Mathematics of Tax-Advantaged Retirement Savings

(NOTE: The content of this blog is intended to demonstrate the mathematical analysis of retirement savings and is not intended as investment advice.)

How do you imagine most people pay expenses during their retirement, after they have stopped working and no longer receive a wage?

Almost everyone needs to save for retirement. There are a number of ways to do this. How can we make sense of the options we have and choose the best one? First, we need to provide some background and then we will get into the mathematics that can help us make these important decisions.

Background on Saving for Retirement

Saving for retirement is an important financial goal for most people.

  • Social Security benefits are not likely to be enough to cover the expenses most people will face in retirement.
  • Pensions that pay an annual amount for one’s entire retirement used to be common but are generally not provided by employers anymore.

To encourage retirement savings the government offers tax-based incentives for people to contribute to their retirement accounts.

Tax is an especially important consideration when saving for retirement. You potentially face two kinds of tax: first, a tax on your wages when they are paid and, second, a tax on the interest your investments earn.

Example: You invest $5000 for 20 years, earning an annual interest of 7%. How much would you have if there were no taxes?

  • FV = 5000(1.07)20 = 19,300

Unfortunately, there is almost always tax! What how much would you have if a tax of 20% is applied to both the initial investment (your wages) and to the interest you earn each year?

  • Your initial investment will now be reduced in value by 20%, 5000 • 0.8 = $4000
  • After one year, your investment will have value 4000 • 1.07 = $4280 before taxes
  • You now pay tax on the interest income – that’s the amount it increased by that year, not the entire amount, in this example your interest income in the first year will be 4280 – 4000 = $280. So you are taxed 20% of $280 which is $56 in taxes.
  • After one year the final value of your investment will 4280 – 56 = $4224.
  • If we work backwards, we can break down all our work, and rearrange the mathematics to see what’s going on:


= 4280 – 56

= 4280 – (280 • 0.2)

= 4000 • 1.07 – 4000 • 0.07 • 0.2

= 4000 • (1.07 – 0.07 • 0.2)

= 4000 • (1 + 0.07(1 – 0.2))

= (5000 • 0.8) • (1 + 0.07(1 – 0.2))

  • Notice that whatever the initial amount c invested, after one year, the value after tax will be c • (1 + 0.07(1 – 0.2)), in other words the tax makes you effectively earn interest at a lower rate, 0.07(1 – 0.2) = 0.056. So even though you are earning 7%, you are only really earning 5.6% after taking taxes into account.
  • This means that after 20 years, the value of your investment will be worth FV = 4000 • 1.05620 = $11,900

Tax brings down the future value of your investment from $19,300 to $11,900 – a significant drop-off! In order to encourage retirement savings, the government provides some tax breaks which allow individuals to avoid paying some of these taxes provided they are saving that money to use in their retirement.


Tax-Incentive Based Retirement Accounts

There are two types of tax-incentive based retirement accounts we will discuss initially:

  1. Individual Retirement Account (IRA, often known as a “traditional IRA”)
  • You pay no tax on your salary when it goes into the account.
  • You pay no tax on income from interest earned in this account.
  • You will pay tax when you withdraw from this account during retirement.
  1. The Roth IRA
  • You will pay tax on your salary, even when it goes into the account.
  • You pay no tax on income from interest earned in this account.
  • You will pay no tax when you withdraw from this account during retirement.

Key facts to note:

  • In all the tax-incentive based retirement accounts you pay no tax on income from interest.
  • In all tax-incentive based retirement accounts withdrawals prior to retirement are taxed as income and also face an additional penalty for early withdrawal.
  • In a traditional IRA, you pay no tax on your salary when it goes into the account but you pay tax when you withdraw it at retirement.
  • In a Roth IRA, you pay tax on your salary when it goes into the account but you pay no tax when you withdraw it at retirement.

See for more details and other types of retirement plans.

While we use the terms pre-tax and post-tax, others use the terms before-tax and after-tax to refer to the same concepts.


 How Big of an Advantage Do Tax-Advantaged Retirement Accounts Provide?

