### Archives For Financial Math Basics

I came across an article last night that stated that the average annual return for the S&P 500 was 12% from 1926-2014.

The way this author arrived at 12% was to add up all the yearly returns over the last 89 years and divide them by 89. When I do that with my numbers, I get 11.99%. Going further, using that return, \$100 invested at the beginning of 1926 would have grown to nearly \$2.4 million by the end of 2014. The math looks like this:

\$100 x (1 + .1199)89

\$100 x 23799.63867

\$2,379,964

Unfortunately, that’s not reality. Let me try to explain it with a simple example.

Say an investor received the following returns over three years:

Year 1: 25%
Year 2: -50%
Year 3: 25%

The average of those three years would be 0%. Now let’s calculate how it would have looked had we invested \$100 at the beginning of the first year and held it through year three:

Year 1: \$125.00 (\$100 + (\$100 x .25)) OR \$100 X (1 + .25)
Year 2: \$62.50 (\$125 + (\$125 X -.50)) OR \$125 X (1 – .50)
Year 3: \$78.13 (\$62.50 + (\$62.50 x .25)) OR \$62.50 x (1 + .25)

ONE QUICK NOTE: Some people have a difficult time with math and since this is a “Basics” post, I’ll spend a little extra time talking about how to think about returns. The reason we add 1 to return expressed as a decimal (1 + .25) is because 1 represents the original investment. So, we are getting our \$100 back plus \$25. When we have a negative return, we subtract the negative return from 1 (1 – .50) because we are losing part of our original investment. In this case we are losing half of our \$125. Does that make sense?

Based on the average return we calculated earlier, the \$100 should have been worth \$100 at the end of the third year because our numbers showed a 0% return, but this shows that our \$100 investment is only worth \$78.13 (a 21.88% loss). What accounts for the difference?

The difference is the first average used was simply the arithmetic mean (average) and doesn’t take into account an actual investment. To find out our actual return, we need to use something called the geometric mean (average). To do that, we need to change our formula up a bit. First, we need to express the returns as multipliers by adding 1 to them. Like this:

Year 1: 1.25
Year 2: 0.50
Year 3: 1.25

Now, we simply multiply those factors together and raise them to the power of 1/3 (or find the 3rd root). Like this:

(1.25 x .50 x 1.25)1/3

.78125.3333
.921008

Now, in order to get this to a percentage, we have to subtract 1 from .921008, which leaves us with -.07899 or -7.899%.

We can check our answer by plugging our finding back into the following equation:

\$100 x (1 – .07899)3

\$100 x .78125

\$78.13

IT WORKS!

So…

Back to the opening paragraph, which stated that the average return from the S&P 500 Index was 12% from 1926-2014. The geometric average is 10.04%. “Not much different,” you might think. The difference is huge when we are talking about 89 years. Let me show you:

\$100 x (1 + .1004)89

\$100 x 4973.8378

\$497,384

That’s nearly \$1.9 MILLION less than the number we get using the average return. Which brings up an important question: why do people still talk about averages?

Extra credit: Here are a couple of useful YouTube videos I found that talk about the geometric average (or mean)…

Last week I was part of a rather heated discussion on facebook about Texas’ budget woes and proposed cuts to its education budget. One of the people involved in the discussion was a school adminstrator. I asked him how he would solve the supposed education cuts. He wrote this:

“A .01% increase in taxes would prevent all this. That’s \$150 a year on a \$150,000 house.”

.01% increase in taxes…

Seems really low, doesn’t it? It’s not. Here is how it translates into taxes:

My house is worth roughly \$150,000 on the tax rolls. We paid \$3,500 in property taxes last year. If they were to increase another \$150, it would represent a 4.3% increase as I show in my math:

150 ÷ 3,500 = .0428 or 4.3%

The reason for the difference is that the .01% increase is PER \$100 of value.

Politicians framed this issue in such a way to make it seem like the tax increase isn’t that much when in fact it’s MUCH more than they make it seem.

