Archives For Financial Math

According to the Vanguard website, the Vanguard S&P 500 Index Fund is down 12.07% YTD as of yesterday’s close. To get an idea of what that return would look like if it were to continue for an entire year, you can annualize the YTD return.

It’s a fairly simple calculation to perform as long as you have the following information:

1. Number of days that have elapsed so far this year. This is easy to calculate if you have access to Excel.

2. The YTD return of the investment that you want to annualize.

The formula for annualizing a ROR is pretty straight forward:

[(1 + YTD ROR)1/(#of days/365)] – 1

The YTD ROR should be expressed as a decimal. Plugging in the Vanguard S&P 500 Index Fund information from above, the equation looks like this:

[(1 – .1207)1/(204/365)] – 1

[.87931/(0.55890411)] – 1

[.87931.7892] – 1

0.7944 – 1

-.2056 or -20.56%

So, a 12.07% loss for the first 204 days of the year equates to a 20.56% loss on an annualized basis.

Now let’s say you are down 12.07% but you purchased this fund on December 31, 2006. How do you annualize that return? The only input that changes in the above formula is the number of days, which is now 570.

[(1 – .1207)1/(570/365)] – 1

[.87931/1.5616] – 1

[.87930.640350877] – 1

0.9209 – 1

-.0791 or -7.91%

Had you purchased an investment on December 31, 2006 that is currently down 12.07% since the time of purchase, your annualized rate of return on that investment would be -7.91%. Not much of a return is it? Anyway, now you know how to annualize your returns. Fun stuff!

The other day I was looking at the amortization for the loan I took out to buy our 2007 Honda Civic. The balance on the loan is now below $10,000 and I immediately thought about paying more on the loan just to get it paid off earlier.

But then I noticed something that I never really noticed before (I noticed it before but never really thought about it): My extra payments wouldn’t save me that much in interest. Why? Well, it has to do with the way loans are structured. When you take out a loan, the payment is calculated based on the length of the loan, the interest rate on the loan, and the amount of the loan.

For example:

I’ll use our Honda loan as an example. Here’s the necessary information:

Interest Rate: 7.3645%
Period Rate: 0.6137%
Loan Term (Years): 3
Payments per Year: 12
Total Number of Payments: 36
Amount Financed: $15,019
Payment Amount: $466.26

Here’s what the amortization for this loan looks like (you can click on it to see a larger version):

Take a look at the first payment:

The beginning balance is $15,019. The interest portion of the $466.26 payment is $92.18 which is calculated by multiplying the beginning balance ($15,019) by the periodic rate (0.6137%). The remainder of the payment is applied to the principle, which becomes the beginning balance of the following month.

Each month the balance on the loan decreases, which makes the periodic interest payment smaller. This leads us to the point of this post:

In order get the most benefit from making extra payments on a loan, you need to make them at the beginning of the loan. How much difference does it make? Let’s see.

I ran two scenarios. The first one assumed an extra $50 each month for the final 18 payments and the second scenario the extra $50 was applied to the first 18 payments. Here’s what the two amortizations look like:

Under the normal amortization for this loan, the total interest charges for this loan would be $1,766. By making extra payments at the end of the loan, you would pay $1,720 in interest, giving you savings of about $46. By making extra payments at the beginning of the loan you save $155 in interest. No, it’s not a lot of money, but this is a short-term loan. Imagine how much the saving would be if this were applied to a mortgage.

Of course there are other things to consider when doing this math. For instance, you have to look at the opportunity cost of the $50 you are putting towards paying off the loan early. Could you put that money to better use elsewhere? That’s something you have to ask yourself.

Anyway, the next time you are tempted to accelerate the payments on a loan, ask yourself how much you are actually going to save by paying it off quickly. You might be surprised to find out it’s not as much as you thought.

Quiz time.

Last week while on vacation, I read Brian Tracy’s Flight Plan*, a short little book about success. On page 14 of the book, Brian has this to say about financial independence:

Here’s a simple exercise: Determine how much it would cost you per month to live comfortably even if you had no income at all. Include all your costs of housing, food, travel, medical expenses, vacations, and entertainment. Mulitply that number by 12 (the number of months in a year), and then mulitply that result by 20 (the number of years you will probably live after you retire). The total represents your retirement goal. This is how much you will have to accumulate to be financially independent.

He left out two very important details in his exercise. Do you know what they are?

I know what it is but I can’t say it here because it would defeat the purpose of the quiz. The first person to answer the question correctly (I’m the judge of what is or is not the correct answer) will win a copy of Flight Plan*. Just remember this little contest is only open to U.S. residents.

Good luck.

[Now begin playing “Zeopardy” music…]

*Affiliate Link