The Real Interest Rate and Annuities

Suppose that you are 70 years old, with $300,000 in savings but no income. How much can you afford to spend each year? If you spend too much and live a long time, you will run out of money. If you spend too little and die unexpectedly early, you will leave behind a large estate, but you will have cheated yourself out of a better quality of life.

Inflation and the Real Interest Rate

One thing that you would have to worry about with your $300,000 in savings is having the value of your money eaten away by inflation. If the average cost of the goods that you buy rises by 5 percent per year, then the purchasing power of your savings falls by that same 5 percent per year.

An increase in the average cost of goods that people buy is called inflation. There are many ways to measure inflation, depending on how you assign weights to different goods. The most well-known measure of inflation in the United States is called the Consumer Price Index, which uses a weighted average of the cost of food, gasoline, utilities, medical care, and other goods and services that people commonly consume.

Each year, the purchasing power of our savings declines at the rate of inflation. Economists use the symbol p to stand for the rate of inflation.

The interest rate that you earn on your savings usually is higher than the rate of inflation. If the inflation rate is 6 percent, then the interest rate would typically be somewhere between 8 and 10 percent. If the inflation rate is 2 percent, the interest rate typically would be somewhere between 4 and 6 percent.

Economists think of the interest rate, i, as consisting of two components: a real interest rate, r, and the rate of inflation, p. We write

i = r + p

When you are trying to forecast the purchasing power of your savings, you care about the real interest rate, r. It is the real interest rate that determines how your purchasing power will change over time.

Until very recently, it was not possible for a consumer to "lock in" a real rate of interest. However, during the Clinton Administration, the U.S. Treasury began to issue "inflation-indexed" bonds, called TIPS. Like all government bonds, you purchase an amount, called the principal, and over a period of years the government repays the principal plus interest. With TIPS, the principal automatically is adjusted by the rate of inflation. The interest rate on TIPS therefore represents the "real" rate of interest. Currently, this stands at about 3.5 percent.

For example, if you invest $10,000 in a TIP and the inflation rate next year is 2 percent, then next year you will receive 3.5 percent interest, and in addition the principal on your TIP will rise by 2 percent to $10,200. Instead of being eaten away by inflation, your real principal is unchanged.

An Annuity

Suppose you knew that you were going to live 15 years. If you were to put the money into an account that earns a real interest rate of 3 percent per year, then each year your savings balance would increase by 3 percent and decrease by the amount that you spend.

Bt = Bt-1(1 + .03) - C

where Bt is the balance in year t and C is the amount that you spend each year.

Is Bt a stock or a flow?
Is C a stock or a flow?

To see how the equation works, suppose that we spend $20,000 a year. After one year, our balance will be
$300,000 (1.03) - $20,000 = $289,000. After the second year, our balance will be
$289,000 (1.03) - $20,000 = $277,670. And so on.

The formula for an annuity

Modern spreadsheet programs have annuity calculations programmed into them. So chances are, you never will have to know how to calculate an annuity. But just for the record, here is how the formula is developed.

Suppose you start out with an intial balance, B0 that increases at a rate a per year, minus a constant amount C per year, for n years. We have

B1 = aB0 - C

B2 = aB1 - C = a2B0 + C(1 +a)


Bn = anB0 - C(1 + a + a2 +

We see that C is multiplied by the sum of a geometric series. The formula for that sum is

For the annuity to exactly exhaust itself after n years, we set Bn equal to zero. So we have

0 = anB0 - C[(1-an)/(1-a)]

Multiplying all the way through by (1-a), dividing all the way through by an, and solving for C in terms of a and B0, we have

C = (1-a)B0/(1/an - 1)

In fact, a equals 1+r, so we have

C = rB0/(1-1/[1+r]^n)

Suppose that our goal is to spend as much as we can over the 15 years. That means that we want to choose the level of consumption, C, so that when t = 15, the balance should be exactly zero.

The formula for an annuity, which is derived in the box on the right, provides the answer. If B0 = $300,000, r = .03, and n=15, then C = $25,130. This says that we can spend $25,130 a year for 15 years. After 15 years, our savings will have gone to zero.

It is important to note that $25,130 is in current dollars. If there is inflation, then C will go up with the rate of inflation. For example, suppose that the real interest rate is three percent (.03) and inflation is five percent (.05). Then the nominal interest rate on the annuity will be eight percent. Our annual spending, C, will increase at a rate of 5 percent per year. That way, every year we will be able to afford the same basket of goods and services that we can afford today for $25,130.

An Insured Annuity

The annuity formula only works if we know exactly how long we are going to live. That is, the annuity that we just calculated will work if we live exactly 15 years. However, if we die early, then we will leave a positive savings balance. If we live longer than 15 years, we will have nothing left to spend.

A solution to this problem is to use our $300,000 to purchase an annuity from a life insurance company. The insurance company can spread its risk over many individuals. It will pay, say, $23,000 a year regardless of how long the person lives. Some people will live longer than 15 years, causing losses for the insurance company. Others will die early, giving the insurance company gains. The gains will offset the losses, and in fact the insurance company will price the contract so that it makes a profit on the average person.

When you purchase an insured annuity, you are trading a risk with the insurance company. If you live a long time, you will wind up better off than if you had simply tried to invest in bonds yourself to create an annuity. If you die soon, then you could have enjoyed a higher standard of living while you were alive by creating your own annuity. The insurance company takes care of you if you live longer than expected. In return, you give the insurance company a profit it you do not live longer than expected.

Suppose that the insurance company makes 0.5 percent on an annuity. That means that if the real interest rate is 3 percent, the insurance company makes 0.5 percent and you effectively receive only 2.5 percent interest. If you pay $300,000 for an insured annuity with an expected life of 15 years, how much will the insurance company give you to spend each year?

(difficult question) Using the spending rate that you just found and an interest rate of 3 percent, calculate how long $300,000 would last. This tells you how long you would need to live in order to be better off with the insured annuity than creating your own annuity. What is the answer?

Pensions and Social Security

A pension is a retirement savings vehicle that is tied in with your employer. During your working life, you and your employer contribute to a pension plan. After you retire, the pension plan pays you either a lump-sum benefit or an annuity. (With a lump-sum benefit, you get all of your benefits at once, and then it is up to you to spread out your spending over the course of your retirement.)

