Compound Interest and Real Annuities

Recall the formula for compound interest. We deposit an initial principal, $1000, into a bank account. If we earn an interest rate of 12 percent per year, then after four years we will have $1000(1 + 0.12)⁴ = $1573.52 in the bank. If interest is compounded monthly, then we earn 1 percent per month for 48 months, which gives us slightly more, due to more compounding. If interest is compounded continuously, then we use the "Pert" formula. We multiply the initial principal, P, by e^rt, where e is the base of the natural logarithms, r is the interest rate, and t is the number of years.

By making calculations using compound interest, we can answer questions like the following: If we save $5,000 a year for 10 years, and the interest rate that we earn is 5 percent, how much will we have after 10 years? If we believe that in 5 years we will need $100,000 for our child's education, who much do we need to save each year in order to achieve that goal, assuming an interest rate of 8 percent per year? etc.

One of the most important issues in personal finance is how much to save for retirement. When you retire, you no longer earn income, and the question of how long your savings will last becomes paramount. People often pay financial planners to help them with these issues. By learning to make the calculations yourself, you can help save your parents the cost of going to a financial planner.

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

One thing that she has to worry about with her $300,000 in savings is having the value of her money eaten away by inflation. If the average cost of the goods that she buys rises by 5 percent per year, then the purchasing power of her 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. We subtract inflation from the interest rate to obtain what we call the real interest rate.

r = i - p

In this equation, r stands for the real interest rate, i stands for the nominal interest rate, and p stands for the rate of inflation. If Arlene can earn a nominal interest rate of 5 percent on a bank certificate of deposit (CD), and the rate of inflation is 2 percent, then the real interest rate is 5 - 2 = 3 percent.

Until recently, the real rate of interest was an abstract concept. There was no way 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. In 2000, the real interest rate was over 3.5 percent. Since that time, the real rate has come down somewhat, to somewhere around 2.5 percent.

For example, if Arlene invests $10,000 in a TIP with a real interest rate of 2.5 percent and the inflation rate next year is 2 percent, then next year she will receive 2.5 percent interest, and in addition the principal on her TIP will rise by 2 percent to $10,200. Instead of being eaten away by inflation, her principal increases. In terms of purchasing power, the principal on a TIP remains constant.

Suppose that Arlene has invested $300,000 at a real interest rate of 3.0 percent per year. For how long can she maintain a consumption rate of $20,000 a year?

After one year, she will have earned 3 percent on $300,000, or $9000 in interest. However, she will have spent $20,000. So she will have left $300,000 plus $9000 in interest minus $20,000 in consumption, or $289,000. The second year, she will earn only $8670 in interest. Assuming that she still spends $20,000, Arlene will have $277,670 after the second year. Using a calculator or a spreadsheet, you can project that she will run out of money after about 20 years.

If Arlene thinks in terms of living exactly 15 years from today, how much money should she spend per year? It turns out that we can calculate this, using a loan amortization formula. We can think of Arlene as lending the bank $300,000 for 15 years, and the bank paying her back in equal annual installments at a rate of 3 percent interest. When a loan is repaid in equal installments, part of the payment covers interest and the rest covers principal. The formula for paying back a loan in equal installments is known as the amortization formula. The amortization formula is

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

where C is the annual installment, r is the annual interest rate, B is the initial loan balance, and n is the number of years to repay the loan. Plugging in $300,000 for B, .03 for r, and 15 for n, we have C = $25,130. This says that by lending (investing) her $300,000 at an interest rate of 3 percent, Arlene can live for 15 years on $25,130 per year.

Of course, Arlene does not know exactly how long she is going to live. What can she do? In theory, a life insurance company could offer Arlene a real annuity, meaning an annuity that adjusts for inflation. Arlene would give the $300,000 to the insurance company, which would invest the money at a real interest rate of 3 percent per year. The insurance company would estimate how long she is expected to live, using actuarial tables, which are statistical tables that project life expectancy. The insurance company then calculates how much it can pay her each year. For example, if the insurance company estimated that Arlene were going to live 15 years and it only wanted to break even on Arlene (that is, earn no profit), at an interest rate of 3 percent it would pay Arlene an annuity of $25,130 a year.

If there were no inflation, then Arlene would receive exactly $25,130 a year. If there is inflation of, say, 2 percent per year, then the nominal interest rate will be 5 percent and the real interest rate will be 3 percent. Arlene will receive $25,130 the first year, $25,130 (1 + .02) the second year, and so on. That is, each year, her annuity payment will rise 2 percent, in order to keep up with inflation. Adjusting for inflation is what makes this a real annuity.

In the real world, there are some complications. First, not all annuities are adjusted for inflation. Although inflation is important, all too often the elderly live on fixed incomes, which are annuities that do not adjust for inflation. Second, insurance companies need to earn a profit. If the insurance company earns 0.5 percent, then Arlene will receive an annuity based on 3.0 - 0.5, or 2.5 percent real interest. This will reduce the amount of her annuity.

Finally, converting a fixed sum of money to an annuity leads to an exchange of risk between Arlene and the insurance company. If Arlene dies early, say in 5 years, she will not have collected her annuity and the insurance company earns a windfall gain. Conversely, if she defies the actuarial tables and lives for 25 years, the insurance company may take a loss, because the $300,000 will not earn enough interest to cover the additional payments.