than doubled over this time at these rates. I chose 8% for 9 years, and 9% for 8 years for a reason…because tomorrow we will explore The Rule of 72, as it applies to investing money. I hope these examples help explain the formula from yesterday, and that it takes some of the mystery out of how much money the bank gives every month for keeping your principle there. CD

Interest Rate vs Annual Percentage Yield

Well, I am back! I was out of the country for a while and catching up on work for the other part, but I am hopefully back to writing three times a week or more. I figured a good place to start anew would be an interest of mine: interest – compound interest to be precise.

Today we will explore the difference and connection between an annual percentage yield a bank advertises and the interest rate that adds to your account every month. These two terms apply to your checking accounts, savings accounts, certificates of deposit, almost all investment products that banks offer.

When you sign up for a savings account, you are allowing the bank to hold your money for a period of time in exchange for an extra percentage of money over that period of time. This percentage is your interest rate. The banks usually give you an interest payment every month (they call it a period – you will see why later.) They calculate this by taking the interest rate and dividing it by 12 (because there are 12 months – or periods – in a year) and multiplying it by the amount of money in the account or the principle. This gives you your interest payment for the month. This happens quite the same next month, but the amount of money in your account is higher – because of last month’s interest payment. In addition to the interest on your initial principle, you are now earning interest on the interest you earned last month. This concept is what is known as compounding interest. Your money is now working hard to earn you more money. Over the course of a year, you earn interest on your interest (in addition to the interest on your initial principle) over and over again as the months go by. The annual percentage yield is the percentage your principle has increased over the course of one year compared to the initial principle.We are going to figure out what the annual percentage yield would be for a 10% Interest Rate.We need to organize our variables first:

Interest Rate (I) = 10.00% per year

Period (P) = 1 month

Number of Periods (N)= 12 (12 months in a year)

Principle ($) = Does not matter right now…we will just use $ as a placeholder.

Annual Percentage Yield (APY) = ???

This math may look daunting if you scan though it, but if you take it step by step, it will make sense.

OK. Let’s start by figuring out the first month. It will look something like this:

We start with the initial principle: $ At the end of the month the bank will give you interest (I) on this money, but they won’t give you 10%, because it is 10% per year. Instead they will divide 10% by 12 (because there is 12 months or periods (P) in a year.) They will then take that monthly interest percentage and multiply it by your principle. So our equation will look like this:

$ + $*(10%/12) = New1$ (New Principle after 1 month/period) Or with only variables from above, it will look like this:

$ + $(I/P) = New1$ We can factor out the principle ($)

$ * (1 + (I/P)) = New1$

Next month/period it will look like this because our $ after this month now equals $*(1 + (I/P)):

New1$ * (1 + (I/P)) = New2$ (New Principle after 2 months). But since we know what New1$ equals, let’s substitute it in and simplify:

[$ * (1 + (I/P))] * (1 + (I/P)) = New2$ Simplify.

$ * (1 + (I/P))^2 = New2$. (^2 means squared or to the power of 2)

The next month will look almost the same:

New2$ * (1 + (I/P)) = New3$ (New Principle after 3 months). Since we know what New2$ equals, let’s substitute it in and simplify:

[$ * (1 + (I/P))^2] * (1 + (I/P)) = New3$ Simplify.

$ * (1 + (I/P))^3 = New3$. (^3 means cubed or to the power of 3)

We can see a pattern:The (1 + (I/P)) part is raised to the power of the number of periods that you wish to calculate: $ * (1 + (I/P))^N This is our formula! The best thing about this formula is that it is a simple formula for complex mathematics. The initial principle stays the same from beginning to end and only the percentage changes over N periods. (1 + (I/P))^N will give you the yield for any number of periods you desire. If we want to see the rate for 12 months or periods we need to raise it to the power of 12. We started out trying to find the annual percentage yield for 10%. Lets give it a shot:

Since 10% = .10,$ * (1 + (.10/12))^12 We need a calculator for this. .10/12 = .008333. So:

$ * (1 + .008333)^12 or:

$ * (1.008333)^12

Using a calculator we see that 1.008333^12 = 1.10471.

So we would have $ * 1.10471, or an increase of $ * .10471 over our original principle.

This means that we have now 110.471% of the money we had initially, thereby making our Annual Percentage Yield 10.471%!

In other words, an interest rate of 10% earns you 10.471% over the course of one year employing the compound interest.

Thanks for stopping by again after my short hiatus. Next time, we will test this formula to ensure it’s accuracy, and we will use this equation to calculate how a given principle will grow over a certain number of periods with a certain interest rate. If anything was unclear about today’s long post, then hopefully the next post will clear it up. CD