Financial Calculations, Personal

Overview

Unlike calculus, geometry, and many other types of math, basic financial calculations can be performed by almost anyone. These simple financial equations address practical questions such as how to get the most music for the money, where to invest for retirement, and how to avoid bouncing a check. Best of all, the math is real life and simple enough that anyone with a calculator can do it.

Fundamental Mathematical Concepts and Terms

Financial math covers a wide range of topics, broken into three major sections: Spending decisions deals with choices such as how to choose a car, how to load an MP3 player for the least amount of cash, and how to use credit cards without getting taken to the bank; Financial toolbox looks at the basics of using a budget, explains how income taxes work, and walks through the process of balancing a checkbook; Investing introduces the essentials of how to invest successfully, as well as sharing the bottom line on what it takes to retire as a millionaire (almost anyone can do it).

Real-life Applications

BUYING MUSIC

Today's music lover has more choices than ever before. Faced with hundreds of portable players, a dozen file formats, and millions of songs available for instant download, the choices can become a bit overwhelming. These choices do not just impact what people listen to, they can also impact the buyer's finances for years to come. Additionally, in many cases, comparing the different offers can be difficult.

One well-known music service ran commercials during the 2005 Super Bowl, urging music buyers to simply "Do the math" and touting its offer as an unparalleled bargain. The reasoning is that the top-selling music player in 2005 held up to 10,000 songs and allowed users to download songs for about a dollar apiece; buying that player along with 10,000 songs to fill it up would cost around $10,000. But the music service's ad offered a seemingly better deal: unlimited music downloads for just $14.95 per month. While this deal sounds much better, a little math is needed to uncover the real answer.

A good starting point is calculating the "break-even" point: how many monthly payments do we make before we actually spend the same $10,000 charged by the other firm. This calculation is simple: divide the $10,000 total by the $14.95 monthly fee to find out how many months it takes to spend $10,000. Not surprisingly, it takes quite a few: 668.9 months, to be exact, or about 56 years, which is the break-even point. This result means that if we plan to listen to our downloaded songs for fewer than 56 years, we will spend less with the monthly payment plan. For example, if we plan to use the music for 20 years, we will spend less than $3,600 during that time (20 years × $14.95 per month), a significant savings when compared to $10,000.

One question raised by this ad is, "How many songs does a typical listener really own?" Assuming the user actually does download 10,000 songs, the previous analysis is correct. But 10,000 songs may not be very realistic; in order to listen to all 10,000 songs just one time, a person would have to listen to music eight hours a day for two full months. In fact, most listeners actually listen to playlists much shorter than 10,000 tracks. So if a listener doesn't want all 10,000 tunes, is the $14.95 per month still the better buy?

Again, the calculations are fairly simple. Let's assume we want to listen to music four hours per day, seven days per week, with no repeats each week. By multiplying the hours times the days, we find that we need 28 hours of music. If a typical song is 3 minutes long, then we divide 60 minutes by 3 minutes to find that we need 20 songs per hour, and by multiplying 20 songs by the 28 hours we need to fill, we find that we need 560 songs to fill our musical week without any repeats. Using these new numbers, the break-even calculation lets us ask the original question again: how long, at $14.95 per month, will it take us to break-even compared to the cost of 560 songs purchased outright? In this case, we divide the $560 we spend to buy the music by the $14.95 monthly cost, and we come up with 37.5 months, or just over three years. In other words, at the end of three years, those low monthly payments have actually equaled the cost of buying the songs to start with, and as we move into the fourth and fifth year, the monthly payments begin to cost us more. Plus, for users whose music library includes only 200 or 300 songs, the break-even time becomes even shorter, making the decision even less obvious than before.

Several other important questions also impact the decision, including, "What happens to downloaded music if we miss a monthly payment?" Since subscription services typically require an ongoing membership in order to download and play music, their music files are designed to quit playing if a user quits paying. The result is generally a music player full of unplayable files. A second consideration is the wide array of file formats currently in

use. Some services dictate a specific brand of player hardware, while others work with multiple brands. Most users feel that the freedom to use multiple brands offers them better protection for their musical investment. Since some players will play songs stored in multiple formats, they offer users the potential to shop around for the best price at various online stores. A final question deals with musical taste and habits. For listeners whose libraries are small, or who expect their musical tastes to remain fairly constant, buying tracks outright is probably less expensive. For listeners who demand an enormous library full of the latest hits and who enjoy collecting music as a hobby, or for those whose music tastes change frequently, a subscription plan may provide greater value.

In the end, this decision is actually similar to other financial choices involving the question of whether to rent or buy (see sidebar "Rent or Buy?"), since the monthly subscription plan is somewhat like renting music. Math provides the tools to help users make the right choice.

CREDIT CARDS

Although the average American already carries eight credit cards, offers arrive in the mail almost every week encouraging us to apply for and use additional cards. Why are banks so eager to issue additional credit cards to consumers who already have them? Answering this question requires an examination of how credit cards work.

In its simplest possible form, a credit card agreement allows consumers to quickly and easily borrow money for daily purchases. Typically, we swipe our card at the store, sign the charge slip or screen, and leave with our goods. At this point in the process, we have our merchandise, paid for with a "loan" from the credit card issuer. The store has its money, less the fee it paid to the credit card company, and the credit card has paid our bill in exchange for a 2–3% fee and for a promise of payment in full at a later date. At the end of the month, we will receive a statement, pay the entire credit card bill on time to avoid interest or late charges, and this simplest type of transaction will be complete.

