Here is a problem that demonstrates the value of investing at a young age.

Alice and Bob are the same age, and they both plan on saving for retirement.

Alice starts at age 25 and invests $1,000 at the beginning of each year until age 65.

Bob plans to do the same thing, but he waits until age 35 to start.

How much does Bob have to contribute each year to have the same amount as Alice at age 65?

Assume both get a fixed investment return of 6 percent each year.

I will get to how to solve this problem in a second. But let me cut to the chase about the answer.

The answer: Bob would have to invest about $1,946 per year–nearly DOUBLE the amount Alice invested each year.

Furthermore, Bob ends up having to invest more money too: he ends up contributing over $60,000 whereas Alice puts in $41,000.

The lesson is that investing at a young age eases the burden of saving for retirement considerably. Here’s a small chart of their investment totals.

(This lesson is often taught as “the power of compounding,” which I believe is missing the point. I think this is more a lesson that having a head start gives you a big lead. Note that even if markets stagnate, Alice still wins out. That is, even if Alice gets a 0 percent return for the first decade, she starts out with a $10,000 lead over Bob. If both expect a 6 percent return from then on, Bob would still need to invest a lot more ($1,635) to catch up by age 65–that’s 60 percent more than Alice contributes annually. Having to save more of his paycheck makes it harder for Bob to save.)

Optional: math

I used a numerical solver in my spreadsheet to play around with the numbers.

But I also came up with an explicit formula for fun.

The amount Alice invests can be written as a geometrical series. The first investment she makes at age 25 will compound for 41 years, then the contribution at age 26 compounds for 40 years, and so on, with the final contribution at age 65 compounding for 1 year (she invests at the beginning of each year):

$1,000(1.06)⁴¹ + $1,000(1.06)⁴⁰ + … +$1,000(1.06)

Bob will invest an amount x each year, but he starts at age 35, meaning his initial contribution compounds for 31 years, and each subsequent one for 1 year less:

$x (1.06)³¹ + $x (1.06)³⁰ + … +$x (1.06)

To make Bob’s investment catch up to Alice’s, we set these two equations equal to each other and solve for x.

The two series can be summed using the formula for the sum of a geometrical series, as follows:

This formula can be generalized for a return of r, Alice investing for T number of year, Bob investing for t years, and Alice contributing A dollars per year. This is the amount Bob would need to contribute each year to catch up at age 65:

I got an email the other day from someone who had some questions about opening a bank account. Specifically, the person was curious about how the APY was calculated and how that related to interest earned:

Presh, could you help me understand what an APY of 0.05% means? I want to start a money market account, and the bank says that their annual percentage yield is 0.05% and that the interest is compounded daily and paid monthly. The amount of interest earned is based on the daily collected balances in the account. I really don’t understand what all this means.

If I deposit $5000 into the account, how much interest can I expect? That’s all I want to know. I don’t understand how to figure out the interest using the above information that the bank has provided. Thanks

This is a very good question, and I felt it would be educational to discuss the details of the bank account.

The big picture: how much interest

Carol primarily wanted to know how much money she would earn on a $5,000 deposit.

The APY is the relevant piece of information here. APY stands for annual percentage yield, and it represents the percent interest one can expect to earn, taking into account for the effects of compounding.

In this case, the 0.05% APY means that you earn 0.05% interest on your money in a year. So if you deposit $5,000, then you can expect to earn $2.50 in interest (0.05% * $5000) by the year end.

(This is a really small sum of money. As you are probably aware, there are online savings accounts with higher APY. Banks like ING and Ally Bank are currently offering something like 0.8%, which would translate into $5,000 earning $40 in interest per year.)

The APY is the most important detail in the email. But there are a few subtle points that also merit discussion.

Payment frequency: interest paid monthly

This is a very important detail. This tells you the bank only credits the interest to your account once a month on a specified day.

It helps to know when the interest is paid, especially if you are closing the account. You can lose out on interest if you close out just before a payment. For instance, if you expect an interest payment of say $50 on the 15th of the month, but you happen to close the account on the 14th, then you would lose out on that $50.

In the account mentioned in the email, we expect to earn $2.50 a year, so that comes out to about 20 cents each month paid on a specific day.

Balance calculation: daily collected balances

This is another important detail. Unlike a CD where you invest a fixed amount of money, a money market account will change depending on withdrawals and deposits.

Because your balance can change daily, the bank has to specify which balance it uses for computing interest.

The daily collected balance is one method banks often use. The daily collected balance method says the following: you take the closing balances for each day of the month for collected funds (deposits that have not cleared are uncollected funds), and then take the average.