We saw above that if there is a tax rate of 20% and return of 7%, the future value of a $5000 taxable investment is:

FV = (5000 • 0.8) (1 +0.07 • 0.8)20=$11,900

With a traditional IRA, there is no tax on the investment income and no tax at the time of your contribution, so prior to withdrawal, the value is 5000(1.07)20 = 19,300, but you are taxed at 20% when you withdraw the money at retirement, so the amount you have to spend in retirement after taxes is:

FV = 5000(1.07)20 • 0.8 = $15,500

How much more do you have with a traditional IRA than a taxable investment? We can find how much more, as a percent, by taking the ratio of the two values:

In other words, in the above example, your investment is worth approximately 30% more with a traditional IRA than a taxable investment.

We can also arrive at this number using the original algebraic expressions and simplifying:

Again, arriving at the same ratio, showing that in this example your investment would be worth approximately 30% more with a traditional IRA than a taxable investment.

However, this is just one example. Using a similar approach we can explore these same ideas to generalize…

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Which is better, odds or probability?

Converting from odds to probabilities is not complicated. Suppose the odds are A to B.  Then the probability will be given by A/(A+B). Converting from probability to odds is easy mechanically but then needs to be converted to whole numbers, which involves some judgment. If the probability of an event is P, then the odds of that event are P/(1-P). Let’s say the probability is 60%. Then the odds would be 60%/(1-60%) = .60/.40, which equals 1.5 to 1, or, as is more commonly used, 3 to 2.

The advantage of probability is that it is much easier to work with mathematically.

The probability of either of two independent events occurring is the probability of event A, pA, plus the probability of event B, pB, or pA + pB.

What are the odds of either a 1 to 3 odds event or a 5 to 4 event occurring if they are independent?

The probability of an event is 33%, and the probability of another independent event is 55%. What is the probability that either of the events happen?

Independent means that the events do not overlap, like horses in different races.

The probability of two independent events occurring is pA * pB.

What are the odds of both a 1 to 3 odds and a 5 to 4 odds event both occurring?

The probability of an event is 33%, and the probability of another event is 55%. What is the probability that both of these events happen?

The easiest way to solve many odds problems is to convert everything to probabilities, compute the result, and then convert it back to odds.

If probabilities are easier, then why are odds used at all? It is likely that odds are used in gambling because they better reflect the payoff of a bet.  If the odds are 3 to 2, in a “fair” bet you would put up $3 on the favorite and your friend would put up $2 on the other team.  If your team wins you would get $2 dollars for your $3 bet, and if your team lost, your friend would get $3 for their $2 bet.  Odds, as we shall see below, also make it harder to see how much the “house” is making and how much the bettors are paying to play.

Odds in the real world

Somewhat counterintuitively, horse racing odds are actually the odds that your horse will lose.  6 to 1 odds means that your horse is expected to lose 6 out of 7 times or win just 1 out of 7 times.

It might seem that the probability of winning would be 1 in 6, but that would be missing the one time that your horse wins. Six out of seven times your horse loses and one out of seven times it wins.

Odds are Wins/Losses.

Probability is Wins/(Wins + Losses).

The odds reflect your payout for winning.  If your horse wins you receive 6 times your bet, plus getting your original bet amount back.  Thus, even a horse with a 3 to 5 payout (or 3 to 5 odds of losing), will provide you with a profit if your horse wins, as you will get back $1.20 plus your original $2 for a typical $2 bet.

What about moneyline odds that are frequently used for sporting events?  You might see moneyline odds like:

Team A -140, Team B 120.

These are quoted as numbers, not as ratios.  In sports, the odds for the favorite are quoted as a negative number, and the odds for the underdog are quoted as a positive number.  The absolute value of both numbers is always greater than or equal to 100.

Absolute value is the magnitude of the number without the negative sign. The symbol |  |, represents absolute value.  For example: The absolute value of -3, written |-3| = 3 and the absolute value of 3, written|3| = 3.

The negative moneyline odds are how much you need to bet to win $100.  While the positive number is the amount you will win if you bet $100.  Since these have two different definitions, there are two different formulas to convert to implied probabilities.  (These are called implied probabilities because they are the probabilities implied by the odds and may not be the true probabilities of the outcomes.)

Negative moneyline odds represent probabilities greater than 50% and the conversion is:

Probability = |odds|/(100 + |odds|), where |   | means absolute value.

If the odds are -150, then the probability is 150/250 or 3/5 or 60% chance of winning.

The more negative the odds, the higher the implied probability:

If the odds are -300, then the probability is 300/400 or 75%.

For positive odds,

Probability = 100/(100 + odds)

If the odds are +150, then the probability is 100/250 or 2/5 or 40%.

The more positive the odds, the lower the implied probability.

