## Supplemental notes to "I Bonds: A no-brainer alternative to fixed annuities?"

#### Note 1

For 1985-1989 results, see Glenn S. Daily, "Insurance Products: How Reliable Are Policy Illustrations?", AAII Journal, January 1991. The results for 1994-1998 are below.

Expected vs. Actual Performance for Single Premium Deferred Annuities Analysis of Rank by Quartile for 52 Products: 1994-1998 |
|||||
---|---|---|---|---|---|

Actual rank by quartile | |||||

Top | 2nd | 3rd | Bottom | ||

Expected rank by quartile | Top | 6 | 2 | 5 | 0 |

2nd | 5 | 3 | 3 | 2 | |

3rd | 1 | 6 | 5 | 1 | |

Bottom | 1 | 2 | 0 | 10 | |

Rankings are based on five-year accumulation values for a $10,000 single premium deposited on 1/1/94. Actual ranks are derived from data in the October 1999 Best's Policy Reports. Expected ranks are based on a five-year projection of the initial interest rate, taking account of expense charges when applicable. Spearman rank correlation coefficient: 0.56. |

#### Note 2

Why does the formula for the I Bond earnings rate use multiplication rather than addition? The answer is that the real rate of return measures changes in purchasing power, and that requires multiplication, not addition. To see why, consider this example:

Suppose your favorite candy costs 25 cents an ounce. For $10, you can buy 40 ounces of candy. Now suppose that the price of candy will rise next year by 4% to 26 cents an ounce, and suppose that you can invest your $10 at a 7% nominal rate of return. Next year you’ll have $10.70, which will buy 41.1538 (i.e., 1070/26) ounces of candy. That’s a 2.88% increase (41.1538/40), and that’s the real return that you earned on your money. You’ll get the same result if you compute the real return directly: 1.07/1.04 – 1 = .0288.

For convenience, many people estimate the real return by subtracting the rate of inflation from the nominal return. Indeed, you can find that formula in some widely-used textbooks, such as N. Gregory Mankiw’s Principles of Economics (1998, p. 507). In this example, 7% minus 4% equals 3%, but we just saw that 3% overstates the real rate of return. You can’t buy 41.20 (i.e., 40 x 1.03) ounces of candy next year; you can buy only 41.1538 ounces.

When inflation is low, the simple approximation does no harm. But if you live in a country plagued by hyperinflation, you’ll want to use the correct formula.

If you want to read more about this subject, see Peter E. Kennedy, "Eight Reasons Why Real versus Nominal Interest Rates Is the Most Important Concept in Macroeconomics Principles Courses," American Economic Review, May 2000 and other references in the More information about I Bonds and TIPS section.