For example, if you started with $5,000 and then on the 15th day withdrew $1000, then you’ll have $4000 for another 15 days. So if you have $5000 for half the month, then $4000 for half the month, your average daily collected balance would be $4,500.

Compounding frequency: interest compounded daily

This is a final point that I wanted to talk about from the email, though it is not as important.

In theory, it matters whether interest is compounded on a daily or monthly basis. The shorter the compounding period, the more interest that will get accumulated over a year. Here is an article that shows how daily compounding results in slightly more interest than monthly compounding: daily vs monthly compounding.

But we don’t really need to concern ourselves with this detail. The reason is that the bank account quoted an APY, which already takes into account the effect of compounding. If you have two accounts with the same APY, they can expect to give you the same interest at the year end. The effect of monthly versus daily compounding is already accounted for in the APY.

The only time you care about the compounding frequency is when you are comparing APR (annual percentage rate) rather than APY (annual percentage yield).

Typically the APR is quoted for credit cards, and APY for savings accounts, a practice that I have explained before in this article: APR vs APY: what is the difference?.

It got me thinking that I needed a better system for remembering my ATM PIN. I did a bit of research and came up with the following tips.

A preliminary: don’t use your birthday

It should go without saying, but it’s a bad idea to use your birthday for your PIN. Yes, it is easy to remember. But that also makes it easy for thieves to decipher. This is even more pertinent nowadays because people often share their birthday publicly on their Facebook profiles.

Still, a lot of people ignore the warning. A recent study found that as many as one in five people used their birthday as a PIN.

Be a bit smarter and use a PIN that thieves cannot figure out as easily. Try one of the tips below to help you remember the PIN.

1. Find out what the numbers spell

Lets say your password is the seven digit number 6363686.

Rather than remembering that jumble, you could just remember the letters correspond to numbers on the keypad and make a word out of it.

The numbers 6363686 happen to correspond to the word MEMENTO. You have a better chance of remembering this word, or you can even put that word on your phone in a memo for a more permanent solution.

You can find out what words your PIN spells by using the website PhoneSpell.org

2. Make a sentence out of the numbers

This is another tip that associates numbers with words.

The idea is to convert the numbers into words, and then make a list out of it.

An example will help to clarify the idea. Let’s say your PIN is 1250. If you spell out your PIN in words, you would write “One, Two, Five, Zero.” The starting letters of those words are O, Tw, Fi, and Z.

Now what you do is you make a list out where each word starts with those letters. For the PIN of 1250, you might write the list: Orange, Twix bars, Figs, Zucchini.

You will find it is easy to go from the list of words back into your pin. The reverse process is as follow:

Orange, Twix bars, Figs, Zucchini –> O, Tw, Fi, Z –> 1, 2, 5, 0 –> 1250

And that’s it, you’ve got your password at all time.

The only tricky part is a couple of the numbers start with the same letter, like five and four, so you will want to use the first two letters (FI and FO) in your list to distinguish.

Here is a cheat sheet with some suggested words:

zero –> Z Zuccini, Zesty salad, Ziti pasta

one —> O Orange, Olive, Oreo cookies

two —> Tw Twix bars, Twinkies, Tweezers

three —> Th Thin cheese slices, Thick bread

four —> Fo Focaccia bread, Food processor

five —> Fi Figs, Filet fish, Fiber cereal

six —> Si Simply potatoes, Single serve pizza

seven —> Se Secret seasoning, Semolina pasta

eight —> E Eggplant, Eggs

nine —> N Nectarine, Naan

Now all you have to do is make a sentence or list and store it on your phone, and you’ll always be able to get to your PIN.

3. Encrypt your password by appending another number

A trick that I like to use is to append some letters that are memorable to me.

For instance, let’s say your birthday is January 23, which is written as 01 for January and 23 for the day.

Now what you do is you add the memorable number 0123 to your PIN and store it on your phone.

If your PIN is 1250, then you can put the number 12500123 in your phone. Or you may write it as 01231250. Either way you know that four digits are to be ignored and you can figure out your PIN.

4. Use sports players jersey numbers

You have to be a big sports fan for this one, but it is a very good trick.

Let’s say your number was 2333. For me I could split this up into two numbers of 23 and 33, and then I could associate the numbers with the jersey numbers of players I know. For this case, I could write down Michael Jordan and Scottie Pippen and the numbers 23 and 33 would quickly come to mind.

You have to be really into sports for this to work, but undoubtedly there are people who know sports trivia so well that it could be useful.

5. Use modulo arithmetic to encrypt your password

This is the most complicated of the systems but it does have a chance to help you encrypt other passwords. Here is a detailed guide from which I will summarize the method.

