# Financial Calculations

How much money will you have when you retire? How much money should you deposit into your account each month in order to have \$10,000 in five years? What return on your money is necessary to double your account in five years?

These, and many others, are important questions for financial planning. In order to reach your goals, you must have some idea of what is necessary for that to happen. Fortunately, there are many financial calculations that can tell you exactly what to expect – and that’s what you’re going to learn in this reference guide.

Return on Investment (ROI)

Before we begin, we need to clarify the financial term “return on investment” more commonly called the “return on your money” or, more simply, the “return.”

Whenever you make an investment, you place a fixed amount of money into an investment. The initial amount you deposit is called the “principal.”

In exchange, you hope to get the principal back along with an additional amount “returned” to you for your efforts. This amount is the “return on the investment” or ROI.

For instance, if you deposit \$100 into an investment and it later increases to \$110 then the investment “returned” \$10 to you. As a matter of standardization, the return is expressed as a percentage of the original investment so we would say your return on investment was \$10/\$100 = 10%.

Further, this percentage should be expressed as an annualized figure in order to make fair comparisons across investments. If the above investment returned \$10 in exactly one year, your return is 10%. If, however, it returned \$10 in only six months then, effectively, you earned at the rate of 20% per year since you picked up the same amount of money in only half the time.

This relationship between principal, interest, and time is often expressed in the simple formula:

I = P * R * T

Where:

I = interest (actual dollar amount)

P = principal

R = rate (interest rate expressed as a percent so that 5% = .05)

T = time (expressed in years so that six months = 0.5 years, nine months = 0.75 years)

It states that the actual amount of dollars you earn on any investment (the interest, I) equals the principal (P) multiplied by the rate of interest (R) multiplied by the amount of time (T).

In the previous example, we know the principal is \$100, the interest rate is 10% and the time is one year. Therefore, the “interest” or the amount returned from the investment is \$10, which is found as follows:

I = P * R * T

I = \$100 * 0.10 * 1 year

I = \$10

Naturally, we don’t need to use the formula to only solve for the interest rate (I). If you know any of the three variables, you can solve for the missing fourth.

For example, if the \$10 interest was returned in only six months (0.5 years) then what the effective interest rate (R)? We can easily find that by plugging in I= \$10, P = \$100, and T = six months, or half a year (0.5 years) and solve for R as follows:

\$10 = \$100 * R * 0.5 years

R = \$10/(\$100 * 0.5)

R = \$10/\$50

R = 0.20, or 20%

If you’re not too comfortable with algebra, there is a simple way to solve for any of the variables without having to rearrange the equation. Rather than write the formula as I = P*R*T, write it down on a piece of paper as follows:

Next, place your finger over the variable you’re trying to solve for and voila – the remaining uncovered letters give you the formula! Using our previous example, if you want to solve for the rate (R) that was earned then you’d simply cover up the R and you’d see this:

It immediately tells you that the formula is “interest divided by (principal * time)”, which is \$10/(\$100*0.5) = 0.20, or 20% as we found before.

If, on the other hand, you wanted to find out the length of time (T) required to produce that 20% return, you’d cover up the letter T and you see this:

You can see that solving for “T” requires you to take the interest amount and divide by (P *R), which is \$10/(\$100 * .20) = 0.5 years, or six months which also corresponds to our earlier answer.

No matter which variable you’re trying to solve for, just cover that letter and the formula takes care of itself. With a little bit of work, you’ll start to visualize the correct formulas without having to resort to sheer memorization or algebraic manipulations.

Interest Doesn’t Necessarily Mean Risk-Free

One of the main points to understand from this section is that the term R (interest rate) does not necessarily mean interest in the traditional sense such as the amount paid by a bank for the money you have on deposit.

In most cases, people are used to using the word “interest” to mean that money paid by another person or institution (such as a bank) for the use of their money. By this definition, interest usually means risk-free.

However, as you read through this reference guide, if a calculation states that the interest rate necessary to accomplish your goal is 20%, for example, don’t think this means you must find a bank paying 20%. The proper interpretation of interest in this guide is the “effective” interest rate or the “return” on your money.