If this transaction were the norm, very few companies would enter the credit card business, as the 2–3% transaction fees would not offset their overhead costs. In reality, a minority of consumers actually pay their entire bills at the end of the month, and any unpaid balances begins accruing interest for the credit card issuer. These interest charges are where credit card companies actually earn their profits, as they are, in effect, making loans to thousands of consumers at rates that typically run from 9–14% for the very best customers, from 16–21% for average borrowers, and in the case of customers with poor credit histories, even higher rates. Countless individuals who would never consider financing a car loan or home mortgage at an interest of 16% routinely borrow at this and higher rates by charging various monthly expenses on credit cards, and consequently carrying a balance on their bill.

The average American household with at least one credit card in 2004 carried a credit card balance of $8,400 and as a result paid lenders more than __BODY__,000 in interest and finance charges alone, making the credit card business the most profitable segment of the banking industry today. This fact alone answers the original question of why so many credit cards are issued each year: because they are highly profitable to the lenders. Card issuers mailed out three billion credit card offers in 2004 (an average of ten invitations for every man, woman, and child in the United States) because they know their math: half of all credit card users carry a balance and pay interest, so the more new cards the lenders issue, the greater their profits will be.

Loaning money in exchange for interest is an ancient practice, discussed in numerous historical documents, including the Jewish Torah and the Muslim Koran, which both discuss the practice of usury, or charging exorbitantly high interest rates. Modern U.S. law restricts excessive interest charges, and most states have usury laws on their books that limit the rate that an individual may charge another individual. These rates vary widely from state to state; as of 2005, the usury rate, defined as the highest simple interest rate one individual may legally charge another for a loan, is 9% in the state of Illinois. In contrast, Florida's rate is 18%, Colorado's rate is 45%, and Indiana has no stated usury rate at all. Ironically, these laws do not apply to entities such as pawn brokers, small loan companies, or auto finance companies, explaining why these firms frequently charge rates far in excess of the legal maximums for individuals. Credit card issuers, in particular, have long been allowed to charge interest rates above state limits, making them typically one of the most expensive avenues for consumer borrowing.

How much does it really cost to use credit cards for purchases? The answer depends on several factors, including how much is paid each month and what interest rate is being charged. For this example, we'll assume a credit card purchase of $400, an interest rate of 17%, and a minimum monthly payment of $10. After the purchase and making six months of minimum payments, the buyer has paid $60 (six months × $10 per month). But because more than half that amount, $33.06 has gone to pay the 17% interest, only $26.94 has been paid on the original $400 purchase. At this point, even though the buyer has paid out $60 of the original bill, in reality $373.06 is still owed ($400–$26.94).

This pattern will continue until the original purchase is completely paid off, including interest. If the buyer continues making only the required $10 monthly payment, it will take five full years, or 60 payments, to retire the original debt. Over the course of those five years, the buyer will pay a total of $194 in interest, swelling the total purchase price from $400 to almost $600. And if the item originally purchased was an airline ticket, a vacation, or a trendy piece of clothing, the buyer will still be paying for the item long after it's been used up and forgotten. While many factors influence the final cost of saying "charge it," a simple rule of thumb is this: Buyers who pay off their charges over the longest time allowed can expect to pay about 50% more in total cost when putting a purchase on the credit card, pushing a $10 meal to an actual cost of $15. Similarly, a $200 dress will actually cost $300, and a __BODY__,000 trip will actually consume __BODY__,500 in payments.

Credit cards are valuable financial tools for dealing with emergencies, safely carrying money while traveling, and in situations such as renting a car when required to do business. They can also be extremely convenient to use, and in most cases are free of fees for those customers who pay their balance in full each month. Only by doing the math and knowing one's personal spending habits can one know if credit cards are simply a convenient financial tool, or a potential financial time bomb.

CAR PURCHASING AND PAYMENTS

For most consumers, an automobile represents the second largest purchase they will ever make, which makes understanding the car buying process critically important. Several important questions should be considered before buying a new car. First, a potential buyer should calculate how much he can spend. Most experts recommend keeping car payments below 20% of take-home pay, so if a worker receives a check for $2,000 each month (after taxes and other withholding), then he should plan to keep his car payments below $400 (20% × $2,000). This figure is for all car payments, so if he already has a $150 payment for another car, he will be shopping in the $250 per month payment range.

Using this $250 monthly payment, the buyer can consult any of several online payment calculators to determine how much he can spend. For example, if the buyer is willing to spend five years (60 months) paying off his vehicle, this might mean he could afford to borrow about $13,000 for a vehicle (this number varies depending on the actual interest rate at the time of the loan). However this value must pay not just for the car, but also for additional fees such as sales tax, license fees, and registration, which vary from state to state and which can easily add hundreds or thousands of dollars to the price of a new vehicle. For this example, we will estimate sales tax at 6%, license fees at $200, and registration at $100; so a car priced at $12,000 will wind up costing a total of $13,020 (12,000 + .06 × 12,000 + $200 + $100), which is right at the target value of $13,000.

The second aspect of the buying equation is the down payment. A down payment is money paid at the time of sale, and reduces the amount that must be borrowed and financed. In the case of the previous example, a down payment of $2,000 would mean that instead of shopping in the $12,000 price range, the buyer could now shop with $14,000 as the top price.

Many buyers have a used car to sell when they are buying a new vehicle, and in many cases they sell this car to the dealer at the same time, a process known as "trading-in." A trade-in involves the dealer buying a car from the customer, usually at a wholesale price, with the intent to resell it later. A trade-in is a completely separate transaction from the car purchase itself, although dealers often try to bundle the two together. Here again, securing information such as the car's fair trade value will allow the savvy customer to receive a fair price for the trade.