Fifty years ago, people believed in lifetime employment. The thinking was that if you went to work for a major corporation, nonprofit institution, or government agency, you would stay there until you retire. The concept of lifetime employment meant that a single worker would have one employer over his (this was before the women's movement) lifetime. Whether or not this thinking was valid then, it certainly is not valid now.

The combination of the myth of lifetime employment and the reality that many people change employers several times over the course of their working lives makes a hash out of the pension system.

Broadly speaking, there are two types of pension plans. In a defined-contribution plan, the contributions that are made by you and on your behalf to the pension plan are managed by the pension plan managers. Based on the size of contributions and the performance of their investments, your pension balance grows until you retire and start do draw it down. At that point, your pension income is based on this balance.

In contrast, with a defined-benefit plan, your benefits are based on a formula that is not directly tied to your contributions. The formula may include your length of service, your highest salary, the salary in your last three years, or other factors. The benefits under a defined-benefit plan could turn out to be higher, lower, or the same as those in a defined-contribution plan.

With a defined-benefit plan, the company stands to gain or lose, based on how well it manages the pension money. If the returns are high, then the plan's assets may exceed what is necessary to pay benefits, and the firm earns a windfall. Conversely, if the pension plan is mismanaged, the benefits must still be paid. In this situation, if the firm goes bankrupt, it might be unable to pay benefits. In the United States, the government has set up a pension insurance fund, called the Pension Benefit Guaranty Corporation, that charges companies a fee (defined-benefit plans pay the largest fees) and in return pays the pension obligations of defined-benefit pension plans that succumb to bankruptcy.

Suppose that this year's contributions from you and your employer to your pension plan are $4000, and that this is fully vested. You quit in order to start your own business, with ten years left until retirement. If the pension plan invests the money at 5 percent per year, how much will be available for you as a lump some in ten years? If the company has a defined-benefit plan that pays out $6500 instead of a defined-contribution plan, who comes out ahead--you or the company?

Defined-benefit plans sometimes use a formula that is based on the salary in the year before you leave a company. If you left a company twenty years ago, this salary is likely to be low relative to salaries today, and a defined-benefit plan would be worth little. On the other hand a defined-contribution plan with that same company might be worth a lot, if your contributions were invested in securities that grew in value over the past twenty years.

On the other hand, suppose that you change firms a few years before you retire. If the new firm uses a defined-benefit plan and you are vested, you will do very well. If it has only a defined-contribution plan, then you will not have time to accumulate very much in terms of wealth.

In other words, if you are early in your career and you are likely to change jobs before you retire, a defined-contribution plan probably will be to your advantage. If you are late in your career, then a defined-benefit plan will be more to your advantage.

A possible reform for pension legislation would be to try to make pensions more "portable," so that you can change jobs without distorting benefits. However, the most portable pensions are those that individuals fund themselves, rather than using their employer as an intermediary.

Social Security

Social Security is a defined-benefit pension plan managed by the government. The government collects "contributions" in the form of social security taxes. It pays benefits according to formulas set by Congress. As with any defined-benefit plan, there is no assurance that the value of what you pay in will equal, exceed, or fall short of what you could have obtained had you taken your contributions and saved them. On average over the history of Social Security, the benefits have been greater than they would have been under a defined-contribution allocation. That is a political decision made by Congress, not an economic result.

At any one point in time, Social Security transfers money between two groups of individuals. Taxpayers pay into the system, and beneficiaries take money out of the system. Social security's financial condition depends on the balance between these two groups.

In 1950, when the system was relatively new, there were 16 workers per retiree. Even now, the ratio of taxpayers to beneficiaries is 4.0, which is relatively high. This gives Congress the option of collecting more in taxes than it pays in benefits, with the surplus going to fund other government spending or to pay down government debt.

As the Baby Boomers age, however, the ratio of taxpayers to beneficiaries will fall. According to a report on global aging by Maureen Culhane of Goldman Sachs (my link to it is broken, but you may be able to Google it), this ratio will fall to 3.4 by 2015 and to 2.3 by the year 2050. (In fact, it is likely that the low point for the ratio will come somewhere in between 2030 and 2040, when it may fall below 2.0) Other things equal, if the ratio is 3.0 instead of 4.0, in order to maintain benefits, each taxpayer must pay 4/3 of what he or she would have paid had the ratio remained at 4.0.

As longevity increases relative to the statutory retirement age, people remain Social Security beneficiaries for more and more of their lives. Thus, the outlays for Social Security tend to grow faster than the economy. In 1960, social security outlays were 2.2 percent of GDP. In 2000, they were 4.1 percent of GDP.

The Economics of Social Security

Economists view social security differently than does the public at large. Many journalists and politicians speak of social security as if it were a defined-contribution pension plan. They speak as if the benefits that someone receives from social security reflect that person's contributions into the program.

Economists instead view social security as an ongoing intergenerational transfer mechanism. At any point in time, the working-age population is being taxed to support retirees. This view of social security as an intergenerational transfer has some interesting corollaries.

  1. Holding tax rates constant and longevity constant, the ability to expand benefits in social security depends on the growth rate of the population plus the growth rate of productivity. If the younger generation is 10 percent larger than the older generation and 10 percent more productive, then benefits for the older generation can be 20 percent higher.

  2. In general, a transfer from the young to the old is a transfer from people who are relatively rich to people who are relatively poor. That is because of the phenomenon of economic growth, which makes each generation better off than the previous generation.

  3. Some of the involuntary, impersonal transfers that take place within social security are reversed by voluntary personal transfers. Within familes, the old tend to give money to the young.

Policies and the Social Security Crisis

The looming decline in the ratio of workers to beneficiaries is sometimes referred to as a crisis for Social Security. Economists do not agree on how to deal with this crisis.

Some economists are not convinced that the crisis will appear. This is not a matter of ideology, but of optimism. There are both liberals and conservatives who believe that productivity growth in the next twenty years may be two percent or more per year. If this proves to be the case, then social security transfers as a percent of GDP will increase little, if at all.

On the other hand, there are economists who are concerned about the demographic outlook for social security. Among such economists, liberals tend to support having the Federal government run a budget surplus over the next twenty years, in order to increase national saving. A higher saving rate will raises the capital/output ratio, which in turn they hope will increase GDP, which would make it easier to afford high security outlays.

Conservative economists, myself included, would like to see the retirement age raised and then indexed so that it increases along with longevity. Such a policy would serve to limit social security outlays as a percent of GDP.