If the odds are 300, then the probability is 100/400 = 25%

If the odds are 100 or -100, then you can compute that the probability is 50% in both cases.

Gambling is a losing game

As long as both the positive number and the negative number have the same absolute value, then the sum of the probabilities will be 100%.  If the absolute value of the negative odds for the favorite is higher than the positive odds for the underdog (or if both are negative), then the sum of the implied probabilities will exceed 100%.

In theory, we know that the probability of all possible outcomes should add up to 100%.  With odds, it is more difficult to quickly see if all the payoffs add up to 100% of the bets.  If the implied probabilities add up to more than 100%, that is a sign that the sponsor is making money.  This is often called the vig, which is short for vigorish.

Think of it this way.  Suppose there are four bets with odds of 2 to 1. What if you bet on all four?  Then you should have a 100% chance of winning.  If each bet is $2, you bet a total of $8.  However, when you win, you get back $6 ($4 plus your original $2).  What happened to the other $2?  The other $2 is kept by the “house” or the party sponsoring the betting.

Let’s look at this from the standpoint of probability.  The probability of a win with 2 to 1 odds against, is 1/3.  Since there are four bets the total probability is 4/3.  Since probabilities must add to 1, the extra 1/3 is the amount of profit or vigorish (vig) going to the house.

Let’s compute the vigorish for a moneyline bet where the odds are -160 and 120.

160/260 + 100/220 = 107%.  So 7% is going to the “house.”

In betting, whether online or at the races, the “house” is always making money so that means that on average everyone else is losing.  The next time you see moneyline odds when watching a game check to see if the absolute value of the negative number is greater than the positive number.  The difference is profit to someone else.

In our next blog, we will compare gambling to investing to see why investing has a positive expected value while gambling has a negative expected value.

What are the Odds?

With the growth of online betting, odds are in the media more than ever. While we don’t recommend gambling, for reasons you will soon learn, we believe that understanding odds will improve your understanding of probability and investing.

Odds and probability

In math class, when we study uncertain outcomes we usually talk about probability, but in the popular media, especially when talking about who will win a sports event, they typically talk about “the odds.” Since they both relate to uncertain outcomes there should be a relationship between the two of them. While odds are frequently used in sports, they are also used in other areas. For example, credit scores are based on the odds of the borrower failing to make a payment vs making a payment on a financial obligation.

Studying odds can help improve your understanding of probability and investing. In fact, the earliest known mathematical treatment of probability, by the very famous and talented mathematicians Pascal and Fermat in 1654, was in the context of a problem from gambling.

Odds can be defined as a ratio of the expected or implied frequency of something happening to the expected or implied frequency of that event not happening. If someone says the odds are 3 to 1 that the train will be late today. They mean that it is three times as likely to be late than not be late. Probability on the other hand is the chance of something happening as a percentage of all possible outcomes. Using the above example, there would be a 75% probability of the train being late today.

Often expressed as percentages, probabilities are always between 0 and 1. With 0 meaning it never happens and 1 meaning it always happens. Odds can be any number but are often quoted as one whole number to another whole number. Odds could be quoted as 5 to 4 or 7 to 3.

Ratios and percentages

Suppose your team’s record against the cross-town rival so far this season is three wins and two losses.

If that record is representative of the overall skill of the two teams, then the odds of your team winning the next game is 3 to 2 (or sometimes 3:2). This is the same as if you were asked “What is the ratio of wins to losses?” The answer would be: “3 wins to 2 losses.” Odds are the ratio of the event happening to the event not happening.

What if you were asked, “What is the probability of your team winning?” In this case, you would say “My team won 3 out of 5 games,” or “60% of the time.” If that percentage holds, then there is a 60% chance of my team winning the next game. Notice that probabilities are like percentages in that they represent the percentage of times that the event occurred divided by the total number of events. In this case, the total number of wins is divided by the total number of games played between those two teams. Probabilities are often expressed as a percentage but also can be expressed as fractions.

What is credit?  What is debt?

“Once I was the credit to my credit card. Spent what I hadn’t got, wasn’t hard.”

— Peter Gabriel, Waiting for the Big One.

While we have previously discussed money, an asset, here we focus on the liability side of the balance sheet.

Many of the words to describe borrowing have been distorted by marketing and there is widespread confusion about the words debit and credit. Let’s clarify the concepts and agree on some usage of the words debit, credit, and debt. (Note that these words have multiple meanings in finance. While they are the same words and are connected to related concepts, when talking about transactions from an individual’s perspective, they do not have the same meaning as debits and credits do in accounting).