You need to start out with a number that is memorable to you. Let’s say you will always remember the number 1234 as an easy code.

The next step is to subtract this number from your PIN to create another number. This number you create is basically an encrypted PIN and you can safely carry it around–only you know how to get back to your actual PIN.

Here is how the process would work. Suppose your PIN is 1250.

Step 1: subtract your number 0123 from your PIN

1 – 1 = 0
2 – 2 = 0
5 – 3 = 2
0 – 4 = -4 + 10 = 6

The only tricky part is if you get a negative number, like -4, you will want to add 10 to make it a positive number. (We are basically working with modulo arithmetic for the number 10)

Now you have a number 0026 (we can call it a fake PIN) that you can store in your phone safely.

When you want to go to the bank, you will then ADD your secret number 0123 to arrive back at your PIN.

Step 2: add your number 0123 to your fake PIN

0 + 1 = 1
0 + 2 = 2
2 + 3 = 5
6 + 4 = 10 – 10 = 0

Like in step 1, the method requires working in modulo 10. That means if you come up with a number 10 or larger, you want to subtract 10 to arrive at the correct number.

The process got us to 1250 which is your PIN.

Clearly these tips can be used for another numberical passwords as well.

I know that some of these five tips sound like more work than rote memorization, but the consequence is they are more foolproof than human memory. The goal is that you will never forget your PIN, so I hop these tips help. They have certainly helped me!

Sources: I found the first two tips at OzSoapBox, the third tip I came up with on my own–though someone at OzSoapBox also suggested it, and the fourth and fifth tip are from StraightDope Message Boards.

It’s also a time to take action. Because many financial activities like taxes and insurance are based on the calendar year, it’s important to take care of some things before the new year.

Spending an hour or so on your finances now can help you save a lot of money. I’ve done a bit of research to find out some of the most important tasks you should look at before the year ends.

Here is my 12 item year end financial checklist. Even if you are short on time, I would suggest you try to get a few items done. Most don’t take very long. Here’s the checklist:

Printable PDF

For completeness, I will mention one more item:

(Debatable) 13. Sell investments for tax writeoff: I originally had this item on the list, but this strategy depends on knowing a lot of things about the market and taxes. Also, I recently read an article about how “loss-harvesting” can backfire. It’s something to discuss with your financial advisor.

So that’s my year end financial list.

What items do you have on your list?

(found via Technology Review)

The purpose of the game

Many economists argue that markets are efficient, meaning prices reflect all publicly available information. This implies prices cannot be forecast and will be random and unpredictable like a coin toss. But is this really true?

To find out, the game ARORA was developed and analyzed by researchers Jasmina Hasanhodzic at AlphaSimplex, Andrew W. Lo at MIT Sloan Management, and Emanuele Viola at Northeastern University.

The setup

The goal of the game is to identify the real data between two sets of pictures that display prices.

The real data comes from historical prices of U.S. indices like the Dow Jones Industrial Average or the U.S. spot price of gold.

The ‘fake’ data is a randomized version of the real data. These prices are created by randomly reordering (permuting) the stock returns of historical data. This method keeps the price structure in tact but destroys the time ordering of the data. The beauty of this method is the new series will have the same mean return and standard deviation!

The only thing disrupted is the timing of the returns which is completely randomized, and hence past returns cannot be used to predict future returns.

The gameplay

Basically you are shown two pictures of stock prices. One is real, historical data and the other is a randomized version. You click on a picture with your guess and the game indicates whether your pick was correct or wrong.

You have to make several picks to finish a game.

The study results

The study found that most people could identify the real data better than at random. This was true for all eight tests–the highest p-value was a mere 0.503%! In six of the tests, the p-value was less than 0.001%. Clearly the participants were very good at identifying real data.

The authors suggest the reason is the instant feedback of the game helps the eye spot patterns to distinguish real from randomized data. This skill perhaps relies on our visual recognition system which is something computers cannot mimic yet.

Playing the game

I tested my own skill in the game. I logged in as a guest and then tried the training for the “Lynx” game.

I was a bit disappointed in my results. Out of 10 levels, I was only able to get 6 correct. It’s a little better than half but not by much. But I tried again and I got 7 out of 10 right–I was beginning to see the pattern!

I would imagine someone with a trading or finance background may do well right off the bat. But after some practice things level out–the study did not find any performance difference between those who had finance experience and those who did not.

If you’re up for a challenge, play the game and let me know your experience. How well did you do?

Full study: Is it Real or Randomized? A Financial Turing Test (pdf)