## Appendix Continuous Compounding

Suppose that your money earns interest at an annual nominal percentage rate (APR) of 6% per year compounded semiannually. What is your effective annual rate of return, accounting for compound interest?

We find the answer by first computing the per (compounding) period rate, 3% per half-year, and then computing the future value (FV) at the end of the year per dollar invested at the beginning of the year. In this example, we get

The effective annual rate (REFF), that is, the annual rate at which your funds have grown, is just this number minus 1.0.

PART I Introduction

Table 5A.1 Effective Annual Rates for APR of 6%

Compounding Frequency |
n |
Reff (%) |

Annually |
1 |
6.00000 |

Semiannually |
2 |
6.09000 |

Quarterly |
4 |
6.13636 |

Monthly |
12 |
6.16778 |

Weekly |
52 |
6.17998 |

Daily |
365 |
6.18313 |

The general formula for the effective annual rate is

APR\n

i where APR is the annual percentage rate and n is the number of compounding periods per year. Table 5A.1 presents the effective annual rates corresponding to an annual percentage rate of 6% per year for different compounding frequencies.

As the compounding frequency increases, (1 + APR/n)n gets closer and closer to eAPR, where e is the number 2.71828 (rounded off to the fifth decimal place). In our example, e06 = 1.0618365. Therefore, if interest is continuously compounded, REFF = .0618365, or 6.18365% per year.

Using continuously compounded rates simplifies the algebraic relationship between real and nominal rates of return. To see how, let us compute the real rate of return first using annual compounding and then using continuous compounding. Assume the nominal interest rate is 6% per year compounded annually and the rate of inflation is 4% per year compounded annually. Using the relationship

Real rate

1 + Nominal rate

Inflation rate R) , R

1+i we find that the effective annual real rate is r = 1.06/1.04 - 1 = .01923 = 1.923% per year With continuous compounding, the relationship becomes er = eR/ei = eR-i

Taking natural logarithms, we get r = R - i

Real rate = Nominal rate - Inflation rate all expressed as annual, continuously compounded percentage rates.

Thus if we assume a nominal interest rate of 6% per year compounded continuously and an inflation rate of 4% per year compounded continuously, the real rate is 2% per year compounded continuously.

To pay a fair interest rate to a depositor, the compounding frequency must be at least equal to the frequency of deposits and withdrawals. Only when you compound at least as frequently as transactions in an account can you assure that each dollar will earn the full r interest due for the exact time it has been in the account. These days, online computing for deposits is common, so one expects the frequency of compounding to grow until the use of continuous or at least daily compounding becomes the norm.

SOLUTIONS |
1. |
a. 1 + R = (1 + r)(1 + i ) = (1.03X1.08) = 1.1124 |

TO CONCEPT |
R = 11.24% | |

CHECKS |
b. 1 + R = (1.03)(1.10) = 1.133 R = 13.3% | |

2. |
The mean excess return for the period 1926-1934 is 4.5% (below the historical average), and the standard deviation (dividing by n - 1) is 30.79% (above the historical average). These results reflect the severe downturn of the great crash and the unusually high volatility of stock returns in this period. | |

3. |
r = (.12 - .13)/1.13 = -.00885, or -.885%. When the inflation rate exceeds the nominal interest rate, the real rate of return is negative. |

E-INVESTMENTS: INFLATION AND RATES

The text describes the relationship between interest rates and inflation in section 5.1. The Federal Reserve Bank of St. Louis has several sources of information available on interest rates and economic conditions. One publication called Monetary Trends contains graphs and tabular information relevant to assess conditions in the capital markets. Go to the most recent edition of Monetary Trends at the following site and answer the following questions.

http://www.stls.frb.org/docs/publications/mt/mt.pdf

1. What is the most current level of 3-month and 30-year Treasury yields?

2. Have nominal interest rates increased, decreased or remained the same over the last three months?

3. Have real interest rates increased, decreased or remained the same over the last two years?

4. Examine the information comparing recent U.S. inflation and long-term interest rates with the inflation and long-term interest rate experience of Japan. Are the results consistent with theory?

CHAPTER

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