Debit vs. Credit

When you take money directly from your checking account that is a debit.  When you buy something with a promise to pay soon, that is buying on credit. By credit, we generally mean a purchase without immediate payment, but that full payment is expected soon.  For example, many business-to-business transactions have terms like “net 30.”  That means you pay the bill within 30 days.  While informal store credit was fairly common for retail purchases 25 to 50 years ago, it has now been largely replaced by “credit cards.” In some small towns and some city bodegas, the clerk might still be willing to keep a tab on your purchases that you pay about once a month.

Whether you pay with cash, check, debit card, or credit card, from a net worth standpoint you are in the same position.  You have either reduced your cash asset, or you have incurred a liability.

Once a payment is no longer expected to be paid in the short term it becomes a debt. If you keep a balance on your credit card, you should probably call it a “debt card” but it is pretty clear why that may not be the best name for marketing purposes.

Types of Debt

Generally, debts are either secured or unsecured and recourse or non-recourse. Secured debt means that if payments are not made, the lender has the right to take the asset that is tied to the loan. The asset is called the “collateral” for the loan. An unsecured debt means that there is no particular asset tied to the debt. Recourse means the lender has the right to demand payment from your personal assets. Non-recourse means that they cannot seek repayment from you personally.

  • Mortgages (loans to purchase a home) and auto loans are examples of secured debt with recourse. If you do not make payments the lender can take your home and they can also make a claim on your other assets if the value of the “collateral” is not sufficient. In some cases, mortgages are non-recourse, and the lender cannot make a claim on your other assets or income.
  • Margin on securities purchases is another form of secured debt. If the value of your portfolio falls, the broker will sell assets to reduce the debt to the required level.  [In financial markets, a form of margin are repurchase agreements.  With a repurchase agreement an owner of a security, sells the security today and agrees to buy it back later. The original owner of the security in essence has borrowed money and the other party has made a short-term secured loan. Such repurchase agreements “called repos” make up a substantial part of the funding of fixed-income capital markets.]
  • Credit cards are generally non-secured, but do have recourse. There is no particular asset or collateral, but the lender can make a claim on your personal assets. A variant on the credit card is a secured credit card, where the borrower deposits money with the bank or credit card company and their credit is limited to the deposit.
  • What about an unsecured, non-recourse loan? It would generally be difficult to get such a loan from a financial institution, however, many loans from friends and family are exactly of this type. The lender hopes that you will pay the money back, but they are not likely to seize assets or take you to court for payment. However, non-payment of such debts often leads to family conflict and hard feelings on both sides that can last decades.

Not all debts involve an exchange of money. Providing a guaranty for someone else’s debt makes it your debt. If the borrower is unable to pay, you will be on the hook. Even though you didn’t provide any money to the borrower, providing a guaranty has the same risk as making a loan to the person who did take out the loan.

Getting out of debt.

Of course, the best way to get out of debt is to make the necessary payments. Bankruptcy is another way out, but it comes with significant consequences. For most loans, if you declare bankruptcy, after whatever debts can be satisfied from your assets are paid, you have no further obligation to pay. One exception to this is student loans, which are not discharged in bankruptcy. However, if you declare bankruptcy, the court will work to satisfy as many creditors as possible by taking money from your bank account and selling your assets. It will also be much more difficult to borrow money for five to seven years.

Debt and Interest

Most debts will require payment of interest, generally monthly. The interest rate will reflect the risk of the transaction to the lender. Secured loans tend to have lower rates and unsecured loans have higher rates. The amount of time allowed to pay off the loan also affects the rate, with longer times to pay generally having higher interest rates. The creditworthiness of the borrower, as reflected in their credit score also affects loan interest rates. Borrowers with strong histories of making loan payments generally have higher credit scores and pay lower rates on their loans. Borrowers without credit scores or with very low credit scores may have limited options for borrowing and may pay very high interest rates or face other fees to borrow.


In summary, we would recommend the use of the term debit, to refer to a transaction that deducts from your checking account immediately. Credit should refer to transactions where you do not pay immediately but are expected to make full payment generally in less than a month. Any longer-term obligation to make a payment, whether for a purchase or for a loan, should be considered debt. Debt can be recourse, non-recourse, and secured or unsecured. Interest costs are largely tied to the general level of interest rates in the economy and to the risk to the lender of non-payment. Other than student loans, most debts can be reduced or discharged in bankruptcy, but with considerable consequences. Understanding the nature of debt transactions can help people achieve better financial outcomes.