Liberal politicians have proposed something called a "lockbox" for social security. Their thinking is that while the baby boom population is still working, the social security system should be able to take in more in revenues than it pays in benefits. If these surpluses can be "locked up" and put away, they suggest, then when the baby boomers retire these funds can be drawn down.

Inside the "lockbox" are claims on future output. If the "lockbox" works, then the Baby Boom generation will have the legal means to collect its Social Security benefits, regardless of how high this makes the tax burden on those who are then working.

Conservative politicians have proposed something called "partial privatization," which would allow individuals to direct some of their social security contributions to the stock market. Their thinking is that the high returns from the stock market might make it easier to pay social security benefits.

Like the lockbox, partial privatization does nothing to lower the ratio of social security benefits to GDP. In fact, if stock prices do well, then baby boomers will be able to consume an even higher share of total output than they would if we stuck with the current social security system. Conversely, if the stock market does poorly, it seems likely that politicians would use tax money for a bailout. Thus, neither the lockbox nor partial privatization address the economics of social security.

The Case for Raising the Retirement Age

I favor raising the retirement age and indexing it to longevity. Indexing would affect people who are far away from retirement. Once you reach age 50, your retirement age would be frozen. Up until that point, the statutory retirement age would increase on a year-for-year basis with observed increases in longevity. Ten years from now, if longevity has increased by two years, then the retirement age will have gone up by two years as well.

An increase in the retirement age should apply to people currently aged 50 and under. It need not apply to people in their 50's and older, who already are planning for their retirement. Also, no one would be forced to wait until the statutory retirement age to retire. You can always retire earlier and live off of savings, with Social Security kicking in at the statutory retirement age.

Instead, by keeping the retirement age constant in spite of higher longevity, we are automatically expanding the role of social security in the economy. We are creating a situation in which improvements in health and medicine lead people to spend a large and growing fraction of their lives dependent on the government for income.

The proportion of the population that obtains benefits from social security might be an arbitrary political decision if the process of collecting taxes to pay social security benefits did not impose any additional cost on the economy. However, as we will see when we study the microeconomics of markets, in addition to collecting revenue, taxes impose a "deadweight loss," meaning that they reduce the economy's output. In other words, using taxes and transfers to try to redistribute the pie has its limits, because the pie gets smaller when tax rates are higher. The social security tax punishes work and thrift, so that we get less of those two activities the more we raise social security taxes.

The social security tax is a particularly evil tax, because it is regressive, meaning that high earners pay a lower proportion of their income for social security taxes than do low earners. Some of the seniors receiving Social security checks are quite affluent, while some of the workers paying taxes to fund those checks may be lower down on the wealth scale. This is not the sort of transfer scheme that we should be leaving on autopilot to grow at an exponential rate.

Basic Financial Calculations

Discounting Future Cash Flows

Stocks, bonds, and other instruments that are discussed in the business section of newspapers and on financial web sites represent claims on future cash flows. If I buy a stock or a bond, then I receive payments in the future. The most basic concept for determining the values of those cash flows is called discounting.

If you ask me whether I would prefer $100 today or $100 a year from now, I will say that I want the $100 today. If I can earn 5 percent interest on money in a savings deposit, then with $100 today I could put the money in a savings account and have $105 a year from now.

If the interest rate is 5 percent, then I would view $105 a year from now as being equivalent to $100 today. Another way of saying this is that the discounted present value of $105 a year from now is $100. When we want to know what a future cash flow is worth today, we calculate its discounted present value.

To calculate the discounted present value of a cash flow to be received one year from now, divide by (1+i), where i is the interest rate expressed as a decimal. If the future cash flow is $105 and the interest rate is 5 percent, or .05, then the discounted present value is $105/(1.05) = $100.

What is the value of a cash flow of $105 that you will receive two years from now? Assuming the same interest rate of 5 percent, we discount twice. That is, we take $105 and divide by 1.05, and then divide by 1.05 once more, for a discounted present value of $95.24. More generally, if the interest rate is constant, we have for a cash flow C that will arrive t years from now,

discounted present value = C/(1+i)t

You may remember the way that compound interest behaves. If you have a savings balance of $100,000 and the interest compounds annually, then after six years of earning interest at a rate of 5 percent per year, your balance will equal
$100,000 (1.05)6 = $134,009.60. Discounted present value is like compounding, except that you work backwards in time. Instead of taking compound interest from today to calculate a value in the future, you start with a value in the future and discount back to the present.

Calculate the discounted present value of a payment of $100 two years from now, if the interest rate is 8 percent per year.

Calculate the present value of a payment of $100 three years from now and a payment of $200 five years from now, if the interest rate is 6 percent per year.

Does a cash flow of $100 to be received eight years from now have a discounted present value that is lower or higher than a cash flow of $100 to be received four years from now? In general, what effect does the length of time until you will receive a cash flow have on the present value of that cash flow?

For a cash flow of $100 to be received one year from now, will the discounted present value be higher if the interest rate is 5 percent or the interest rate is 10 percent? In general, what effect does a higher interest rate (sometimes called the discount rate) have on the value of a future cash flow?

Forward Interest Rates and the Yield Curve

If you check interest rates in the newspaper, you may find that the interest rate on a ten-year bond is 5 percent, while the interest rate on a one-year bond is only 3 percent. Financial pundits refer to the different interest rates for different time periods as the yield curve.

The yield curve consists of the immediate short-term interest rate as well as short-term interest rates that are expected in the future. The latter are called forward interest rates.

Here is an example of a simple two-year yield curve, consisting of the current one-year rate and next year's forward rate. Suppose that the interest rate this year is 4 percent, and the forward rate is 6 percent. What is the discounted present value of $100 to be received two years from now?

To discount $100 back to one year from now, we take $100/(1.06) = $94.34. To discount this back to the present, we take $94.34 and divide by 1.04, to obtain $90.71.

If the current one-year rate is 4 percent, the one-year forward rate is 6 percent, and the next year's forward rate is 5 percent, what is the present value of $100 to be received three years from now?

A ten-year bond pays a single interest rate for its entire term. This interest rate is something like the average of the current one-year rate and the forward rates for the following nine years. Technically, it is closer to a geometric weighted average than an arithmetic weighted average.

On July 3, 2002, the interest rate on 10-year notes issued by the U.S. Treasury was 4.75 percent. The rate on two-year notes was 2.77 percent, and the rate on the three-month bill was 1.68 percent. Thus, interest rates were much higher on long-term bonds than on short-term bonds. We say that the yield curve was steeply upward-sloping. If long-term rates are only modestly higher than short-term rates, then we say that the yield curve is mildly upward-sloping (which is normal). When long-term rates are below short-term rates, we say that they yield curve is inverted.