The guy got rather testy with me when I told him that I would not be willing to fork over another \$150 per year. My reasoning is that although it’s “only” \$150, it’s a tax increase. Once taxes go up (especially property taxes), they seldom if ever come back down again. Once this particular budget shortfall is cured by a tax increase, they’ll find another place to use the tax increase. Not only that, the \$150 is based on property value, which will most likely increase in the future.

Larry Swedroe posted an interesting article on how to do the math on a mortgage refinancing to find out whether or not it’s to your benefit to refinance. He used the following example:

• 12 years left on a 15-year mortgage.

• Current mortgage rate is 4.75%.

• Current monthly payment \$1,369

• Current balance on the mortgage around \$150,000 (this is the amount to be refinanced).

This couple is looking to refinance into:

• 15-year mortgage.

• \$150,000 loan.

• \$3,000 in closing costs (to be paid upfront).

This couple is in the 25% income tax bracket.

So…based on those numbers, would it be a good idea to refinance?

Well, as Larry points out, there are a lot of things to consider before jumping into a refinance.

1. In this example, the spread between the two interest rates is less than a 100 basis points (less than one percent). Naturally, the bigger the spread, the more advantageous it is to refinance.

2. There is more to the calculation than simply looking at the difference in payments since the payment includes principal, which is your own money. So, we have to look at the difference in the interest portion of the payment.

3. As Larry also points out, interest payments are tax deductible (if you itemize your deductions). Larry uses the 25% income tax bracket for his example. Based on that, the couple is paying \$.75 for each \$1.00 of interest. Basically, what this means is that this makes the refinance less advantageous (you’ll see this in the spreadsheet).

4. This couple is already three years into their loan. They are looking to refnance into a new 15-year mortgage. That means they have paid 3 years of interest on the old loan and will be paying 15 years of interest on the new loan for a total of 18 years of interest.

5. The closing costs are paid upfront.

After running the example, I came to the conclusion that refinancing this loan will cost an additional \$702 in after-tax interest. I arrived at this number by adding up the three year’s of interest paid on the original loan plus the 15 year’s of interest on the new loan. Were they to continue with the old loan, they would have paid a total of \$70,417 in interest (\$52,812 after-tax in a 25% income tax bracket).

What Larry leaves out, in my opinion, is a discussion of the opportunity cost between the two loans. By choosing to refinance, this couple would be freeing up cashflow that could be put to work elsewhere (unless they are using the cashflow to shore up their budget). The payment difference of \$296 per month could be invested elsewhere for the next 12 years. Using a monthly total return on the S&P 500 Index of .75% (including a management fee), that \$296 per month payment difference could grow to more than \$112,000 in 15 years. If they invested the \$3,000 plus the entire \$1,369 monthly payment for 3 years after the end of the original mortgage, they would have over \$60,000 at the end of 15 years. Another thing worth mentioning is that all of the \$296 per month could be put into a Roth IRA where only a portion of the \$1,369 payments could be put in a Roth because they would exceed the Roth limits.

Based on those numbers, the refinance looks like a no-brainer. But, I left out three things: 1. Investing in the stock market is not a sure thing and 2. I didn’t make adjustments for taxes, which favored the refinance. 3. In order for the scenario to work, the payment difference MUST be invested and not spent.

With that said, I am making available the spreadsheet I used for this post, which you can download here: Mortgage Amortization Comparison (Two 15-year Mortgages). I didn’t spend a lot of time making it user-friendly but if you understand the basics of Excel, you can get in there and change up the variables yourself.

I was cleaning out my email inbox and found this email from way back in July:

Hello,

I have a question. If a person owed money to someone in 1998, but did not repay that amount until 2010, how can he calculate what amount to return now (in 2010) considering that much inflation has taken place since 1998? Should he use the yearly CPI inflation rate for his country? Will appreciate your help. Thanks.