If you can’t find a bank paying 20% then you will have to find another source – perhaps the stock market – and any profits effectively represent the interest paid on your money, or your return on investment (ROI).

If you invest \$10,000 in the stock market and it’s worth \$12,000 in one year then your “interest” on your money is \$2,000, or 20%, even though it is technically a capital gain. In most areas of finance, the word “interest” is used generically to mean the return on your money whether it is risk-free or not.

The formulas in this guide can greatly help you with your financial decisions. For example, if a formula shows that 5% is necessary to accomplish your goal and the risk-free interest rate is 3%, you have some choices to make. First, if you insist on the risk-free rate then you must extend the time period you are willing to wait for that money. On the other hand, if you cannot extend the time, you’ll have to accept a little more risk.

Will you need to invest in stocks? Perhaps not but you may need to consider commercial paper or other types of investments that have higher expected returns. Of course, the fact that risk is present for all investments (other than risk-free investments) means there is always the chance that the desired return won’t be realized. That’s why it is so critical to define your goals, find your risk tolerances, and hedge investments accordingly. Only then can you have a realistic chance of reaching those goals.

Let’s take a look at some very helpful financial planning calculations.

Rules of Thumb

Rule of 72 – How long to double your account

Rule of 114 – How long to triple your account

The benefit to these calculations is that they are fast, simple and many times can be performed in your head. The drawback is that they are not exact; however, they do produce reasonably close results. Use these as guidelines but always double check with precise formulas (described later) if you are using to make important investment decisions that require an exact answer.

# Rule of 72 – How Long to Double Your Account?

To find how long it will take to double your account at x% interest rate, take 72/x.

Example 1:  How long will it take to double your account at 10% interest?

Answer: It will take approximately 72/10 = 7.2 years.

We can also solve the problem backwards and find out the interest rate. If you want to double your account in 7.2 years, it will require 72/7.2 = 10% interest.

Example 2:  How long will it take to double your account at 12% interest?

Example 3:  What interest rate is necessary to double your account in 7 years?

How accurate are the results? The following table shows a few interest rates and the corresponding answers generated by the Rule of 72 compared to the exact calculation (described later).

You can see that the Rule of 72 slightly overestimates the amount of time for low interest rates while underestimating for higher rates. Still, considering the ease of calculation, it provides a very good estimate.

Rule of 114 – How Long to Triple Your Account?

To find out how long it will take to triple your account at x% interest rate, take 114/x.

Example 1:  How long will it take to triple your account at 10% interest?

Answer: It will take approximately 114/10 = 11.4 years.

As before, we can solve this problem for the interest rate as well. What interest rate is necessary to triple your account in 11.4 years? Answer: 114/11.4 = 10%

Example 2:  How long will it take to triple your account at 12% interest?

Example 3:  What interest rate is necessary to triple your account in 12 years?

As with the previous rule, the Rule of 114 slightly overestimates the amount of time for low interest rates while underestimating for higher rates but the estimates are still quite good.

To find out how long it will take to quadruple your account at x% interest rate, take 144/x.

Example 1:  How long will it take to quadruple your account at 10% interest?

Answer: It will take approximately 144/10 = 14.4 years.

What interest rate is required to quadruple your account in 14.4 years?

Example 2:  How long will it take to quadruple your account at 12% interest?

Example 3:  What interest rate is necessary to quadruple your account in 18 years?

How accurate are the results? The following table shows a few interest rates and the corresponding answers generated by the Rule of 144 compared to the exact calculation (described later).

One of the most significant insights that can be drawn from all three rules is that the amount of time necessary to double, triple, or quadruple your account value is probably less than most people would intuitively expect. The moral: You don’t need to take large risks to attain great wealth – if you’re willing to wait just a little bit.

Future Value of Money

The previous rules of thumb gave us approximations for the amount of time to double, triple, or quadruple your account value. Now it’s time to look at exact formulas.