Many consumers find the car-buying experience frustrating, and they worry that they are being taken advantage of. Automobile dealerships are among the only places in the United States where every piece of merchandise has a price tag clearly attached, but both the seller and the buyer know the price on the tag means very little. Most cars today are sold at a significant discount, meaning that a sticker price of $20,000 could easily translate to an actual sales price of $18,000. Incentives, commonly in the form of rebates (money paid back to the buyer by the manufacturer), can chop another $2,000-$5,000 off the actual price, depending on the model and how late in the season one shops. While dealers are willing to negotiate and offer lower prices when they must, they are also going to try to sell at a higher price whenever possible, which places the burden on the buyer to do the homework before shopping. Numerous websites and printed manuals provide actual dealer costs for every vehicle sold in the United States, as well as advice on how much to offer and when to walk away.

CHOOSING A WIRELESS PLAN

Comparing cellular service plans has become an annual ritual for most consumers, as they wrestle with whether to stay with their current cell phone and provider or make the jump to a new company. Beyond the questions of which service offers the best coverage area and which phone is the most futuristic-looking, some basic calculations can help determine the best value for the money.

There are normally three segments to wireless plans. The first segment consists of a set quantity of included minutes that can be used without incurring additional charges. These are typically described as "anytime" minutes, and are the most valuable minutes because they can be used during daytime hours. These minutes are typically offered on a use-it-or-lose-it basis, meaning that if a plan includes 400 minutes and the customer uses only 150, the other 250 minutes are simply lost. Some plans now offer rollover minutes, which means that in the previous example, the 250 minutes would roll to the next month and add to that month's original 400 minutes, providing a total of 650 minutes that could be used without additional charges.

Another segment is that many wireless plans include large blocks of so-called free time, during which calls can be made without using any of the plan's included minutes. These free periods are usually offered during times when the phone network is lightly used, such as late at night and on weekends when most businesses are closed. Users may talk non-stop during these free periods without paying any additional fees.

The third major component of a wireless plan is its treatment of any additional minutes used during non-free periods. In many cases, these additional minutes are billed at fairly high rates, and using additional minutes past those included in a plan's base contract can potentially double or triple the monthly bill.

Other features are sometimes offered, including perks such as free long-distance calling, premium features such as caller identification, and free voicemail. In other cases, providers allow free calls between their own members as part of so-called affinity plans. Cellular plans are typically sold in one- or two-year contracts.

Choosing a wireless plan can be challenging, since there are so many options, and choosing the wrong plan can be a costly choice. A few guidelines can help simplify this choice. First, users should estimate how many minutes will be needed during non-free periods, and then add 10–15% to this estimate in order to provide a margin of error. Next, users can consider whether an affinity plan or free long distance can impact their choices; in cases where most calls are made between family members, plans with these features can offer significant savings.

Finally, users can compare options among the several providers, paying careful attention to coverage areas. For most users, saving a few dollars per month by choosing a carrier with less coverage winds up being an unsatisfying choice. In addition, users should carefully weigh whether to sign a two-year contract, which may offer lower rates, or a one-year plan. One-year plans provide the most flexibility, since rates generally fall over time and a shorter contract allows one to reevaluate alternative plans more often. In addition, wireless providers are now required to let customers keep their cell numbers when they change providers (a feature called "portability"), simplifying the change-over process.

For users needing very few minutes each month, or those on extremely tight budgets, pay-as-you-go plans offer a thrifty alternative. These plans do not normally include free phones or bundles of minutes; instead, a user recharges the account by buying minutes in credit card form at a convenience store or similar outlet. For users who talk 30 minutes or less each month, these plans can be ideal.

When purchasing a wireless plan, add-ons will inevitably increase the final cost. A plan advertised at $39.95 per month will typically generate bills of $43.00 or more when all the taxes and fees are added in, so plan accordingly.

BUDGETS

Personal budgets fill two needs. First, they measure or report, allowing people to assess how much they are spending and what they are spending on. Second, budgets forecast or predict, allowing people to evaluate where their finances are headed and make changes, if necessary. A budget is much like an annual checkup for finances, and can be simple or complex. The simplest budget consists of two columns, labeled "In" and "Out."

The first step in the budgeting process consists of filling the in column with all sources of income, including wages, bonuses, interest, and miscellaneous income. In the case of income that is received more frequently, such as weekly paychecks, or less frequently, such as a quarterly bonus, one must convert the income to a monthly basis for budget purposes, with quarterly items being divided by three and weekly items being multiplied by four. In the case of semiannual items, such as auto insurance premiums, the amount is divided by six.

Next, in the out column, all identifiable outflows should be listed, such as mortgage/rent payments, utilities (electricity, gas, water), car payments and gasoline, interest expense (i.e., credit card charges), health care, charitable donations, groceries, and eating out. The details of this list will vary from person to person, but an effort should be made to include all expenditures, with particular attention paid to seemingly small purchases, such as soft drinks and snacks, cigarettes, and small items bought with cash. For accuracy, any purchase costing over __BODY__ should be included.

The third step is to add up each column, and find the difference between them; in simplest terms, if the out column is larger than the in column, more money is flowing out than in, the budget is out of balance and the family's financial reserves are being depleted. If more money is flowing in than out, the family's budget is working, and attention should be paid to maintaining this state.

The fourth step in this process is evaluating each of the specific spending categories to determine whether it is consuming a reasonable proportion of the spendable income. For instance, each individual category can be divided by the total to determine the percentage spent; a family spending $700 of their monthly $2,000 on car payments, gas, and insurance should probably conclude that this expenditure (700/2000 = 35%) is excessive and needs to be adjusted. In many cases, families creating a first-time budget find that they are spending far more than they realized at restaurants, and that by cooking more of their own meals they can almost painlessly reduce their monthly deficits.