The amount to be paid back and the term of the loan should have been spelled out when the money was initially borrowed. Although the CPI can be used, it should be the beginning point. The person who loaned the money was out that money for the last 12 years. Paying back money with only an adjustment for the CPI is basically paying them back the original amount. To compensate them for the usage of the money, the borrower would need to include some percentage above the rate of inflation (the CPI in this case). I would say a good starting point is 2% over the CPI.

Basically, what that means is if this person borrowed \$1,000 back in 1998, they should pay back around \$1,726.93 at CPI +2% (or \$1,351.59 just using the CPI for the last 12 years).

One of the most important areas of personal finance is setting and reaching financial goals. Why are financial goals important? Without them, it’s likely you won’t save and invest your money wisely. Having goals tends to help us focus on what’s important. Without them, we tend to allow life to just happen to us.

What Are Financial Goals?

There are many different kinds of financial goals:

• Get out of debt

• Create an emergency fund

• Pay cash for a new (or used) car

• Downpayment on a house

• College fund

• Retirement

The Goal-Setting Process

I’m not a goal-setting expert but I was able to come up with six steps in the goal-setting process:

1. Determine your goal and the amount of money needed to meet the goal.

2. Set a due date for meeting the goal.

3. Decided what investment vehicle that will be used to meet the goal.

4. Calculate the lump sum or periodic payment that will be needed to meet the goal.

Example

Let’s look at what the process looks like for someone saving up for a downpayment on a house. Let’s say in 5 years you desire a 20% downpayment on a \$200,000 house (\$40,000).

1. \$40,000

2. 5 years (60 months)

3. Since the goal is relatively short-term, the savings will be kept in an interest-bearing savings account. For this exercise, we’ll use an annual interest rate of 1.28%.

4. To determine the lump sum or monthly payment necessary to meet this goal, you can use any number of online calculators, a regular calculator, or you can download this simple Excel Spreadsheet I put together for this post. Because interest rates on savings accounts are so low, the lump sum needed to meet a \$40,000 goal in 5 years is really high at \$37,500. If you’re going to reach the goal with monthly savings, you’ll need to save \$645 per month.

5. For short-term goals, you’ll want to track your performance on a regular basis (monthly or quarterly) and make adjustments as necessary.

6. If all goes to plan, this goal should be met in five years (sooner if interest rates are higher or you can add more to your savings).

If you’ve never set and reached a financial goal, I urge you to give it a try.

Thoughts?

Just received this comment on this post. The question:

Came across this post and had a question….how do you calculate your PRR for the time period when the Annualized PRR is a negative number??

When you enter a negative number into the Excel IRR function you get an error because you can’t raise a negative number to a power (i.e. -8.43% ^ (180/365) = #NUM error).

Okay, this is easy enough to address. The actual formula for calculating a periodic return is…

[(1 + ROR)# of days/365] – 1

In this case, it would like this:

[(1 – .0843)180/365] -1

[.9157.49315] -1

.9575 -1

-.0425 or -4.25%

The readers problem was that he tried to perform the calculation using -8.43% rather than its decimal equivalent.

Hope this helps.

Related:

Reader Question Regarding Personal Rate of Return

How to Annualize a Rate of Return

Reader Question: How Do You Calculate Compound Growth (or Interest)?

Let’s say you want to buy a house. You need to finance \$200,000 for 30 years and your interest rate is 5% per year. Do you know how to calculate your monthly payment?

Sure, you could go to an online calculator but it’s also good to know the formula. To make this particular calculation, you need this formula:

Where…

Principal = \$200,000
Rate = .05 ÷ 12 or .0041667
N = Number of payments (360)

Now I’ll walk you through the calculation:

I ran the calculation in Excel, and it game me a payment of \$1,073.64. The one penny difference is due to rounding.

The cool thing about this formula is that it works on any kind of loan as long as you make the proper adjustments. You would do yourself well to learn this formula or at least know how to use a financial calculator the next time you purchase a car.