Any amount of money you have right now is called the “present value.” The value of that money in the future assuming a constant growth rate (i.e., the interest rate stays constant) is called the “future value.”

To find the future value of money, you must take the principal amount multiplied by 1+interest rate (expressed as a decimal) raised to the number of years:

Principal * (1+i) years

Example 1: You have \$1,000 in the bank and it will grow at 5% per year.  How much will it be worth in 5 years?

\$1,000 * (1.05)5 = \$1,276.28.

Note that we could have also found this answer the long way by multiplying \$1,000 * 1.05, which would give us the future value in one year (\$1,050). We could then multiply that answer by 1.05, which gives us the future value in two years. If we repeat this procedure five times, we’d end up with the same answer. Mathematically, multiplying any number by itself x times is the same thing as raising it to the x power. In other words, 1.05 * 1.05 * 1.05 * 1.05 * 1.05 is the same thing as (1.05)5.

This formula assumes that interest is paid at the end of each year. What if it is paid monthly?

In this case, the monthly interest rate is .05/12. We also know that the number of payment periods must be five years * 12 months, or 60 periods. Therefore, if interest is paid monthly at the rate of 5% per year, the \$1,000 initial investment would be worth:

\$1,000 * (1 + .05/12) 60 = \$1,283.36

We can extend the above line of thinking indefinitely and assume that interest is paid as fast as possible – every micro-second and faster. In this case, we say the interest is paid “continuously.” Despite intuition, the principal amount will not grow to an infinite amount. Instead, it will not be too much greater than if the interest is paid monthly.

How do we calculate interest paid continuously? Let’s put things in simple terms and see if we can unlock a pattern. Assume that interest rates are 100% and paid annually. The future value formula is then:

(1 + 1)1

In other words, if you start with \$100, you will have \$100 * (1 + 1)1 = \$200 at the end of one year.

However, if the interest is paid monthly then the formula would be:

(1 + 1/12)12

Your \$100 investment would now be worth \$100 * (1 + 1/12)12 = \$261.30

If interest is paid daily, the formula is:

(1 + 1/360)360

The \$100 investment would be worth \$100 * (1 + 1/360)360 = \$271.45

Are you starting to see the pattern? We take our interest rate and divide by the number of periods, add one to it, and then raise it to the number of periods. If we were to take this to a very small time frame, say one million units per year, the formula would reduce to:

(1 + 1/1,000,000)1,000,000

Your investment would now be worth \$100 * (1 + 1/1,000,000)1,000,000 = \$271.82, which isn’t too greater than the previous answer of \$271.45.

If we make two million pay periods per year, the future value becomes:

\$100 (1 + 1/2,000,000)2,000,000 = \$271.82, which exactly matches the previous answer with only half the pay periods! It seems as though we’re closing in on a maximum limit.

What is that limit? In finance and other areas of mathematics, it has a special name “e” which stands for the “exponential” function. The bigger we make the number of pay periods, the closer we’ll get to the true value of e. Let’s pick an arbitrarily large number, say ten million:

\$100 (1 + 1/10,000,000)10,000,000 = 2.718281828...

The trailing dots just means the number continues on forever, never ending, never repeating.

Most financial and scientific calculators have the “e” function built in, which is a key that usually looks like ex. This key allows you to raise e to any number you’d like. If you hit the number “1” and then the ex key, the “=” key you’ll find the calculator gives you 2.718281828…, which is the value of e.

With the aid of e, we have a simplified way of finding the future value for any interest rate or time period. All you have to do is multiply the interest rate by the number or pay periods, raise e to that number, and multiply by the principal.

For instance, let’s go back to our previous example and see if we can simplify it. Assume you have \$100 that it continuously compounded at 100% for one year. How much will you have at the end of the year?

\$100e1 = \$271.83

Now that you understand e, let’s use a little more realistic example.

Example 2:

Assume you have \$1,000 invested for five years at 5% and compounded continuously. How much will you have at the end of five years?

\$1,000 * e.05 * 5 = \$1,284.03

Example 3:

You have invested \$3,000 for three years at 7% and compounded continuously. How much will you have at the end of three years?