For this course, you will not need to know anything about doing calculations involving the forward rate and the yield curve. For teaching purposes they make things unnecessarily complicated. However, for investors on Wall Street, the forward rate and the yield curve matter a lot. The real world, unfortunately, is complex.

Interest, Rent, and Capital Gains

Suppose that there are two identical condominiums, one of which is for rent with the other one for sale. Financially, will it be to your advantage to live in the rental or to buy the other condo?

We can think in terms of borrowing the money to buy the condo and then selling it after one year. Suppose that it costs $200,000 and that after one year we can sell it for $204,000. Houses suffer from wear and tear, like lawnmowers. However, unlike lawnmowers they tend to increase in value. This is because of general inflation as well the fact that land is scarce and tends to become more valuable over time. An increase in the value of an asset is called a capital gain.

If the interest rate is 6 percent, then our cost will be
$200,000 (1.06) - $204,000 = $8000. If instead we paid $8000 in rent, that would work out to a rent of $666.67 per month. Therefore, if the rental is for $700 per month, then it would be better to buy the condo. If the rental is $600 a month, we would be better off living in the rental. (To keep things from getting too complicated, I am leaving out some factors that matter in the real world, including taxes and the closing costs involved in buying and selling a home.)

We could arrive at the same rent by looking at the condo from the perspective of an investor. Suppose that we are thinking of buying the condo and renting it out to someone else. If we can rent the condo for more than $8000 per year, then we can make a profit by buying it for $200,000. Otherwise, we cannot.

Recall the formula that we use for the profitability of owning a capital asset:

profitability = rental rate + appreciation - interest rate

The general relationship between interest, rental income, and capital gains is

Using i to stand for the interest rate, r to stand for the rental rate (the ratio of rent to price) and p to stand for the rate of capital gain (the average annual rate of appreciation), we have

profitability = r + p - i

In our example, the ratio of rent to price, r is $8000/$200,000 = .04, the interest rate, i is .06, and , the rate of capital gain, p is $204,000/$200,000 = .02. Thus, we have

profitability = .04 + .02 - .06 = 0

When profitability is zero, we are indifferent between owning and renting. There is a tendency for the price of an asset to adjust up or down so that owning and renting provide equivalent net benefits. In terms of our formula, there is a tendency for profitability to be zero. If profitability were clearly positive, people would bid up the price of the asset, which causes the ratio of rent to price (the rental rate) to go down, which brings profitability back toward zero. The opposite would happen if profitability were clearly negative.

Common Stock and the Price/Earnings Ratio

The basic relationship between the interest rate, the rental rate, and the capital gains rate that holds for a condo also holds for other capital assets. For example, for Josh's lawn mowing business, the "rent" that he derives from an additional lawnmower is equal to the value that he gets from the increase in lawns mowed. The capital gain (in this case a loss) on the lawnmower is equal to its rate of depreciation.

If you buy shares of stock, the regular income that you receive in lieu of rent consists of dividends. The ratio of a stock's dividends to its price can be used as r in the basic equation relating i, r, and p. For example, on Wednesday, July 3, 2002, the stock of Freddie Mac closed at $59.30 a share. It paid a dividend of $.88 per share, for a dividend rate or r of .0148, or 1.48 percent. If investors were using the interest rate on the 10-year Treasury note as a benchmark for pricing Freddie Mac stock, then we would set i = 4.75 percent. That means that the expected rate of capital gain on Freddie Mac stock, p, would have to equal
4.75 - 1.48 = 3.27 percent.

A stock does not have to pay dividends in order to be valuable. If the company is profitable, it brings in more in revenue than in expenses. These profits are called earnings. If a company does not distribute earnings as dividends, it can use them in other ways to enhance shareholder value. For example, a company can go into the market and buy back its own shares, increasing the demand for the stock and raising its price.

Some companies pay relatively high dividends, and others pay relatively low dividends. However, if their earnings are similar, investors would see the stocks as having similar value. Therefore, many investors prefer to use the ratio of earnings per share in place of the ratio of dividends per share. Thus, earnings per share becomes r and the overall rate of inflation becomes p in the basic equation. Economist Edward Yardeni calls this the "Fed model," because he believes that the Federal Reserve Board uses this equation to determine whether the stock market as a whole is overvalued, undervalued, or valued correctly.

[Fed model] 10-year interest rate = earnings/price ratio + inflation

On July 3, 2002, the ten-year note rate was 4.75 percent, and overall inflation appeared to be around 1 or 2 percent. Using 1.5 percent inflation, the earnings/price ratio should be:

4.75 - 1.50 = 3.25 percent

In the newspaper, what gets reported is the inverse of the ratio of earnings to price. That is, the financial press reports the ratio of price to earnings (P/E). Therefore, if the Fed model tells us that on July 3 the earnings/price ratio should have been 3.25 percent, or .0325, then the P/E ratio should have been the inverse of that, or 1/.0325, which is about 30. In fact, the market P/E ratio was slightly below 30, so that on July 3, 2002, the Fed model suggested that stocks were undervalued.

The P/E ratios for individual stocks can be all over the map. For example, on July 3, the P/E for Freddie Mac was just 9. For Coca-cola, the P/E ratio was 47. When a P/E ratio is low, that is because investors do not expect earnings to grow as fast as the overall economy. When a P/E ratio is high, investors think that earnings for the company can grow faster than the economy as a whole. For this reason, stocks with high P/E ratios are called "growth stocks" and stocks with low P/E ratios are called "value stocks."

Efficient Markets and the Portfolio Theorem

In the 1960's, economists developed a theory of efficient markets, also known as the "random walk" hypothesis. The efficient markets hypothesis states that the best information about a tradable security, such as a company's common stock, is always embedded in the security's price. Therefore, you cannot take advantage of any known information to get an excess return from trading stocks.

Suppose that on the basis of widely-known information it became possible to predict that stock X would earn an especially high rate of return. In that case, traders would sell other securities to buy X, causing the price of X to go up until the expected excess return disappears. In an efficient market, this adjustment would take place in an instant.

There is a joke about an economist walking down the street, and his friend tells him that there is a $20 bill on the sidewalk. "Can't be there," says the economist. In an efficient market, there are no $20 bills lying around waiting to be picked up.