With a rapidly changing financial system, even a simple question like, ‘what is money?’ can become complex. With new financial apps and rise and fall of various flavors of digital currency, it seems that anything or maybe everything could be money.

Abstractly, one can think of money as a medium that stores value that is readily accepted as payment for products, services, or other items of value.  With this definition, almost anything could be considered money, and throughout history and around the world, almost everything has been a form of money.  Gold, silver, sheep, goats, grain, shells, and unfortunately, people.

Such a broad approach to thinking about money may lead to confusion. A better approach might be to understand the role of various potential candidates for money in our current financial system and how they are related to each other.

In the US there are only two basic forms of money, as defined by the Federal Reserve. These are physical currency (coins and bills, issued by the US Government) and digitally recorded deposits kept at regulated banks. With a few minor exceptions that’s it. If these are the only things that are money, what is everything else?

We can break various forms of money-like things into three categories:
Commodities, IOUs (I-Owe-You’s), and Payment Systems. Most things that we might think of as money fall into one or more of these three categories.


For much of financial history, money was things, physical things. The best type of things for commercial transactions are things that have an agreed upon value, are easily transportable, and difficult to counterfeit. The classic commodity of this sort is gold, and even until 1971, gold was the basis for the international monetary system and for many years the US had paper certificates that could be converted into gold. Silver is another commodity that has been used to back currencies and as a basis for trade (Growing up I had a silver certificate that I may still have stored in a closet somewhere in my house). However, any commodity (any “thing”) that people are willing to use to denominate transactions is a potential form of money.

While physical commodities may have certain advantages as a means of exchange, they face severe limitations relative to the basic forms of money: currency and bank deposits. Perhaps the most significant disadvantage is the need to store and transfer commodities. For physical commodities such as gold or goats, the difficulty of storage and transfer may be obvious (imagine bringing goats to the mall).

IOUs (“I Owe You”)

Another form of a money-like instrument is an IOU, that is a promise to make a future payment.  Before the widespread use of credit cards and in many small towns, people who went to a store often didn’t pay when they left with merchandise, rather the store would keep track of the amount owed. In some cases, the buyer would sign a small note or “chit” stating the amount owed.  Merchants might even exchange these IOUs as a form of payment. Banks also issue IOUs in various forms such as cashier’s checks, Certificates of Deposit and Letters of Credit, all of which can be used to facilitate transactions. IOUs can also be from companies and governments. A major advance in the monetary system was establishing deposit insurance, so that specific IOUs (records of deposits) created by banks are backed by the US government, substantially reducing the risk of keeping your money at a bank.

In some cases, non-bank IOUs, promises that can be exchanged for cash on demand or mature within short periods of time can also be a form of money when held by regulated money market funds or on the balance sheet of corporations. (This was the exception to the two types of money described above.) Chips at a casino or Dave and Busters are also IOUs, but most people don’t consider them to be money as they can only be exchanged at one location.

Payment Systems

Many of the other things that we consider to be money are not money, but rather are “payment systems.”  If the primary forms of money are currency and bank deposits, how can you use money for a purchase?  ATMs, or bank tellers, provide a way to convert money in bank accounts to cash, and the other way around.  Currency can be exchanged in real time for a transaction. At some point, however, the amount of cash required can be prohibitive to carry around: Would you like to buy a car with a stack of $20 dollar bills? Would you like to deliver your rent, in cash, to your landlord each month?

As a result, there are several ways to transfer money from your bank account to the seller.  Traditionally, checks, which are fundamentally a physical means of transfer of money from one checking account to another, were a widely used form of payment. However, the check itself is not money, rather it is a way of transferring money from one account to another.  However, these days the use of checks is declining rapidly and being replaced by electronic means of transfer. For example, many people use some form of bill payment app from their bank.  While this may seem to be an electronic transfer system, some electronic bill payments are just checks in disguise. In some cases, when you enter a bill to be paid, the bank still sends a check.

Other forms of payment as ACH and wire transfers are electronic transfers of money, without the need to deliver physical checks from one party to another. The entire transfer takes place electronically without the need to create a physical check as an intermediary.