\$3,000 * e.07 *3 = \$3,701.03

In other words, the very most a three-year investment at 7% could ever produce is \$3,701.03. If the interest rate is paid at anything less than “continuously” the future value will be somewhat less. Using e is a great way to define potential limits of investments.

Now that you understand the future value of money as well as the varying rates at which interest could be paid, let’s take a look at some other examples.

Example 4:  You have \$20,000 in your IRA that is expected to grow at 12% per year.  How much will it be worth in 30 years?

Solving For Time

As with any formula, we can rearrange it to solve for other variables  that are present in the formula. Using the previous future value  formula, we can rework it and solve for time.

Example 1:  You have \$20,000 in your IRA that is expected to grow at grow at 12% per year.  How long will it take before it's worth \$1 million?

Before we can answer this, we need to turn to a mathematical function called logarithms. A logarithm, or log for short, is really not that complicated. The log of any number is simply the exponent on base 10 that creates the number in question.

For example, log (100) = 2

The interpretation here is that 10 (the base) raised to what number gives the answer 100? In other words, if 10x = 100, what does x equal? It must equal two. Recall that whenever we raise a number to a power, it means that we are multiplying the base number by itself that many times. Therefore, 102 = 10*10 = 100. Now it should make better sense why log (100) = 2.

If you notice in the future value formula, the “time” is represented as the exponent. Logarithms solve for exponents and that’s what makes them so powerful. If you want to solve for exponents, you’ll need to use logarithms.

Now, there are a couple of mathematical tricks using logs that you must understand before you can solve for time. There is a property of logarithms that says a = log 10a. This is nothing more than an identity. It is asking for the exponent on base 10 that gives the answer “a.” Obviously, if 10 is raised to “a” then the answer must be “a.”

Using our previous example, 2 = log 102. If 102 = 100 then log 100 = 2.

Using the above identity, we can multiply number together using logs. For instance, how can we solve for a * b using logs?

Step 1:

Recognize that a = 10log a and b = 10log b

Step 2:

If Step 1 is true then a * b = 10log a * 10log b

Step 3:

A basic property of exponents states that if you are multiplying two numbers together with the same base then we can just add the exponents together and raise the sum to the same base. For instance, 102 * 102 = 104

Therefore, if we take the log of both sides of Step 2, we get:

log a * b = log a + log b

In other words, if we want to multiply two numbers together we can simply add the logs of those two numbers. Now, you’re probably wondering why we wouldn’t just multiply the two numbers together in the first place! Why bother taking the logs and adding them?

The answer is that many times the numbers we are multiplying are too big. Sometimes it’s easier to add. Further, for our purposes, we’re going to use this trick to solve for time.

The problem with solving for time is that we are trying to solve for an unknown exponent. In order to do so, we must get the exponent into the equation by some mathematical manipulation – and that’s what logs allow us to do.

For instance, how do you suppose we can “rearrange” the expression 52 so there is no exponent? It’s simple with logs.

First, understand that 52 = 5 * 5 by definition. Therefore, if we take the log of both sides we get:

log 52 = log (5 * 5) = log 5 + log 5 =  2 * log 5

More generally, how can we rearrange am?

If we take the log of both sides, we get:

log (am) = log (a1 * a2 * …am) = log a1 + log a2 + log am

In other word, there are “m” factors of “log a” so we can rewrite that as m * log a.

So for the log of any number, we can take the exponent multiplied by the log of the base. For instance, log (37) must equal 7 * log (3).

log (37) = log (2,187) = 3.34

and secondly,

7* log (3) = 7 * 0.48 = 3.34

Armed with the powerful log function, we can now solve for exponents. Let’s start with an easy one. If 5 x = 25, what does x equal?

Simple, take the log of both sides:

log 5x =  log 25

x * log 5 = log 25

x = log 25/log 5

x = 2

If you have a financial or scientific calculator, try it! Take log (25) and divide it by log (5) and you’ll see the answer is two. Therefore, 52 = 25.