The efficient markets hypothesis says that all new information arrives randomly. If it were predictable, it would not be new information! If new information arrives randomly, and stock prices only respond to new information, then stock prices will move randomly. Every instant, regardless of past behavior, a stock price has about a 50-50 chance of going up or going down. If someone predicts that a particular stock will go up, that prediction has a 50-50 chance of coming true.

A statistical process in which something has a 50-50 chance of going up or down is called a random walk. The efficient markets hypothesis says that stock prices will follow a random walk. Actually, there is a slight upward bias in stock prices, because over a long period of time, stocks earn a positive average return. This is called a "random walk with upward drift."

Individual investors and Wall Street analysts put a lot of effort and resources into picking stocks. They argue over whether a strategy of investing in "value" stocks works better than a strategy of investing in "growth" stocks. They argue about whether you should buy stocks with "momentum" (i.e., their prices have risen recently) or instead take a "contrarian" position (buying stocks whose prices have fallen recently).

According to the efficient markets hypothesis, all of this effort to come up with trading strategies is wasted. You can do just as well picking stocks by throwing a dart at the list of stocks on the financial page as you can be spending time and effort doing research or listening to a particular adviser, broker, or strategist.

The efficient markets hypothesis also says that there is no point in trying to time the market. Market timing means trying to guess when stocks are undervalued or overvalued, and to buy and sell accordingly. The "Fed model" to which we referred in the previous section is a formula for trying to time the market. If the efficient markets hypothesis holds, then the Fed model and other market timing formulas are of no value in predicting the future course of the market. When the model signals "buy," the stock market only has a 50-50 chance of going up. When the model signals "sell," the stock market still has a 50-50 chance of going up.

To test the efficient markets hypothesis, you have to compare it to an alternative. An alternative might be the hypothesis that the stock recommendations of the most highly-paid analysts will outperform the market. Another alternative might be the hypothesis that using the Fed model to trigger buying and selling decisions will outperform the market.

In almost all tests, the efficient markets hypothesis holds up well relative to the alternatives. There is no formula or strategy that can enable a trader to make excess profits reliably.

Markets tend to be efficient where there are many, well-informed traders. This would include anything traded on the major exchanges, including stocks, bonds, international currencies, and farm commodities (wheat, soybeans, etc.). The market for a particular type of used guitar might not be so efficient.

For each of the following markets, comment on whether you think it will be efficient. Explain your reasoning.

An economist gets into a traffic jam and says, "I'm not even going to try to change lanes." Explain the economist's behavior in terms of the theory of efficient markets.

In an efficient market, expected events are already factored into stock prices. Everyone knows that retail stores earn much of their profits during the Christmas season. You cannot make an excess return by buying stock in Circuit City in October. On the other hand, if Circuit City's sales are much higher (or lower) than expected this November, its share price will rise (or fall) as soon as investors become aware of the news.

Risk, Return, and the Portfolio Theorem

Bonds issued by the U.S. Treasury have a return that is essentially guaranteed. Other securities tend to be riskier. For example, corporate bonds suffer from the risk that a corporation could go bankrupt. As of July, 2002, the risk of bankruptcy appeared to be low for Freddie Mac, but it appeared to be high for WorldCom. The rate of interest on bonds issued by Freddie Mac was close to the rate on Treasuries. The rate of interest on Worldcom bonds was much higher.

Stocks also are risky securities. In any given year, some stocks will go up sharply while others will decline dramatically.

An economic measure of risk is based on the statistical concept of variance. The variance is the average squared deviation from the mean outcome. For example, if we flip two coins, we could get HH, HT, TH, or TT. If we score one point for each head, then on average we will get 4/4 = 1 point. The variance will be

variance = (1/4)[(2-1)2 + (1-1)2 + (1-1)2 + (0-1)2] = 1/2

Formulas aside, the issue is that you do not know ahead of time exactly what the rate of return will be when you invest in a stock. There are many plausible scenarios, some of which will be good for the company and some of which will be bad for the company. If you take an average across all of the plausible scenarios, that average is a measure of the mean return or expected return. If you look at how widely the scenarios differ from one another, that is a measure of the risk or the variance of the return.

For example, if you buy stock in a major utility, such as Pepco, you take relatively little risk. Pepco is unlikely to turn in a spectacular performance, and it is unlikely to go bankrupt. Pepco is a low-variance stock.

On the other hand, you buy stock in a young biotech firm, you take a large risk. If the firm can bring valuable drug treatments to market within ten years, it could be a big winner. If not, then you stand to lose every dollar that you invest. The biotech firm is a high-variance stock.

As investors, we prefer lower variance. That is, if we were given the choice between a sure $10,000 or a coin flip with a 50-50 chance of winning $20,000, we would take the sure $10,000.

Of course, we also prefer higher expected return. If we were offered a coin flip where the payoffs are $25,000 for heads and $5,000 for tails, we would prefer that to a coin flip where the payoffs are $20,000 for heads and $0 for tails.

The interesting decisions are where in order to get a higher return you need to take a larger risk. For example, would you prefer a sure $10,000 or a coin flip where you get $25,000 for heads and $1000 for tails?

If two stocks have the same expected rate of return, a portfolio consisting of both stocks will have the same return but a lower variance, as long as the stocks are not perfectly correlated with each other. If the stocks were perfectly correlated, then they would always move up or down together by the same amount. In reality, stocks are influenced by different factors, and their prices do not all move together. The lower the rate of correlation among stocks in a portfolio, the lower will be the variance of the portfolio compared with the variance of the individual stocks. Because the variance of a portfolio is less than the variance of an individual stock, diversification reduces the risk of a stock portfolio.

Many people understand that diversification reduces risk. However, some people interpret this as meaning that diversification deprives you of a chance of earning a large return. You will see financial journalists and others argue that the way to get the best return is to concentrate on a few stocks.

In fact, the portfolio theorem says that a diversified portfolio is optimal for any desired combination of risk and return. It is usually called the portfolio separation theorem, because it separates the decision about how much risk to take from the decision of which stocks to hold in a portfolio. The theorem states that everyone should have holdings that combine some amount of risk-free securities (short-term Treasuries) and some amount of the well-diversified stock market portfolio. People who prefer low risk should put relatively more of their wealth into the risk-free component; those who are willing to take more risk in order to achieve a higher return should hold relatively more of their wealth in the diversified stock portfolio, and not as much in the risk-free securities.