The previous four steps of this process ask "What is being spent?" The fifth and final step asks, "What should be spent?" or "What is the spending goal?" At a minimum, efforts should be made to bring the entire budget into balance by adjusting specific categories of spending. Ideally, goals can be set for each category and reevaluated at the end of each month. A budget provides a simple, inexpensive tool to begin taking control of one's personal finances. W. Edwards Deming, the genius who transformed the Japanese from makers of cheap trinkets into the worldwide experts on quality manufacturing, is often paraphrased as saying, "You can't change what you can't measure." A simple three-column budget provides the basic measurement tool to begin measuring one's financial health and changing one's financial future.

UNDERSTANDING INCOME TAXES

The United States Treasury Department collects around __BODY__ trillion in individual income taxes each year from U.S. workers, most of it subtracted from paychecks. While income tax software has taken much of the agony out of tax preparation each April, most workers still have to interact with the Internal Revenue Service, or IRS, from time to time, especially in the area of filling out tax paperwork.

Employers are required by law to withhold money from employee paychecks to pay income taxes. But because each person's tax situation is different, the IRS has a specific form designed to tell employers how much to withhold from each employee. This form, the W-4, asks taxpayers a series of questions, such as how many children they have and whether they expect to file specific tax forms or not. By supplying this form to new employees, companies can ensure that they withhold the proper amount from each paycheck, as well as protect employees from penalties that apply if they do not have enough of their taxes withheld. In cases where family information changes, or where the previous year's withholding amount was too high or too low, a new form can be filed with the employer at any time during the year.

At the end of the calendar year, employers issue a report to each employee called a W-2. Form W-2 is a summary of an employee's earnings for the entire year, including the total amount earned, or gross pay and, amounts withheld for income tax, social security, unemployment insurance, and other deductions. The information from the W-2 is used by the employee when filing federal and state income returns each year. W-2 forms are required to be mailed to employees by January 31; if a W-2 is not received by the first week in February, the employee should contact the employer.

Other forms are used to report other types of income. The 1099 form is similar to W-2s and is sent to individuals who received various types of non-wage income during the year. For example, form 1099-INT is used by banks to provide account holders with a record of interest earned, form 1099-DIV is used to report dividend income, and form 1099-MISC is used to report monetary winnings such as contest prizes, as well as other types of miscellaneous income. These forms should not be discarded, as the amounts on them are reported to the IRS, which matches these reported amounts with individual tax returns to make sure the income was reported and taxes were paid on it. Failure to report income and payroll taxes could lead to penalties and the possibility of a tax audit, in which the taxpayer is required to document all aspects of the tax return to an IRS official.

BALANCING A CHECKBOOK

Balancing a checkbook is an important chore that few people enjoy. A correctly balanced checkbook provides several distinct benefits, including the knowledge of where one's money is being spent, and the avoidance of embarrassing and costly bounced checks. A balanced account also allows one to catch any mistakes, made either by the bank or by the individual, before they create other problems. Balancing a checkbook is actually quite simple and can usually be accomplished in less than half an hour. Whether one uses software or the traditional paper- and-pencil method, the general approach is the same.

Balancing a checkbook begins with good recordkeeping, which means correctly writing down each transaction, including every paper check written, deposit made, ATM withdrawal taken, or check-card purchase made. Bad recordkeeping is a major cause of checkbook balancing problems.

Determining whether all of one's transactions have cleared the checking account is described as the process of a paper check winding its way through the financial system from the merchant to the bank, which can take several days. It also refers to deposits or withdrawals made after the statement date. The net effect of clearing delays is that most consumers will have records of transactions that are not in the latest bank statement, meaning this statement balance may appear either too high or too low. Determining whether all items have cleared involves a review of the records collected in the previous step. A checkmark is placed next to the item on the bank statement for each check, ATM receipt, or other record. Once this process is complete, and assuming good records have been kept, all the items in the bank statement will be checked, and several items that were not in the statement at all will remain. The process of adjusting for these uncleared items is called reconciling the statement.

To reconcile a check register with the bank statement, all the uncleared items must be accounted for, since these transactions appear in the personal check register but not in the statement. Specifically, deposits and other uncleared additions to the account must be subtracted, while withdrawals, check-card transactions, written checks, and other uncleared subtractions from the account must be added back in. The net effect of this process is to back the records up to the date of the bank statement, at which time the two totals, the check register and the bank statement, should match. Many banks include a simple form on the back of the printed bank statement to simplify this process.

For most customers, a day will arrive when the account simply does not balance. Since bank errors are fairly rare, the most common explanation is an error by the customer. A few simple steps to take include scanning for items entered twice, or not entered at all; data entry errors, such as a withdrawal mistakenly entered as a deposit; simple math errors; and forgetting to subtract monthly service charges or fees. Most balancing errors fall into one of these categories, and as before, good recordkeeping will simplify the process of locating the mistake.

Balancing a checkbook is not difficult. The time invested in this simple exercise can often pay for itself in avoided embarrassment and expense.

SOCIAL SECURITY SYSTEM

The Social Security system was established by President Franklin Roosevelt in 1935, creating a national system to provide retirement income to American workers and to insure that they have adequate income to meet basic living expenses. Due largely to this program, nine in ten American senior citizens now live above the official poverty line.

But a Social Security number is important long before one retires. Because the United States does not have an official, government-issued identification program, Social Security numbers are frequently used as personal identification numbers by universities, employers, and banks. U.S. firms are also required by law to verify an applicant's Social Security number as part of the hiring process, making a Social Security card a necessity for anyone wanting to work. For this reason, most Americans apply for and receive a Social Security number and card while they are still minors.

Social Security numbers and cards are issued free of charge at all Social Security Administration offices. An applicant must present documents such as a birth certificate, passport, or school identification card in order to verify the person's identity. After these documents are verified, a number will be issued. A standard Social Security number is composed of three groups of digits, separated by dashes, such as 123–45–6789, and always contains a total of nine digits. Each person's number is unique, and in some cases, the first three digits may indicate the region in which the card was issued. The simplest way for a child to receive a Social Security number is for the parents to apply at birth, at the same time they apply for a birth certificate. After age 12, a child applying for a card, in addition to providing documentation of age and citizenship, must also complete an in-person interview to explain why no card has been previously issued.