Note: There are two log functions: common log (base 10) and natural log (base e), and most financial and engineering calculators have both. It does not matter which one you use as long as you're consistent. Pick one or the other when doing the calculations.

Example 1:  You have \$20,000 in your IRA that is expected to grow at grow at 12% per year.  How long will it take before it's worth \$1 million?

The question is asking you to solve for “n” for the following equation:

\$20,000 * (1.12)n = \$1,000,000

Step 1:

Divide both sides by \$20,000

(1.12)n = 50

Step 2:

Take log of both sides

n * log (1.12) = log (50)

Step 3:

Get “n” by itself on the left so divide both sides by log (1.12)

n = log (50)/log (1.12)

n = 34.5

It will take 34.5 years for your \$20,000 to grow to \$1,000,000 assuming you can get a constant growth rate of 12% per year.

Once you understand the math behind the solution, you can quickly get the answer on a calculator by taking the log of the future value (\$1,000,000) and dividing it by the log of the present value (\$20,000). Once you have that answer, divide it by the log of “1 + interest rate” as follows:

log (\$1,000,000/\$20,000)  = 34.5 years

log 1.12

Note that we must express the interest rate as a decimal. If we are assuming 12% then that is 0.12 as a decimal. If we add that to one the answer is 1.12.

Example 2: You have \$20,000 in your IRA that is expected to grow at grow at 12% per year.  How long will it take before it doubles?

In other words, solve the following \$20,000 * (1.12)n = \$40,000

log (\$40,000/\$20,000)  = 6.11 years

log 1.12

We can check this using our earlier "Rule of 72":

72/12 = 6 years

How long before it triples?

Again, set up the equation as \$20,000 * (1.12)n = \$60,000 and solve for “n”:

log (\$60,000/\$20,000)  = 9.7 years

log 1.12

We can check this using our earlier "Rule of 114":

114/12 = 9.5 years, which is pretty close to the exact answer.

log (\$80,000/\$20,000)  = 12.2 years

log 1.12

According to the Rule of 144 the answer is 144/12 = 12 years.

Present Value of Money

Once you understand the future value formula, you can easily solve for other variables. For instance, rather than solve for the future value, we may wish to know the present value – the amount of money needed today to reach a particular goal.

Example 1:

You wish to deposit money in your bank at 5% in order to have \$15,000 in 5 years. You will make only one deposit. How much do you need to deposit today?

This is asking you to solve the following equation:

x * (1.05)5 = \$15,000

This is easy since we know the exponent so we won’t need to use logs. In order to solve for “x” just divide both sides by (1.05)5:

x = \$15,000/1.055 = \$11,752.89

Because this is just working the future value formula in reverse, we can use the future value formula to check the answer:

\$11,752.89 * (1.05)5 = \$15,000

Therefore, if you deposit \$11,752.80 in an account and earn 5% per year, your account will be worth \$15,000 in five years. Remember, this 5% “interest” could be a risk-free rate from a checking account or the capital gains from the stock market. It’s just stating that this is the rate of return you’d need on your money.

Solving for Interest

Now that you understand how to set up the equations and solve for future and present values, let’s find out how to solve for a missing interest rate.

To do so, you must understand another simple mathematical trick. If you have a number raised to some number and wish to get rid of the exponent, you must raise the entire expression to 1/exponent.

For instance, if 52 is raised to the ½ power, you’d get five:

(52)1/2 = 5

Let’s try one more. What is (74)1/4? It must equal seven.

In essence, this trick just “erases” the exponent without changing the value of the equation.

With that simple trick, we can now solve problems for an unknown interest rate. Let’s work the previous problem assuming we don’t know the interest rate.

Example 1:

You wish to deposit \$11,752.80 in a bank in order to have \$15,000 in 5 years. You will make only one deposit. What rate of interest is required?