For example, suppose that the risk-free rate of return is 1 percent, and the expected rate of return on the diversified portfolio of risky stocks is 3 percent. If you put 50 percent of your wealth into the risk-free asset and the other 50 percent intothe diversified portfolio, then your expected rate of return is

(.5)1 + (.5)3 = 2 percent

Now,suppose that you are willing to take more risk to get a higher return. You could shift to having 20 percent of your wealth in the risk-free security and 80 percent of your wealth in the diversified portfolio. Thus, your expected return would be

(.2)1 + (.8)3 = 2.6 percent

The portfolio theorem says that of all the ways of increasing your return, shifting wealth from the risk-free asset to the diversified portfolio is the most efficient. Any other method, including buying a concentrated portfolio of just a few stocks, will produce more risk for the given level of expected return.

The portfolio theorem says that the best trade-off that you can get between risk and return is to exchange some of the riskless asset for shares in the broad market portfolio. An efficient market should ensure that this is the case.

Suppose that moving 10 percent of your portfolio from the riskless asset to the market portfolio increases your expected return by 0.2 percent and increases your risk by 5 percent. This says that the market ratio of risk to return is 0.2/5.0. Now, suppose that you could get a better trade-off by taking stock X out of the market portfolio. That would mean that by selling stock X and buying more of the other stocks, you could raise your expected return by more than 0.2 percent for every 5 percent increase in risk. If this were possible, then everyone would sell stock X and its price would fall until the expected return rose to the point where you no longer could get a good risk-return trade-off for taking X out of the market portfolio.

The portfolio theorem is really an elaboration of the theory of efficient markets. It says that in an efficient market, the trade-off between risk and return will always be equal to the trade-off that you would get from exchanging some of the risk-free asset for some of the market portfolio of risky securities. Each individual stock price will be bid up or down until this trade-off holds at the margin for every stock.

The portfolio theorem says that there is no reason for different people to hold different stock portfolios. Everyone's stock portfolio should consist of the same diversified basket of stocks. The only differences should be that individuals with high risk aversion will hold relatively more of the risk-free asset (short-term Treasury securities) and relatively less of the basket of stocks. Individuals with more tolerance for risk will hold relatively more stocks and relatively less of the risk-free asset.


Combining the efficient markets hypothesis with the portfolio theorem leads to a strategy known as Indexing. The concept of Indexing is to buy all stocks in proportion to their weights in a broad market index, such as the S&P 500 or the Russell 2000. This gives you a diversified portfolio, where you do not try to time the market or to select particular stocks. Indexing means that you "buy and hold" the stocks in the index, which means that you engage in relatively little active stock trading.

When economists developed these theories, they predicted that Indexing would outperform traditional mutual funds and other managed portfolios. Studies have borne out this prediction. As a result, a large share of investor wealth has been shifted into index funds, which use the Indexing strategy. However, there is still a lot of wealth invested in traditional managed funds, which continue to underperform the Index funds. This suggests that the portfolio theorem and the theory of efficient markets still are not sufficiently understood and accepted by investors.

Futures and Options

In modern financial markets, instruments known as "derivatives" play an important role. The most important derivatives are futures and options.

A futures contract is a promise to deliver a well-specified commodity at some future date. A classic user of a futures contract would be a farmer trying to lock in the price of corn that is not yet ready for market. Suppose that today is the first of June, and the farmer sees that in the futures market the price of corn for delivery in August is $4 a bushel. The farmer, whose corn will be ready for delivery in August, could wait until then to sell the corn. However, the price of corn could go up or down between now and August.

If the farmer likes the price of $4 a bushel, the farmer can sell an August futures contract to deliver, say 10,000 bushels of corn. Such contracts are traded on the Chicago Mercantile Exchange, which is a bit like the New York Stock Exchange.

In theory, the farmer has promised to deliver corn in August to whoever purchased the futures contract. However, when August rolls around, the farmer is not going to come waltzing onto the floor of the Merc with 10,000 bushels of corn. In August, the farmer buys back the futures contract at the price that prevails in August. If the price of corn has fallen to, say, $3 a bushel, the farmer makes a profit of $10,000 on the futures market transactions. Conversely, if the price of corn goes up to $5 a bushel in August, the farmer loses $10,000 on the futures contract transactions.

Also in August, the farmer sells the 10,000 bushels of corn in a local market at the prevailing price. If the price is $3 a bushel, the farmer gets $30,000. If the price is $5 a bushel, the farmer gets $50,000.

Notice that no matter what happens to the price of corn, when the farmer sells a futures contract the farmer is bound to receive $40,000 one way or the other. The value of the corn that is sold in August plus (or minus) the profit (or loss) on the futures contract will equal $40,000. The farmer's transactions in the futures market are known as hedging. A hedge is a way of reducing your losses if prices move against you while limiting your gains if prices move favorably.

Today's price of corn is known as the spot price. The relationship between futures prices and spot prices is governed by the cost of storage. If the cost of storing corn for two months, which includes the interest cost of borrowing money to buy and hold corn for two months as well as the cost of renting a silo, is one percent of the spot price of corn, then the futures price should be one percent over the spot price. If the futures price were higher than that, then speculators would sell corn futures contracts, buy corn in the spot market and store it, making a sure profit. Conversely, if the futures price were too low, speculators could buy in the futures market and sell in the spot market and make a sure profit. These sorts of transactions that force prices at different times and places to stay in line with one another are called arbitrage.

Arbitrage between cash and futures prices ensures that something that affects the price of a commodity at any point in time will affect it at all points in time. For example, if information that arrives in April suggests that the demand for gasoline will be particularly high in June, this will cause the June futures price to rise. Speculators will see that it is profitable to buy gasoline now and store it until June, so that they will bid up the price of gasoline now. Thus, the spot price of gasoline will rise right away.


An option is the right to buy something at a particular price, known as the strike price. For example, suppose that Freddie Mac stock is trading for $60 a share in July, and I pay $50 for an August call option to buy 100 shares at a strike price of $62 a share. Between now and August, if the price of Freddie Mac stock goes above $62 a share, I may choose to exercise my option. If the price reaches $63 a share, then I can exercise my option to buy 100 shares at $62 a share and then immediately sell those 100 shares for $63 a share, for a profit of $100 (minus the $50 that I paid for the option).

On the other hand, if the price of the stock goes below the strike price, I do not have to exercise my option to buy it at the strike price. Instead, I leave the option unexercised.