When a person begins working, the employer withholds part of the worker's earnings to be deposited into the Social Security system; as of 2005, these contributions are taken out of the first $90,000 in earned income each year at a rate of 7.65%. Starting at age 25, each worker receives an annual statement listing their income for the previous year; this information should be carefully checked for accuracy. While taking one's Social Security card to job interviews or loan applications is a good idea, the Social Security Administration recommends that cards be kept in a safe place, rather than carried on one's person. In the event that a Social Security card is lost or stolen, a new card can be requested at no charge by completing the proper form and submitting verification of identity. The new card will have the same number on it as the old card. In the case of a name change due to marriage, divorce, or similar events, a new card can be issued with the same number and the cardholder's new name. This process requires documentation showing both the previous name and the new name.

The Social Security system remains the largest single retirement plan in the country, is mandatory for most workers, and is expected to remain in place for the fore-seeable future.

INVESTING

Investing simply means applying money in such a way that it grows, or increases, over time. In a certain sense, investing is somewhat like renting money to someone else, and in return, receiving a rental fee for the privilege. Investments come in an almost endless variety of forms, including stocks, bonds, real estate, commodities, precious metals, and treasuries. While this array of options may seem bewildering at first, all investment decisions are ultimately governed by a simple principle: "risk equals reward."

Risk is the potential for loss in any investment. The least risky investments are generally government-backed investments, such as Treasury bills and Treasury bonds issued by the United States government. These investments are considered extremely safe because they are backed by the U.S. Treasury and, barring the collapse of the government, will absolutely be repaid. For this reason, these investments are sometimes described as riskless. At the other end of the risk spectrum might be an investment in a company that is already bankrupt and is trying to pull itself out of insolvency. Because the risk of losing one's investment in such a firm is extremely high, this type of investment is often referred to as a junk bond, since its potential for loss is high. Between riskless and highly risky investments are a variety of other options that provide various levels of risk. Risk is generally considered higher when money is invested for longer periods of time, so short-term investments are inherently less risky than long-term ones.

Reward is the return investors hope to receive in exchange for the use of their money. Most investors are only willing to lend their money to someone for something in return. Investors who buy a rare coin or a piece of real estate are hoping that the value of the coin or house will rise, so they can reap a reward when they sell it. Likewise, investors who buy shares of a company's stock is betting that the company will make money, which it will then pass along to them as a dividend. Investors also hope that as the company grows, other investors will see its value and the stock price itself will rise, allowing them to profit a second time when they sell the stock. Investment rewards take many different forms, but financial returns are the main incentive for people to invest.

The principle "risk equals reward" states that investments with higher levels of risk will normally offer higher returns, while safer (less risky) investments will normally return smaller rewards. For this reason, the very safest investments pay very low rates. An insured deposit in a savings account at a typical U.S. bank earns about 1–2% per year, since these funds are insured and can be withdrawn at any time. Other safe investments, such as U.S. Treasury bills and U.S. savings bonds, pay low interest rates, typically 3–4% for a one-year investment.

Corporate bonds and stocks are two tools that allow public corporations to raise money. Bonds are considered a less risky investment than stocks, and hence pay lower returns, generally a few percentage points higher than Treasury bills. Historically, stocks in U.S. firms have returned an average of 9–10% per year over the long-term. However, this average return conceals considerable volatility, or swings, in value. This volatility means in a given year the stock market might rise by 30-40%, decline by the same amount, or experience little or no change. This variation in annual rates of return is one reason stocks are considered more risky than Treasuries, and hence pay a higher rate of return. Most financial experts recommend that those investing for periods longer than ten years place most of their funds in a variety of different kinds of stocks.

Among the riskiest investments are stock options and commodity futures. Because these types of investments are complex and can potentially lead to the loss of one's entire investment, they are generally appropriate only for experienced, professional investors. Other investments, such as rental real estate, can offer substantial returns in exchange for additional work required to maintain, repair, and manage the property.

A few tricks can help young investors take advantage of certain laws to invest their money. Because the government taxes most forms of income, any investment vehicle that allows the investor to defer (delay) paying taxes will generally produce higher returns with no increase in risk. As an example, consider a worker who begins investing $3,000 per year in a retirement account at age 29. If the worker deposits this money in a normal, taxable savings account or investment fund, each year he will have to pay income tax on the earnings, meaning that his net return will be lower. But if this same amount of money is invested in a tax-sheltered account, the money can grow tax-free, meaning the income each year is higher. Over the course of a career, this difference can become enormous. In this example, the worker's contributions to the taxable account will grow to $450,000 by age 65. But in a tax-sheltered account, those very same contributions would swell to more than $770,000, a 70% advantage gained simply by avoiding tax payments on each year's earnings.

One of the simplest ways to begin a tax-deferred retirement plan is with a Roth Individual Retirement Account (IRA). Available at most banks and investment firms, Roth accounts allow any person with income to open an account and begin saving tax-free. Beginning in 2005, the maximum annual contribution to a Roth IRA is $4,000, which will increase again in 2008 to $5,000. One notable feature of IRAs is the hefty 10% penalty paid on withdrawals made before retirement. While this may seem like a disadvantage, this penalty provides strong incentive to keep retirement funds invested, rather than withdrawing them for current needs.

