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 + ...an-1)

We see that C is multiplied by the sum of a geometric series. The formula for that sum is
(1-an)/(1-a)

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?