Options only last a finite amount of time. The last day that I can exercise my option is called the expiration date. On the expiration date, if the price of the stock is below the strike price, then my call option expires worthless. Otherwise, the value of the option on the expiration date will be proportional to the difference between the price of the stock and the strike price of the option.

The cost of an option is fixed at the time that I buy it. I cannot lose more than what I pay for the option. If the price of the stock falls, I am not obligated to exercise the option. Thus, unlike a futures contract, my potential loss is limited.

Thus, the payoff from an option is asymmetric. For a call option, a big rise in the stock price can generate a big profit. However, an equally large drop in the stock price only means that the option expires worthless and I am out the cost of the option.

Because of the asymmetric payoff, options are more valuable when the underlying securities have prices that are highly volatile. The sharper the up-and-down swings in a stock price, the more valuable are options on that stock.

Other the factors that make an option valuable are:

  1. Time to expiration. The more time that remains until an option expires, the more valuable is the option.

  2. Strike price. Suppose that I have a call option on Freddie Mac stock with a strike price of 61. If the price of the stock is already at 62, then my option is "in the money." It is particularly valuable. If the stock price is at 61, then my option is "at the money." If the stock price is at 60, then the option is "out of the money." It still has value, but not as much as if it were at the money or in the money.

The option to sell a security at a strike price is called a put option. If I have a put option on Freddie Mac stock with a strike price of 61, and the price of the stock falls to 60, then I can exercise my option. I buy the stock for 60, then use the option to sell it for 61.

Selling an option also is known as writing an option or being short an option. Buying an at-the-money call option on a stock and writing an at-the-money put option on the same stock is like owning the stock. If the stock price goes down, then my call option will expire worthless and my short put will be costly. If the stock price goes up, then my short put will be unexercised and my call option will be profitable. Overall, the equivalence between owning a stock and being long an at-the-money call and short an at-the-money put is known as put-call parity.

There is a formula for determining the fair price of an option, taking into account all of the factors. The formula was developed by Myron Scholes and Fischer Black (Scholes won the Nobel Prize, but Black died several years before the award was given). The Black-Scholes formula is too complex to go into in this course.

The significance of market-traded options is that they tell us how investors estimate volatility. Using the Black-Scholes formula and observing the price of an option, you can determine the estimate of volatility that makes the price of the option fair. That is the market's estimate of volatility. When the at-the-money options on a particular stock are expensive, that suggests that the market views the stock as volatile, meaning that its price could go up or down a great deal. If the at-the-money options are less expensive, this means that the market expects the stock's price to be fairly stable.

Certain trading strategies also can be interpreted in terms of options, based on how they will be affected by volatility. For example, there is an aphorism that goes "ride the winners, sell the losers." This means that if a stock that you own goes up, you buy more of it, and if a stock that you own goes down, you sell it. Following this strategy is like owning a portfolio of call options--you are "long" volatility. Higher volatility will give you more profits.

Conversely, suppose you use price limits to determine when to buy and sell stocks. You have a low price that is a "buy" trigger and a high price that is a "sell" trigger. With this approach, it is as if you are writing options on your portfolio. Writing options on stocks that you own is known as writing covered options. You are "short" volatility, and you stand to do well in a market that does not fluctuate too widely.

In an efficient market, it does not pay to be either long volatility or short volatility. No trading strategy is superior to just buying and holding the market portfolio. Being short volatility (or writing covered options) is just another way of putting a larger share of your wealth in the risk-free asset. Being long volatility is just another way of putting a larger share of your wealth in the risky portfolio that offers higher expected returns.

Personal Finance

The economics of finance can be used to analyze some common personal financial issues. Most of these issues will not affect you until you are older, at which point it will help to be able to remember the economic perspective.

Compound Interest

Suppose that you are 30 years old, with a family income of $50,000 a year. If you save $3000 a year, how much will you have after 35 years? The principal alone, without earning interest, will be 35 times $3000, or just $105,000. However, if you invest the money and earn a real annual return of 4 percent, at the end of 35 years you will have $221,000. The difference between $221,000 and $105,000 is the power of compound interest.

Below is a table that shows part of this calculation

Yearsavingsinterest on last year's balancethis year's balance
1$3000 -- $3000
2$3000 $120 $6120
3$3000 $245 $9365
4$3000 $374 $12,739
...33$3000 $7524 $198,629
34$3000 $7945 $209,574
35$3000 $8383 $220,957

In the real world, inflation tends to distort this sort of calculation. Because of inflation, you might be able to earn a return of 7 percent, ending up with a balance of over $400,000. Moreover, if inflation is, say, 3 percent per year, then you should be able to save more than $3000 a year in later years. In fact, your savings should go up by 3 percent per year, which will further increase your final balance. However, if inflation is 3 percent per year, that means that the cost of living after 35 years will be almost triple what it is today. Overall, a world in which the nominal interest rate is 7 percent and inflation is 3 percent is like one in which the nominal interest rate is 4 percent and inflation is zero.

The point to appreciate about compound interest is that over a long period of time a relatively small annual rate of savings can add up. On the other hand, living beyond your means and going into debt means that compound interest works against you. Adding a little more debt each year can lead you into a very deep hole. The compounding effect is even stronger, because interest rates for consumers tend to be high (over 10 percent on many credit cards). If you do not pay your full credit card balance every month, you end up fighting a very strong current of compound interest flowing against you.

Buying a Home

Our general formula for the profitability of buying a home is based on a comparison of the purchase price to the rent on an equivalent home. The formula is

profitability = rental rate + appreciation - interest rate

For example, if a house costs $150,000 and it could be rented for $6,000 a year, then the rental rate is $6,000/$150,000 = .04 or 4 percent. If it appreciates at a rate of 4 percent per year and the interest rate is 7 percent, then the profitability is 4+4-7=1 percent, which means that it is profitable to buy the house rather than rent.

A major complicating factor with buying a house is that it costs a lot to buy and sell. Real estate sales commissions are around 6 percent. There are also a number of fees charged by lenders, title insurance companies, and other service providers. Finally, the local government often collects transfer taxes and fees for recording the transaction.

The costs of buying and selling a home affect the home-buying decision in many ways. Basically, the sooner you have to sell a house, the less likely it is to be profitable to buy rather than to rent.

If you are likely to be moving soon because of a job change or a change in family status, it can be unwise to buy a house. When you are starting a family, it may be better to buy a house with an extra bedroom now, rather than buy one house this year and another in two years when you have more children.