Another outstanding investment option is a 401(k) plan, offered by many large employers under a variety of names. These plans not only allow earnings to grow tax-deferred like an IRA, they offer other advantages as well. For instance, most firms will automatically withdraw 401(k) contributions from an employee's paycheck, meaning he doesn't have to make the decision each month whether to invest or not. Also, some companies offer to match employee contributions with additional contributions. In a case where a company offers a 1:1 match on the first $2,000 an employee saves, the employee's $2,000 immediately becomes $4,000, equal to a 100% return on the investment the first year, with no added risk. In the case of a 50% match on the first $3,000, the firm would contribute __BODY__,500. Company matches are among the best deals available and should always be taken advantage of.

Investing is a complex subject, and investing in an unfamiliar area is a chance for losses. By choosing a variety of investments, most investors can generate good returns without exposing themselves to excessive risk. And by taking time to learn more about investment options, most investors can increase their returns without unduly increasing their risk.

RETIRING COMFORTABLY BY INVESTING WISELY

Who wants to be a millionaire? More importantly, what chance does an average 18-year-old person have of actually reaching that lofty plateau? Surprisingly, almost anyone who sets that as a goal and makes a few smart choices and exercises self-discipline along the way can fully expect to be a millionaire by the time he retires. In fact, there are so many millionaires in the United States today that most people already know one or two, even though they are tough to pick out since few of them fit the common stereotype (see sidebar: Millionaire Myths).

Is a million dollars enough to retire comfortably on? Most people would scoff at the question, but the answer may not be as obvious as it first seems. Most members of the World War II generation clearly remember an era of $5,000 houses, $500 cars, and 5-cent soft drinks. What they may not recall so clearly is that in 1951, the average American worker earned only $56.00 per week, meaning that while prices are much higher today, wages have risen substantially as well.

This gradual rise in prices (and the corresponding fall in the purchasing power of a dollar) is called inflation. When inflation is low, and prices and wages increase 3–4% per year, most economists feel the economy is growing at a healthy pace. When inflation reaches higher levels, such as the double-digit rates experienced in the late 1970s, the national economy begins to collapse. And in rare situations, a disastrous phenomena known as hyperinflation takes over. In 1922, Germany experienced an inflation rate of 5,000%. This staggering rate meant that in a two-year period, a fortune of 20 billion German marks would have been reduced in value to the equivalent of one mark. One anecdotal account of hyperinflation in Germany tells of individuals buying a bottle of wine in the expectation that the following day the empty bottle could be sold for more than the full bottle originally cost. Hyperinflation has occurred more recently as well: Peru, Brazil, and Ukraine all experienced hyperinflation during the 1990s; with prices rising quickly, sometimes several times each day, workers began demanding payment daily so they could rush out and spend their earnings before the money lost much of its value.

While hyperinflation can destroy a nation's economy, it is a rare event. A far more realistic concern for workers intent on retiring comfortably is the slow but steady erosion of their money's value by inflation. In the same way that the 5-cent sodas of the 1950s now cost more than a dollar, an increase of twenty-fold, one must assume that the one-dollar sodas of today may well cost $20 by the middle of the twenty-first century. And as costs continue to climb, the value of a dollar, or a million dollars, will correspondingly fall.

The million dollar question (will a million dollars be enough?) can be answered fairly simply using a mathematical approach and several steps. The first question: how much money will be needed in 50 years to equal the value of __BODY__ million today? The first step of this process is determining how much buying power __BODY__ million loses in one year. If the rate of inflation is 3%, a reasonable guess, then over the course of one year __BODY__ million is reduced in buying power by 3%. At the end of the first year, it has buying power equal to __BODY__,000,000 × 97%, or $970,000. This is still a fantastic sum of money to most people, but the true impact of inflation is not felt in the first year, but in the last.

These calculations could continue indefinitely, multiplying $970,000 × 97% to get the value at the end of the second year, and so forth. If this were done for 50 years, we could eventually produce an inflation "multiplier," a single value by which we multiply our starting value to find the predicted future buying power of that sum. In this example, the inflation multiplier is .22, which we multiply by our starting sum of __BODY__ million to find that at retirement in 50 years the nest-egg will have the buying power of only $220,000 today. And while $220,000 is a nice sum of money, it may not be enough to support a comfortable retirement for very many years.

This raises another obvious question: how much will it take in 50 years to retain the buying power of __BODY__ million today? This calculation is basically the inverse of the previous one. To determine how much is required one year hence to have the buying power of __BODY__ million today, we simply multiply by 1.03 (based on our 3% inflation assumption), giving a need next year for __BODY__,030,000. Again, we can carry this out for 50 years and produce a multiplier value, which in this case turns out to be 4.5. We then multiply that value times the base of __BODY__ million to learn that in order to have the buying power of __BODY__ million today will require one to have accumulated more than $4 million by retirement.

In summary, the answer to the question is simple: If a retirement fund of $220,000 would be adequate for today, then __BODY__ million will be adequate in 50 years. But if it would take __BODY__ million to meet one's retirement needs today, the goal will need to be quite a bit higher, since today's college students will likely retire in an era when a bottle of drinking water will set them back $20.

This example requires that we picture our bank account as a swimming pool and the money we save as water. The goal is to fill the pool completely by the time of retirement. Because the pool begins completely empty, the task may seem daunting. But like most challenging goals, this one can be achieved with the right approach.

In order to fill the pool, one must attach a pipe that allow water to flow in, and the first decision relates to the size of this pipe, since the larger the pipe, the more water it can carry and the faster the pool will fill. The size of the pipe equates to income level, or for this illustration, the total amount we expect to earn over an entire career. This first decision may be the single most important choice one makes on the road to millionaire status, since this first decision will largely determine the size of the pipe and the size of one's income.