Step 1: Set up the problem

\$11,752.80 * (1+i)5 = \$15,000

Step 2: Get the 1+i by itself

(1+i)5 = \$15,000/\$11,752.80, or

(1+i)5 = 1.28

Step 3: Raise both sides to the 1/5 power

((1+i)5 )1/5 = 1.281/5

This effectively “erases” the “5” exponent on the left side and you’re left with

(1+i) = 1.281/5

1+i = 1.05

i = .05, or 5% interest, which is exactly what we used for the previous problem.

Present Value of an Annuity

An annuity is just a stream of steady payments, rather than just one lump sum amount. Examples of common annuities are home and car loans.

For example, you may spend \$300 per month for a car payment over a four year period. If you do, that cash obviously has a lot of value over time. Rather than buy the car, you could elect to deposit \$300 per month into a savings account paying a given interest rate, say 5%. After the first month, the \$300 earns a little bit of interest at which time another \$300 hits your account. After the second month, the first \$300 deposit has earned interest for two months while the second deposit earned interest for only one. This process continues for four years. How much money would you have in the bank at the end? That’s what the future value of an annuity tells us.

Conversely, we can take a future value and find the present value of those cash flows – the present value of an annuity.

Example 1: You are willing to spend \$500 per month over 5 years on a new car. Interest rates are 6% per year. How much can you spend on the car?

We are exchanging a stream of payments (the monthly car payment) in exchange for a lump sum amount today to buy the car – the present value of the annuity.

Notice that the periodic interest rate is 6%/12 = 0.5%, which is .005 as a decimal. Also, you are making monthly payments over five years, which is equal to 60 payments.

Answer: \$500 *  1 - (1/1.005)60  = \$25,862

.005

You can shop for a car worth about \$26,000.

We can also rework this formula to solve for the payment or the monthly (or other time period) cash flow. A typical use is to find how much you can periodically with draw from an account over a given time as in the following example.

Example 2: You have accumulated an IRA worth \$1,000,000 and are ready to retire.  You wish to deplete the account in 20 years. Assuming your account continues to grow at 8% per year, how much can you withdraw per month in order to exactly deplete the account in 20 years?

Note: the periodic interest rate is 8%/12 = .67%, which is .0067 as a decimal. If you are making monthly withdrawals over 20 years, that's 20*12 = 240 periods.

\$1,000,000

(1-(1/1.0067)240)/.0067

= \$8,389.62 per month

Future Value of an Annuity

The future value of an annuity is the value in the future that is a result of a steady stream of payments today. Obviously, there are two ways that payments could be made. First, you could make the payment on the first of each month (or other time period). Second, you could make them at the end of the month. While it may not sound like a big difference, over time it can really add up as shown in the following two examples.

Example: You wish to deposit \$2,000 per year and the end of each year for the next 30 years. You expect to earn an average of 12% per year. How much will you have saved at the end of 30 years?

Because the payments are made at the end, this is called an "ordinary annuity."

\$2,000 * (1.1230 - 1)/.12 = \$482,665

Example 2: Instead, what if you made contributions of \$2,000 per year but did so by depositing \$500 at the end of each quarter? Now the periodic interest rate is 12%/4 = 3% (.03 as a decimal) and there are now four payments per year for a total of 120 payments.

\$500 * (1.03120 -1)/.03 = \$561,849

Notice the difference in the results. It pays to get the money in early! If you plan to put \$2,000 per year in an IRA, it really pays to split up those payments and get the money working sooner.

What if you made the contributions at the beginning of each period instead of at the end?  This would be called an "annuity due" and is the same answer you get from above just multiplied by 1 plus the periodic interest rate.

Example 1: A \$2,000 contribution at the beginning of each year for 30 years at 12% would be worth \$482,665 * (1.12) = 540,584

Example 2: A \$500 contribution at the beginning of each quarter for 30 years at 12% (periodic interest rate of 12%/4 = 3% or .03) would be worth \$561,849 * 1.03 = 578,704.

This guide presents most of the financial calculations you will ever need to make your investment decisions. Most financial calculators and many websites provide programs that calculate the answers for you. But as with any calculation, you’ll make better decisions if you understand how to interpret the results. Hopefully this guide has provided the necessary understanding.