Because profitability depends so much on home price appreciation, one does not want to buy a house when prices are too high. In an efficient market, there should be no way of telling when prices are out of line. However, house prices sometimes seem to reach irrational levels for short periods of time in specific markets. A sign that prices are too high is when the ratio of annual rent to purchase price is unusually low, say less than 2 percent.

The riskiest properties to own are condominiums. Condos tend to be the shock absorbers of the housing market. When demand is high, condo prices soar. When demand falls off, condo prices drop the most.

Most young people do not have enough cash to buy a home. Therefore, you typically have to borrow most of the money to buy a house. The money that you borrow is called a mortgage loan. If you default on a mortgage loan, the lender can take possession of your house. We say that the house is collateral for the mortgage loan. The collateral reduces the lender's risk, so that a mortgage loan costs you less than any other loan that you might obtain.

A typical mortgage loan has a 30-year term, with payments made monthly. The monthly payment is designed to gradually reduce the mortgage balance to zero, just as an annuity payment is designed to gradually exhaust savings. In fact, the formula for calculating a mortgage payment is pretty much the same as the formula for an annuity. The main difference is that the mortgage payment is monthly, so that the interest rate has to be converted to a monthly rate.

Most mortgages are paid off before the 30 year term expires.

The reason that the thirty-year term is popular is that by stretching out the payments over that period the monthly payments are kept low. However, if you are likely to pay off a mortgage loan in ten years or less, it makes sense to take an adjustable-rate mortgage, where the interest rate can change after 3 years or 5 years. These loans carry lower interest rates than the standard thirty-year fixed rate, but the rate can increase. If you were keeping the loan for ten years or more, the rate increase could be a big issue. However, few people keep their mortgage loans that long.

When you have a mortgage loan on your residence, you can deduct the interest expense from your income. You will hear it said that a mortgage loan is a great tax deduction, and some financial advice gurus even recommend taking out the largest mortgage loan that you can. This is flawed advice, for several reasons.

  1. The deductibility of home mortgage interest does not mean that taking out a mortgage loan puts money in your pocket. At best, it reduces the cost of the loan. If your mortgage rate is 7 percent, then on an after-tax basis it might be closer to 5 percent.

  2. The tax deduction has many limitations and restrictions. For many people, a mortgage ends up making only a small difference in tax liability.

  3. If you take out a larger mortgage than you need, then that gives you money to invest. When you invest that money, you earn taxable income. The taxes on that income tend to cancel out the tax savings from the larger mortgage.

Taxes, IRA's, and 401(K) plans

Income taxes do affect personal financial decisions. Other things equal, an investment with tax-exempt income is better than an investment where the income may be taxed. When the income is tax exempt, your savings accumulate more effectively.

The best investment vehicles go even further to save on taxes, because the the money you put into the accounts is tax deductible. If you earn $50,000 and put $2000 into an Individual Retirement Account (IRA), you can deduct the $2,000 from your taxable income as well as accumulate investment earnings tax-free until you retire. Thus, it pays to put money in an IRA.

Similarly, there are employer-sponsored retirement savings plans, called 401(K) plans, because the provision in the tax code is called 401(K). Like IRA's, they allow you to take an income tax deduction for your savings. In addition, many companies have matching programs, where they will kick in additional money in proportion to what you save.

There is almost no valid reason not to take maximum advantage of 401(k) plans and IRA's. Because of the tax advantages, these are the best savings vehicles.


People who want to get the best returns over a long period should put some of their investment portfolio in the stock market. Your stock market portfolio should be in mutual funds that replicate the performance of a major stock index. This approach is known as indexing.

Inside a tax-exempt account, such as an IRA, regular mutual funds, called index funds, provide excellent diversification at low cost. However, regular mutual funds are required to distribute income each year, which has tax consequences. A new instrument, known as exchange-trade mutual funds, allows you to defer taxes until you sell your shares in the funds. An exchange-traded index fund is the best investment vehicle when you are putting savings into a taxable account.

Corporate Finance: Leverage and the Modigliani-Miller Theorem

Corporate finance is a broad, complex subject. Here, we will just touch on a couple of significant concepts.

Corporations own long-lived assets, like the fruit tree that we used as an example in basic financial calculations. These long-lived assets typically require a large up-front expenditure, and they then yield income over time. Corporations cannot pay for these assets out of current income, so they must raise funds in the capital markets. The two main classes of funds are equity (common stock) and debt (corporate bonds, commercial paper, and bank loans).

Using a high ratio of debt to equity is called leverage. It is called leverage because the debt acts like a lever in that it allows a given amount of equity to carry more than its weight in assets.

A firm that raises $100 million in equity capital and $100 million in debt can purchase $200 million in capital assets. A firm that raises $100 million in debt and $900 million in debt can purchase $1 billion in capital assets.

The highly-levered company, or highly-leveraged (sic) company, will have a return on equity that is very sensitive to changes in the value of its capital assets. If you own a baseball team that is financed mostly with debt, then a good attendance will mean a high return, but bad attendance will mean a low return or loss.

The Modigliani-Miller Theorem

Fifty years ago, Wall Street practitioners thought that companies with higher leverage would have higher stock prices. However, in 1958, Franco Modigliani and Merton Miller made an argument that leverage should not matter.

The Modigliani-Miller argument is that it is individuals, not corporations, that have the final say on leverage. The local phone company, Verizon, has physical assets, including a network of wires and switches. If I would like to take a position in those assets that is more levered than Verizon's debt-equity ratio, I could borrow money to buy stock in Verizon. My borrowing is personal leverage. Conversely, someone who wants to take a position in phone company assets that is less levered than Verizon's debt-equity ratio, that person could have a portfolio that combines some Verizon stock with some interest-bearing securities.

The Modigliani-Miller theorem states that the total value of a firm depends on the underlying value of its assets, not on how they are financed. Investors are said to "pierce the corporate veil" and to evaluate the physical (and intellectual) assets of the company, regardless of its financial structure.

Merton Miller tells a funny story about being asked to explain the Modigliani-Miller theorem, which helped win a Nobel Prize, in a 10-second sound bite.

The way I heard the story, Miller was asked by a reporter to explain what made the theory worthy of a Nobel Prize. Miller responded, "Say you have a pizza, and it is divided into four slices. If you cut it into eight slices, you still have the same amount of pizza. We proved that! Rigorously!"