Educational level and income are highly correlated, and not surprisingly, less education generally equates to less income. A report by the U.S. Census Office provides the details to support this claim, finding that students who leave high school before completion can expect to earn about __BODY__ million over their careers. While this sounds like a hefty amount, it is far below what most families need to live, and almost certainly not enough to amass a million dollars in retirement savings. Just for comparison, this value equates to annual earnings of less than $24,000 per year. In our current illustration, this equates to a tiny pipe, and means the swimming pool will probably wind up empty.

The good news from the report is that each step along the educational path makes the pipe a little larger, and fills the pool a little faster. For high school students who stay enrolled until graduation, lifetime earnings climb by 20%, to __BODY__.2 million, meaning that a high school junior who chooses to finish school rather than dropping out will earn almost a quarter of a million dollars for his or her efforts. And with each diploma comes additional earning power. An associate's degree raises average lifetime earnings to __BODY__.5 million, while a bachelor's degree pushes average lifetime earnings to $2.1 million, more than double the amount earned by the high school dropout. Master's degrees, doctorates, and professional degrees such as law and medical degrees each raise expected earnings as well, increasing the size of the pipe and filling the pool faster. Simple logic dictates that when the pipe is two to four times as large, the pool will fill far more quickly. For this reason, one of the best ways to predict an individual's retirement income level is simply to ask, "How long did you stay in school?"

Retirement savings are impacted by income level in multiple ways. First, since every household has to pay for basics such as food, housing, clothing, and transportation, total income level determines how much is left over after these expenses are paid each month, and therefore how much is available to be invested for retirement. Second, as detailed in the Social Security system section, Social Security pays retirement wages based on one's earnings while working, so those who earn more during their career will also receive larger Social Security payments after retirement. Third, employers frequently contribute to retirement plans for their workers, and the level of these contributions is also tied directly to how much the worker earns, with higher earnings equating to higher contributions and greater retirement income. Because each of these pieces of the retirement puzzle is tied to income level, each one adds to the size of the pipe, and helps fill the pool more quickly. Again, education is a primary predictor of income level.

Of course a few people do manage to strike it rich in Las Vegas or win the state lottery, which is roughly equivalent to backing a tanker truck full of water up to the pool and dumping it in. For these few people, the size of the income pipe turns out to be fairly unimportant, since they have beaten some of the longest odds around. To get some idea just how unlikely one is to actually win a lottery, consider other possibilities. For example, most people don't worry about being struck by lightning, and this is reasonable, since a person's odds of being struck by lightning in an entire lifetime are about one in 3,000, meaning that on average if he lived 3,000 lifetimes, he would probably be struck only once. And even though shark attacks make the news virtually every year, the odds of being attacked by a shark are even lower, around one in 12,000.

Since most people fully expect to live their entire lives without being attacked by a shark or being struck by lightning, it seems far-fetched that many would play the lottery each week, given that the odds of winning are astronomically worse. As an example, the Irish Lotto game, which offers some of the best odds of any national lottery on the planet, gives buyers a 1-in-5 million chance of winning, meaning a player is 1,600 times more likely to be struck by lightning than to win the jackpot. And the U.S. Power Ball game offers larger jackpots, but even lower odds of winning: a player in this game is 16 times less likely to win than in the Irish Lotto, meaning the average Power Ball player should expect to be struck by lightning 26,000 times as often as he wins the jackpot. Of all the unlikely events that might occur, winning the lottery is among the most unlikely.

Once the pipe is turned on, which means we have begun making money, one may find the pool filling too slowly, which means assets and savings are accumulating too slowly. At this point it becomes necessary to notice that the pool includes numerous drains in the floor, some large and others small. Water is continually flowing out these drains, which represent financial obligations such as utility bills, tuition payments, mortgages, and grocery costs. In some cases, the water may flow out faster than the pipe can pump it in, causing the water level to drop until the pool runs dry, meaning the employee runs out of money, and bankruptcy follows. In most families, the inflow and outflow of money roughly balance each other, and each month's bills are paid with a few dollars left, but the pool never really fills up. In either case, retirement will arrive with little or nothing saved, and retirement survival will depend largely on the generosity of the Social Security system.

A more pleasant alternative involves closing some of the drains in the pool, or reducing some expenditures. For most families, the largest drains in the pool will be monthly items such as mortgage and car loan payments that are set for periods of several years and may not be easily changed over the short-run. For these items, decisions can only be made periodically, such as when a new car or home is purchased.

However, some seemingly small items may create huge drains in the family financial pool. For most families, eating out consumes a majority of the food budget, even though eating at home is typically both cheaper. Numerous small bills such as cable, wireless, and internet access can add up to take quite a drink out of the pool, even though each one by itself seems small. Yet, while the total dollar value of such items may seem insignificant, their impact over time can be enormous. By removing just $50 from consumption and investing it at 8% each month during the 50 years of a career, this trivial amount will grow to almost $350,000. These types of choices are among the most difficult to make, but can be among the most significant, especially considering that $50 per month represents what many Americans spend on soft drinks or gourmet coffee. A good rule of thumb for this calculation is to multiply the monthly contribution times 7,000 to find its future value at retirement, assuming one begins at age 20 and retires at age 70.

The other major factor in retiring comfortably is time. To put it simply, the final value of one dollar invested at age 20 will be greater than the final value of four dollars invested at age 50. This means that $10,000 invested at age 20 will grow to $143,000 by age 75, while $40,000 invested at age 50 will be worth only $134,000 at the same time. In fact, a good general rule of thumb is for each eight years that pass, the final value of the retirement nest egg will be reduced by 50%. It is never too early to start saving for retirement.

Millionaire Myths

Say the word "millionaire," and most Americans picture Donald Trump, fully decked out in expensive designer suits and heavy gold jewelry. To most Americans, yachts, mansions, lavish vacations, and fine wines are the sure signs that a person has made it big and has accumulated a seven-figure net worth. But recent research paints a very different picture: most millionaires live fairly frugal lives and tend to prefer saving over spending, even after they've made it big. In fact, the most surprising fact about real millionaires is this: they don't look or act at all like TV millionaires.