There has also been substantial growth in other forms of payment including Debit Cards, Credit Cards, Digital Wallets, and other digital payment systems such as Venmo and Paypal.  Each of these provides another way of accessing the money you hold in your bank account.  Debit cards and Venmo take money directly from your bank account. Credit cards create a temporary IOU that you can pay off monthly, or if you don’t make the full payment convert into revolving debt, generally with very high interest charges. Digital wallets are another way of using a credit cards, with the credit card data being delivered by your phone rather than from a plastic card or a chip imbedded in the card.  Despite the fancy exterior, most of these innovations still utilize checking accounts for the underlying storage of money.

What about Crypto?

If Cryptocurrencies aren’t money; what are they.  While various forms of cryptocurrency have money-like features, they are best thought of as digital commodities. While they are not physical, they still represent an item in limited supply that people can agree to use as a basis for transactions. While storage and transfer are always issues for physical commodities, for digital commodities, crypto assets were supposedly designed to eliminate these problems.  However due to the complexity of operating directly in crypto currencies many people rely on intermediaries to transact and store their crypto assets. Unfortunately, as we have seen, it is possible these entities may be incompetent or corrupt, leading to losses of assets and confidence.

Another characteristic of a good candidate for money is that it has “unit value.” Unit value means that the “price” of something should only change if the cost of the inputs (including profits) change. Thus, anything that changes value on its own, like a commodity, may be a poor choice for money. Since Crypto currency prices vary wildly; they are speculative assets, and not good candidates for money. (Stable coins have been developed to address this issue, but maintaining stability creates other problems.) Although many things have been used as a basis for exchange, governments have been using coinage as a standard of value since ancient times. Throughout history, societies have recognized that using commodities as money can create havoc, and that government issued currencies provide greater value to trade and government management of the economy.

Commodities often become used as currency when governments fail or when they destabilize their own currencies through excessive spending or excessive printing of currency.  The current popularity of crypto may be related to the economic disruptions of 2007 and 2008 and a loss of confidence in the banking system.

Crypto assets, therefore, are best thought of as crypto-commodities and should be evaluated and utilized in much the same way as other commodities. They may be a part of a portfolio, but they are not income producing assets and their value may fluctuate widely. They may provide a hedge against certain economic outcomes, but the relationships may be difficult to assess. In addition, extreme care is required in choosing how to store and transact in digital commodities.

Money, money substitutes, and payment systems are likely to continue to evolve rapidly. This creates the risks for individuals, who may face false claims and failed innovations. While skepticism about government may be warranted, especially in underserved communities, the alternatives to government regulated financial entities do not have a good track record, and government regulation is generally a requirement to prevent misuse of customer assets and protection against predatory practices. Best practice, in a period of rapid change, may be to keep most money in traditional bank accounts and be cautious about the use of new payment systems until they are widely-used and a regulatory regime has been established and tested.

Money and FiCycle

Money plays an important role in the FiCycle Math curriculum. First there is an assumed role, which we do not state explicitly. That is, all transactions in our workbook, spreadsheets and projects are denominated in dollars and that transactions, other than a few exceptions, will use money as the means for exchange. We don’t say if payment is made with coins, bills, by check or with an App, but we assume that students know how to purchase something or send money to another person using money.

Second, we focus extensively on distinguishing cash from wealth.  Cash, which can refer to currency either as bills or coins, or money held in a checking account, is one aspect of wealth. But wealth also includes other assets and is reduced by debts. Understanding the role of wealth in financial transactions is a central theme of FiCycle.

Third, even while focusing on wealth, access to money is essential for paying bills and lack of money can lead to severe financial consequences, even if a person has other forms of wealth. Budgeting can be viewed as a process for making sure that you have sufficient cash (that is access to money) to make necessary payments.



M1 consists of (1) currency outside the U.S. Treasury, Federal Reserve Banks, and the vaults of depository institutions; (2) demand deposits at commercial banks (excluding those amounts held by depository institutions, the U.S. government, and foreign banks and official institutions) less cash items in the process of collection and Federal Reserve float; and (3) other liquid deposits, consisting of other checkable deposits (or OCDs, which comprise negotiable order of withdrawal, or NOW, and automatic transfer service, or ATS, accounts at depository institutions, share draft accounts at credit unions, and demand deposits at thrift institutions) and savings deposits (including money market deposit accounts).

The Fed – Developments in Noncash Payments for 2019 and 2020: Findings from the Federal Reserve Payments Study