The average millionaire in the United States today buys clothes at J.C. Penney's, drives an American made car (or a pickup), and has never spent more than $250 on a wristwatch. He or she inherited little or nothing from parents and has built the fortune in such industries as rice farming, welding contracting, or carpet cleaning. This person is frugal, remains married to the first spouse, has been to college (but frequently was not an A student), and lives in a modest house bought 20 years ago.

In short, while most millionaires are gifted with vision and foresight, there is little they have done that cannot be duplicated by any hard-working, dedicated young person today. The basic principles of accumulating wealth are not hard to understand, but they require hard work and self-discipline to apply.

CALCULATING A TIP

After the meal is over and everyone is stuffed, it's time to pay the bill and make one of the most common financial calculations: deciding how much to tip a server. Some diners believe that the term "tips" is an acronym for "to insure prompt service," hence they believe that the size of the tip should be tied to the level of service, with excellent service receiving a larger tip and poor service receiving less, or none. Others recognize that servers often make sub-minimum wage salaries (as of 2005, this could be as little as $2.13 per hour) and depend on tips for most of their income, hence they generally tip well regardless of the level of service. Another important consideration is that servers are often the victims of kitchen mistakes and delays, and therefore penalizing them for these problems seems unreasonable. A good general rule of thumb is to tip 20% for outstanding service, 15% for good service, and 10% or less for poor service. Regardless of which tipping philosophy one adopts, some basic math will help calculate the proper amount to leave.

For example, imagine that the bill for dinner is $56.97, which includes sales tax. By looking at the itemized bill, we determine that the pre-tax total is $52.75, since most people calculate the tip on the food and drink total, not including tax. Since the service was excellent we choose to tip 20%. Most tip calculations begin by figuring the simplest calculation, 10%, since this figure can be determined using no real math at all. Ten percent of any number can be found simply by moving the decimal point one place to the left. In the case of our bill of $52.75, we simply shift the decimal and wind up with 10% being $5.275, or five dollars twenty seven and one-half cents. Then to get to 20%, we simply double this figure and wind up with a tip of $10.55.

In real life, we are not concerned about making our tip come out to an exact percentage, so we generally round up or down in order to simplify the calculations. In this case, we would round the $5.275 to $5.25, which is then easily doubled to $10.50 for our 20% tip. Finding the amount of a 15% tip can be accomplished either of two ways. First, we can take the original 10% value and add half again to it. In this case, half of the original $5.27 is about $2.50, telling us that our final 15% tip is going to be around $7.75, which we might leave as-is or round up to $8.00 just to be generous. A second, less-obvious approach involves our two previous calculations of 10% and 20%. Since 15% is midway between these two values, we could take these two numbers and choose the midway point (a process that mathematicians call "interpolation"). In other words, 10% is $5.27 (or about $5.00) and 20% is $10.55 (or about $11.00), so the midway point would be somewhere in the $7.00–8.00 range. Either of these two methods will allow us to quickly find an approximate amount for a 15% tip.

CURRENCY EXCHANGE

Because most nations issue their own currency, traveling outside the United States often requires one to

exchange U.S. dollars for the destination nation's currency. But this process is complicated by the fact that one unit of a foreign currency is not worth exactly one U.S dollar, meaning that one U.S. dollar may buy more or fewer units of the local currency. Currency can be exchanged at many banks and at most major airports, normally for a small fee. Banks generally offer better exchange rates than local merchants, so travelers who plan to stay for some time typically exchange larger amounts of money at a bank when they first arrive, rather than smaller amounts at various shops or hotels during their stay.

Consider a person who wishes to travel from the United States to Mexico and Canada. Before leaving the States, the traveler decides to convert $100 into Mexican currency and $100 into Canadian currency. At the currency exchange kiosk, there is a large board that displays various currencies and their exchange rates.

The official unit of currency in Mexico is the peso, and the listed exchange rate is 11.4, meaning that each

U.S. dollar is worth 11.4 pesos. Multiplying 100 × 11.4, the person learns that one is able to purchase 1,140 pesos with $100. Canadians also use dollars, but Canadian dollars have generally been worth less than U.S. dollars. On the day of the exchange, the rate is 1.3, meaning that each U.S. dollar will buy __BODY__.3 Canadian dollars, so with $100 the person is able to purchase 130 Canadian dollars. At this point, the shopper might wonder about the exchange rate between Canadian dollars and pesos. Since it is known that 130 Canadian dollars equals the value of 1,140 pesos, the person can simply divide 1,140 by 130 to determine that on this date, the exchange rate is 8.77 pesos to one Canadian dollar.

Exchange rates fluctuate over time. On a business trip one year later, this same person might find that the $100 would now buy 2,000 pesos, meaning that the U.S. dollar has become stronger, or more valuable, when compared to the peso. Conversely, it might be that the dollar has weakened, and will now purchase only 800 pesos. These fluctuations in exchange rates can impact travelers, as the changing rates may make an overseas vacation more or less expensive, but they can be particularly troublesome for large corporations that conduct business across the globe. In their situation, products made in one country are often exported for sale in another, and changing exchange rates may cause profits to rise or fall as the amount of local currency earned goes up or down.

In addition to U.S. dollars, other well-known national currencies (along with their exchange rates in early 2005) include the British pound (.52), the Japanese yen (105), the Chinese yuan (8.3), and the Russian ruble (27.7). Beginning in 2002, 12 European nations, including Germany, Spain, France, and Italy, merged their separate currencies to form a common European currency, the Euro (.76). Designed to simplify commerce and expand trade across the European continent, conversion to the Euro was the largest monetary changeover in world history.